Introduction
The Schwarzschild metrics [69 ] , which in Boyer–Lindquist coordinates ( t , r , ϑ , φ ) (t,r,\vartheta,\varphi) ( t , r , ϑ , φ ) [5 ] take the form
g M = − ( 1 − 2 M r ) d t 2 + ( 1 − 2 M r ) − 1 d r 2 + r 2 ( d ϑ 2 + sin 2 ϑ d φ 2 ) , g_M=-\left(1-\frac{2M}{r}\right)dt^2+\left(1-\frac{2M}{r}\right)^{-1}dr^2+r^2\left(d\vartheta^2+\sin^2\vartheta\,d\varphi^2\right), g M = − ( 1 − r 2 M ) d t 2 + ( 1 − r 2 M ) − 1 d r 2 + r 2 ( d ϑ 2 + sin 2 ϑ d φ 2 ) , constitute the simplest black hole solutions to the Einstein vacuum equations [27 ]
R μ ν = 0 , R_{\mu\nu}=0, R μν = 0 , where R μ ν R_{\mu\nu} R μν M > 0 M>0 M > 0 T : = ∂ t T:=\partial_t T := ∂ t g a , M g_{a,M} g a , M a a a
The study of the stability of these solutions is a central problem in general relativity, and substantial developments have been made over the past decades. In particular, [20 ] and [44 , 21 ] established linear and nonlinear stability for the exterior of Schwarzschild black holes, employing techniques such as the vector field method, originally introduced by Klainerman [40 ] . More recently, [49 , 47 , 48 , 33 , 71 ] established the nonlinear stability of slowly rotating Kerr black holes, while in [59 , 60 , 72 , 73 ] the nonlinear stability has been extended to the full subextremal range. These results provide important evidence towards the following conjecture.
Conjecture (Black Hole Stability Conjecture, [46 ] ). "Generic" small perturbations of a Kerr black hole evolve into a spacetime that remains regular outside the event horizon and eventually approaches, globally, a nearby Kerr black hole.
Noting that the results mentioned above are all established in vacuum, the notion of “generic” in the conjecture refers primarily to the analytic properties of the perturbations. One may, however, interpret the question of stability in a broader sense by also varying the nature of the perturbation , namely by considering black holes as solutions of Einstein–matter systems. This leads to the study of the stability of black hole families in the presence of different types of matter fields and, in particular, to the question of whether analogous stability and asymptotic convergence results hold beyond the vacuum setting. It is therefore natural to couple the Einstein equations to, for instance, scalar, Klein–Gordon, Vlasov, Maxwell, or Dirac fields, depending on the physical phenomenon of interest. Among these, the Dirac equation provides the fundamental relativistic model for 1/2-spin matter and is therefore the natural framework for describing particles such as electrons and neutrinos, complementing the scalar and kinetic models more commonly considered in the literature.
In this work, we shall consider the Einstein–massless Dirac system
R μ ν − 1 2 R g g μ ν = 8 π T μ ν , (1) R_{\mu\nu} - \frac{1}{2}R_g g_{\mu\nu} = 8\pi T_{\mu\nu}, \tag*{(1)} R μν − 2 1 R g g μν = 8 π T μν , (1) γ μ D μ Ψ = 0 , (2) \gamma^\mu D_\mu\Psi= 0, \tag*{(2)} γ μ D μ Ψ = 0 , (2) where ( M , g ) (\mathcal{M},g) ( M , g ) R g R_g R g Ψ \Psi Ψ C 4 \mathbb{C}^4 C 4 energy-momentum tensor is given by
T μ ν : = i 4 ( Ψ ‾ γ μ D ν Ψ − D ν Ψ ‾ γ μ Ψ ) + i 4 ( Ψ ‾ γ ν D μ Ψ − D μ Ψ ‾ γ ν Ψ ) . T_{\mu\nu} := \frac{i}{4}\left(\overline{\Psi}\gamma_\mu D_\nu\Psi- D_\nu\overline{\Psi}\gamma_\mu\Psi\right) + \frac{i}{4}\left(\overline{\Psi}\gamma_\nu D_\mu\Psi- D_\mu\overline{\Psi}\gamma_\nu\Psi\right). T μν := 4 i ( Ψ γ μ D ν Ψ − D ν Ψ γ μ Ψ ) + 4 i ( Ψ γ ν D μ Ψ − D μ Ψ γ ν Ψ ) . Here, D μ D_\mu D μ spinorial covariant derivative , while γ μ ∈ M 4 ( C ) \gamma^\mu\in M_4(\mathbb{C}) γ μ ∈ M 4 ( C ) curved Dirac matrices . Their definition depends on the choice of a tetrad and the associated spin connection , and is such that the Dirac equation is compatible with changes of coordinates and local Lorentz transformations. We refer to Appendix § 10 for a discussion of this construction.
We restrict our attention to the massless case, for which a well-developed decay theory is available on a fixed Schwarzschild background [61 ] and provides a starting point for our nonlinear analysis. The massive case, on the other hand, entails additional analytical challenges, particularly in controlling the long-time behaviour of solutions and the intricate interactions between geometry and matter. We refer to [52 ] , where these difficulties are addressed in the context of the nonlinear stability of Minkowski spacetime for the Einstein–massive Dirac system.
From an analytical perspective, the Dirac field introduces at least two features that distinguish the problem from the corresponding Einstein–scalar field system. First, the Dirac operator is first order, and its square is related to the wave operator through the Lichnerowicz formula [70 ] ,
( γ μ D μ ) 2 = − D μ D μ + 1 4 R g . (3) (\gamma^\mu D_\mu)^2 = -D^\mu D_\mu+ \frac{1}{4}R_g. \tag*{(3)} ( γ μ D μ ) 2 = − D μ D μ + 4 1 R g . (3) We stress that the second-order operator on the right-hand side may be viewed as a spinorial wave operator , since it acts through the spinorial covariant derivatives D μ D_\mu D μ Dirac energy-momentum tensor does not satisfy the positive energy condition [18 ] . This may be related to the fact that the Dirac field is more naturally interpreted in the framework of quantum field theory, after second quantization. In this setting, pointwise energy conditions are replaced by weaker constraints, known as quantum energy inequalities, see [45 ] . In this context, we also mention [39 ] , which studies the spherically symmetric Einstein–Dirac system in semiclassical gravity, where the Einstein field equations are sourced by the expectation value of the stress-energy tensor.
Stability of Schwarzschild for the Einstein–Dirac system in spherical symmetry
The presence of an external field genuinely complicates the dynamics, as it introduces a backreaction mechanism between matter and geometry. To study this coupling in a tractable setting, we impose spherical symmetry, which reduces the problem to a ( 1 + 1 ) (1+1) ( 1 + 1 ) Birkhoff’s theorem [38 ] implies that every spherically symmetric vacuum solution is locally Schwarzschild.
The spherical reduction of the Dirac equation requires some care, since a spinor field itself cannot be invariant under rotations in the same sense as a scalar field. To understand the distinction, recall that on a spherically symmetric background, the wave operator preserves each eigenspace of the orbital angular momentum. On Kerr, by contrast, the equation separates in spheroidal harmonics rather than spherical harmonics, leading to coupling between spherical harmonic modes [74 ] . A somewhat reminiscent structure already appears for the Dirac operator in spherical symmetry: the invariant subspaces are labeled by the total angular momentum and mix different orbital modes, so that individual modes do not give rise to a spherically symmetric stress-energy tensor. Thus, to obtain a source compatible with spherical symmetry, one decomposes the angular dependence into spinorial spherical harmonics and restricts to the lowest angular modes. We refer to Appendix § 10 for the explicit reduction used throughout the paper, and to [11 , 2 ] for the decomposition of the Dirac operator on spheres.
In this setting, the reduced Dirac field is described by two complex-valued functions, denoted by ψ + \psi_+ ψ + ψ − \psi_- ψ − coupled ( 1 + 1 ) (1+1) ( 1 + 1 ) dimensional system consisting of the Dirac equations for the two spinor components and the Einstein equations governing the geometry. See already § 4.1 for the precise equations of the model. In this paper, we establish the nonlinear stability of the Schwarzschild exterior within the spherically symmetric Einstein–massless Dirac system. Let us now state a preliminary version resuming our main result: we refer to § 5 , and in particular to Theorem 2 , for a detailed statement.
Theorem 1 (Stability of Schwarzschild for Einstein–massless Dirac in spherical symmetry, first version). Let us consider a characteristic initial data set for the spherically symmetric Einstein–massless Dirac system posed on a bifurcate null hypersurface C ^ i n ∪ C ^ o u t \widehat{\mathcal{C}}_{\mathrm{in}} \cup\widehat{\mathcal{C}}_{\mathrm{out}} C in ∪ C out Figure 1 , which lies close to Schwarzschild in an appropriate norm. Then, the initial data evolve into a spacetime with the following properties:
Existence of a black hole region: Future null infinity I + \mathcal{I}^{+} I + I − ( I + ) \mathcal{I}^{-}(\mathcal{I}^{+}) I − ( I + ) H + \mathcal{H}^{+} H + B H : = M ∖ J − ( I + ) \mathcal{B}\mathcal{H}:=\mathcal{M}\setminus\mathcal{J}^{-}(\mathcal{I}^{+}) B H := M ∖ J − ( I + )
Orbital stability: The metric remains close to the initial Schwarzschild metric in the domain of outer communication and appropriately defined fluxes and pointwise C 1 C^{1} C 1 C ^ i n ∪ C ^ o u t \widehat{\mathcal{C}}_{\mathrm{in}} \cup\widehat{\mathcal{C}}_{\mathrm{out}} C in ∪ C out
Asymptotic stability: The metric decays polynomially in C 1 C^{1} C 1 (5) ) converges uniformly to the final mass of the black hole and the Dirac field decays polynomially to zero pointwise and in an appropriate energy norm.
Figure 1. The Penrose diagram of the maximal development arising from an admissible initial data set as in Theorem 1 . For an introduction to Penrose diagrams, see [24 Appendix C] .
We note that the decay rates obtained for the Dirac field are not expected to be optimal . Indeed, the argument does not require higher-order regularity estimates for the derivatives of the spinor field, which would allow for a more precise characterization of its decay. Nevertheless, the rates we obtain are consistent with those known for the linear Dirac equation on a Schwarzschild background [61 ] .
Moreover, the failure of the positive energy condition has a concrete consequence in spherical symmetry: the Hawking mass is not necessarily monotone and may undergo a small decrease along the evolution. This contrasts with the standard setting of spherically symmetric Einstein–matter systems (e.g. scalar field, Maxwell, Vlasov), for which this condition holds and monotonicity arguments play a central role in the analysis of black-hole formation and stability, see for instance [19 , 43 ] . In our model, this failure of monotonicity also means that the apparent horizon is not necessarily contained behind the final event horizon. A related obstruction to standard monotonicity arguments arises in the nonlinear stability analysis of Schwarzschild outside spherical symmetry [21 ] . Thus, the Einstein–Dirac system provides an interesting model in which the interaction between a black hole and the matter field may lead to a decrease in its mass already in spherical symmetry. This is reminiscent of the semiclassical picture of black hole evaporation through Hawking radiation, see for instance [57 , 54 ] .
Recent progress on the formation and stability of extremal black holes [50 , 51 , 3 ] raises the question of whether the corresponding gluing constructions and threshold results can be extended to the Einstein–Dirac and Einstein–Dirac–Maxwell systems. Indeed, the techniques developed so far rely in part on energy estimates and monotonicity arguments made possible by the positivity properties of the matter fields. It would therefore be interesting to investigate whether the geometric structure and conservation laws of the Dirac operator can compensate, at least partially, for the lack of positivity and provide alternative mechanisms to establish analogous results.
Overview of the proof
The proof of Theorem 1 relies on a bootstrap argument involving both the geometry and the Dirac field. A key ingredient is the construction of teleological gauges together with anchored Schwarzschild backgrounds, which allow us to identify the Schwarzschild black hole towards which the solution converges. In this section, we review the main elements of our strategy.
Linear Dirac equation on Schwarzschild black holes
Let us briefly review the approach of [61 ] , which provides the main source of inspiration for this work. In that paper, the vector-field method is used to establish a hierarchy of weighted energy estimates yielding boundedness and decay for the Dirac field for any spin-weighted spherical modes on hyperboloidal foliations. To make the discussion more closely aligned with our framework, we schematically present the corresponding argument for the lowest spherical mode in double-null Eddington–Finkelstein coordinates ( u , v ) (u,v) ( u , v ) [20 ] for an introduction to this coordinate system. Our precise conventions are given in § 4 .
Let ψ ± \psi_{\pm} ψ ± hierarchy of weighted energy estimates , consisting of a flux estimate along the outgoing and ingoing null hypersurfaces and a corresponding integrated energy decay estimate :
∫ C ( τ 2 ) ( r p ∣ ∂ v ( r ψ + ) ∣ 2 + ∣ ψ − ∣ 2 ) d v + ∫ C ( τ 2 ) ( ∣ ∂ u ψ − ∂ u r ∣ 2 + ∣ ψ + ∣ 2 ) ( − ∂ u r ) d u + ∫ τ 1 τ 2 ∫ C ( τ ) ( r p − 1 ∣ ∂ v ( r ψ + ) ∣ 2 + ∣ ψ − ∣ 2 ) d v d τ + ∫ τ 1 τ 2 ∫ C ( τ ) ( ∣ ∂ u ψ − ∂ u r ∣ 2 + ∣ ψ + ∣ 2 ) ( − ∂ u r ) d u d τ ≲ ∫ C ( τ 1 ) ( r p ∣ ∂ v ( r ψ + ) ∣ 2 + ∣ ψ − ∣ 2 ) d v + ∫ C ( τ 1 ) ( ∣ ∂ u ψ − ∂ u r ∣ 2 + ∣ ψ + ∣ 2 ) ( − ∂ u r ) d u + l . o . t . , \begin{aligned}
&\int_{\mathcal{C}(\tau_{2})} \left(r^{p}\lvert\partial_{v}(r\psi_{+})\rvert^{2}+\lvert\psi_{-}\rvert^{2}\right)\,dv
+\int_{\mathcal{C}(\tau_{2})}\left(\left\lvert\frac{\partial_{u}\psi_{-}}{\partial_{u}r}\right\rvert^{2}+\lvert\psi_{+}\rvert^{2}\right)(-\partial_{u}r)\,du \\
&\quad+\int_{\tau_{1}}^{\tau_{2}}\int_{\mathcal{C}(\tau)}\left(r^{p-1}\lvert\partial_{v}(r\psi_{+})\rvert^{2}+\lvert\psi_{-}\rvert^{2}\right)\,dv\,d\tau
+\int_{\tau_{1}}^{\tau_{2}}\int_{\mathcal{C}(\tau)}\left(\left\lvert\frac{\partial_{u}\psi_{-}}{\partial_{u}r}\right\rvert^{2}+\lvert\psi_{+}\rvert^{2}\right)(-\partial_{u}r)\,du\,d\tau\\
&\quad\lesssim\int_{\mathcal{C}(\tau_{1})}\left(r^{p}\lvert\partial_{v}(r\psi_{+})\rvert^{2}+\lvert\psi_{-}\rvert^{2}\right)\,dv
+\int_{\mathcal{C}(\tau_{1})}\left(\left\lvert\frac{\partial_{u}\psi_{-}}{\partial_{u}r}\right\rvert^{2}+\lvert\psi_{+}\rvert^{2}\right)(-\partial_{u}r)\,du+\mathrm{l.o.t.},
\end{aligned} ∫ C ( τ 2 ) ( r p ∣ ∂ v ( r ψ + ) ∣ 2 + ∣ ψ − ∣ 2 ) d v + ∫ C ( τ 2 ) ( ∂ u r ∂ u ψ − 2 + ∣ ψ + ∣ 2 ) ( − ∂ u r ) d u + ∫ τ 1 τ 2 ∫ C ( τ ) ( r p − 1 ∣ ∂ v ( r ψ + ) ∣ 2 + ∣ ψ − ∣ 2 ) d v d τ + ∫ τ 1 τ 2 ∫ C ( τ ) ( ∂ u r ∂ u ψ − 2 + ∣ ψ + ∣ 2 ) ( − ∂ u r ) d u d τ ≲ ∫ C ( τ 1 ) ( r p ∣ ∂ v ( r ψ + ) ∣ 2 + ∣ ψ − ∣ 2 ) d v + ∫ C ( τ 1 ) ( ∂ u r ∂ u ψ − 2 + ∣ ψ + ∣ 2 ) ( − ∂ u r ) d u + l.o.t. , where τ \tau τ Γ \Gamma Γ τ 1 ≤ τ 2 \tau_1 \leq\tau_2 τ 1 ≤ τ 2 p ∈ [ 0 , 3 − δ ] p \in[0, 3-\delta] p ∈ [ 0 , 3 − δ ] δ > 0 \delta> 0 δ > 0 C ( τ ) \mathcal{C}(\tau) C ( τ ) C ( τ ) \mathcal{C}(\tau) C ( τ ) outgoing and ingoing null hypersurfaces, respectively. For a schematic picture summarizing this construction, see Figure 2 .
Figure 2. The Penrose diagram depicting the foliations C ( τ ) \mathcal{C}(\tau) C ( τ ) C ( τ ) \mathcal{C}(\tau) C ( τ ) τ \tau τ
Since the components ψ ± \psi_{\pm} ψ ± coupled first-order system , the authors of [61 ] introduce suitable reweighted variables Φ ± : = f ± ( r ) ψ ± \Phi_{\pm} := f_{\pm}(r)\psi_{\pm} Φ ± := f ± ( r ) ψ ±
∂ u Φ + = − a ( r ) Φ − , ∂ v Φ − = a ( r ) Φ + , \partial_u\Phi_+ = -a(r)\Phi_-, \qquad\partial_v\Phi_- = a(r)\Phi_+, ∂ u Φ + = − a ( r ) Φ − , ∂ v Φ − = a ( r ) Φ + , with Φ − \Phi_- Φ − choice of a vector field , adapted to the region under consideration, and suitable reweightings to deal with the coupling between the two components. Let us briefly illustrate how this interplay enters the argument, focusing in particular on the Morawetz, red-shift and r p r^p r p
The first step is to establish an integrated local energy decay estimate by combining the Killing vector field ∂ t \partial_t ∂ t f ∂ r + q f\partial_r+q f ∂ r + q f ( r ) f(r) f ( r ) q ( r ) q(r) q ( r ) (3) expressed in terms of Φ ± \Phi_{\pm} Φ ±
∂ u ∂ v Φ ± + b Φ ± + c Φ ∓ = 0. \partial_u\partial_v\Phi_{\pm}+b\Phi_{\pm}+c\Phi_{\mp}=0. ∂ u ∂ v Φ ± + b Φ ± + c Φ ∓ = 0. This symmetric structure is essential, as it plays the role of Chandrasekhar's transformation [12 ] , which reduces the Teukolsky equations for integer spin fields to scalar wave-like equations. Since the resulting estimate degenerates towards the event horizon and has non-optimal r r r
Close to the horizon, a fundamental role is played by the red-shift estimate , which captures the well-known red-shift effect and provides non-degenerate control of the field towards the event horizon. This mechanism is obtained by considering the vector field ( − ∂ u r ) − 1 ∂ u (-\partial_u r)^{-1}\partial_u ( − ∂ u r ) − 1 ∂ u [25 ] , which is non-degenerate at the horizon. Indeed, in ingoing Eddington–Finkelstein coordinates ( v , r ) (v,r) ( v , r ) − ∂ r -\partial_r − ∂ r Φ ~ − \widetilde{\Phi}_- Φ − Φ − \Phi_- Φ −
Towards infinity, the celebrated r p r^p r p [26 ] are adapted to the Dirac field, obtaining the above hierarchy for p ∈ [ 0 , 3 ) p \in[0,3) p ∈ [ 0 , 3 ) r p ∂ v r^p\partial_v r p ∂ v [61 ] instead start with a suitable reweighting Φ ~ + \widetilde{\Phi}_+ Φ + r r r p p p 3 3 3 Φ − \Phi_- Φ −
Once these estimates are obtained, the pigeonhole principle [26 ] yields
∫ C ( τ ) ( r p ∣ ∂ v ( r ψ + ) ∣ 2 + ∣ ψ − ∣ 2 ) d v + ∫ C ( τ ) ( ∣ ∂ u ψ − − ∂ u r ∣ 2 + ∣ ψ + ∣ 2 ) ( − ∂ u r ) d u ≲ τ − 3 + δ + p . \int_{\mathcal{C}(\tau)} \left(r^p|\partial_v(r\psi_+)|^2+|\psi_-|^2\right)\,dv+\int_{\mathcal{C}(\tau)}\left(\left|\frac{\partial_u\psi_-}{-\partial_u r}\right|^2+|\psi_+|^2\right)(-\partial_u r)\,du\lesssim\tau^{-3+\delta+p}. ∫ C ( τ ) ( r p ∣ ∂ v ( r ψ + ) ∣ 2 + ∣ ψ − ∣ 2 ) d v + ∫ C ( τ ) ( − ∂ u r ∂ u ψ − 2 + ∣ ψ + ∣ 2 ) ( − ∂ u r ) d u ≲ τ − 3 + δ + p . where the implicit constant only depends on δ \delta δ
The nonlinear case: the bootstrap argument
The approach described in the previous section, based on identifying and working with appropriate Dirac reweightings, will also be crucial in our nonlinear perturbative setting. In this case, however, due to the coupling between the geometry and the Dirac field, we set up a bootstrap argument , assuming suitable estimates for both and improving them along the proof, thereby extending these estimates globally. Our setup is inspired by [21 ] , and by the works [53 , 4 ] in the context of Reissner–Nordström black holes in spherical symmetry, in both the subextremal and extremal regimes.
Given a Schwarzschild black hole of mass M 0 > 0 M_0>0 M 0 > 0 seed data sets describing perturbations of the Schwarzschild initial data, prescribed on an initial bifurcate null hypersurface C ^ i n ∪ C ^ o u t \widehat{\mathcal{C}}_{\mathrm{in}}\cup\widehat{\mathcal{C}}_{\mathrm{out}} C in ∪ C out [41 ] . The bifurcation sphere is located at an area radius Λ : = 100 M 0 \Lambda:=100M_0 Λ := 100 M 0 ψ + ∘ o u t \psi^{\circ\mathrm{out}}_+ ψ + ∘ out ψ − ∘ i n \psi^{\circ\mathrm{in}}_- ψ − ∘ in C ^ o u t \widehat{\mathcal{C}}_{\mathrm{out}} C out C ^ i n \widehat{\mathcal{C}}_{\mathrm{in}} C in M 0 M_0 M 0 seed data norm
D [ S ] ∼ sup C ^ i n ( ∣ ψ − ∘ i n ∣ + ∣ ∂ u ^ ψ + ∘ i n ∣ ) + sup C ^ o u t ( r ∣ ψ + ∘ o u t ∣ + r 2 ∣ ∂ v ^ ( r ψ + ∘ o u t ) ∣ ) ≤ ε . \mathcal{D}[\mathcal{S}]\sim\sup_{\widehat{\mathcal{C}}_{\mathrm{in}}}\left(\left|\psi^{\circ\mathrm{in}}_-\right|+\left|\partial_{\widehat{u}}\psi^{\circ\mathrm{in}}_+\right|\right)+\sup_{\widehat{\mathcal{C}}_{\mathrm{out}}}\left(r\left|\psi^{\circ\mathrm{out}}_+\right|+r^2\left|\partial_{\widehat{v}}\left(r\psi^{\circ\mathrm{out}}_+\right)\right|\right)\le\varepsilon. D [ S ] ∼ C in sup ( ψ − ∘ in + ∂ u ψ + ∘ in ) + C out sup ( r ψ + ∘ out + r 2 ∂ v ( r ψ + ∘ out ) ) ≤ ε . Since ε \varepsilon ε ε \varepsilon ε ( u ^ , v ^ ) (\widehat{u},\widehat{v}) ( u , v ) initial-data-normalized coordinates , chosen such that ∂ u ^ r = − 1 \partial_{\widehat{u}}r=-1 ∂ u r = − 1 C ^ i n \widehat{\mathcal{C}}_{\mathrm{in}} C in ∂ v ^ r = 1 \partial_{\widehat{v}}r=1 ∂ v r = 1 C ^ o u t \widehat{\mathcal{C}}_{\mathrm{out}} C out u ^ \widehat{u} u
As depicted in Figure 3 , we consider the timelike curve Γ : = { r = Λ } \Gamma:=\{r=\Lambda\} Γ := { r = Λ } τ \tau τ τ f \tau_f τ f bootstrap domain D τ f \mathcal{D}_{\tau_f} D τ f Γ ( τ f ) \Gamma(\tau_f) Γ ( τ f ) D τ f : = J − ( Γ ( τ f ) ) \mathcal{D}_{\tau_f}:=J^-(\Gamma(\tau_f)) D τ f := J − ( Γ ( τ f )) D τ f \mathcal{D}_{\tau_f} D τ f τ f \tau_f τ f ( u , v ) (u,v) ( u , v ) teleologically normalized according to the conditions ∂ u r = − ( 1 − 2 m r ) \partial_u r=-\left(1-\frac{2m}{r}\right) ∂ u r = − ( 1 − r 2 m ) D τ f \mathcal{D}_{\tau_f} D τ f m m m (5) , and ∂ v r = 1 − 2 m r \partial_v r=1-\frac{2m}{r} ∂ v r = 1 − r 2 m D τ f \mathcal{D}_{\tau_f} D τ f T T T time function on D τ f \mathcal{D}_{\tau_f} D τ f Γ \Gamma Γ
Figure 3. The Penrose diagram showing the bootstrap domain D τ f : = J − ( Γ ( τ f ) ) \mathcal{D}_{\tau_f}:=J^-(\Gamma(\tau_f)) D τ f := J − ( Γ ( τ f )) Γ : = { r = Λ } \Gamma:=\{r=\Lambda\} Γ := { r = Λ } ( u , v ) (u,v) ( u , v ) [53 ] .
To identify the Schwarzschild black hole towards which the solution converges, we introduce a τ f \tau_f τ f Schwarzschild background ( M τ f , r ‾ τ f ) (M_{\tau_f},\overline{r}_{\tau_f}) ( M τ f , r τ f ) Γ \Gamma Γ
M τ f : = m ( Γ ( τ f ) ) . M_{\tau_f}:=m(\Gamma(\tau_f)). M τ f := m ( Γ ( τ f )) . We then fix the area radius coordinate r ˉ τ f \bar{r}_{\tau_f} r ˉ τ f
r ˉ τ f ( Γ ( τ f ) ) = Λ . \bar{r}_{\tau_f}\bigl(\Gamma(\tau_f)\bigr)=\Lambda. r ˉ τ f ( Γ ( τ f ) ) = Λ. Thus, the reference background is updated as the bootstrap domain is extended. The precise construction and its role in the argument are described in § 5.1.3 .
Motivated by the linear case discussed above, we introduce energy norms such as
E p ( τ ) : = ∫ C ( τ ) ( r p ∣ ∂ v ( r ψ + ) ∣ 2 + ∣ ψ − ∣ 2 + … ) d v , E ( τ ) : = ∫ C ( τ ) ( ∣ ∂ u ψ − − ∂ u r ˉ τ f ∣ 2 + ∣ ψ + ∣ 2 + … ) ( − ∂ u r ˉ τ f ) d u , \mathcal{E}_{p}(\tau):=\int_{\mathcal{C}(\tau)}\left(r^{p}\lvert\partial_{v}(r\psi_{+})\rvert^{2}+\lvert\psi_{-}\rvert^{2}+\ldots\right)\,dv,\qquad\mathcal{E}(\tau):=\int_{\mathcal{C}(\tau)}\left(\left\lvert\frac{\partial_{u}\psi_{-}}{-\partial_{u}\bar{r}_{\tau_f}}\right\rvert^{2}+\lvert\psi_{+}\rvert^{2}+\ldots\right)(-\partial_{u}\bar{r}_{\tau_f})\,du, E p ( τ ) := ∫ C ( τ ) ( r p ∣ ∂ v ( r ψ + ) ∣ 2 + ∣ ψ − ∣ 2 + … ) d v , E ( τ ) := ∫ C ( τ ) ( − ∂ u r ˉ τ f ∂ u ψ − 2 + ∣ ψ + ∣ 2 + … ) ( − ∂ u r ˉ τ f ) d u , and formulate the corresponding bootstrap assumptions on D τ f D_{\tau_f} D τ f
∣ r − r ˉ τ f ∣ ≲ ε 3 / 2 τ − 2 + δ , ∣ m − M τ f ∣ ≲ ε 3 / 2 τ − 3 + δ , \lvert r-\bar{r}_{\tau_f}\rvert\lesssim\varepsilon^{3/2}\tau^{-2+\delta},\qquad\lvert m-M_{\tau_f}\rvert\lesssim\varepsilon^{3/2}\tau^{-3+\delta}, ∣ r − r ˉ τ f ∣ ≲ ε 3/2 τ − 2 + δ , ∣ m − M τ f ∣ ≲ ε 3/2 τ − 3 + δ , while for the Dirac field we assume
E p ( τ ) ≲ ε 2 τ − 3 + δ + p , E ( τ ) ≲ ε 2 τ − 3 + δ , \mathcal{E}_{p}(\tau)\lesssim\varepsilon^{2}\tau^{-3+\delta+p},\qquad\mathcal{E}(\tau)\lesssim\varepsilon^{2}\tau^{-3+\delta}, E p ( τ ) ≲ ε 2 τ − 3 + δ + p , E ( τ ) ≲ ε 2 τ − 3 + δ , for τ ∈ [ 1 , τ f ] \tau\in[1,\tau_f] τ ∈ [ 1 , τ f ]
Improving the bootstrap assumptions
Let us now outline the main elements needed to improve the bootstrap assumptions.
On the geometric side, the double-null gauge together with spherical symmetry reduces the Einstein equations to a system of transport equations in terms of geometric quantities such as
κ : = ∂ v r 1 − 2 m r , γ : = − ∂ u r 1 − 2 m r . \kappa:=\frac{\partial_{v}r}{1-\frac{2m}{r}},\qquad\gamma:=-\frac{\partial_{u}r}{1-\frac{2m}{r}}. κ := 1 − r 2 m ∂ v r , γ := − 1 − r 2 m ∂ u r . The strategy is then to integrate these equations and compare the dynamical solution with the anchored Schwarzschild background of mass M τ f M_{\tau_f} M τ f
∂ v r ˉ τ f = − ∂ u r ˉ τ f = 1 − 2 M τ f r ˉ τ f , γ ˉ = κ ˉ = 1. \partial_{v}\bar{r}_{\tau_f}=-\partial_{u}\bar{r}_{\tau_f}=1-\frac{2M_{\tau_f}}{\bar{r}_{\tau_f}},\qquad\bar{\gamma}=\bar{\kappa}=1. ∂ v r ˉ τ f = − ∂ u r ˉ τ f = 1 − r ˉ τ f 2 M τ f , γ ˉ = κ ˉ = 1. Recalling that the teleological gauges imposed above correspond to κ = 1 \kappa=1 κ = 1 γ = 1 \gamma=1 γ = 1 κ ~ : = κ − 1 \tilde{\kappa}:=\kappa-1 κ ~ := κ − 1 γ ~ : = γ − 1 \tilde{\gamma}:=\gamma-1 γ ~ := γ − 1 κ ~ \tilde{\kappa} κ ~ γ ~ \tilde{\gamma} γ ~ ∂ v r = κ ( 1 − 2 m r ) \partial_{v}r=\kappa(1-\frac{2m}{r}) ∂ v r = κ ( 1 − r 2 m ) ∂ u r = − γ ( 1 − 2 m r ) \partial_{u}r=-\gamma(1-\frac{2m}{r}) ∂ u r = − γ ( 1 − r 2 m )
Let us now emphasize an important point concerning the control of the null lapse Ω \Omega Ω (4) . As already mentioned, once "squared", the Dirac equation contains additional curvature terms that do not appear in the scalar field case and that are expressed in terms of derivatives of Ω \Omega Ω ∂ u log Ω \partial_{u}\log\Omega ∂ u log Ω ∂ v log Ω \partial_{v}\log\Omega ∂ v log Ω Lemma 5.2 , we prove the corresponding estimates and discuss how these quantities are sensitive to the direction of integration relative to the teleological gauge conditions, which are imposed on the final cones.
On the other hand, improving the bootstrap assumptions on the Dirac field consists in adapting the strategy pursued in [61 ] , discussed above, to the nonlinear perturbative setting. In particular, the reweightings used in the linear setting have to be carefully adapted to the present problem. Indeed, a direct linearization would not in general be sufficient, as the resulting nonlinear error terms may be too large to close the bootstrap argument. We therefore identify suitable Dirac reweightings in each region of interest, allowing us to exploit the same underlying mechanism as in the linear case and derive a hierarchy of the form
E p ( τ 2 ) + E ( τ 2 ) ≲ E p ( τ 1 ) + E ( τ 1 ) + E e r r , \mathcal{E}_{p}(\tau_{2})+\mathcal{E}(\tau_{2})\lesssim\mathcal{E}_{p}(\tau_{1})+\mathcal{E}(\tau_{1})+\mathcal{E}_{\mathrm{err}}, E p ( τ 2 ) + E ( τ 2 ) ≲ E p ( τ 1 ) + E ( τ 1 ) + E err , where E r r \mathit{Err} Err 1 ≤ τ 1 ≤ τ 2 ≤ τ f 1 \le\tau_1 \le\tau_2 \le\tau_f 1 ≤ τ 1 ≤ τ 2 ≤ τ f p ∈ [ 0 , 3 − δ ] p \in[0,3-\delta] p ∈ [ 0 , 3 − δ ]
∫ τ 1 τ 2 ( E p − 1 ( τ ) + E ( τ ) ) d τ ≲ E p ( τ 1 ) + E ( τ 1 ) + E r r , \int_{\tau_1}^{\tau_2} \left(\mathcal{E}_{p-1}(\tau) + \mathcal{E}(\tau)\right)\,\mathrm{d}\tau\lesssim\mathcal{E}_p(\tau_1) + \mathcal{E}(\tau_1) + \mathit{Err}, ∫ τ 1 τ 2 ( E p − 1 ( τ ) + E ( τ ) ) d τ ≲ E p ( τ 1 ) + E ( τ 1 ) + Err , for p ∈ [ 1 , 3 − δ ] p \in[1,3-\delta] p ∈ [ 1 , 3 − δ ]
Finally, once the bootstrap assumptions have been improved and hence established globally in the exterior of the black hole, one last technical step remains: controlling the teleological gauges and proving their convergence to a well-defined eschatological gauge as τ f → + ∞ \tau_f \to+\infty τ f → + ∞ 1 − 2 m r > 0 1-\frac{2m}{r}>0 1 − r 2 m > 0 § 9 , where we establish the convergence to the asymptotic Schwarzschild black hole with mass M ∞ M_\infty M ∞
Related works
Let us conclude by mentioning some results in the literature that are relevant from the perspective of their setting and techniques. The stability of spherically symmetric black holes as solutions to various Einstein–matter systems has already been studied. We mention, for instance, [53 , 4 ] , which establish stability results for sub-extremal and extremal Reissner–Nordström black holes, respectively, in the Einstein–Maxwell–neutral scalar field system, as well as [76 ] for the stability of Schwarzschild in the Einstein–massless Vlasov system.
We next recall some results concerning the Dirac equation on curved spacetimes. The study of Dirac fields on rotating black-hole backgrounds was significantly advanced by the separability results of Teukolsky [74 ] for the massless neutrino equation on Kerr and Chandrasekhar [13 ] in the massive case. Related results were obtained for Kerr–Newman spacetimes and in alternative coordinate systems [63 , 67 ] .
From the perspective of dispersive PDEs, the properties of the Dirac operator have been studied on a variety of curved backgrounds. In particular, local smoothing estimates on asymptotically flat and warped product spaces were established in [9 ] , while Strichartz estimates were obtained on spherically symmetric spaces and asymptotically flat manifolds in [8 , 10 ] . Scattering for both massless and massive Dirac fields on spherically symmetric black holes was studied in [62 Mel03] , with related results on the exterior of subextremal Kerr [35 ] and in the interior of Reissner–Nordström black holes [34 ] .
Turning to the coupled Einstein–Dirac system, we mention the works of Finster, Smoller, and Yau, who constructed particle-like solutions for the Einstein–Dirac [30 ] and Einstein–Dirac–Maxwell [29 ] systems, which were later studied perturbatively by [66 , 65 ] . Finster, Smoller, and Yau [31 ] , as well as Dafermos [18 ] , also studied the non-existence of time-periodic solutions; we refer to [32 ] for a discussion of these results.
More recently, the already mentioned work [61 ] established Price’s laws for the massless Dirac equation on a fixed Schwarzschild background using weighted energy estimates and the vector-field method. In the nonlinear setting, global stability of Minkowski spacetime for the Einstein–Dirac system has been established in both the massless [14 ] and massive [52 ] cases. We also refer to [22 ] , which proves a nonlinear stability result for a coupled system involving a massive Dirac field on Minkowski spacetime. Local well-posedness for the characteristic initial value problem for the Einstein–Dirac system was studied in [80 ] , and the formation of trapped surfaces has been investigated for the closely related Einstein–Weyl system [79 ] . Finally, [36 , 37 ] has established global small-data stability for nonlinear Dirac systems near Minkowski spacetime and on fixed Schwarzschild backgrounds.
Outline
The paper is organized as follows. § 4 is devoted to presenting all the preliminaries regarding the spherically symmetric Einstein–massless Dirac system, from the explicit formulation of the equations (see also Appendix § 10 ) to the definition of the characteristic initial value problem and its well-posedness (we refer to Appendix § 11 ). In § 5 , we detail the framework of our stability result. In particular, we give a rigorous statement of the main result, describe the proof and the bootstrap argument, and introduce the corresponding teleological gauges and energy norms. Sections 4 to 6 focus on the improvement of the bootstrap assumptions for the geometry and the Dirac field. In § 9 we finally put all pieces together and conclude the main proof.
Acknowledgments
The author would like to thank his advisors, Anne-Sophie de Suzzoni and Cécile Huneau, for their support and the many helpful discussions. Special thanks also go to the General Relativity Group in Jussieu for the meaningful monthly exchanges and conversations. The author also acknowledges the financial support received through the S. S. Chern Young Faculty Award, funded by AX and awarded to Anne-Sophie de Suzzoni.
Declaration of AI use
The author used ChatGPT (OpenAI) to assist with English language editing, the correction of typos, and the improvement of the clarity and presentation of the manuscript. All AI-assisted suggestions were reviewed and revised by the author, who takes full responsibility for the mathematical content and the final manuscript.
Preliminaries
In this section we precisely introduce the framework in which we will be working. § 4.1 contains the explicit formulation of the spherically symmetric Einstein–massless Dirac system, and we refer to Appendix § 10 for a detailed derivation. The main properties of the Schwarzschild solution are recalled in § 4.2 , while in § 4.3 we set the characteristic initial value problem, whose well-posedness is tackled in Appendix § 11 .
Einstein–Dirac system in spherical symmetry
Let ( M , g ) (\mathcal{M},g) ( M , g ) g g g ( − , + , + , + ) (-,+,+,+) ( − , + , + , + ) spin structure . We say that ( M , g ) (\mathcal{M},g) ( M , g ) spherically symmetric if it is diffeomorphic to Q × S 2 Q \times\mathbb{S}^{2} Q × S 2
g = g Q + r 2 d σ S 2 , g = g_{Q} + r^{2}d\sigma_{\mathbb{S}^{2}}, g = g Q + r 2 d σ S 2 , where ( Q , g Q ) (Q,g_{Q}) ( Q , g Q ) ( 1 + 1 ) (1+1) ( 1 + 1 ) d σ S 2 : = d ϑ 2 + sin 2 ϑ d φ 2 d\sigma_{\mathbb{S}^{2}} := d\vartheta^{2} + \sin^{2}\vartheta\,d\varphi^{2} d σ S 2 := d ϑ 2 + sin 2 ϑ d φ 2 r r r Q Q Q area radius of the orbits of the isometric S O ( 3 ) SO(3) S O ( 3 ) ( M , g ) (\mathcal{M},g) ( M , g ) ( Q , g Q ) (Q,g_{Q}) ( Q , g Q ) global double-null coordinate system ( u , v ) (u,v) ( u , v ) g g g
g = − 4 Ω 2 ( u , v ) d u d v + r 2 ( u , v ) d σ S 2 , (4) g = -4\Omega^{2}(u,v)\,du\,dv + r^{2}(u,v)\,d\sigma_{\mathbb{S}^{2}}, \tag*{(4)} g = − 4 Ω 2 ( u , v ) d u d v + r 2 ( u , v ) d σ S 2 , (4) where the null lapse Ω 2 \Omega^{2} Ω 2 Q Q Q u u u v v v ( Q , g Q ) (Q,g_{Q}) ( Q , g Q ) ( Q , r , Ω 2 ) (Q,r,\Omega^{2}) ( Q , r , Ω 2 )
We next introduce the notation for the geometric quantities used throughout the paper. We define the Hawking mass m : M → R m:\mathcal{M}\to\mathbb{R} m : M → R
m : = r 2 ( 1 − g ( ∇ r , ∇ r ) ) = r 2 ( 1 + ∂ u r ∂ v r Ω 2 ) , (5) m := \frac{r}{2}\left(1-g(\nabla r,\nabla r)\right)=\frac{r}{2}\left(1+\frac{\partial_{u}r\,\partial_{v}r}{\Omega^{2}}\right), \tag*{(5)} m := 2 r ( 1 − g ( ∇ r , ∇ r ) ) = 2 r ( 1 + Ω 2 ∂ u r ∂ v r ) , (5) which is clearly independent of the choice of double-null gauge, and the mass aspect function
μ : = 2 m r . (6) \mu:= \frac{2m}{r}. \tag*{(6)} μ := r 2 m . (6) Moreover, we denote
ν : = ∂ u r , λ : = ∂ v r , κ : = Ω 2 − ν = λ 1 − μ , γ : = Ω 2 λ = − ν 1 − μ . (7) \nu:= \partial_u r,\quad\lambda:= \partial_v r,\quad\kappa:= \frac{\Omega^2}{-\nu} = \frac{\lambda}{1-\mu},\quad\gamma:= \frac{\Omega^2}{\lambda} = \frac{-\nu}{1-\mu}. \tag*{(7)} ν := ∂ u r , λ := ∂ v r , κ := − ν Ω 2 = 1 − μ λ , γ := λ Ω 2 = 1 − μ − ν . (7) We now turn to the Dirac field. As detailed in Appendix § 10 .§ 4 , we fix the regular orthonormal frame
e 0 : = 1 2 ( 1 κ ∂ v + 1 − ν ∂ u ) , e 1 : = 1 r ∂ ϑ , e 2 : = 1 r sin ϑ ∂ φ , e 3 : = 1 2 ( 1 κ ∂ v − 1 − ν ∂ u ) , e_0 := \frac{1}{2}\left(\frac{1}{\kappa}\partial_v+\frac{1}{-\nu}\partial_u\right),\quad e_1 := \frac{1}{r}\partial_\vartheta,\quad e_2 := \frac{1}{r\sin\vartheta}\partial_\varphi,\quad e_3 := \frac{1}{2}\left(\frac{1}{\kappa}\partial_v-\frac{1}{-\nu}\partial_u\right), e 0 := 2 1 ( κ 1 ∂ v + − ν 1 ∂ u ) , e 1 := r 1 ∂ ϑ , e 2 := r sin ϑ 1 ∂ φ , e 3 := 2 1 ( κ 1 ∂ v − − ν 1 ∂ u ) , and restrict the Dirac spinor to the lowest angular mode compatible with spherical symmetry, denoted by H ~ 1 / 2 , 1 / 2 − \widetilde{\mathcal{H}}_{1/2,1/2}^{-} H 1/2 , 1/2 − Ψ \Psi Ψ (2) decomposes into two components ψ ± ( u , v ) \psi_{\pm}(u,v) ψ ± ( u , v ) reduced Dirac equations
( 1 − ν ∂ u + ∂ u κ 2 Ω 2 ) ψ + = − ψ − , (8) \left(\frac{1}{-\nu}\partial_u+\frac{\partial_u\kappa}{2\Omega^2}\right)\psi_+ = -\psi_-, \tag*{(8)} ( − ν 1 ∂ u + 2 Ω 2 ∂ u κ ) ψ + = − ψ − , (8) ( 1 κ ∂ v − ∂ v ν 2 Ω 2 + 1 − μ r ) ψ − = 1 r 2 ψ + . (9) \left(\frac{1}{\kappa}\partial_v-\frac{\partial_v\nu}{2\Omega^2}+\frac{1-\mu}{r}\right)\psi_- = \frac{1}{r^2}\psi_+. \tag*{(9)} ( κ 1 ∂ v − 2 Ω 2 ∂ v ν + r 1 − μ ) ψ − = r 2 1 ψ + . (9) On the other hand, Einstein equations (1) reduce to a hyperbolic-transport system for the metric components r , Ω r,\Omega r , Ω
∂ u ∂ v r = − 2 Ω 2 m r 2 + 4 π r T u v , (10) \partial_u\partial_v r = -\frac{2\Omega^2m}{r^2}+4\pi rT_{uv}, \tag*{(10)} ∂ u ∂ v r = − r 2 2 Ω 2 m + 4 π r T uv , (10) 1 2 ∂ u ∂ v log Ω 2 = 2 Ω 2 m r 3 − 8 π T u v , (11) \frac{1}{2}\partial_u\partial_v\log\Omega^2 = \frac{2\Omega^2m}{r^3}-8\pi T_{uv}, \tag*{(11)} 2 1 ∂ u ∂ v log Ω 2 = r 3 2 Ω 2 m − 8 π T uv , (11) ∂ u ( Ω − 2 ∂ u r ) = − 4 π r Ω 2 T u u , (12) \partial_u(\Omega^{-2}\partial_u r) = -\frac{4\pi r}{\Omega^2}T_{uu}, \tag*{(12)} ∂ u ( Ω − 2 ∂ u r ) = − Ω 2 4 π r T uu , (12) ∂ v ( Ω − 2 ∂ v r ) = − 4 π r Ω 2 T v v , (13) \partial_v(\Omega^{-2}\partial_v r) = -\frac{4\pi r}{\Omega^2}T_{vv}, \tag*{(13)} ∂ v ( Ω − 2 ∂ v r ) = − Ω 2 4 π r T v v , (13) where the components of the Dirac energy-momentum tensor are given by
T u u = − ν 2 π ℑ [ ψ − ‾ ∂ u ψ − ] , T v v = κ 2 π r 2 ℑ [ ψ + ‾ ∂ v ψ + ] , T u v = Ω 2 2 π r 2 ℑ [ ψ − ‾ ψ + ] . (14) T_{uu} = -\frac{\nu}{2\pi}\Im[\overline{\psi_-}\partial_u\psi_-],\quad T_{vv} = \frac{\kappa}{2\pi r^2}\Im[\overline{\psi_+}\partial_v\psi_+],\quad T_{uv} = \frac{\Omega^2}{2\pi r^2}\Im[\overline{\psi_-}\psi_+]. \tag*{(14)} T uu = − 2 π ν ℑ [ ψ − ∂ u ψ − ] , T v v = 2 π r 2 κ ℑ [ ψ + ∂ v ψ + ] , T uv = 2 π r 2 Ω 2 ℑ [ ψ − ψ + ] . (14) We therefore refer to ( r , Ω , ψ + , ψ − ) (r,\Omega,\psi_+,\psi_-) ( r , Ω , ψ + , ψ − ) spherically symmetric solution to the Einstein–massless Dirac system if equations (8) –(13) are satisfied.
From definitions (5) –(7) , we may rearrange (10) –(13) and derive the equivalent system
∂ u κ = − 4 π r κ 2 Ω 2 T u u = − 2 r κ ℑ [ ψ − ‾ ∂ u ψ − ] , (15) \partial_u \kappa= -\frac{4\pi r\kappa^2}{\Omega^2}T_{uu}=-2r\kappa\Im[\overline{\psi_-}\partial_u\psi_-], \tag*{(15)} ∂ u κ = − Ω 2 4 π r κ 2 T uu = − 2 r κ ℑ [ ψ − ∂ u ψ − ] , (15) ∂ v γ = 4 π r γ 2 Ω 2 T v v = 2 γ r ( 1 − μ ) ℑ [ ψ + ‾ ∂ v ψ + ] , (16) \partial_v\gamma= \frac{4\pi r\gamma^2}{\Omega^2}T_{vv}=\frac{2\gamma}{r(1-\mu)}\Im[\overline{\psi_+}\partial_v\psi_+], \tag*{(16)} ∂ v γ = Ω 2 4 π r γ 2 T v v = r ( 1 − μ ) 2 γ ℑ [ ψ + ∂ v ψ + ] , (16) ∂ u m = 2 π r 2 Ω − 2 ( T u v ∂ u r − T u u ∂ v r ) = ν ℑ [ ψ − ‾ ψ + ] − r 2 ( 1 − μ ) ℑ [ ψ − ‾ ∂ u ψ − ] , (17) \partial_u m=2\pi r^2\Omega^{-2}(T_{uv}\partial_u r-T_{uu}\partial_v r)=\nu\Im[\overline{\psi_-}\psi_+]-r^2(1-\mu)\Im[\overline{\psi_-}\partial_u\psi_-], \tag*{(17)} ∂ u m = 2 π r 2 Ω − 2 ( T uv ∂ u r − T uu ∂ v r ) = ν ℑ [ ψ − ψ + ] − r 2 ( 1 − μ ) ℑ [ ψ − ∂ u ψ − ] , (17) ∂ v m = 2 π r 2 Ω − 2 ( T u v ∂ v r − T v v ∂ u r ) = λ ℑ [ ψ − ‾ ψ + ] + ℑ [ ψ + ‾ ∂ v ψ + ] . (18) \partial_v m=2\pi r^2\Omega^{-2}(T_{uv}\partial_v r-T_{vv}\partial_u r)=\lambda\Im[\overline{\psi_-}\psi_+]+\Im[\overline{\psi_+}\partial_v\psi_+]. \tag*{(18)} ∂ v m = 2 π r 2 Ω − 2 ( T uv ∂ v r − T v v ∂ u r ) = λ ℑ [ ψ − ψ + ] + ℑ [ ψ + ∂ v ψ + ] . (18) Throughout the paper, we will employ several reweightings of the spinor components to highlight useful structures and simplify the equations. Of particular importance is the following
Φ + : = κ 1 / 2 ψ + , Φ − : = r ( − ν ) 1 / 2 ψ − , (19) \Phi_+ := \kappa^{1/2}\psi_+,\qquad\Phi_- := r(-\nu)^{1/2}\psi_-, \tag*{(19)} Φ + := κ 1/2 ψ + , Φ − := r ( − ν ) 1/2 ψ − , (19) which may be viewed as a dynamical generalization of the transformation introduced in [61 Definition 3.1] for the Dirac equation on a fixed Schwarzschild background. In terms of these components, (8) –(9) take the particularly symmetric form
∂ u Φ + = − Ω r Φ − , (20) \partial_u\Phi_+ = -\frac{\Omega}{r}\Phi_-, \tag*{(20)} ∂ u Φ + = − r Ω Φ − , (20) ∂ v Φ − = Ω r Φ + . (21) \partial_v\Phi_- = \frac{\Omega}{r}\Phi_+. \tag*{(21)} ∂ v Φ − = r Ω Φ + . (21) Applying ∂ v \partial_v ∂ v (20) and ∂ u \partial_u ∂ u (21) , one readily finds the following wave-like coupled system
∂ u ∂ v Φ + + Ω 2 r 2 Φ + + Ω r ( ∂ v log Ω − λ r − 1 ) Φ − = 0 , (22) \partial_u\partial_v\Phi_+ + \frac{\Omega^2}{r^2}\Phi_+ + \frac{\Omega}{r}(\partial_v\log\Omega-\lambda r^{-1})\Phi_- = 0, \tag*{(22)} ∂ u ∂ v Φ + + r 2 Ω 2 Φ + + r Ω ( ∂ v log Ω − λ r − 1 ) Φ − = 0 , (22) ∂ u ∂ v Φ − + Ω 2 r 2 Φ − − Ω r ( ∂ u log Ω − ν r − 1 ) Φ + = 0. (23) \partial_u\partial_v\Phi_- + \frac{\Omega^2}{r^2}\Phi_- - \frac{\Omega}{r}(\partial_u\log\Omega-\nu r^{-1})\Phi_+ = 0. \tag*{(23)} ∂ u ∂ v Φ − + r 2 Ω 2 Φ − − r Ω ( ∂ u log Ω − ν r − 1 ) Φ + = 0. (23) Remark 2.4. Recalling that the wave equation for a spherically symmetric scalar field ϕ \phi ϕ
∂ u ∂ v ϕ + λ r ∂ u ϕ + ν r ∂ v ϕ = 0 , \partial_u\partial_v\phi+\frac{\lambda}{r}\partial_u\phi+\frac{\nu}{r}\partial_v\phi=0, ∂ u ∂ v ϕ + r λ ∂ u ϕ + r ν ∂ v ϕ = 0 , we may view the Einstein–Dirac system, roughly speaking, as the Einstein equations coupled to two nonlinear coupled wave equations. In contrast to the scalar field case, however, the resulting equations contain additional geometric coupling terms, including derivatives of the dynamical null lapse Ω \Omega Ω
The Schwarzschild family of black holes
Let us now briefly recall the geometric features of the Schwarzschild family that will be relevant for our analysis. As already mentioned, for M > 0 M>0 M > 0
g M = − X ( r ) d t 2 + X ( r ) − 1 d r 2 + r 2 d σ S 2 , X ( r ) : = 1 − 2 M r . g_M=-X(r)dt^2+X(r)^{-1}dr^2+r^2d\sigma_{\mathbb{S}^2},\qquad X(r):=1-\frac{2M}{r}. g M = − X ( r ) d t 2 + X ( r ) − 1 d r 2 + r 2 d σ S 2 , X ( r ) := 1 − r 2 M . These coordinates cover only the domain of outer communication , corresponding to ( t , r , ϑ , φ ) ∈ R × ( 2 M , + ∞ ) × S 2 (t,r,\vartheta,\varphi)\in\mathbb{R}\times(2M,+\infty)\times\mathbb{S}^2 ( t , r , ϑ , φ ) ∈ R × ( 2 M , + ∞ ) × S 2 H + : = { r = 2 M } \mathcal{H}^+:=\{r=2M\} H + := { r = 2 M } tortoise coordinate
r ∗ ( r ) : = ∫ 3 M r d r ′ X ( r ′ ) , r_*(r):=\int_{3M}^{r}\frac{dr'}{X(r')}, r ∗ ( r ) := ∫ 3 M r X ( r ′ ) d r ′ , and the advanced time coordinate v : = 1 2 ( t + r ∗ ) v:=\frac{1}{2}(t+r_*) v := 2 1 ( t + r ∗ ) ingoing Eddington–Finkelstein coordinates , the metric rewrites
g M = − 4 X d v 2 + 4 d v d r + r 2 d σ S 2 , g_M=-4X\,dv^2+4\,dv\,dr+r^2d\sigma_{\mathbb{S}^2}, g M = − 4 X d v 2 + 4 d v d r + r 2 d σ S 2 , Figure 4. The Penrose diagram of a Schwarzschild black hole. Our stability result concerns the exterior region, which is here colored in a darker gray shade.
which extends smoothly across H + \mathcal{H}^{+} H + retarded time coordinate u : = 1 2 ( t − r ∗ ) u := \frac{1}{2}(t-r_{*}) u := 2 1 ( t − r ∗ ) double-null Eddington–Finkelstein coordinates
g M = − 4 X ( u , v ) d u d v + r 2 ( u , v ) d σ S 2 , g_{M}=-4X(u,v)dudv+r^{2}(u,v)d\sigma_{\mathbb{S}^{2}}, g M = − 4 X ( u , v ) d u d v + r 2 ( u , v ) d σ S 2 , where we stressed that now X X X r r r u u u v v v H + \mathcal{H}^{+} H + I + \mathcal{I}^{+} I + u = + ∞ u=+\infty u = + ∞ v = + ∞ v=+\infty v = + ∞ (5) –(7) in these coordinates are given by
m = M , λ = − ν = 1 − μ = Ω 2 = X , κ = γ = 1 , m=M,\qquad\lambda=-\nu=1-\mu=\Omega^{2}=X,\qquad\kappa=\gamma=1, m = M , λ = − ν = 1 − μ = Ω 2 = X , κ = γ = 1 , while, using r ∗ = v − u r_{*}=v-u r ∗ = v − u T = ∂ t = ∂ v + ∂ u T=\partial_{t}=\partial_{v}+\partial_{u} T = ∂ t = ∂ v + ∂ u
Although the normalization r ∗ ( 3 M ) = 0 r_{*}(3M)=0 r ∗ ( 3 M ) = 0 [4 Lemma 2.2] , and apply it here to the Schwarzschild case. The lemma will be useful in Definition 3.6 below.
Lemma 2.5. Let M > 0 M>0 M > 0 R 0 ∈ ( 2 M , + ∞ ) R_{0}\in(2M,+\infty) R 0 ∈ ( 2 M , + ∞ ) ( u 0 , v 0 ) ∈ R 2 (u_{0},v_{0})\in\mathbb{R}^{2} ( u 0 , v 0 ) ∈ R 2 r ˉ : R 2 → ( 2 M , + ∞ ) \bar{r}:\mathbb{R}^{2}\to(2M,+\infty) r ˉ : R 2 → ( 2 M , + ∞ )
g ˉ : = − 4 X ( r ˉ ) d u d v + r ˉ 2 ( u , v ) d σ S 2 (24) \bar{g}:=-4X(\bar{r})dudv+\bar{r}^{2}(u,v)d\sigma_{\mathbb{S}^{2}} \tag*{(24)} g ˉ := − 4 X ( r ˉ ) d u d v + r ˉ 2 ( u , v ) d σ S 2 (24) on R 2 × S 2 \mathbb{R}^{2}\times\mathbb{S}^{2} R 2 × S 2 M M M
m ˉ = M , ∂ v r ˉ = − ∂ u r ˉ = X ( r ˉ ) , r ˉ ( u 0 , v 0 ) = R 0 . \bar{m}=M,\qquad\partial_{v}\bar{r}=-\partial_{u}\bar{r}=X(\bar{r}),\qquad\bar{r}(u_{0},v_{0})=R_{0}. m ˉ = M , ∂ v r ˉ = − ∂ u r ˉ = X ( r ˉ ) , r ˉ ( u 0 , v 0 ) = R 0 . Even though, as previously noted, the event horizon H + \mathcal{H}^{+} H + { u = + ∞ } \{u=+\infty\} { u = + ∞ } (7) up to the horizon as follows:
r ˉ ( ∞ , ⋅ ) = 2 M , λ ˉ ( ∞ , ⋅ ) = 0 , μ ˉ ( ∞ , ⋅ ) = 1. \bar{r}(\infty,\cdot)=2M,\qquad\bar{\lambda}(\infty,\cdot)=0,\qquad\bar{\mu}(\infty,\cdot)=1. r ˉ ( ∞ , ⋅ ) = 2 M , λ ˉ ( ∞ , ⋅ ) = 0 , μ ˉ ( ∞ , ⋅ ) = 1. Remark 2.6 (Dirac on Schwarzschild). On a fixed Schwarzschild background, (22) –(23) therefore reduce to the symmetric hyperbolic system
∂ u ∂ v Φ ± + X r ˉ 2 Φ ± + X r ˉ 3 ( 3 M − r ˉ ) Φ ∓ = 0. \partial_{u}\partial_{v}\Phi_{\pm}+\frac{X}{\bar{r}^{2}}\Phi_{\pm}+\frac{\sqrt{X}}{\bar{r}^{3}}(3M-\bar{r})\Phi_{\mp}=0. ∂ u ∂ v Φ ± + r ˉ 2 X Φ ± + r ˉ 3 X ( 3 M − r ˉ ) Φ ∓ = 0. As highlighted in [61 Remark 4.2] , this feature is crucial in the proof of the Morawetz estimate and it may be regarded as the analogue of Chandrasekhar’s transformation [12 ] for integer spin fields, which recasts the corresponding Teukolsky equations as scalar wave-like equations. To retain this structure in the dynamical setting, we view (22) –(23) as a perturbation of the system above, which requires error estimates for the terms involving Ω \Omega Ω
The characteristic initial value problem
We conclude this section by briefly recalling the characteristic initial value problem in general relativity. Given U 0 < U 1 U_0<U_1 U 0 < U 1 V 0 < V 1 V_0<V_1 V 0 < V 1 bifurcate null hypersurface and the associated diamond-shaped region by
C ( U 0 , U 1 , V 0 , V 1 ) : = ( C i n ( U 0 , U 1 ; V 0 ) ∪ C o u t ( V 0 , V 1 ; U 0 ) ) : = ( [ U 0 , U 1 ] × { V 0 } ) ∪ ( { U 0 } × [ V 0 , V 1 ] ) , \mathcal{C}(U_0,U_1,V_0,V_1):=(\mathcal{C}_{\mathrm{in}}(U_0,U_1;V_0)\cup\mathcal{C}_{\mathrm{out}}(V_0,V_1;U_0)):=([U_0,U_1]\times\{V_0\})\cup(\{U_0\}\times[V_0,V_1]), C ( U 0 , U 1 , V 0 , V 1 ) := ( C in ( U 0 , U 1 ; V 0 ) ∪ C out ( V 0 , V 1 ; U 0 )) := ([ U 0 , U 1 ] × { V 0 }) ∪ ({ U 0 } × [ V 0 , V 1 ]) , R ( U 0 , U 1 , V 0 , V 1 ) : = [ U 0 , U 1 ] × [ V 0 , V 1 ] . \mathcal{R}(U_0,U_1,V_0,V_1):=[U_0,U_1]\times[V_0,V_1]. R ( U 0 , U 1 , V 0 , V 1 ) := [ U 0 , U 1 ] × [ V 0 , V 1 ] . For simplicity, we will omit the dependence on ( U 0 , U 1 , V 0 , V 1 ) (U_0,U_1,V_0,V_1) ( U 0 , U 1 , V 0 , V 1 ) C \mathcal{C} C R \mathcal{R} R f f f C \mathcal{C} C f o u t : = f ∣ C o u t f_{\mathrm{out}}:=f|_{\mathcal{C}_{\mathrm{out}}} f out := f ∣ C out f i n : = f ∣ C i n f_{\mathrm{in}}:=f|_{\mathcal{C}_{\mathrm{in}}} f in := f ∣ C in
Definition 2.7. A smooth bifurcate characteristic initial data set for the spherically symmetric Einstein–massless Dirac system consists of a quadruple of smooth functions ( r ˚ , Ω ˚ , ψ ˚ + o u t , ψ ˚ − i n ) (\mathring{r},\mathring{\Omega},\mathring{\psi}^{\mathrm{out}}_{+},\mathring{\psi}^{\mathrm{in}}_{-}) ( r ˚ , Ω ˚ , ψ ˚ + out , ψ ˚ − in )
r ˚ : C → R + , Ω ˚ 2 : C → R + , ψ ˚ + o u t : C o u t → C , ψ ˚ − i n : C i n → C , \mathring{r}:\mathcal{C}\to\mathbb{R}^{+},\qquad\mathring{\Omega}^{2}:\mathcal{C}\to\mathbb{R}^{+},\qquad\mathring{\psi}^{\mathrm{out}}_{+}:\mathcal{C}_{\mathrm{out}}\to\mathbb{C},\qquad\mathring{\psi}^{\mathrm{in}}_{-}:\mathcal{C}_{\mathrm{in}}\to\mathbb{C}, r ˚ : C → R + , Ω ˚ 2 : C → R + , ψ ˚ + out : C out → C , ψ ˚ − in : C in → C , such that ν ˚ < 0 \mathring{\nu}<0 ν ˚ < 0 C \mathcal{C} C C i n \mathcal{C}_{\mathrm{in}} C in C o u t \mathcal{C}_{\mathrm{out}} C out
Remark 2.8. The choice of the free spinor data on each initial hypersurface is dictated by the transport structure of the reduced Dirac equations (2.5)–(2.6). On the other hand, the assumption ν ˚ < 0 \mathring{\nu}<0 ν ˚ < 0 C \mathcal{C} C initial absence of anti-trapped spheres , is needed to define the renormalized components Φ ± \Phi_{\pm} Φ ±
We next state the local existence result for the characteristic initial value problem for the Einstein–massless Dirac system.
Proposition 2.9. For any U 0 < U 1 U_0<U_1 U 0 < U 1 V 0 < V 1 V_0<V_1 V 0 < V 1 and B > 0 B>0 B > 0 there exists a constant ε s l a b > 0 \varepsilon_{\mathrm{slab}}>0 ε slab > 0 with the following property. Let ( r ˚ , Ω ˚ , ψ ˚ + o u t , ψ ˚ − i n ) (\mathring{r},\mathring{\Omega},\mathring{\psi}^{\mathrm{out}}_{+},\mathring{\psi}^{\mathrm{in}}_{-}) ( r ˚ , Ω ˚ , ψ ˚ + out , ψ ˚ − in ) be a characteristic initial data set for the spherically symmetric Einstein–massless Dirac system on C ( U 0 , U 1 , V 0 , V 1 ) \mathcal{C}(U_0,U_1,V_0,V_1) C ( U 0 , U 1 , V 0 , V 1 ) satisfying
∥ log r ˚ ∥ C 2 ( C ) + ∥ log Ω ˚ ∥ C 2 ( C ) + ∥ ψ ˚ + o u t ∥ C 0 ( C o u t ) + ∥ ∂ v ψ ˚ + o u t ∥ C 0 ( C o u t ) + ∥ ψ ˚ − i n ∥ C 0 ( C i n ) + ∥ ∂ u ψ ˚ − i n ∥ C 0 ( C i n ) ≤ B . \lVert\log\mathring{r}\rVert_{C^{2}(\mathcal{C})}+\lVert\log\mathring{\Omega}\rVert_{C^{2}(\mathcal{C})}+\lVert\mathring{\psi}^{\mathrm{out}}_{+}\rVert_{C^{0}(\mathcal{C}_{\mathrm{out}})}+\lVert\partial_v\mathring{\psi}^{\mathrm{out}}_{+}\rVert_{C^{0}(\mathcal{C}_{\mathrm{out}})}+\lVert\mathring{\psi}^{\mathrm{in}}_{-}\rVert_{C^{0}(\mathcal{C}_{\mathrm{in}})}+\lVert\partial_u\mathring{\psi}^{\mathrm{in}}_{-}\rVert_{C^{0}(\mathcal{C}_{\mathrm{in}})}\leq B. ∥ log r ˚ ∥ C 2 ( C ) + ∥ log Ω ˚ ∥ C 2 ( C ) + ∥ ψ ˚ + out ∥ C 0 ( C out ) + ∥ ∂ v ψ ˚ + out ∥ C 0 ( C out ) + ∥ ψ ˚ − in ∥ C 0 ( C in ) + ∥ ∂ u ψ ˚ − in ∥ C 0 ( C in ) ≤ B . Then there exists a unique smooth solution ( r , Ω , ψ + , ψ − ) (r,\Omega,\psi_{+},\psi_{-}) ( r , Ω , ψ + , ψ − ) to the spherically symmetric Einstein–Dirac system on the slab
D : = R ( U 0 , U 0 + min { ε s l a b , U 1 − U 0 } , V 0 , V 1 ) ∪ R ( U 0 , U 1 , V 0 , V 0 + min { ε s l a b , V 1 − V 0 } ) \mathcal{D}:=\mathcal{R}(U_0,U_0+\min\{\varepsilon_{\mathrm{slab}},U_1-U_0\},V_0,V_1)\cup\mathcal{R}(U_0,U_1,V_0,V_0+\min\{\varepsilon_{\mathrm{slab}},V_1-V_0\}) D := R ( U 0 , U 0 + min { ε slab , U 1 − U 0 } , V 0 , V 1 ) ∪ R ( U 0 , U 1 , V 0 , V 0 + min { ε slab , V 1 − V 0 }) satisfying ν < 0 \nu<0 ν < 0 on D \mathcal{D} D and which extends the initial data. Moreover, the norms
∥ log r ∥ C k ( D ) , ∥ log Ω ∥ C k ( D ) , ∥ ψ + ∥ C k ( D ) , ∥ ψ − ∥ C k ( D ) \lVert\log r\rVert_{C^{k}(\mathcal{D})},\quad\lVert\log\Omega\rVert_{C^{k}(\mathcal{D})},\quad\lVert\psi_{+}\rVert_{C^{k}(\mathcal{D})},\quad\lVert\psi_{-}\rVert_{C^{k}(\mathcal{D})} ∥ log r ∥ C k ( D ) , ∥ log Ω ∥ C k ( D ) , ∥ ψ + ∥ C k ( D ) , ∥ ψ − ∥ C k ( D ) are finite for any k k k and can be bounded in terms of appropriate higher order initial data norms.
Proof. We briefly comment on the proof and refer to Appendix § 11 for the details. Local existence for the characteristic initial value problem in small rectangles follows from a standard iteration argument, applied to the Einstein–Dirac system (2.5)–(2.10), after rewriting the Dirac equations in terms of the renormalized variables Φ ± \Phi_{\pm} Φ ± [23 ] . The extension to thin slabs is then obtained by working similarly to [50 Proposition 3.17] . For more general arguments exploiting the underlying null structure, both for the vacuum Einstein equations and the Einstein–Dirac system without spherical symmetry, we refer to [55 , 80 ] .
Thanks to [7 , 68 ] , we have a natural notion of maximal globally hyperbolic development. In spherical symmetry, this is a subset of the domain of dependence of C \mathcal{C} C ( 1 + 1 ) (1+1) ( 1 + 1 )
Proposition 2.10. Let ( r ˚ , Ω ˚ , ψ ˚ + o u t , ψ ˚ − i n ) (\mathring r,\mathring\Omega,\mathring\psi^{\mathrm{out}}_{+},\mathring\psi^{\mathrm{in}}_{-}) ( r ˚ , Ω ˚ , ψ ˚ + out , ψ ˚ − in ) C ( U 0 , U 1 , V 0 , V 1 ) C(U_0,U_1,V_0,V_1) C ( U 0 , U 1 , V 0 , V 1 ) U 1 U_1 U 1 V 1 V_1 V 1 + ∞ +\infty + ∞ Q max ⊂ R ( U 0 , U 1 , V 0 , V 1 ) \mathcal{Q}_{\max}\subset\mathcal{R}(U_0,U_1,V_0,V_1) Q m a x ⊂ R ( U 0 , U 1 , V 0 , V 1 )
Q max \mathcal{Q}_{\max} Q m a x R u , v 1 + 1 \mathbb{R}^{1+1}_{u,v} R u , v 1 + 1 C C C
The solution ( r , Ω , ψ + , ψ − ) (r,\Omega,\psi_{+},\psi_{-}) ( r , Ω , ψ + , ψ − ) Q max \mathcal{Q}_{\max} Q m a x
Q max \mathcal{Q}_{\max} Q m a x
Stability of Schwarzschild for the Einstein–Dirac system
In this section, we present the notation and framework needed to formalize the statement of Theorem 1 . In particular, in § 5.1 we set up the main definitions, from the seed data to the energy hierarchies employed in the bootstrap argument. § 5.2 is finally devoted to the detailed statement of our main result, including the corresponding bootstrap assumptions. Our setup is inspired by [4 ] .
Setup for the statement of stability
Let us start by fixing the mass parameter M 0 > 0 M_0>0 M 0 > 0
C ^ : = C ^ i n ∪ C ^ o u t = C ( 0 , U ∗ , 0 , + ∞ ) , U ∗ : = 98.5 M 0 , \widehat{C}:=\widehat{C}_{\mathrm{in}}\cup\widehat{C}_{\mathrm{out}}=C(0,U_*,0,+\infty),\qquad U_*:=98.5M_0, C := C in ∪ C out = C ( 0 , U ∗ , 0 , + ∞ ) , U ∗ := 98.5 M 0 , with null coordinates denoted by u ^ \widehat{u} u v ^ \widehat{v} v
m ( 0 , 0 ) = M 0 , r ( 0 , 0 ) = Λ : = 100 M 0 . m(0,0)=M_0,\qquad r(0,0)=\Lambda:=100M_0. m ( 0 , 0 ) = M 0 , r ( 0 , 0 ) = Λ := 100 M 0 . In particular, the bifurcation sphere lies in the exterior region of the reference Schwarzschild solution, since 1 − 2 M 0 Λ > 0 1-\frac{2M_0}{\Lambda}>0 1 − Λ 2 M 0 > 0 M 0 M_0 M 0
Gauge-normalized seed data
As already remarked, the standard monotonicity arguments that provide a priori control of the signs of geometric quantities such as λ \lambda λ ν \nu ν D \mathcal{D} D C ^ \widehat{C} C a posteriori through the bootstrap argument.
Definition 3.1 (ε \varepsilon ε seed data set on C ^ \widehat{C} C
S : = ( ψ ˚ + o u t , ψ ˚ − i n ) , \mathcal{S}:=(\mathring\psi^{\mathrm{out}}_{+},\mathring\psi^{\mathrm{in}}_{-}), S := ( ψ ˚ + out , ψ ˚ − in ) , where ψ ˚ + o u t : C ^ o u t → C \mathring\psi^{\mathrm{out}}_{+}:\widehat{C}_{\mathrm{out}}\to\mathbb{C} ψ ˚ + out : C out → C ψ ˚ − i n : C ^ i n → C \mathring\psi^{\mathrm{in}}_{-}:\widehat{C}_{\mathrm{in}}\to\mathbb{C} ψ ˚ − in : C in → C ε > 0 \varepsilon>0 ε > 0 ε \varepsilon ε seed data norm :
D [ S ] : = sup C ^ i n ( ∣ ψ ˚ − i n ∣ + ∣ ∂ u ^ ψ ˚ − i n ∣ ) + sup C ^ o u t ( ( 1 + v ^ ) ∣ ψ ˚ + o u t ∣ + ( 1 + v ^ ) 2 ∣ ∂ v ^ ψ ˚ + o u t ∣ + ( 1 + v ^ ) 2 ∣ ∂ v ^ ( v ^ ψ ˚ + o u t ) ∣ ) ≤ ε . (25) \mathcal{D}[\mathcal{S}]:=\sup_{\widehat{C}_{\mathrm{in}}}\left(|\mathring\psi^{\mathrm{in}}_{-}|+|\partial_{\widehat{u}}\mathring\psi^{\mathrm{in}}_{-}|\right)+\sup_{\widehat{C}_{\mathrm{out}}}\left((1+\widehat{v})|\mathring\psi^{\mathrm{out}}_{+}|+(1+\widehat{v})^2|\partial_{\widehat{v}}\mathring\psi^{\mathrm{out}}_{+}|+(1+\widehat{v})^2|\partial_{\widehat{v}}(\widehat{v}\mathring\psi^{\mathrm{out}}_{+})|\right)\leq\varepsilon. \tag*{(25)} D [ S ] := C in sup ( ∣ ψ ˚ − in ∣ + ∣ ∂ u ψ ˚ − in ∣ ) + C out sup ( ( 1 + v ) ∣ ψ ˚ + out ∣ + ( 1 + v ) 2 ∣ ∂ v ψ ˚ + out ∣ + ( 1 + v ) 2 ∣ ∂ v ( v ψ ˚ + out ) ∣ ) ≤ ε . (25) Let us now show that for suitably small ε > 0 \varepsilon>0 ε > 0 ε \varepsilon ε Definition 2.7 .
Proposition 3.2. There exists an ε l o c > 0 \varepsilon_{\mathrm{loc}}>0 ε loc > 0 S = ( ψ ˚ + o u t , ψ ˚ − i n ) \mathcal{S}=(\mathring{\psi}_{+}^{\mathrm{out}},\mathring{\psi}_{-}^{\mathrm{in}}) S = ( ψ ˚ + out , ψ ˚ − in ) ε \varepsilon ε C ^ \widehat{\mathcal{C}} C ε ≤ ε l o c \varepsilon\leq\varepsilon_{\mathrm{loc}} ε ≤ ε loc ( r ˚ , Ω ˚ , ψ ˚ + , ψ ˚ − ) (\mathring{r},\mathring{\Omega},\mathring{\psi}_{+},\mathring{\psi}_{-}) ( r ˚ , Ω ˚ , ψ ˚ + , ψ ˚ − ) C ^ \widehat{\mathcal{C}} C ( Q ^ max , r , Ω , ψ + , ψ − ) (\widehat{Q}_{\max},r,\Omega,\psi_{+},\psi_{-}) ( Q m a x , r , Ω , ψ + , ψ − )
r ( 0 , 0 ) = Λ r(0,0)=\Lambda r ( 0 , 0 ) = Λ
m ( 0 , 0 ) = M 0 m(0,0)=M_{0} m ( 0 , 0 ) = M 0
ν = − 1 \nu=-1 ν = − 1 C ^ i n \widehat{\mathcal{C}}_{\mathrm{in}} C in
λ = 1 \lambda=1 λ = 1 C ^ o u t \widehat{\mathcal{C}}_{\mathrm{out}} C out
We refer to characteristic initial data sets obtained from S \mathcal{S} S gauge-normalized .
Proof of Proposition 3.2. For ( u ^ , v ^ ) ∈ C ^ (\widehat{u},\widehat{v})\in\widehat{\mathcal{C}} ( u , v ) ∈ C
r ˚ i n ( u ^ ) : = Λ − u ^ on C ^ i n , r ˚ o u t ( v ^ ) : = Λ + v ^ on C ^ o u t , r ˚ : = { r ˚ i n on C ^ i n , r ˚ o u t on C ^ o u t , \mathring{r}_{\mathrm{in}}(\widehat{u}):=\Lambda-\widehat{u}\quad\text{on }\widehat{\mathcal{C}}_{\mathrm{in}},\qquad\mathring{r}_{\mathrm{out}}(\widehat{v}):=\Lambda+\widehat{v}\quad\text{on }\widehat{\mathcal{C}}_{\mathrm{out}},\qquad\mathring{r}:=\begin{cases}\mathring{r}_{\mathrm{in}}&\text{on }\widehat{\mathcal{C}}_{\mathrm{in}},\\\mathring{r}_{\mathrm{out}}&\text{on }\widehat{\mathcal{C}}_{\mathrm{out}},\end{cases} r ˚ in ( u ) := Λ − u on C in , r ˚ out ( v ) := Λ + v on C out , r ˚ := { r ˚ in r ˚ out on C in , on C out , which implies by construction conditions 1., 3., 4. and r ˚ > 0 \mathring{r}>0 r ˚ > 0
min C ^ r ˚ = r ˚ ( U ∗ , 0 ) = r ( 0 , 0 ) − U ∗ = Λ − U ∗ = 1.5 M 0 + O ( ε ) > 0 , \min_{\widehat{\mathcal{C}}}\mathring{r}=\mathring{r}(U_{*},0)=r(0,0)-U_{*}=\Lambda-U_{*}=1.5M_{0}+O(\varepsilon)>0, C min r ˚ = r ˚ ( U ∗ , 0 ) = r ( 0 , 0 ) − U ∗ = Λ − U ∗ = 1.5 M 0 + O ( ε ) > 0 , for ε \varepsilon ε C ^ i n \widehat{\mathcal{C}}_{\mathrm{in}} C in
Ω ˚ i n 2 ( u ^ ) : = ( 1 − 2 M 0 Λ ) − 1 exp ( − 2 ∫ 0 u ^ ( r ˚ i n ℑ [ ψ ˚ − i n ‾ ∂ u ^ ψ ˚ − i n ] ) ( u ^ ′ ) d u ^ ′ ) , \mathring{\Omega}_{\mathrm{in}}^{2}(\widehat{u}):=\left(1-\frac{2M_{0}}{\Lambda}\right)^{-1}\exp\left(-2\int_{0}^{\widehat{u}}\left(\mathring{r}_{\mathrm{in}}\Im[\overline{\mathring{\psi}_{-}^{\mathrm{in}}}\partial_{\widehat{u}}\mathring{\psi}_{-}^{\mathrm{in}}]\right)(\widehat{u}')\,\mathrm{d}\widehat{u}'\right), Ω ˚ in 2 ( u ) := ( 1 − Λ 2 M 0 ) − 1 exp ( − 2 ∫ 0 u ( r ˚ in ℑ [ ψ ˚ − in ∂ u ψ ˚ − in ] ) ( u ′ ) d u ′ ) , and
ψ ˚ + i n ( u ^ ) : = 1 Ω ˚ i n 2 ( u ^ ) ( Ω ˚ i n 2 ( 0 ) ψ ˚ + o u t ( 0 ) − ∫ 0 u ^ ( Ω ˚ i n 2 ψ ˚ − i n ) ( u ^ ′ ) d u ^ ′ ) \mathring{\psi}_{+}^{\mathrm{in}}(\widehat{u}):=\frac{1}{\mathring{\Omega}_{\mathrm{in}}^{2}(\widehat{u})}\left(\mathring{\Omega}_{\mathrm{in}}^{2}(0)\mathring{\psi}_{+}^{\mathrm{out}}(0)-\int_{0}^{\widehat{u}}\left(\mathring{\Omega}_{\mathrm{in}}^{2}\mathring{\psi}_{-}^{\mathrm{in}}\right)(\widehat{u}')\,\mathrm{d}\widehat{u}'\right) ψ ˚ + in ( u ) := Ω ˚ in 2 ( u ) 1 ( Ω ˚ in 2 ( 0 ) ψ ˚ + out ( 0 ) − ∫ 0 u ( Ω ˚ in 2 ψ ˚ − in ) ( u ′ ) d u ′ ) which are both well-defined by (25) .
On the other hand, on C ^ o u t \widehat{\mathcal{C}}_{\mathrm{out}} C out Ω ˚ o u t 2 \mathring{\Omega}_{\mathrm{out}}^{2} Ω ˚ out 2 ν ˚ o u t \mathring{\nu}_{\mathrm{out}} ν ˚ out m ˚ o u t \mathring{m}_{\mathrm{out}} m ˚ out ψ ˚ − o u t \mathring{\psi}_{-}^{\mathrm{out}} ψ ˚ − out
∂ v ^ Ω ˚ o u t 2 = 2 Ω ˚ o u t 2 r ˚ ( 1 − μ ˚ o u t ) ℑ [ ψ ˚ + o u t ‾ ∂ v ^ ψ ˚ + o u t ] , Ω ˚ o u t 2 ( 0 ) = Ω ˚ i n 2 ( 0 ) = ( 1 − 2 M 0 Λ ) − 1 , ∂ v ^ ν ˚ o u t = − 2 Ω ˚ o u t 2 r ˚ 2 ( m ˚ o u t − r ˚ ℑ [ ψ ˚ − o u t ‾ ψ ˚ + o u t ] ) , ν ˚ o u t ( 0 ) = ν ˚ i n ( 0 ) = − 1 , ∂ v ^ m ˚ o u t = ℑ [ ψ ˚ − o u t ‾ ψ ˚ + o u t ] + ℑ [ ψ ˚ + o u t ‾ ∂ v ^ ψ ˚ + o u t ] , m ˚ o u t ( 0 ) = M 0 , ∂ v ^ ψ ˚ − o u t = − ( ∂ v ^ ν ˚ o u t 2 ν ˚ o u t + 1 r ˚ ) ψ ˚ − o u t + 1 r ˚ 2 ψ ˚ + o u t , ψ ˚ − o u t ( 0 ) = ψ ˚ − i n ( 0 ) . (27) \begin{aligned}
\partial_{\widehat{v}}\mathring{\Omega}_{\mathrm{out}}^{2}&=\frac{2\mathring{\Omega}_{\mathrm{out}}^{2}}{\mathring{r}(1-\mathring{\mu}_{\mathrm{out}})}\Im[\overline{\mathring{\psi}_{+}^{\mathrm{out}}}\partial_{\widehat{v}}\mathring{\psi}_{+}^{\mathrm{out}}],&\mathring{\Omega}_{\mathrm{out}}^{2}(0)&=\mathring{\Omega}_{\mathrm{in}}^{2}(0)=\left(1-\frac{2M_{0}}{\Lambda}\right)^{-1},\\
\partial_{\widehat{v}}\mathring{\nu}_{\mathrm{out}}&=-\frac{2\mathring{\Omega}_{\mathrm{out}}^{2}}{\mathring{r}^{2}}\left(\mathring{m}_{\mathrm{out}}-\mathring{r}\Im[\overline{\mathring{\psi}_{-}^{\mathrm{out}}}\mathring{\psi}_{+}^{\mathrm{out}}]\right),&\mathring{\nu}_{\mathrm{out}}(0)&=\mathring{\nu}_{\mathrm{in}}(0)=-1,\\
\partial_{\widehat{v}}\mathring{m}_{\mathrm{out}}&=\Im[\overline{\mathring{\psi}_{-}^{\mathrm{out}}}\mathring{\psi}_{+}^{\mathrm{out}}]+\Im[\overline{\mathring{\psi}_{+}^{\mathrm{out}}}\partial_{\widehat{v}}\mathring{\psi}_{+}^{\mathrm{out}}],&\mathring{m}_{\mathrm{out}}(0)&=M_{0},\\
\partial_{\widehat{v}}\mathring{\psi}_{-}^{\mathrm{out}}&=-\left(\frac{\partial_{\widehat{v}}\mathring{\nu}_{\mathrm{out}}}{2\mathring{\nu}_{\mathrm{out}}}+\frac{1}{\mathring{r}}\right)\mathring{\psi}_{-}^{\mathrm{out}}+\frac{1}{\mathring{r}^{2}}\mathring{\psi}_{+}^{\mathrm{out}},&\mathring{\psi}_{-}^{\mathrm{out}}(0)&=\mathring{\psi}_{-}^{\mathrm{in}}(0). \tag*{(27)}
\end{aligned} ∂ v Ω ˚ out 2 ∂ v ν ˚ out ∂ v m ˚ out ∂ v ψ ˚ − out = r ˚ ( 1 − μ ˚ out ) 2 Ω ˚ out 2 ℑ [ ψ ˚ + out ∂ v ψ ˚ + out ] , = − r ˚ 2 2 Ω ˚ out 2 ( m ˚ out − r ˚ ℑ [ ψ ˚ − out ψ ˚ + out ] ) , = ℑ [ ψ ˚ − out ψ ˚ + out ] + ℑ [ ψ ˚ + out ∂ v ψ ˚ + out ] , = − ( 2 ν ˚ out ∂ v ν ˚ out + r ˚ 1 ) ψ ˚ − out + r ˚ 2 1 ψ ˚ + out , Ω ˚ out 2 ( 0 ) ν ˚ out ( 0 ) m ˚ out ( 0 ) ψ ˚ − out ( 0 ) = Ω ˚ in 2 ( 0 ) = ( 1 − Λ 2 M 0 ) − 1 , = ν ˚ in ( 0 ) = − 1 , = M 0 , = ψ ˚ − in ( 0 ) . (27) for which the solution exists locally for ( Ω 2 , ν , m , ψ − ) (\Omega^{2},\nu,m,\psi_{-}) ( Ω 2 , ν , m , ψ − ) Ω 2 , − ν , 1 − μ > 0 \Omega^{2},-\nu,1-\mu>0 Ω 2 , − ν , 1 − μ > 0 C ^ o u t \widehat{\mathcal{C}}_{\mathrm{out}} C out
1 2 ≤ 1 − μ ˚ o u t ≤ 2 , ( 1 + v ^ ) ∣ ψ ˚ − o u t ∣ ≤ 2 , (28) \frac{1}{2}\leq1-\mathring{\mu}_{\mathrm{out}}\leq2,\qquad(1+\widehat{v})\lvert\mathring{\psi}_{-}^{\mathrm{out}}\rvert\leq2, \tag*{(28)} 2 1 ≤ 1 − μ ˚ out ≤ 2 , ( 1 + v ) ∣ ψ ˚ − out ∣ ≤ 2 , (28) on { 0 } × [ 0 , v ^ 0 ] \{0\}\times[0,\widehat{v}_{0}] { 0 } × [ 0 , v 0 ] ε \varepsilon ε v ^ 0 \widehat{v}_{0} v 0 (27) , we have
Ω ˚ o u t 2 ( v ^ ) : = ( 1 − 2 M 0 r 0 ) − 1 exp ( ∫ 0 v ^ 2 r ˚ o u t ( 1 − μ ˚ o u t ) ℑ [ ψ ˚ + o u t ‾ ∂ v ^ ψ ˚ + o u t ] d v ^ ′ ) \mathring{\Omega}_{\mathrm{out}}^{2}(\widehat{v}):=\left(1-\frac{2M_{0}}{r_{0}}\right)^{-1}\exp\left(\int_{0}^{\widehat{v}}\frac{2}{\mathring{r}_{\mathrm{out}}(1-\mathring{\mu}_{\mathrm{out}})}\Im[\overline{\mathring{\psi}_{+}^{\mathrm{out}}}\partial_{\widehat{v}}\mathring{\psi}_{+}^{\mathrm{out}}]\,\mathrm{d}\widehat{v}'\right) Ω ˚ out 2 ( v ) := ( 1 − r 0 2 M 0 ) − 1 exp ( ∫ 0 v r ˚ out ( 1 − μ ˚ out ) 2 ℑ [ ψ ˚ + out ∂ v ψ ˚ + out ] d v ′ ) which satisfies 1 2 ≤ Ω ˚ o u t 2 ( v ^ ) ≤ 2 \frac{1}{2} \le\mathring{\Omega}_{\mathrm{out}}^{2}(\hat{v}) \le2 2 1 ≤ Ω ˚ out 2 ( v ^ ) ≤ 2 (25) and (28) . Furthermore, combining again the ε \varepsilon ε v ^ ∈ [ 0 , + ∞ ) \hat{v} \in[0,+\infty) v ^ ∈ [ 0 , + ∞ )
∣ m ˚ o u t ( v ^ ) − M 0 ∣ ≲ ∫ 0 v ^ ( ε ( 1 + v ^ ′ ) − 2 + ε 2 ( 1 + v ^ ′ ) − 3 ) d v ^ ′ ≲ ε , \left|\mathring{m}_{\mathrm{out}}(\hat{v})-M_{0}\right| \lesssim\int_{0}^{\hat{v}}\left(\varepsilon(1+\hat{v}')^{-2}+\varepsilon^{2}(1+\hat{v}')^{-3}\right)\,\mathrm{d}\hat{v}' \lesssim\varepsilon, ∣ m ˚ out ( v ^ ) − M 0 ∣ ≲ ∫ 0 v ^ ( ε ( 1 + v ^ ′ ) − 2 + ε 2 ( 1 + v ^ ′ ) − 3 ) d v ^ ′ ≲ ε , which allows to improve and propagate 1 2 ≤ 1 − μ ˚ o u t ≤ 2 \frac{1}{2} \le1-\mathring{\mu}_{\mathrm{out}} \le2 2 1 ≤ 1 − μ ˚ out ≤ 2 C ^ o u t \hat{C}_{\mathrm{out}} C ^ out (27) , we integrate from the bifurcation sphere and bound
∣ ν ˙ ˚ o u t ( v ^ ) + 1 ∣ ≲ ∫ 0 v ^ ∣ 2 Ω ˚ o u t 2 r ˚ o u t 2 ∣ ( m ˚ o u t + r ˚ o u t ∣ ψ ˚ o u t − ∣ ∣ ψ ˚ o u t + ∣ ) d v ^ ′ ≲ 1 2 , \left|\mathring{\dot{\nu}}_{\mathrm{out}}(\hat{v})+1\right| \lesssim\int_{0}^{\hat{v}}\left|\frac{2\mathring{\Omega}_{\mathrm{out}}^{2}}{\mathring{r}_{\mathrm{out}}^{2}}\right|\left(\mathring{m}_{\mathrm{out}}+\mathring{r}_{\mathrm{out}}|\mathring{\psi}_{\mathrm{out}}^{-}||\mathring{\psi}_{\mathrm{out}}^{+}|\right)\,\mathrm{d}\hat{v}' \lesssim\frac{1}{2}, ν ˙ ˚ out ( v ^ ) + 1 ≲ ∫ 0 v ^ r ˚ out 2 2 Ω ˚ out 2 ( m ˚ out + r ˚ out ∣ ψ ˚ out − ∣∣ ψ ˚ out + ∣ ) d v ^ ′ ≲ 2 1 , implying that − 3 2 ≤ ν ˙ ˚ o u t ≤ − 1 2 -\frac{3}{2} \le\mathring{\dot{\nu}}_{\mathrm{out}} \le-\frac{1}{2} − 2 3 ≤ ν ˙ ˚ out ≤ − 2 1 C ^ o u t \hat{C}_{\mathrm{out}} C ^ out ν ˙ < 0 \dot{\nu}<0 ν ˙ < 0 C ^ \hat{C} C ^
Φ ˚ − o u t : = r ˚ o u t ( − ν ˙ ˚ o u t ) 1 / 2 ψ ˚ − o u t \mathring{\Phi}_{-}^{\mathrm{out}}:=\mathring{r}_{\mathrm{out}}(-\mathring{\dot{\nu}}_{\mathrm{out}})^{1/2}\mathring{\psi}_{-}^{\mathrm{out}} Φ ˚ − out := r ˚ out ( − ν ˙ ˚ out ) 1/2 ψ ˚ − out and combining (27) with (25) and the previous estimates, we readily bound
( 1 + v ^ ) ∣ ψ ˚ − o u t ( v ^ ) ∣ ≲ ∣ Φ ˚ − o u t ( v ^ ) ∣ ≲ Λ ∣ ψ ˚ − i n ( 0 ) ∣ + ∫ 0 v ^ ∣ ∂ v ^ ′ Φ ˚ − o u t ∣ d v ^ ′ ≲ Λ ε + ∫ 0 v ^ ( 1 + v ^ ′ ) − 1 ∣ ψ ˚ + o u t ∣ d v ^ ′ ≤ 3 2 , (1+\hat{v})|\mathring{\psi}_{-}^{\mathrm{out}}(\hat{v})| \lesssim|\mathring{\Phi}_{-}^{\mathrm{out}}(\hat{v})| \lesssim\Lambda|\mathring{\psi}_{-}^{\mathrm{in}}(0)|+\int_{0}^{\hat{v}}|\partial_{\hat{v}'}\mathring{\Phi}_{-}^{\mathrm{out}}|\,\mathrm{d}\hat{v}' \lesssim\Lambda\varepsilon+\int_{0}^{\hat{v}}(1+\hat{v}')^{-1}|\mathring{\psi}_{+}^{\mathrm{out}}|\,\mathrm{d}\hat{v}' \le\frac{3}{2}, ( 1 + v ^ ) ∣ ψ ˚ − out ( v ^ ) ∣ ≲ ∣ Φ ˚ − out ( v ^ ) ∣ ≲ Λ∣ ψ ˚ − in ( 0 ) ∣ + ∫ 0 v ^ ∣ ∂ v ^ ′ Φ ˚ − out ∣ d v ^ ′ ≲ Λ ε + ∫ 0 v ^ ( 1 + v ^ ′ ) − 1 ∣ ψ ˚ + out ∣ d v ^ ′ ≤ 2 3 , for ε \varepsilon ε
In this manner, one may assemble the functions built along the proof to a full initial data set, where r ˚ \mathring{r} r ˚ Ω ˚ 2 \mathring{\Omega}^{2} Ω ˚ 2 ν ˙ ˚ < 0 \mathring{\dot{\nu}}<0 ν ˙ ˚ < 0 (5) , one verifies that condition 2. is also attained, concluding the proof.
Teleological gauges
Let us now introduce the first geometric ingredients that formalize how and in which sense the dynamic evolution will converge to a Schwarzschild spacetime, that is the construction of a teleological gauge and of a proper time function . We follow here the constructions introduced in [4 , 53 ] . The assumptions required for these constructions to be well-defined will be later formalized in the definition of the bootstrap set, see already Definition 3.11. We refer to Figure 5 for a schematic overview of the gauge choice and the notation.
Figure 5. The Penrose diagram of the bootstrap set D τ f D_{\tau_f} D τ f
Let S \mathcal{S} S ε \varepsilon ε ε ≤ ε l o c \varepsilon\le\varepsilon_{\mathrm{loc}} ε ≤ ε loc ( Q ^ max , r , Ω , ψ + , ψ − ) (\hat{Q}_{\max},r,\Omega,\psi_{+},\psi_{-}) ( Q ^ m a x , r , Ω , ψ + , ψ − )
Γ : = { r = Λ } , \Gamma:=\{r=\Lambda\}, Γ := { r = Λ } , which, for ε \varepsilon ε C ^ \hat{C} C ^ Γ \Gamma Γ Q ^ max \hat{Q}_{\max} Q ^ m a x Γ \Gamma Γ τ \tau τ Γ ( 1 ) = ( 0 , 0 ) \Gamma(1)=(0,0) Γ ( 1 ) = ( 0 , 0 ) Γ \Gamma Γ Γ ( τ ) = ( Γ u ^ ( τ ) , Γ v ^ ( τ ) ) \Gamma(\tau)=(\Gamma^{\hat{u}}(\tau),\Gamma^{\hat{v}}(\tau)) Γ ( τ ) = ( Γ u ^ ( τ ) , Γ v ^ ( τ )) ( u ^ , v ^ ) (\hat{u},\hat{v}) ( u ^ , v ^ ) Q ^ max \hat{Q}_{\max} Q ^ m a x
For τ f ∈ [ 1 , ∞ ) \tau_{f}\in[1,\infty) τ f ∈ [ 1 , ∞ ) [ 1 , τ f ] [1,\tau_{f}] [ 1 , τ f ] Γ \Gamma Γ
D ^ τ f : = [ 0 , Γ u ^ ( τ f ) ] × [ 0 , Γ v ^ ( τ f ) ] . \hat{D}_{\tau_{f}}:=[0,\Gamma^{\hat{u}}(\tau_{f})]\times[0,\Gamma^{\hat{v}}(\tau_{f})]. D ^ τ f := [ 0 , Γ u ^ ( τ f )] × [ 0 , Γ v ^ ( τ f )] . Assuming that γ ^ > 0 \hat{\gamma}>0 γ ^ > 0 κ ^ > 0 \hat{\kappa}>0 κ ^ > 0 D ^ τ f \hat{D}_{\tau_{f}} D ^ τ f
Definition 3.4 (Teleologically normalized coordinates). Define u τ f : [ 0 , Γ u ^ ( τ f ) ] → R \mathfrak{u}_{\tau_{f}}:[0,\Gamma^{\hat{u}}(\tau_{f})]\to\mathbb{R} u τ f : [ 0 , Γ u ^ ( τ f )] → R v τ f : [ 0 , Γ v ^ ( τ f ) ] → R \mathfrak{v}_{\tau_{f}}:[0,\Gamma^{\hat{v}}(\tau_{f})]\to\mathbb{R} v τ f : [ 0 , Γ v ^ ( τ f )] → R
u τ f ( u ^ ) : = ∫ 0 u ^ γ ^ ( u ^ ′ , Γ v ^ ( τ f ) ) d u ^ ′ , v τ f ( v ^ ) : = ∫ 0 v ^ κ ^ ( Γ u ^ ( τ f ) , v ^ ′ ) d v ^ ′ . (29) \mathfrak{u}_{\tau_{f}}(\hat{u}):=\int_{0}^{\hat{u}}\hat{\gamma}(\hat{u}',\Gamma^{\hat{v}}(\tau_{f}))\,\mathrm{d}\hat{u}',\qquad
\mathfrak{v}_{\tau_{f}}(\hat{v}):=\int_{0}^{\hat{v}}\hat{\kappa}(\Gamma^{\hat{u}}(\tau_{f}),\hat{v}')\,\mathrm{d}\hat{v}'. \tag*{(29)} u τ f ( u ^ ) := ∫ 0 u ^ γ ^ ( u ^ ′ , Γ v ^ ( τ f )) d u ^ ′ , v τ f ( v ^ ) := ∫ 0 v ^ κ ^ ( Γ u ^ ( τ f ) , v ^ ′ ) d v ^ ′ . (29) which assemble into the map
Φ τ f : D ^ τ f → R 2 \Phi_{\tau_{f}}:\hat{D}_{\tau_{f}}\to\mathbb{R}^{2} Φ τ f : D ^ τ f → R 2 ( u ^ , v ^ ) ↦ ( u τ f ( u ^ ) , v τ f ( v ^ ) ) , (30) (\hat{u},\hat{v})\mapsto(\mathfrak{u}_{\tau_{f}}(\hat{u}),\mathfrak{v}_{\tau_{f}}(\hat{v})), \tag*{(30)} ( u ^ , v ^ ) ↦ ( u τ f ( u ^ ) , v τ f ( v ^ )) , (30) which is a diffeomorphism onto its image. We denote the image of Φ τ f \Phi_{\tau_{f}} Φ τ f D τ f D_{\tau_{f}} D τ f ( u τ f , v τ f ) = Φ τ f ( u ^ , v ^ ) (u_{\tau_{f}},v_{\tau_{f}})=\Phi_{\tau_{f}}(\hat{u},\hat{v}) ( u τ f , v τ f ) = Φ τ f ( u ^ , v ^ ) teleologically normalized coordinates , and we denote by Φ τ f − 1 \Phi_{\tau_{f}}^{-1} Φ τ f − 1
From Remark 2.3, the solution ( r , Ω ^ , ψ + , ψ − ) (r,\widehat{\Omega},\psi_+,\psi_-) ( r , Ω , ψ + , ψ − ) ( u τ f , v τ f ) (u_{\tau_f},v_{\tau_f}) ( u τ f , v τ f ) ( r τ f , Ω τ f , ψ + , τ f , ψ − , τ f ) (r_{\tau_f},\Omega_{\tau_f},\psi_{+,\tau_f},\psi_{-,\tau_f}) ( r τ f , Ω τ f , ψ + , τ f , ψ − , τ f )
r τ f : = r ∘ Φ ^ τ f , Ω τ f : = 1 u τ f ′ ∘ u τ f − 1 1 v τ f ′ ∘ v τ f − 1 Ω ^ ∘ Φ ^ τ f , ψ ± , τ f : = ψ ± ∘ Φ ^ τ f . r_{\tau_f}:=r\circ\widehat{\Phi}_{\tau_f},\qquad\Omega_{\tau_f}:=\frac{1}{u_{\tau_f}'\circ u_{\tau_f}^{-1}}\frac{1}{v_{\tau_f}'\circ v_{\tau_f}^{-1}}\widehat{\Omega}\circ\widehat{\Phi}_{\tau_f},\qquad\psi_{\pm,\tau_f}:=\psi_\pm\circ\widehat{\Phi}_{\tau_f}. r τ f := r ∘ Φ τ f , Ω τ f := u τ f ′ ∘ u τ f − 1 1 v τ f ′ ∘ v τ f − 1 1 Ω ∘ Φ τ f , ψ ± , τ f := ψ ± ∘ Φ τ f . To simplify the notation, we omit the dependence τ f \tau_f τ f ( u , v ) (u,v) ( u , v ) ( r , Ω , ψ + , ψ − ) (r,\Omega,\psi_+,\psi_-) ( r , Ω , ψ + , ψ − ) τ f \tau_f τ f
In the ( u τ f , v τ f ) (u_{\tau_f},v_{\tau_f}) ( u τ f , v τ f ) Γ \Gamma Γ Γ ( τ ) = ( Γ u τ f ( τ ) , Γ v τ f ( τ ) ) \Gamma(\tau)=(\Gamma^{u_{\tau_f}}(\tau),\Gamma^{v_{\tau_f}}(\tau)) Γ ( τ ) = ( Γ u τ f ( τ ) , Γ v τ f ( τ )) τ \tau τ D τ f \mathcal{D}_{\tau_f} D τ f
τ ( u τ f , v τ f ) : = { τ : Γ u τ f ( τ ) = u τ f if r ( u τ f , v τ f ) ≥ Λ , τ : Γ v τ f ( τ ) = v τ f if r ( u τ f , v τ f ) < Λ , (31) \tau(u_{\tau_f},v_{\tau_f}):=
\begin{cases}
\tau:\Gamma^{u_{\tau_f}}(\tau)=u_{\tau_f} & \text{if }r(u_{\tau_f},v_{\tau_f})\ge\Lambda,\\
\tau:\Gamma^{v_{\tau_f}}(\tau)=v_{\tau_f} & \text{if }r(u_{\tau_f},v_{\tau_f})<\Lambda,
\end{cases} \tag*{(31)} τ ( u τ f , v τ f ) := { τ : Γ u τ f ( τ ) = u τ f τ : Γ v τ f ( τ ) = v τ f if r ( u τ f , v τ f ) ≥ Λ , if r ( u τ f , v τ f ) < Λ , (31) whose behaviour qualitatively reflects that of Bondi time in the far region and of Eddington–Finkelstein time near the event horizon. Next, we define the following null segments on D τ f \mathcal{D}_{\tau_f} D τ f
C ‾ v τ f : = ( [ 0 , Γ u τ f ( τ f ) ] × { v } ) ∩ { r ≤ Λ } , C u τ f : = ( { u } × [ 0 , Γ v τ f ( τ f ) ] ) ∩ { r ≥ Λ } , \underline{C}^{\tau_f}_v:=([0,\Gamma^{u_{\tau_f}}(\tau_f)]\times\{v\})\cap\{r\le\Lambda\},\qquad C^{\tau_f}_u:=\bigl(\{u\}\times[0,\Gamma^{v_{\tau_f}}(\tau_f)]\bigr)\cap\{r\ge\Lambda\}, C v τ f := ([ 0 , Γ u τ f ( τ f )] × { v }) ∩ { r ≤ Λ } , C u τ f := ( { u } × [ 0 , Γ v τ f ( τ f )] ) ∩ { r ≥ Λ } , H v τ f : = ( [ 0 , Γ u τ f ( τ f ) ] × { v } ) ∩ { r ≥ Λ } , H u τ f : = ( { u } × [ 0 , Γ v τ f ( τ f ) ] ) ∩ { r ≤ Λ } . H^{\tau_f}_v:=([0,\Gamma^{u_{\tau_f}}(\tau_f)]\times\{v\})\cap\{r\ge\Lambda\},\qquad H^{\tau_f}_u:=\bigl(\{u\}\times[0,\Gamma^{v_{\tau_f}}(\tau_f)]\bigr)\cap\{r\le\Lambda\}. H v τ f := ([ 0 , Γ u τ f ( τ f )] × { v }) ∩ { r ≥ Λ } , H u τ f := ( { u } × [ 0 , Γ v τ f ( τ f )] ) ∩ { r ≤ Λ } . where again we will omit the dependence on τ f \tau_f τ f Γ = { r = Λ } \Gamma=\{r=\Lambda\} Γ = { r = Λ } v v v ( R × { v } ) ∩ Γ (\mathbb{R}\times\{v\})\cap\Gamma ( R × { v }) ∩ Γ ( u Λ ( v ) , v ) (u^\Lambda(v),v) ( u Λ ( v ) , v ) u u u ( { u } × R ) ∩ Γ (\{u\}\times\mathbb{R})\cap\Gamma ({ u } × R ) ∩ Γ ( u , v Λ ( u ) ) (u,v^\Lambda(u)) ( u , v Λ ( u ))
C ‾ τ f ( τ ) : = C ‾ Γ v τ f ( τ ) τ f , C τ f ( τ ) : = C Γ u τ f ( τ ) τ f . \underline{C}^{\tau_f}(\tau):=\underline{C}^{\tau_f}_{\Gamma^{v_{\tau_f}}(\tau)},\qquad C^{\tau_f}(\tau):=C^{\tau_f}_{\Gamma^{u_{\tau_f}}(\tau)}. C τ f ( τ ) := C Γ v τ f ( τ ) τ f , C τ f ( τ ) := C Γ u τ f ( τ ) τ f . Let us now comment on this construction, giving some intuition for its meaning and emphasizing its teleological aspect in the overall picture. For the seed data sets under consideration, Γ \Gamma Γ τ ∈ [ 1 , + ∞ ) \tau\in[1,+\infty) τ ∈ [ 1 , + ∞ ) τ ↦ Γ u ^ ( τ ) \tau\mapsto\Gamma^{\widehat{u}}(\tau) τ ↦ Γ u ( τ )
u ^ H + : = lim τ → + ∞ Γ u ^ ( τ ) \widehat{u}_{\mathcal{H}^+}:=\lim_{\tau\to+\infty}\Gamma^{\widehat{u}}(\tau) u H + := τ → + ∞ lim Γ u ( τ ) is well-defined and satisfies the strict inequality
u ^ H + < U ∗ , \widehat{u}_{\mathcal{H}^+}<U_*, u H + < U ∗ , see already Lemma 7.5 below. We therefore set
D ^ ∞ : = [ 0 , u ^ H + ) × [ 0 , + ∞ ) , \widehat{\mathcal{D}}_\infty:=[0,\widehat{u}_{\mathcal{H}^+})\times[0,+\infty), D ∞ := [ 0 , u H + ) × [ 0 , + ∞ ) , which represents the final asymptotic black hole exterior. Moreover, there exist surjective and strictly increasing C 1 C^1 C 1 ( u ∞ , v ∞ ) : [ 0 , u ^ H + ) × [ 0 , + ∞ ) → [ 0 , + ∞ ) × [ 0 , + ∞ ) (u_\infty,v_\infty):[0,\widehat{u}_{\mathcal{H}^+})\times[0,+\infty)\to[0,+\infty)\times[0,+\infty) ( u ∞ , v ∞ ) : [ 0 , u H + ) × [ 0 , + ∞ ) → [ 0 , + ∞ ) × [ 0 , + ∞ ) ∂ u ∞ r → − 1 \partial_{u_\infty}r\to-1 ∂ u ∞ r → − 1 ∂ v ∞ r → 1 \partial_{v_\infty}r\to1 ∂ v ∞ r → 1 I + \mathcal{I}^+ I + D ∞ \mathcal{D}_\infty D ∞ D ^ ∞ \widehat{\mathcal{D}}_\infty D ∞ eschatological gauge . Their construction will be given in § 9.2 .
Furthermore the geometric estimates established below will imply that, for R > 2 M 0 + O ( ε ) R>2M_0+O(\varepsilon) R > 2 M 0 + O ( ε ) Γ R : = { r = R } \Gamma_R:=\{r=R\} Γ R := { r = R } D τ f \mathcal{D}_{\tau_f} D τ f τ f ∈ [ 1 , + ∞ ) \tau_f\in[1,+\infty) τ f ∈ [ 1 , + ∞ )
Anchored background solution
Since we are aiming to converge to a Schwarzschild black hole with a final mass M ∞ M_\infty M ∞
Definition 3.6. Assume τ f < ∞ \tau_f < \infty τ f < ∞ ( r τ f , Ω τ f , ψ + , τ f , ψ − , τ f ) (r_{\tau_f}, \Omega_{\tau_f}, \psi_{+,\tau_f}, \psi_{-,\tau_f}) ( r τ f , Ω τ f , ψ + , τ f , ψ − , τ f ) D τ f D_{\tau_f} D τ f (29) , we define the τ f \tau_f τ f ( r ˉ τ f , Ω ˉ τ f ) (\bar{r}_{\tau_f}, \bar{\Omega}_{\tau_f}) ( r ˉ τ f , Ω ˉ τ f )
M τ f : = m τ f ( Γ ( τ f ) ) (32) M_{\tau_f} := m_{\tau_f}(\Gamma(\tau_f)) \tag*{(32)} M τ f := m τ f ( Γ ( τ f )) (32) in Eddington–Finkelstein double-null form (24) which is uniquely determined, according to Lemma 2.5, by the condition
r ˉ τ f ( Γ ( τ f ) ) = Λ . \bar{r}_{\tau_f}(\Gamma(\tau_f)) = \Lambda. r ˉ τ f ( Γ ( τ f )) = Λ. Given the anchored background solution, from now on we denote with barred quantities (e.g. m ˉ τ f = M τ f \bar{m}_{\tau_f}=M_{\tau_f} m ˉ τ f = M τ f λ ˉ τ f = − ν ˉ τ f = Ω ˉ τ f 2 = 1 − 2 M τ f r ˉ τ f − 1 \bar{\lambda}_{\tau_f}=-\bar{\nu}_{\tau_f}=\bar{\Omega}_{\tau_f}^{2}=1-2M_{\tau_f}\bar{r}_{\tau_f}^{-1} λ ˉ τ f = − ν ˉ τ f = Ω ˉ τ f 2 = 1 − 2 M τ f r ˉ τ f − 1 r ~ τ f : = r τ f − r ˉ τ f \widetilde{r}_{\tau_f}:=r_{\tau_f}-\bar{r}_{\tau_f} r τ f := r τ f − r ˉ τ f m ~ τ f : = m τ f − M τ f \widetilde{m}_{\tau_f}:=m_{\tau_f}-M_{\tau_f} m τ f := m τ f − M τ f κ ~ τ f : = κ τ f − 1 \widetilde{\kappa}_{\tau_f}:=\kappa_{\tau_f}-1 κ τ f := κ τ f − 1
Since in the proof of stability, τ f → ∞ \tau_f \to\infty τ f → ∞ τ f = ∞ \tau_f=\infty τ f = ∞ [4 Section 3.3] , instead of anchoring at "τ f = ∞ \tau_f=\infty τ f = ∞ Γ ∩ C ^ = { ( 0 , 0 ) } \Gamma\cap\widehat{C}=\{(0,0)\} Γ ∩ C = {( 0 , 0 )} Lemma 7.10 that the following limit exists
lim τ f → + ∞ m τ f = M ∞ , \lim_{\tau_f\to+\infty}m_{\tau_f}=M_\infty, τ f → + ∞ lim m τ f = M ∞ , we define the ∞ \infty ∞ ( r ˉ ∞ , Ω ˉ ∞ 2 ) (\bar{r}_\infty,\bar{\Omega}_\infty^2) ( r ˉ ∞ , Ω ˉ ∞ 2 ) M ∞ M_\infty M ∞ (24) with the following property (see already Proposition 7.12 ):
r ˉ ∞ ( 0 , 0 ) = r ˉ ⋆ : = lim τ f → + ∞ r ˉ τ f ( 0 , 0 ) , \bar{r}_\infty(0,0)=\bar{r}_\star:=\lim_{\tau_f\to+\infty}\bar{r}_{\tau_f}(0,0), r ˉ ∞ ( 0 , 0 ) = r ˉ ⋆ := τ f → + ∞ lim r ˉ τ f ( 0 , 0 ) , as depicted in Figure 6 . Moreover, (180) will imply that this background solution is indeed anchored "at τ f = ∞ \tau_f=\infty τ f = ∞
Figure 6. The Penrose diagram of a Schwarzschild black hole, where we draw the foliations C ( τ ) \mathcal{C}(\tau) C ( τ ) C ( τ ) \mathcal{C}(\tau) C ( τ ) τ \tau τ
lim τ → + ∞ r ˉ ∞ ( Γ ( τ ) ) = Λ . \lim_{\tau\to+\infty}\bar{r}_\infty(\Gamma(\tau))=\Lambda. τ → + ∞ lim r ˉ ∞ ( Γ ( τ )) = Λ. Energy fluxes for the Dirac field
We now introduce the final ingredients needed for the precise statement of our main result, namely the quantities used to measure and control the Dirac field along the null hypersurfaces introduced above.
Definition 3.7. Let ( r τ f , Ω τ f , ψ + , τ f , ψ − , τ f ) (r_{\tau_f},\Omega_{\tau_f},\psi_{+,\tau_f},\psi_{-,\tau_f}) ( r τ f , Ω τ f , ψ + , τ f , ψ − , τ f ) D τ f D_{\tau_f} D τ f ( u τ f , v τ f ) (u_{\tau_f},v_{\tau_f}) ( u τ f , v τ f ) τ f ∈ [ 1 , ∞ ] \tau_f\in[1,\infty] τ f ∈ [ 1 , ∞ ] τ , τ ′ ∈ [ 1 , τ f ] \tau,\tau'\in[1,\tau_f] τ , τ ′ ∈ [ 1 , τ f ] p ∈ [ 0 , 3 ) p\in[0,3) p ∈ [ 0 , 3 ) ( u , v ) ∈ D τ f (u,v)\in D_{\tau_f} ( u , v ) ∈ D τ f
The energy flux to the horizon
E τ f ( τ ) : = ∫ C τ f ( τ ) ( − ν ˉ τ f ∣ ψ + , τ f ∣ 2 − ν ˉ τ f ∣ ψ − , τ f ∣ 2 + ∣ ∂ u τ f ψ − , τ f ∣ 2 − ν ˉ τ f ) d u τ f ; \mathcal{E}^{\tau_f}(\tau):=\int_{\mathcal{C}^{\tau_f}(\tau)}\left(-\bar{\nu}_{\tau_f}\lvert\psi_{+,\tau_f}\rvert^2-\bar{\nu}_{\tau_f}\lvert\psi_{-,\tau_f}\rvert^2+\frac{\lvert\partial_{u_{\tau_f}}\psi_{-,\tau_f}\rvert^2}{-\bar{\nu}_{\tau_f}}\right)\,\mathrm{d}u_{\tau_f}; E τ f ( τ ) := ∫ C τ f ( τ ) ( − ν ˉ τ f ∣ ψ + , τ f ∣ 2 − ν ˉ τ f ∣ ψ − , τ f ∣ 2 + − ν ˉ τ f ∣ ∂ u τ f ψ − , τ f ∣ 2 ) d u τ f ; The r p r^p r p
E p τ f ( τ ) : = ∫ C τ f ( τ ) ( r τ f p ∣ ∂ v τ f ( r τ f ψ + , τ f ) ∣ 2 + c p ∗ r τ f p ∗ + 2 ∣ ∂ v τ f ψ + , τ f ∣ 2 + c p ∗ r τ f p ∗ ∣ ψ + , τ f ∣ 2 + ∣ ψ − , τ f ∣ 2 ) d v τ f , \mathcal{E}^{\tau_f}_{p}(\tau):=\int_{\mathcal{C}^{\tau_f}(\tau)}\left(r_{\tau_f}^{p}\lvert\partial_{v_{\tau_f}}(r_{\tau_f}\psi_{+,\tau_f})\rvert^2+c_{p_*}r_{\tau_f}^{p_*+2}\lvert\partial_{v_{\tau_f}}\psi_{+,\tau_f}\rvert^2+c_{p_*}r_{\tau_f}^{p_*}\lvert\psi_{+,\tau_f}\rvert^2+\lvert\psi_{-,\tau_f}\rvert^2\right)\,\mathrm{d}v_{\tau_f}, E p τ f ( τ ) := ∫ C τ f ( τ ) ( r τ f p ∣ ∂ v τ f ( r τ f ψ + , τ f ) ∣ 2 + c p ∗ r τ f p ∗ + 2 ∣ ∂ v τ f ψ + , τ f ∣ 2 + c p ∗ r τ f p ∗ ∣ ψ + , τ f ∣ 2 + ∣ ψ − , τ f ∣ 2 ) d v τ f , where p ∗ : = p p_*:=p p ∗ := p p ∈ [ 0 , 1 ) p\in[0,1) p ∈ [ 0 , 1 ) c p ∗ : = ( p ∗ − 1 ) 2 c_{p_*}:=(p_*-1)^2 c p ∗ := ( p ∗ − 1 ) 2
The flux along ingoing cones in the far region
F τ f ( v , τ ′ ) : = ∫ H v τ f ∩ { τ ≥ τ ′ } ( r τ f − 2 ∣ ψ + , τ f ∣ 2 + r τ f 2 ∣ ψ − , τ f ∣ 2 + r τ f 2 ∣ ∂ u τ f ψ − , τ f ∣ 2 ) d u τ f ; \mathcal{F}^{\tau_f}(v,\tau'):=\int_{\mathcal{H}^{\tau_f}_v\cap\{\tau\ge\tau'\}}\left(r_{\tau_f}^{-2}\lvert\psi_{+,\tau_f}\rvert^2+r_{\tau_f}^{2}\lvert\psi_{-,\tau_f}\rvert^2+r_{\tau_f}^{2}\lvert\partial_{u_{\tau_f}}\psi_{-,\tau_f}\rvert^2\right)\,\mathrm{d}u_{\tau_f}; F τ f ( v , τ ′ ) := ∫ H v τ f ∩ { τ ≥ τ ′ } ( r τ f − 2 ∣ ψ + , τ f ∣ 2 + r τ f 2 ∣ ψ − , τ f ∣ 2 + r τ f 2 ∣ ∂ u τ f ψ − , τ f ∣ 2 ) d u τ f ; The flux along outgoing cones in the near region
F τ f ( u , τ ′ ) : = ∫ H u τ f ∩ { τ ≥ τ ′ } ( ∣ ψ + , τ f ∣ 2 + ∣ ψ − , τ f ∣ 2 + ∣ ∂ v τ f ψ + , τ f ∣ 2 ) d v τ f . \mathcal{F}^{\tau_f}(u,\tau'):=\int_{\mathcal{H}^{\tau_f}_u\cap\{\tau\ge\tau'\}}\left(\lvert\psi_{+,\tau_f}\rvert^2+\lvert\psi_{-,\tau_f}\rvert^2+\lvert\partial_{v_{\tau_f}}\psi_{+,\tau_f}\rvert^2\right)\,\mathrm{d}v_{\tau_f}. F τ f ( u , τ ′ ) := ∫ H u τ f ∩ { τ ≥ τ ′ } ( ∣ ψ + , τ f ∣ 2 + ∣ ψ − , τ f ∣ 2 + ∣ ∂ v τ f ψ + , τ f ∣ 2 ) d v τ f . Remark 3.8. The choice of derivative terms in the above fluxes exploits the transport structure of the reduced Dirac equations (8) –(9) . In particular, the null derivatives not appearing explicitly in the definitions can be recovered from the Dirac equations in terms of the spinor components and the geometric coefficients. Thus, the fluxes only involve the derivatives needed to control the Dirac field along the corresponding null hypersurfaces.
Remark 3.9. The dependence of E p τ f \mathcal{E}^{\tau_f}_{p} E p τ f p ∗ p_* p ∗ c p ∗ c_{p_*} c p ∗ [4 Remark 3.6] and we report here the reason for the sake of completeness. For p ≥ 1 p\ge1 p ≥ 1 r τ f p + 2 ∣ ∂ v τ f ψ + , τ f ∣ 2 + r τ f p ∣ ψ + , τ f ∣ 2 r_{\tau_f}^{p+2}\lvert\partial_{v_{\tau_f}}\psi_{+,\tau_f}\rvert^2+r_{\tau_f}^{p}\lvert\psi_{+,\tau_f}\rvert^2 r τ f p + 2 ∣ ∂ v τ f ψ + , τ f ∣ 2 + r τ f p ∣ ψ + , τ f ∣ 2 v τ f v_{\tau_f} v τ f τ f \tau_f τ f c p ∗ c_{p_*} c p ∗ r τ f p ∣ ∂ v τ f Φ ~ + , τ f ∣ 2 r_{\tau_f}^{p}\lvert\partial_{v_{\tau_f}}\widetilde{\Phi}_{+,\tau_f}\rvert^2 r τ f p ∣ ∂ v τ f Φ + , τ f ∣ 2 p < 1 p<1 p < 1 p ↗ 1 p\nearrow1 p ↗ 1 Lemma 6.13 ).
Precise statement of the main result and the bootstrap set
We are now in position to give the precise formulation of Theorem 1 stated in the Introduction.
Theorem 2 (Stability of Schwarzschild for Einstein–massless Dirac in spherical symmetry). Let M 0 > 0 M_0>0 M 0 > 0 0 < δ < 0.01 0<\delta<0.01 0 < δ < 0.01 ε s t a b ( M 0 , δ ) > 0 \varepsilon_{\mathrm{stab}}(M_0,\delta)>0 ε stab ( M 0 , δ ) > 0 C ( M 0 , δ ) C(M_0,\delta) C ( M 0 , δ ) ≲ \lesssim ≲ ε \varepsilon ε S S S 0 < ε ≤ ε s t a b 0<\varepsilon\le\varepsilon_{\mathrm{stab}} 0 < ε ≤ ε stab
Existence of a black hole region: Denoting by ( Q ^ max , r , Ω 2 , ψ + , ψ − ) (\widehat{Q}_{\max},r,\Omega^2,\psi_+,\psi_-) ( Q m a x , r , Ω 2 , ψ + , ψ − ) S S S u ^ H + ∈ ( 0 , U ∗ ) \widehat{u}_{\mathcal{H}^+}\in(0,U_*) u H + ∈ ( 0 , U ∗ ) M ∞ > 0 M_\infty>0 M ∞ > 0
[ 0 , u ^ H + ) × [ 0 , + ∞ ) ⊂ Q ^ max , lim v ^ → + ∞ r ( u ^ , v ^ ) = + ∞ for every u ^ ∈ [ 0 , u ^ H + ) , lim v ^ → + ∞ r ( u ^ H + , v ^ ) = 2 M ∞ . [0,\widehat{u}_{\mathcal{H}^+})\times[0,+\infty)\subset\widehat{Q}_{\max},\qquad\lim_{\widehat{v}\to+\infty}r(\widehat{u},\widehat{v})=+\infty\ \text{for every }\widehat{u}\in[0,\widehat{u}_{\mathcal{H}^+}),\qquad\lim_{\widehat{v}\to+\infty}r(\widehat{u}_{\mathcal{H}^+},\widehat{v})=2M_\infty. [ 0 , u H + ) × [ 0 , + ∞ ) ⊂ Q m a x , v → + ∞ lim r ( u , v ) = + ∞ for every u ∈ [ 0 , u H + ) , v → + ∞ lim r ( u H + , v ) = 2 M ∞ . Therefore, [ 0 , u ^ H + ) × { v ^ = + ∞ } [0,\widehat{u}_{\mathcal{H}^+})\times\{\widehat{v}=+\infty\} [ 0 , u H + ) × { v = + ∞ } I + \mathcal{I}^+ I +
B H : = Q ^ max ∖ J − ( I + ) \mathcal{BH}:=\widehat{Q}_{\max}\setminus\mathcal{J}^{-}(\mathcal{I}^+) B H := Q m a x ∖ J − ( I + ) and H + : = ∂ J − ( I + ) = { u ^ H + } × [ 0 , + ∞ ) \mathcal{H}^{+} := \partial J^{-}(\mathcal{I}^{+}) = \{\hat{u}_{\mathcal{H}^{+}}\} \times[0,+\infty) H + := ∂ J − ( I + ) = { u ^ H + } × [ 0 , + ∞ ) [15 ] . Moreover, there exist C 2 C^{2} C 2 ( u ∞ , v ∞ ) (u_{\infty},v_{\infty}) ( u ∞ , v ∞ ) [ 0 , u ^ H + ) × [ 0 , + ∞ ) [0,\hat{u}_{\mathcal{H}^{+}}) \times[0,+\infty) [ 0 , u ^ H + ) × [ 0 , + ∞ ) u ∞ u_{\infty} u ∞ H + \mathcal{H}^{+} H + { u ∞ = + ∞ } \{u_{\infty}=+\infty\} { u ∞ = + ∞ } ∂ u ∞ r → − 1 \partial_{u_{\infty}}r \to-1 ∂ u ∞ r → − 1
Orbital stability: There exists an ∞ \infty ∞ r ˉ ∞ \bar{r}_{\infty} r ˉ ∞ ( u ∞ , v ∞ ) (u_{\infty},v_{\infty}) ( u ∞ , v ∞ )
∣ M ∞ − M 0 ∣ ≲ ε . (33) |M_{\infty}-M_{0}| \lesssim\varepsilon. \tag*{(33)} ∣ M ∞ − M 0 ∣ ≲ ε . (33) Relative to this background solution, the energy of the Dirac field is bounded by its initial value
sup τ ∈ [ 1 , + ∞ ) ( E ∞ ( τ ) + E 3 − δ ∞ ( τ ) ) ≲ E ∞ ( 1 ) + E 3 − δ ∞ ( 1 ) ≲ ε 2 , (34) \sup_{\tau\in[1,+\infty)}\left(\mathcal{E}^{\infty}(\tau)+\mathcal{E}^{\infty}_{3-\delta}(\tau)\right) \lesssim\mathcal{E}^{\infty}(1)+\mathcal{E}^{\infty}_{3-\delta}(1) \lesssim\varepsilon^{2}, \tag*{(34)} τ ∈ [ 1 , + ∞ ) sup ( E ∞ ( τ ) + E 3 − δ ∞ ( τ ) ) ≲ E ∞ ( 1 ) + E 3 − δ ∞ ( 1 ) ≲ ε 2 , (34) and the Dirac field is pointwise bounded in C 1 C^{1} C 1
sup J − ( I + ) ( r ∣ ψ + ∣ + r ∣ ψ − ∣ + r 2 ∣ ∂ v ^ ψ + ∣ + r 2 ∣ ∂ v ^ ( r ψ + ) ∣ + r ∣ ∂ u ^ ψ − − v ^ ∣ ) \sup_{J^{-}(\mathcal{I}^{+})}\left(r|\psi_{+}|+r|\psi_{-}|+r^{2}|\partial_{\hat{v}}\psi_{+}|+r^{2}|\partial_{\hat{v}}(r\psi_{+})|+r\left|\frac{\partial_{\hat{u}}\psi_{-}}{-\hat{v}}\right|\right) J − ( I + ) sup ( r ∣ ψ + ∣ + r ∣ ψ − ∣ + r 2 ∣ ∂ v ^ ψ + ∣ + r 2 ∣ ∂ v ^ ( r ψ + ) ∣ + r − v ^ ∂ u ^ ψ − ) ≲ sup J − ( I + ) ∩ C ^ ( r ∣ ψ + ∣ + r ∣ ψ − ∣ + r 2 ∣ ∂ v ^ ψ + ∣ + r 2 ∣ ∂ v ^ ( r ψ + ) ∣ + r ∣ ∂ u ^ ψ − − v ^ ∣ ) ≲ ε . (35) \lesssim\sup_{J^{-}(\mathcal{I}^{+})\cap\hat{\mathcal{C}}}\left(r|\psi_{+}|+r|\psi_{-}|+r^{2}|\partial_{\hat{v}}\psi_{+}|+r^{2}|\partial_{\hat{v}}(r\psi_{+})|+r\left|\frac{\partial_{\hat{u}}\psi_{-}}{-\hat{v}}\right|\right)\lesssim\varepsilon. \tag*{(35)} ≲ J − ( I + ) ∩ C ^ sup ( r ∣ ψ + ∣ + r ∣ ψ − ∣ + r 2 ∣ ∂ v ^ ψ + ∣ + r 2 ∣ ∂ v ^ ( r ψ + ) ∣ + r − v ^ ∂ u ^ ψ − ) ≲ ε . (35) Asymptotic stability: The geometry decays towards the ∞ \infty ∞
∣ γ ∞ − 1 ∣ ≲ ε 2 r − 2 τ − 3 + δ , (36) |\gamma_{\infty}-1| \lesssim\varepsilon^{2}r^{-2}\tau^{-3+\delta}, \tag*{(36)} ∣ γ ∞ − 1∣ ≲ ε 2 r − 2 τ − 3 + δ , (36) in D ∞ ∩ { r ≥ Λ } \mathcal{D}_{\infty}\cap\{r\geq\Lambda\} D ∞ ∩ { r ≥ Λ }
∣ ν ∞ ν ˉ ∞ − 1 ∣ ≲ ε 2 τ − 1 + δ (37) \left|\frac{\nu_{\infty}}{\bar{\nu}_{\infty}}-1\right| \lesssim\varepsilon^{2}\tau^{-1+\delta} \tag*{(37)} ν ˉ ∞ ν ∞ − 1 ≲ ε 2 τ − 1 + δ (37) in D ∞ \mathcal{D}_{\infty} D ∞
∣ r − r ˉ ∞ ∣ ≲ ε 2 τ − 2 + δ , ∣ λ ∞ − λ ˉ ∞ ∣ ≲ ε 2 τ − 2 + δ , ∣ κ ∞ − 1 ∣ ≲ ε 2 τ − 3 + δ , ∣ m − M ∞ ∣ ≲ ε 2 τ − 3 + δ , (38) |r-\bar{r}_{\infty}| \lesssim\varepsilon^{2}\tau^{-2+\delta},\quad|\lambda_{\infty}-\bar{\lambda}_{\infty}| \lesssim\varepsilon^{2}\tau^{-2+\delta},\quad|\kappa_{\infty}-1| \lesssim\varepsilon^{2}\tau^{-3+\delta},\quad|m-M_{\infty}| \lesssim\varepsilon^{2}\tau^{-3+\delta}, \tag*{(38)} ∣ r − r ˉ ∞ ∣ ≲ ε 2 τ − 2 + δ , ∣ λ ∞ − λ ˉ ∞ ∣ ≲ ε 2 τ − 2 + δ , ∣ κ ∞ − 1∣ ≲ ε 2 τ − 3 + δ , ∣ m − M ∞ ∣ ≲ ε 2 τ − 3 + δ , (38) ∣ ∂ v ∞ log Ω ∞ − M ∞ r ˉ ∞ 2 ∣ ≲ ε 2 τ − 2 + δ , ∣ ∂ u ∞ log Ω ∞ + M ∞ r ˉ ∞ 2 ∣ ≲ ε 2 r − 3 / 4 + δ , (39) \left|\partial_{v_{\infty}}\log\Omega_{\infty}-\frac{M_{\infty}}{\bar{r}_{\infty}^{2}}\right| \lesssim\varepsilon^{2}\tau^{-2+\delta},\quad\left|\partial_{u_{\infty}}\log\Omega_{\infty}+\frac{M_{\infty}}{\bar{r}_{\infty}^{2}}\right| \lesssim\varepsilon^{2}r^{-3/4+\delta}, \tag*{(39)} ∂ v ∞ log Ω ∞ − r ˉ ∞ 2 M ∞ ≲ ε 2 τ − 2 + δ , ∂ u ∞ log Ω ∞ + r ˉ ∞ 2 M ∞ ≲ ε 2 r − 3/4 + δ , (39) on D ∞ \mathcal{D}_{\infty} D ∞ H + \mathcal{H}^{+} H +
E ∞ ( τ ) + E p ∞ ( τ ) ≲ ε 2 τ − 3 + δ + p , (40) \mathcal{E}^{\infty}(\tau)+\mathcal{E}_{p}^{\infty}(\tau) \lesssim\varepsilon^{2}\tau^{-3+\delta+p}, \tag*{(40)} E ∞ ( τ ) + E p ∞ ( τ ) ≲ ε 2 τ − 3 + δ + p , (40) for every τ ∈ [ 1 , + ∞ ) \tau\in[1,+\infty) τ ∈ [ 1 , + ∞ ) p ∈ [ 0 , 3 − δ ] p\in[0,3-\delta] p ∈ [ 0 , 3 − δ ]
r ˉ ∞ 1 / 2 ∣ ψ + ∣ ≲ ε τ − 3 / 2 + δ / 2 , r ˉ ∞ ∣ ψ − ∣ ≲ ε τ − 3 / 2 + δ / 2 , (41) \bar{r}_{\infty}^{1/2}|\psi_{+}| \lesssim\varepsilon\tau^{-3/2+\delta/2},\quad\bar{r}_{\infty}|\psi_{-}| \lesssim\varepsilon\tau^{-3/2+\delta/2}, \tag*{(41)} r ˉ ∞ 1/2 ∣ ψ + ∣ ≲ ε τ − 3/2 + δ /2 , r ˉ ∞ ∣ ψ − ∣ ≲ ε τ − 3/2 + δ /2 , (41) on D ∞ \mathcal{D}_{\infty} D ∞ H + \mathcal{H}^{+} H +
Remark 3.10. The decay rates obtained for the geometric quantities are very similar to those in [4 ] , which reflects the fact that our bootstrap scheme is adapted from their work. The decay rates obtained for the Dirac field, on the other hand, are not expected to be sharp. As already mentioned, unlike [61 ] we do not commute the Dirac equation with higher-order differential operators, since this is not needed to close the bootstrap argument, and we therefore do not recover the sharp decay rates established there.
The proof of Theorem 2 is given at the end of § 9 . Part 1. is shown in Proposition 7.14 , while the eschatological gauge ( u ∞ , v ∞ ) (u_{\infty},v_{\infty}) ( u ∞ , v ∞ ) § 9.2 . On the other hand, Parts 2.–3. rely on the estimates established in § 6 to § 8 .
The core of the stability proof is a continuity argument implemented through a bootstrap scheme. We therefore begin by introducing the bootstrap set and stating the precise bootstrap assumptions underlying the argument, which are adapted to the Dirac setting from [4 ] .
Definition 3.11 (Bootstrap set). Let A ≥ 1 A \ge1 A ≥ 1 0 < ε ≤ ε l o c 0 < \varepsilon\le\varepsilon_{\mathrm{loc}} 0 < ε ≤ ε loc S \mathcal{S} S ε \varepsilon ε ( Q ^ max , r , Ω , ψ + , ψ − ) (\widehat{Q}_{\max},r,\Omega,\psi_{+},\psi_{-}) ( Q m a x , r , Ω , ψ + , ψ − ) ( u ^ , v ^ ) (\hat{u},\hat{v}) ( u ^ , v ^ ) bootstrap set B ( S , ε , A ) \mathcal{B}(\mathcal{S},\varepsilon,A) B ( S , ε , A ) τ f ∈ [ 1 , + ∞ ) \tau_f \in[1,+\infty) τ f ∈ [ 1 , + ∞ )
There exists a timelike curve Γ : [ 1 , τ f ] → Q ^ max \Gamma:[1,\tau_f]\to\widehat{Q}_{\max} Γ : [ 1 , τ f ] → Q m a x
d d τ ( Γ u ^ , Γ v ^ ) = ( 1 − μ ^ − 2 ν ^ , 1 − μ ^ 2 λ ^ ) ∣ Γ ( τ ) , Γ ( 1 ) = ( 0 , 0 ) . \frac{d}{d\tau}(\Gamma^{\hat{u}},\Gamma^{\hat{v}})=\left(\frac{\sqrt{1-\hat{\mu}}}{-2\hat{\nu}},\frac{\sqrt{1-\hat{\mu}}}{2\hat{\lambda}}\right)\bigg|_{\Gamma(\tau)},\qquad\Gamma(1)=(0,0). d τ d ( Γ u ^ , Γ v ^ ) = ( − 2 ν ^ 1 − μ ^ , 2 λ ^ 1 − μ ^ ) Γ ( τ ) , Γ ( 1 ) = ( 0 , 0 ) . By the choice of the tangent vector to Γ \Gamma Γ r ( Γ ( 1 ) ) = r ( 0 , 0 ) = Λ r(\Gamma(1))=r(0,0)=\Lambda r ( Γ ( 1 )) = r ( 0 , 0 ) = Λ r ( Γ ( τ ) ) = Λ r(\Gamma(\tau))=\Lambda r ( Γ ( τ )) = Λ τ ∈ [ 1 , τ f ] \tau\in[1,\tau_f] τ ∈ [ 1 , τ f ] Q ^ max \widehat{Q}_{\max} Q m a x
D ^ τ f : = [ 0 , Γ u ^ ( τ f ) ] × [ 0 , Γ v ^ ( τ f ) ] ⊂ Q ^ max ; \widehat{\mathcal{D}}_{\tau_f}:=[0,\Gamma^{\hat{u}}(\tau_f)]\times[0,\Gamma^{\hat{v}}(\tau_f)]\subset\widehat{Q}_{\max}; D τ f := [ 0 , Γ u ^ ( τ f )] × [ 0 , Γ v ^ ( τ f )] ⊂ Q m a x ; The quantities γ ^ \hat{\gamma} γ ^ κ ^ \hat{\kappa} κ ^ [ 0 , Γ u ^ ( τ f ) ] × { Γ v ^ ( τ f ) } [0,\Gamma^{\hat{u}}(\tau_f)]\times\{\Gamma^{\hat{v}}(\tau_f)\} [ 0 , Γ u ^ ( τ f )] × { Γ v ^ ( τ f )} { Γ u ^ ( τ f ) } × [ 0 , Γ v ^ ( τ f ) ] \{\Gamma^{\hat{u}}(\tau_f)\}\times[0,\Gamma^{\hat{v}}(\tau_f)] { Γ u ^ ( τ f )} × [ 0 , Γ v ^ ( τ f )] ( u τ f , v τ f ) (u_{\tau_f},v_{\tau_f}) ( u τ f , v τ f ) D ^ τ f \widehat{\mathcal{D}}_{\tau_f} D τ f
Given the τ f \tau_f τ f ( r ˉ τ f , Ω ˉ τ f ) (\bar{r}_{\tau_f},\bar{\Omega}_{\tau_f}) ( r ˉ τ f , Ω ˉ τ f ) D τ f \mathcal{D}_{\tau_f} D τ f Definition 3.6 , the following set of bootstrap assumptions hold: for the geometry
∣ ν τ f ν ˉ τ f − 1 ∣ ≤ A ε 3 / 2 τ − 1 + δ , (42) \left|\frac{\nu_{\tau_f}}{\bar{\nu}_{\tau_f}}-1\right|\le A\varepsilon^{3/2}\tau^{-1+\delta}, \tag*{(42)} ν ˉ τ f ν τ f − 1 ≤ A ε 3/2 τ − 1 + δ , (42) ∣ r ~ τ f ∣ ≤ A ε 3 / 2 τ − 2 + δ , (43) |\widetilde{r}_{\tau_f}|\le A\varepsilon^{3/2}\tau^{-2+\delta}, \tag*{(43)} ∣ r τ f ∣ ≤ A ε 3/2 τ − 2 + δ , (43) ∣ m ~ τ f ∣ ≤ A ε 3 / 2 τ − 3 + δ , (44) |\widetilde{m}_{\tau_f}|\le A\varepsilon^{3/2}\tau^{-3+\delta}, \tag*{(44)} ∣ m τ f ∣ ≤ A ε 3/2 τ − 3 + δ , (44) on D τ f \mathcal{D}_{\tau_f} D τ f
E τ f ( τ ) ≤ A ε 2 τ − 3 + δ , (45) \mathcal{E}^{\tau_f}(\tau)\le A\varepsilon^{2}\tau^{-3+\delta}, \tag*{(45)} E τ f ( τ ) ≤ A ε 2 τ − 3 + δ , (45) E p τ f ( τ ) ≤ A ε 2 τ − 3 + p + δ , (46) \mathcal{E}^{\tau_f}_{p}(\tau)\le A\varepsilon^{2}\tau^{-3+p+\delta}, \tag*{(46)} E p τ f ( τ ) ≤ A ε 2 τ − 3 + p + δ , (46) F τ f ( v , τ ) ≤ A ε 2 τ − 3 + δ , (47) \mathcal{F}^{\tau_f}(v,\tau)\le A\varepsilon^{2}\tau^{-3+\delta}, \tag*{(47)} F τ f ( v , τ ) ≤ A ε 2 τ − 3 + δ , (47) F τ f ( u , τ ) ≤ A ε 2 τ − 3 + δ , (48) \mathcal{F}^{\tau_f}(u,\tau)\le A\varepsilon^{2}\tau^{-3+\delta}, \tag*{(48)} F τ f ( u , τ ) ≤ A ε 2 τ − 3 + δ , (48) for every τ ∈ [ 1 , τ f ] \tau\in[1,\tau_f] τ ∈ [ 1 , τ f ] ( u , v ) ∈ D τ f (u,v)\in\mathcal{D}_{\tau_f} ( u , v ) ∈ D τ f p ∈ [ 0 , 3 − δ ] p\in[0,3-\delta] p ∈ [ 0 , 3 − δ ]
Remark 3.12. The bootstrap assumptions on the energy quantities are guided by the non-sharp decay estimates obtained in [61 Proposition 5.15] . Since these estimates do not capture the optimal decay even on a fixed Schwarzschild background, they are expected to be robust under perturbation, which makes them a natural choice in the nonlinear setting considered here.
Our aim is therefore to show that B ( S , ε , A ) \mathcal{B}(\mathcal{S},\varepsilon,A) B ( S , ε , A ) Proposition 7.1 ), thereby yielding a global solution with the expected decays. The key step is to establish openness by improving the bootstrap assumptions. In particular, the estimates for (42) –(44) are obtained in § 6.3 , while those for (45) –(48) are proved in § 8.5 .
Estimates for the geometry
The goal of this section is improving the bootstrap assumptions (42) –(44) . To this end, in § 6.1 we establish useful bounds on the main geometric tilde quantities, such as κ ~ τ f \widetilde{\kappa}_{\tau_f} κ τ f γ ~ τ f \widetilde{\gamma}_{\tau_f} γ τ f λ ~ τ f \widetilde{\lambda}_{\tau_f} λ τ f § 6.2 , some important properties on Γ \Gamma Γ τ \tau τ § 6.3 we prove the following result.
Proposition 4.1 (Improving the bootstrap assumptions for the geometry). There exist positive constants A 0 A_0 A 0 ε 0 \varepsilon_0 ε 0 M 0 M_0 M 0 δ \delta δ A = A 0 A=A_0 A = A 0 ε ≤ ε 0 \varepsilon\leq\varepsilon_0 ε ≤ ε 0 τ f ∈ B ( S , ε , A ) \tau_f\in\mathcal{B}(\mathcal{S},\varepsilon,A) τ f ∈ B ( S , ε , A )
∣ ν τ f ν ‾ τ f − 1 ∣ ≤ 1 2 A ε 3 / 2 τ f − 1 + δ , \left|\frac{\nu_{\tau_f}}{\overline{\nu}_{\tau_f}}-1\right|\leq\frac{1}{2}A\varepsilon^{3/2}\tau_f^{-1+\delta}, ν τ f ν τ f − 1 ≤ 2 1 A ε 3/2 τ f − 1 + δ , ∣ r ~ τ f ∣ ≤ 1 2 A ε 3 / 2 τ f − 2 + δ , |\widetilde{r}_{\tau_f}|\leq\frac{1}{2}A\varepsilon^{3/2}\tau_f^{-2+\delta}, ∣ r τ f ∣ ≤ 2 1 A ε 3/2 τ f − 2 + δ , ∣ m ~ τ f ∣ ≤ 1 2 A ε 3 / 2 τ f − 3 + δ |\widetilde{m}_{\tau_f}|\leq\frac{1}{2}A\varepsilon^{3/2}\tau_f^{-3+\delta} ∣ m τ f ∣ ≤ 2 1 A ε 3/2 τ f − 3 + δ hold on D τ f \mathcal{D}_{\tau_f} D τ f
From § 6.1 to Section 6.3, we will work on a bootstrap domain D τ f \mathcal{D}_{\tau_f} D τ f τ f \tau_f τ f τ f \tau_f τ f ( u τ f , v τ f ) (u_{\tau_f},v_{\tau_f}) ( u τ f , v τ f ) ( r τ f , Ω τ f , ψ + , τ f , ψ − , τ f ) (r_{\tau_f},\Omega_{\tau_f},\psi_{+,\tau_f},\psi_{-,\tau_f}) ( r τ f , Ω τ f , ψ + , τ f , ψ − , τ f ) ( r ‾ τ f , Ω ‾ τ f ) (\overline{r}_{\tau_f},\overline{\Omega}_{\tau_f}) ( r τ f , Ω τ f ) A A A (42) –(48) , we will use the notations a ≲ b a\lesssim b a ≲ b a ≳ b a\gtrsim b a ≳ b a ∼ b a\sim b a ∼ b u u u v v v A A A ε \varepsilon ε τ \tau τ τ f \tau_f τ f A A A ε \varepsilon ε A ε 3 / 2 ≲ ε A\varepsilon^{3/2}\lesssim\varepsilon A ε 3/2 ≲ ε ε \varepsilon ε A A A § 8.5 depending only on M 0 M_0 M 0 δ \delta δ
Initial estimates
Since the bootstrap setup is similar to that of [4 ] , the bounds proved in this section follow a similar strategy. However, the specific features of the Einstein–Dirac system, and in particular the lack of a definite sign for the energy-momentum tensor, require some small modifications to the argument (see for instance Remark 4.4). We therefore include the details of the proofs.
Lemma 4.2. For any A ≥ 1 A\geq1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_f\in\mathcal{B} τ f ∈ B
1 2 M ≤ m ≤ 2 M , (49) \frac{1}{2}M\leq m\leq2M, \tag*{(49)} 2 1 M ≤ m ≤ 2 M , (49) 1 2 ≤ ν ν ‾ ≤ 2 , (50) \frac{1}{2}\leq\frac{\nu}{\overline{\nu}}\leq2, \tag*{(50)} 2 1 ≤ ν ν ≤ 2 , (50) in D τ f \mathcal{D}_{\tau_f} D τ f
1 − μ ≥ 3 4 , (51) 1-\mu\geq\frac{3}{4}, \tag*{(51)} 1 − μ ≥ 4 3 , (51) − 2 ≤ ν ≤ − 1 2 , (52) -2\leq\nu\leq-\frac{1}{2}, \tag*{(52)} − 2 ≤ ν ≤ − 2 1 , (52) in D τ f ∩ { r ≥ Λ } \mathcal{D}_{\tau_f}\cap\{r\geq\Lambda\} D τ f ∩ { r ≥ Λ }
Proof. The bounds (49) and (50) follow directly from (44) and (42) , respectively. Regarding (51) , we notice that
1 − μ ≥ 1 − 4 M Λ ≥ 3 4 , 1-\mu\ge1-\frac{4M}{\Lambda}\ge\frac{3}{4}, 1 − μ ≥ 1 − Λ 4 M ≥ 4 3 , in D τ f ∩ { r ≥ Λ } D_{\tau_f}\cap\{r\ge\Lambda\} D τ f ∩ { r ≥ Λ } (52) , we use (42) together with the fact that − ν ˉ = 1 − 2 M / r ˉ -\bar{\nu}=1-2M/\bar{r} − ν ˉ = 1 − 2 M / r ˉ − 1 ≤ ν ˉ ≤ − 3 4 -1\le\bar{\nu}\le-\frac{3}{4} − 1 ≤ ν ˉ ≤ − 4 3 D τ f ∩ { r ≥ Λ } D_{\tau_f}\cap\{r\ge\Lambda\} D τ f ∩ { r ≥ Λ }
We now use the bootstrap assumptions for the energy decay of the Dirac components to derive bounds for κ ~ \tilde{\kappa} κ ~ γ ~ \tilde{\gamma} γ ~
Lemma 4.3 (Estimate for κ ~ \tilde{\kappa} κ ~ A ≥ 1 A\ge1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_f\in\mathcal{B} τ f ∈ B
∣ κ ~ ∣ ≲ A ε 2 τ − 3 + δ , (53) |\tilde{\kappa}|\lesssim A\varepsilon^2\tau^{-3+\delta}, \tag*{(53)} ∣ κ ~ ∣ ≲ A ε 2 τ − 3 + δ , (53) 1 2 ≤ κ ≤ 2. (54) \frac{1}{2}\le\kappa\le2. \tag*{(54)} 2 1 ≤ κ ≤ 2. (54) in D τ f D_{\tau_f} D τ f
Proof. From (15) and κ = κ ~ + 1 \kappa=\tilde{\kappa}+1 κ = κ ~ + 1
∂ u κ ~ = − 2 r ( κ ~ + 1 ) ℑ [ ψ − ‾ ∂ u ψ − ] . \partial_u\tilde{\kappa}=-2r(\tilde{\kappa}+1)\Im[\overline{\psi_-}\partial_u\psi_-]. ∂ u κ ~ = − 2 r ( κ ~ + 1 ) ℑ [ ψ − ∂ u ψ − ] . Integrating backwards in u u u { Γ u ( τ f ) } × [ 0 , Γ v ( τ f ) ] \{\Gamma^u(\tau_f)\}\times[0,\Gamma^v(\tau_f)] { Γ u ( τ f )} × [ 0 , Γ v ( τ f )] κ ~ = 0 \tilde{\kappa}=0 κ ~ = 0
κ ~ ( u , v ) = − 1 + exp ( ∫ C v ∩ { u ′ ≥ u } 2 r ℑ [ ψ − ‾ ∂ u ψ − ] d u ′ ) . \tilde{\kappa}(u,v)=-1+\exp\left(\int_{C_v\cap\{u'\ge u\}}2r\Im[\overline{\psi_-}\partial_u\psi_-]\,du'\right). κ ~ ( u , v ) = − 1 + exp ( ∫ C v ∩ { u ′ ≥ u } 2 r ℑ [ ψ − ∂ u ψ − ] d u ′ ) . Therefore, when r ( u , v ) ≤ Λ r(u,v)\le\Lambda r ( u , v ) ≤ Λ
∣ κ ~ ( u , v ) ∣ ≲ ∣ − 1 + exp ( ∫ u Γ u ( τ f ) ( − ν ˉ ∣ ψ − ∣ 2 + ∣ ∂ u ψ − ∣ 2 − ν ˉ ) d u ′ ) ∣ ≲ ∣ − 1 + exp ( E ( τ ) ) ∣ ≲ A ε 2 τ − 3 + δ , |\tilde{\kappa}(u,v)|\lesssim\left|-1+\exp\left(\int_u^{\Gamma^u(\tau_f)}\left(-\bar{\nu}|\psi_-|^2+\frac{|\partial_u\psi_-|^2}{-\bar{\nu}}\right)\,du'\right)\right|\lesssim|-1+\exp(\mathcal{E}(\tau))|\lesssim A\varepsilon^2\tau^{-3+\delta}, ∣ κ ~ ( u , v ) ∣ ≲ − 1 + exp ( ∫ u Γ u ( τ f ) ( − ν ˉ ∣ ψ − ∣ 2 + − ν ˉ ∣ ∂ u ψ − ∣ 2 ) d u ′ ) ≲ ∣ − 1 + exp ( E ( τ )) ∣ ≲ A ε 2 τ − 3 + δ , by the bootstrap assumption (45) . On the other hand, when r ( u , v ) > Λ r(u,v)>\Lambda r ( u , v ) > Λ
∣ κ ~ ( u , v ) ∣ ≲ ∣ − 1 + exp ( ∫ C v ( − ν ˉ ∣ ψ − ∣ 2 + ∣ ∂ u ψ − ∣ 2 − ν ˉ ) d u ′ + ∫ u u Λ ( v ) ( r 2 ∣ ψ − ∣ 2 + r 2 ∣ ∂ u ψ − ∣ 2 ) d u ′ ) ∣ |\tilde{\kappa}(u,v)|\lesssim\left|-1+\exp\left(\int_{C_v}\left(-\bar{\nu}|\psi_-|^2+\frac{|\partial_u\psi_-|^2}{-\bar{\nu}}\right)\,du'+\int_u^{u^\Lambda(v)}\left(r^2|\psi_-|^2+r^2|\partial_u\psi_-|^2\right)\,du'\right)\right| ∣ κ ~ ( u , v ) ∣ ≲ − 1 + exp ( ∫ C v ( − ν ˉ ∣ ψ − ∣ 2 + − ν ˉ ∣ ∂ u ψ − ∣ 2 ) d u ′ + ∫ u u Λ ( v ) ( r 2 ∣ ψ − ∣ 2 + r 2 ∣ ∂ u ψ − ∣ 2 ) d u ′ ) ≲ ∣ − 1 + exp ( E ( τ ) + F ( v , τ ( u , v ) ) ) ∣ ≲ A ε 2 τ − 3 + δ , \lesssim|-1+\exp(\mathcal{E}(\tau)+\mathcal{F}(v,\tau(u,v)))|\lesssim A\varepsilon^2\tau^{-3+\delta}, ≲ ∣ − 1 + exp ( E ( τ ) + F ( v , τ ( u , v ))) ∣ ≲ A ε 2 τ − 3 + δ , where we used (45) and (47) to conclude. Finally, (54) readily follows from (53) .
Remark 4.4. From (51) and (54) , and recalling λ = κ ( 1 − μ ) \lambda=\kappa(1-\mu) λ = κ ( 1 − μ )
λ ≤ 2 in D τ f , (55) \lambda\le2\quad\text{in }D_{\tau_f}, \tag*{(55)} λ ≤ 2 in D τ f , (55) λ ≥ 3 8 in D τ f ∩ { r ≥ Λ } . (56) \lambda\ge\frac{3}{8}\quad\text{in }D_{\tau_f}\cap\{r\ge\Lambda\}. \tag*{(56)} λ ≥ 8 3 in D τ f ∩ { r ≥ Λ } . (56) However, since the Raychaudhuri equation (13) has no definite sign, (56) does not by itself imply that λ > 0 \lambda>0 λ > 0 D τ f D_{\tau_f} D τ f
Similarly, from the identity Ω 2 = κ ( − ν ) > 0 \Omega^2=\kappa(-\nu)>0 Ω 2 = κ ( − ν ) > 0
∣ Ω ~ 2 ∣ = ∣ ν ˉ ( κ ~ + κ ( ν ν ˉ − 1 ) ) ∣ ≲ ( − ν ˉ ) A ε 3 / 2 τ − 1 + δ , (57) |\tilde{\Omega}^2|=\left|\bar{\nu}\left(\tilde{\kappa}+\kappa\left(\frac{\nu}{\bar{\nu}}-1\right)\right)\right|\lesssim(-\bar{\nu})A\varepsilon^{3/2}\tau^{-1+\delta}, \tag*{(57)} ∣ Ω ~ 2 ∣ = ν ˉ ( κ ~ + κ ( ν ˉ ν − 1 ) ) ≲ ( − ν ˉ ) A ε 3/2 τ − 1 + δ , (57) in D τ f D_{\tau_f} D τ f
Lemma 4.5 (Estimate for γ ~ \tilde{\gamma} γ ~ A ≥ 1 A\ge1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_f\in\mathcal{B} τ f ∈ B
∣ γ ~ ∣ ≲ A ε 2 r − 2 τ − 3 + δ , (58) |\tilde{\gamma}|\lesssim A\varepsilon^2r^{-2}\tau^{-3+\delta}, \tag*{(58)} ∣ γ ~ ∣ ≲ A ε 2 r − 2 τ − 3 + δ , (58) in D τ f ∩ { r ≥ Λ } D_{\tau_f}\cap\{r\ge\Lambda\} D τ f ∩ { r ≥ Λ }
Proof. From (16) and γ = γ ~ + 1 \gamma=\tilde{\gamma}+1 γ = γ ~ + 1
∂ v γ ~ = 2 ( γ ~ + 1 ) r λ ℜ [ ψ + ∂ v ψ + ] . \partial_v\tilde{\gamma}=\frac{2(\tilde{\gamma}+1)}{r\lambda}\Re[\psi_+\partial_v\psi_+]. ∂ v γ ~ = r λ 2 ( γ ~ + 1 ) ℜ [ ψ + ∂ v ψ + ] . Integrating backwards in v v v [ 0 , Γ u ( τ f ) ] × { Γ v ( τ f ) } [0,\Gamma^u(\tau_f)]\times\{\Gamma^v(\tau_f)\} [ 0 , Γ u ( τ f )] × { Γ v ( τ f )} γ ~ = 0 \tilde{\gamma}=0 γ ~ = 0 r 2 r^2 r 2 ( u , v ) ∈ D τ f ∩ { r ≥ Λ } (u,v)\in D_{\tau_f}\cap\{r\geq\Lambda\} ( u , v ) ∈ D τ f ∩ { r ≥ Λ }
∣ r 2 γ ~ ( u , v ) ∣ ≲ r 2 ∫ v Γ v ( τ f ) ∣ γ ~ ∣ + 1 r ∣ ψ + ∣ ∣ ∂ v ψ + ∣ d v ′ ≲ ∫ v Γ v ( τ f ) r ( ∣ γ ~ ∣ + 1 ) ∣ ψ + ∣ ∣ ∂ v ψ + ∣ d v ′ |r^2\tilde{\gamma}(u,v)|\lesssim r^2\int_v^{\Gamma^v(\tau_f)}\frac{|\tilde{\gamma}|+1}{r}|\psi_+||\partial_v\psi_+|\,dv'\lesssim\int_v^{\Gamma^v(\tau_f)}r(|\tilde{\gamma}|+1)|\psi_+||\partial_v\psi_+|\,dv' ∣ r 2 γ ~ ( u , v ) ∣ ≲ r 2 ∫ v Γ v ( τ f ) r ∣ γ ~ ∣ + 1 ∣ ψ + ∣∣ ∂ v ψ + ∣ d v ′ ≲ ∫ v Γ v ( τ f ) r ( ∣ γ ~ ∣ + 1 ) ∣ ψ + ∣∣ ∂ v ψ + ∣ d v ′ ≲ ∫ v Γ v ( τ f ) ( ∣ γ ~ ∣ + 1 ) ( ∣ ψ + ∣ 2 + r 2 ∣ ∂ v ψ + ∣ 2 ) d v ′ , \lesssim\int_v^{\Gamma^v(\tau_f)}(|\tilde{\gamma}|+1)(|\psi_+|^2+r^2|\partial_v\psi_+|^2)\,dv', ≲ ∫ v Γ v ( τ f ) ( ∣ γ ~ ∣ + 1 ) ( ∣ ψ + ∣ 2 + r 2 ∣ ∂ v ψ + ∣ 2 ) d v ′ , where we used (56) to move the r r r
∣ r 2 γ ~ ( u , v ) ∣ ≲ E 0 ( τ ) exp ( E 0 ( τ ) ) ≲ A ε 2 τ − 3 + δ , |r^2\tilde{\gamma}(u,v)|\lesssim\mathcal{E}_0(\tau)\exp\bigl(\mathcal{E}_0(\tau)\bigr)\lesssim A\varepsilon^2\tau^{-3+\delta}, ∣ r 2 γ ~ ( u , v ) ∣ ≲ E 0 ( τ ) exp ( E 0 ( τ ) ) ≲ A ε 2 τ − 3 + δ , thanks to the bootstrap assumption (3.22).
We now derive Taylor expansions for 1 − μ ~ 1-\tilde{\mu} 1 − μ ~ ∂ u r ~ \partial_u\tilde r ∂ u r ~ ∂ v r ~ \partial_v\tilde r ∂ v r ~ § 6.3 .
Lemma 4.6. For any A ≥ 1 A\geq1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_f\in\mathcal{B} τ f ∈ B
∣ 1 − μ ~ − 2 M r ~ r ~ 2 ∣ ≲ ∣ r ~ ∣ 2 r ~ 3 + ∣ m ~ ∣ r ~ , (59) \left|1-\tilde{\mu}-\frac{2M\tilde r}{\tilde r^2}\right|\lesssim\frac{|\tilde r|^2}{\tilde r^3}+\frac{|\tilde m|}{\tilde r}, \tag*{(59)} 1 − μ ~ − r ~ 2 2 M r ~ ≲ r ~ 3 ∣ r ~ ∣ 2 + r ~ ∣ m ~ ∣ , (59) ∣ 1 − μ ~ ∣ ≲ A ε 3 / 2 τ − 2 + δ , (60) |1-\tilde{\mu}|\lesssim A\varepsilon^{3/2}\tau^{-2+\delta}, \tag*{(60)} ∣1 − μ ~ ∣ ≲ A ε 3/2 τ − 2 + δ , (60) in D τ f D_{\tau_f} D τ f
Proof. From the bootstrap assumption (43) , we readily obtain the Taylor expansion
1 r η = 1 r ~ η − η r ~ r ~ η + 1 + O ( ∣ r ~ ∣ 2 r ~ η + 2 ) (61) \frac{1}{r^\eta}=\frac{1}{\tilde r^\eta}-\frac{\eta\tilde r}{\tilde r^{\eta+1}}+O\left(\frac{|\tilde r|^2}{\tilde r^{\eta+2}}\right) \tag*{(61)} r η 1 = r ~ η 1 − r ~ η + 1 η r ~ + O ( r ~ η + 2 ∣ r ~ ∣ 2 ) (61) for any η > 0 \eta>0 η > 0
1 − μ ~ = − 2 m ~ r + 2 M r = − 2 m ~ r ~ − 2 M ( 1 r − 1 r ~ ) = − 2 m ~ r ~ + 2 M r ~ r ~ 2 + O ( ∣ r ~ ∣ 2 r ~ 3 ) , 1-\tilde{\mu}=-\frac{2\tilde m}{r}+\frac{2M}{r}=-\frac{2\tilde m}{\tilde r}-2M\left(\frac{1}{r}-\frac{1}{\tilde r}\right)=-\frac{2\tilde m}{\tilde r}+\frac{2M\tilde r}{\tilde r^2}+O\left(\frac{|\tilde r|^2}{\tilde r^3}\right), 1 − μ ~ = − r 2 m ~ + r 2 M = − r ~ 2 m ~ − 2 M ( r 1 − r ~ 1 ) = − r ~ 2 m ~ + r ~ 2 2 M r ~ + O ( r ~ 3 ∣ r ~ ∣ 2 ) , which yields (59) . Equation (60) then follows using (43) and (44) .
Lemma 4.7. For any A ≥ 1 A\geq1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_f\in\mathcal{B} τ f ∈ B
∂ u r ~ = − 2 M γ r ~ 2 r ~ + E r r u in D τ f ∩ { r ≥ Λ } , (62) \partial_u\tilde r=-\frac{2M\gamma}{\tilde r^2}\tilde r+\mathop{\rm Err}\nolimits_u\quad\text{in }D_{\tau_f}\cap\{r\geq\Lambda\}, \tag*{(62)} ∂ u r ~ = − r ~ 2 2 M γ r ~ + Err u in D τ f ∩ { r ≥ Λ } , (62) ∂ v r ~ = 2 M κ r ~ 2 r ~ + E r r v in D τ f , (63) \partial_v\tilde r=\frac{2M\kappa}{\tilde r^2}\tilde r+\mathop{\rm Err}\nolimits_v\quad\text{in }D_{\tau_f}, \tag*{(63)} ∂ v r ~ = r ~ 2 2 M κ r ~ + Err v in D τ f , (63) where the error terms satisfy the bounds
∣ E r r u ∣ ≲ A ε 3 / 2 r ˉ − 1 τ − 3 + δ , (64) |\mathop{\rm Err}\nolimits_u|\lesssim A\varepsilon^{3/2}\bar r^{-1}\tau^{-3+\delta}, \tag*{(64)} ∣ Err u ∣ ≲ A ε 3/2 r ˉ − 1 τ − 3 + δ , (64) ∣ E r r v ∣ ≲ A ε 3 / 2 τ − 3 + δ . (65) |\mathop{\rm Err}\nolimits_v|\lesssim A\varepsilon^{3/2}\tau^{-3+\delta}. \tag*{(65)} ∣ Err v ∣ ≲ A ε 3/2 τ − 3 + δ . (65) Proof. Regarding ∂ u r ~ \partial_u\tilde r ∂ u r ~ (59) ,
∂ u r ~ = ν ~ = − γ ( 1 − μ ) + γ ˉ 1 − μ = − γ ~ ( 1 − μ ) − γ ( 1 − μ ~ ) = − γ ~ ( 1 − μ ) − γ [ 2 M r ~ r ~ 2 + O ( ∣ r ~ ∣ 2 r ~ 3 + ∣ m ~ ∣ r ~ ) ] . \partial_u\tilde r=\tilde\nu=-\gamma(1-\mu)+\bar{\gamma}\sqrt{1-\mu}=-\tilde{\gamma}(1-\mu)-\gamma(1-\tilde{\mu})=-\tilde{\gamma}(1-\mu)-\gamma\left[\frac{2M\tilde r}{\tilde r^2}+O\left(\frac{|\tilde r|^2}{\tilde r^3}+\frac{|\tilde m|}{\tilde r}\right)\right]. ∂ u r ~ = ν ~ = − γ ( 1 − μ ) + γ ˉ 1 − μ = − γ ~ ( 1 − μ ) − γ ( 1 − μ ~ ) = − γ ~ ( 1 − μ ) − γ [ r ~ 2 2 M r ~ + O ( r ~ 3 ∣ r ~ ∣ 2 + r ~ ∣ m ~ ∣ ) ] . which implies (62) , where the error term satisfies
∣ E r r u ∣ ≲ ∣ r ~ ∣ 2 r ˉ 3 + ∣ m ~ ∣ r ˉ + ∣ γ ~ ∣ ≲ A ε 3 / 2 r ˉ − 1 τ − 3 + δ \lvert Err_u\rvert\lesssim\frac{\lvert\tilde r\rvert^2}{\bar r^3}+\frac{\lvert\tilde m\rvert}{\bar r}+\lvert\tilde\gamma\rvert\lesssim A\varepsilon^{3/2}\bar r^{-1}\tau^{-3+\delta} ∣ E r r u ∣ ≲ r ˉ 3 ∣ r ~ ∣ 2 + r ˉ ∣ m ~ ∣ + ∣ γ ~ ∣ ≲ A ε 3/2 r ˉ − 1 τ − 3 + δ by the bootstrap assumptions and (58) . On the other hand, we have
∂ v r ~ = λ ~ = κ ( 1 − μ ) − κ ˉ ( 1 − μ ˉ ) = κ ~ ( 1 − μ ) + κ ( 1 − μ ˉ ) = κ ~ ( 1 − μ ) + κ [ 2 M r ~ r ˉ 2 + O ( ∣ r ~ ∣ 2 r ˉ 3 + ∣ m ~ ∣ r ˉ ) ] , \partial_v\tilde r=\tilde\lambda=\kappa(1-\mu)-\bar\kappa(1-\bar\mu)=\tilde\kappa(1-\mu)+\kappa(1-\bar\mu)=\tilde\kappa(1-\mu)+\kappa\left[\frac{2M\tilde r}{\bar r^2}+O\left(\frac{\lvert\tilde r\rvert^2}{\bar r^3}+\frac{\lvert\tilde m\rvert}{\bar r}\right)\right], ∂ v r ~ = λ ~ = κ ( 1 − μ ) − κ ˉ ( 1 − μ ˉ ) = κ ~ ( 1 − μ ) + κ ( 1 − μ ˉ ) = κ ~ ( 1 − μ ) + κ [ r ˉ 2 2 M r ~ + O ( r ˉ 3 ∣ r ~ ∣ 2 + r ˉ ∣ m ~ ∣ ) ] , yielding (63) , where the error term satisfies
∣ E r r v ∣ ≲ ∣ r ~ ∣ 2 r ˉ 3 + ∣ m ~ ∣ r ˉ + ∣ κ ~ ∣ ≲ A ε 3 / 2 τ − 3 + δ \lvert Err_v\rvert\lesssim\frac{\lvert\tilde r\rvert^2}{\bar r^3}+\frac{\lvert\tilde m\rvert}{\bar r}+\lvert\tilde\kappa\rvert\lesssim A\varepsilon^{3/2}\tau^{-3+\delta} ∣ E r r v ∣ ≲ r ˉ 3 ∣ r ~ ∣ 2 + r ˉ ∣ m ~ ∣ + ∣ κ ~ ∣ ≲ A ε 3/2 τ − 3 + δ by the bootstrap assumptions and (53) .
Properties of Γ \Gamma Γ τ \tau τ
In this section, we derive some basic properties of the timelike curve Γ = { r = Λ } \Gamma=\{r=\Lambda\} Γ = { r = Λ } τ ( u , v ) \tau(u,v) τ ( u , v ) (31) , that will be useful throughout the paper. Recall that τ \tau τ Γ \Gamma Γ
g ( Γ ˙ , Γ ˙ ) = − 1 , (66) g(\dot\Gamma,\dot\Gamma)=-1, \tag*{(66)} g ( Γ ˙ , Γ ˙ ) = − 1 , (66) where ˙ \dot{\ } ˙ d d τ \frac{d}{d\tau} d τ d Γ u ( τ ) \Gamma^u(\tau) Γ u ( τ ) Γ v ( τ ) \Gamma^v(\tau) Γ v ( τ ) Γ \Gamma Γ ( u , v ) (u,v) ( u , v )
Lemma 4.8. For any A ≥ 1 A\ge1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_f\in\mathcal{B} τ f ∈ B
Γ ˙ u ( τ ) ∼ Γ ˙ v ( τ ) ∼ 1 (67) \dot\Gamma^u(\tau)\sim\dot\Gamma^v(\tau)\sim1 \tag*{(67)} Γ ˙ u ( τ ) ∼ Γ ˙ v ( τ ) ∼ 1 (67) for τ ∈ [ 1 , τ f ] \tau\in[1,\tau_f] τ ∈ [ 1 , τ f ]
Proof. Since r r r Γ \Gamma Γ ν Γ ˙ u + λ Γ ˙ v = 0 \nu\dot\Gamma^u+\lambda\dot\Gamma^v=0 ν Γ ˙ u + λ Γ ˙ v = 0 Γ \Gamma Γ (51) and (54) yield ∣ λ ∣ Γ = κ ( 1 − μ ) ∣ Γ ∼ 1 \lvert\lambda\rvert_\Gamma=\kappa(1-\mu)\rvert_\Gamma\sim1 ∣ λ ∣ Γ = κ ( 1 − μ ) ∣ Γ ∼ 1 (52) that Γ ˙ u ∼ Γ ˙ v \dot\Gamma^u\sim\dot\Gamma^v Γ ˙ u ∼ Γ ˙ v (66) can be written as
4 ν λ 1 − μ Γ ˙ u Γ ˙ v = − 1 , \frac{4\nu\lambda}{1-\mu}\dot\Gamma^u\dot\Gamma^v=-1, 1 − μ 4 ν λ Γ ˙ u Γ ˙ v = − 1 , yielding Γ ˙ u Γ ˙ v ∼ 1 \dot\Gamma^u\dot\Gamma^v\sim1 Γ ˙ u Γ ˙ v ∼ 1 (51) . Thus, (67) readily follows.
The following lemma describes the relation between τ \tau τ ( u , v ) (u,v) ( u , v )
Lemma 4.9. For any A ≥ 1 A\ge1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_f\in\mathcal{B} τ f ∈ B
τ ( u 1 , v ) − τ ( u 2 , v ) ∼ u 1 − u 2 , (68) \tau(u_1,v)-\tau(u_2,v)\sim u_1-u_2, \tag*{(68)} τ ( u 1 , v ) − τ ( u 2 , v ) ∼ u 1 − u 2 , (68) for ( u 1 , v ) , ( u 2 , v ) ∈ D τ f ∩ { r ≥ Λ } (u_1,v),(u_2,v)\in D_{\tau_f}\cap\{r\ge\Lambda\} ( u 1 , v ) , ( u 2 , v ) ∈ D τ f ∩ { r ≥ Λ }
τ ( u , v 1 ) − τ ( u , v 2 ) ∼ v 1 − v 2 , (69) \tau(u,v_1)-\tau(u,v_2)\sim v_1-v_2, \tag*{(69)} τ ( u , v 1 ) − τ ( u , v 2 ) ∼ v 1 − v 2 , (69) for ( u , v 1 ) , ( u , v 2 ) ∈ D τ f ∩ { r ≤ Λ } (u,v_1),(u,v_2)\in D_{\tau_f}\cap\{r\le\Lambda\} ( u , v 1 ) , ( u , v 2 ) ∈ D τ f ∩ { r ≤ Λ } η > 1 \eta>1 η > 1 1 ≤ τ 1 ≤ τ 2 ≤ τ f 1\le\tau_1\le\tau_2\le\tau_f 1 ≤ τ 1 ≤ τ 2 ≤ τ f
∫ H v ∩ { τ 1 ≤ τ ≤ τ 2 } τ − η d u ≲ η τ 1 − η + 1 , (70) \int_{H_v\cap\{\tau_1\le\tau\le\tau_2\}}\tau^{-\eta}\,du\lesssim_\eta\tau_1^{-\eta+1}, \tag*{(70)} ∫ H v ∩ { τ 1 ≤ τ ≤ τ 2 } τ − η d u ≲ η τ 1 − η + 1 , (70) ∫ H u ∩ { τ 1 ≤ τ ≤ τ 2 } τ − η d v ≲ η τ 1 − η + 1 . (71) \int_{H_u\cap\{\tau_1\le\tau\le\tau_2\}}\tau^{-\eta}\,dv\lesssim_\eta\tau_1^{-\eta+1}. \tag*{(71)} ∫ H u ∩ { τ 1 ≤ τ ≤ τ 2 } τ − η d v ≲ η τ 1 − η + 1 . (71) Proof. It is a straightforward consequence of (67) , the fundamental theorem of calculus, and the change of variables formula.
We conclude this section by showing that the v v v v ^ \hat{v} v ^
Lemma 4.10. For τ f ∈ B \tau_f \in\mathcal{B} τ f ∈ B ε \varepsilon ε A A A v ^ 1 , v ^ 2 ∈ [ 0 , Γ v ^ ( τ f ) ] \hat{v}_1,\hat{v}_2 \in[0,\Gamma^{\hat{v}}(\tau_f)] v ^ 1 , v ^ 2 ∈ [ 0 , Γ v ^ ( τ f )]
v ( v ^ 2 ) − v ( v ^ 1 ) ∼ v ^ 2 − v ^ 1 . (72) v(\hat{v}_2)-v(\hat{v}_1)\sim\hat{v}_2-\hat{v}_1. \tag*{(72)} v ( v ^ 2 ) − v ( v ^ 1 ) ∼ v ^ 2 − v ^ 1 . (72) Moreover, on D ^ τ f \hat{D}_{\tau_f} D ^ τ f
κ ^ ∼ 1. (73) \hat{\kappa}\sim1. \tag*{(73)} κ ^ ∼ 1. (73) Proof. The proof is analogous to [4 Lemma 5.7] . Recalling the definition of the map v ^ ↦ v ( v ^ ) \hat{v}\mapsto v(\hat{v}) v ^ ↦ v ( v ^ ) (29) , the derivative of the inverse map v ↦ v − 1 ( v ) v\mapsto v^{-1}(v) v ↦ v − 1 ( v ) λ ( 0 , v ) \lambda(0,v) λ ( 0 , v ) (72) follows from (55) , (56) , and the fundamental theorem of calculus.
On the other hand, (73) follows from the identity
κ ^ ( u ^ , v ^ ) = κ ( u ( u ^ ) , v ( v ^ ) ) λ ( 0 , v ( v ^ ) ) \hat{\kappa}(\hat{u},\hat{v})=\frac{\kappa(u(\hat{u}),v(\hat{v}))}{\lambda(0,v(\hat{v}))} κ ^ ( u ^ , v ^ ) = λ ( 0 , v ( v ^ )) κ ( u ( u ^ ) , v ( v ^ )) and (54) .
Improving the bootstrap assumptions for the geometry
We are now in position to improve the bootstrap assumptions on the geometry. Let us start with r ~ \tilde{r} r ~ [4 Lemma 5.12] , bounding the difference of the area radius function in three steps: along Γ \Gamma Γ { r ≤ Λ } \{r\leq\Lambda\} { r ≤ Λ } { r ≥ Λ } \{r\geq\Lambda\} { r ≥ Λ }
Proposition 4.11. For any A ≥ 1 A\geq1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_f\in\mathcal{B} τ f ∈ B
∣ r ~ ∣ ≲ A ε 3 / 2 τ − 2 + δ , (74) |\tilde{r}|\lesssim A\varepsilon^{3/2}\tau^{-2+\delta}, \tag*{(74)} ∣ r ~ ∣ ≲ A ε 3/2 τ − 2 + δ , (74) ∣ λ ~ ∣ ≲ A ε 3 / 2 τ − 2 + δ (75) |\tilde{\lambda}|\lesssim A\varepsilon^{3/2}\tau^{-2+\delta} \tag*{(75)} ∣ λ ~ ∣ ≲ A ε 3/2 τ − 2 + δ (75) in D τ f D_{\tau_f} D τ f
Proof. Estimate along Γ \Gamma Γ Using the relation ν Γ ˙ u + λ Γ ˙ v = 0 \nu\dot{\Gamma}^{u}+\lambda\dot{\Gamma}^{v}=0 ν Γ ˙ u + λ Γ ˙ v = 0 Γ \Gamma Γ
d d τ r ~ ( Γ ( τ ) ) = ν Γ ˙ u + λ Γ ˙ v ν ˉ Γ ˙ u + λ ˉ Γ ˙ v = − ν ˉ Γ ˙ u − λ ˉ Γ ˙ v = ( λ ˉ λ ν − ν ˉ ) Γ ˙ u = − ( 1 − μ ) ( κ ˉ κ − γ ˉ γ ) γ Γ ˙ u . \frac{d}{d\tau}\tilde{r}(\Gamma(\tau))=\frac{\nu\dot{\Gamma}^{u}+\lambda\dot{\Gamma}^{v}}{\bar{\nu}\dot{\Gamma}^{u}+\bar{\lambda}\dot{\Gamma}^{v}}=-\bar{\nu}\dot{\Gamma}^{u}-\bar{\lambda}\dot{\Gamma}^{v}=\left(\frac{\bar{\lambda}}{\lambda}\nu-\bar{\nu}\right)\dot{\Gamma}^{u}=-(1-\mu)\left(\frac{\bar{\kappa}}{\kappa}-\frac{\bar{\gamma}}{\gamma}\right)\gamma\dot{\Gamma}^{u}. d τ d r ~ ( Γ ( τ )) = ν ˉ Γ ˙ u + λ ˉ Γ ˙ v ν Γ ˙ u + λ Γ ˙ v = − ν ˉ Γ ˙ u − λ ˉ Γ ˙ v = ( λ λ ˉ ν − ν ˉ ) Γ ˙ u = − ( 1 − μ ) ( κ κ ˉ − γ γ ˉ ) γ Γ ˙ u . Using (53) and (58) , we then bound
∣ κ ˉ κ − γ ˉ γ ∣ ≲ A ε 2 τ − 3 + δ \left|\frac{\bar{\kappa}}{\kappa}-\frac{\bar{\gamma}}{\gamma}\right|\lesssim A\varepsilon^{2}\tau^{-3+\delta} κ κ ˉ − γ γ ˉ ≲ A ε 2 τ − 3 + δ along Γ \Gamma Γ
∣ r ~ ∣ Γ ≲ A ε 2 τ − 2 + δ , (76) |\tilde{r}|_{\Gamma}\lesssim A\varepsilon^{2}\tau^{-2+\delta}, \tag*{(76)} ∣ r ~ ∣ Γ ≲ A ε 2 τ − 2 + δ , (76) by (67) , the fundamental theorem of calculus and the anchoring condition r ~ ( Γ ( τ f ) ) = 0 \tilde{r}(\Gamma(\tau_f))=0 r ~ ( Γ ( τ f )) = 0
Estimate for r ≤ Λ r\leq\Lambda r ≤ Λ We view (63) as an ODE for r ~ \tilde{r} r ~ v v v ( u , v ) ∈ D τ f ∩ { r ≤ Λ } (u,v)\in D_{\tau_f}\cap\{r\leq\Lambda\} ( u , v ) ∈ D τ f ∩ { r ≤ Λ }
r ~ ( u , v ) = exp ( − ∫ v v Λ ( u ) 2 M κ r ˉ 2 d v ′ ) [ r ~ ( u , v Λ ( u ) ) − ∫ v v Λ ( u ) exp ( ∫ v ′ v Λ ( u ) 2 M κ r ˉ 2 d v ′ ′ ) E r r v d v ′ ] = exp ( − ∫ v v Λ ( u ) 2 M κ r ˉ 2 d v ′ ) r ~ ( u , v Λ ( u ) ) − ∫ v v Λ ( u ) exp ( − ∫ v v ′ 2 M κ r ˉ 2 d v ′ ′ ) E r r v d v ′ . (77) \begin{aligned}
\tilde{r}(u,v)=\exp\left(-\int_{v}^{v^{\Lambda}(u)}\frac{2M\kappa}{\bar{r}^{2}}\,dv'\right)\left[\tilde{r}(u,v^{\Lambda}(u))-\int_{v}^{v^{\Lambda}(u)}\exp\left(\int_{v'}^{v^{\Lambda}(u)}\frac{2M\kappa}{\bar{r}^{2}}\,dv''\right)\mathcal{E}rr_{v}\,dv'\right] \\
&=\exp\left(-\int_{v}^{v^{\Lambda}(u)}\frac{2M\kappa}{\bar{r}^{2}}\,dv'\right)\tilde{r}(u,v^{\Lambda}(u))-\int_{v}^{v^{\Lambda}(u)}\exp\left(-\int_{v}^{v'}\frac{2M\kappa}{\bar{r}^{2}}\,dv''\right)\mathcal{E}rr_{v}\,dv'. \tag*{(77)}
\end{aligned} r ~ ( u , v ) = exp ( − ∫ v v Λ ( u ) r ˉ 2 2 M κ d v ′ ) [ r ~ ( u , v Λ ( u )) − ∫ v v Λ ( u ) exp ( ∫ v ′ v Λ ( u ) r ˉ 2 2 M κ d v ′′ ) E r r v d v ′ ] = exp ( − ∫ v v Λ ( u ) r ˉ 2 2 M κ d v ′ ) r ~ ( u , v Λ ( u )) − ∫ v v Λ ( u ) exp ( − ∫ v v ′ r ˉ 2 2 M κ d v ′′ ) E r r v d v ′ . (77) Therefore (65) , (71) , (76) and (77) yield
∣ r ~ ( u , v ) ∣ ≲ A ε 2 τ − 2 + δ ( u , v ) + A ε 3 / 2 τ − 2 + δ ( u , v ) ≲ A ε 3 / 2 τ − 2 + δ ( u , v ) , (78) |\tilde{r}(u,v)|\lesssim A\varepsilon^{2}\tau^{-2+\delta}(u,v)+A\varepsilon^{3/2}\tau^{-2+\delta}(u,v)\lesssim A\varepsilon^{3/2}\tau^{-2+\delta}(u,v), \tag*{(78)} ∣ r ~ ( u , v ) ∣ ≲ A ε 2 τ − 2 + δ ( u , v ) + A ε 3/2 τ − 2 + δ ( u , v ) ≲ A ε 3/2 τ − 2 + δ ( u , v ) , (78) after noting that the exponentials are bounded by 1, as the integrands are non-negative.
**Estimate for r ≥ Λ r \ge\Lambda r ≥ Λ (62) as an ODE for r ~ \tilde r r ~ u u u ( u , v ) ∈ D τ f ∩ { r ≥ Λ } (u,v) \in D_{\tau_f} \cap\{r \ge\Lambda\} ( u , v ) ∈ D τ f ∩ { r ≥ Λ }
r ~ ( u , v ) = exp ( ∫ u u Λ ( v ) 2 M γ r ˉ 2 d u ′ ) [ r ~ ( u Λ ( v ) , v ) − ∫ u u Λ ( v ) exp ( − ∫ u ′ u Λ ( v ) 2 M γ r ˉ 2 d u ′ ′ ) E r r u d u ′ ] = exp ( ∫ u u Λ ( v ) 2 M γ r ˉ 2 d u ′ ) r ~ ( u Λ ( v ) , v ) − ∫ u u Λ ( v ) exp ( ∫ u u ′ 2 M γ r ˉ 2 d u ′ ′ ) E r r u d u ′ . (79) \begin{aligned}
\tilde r(u,v) &= \exp\left(\int_u^{u^\Lambda(v)} \frac{2M\gamma}{\bar r^2}\,du'\right)\left[\tilde r(u^\Lambda(v),v)-\int_u^{u^\Lambda(v)}\exp\left(-\int_{u'}^{u^\Lambda(v)}\frac{2M\gamma}{\bar r^2}\,du''\right)\mathcal E rr_u\,du'\right]\\
&=\exp\left(\int_u^{u^\Lambda(v)} \frac{2M\gamma}{\bar r^2}\,du'\right)\tilde r(u^\Lambda(v),v)-\int_u^{u^\Lambda(v)}\exp\left(\int_u^{u'}\frac{2M\gamma}{\bar r^2}\,du''\right)\mathcal E rr_u\,du'. \tag*{(79)}
\end{aligned} r ~ ( u , v ) = exp ( ∫ u u Λ ( v ) r ˉ 2 2 M γ d u ′ ) [ r ~ ( u Λ ( v ) , v ) − ∫ u u Λ ( v ) exp ( − ∫ u ′ u Λ ( v ) r ˉ 2 2 M γ d u ′′ ) E r r u d u ′ ] = exp ( ∫ u u Λ ( v ) r ˉ 2 2 M γ d u ′ ) r ~ ( u Λ ( v ) , v ) − ∫ u u Λ ( v ) exp ( ∫ u u ′ r ˉ 2 2 M γ d u ′′ ) E r r u d u ′ . (79) Noting that the integrating factor is bounded by (52) , as
∫ u u Λ ( v ) 2 M γ r ˉ 2 d u ′ ≤ ∫ H v r ˉ − 2 d u ′ ≲ 1 , \int_u^{u^\Lambda(v)}\frac{2M\gamma}{\bar r^2}\,du' \le\int_{\mathcal H_v}\bar r^{-2}\,du' \lesssim1, ∫ u u Λ ( v ) r ˉ 2 2 M γ d u ′ ≤ ∫ H v r ˉ − 2 d u ′ ≲ 1 , we obtain by (64) , (70) , (76) and (79) that
∣ r ~ ( u , v ) ∣ ≲ A ε 2 τ − 2 + δ ( u , v ) + A ε 3 / 2 τ − 2 + δ ( u , v ) ≲ A ε 3 / 2 τ − 2 + δ ( u , v ) . (80) |\tilde r(u,v)| \lesssim A\varepsilon^2\tau^{-2+\delta}(u,v)+A\varepsilon^{3/2}\tau^{-2+\delta}(u,v)\lesssim A\varepsilon^{3/2}\tau^{-2+\delta}(u,v). \tag*{(80)} ∣ r ~ ( u , v ) ∣ ≲ A ε 2 τ − 2 + δ ( u , v ) + A ε 3/2 τ − 2 + δ ( u , v ) ≲ A ε 3/2 τ − 2 + δ ( u , v ) . (80) Combining (76) , (78) and (80) yields (74) as desired. Finally, (75) readily follows from (63) and (65) .
Remark 4.12. The favorable sign of the coefficient in (77) could be exploited by integrating backwards in v v v [53 ] , Lemma 8.19] for the subextremal Reissner–Nordström case, and even in [4 ] , Lemma 5.12] for the extremal case, where the usual red-shift degenerates.
We now proceed with the estimate for v ~ \tilde v v ~ { r ≤ Λ } \{r \le\Lambda\} { r ≤ Λ } { r ≥ Λ } \{r \ge\Lambda\} { r ≥ Λ }
Proposition 4.13. For any A ≥ 1 A \ge1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_f \in\mathbb B τ f ∈ B
∣ ν ν ˉ − 1 ∣ ≲ A 2 ε 3 / 2 τ − 1 + δ (81) \left|\frac{\nu}{\bar\nu}-1\right|\lesssim A^2\varepsilon^{3/2}\tau^{-1+\delta} \tag*{(81)} ν ˉ ν − 1 ≲ A 2 ε 3/2 τ − 1 + δ (81) on D τ f D_{\tau_f} D τ f
Proof. **Estimate for r ≥ Λ r \ge\Lambda r ≥ Λ (62) and the bootstrap assumptions, we readily bound
∣ ν ~ ∣ ≲ r ˉ − 2 ∣ r ~ ∣ + A ε 3 / 2 r ˉ − 1 τ − 3 + δ ≲ A ε 3 / 2 r ˉ − 1 τ − 2 + δ , (82) |\tilde\nu|\lesssim\bar r^{-2}|\tilde r|+A\varepsilon^{3/2}\bar r^{-1}\tau^{-3+\delta}\lesssim A\varepsilon^{3/2}\bar r^{-1}\tau^{-2+\delta}, \tag*{(82)} ∣ ν ~ ∣ ≲ r ˉ − 2 ∣ r ~ ∣ + A ε 3/2 r ˉ − 1 τ − 3 + δ ≲ A ε 3/2 r ˉ − 1 τ − 2 + δ , (82) which implies (81) for r ≥ Λ r \ge\Lambda r ≥ Λ ∣ ν ˉ ∣ ∼ 1 |\bar\nu|\sim1 ∣ ν ˉ ∣ ∼ 1
**Estimate for r ≤ Λ r \le\Lambda r ≤ Λ (10) , we obtain
∣ ∂ v log ( ν ν ˉ ) ∣ = ∣ ∂ v ν ν − ∂ v ν ˉ ν ˉ ∣ = ∣ 2 κ m r 2 + 4 π r ν T u v − 2 M r ˉ 2 ∣ ≲ 2 ∣ κ m r 2 − M r ˉ 2 ∣ + ∣ 4 π r ν T u v ∣ ≲ 2 m r 2 ∣ κ ~ ∣ + 2 ∣ m ~ ∣ r 2 + 2 M ∣ 1 r 2 − 1 r ˉ 2 ∣ + 4 π r ν T u v ≲ ∣ κ ~ ∣ + ∣ m ~ ∣ + ∣ r ~ ∣ + ∣ ψ + ∣ ∣ ψ − ∣ ≲ A 2 ε 3 / 2 τ − 2 + δ + ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 , (83) \begin{aligned}
\left|\partial_v\log\left(\frac{\nu}{\bar\nu}\right)\right| &= \left|\frac{\partial_v\nu}{\nu}-\frac{\partial_v\bar\nu}{\bar\nu}\right|=\left|\frac{2\kappa m}{r^2}+\frac{4\pi r}{\nu}T_{uv}-\frac{2M}{\bar r^2}\right|\lesssim2\left|\frac{\kappa m}{r^2}-\frac{M}{\bar r^2}\right|+\left|\frac{4\pi r}{\nu}T_{uv}\right|\\
&\lesssim\frac{2m}{r^2}|\tilde\kappa|+\frac{2|\tilde m|}{r^2}+2M\left|\frac{1}{r^2}-\frac{1}{\bar r^2}\right|+\frac{4\pi r}{\nu}T_{uv}\lesssim|\tilde\kappa|+|\tilde m|+|\tilde r|+|\psi_+|\,|\psi_-|\\
&\lesssim A^2\varepsilon^{3/2}\tau^{-2+\delta}+|\psi_+|^2+|\psi_-|^2, \tag*{(83)}
\end{aligned} ∂ v log ( ν ˉ ν ) = ν ∂ v ν − ν ˉ ∂ v ν ˉ = r 2 2 κm + ν 4 π r T uv − r ˉ 2 2 M ≲ 2 r 2 κm − r ˉ 2 M + ν 4 π r T uv ≲ r 2 2 m ∣ κ ~ ∣ + r 2 2∣ m ~ ∣ + 2 M r 2 1 − r ˉ 2 1 + ν 4 π r T uv ≲ ∣ κ ~ ∣ + ∣ m ~ ∣ + ∣ r ~ ∣ + ∣ ψ + ∣ ∣ ψ − ∣ ≲ A 2 ε 3/2 τ − 2 + δ + ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 , (83) by (44) , (53) and (74) . Since
∣ log ( ν ν ˉ ) ∣ Γ ≲ ∣ r ~ ∣ Γ ≲ A 2 ε 3 / 2 τ − 2 + δ , \left|\log\left(\frac{\nu}{\bar\nu}\right)\right|_\Gamma\lesssim|\tilde r|_\Gamma\lesssim A^2\varepsilon^{3/2}\tau^{-2+\delta}, log ( ν ˉ ν ) Γ ≲ ∣ r ~ ∣ Γ ≲ A 2 ε 3/2 τ − 2 + δ , by (82) , we conclude by integrating (83) backwards in v v v Γ \Gamma Γ (71) and the fact that
∫ v v Λ ( u ) ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 ) d v ′ ≤ F ( u , τ ) ≤ A ε 2 τ − 3 + δ . \int_v^{v^\Lambda(u)}\left(|\psi_+|^2+|\psi_-|^2\right)\,dv'\le\mathcal F(u,\tau)\le A\varepsilon^2\tau^{-3+\delta}. ∫ v v Λ ( u ) ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 ) d v ′ ≤ F ( u , τ ) ≤ A ε 2 τ − 3 + δ . Finally, we bound m ~ \tilde m m ~ ε \varepsilon ε (84) below is strictly better than in the bootstrap assumption (44) .
Proposition 4.14. For any A ≥ 1 A \ge1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_f \in\mathcal{B} τ f ∈ B
∣ m ~ ∣ ≲ A ε 2 τ − 3 + δ , (84) |\tilde m| \lesssim A\varepsilon^2\tau^{-3+\delta}, \tag*{(84)} ∣ m ~ ∣ ≲ A ε 2 τ − 3 + δ , (84) on D τ f D_{\tau_f} D τ f
Proof. We begin by focusing on the final ingoing cone [ 0 , Γ u ( τ f ) ] × { Γ v ( τ f ) } [0,\Gamma^u(\tau_f)] \times\{\Gamma^v(\tau_f)\} [ 0 , Γ u ( τ f )] × { Γ v ( τ f )} (17) ,
∂ u m ~ = ν κ 1 / 2 ℑ [ ψ − ‾ ψ + ] − r 2 ( 1 − μ ) ℑ [ ψ − ‾ ∂ u ψ − ] , \partial_u\tilde m = \frac{\nu}{\kappa^{1/2}}\Im[\overline{\psi_-}\psi_+] - r^2(1-\mu)\Im[\overline{\psi_-}\partial_u\psi_-], ∂ u m ~ = κ 1/2 ν ℑ [ ψ − ψ + ] − r 2 ( 1 − μ ) ℑ [ ψ − ∂ u ψ − ] , we obtain by integrating backwards in u u u ( Γ u ( τ f ) , Γ v ( τ f ) ) (\Gamma^u(\tau_f),\Gamma^v(\tau_f)) ( Γ u ( τ f ) , Γ v ( τ f )) (32) , (70) and the bootstrap assumptions, that
∣ m ~ ( u , Γ v ( τ f ) ) ∣ ≲ F ( Γ v ( τ f ) , τ ( u , Γ v ( τ f ) ) ) ≲ A ε 2 τ ( u , Γ v ( τ f ) ) − 3 + δ , (85) |\tilde m(u,\Gamma^v(\tau_f))| \lesssim\mathcal{F}(\Gamma^v(\tau_f),\tau(u,\Gamma^v(\tau_f))) \lesssim A\varepsilon^2\tau(u,\Gamma^v(\tau_f))^{-3+\delta}, \tag*{(85)} ∣ m ~ ( u , Γ v ( τ f )) ∣ ≲ F ( Γ v ( τ f ) , τ ( u , Γ v ( τ f ))) ≲ A ε 2 τ ( u , Γ v ( τ f ) ) − 3 + δ , (85) for every u ≤ Γ u ( τ f ) u \le\Gamma^u(\tau_f) u ≤ Γ u ( τ f ) v < Γ v ( τ f ) v < \Gamma^v(\tau_f) v < Γ v ( τ f ) (18) , that we write here
∂ v m ~ = λ κ 1 / 2 ℑ [ ψ − ‾ ψ + ] + 1 κ ℑ [ ψ + ‾ ∂ v ψ + ] , \partial_v\tilde m = \frac{\lambda}{\kappa^{1/2}}\Im[\overline{\psi_-}\psi_+] + \frac{1}{\kappa}\Im[\overline{\psi_+}\partial_v\psi_+], ∂ v m ~ = κ 1/2 λ ℑ [ ψ − ψ + ] + κ 1 ℑ [ ψ + ∂ v ψ + ] , and we integrate it backwards in v v v D τ f ∩ { r ≥ Λ } D_{\tau_f} \cap\{r \ge\Lambda\} D τ f ∩ { r ≥ Λ }
∣ m ~ ( u , v ) ∣ ≲ ∣ m ~ ( u , Γ v ( τ f ) ) ∣ + ∫ v Γ v ( τ f ) ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v ′ |\tilde m(u,v)| \lesssim|\tilde m(u,\Gamma^v(\tau_f))| + \int_v^{\Gamma^v(\tau_f)} (|\psi_-|^2 + |\psi_+|^2 + |\partial_v\psi_+|^2)\,dv' ∣ m ~ ( u , v ) ∣ ≲ ∣ m ~ ( u , Γ v ( τ f )) ∣ + ∫ v Γ v ( τ f ) ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v ′ ≲ A ε 2 τ ( u , Γ v ( τ f ) ) − 3 + δ + E 0 ( τ ( u , v ) ) ≲ A ε 2 τ ( u , v ) − 3 + δ , (86) \lesssim A\varepsilon^2\tau(u,\Gamma^v(\tau_f))^{-3+\delta} + \mathcal{E}_0(\tau(u,v)) \lesssim A\varepsilon^2\tau(u,v)^{-3+\delta}, \tag*{(86)} ≲ A ε 2 τ ( u , Γ v ( τ f ) ) − 3 + δ + E 0 ( τ ( u , v )) ≲ A ε 2 τ ( u , v ) − 3 + δ , (86) where we combined the previous geometric estimates with the bootstrap assumptions.
Next, when ( u , v ) ∈ D τ f ∩ { r ≤ Λ } (u,v) \in D_{\tau_f} \cap\{r \le\Lambda\} ( u , v ) ∈ D τ f ∩ { r ≤ Λ } v v v ( u , v Λ ( u ) ) (u,v^\Lambda(u)) ( u , v Λ ( u )) (86) to estimate
∣ m ~ ( u , v ) ∣ ≲ A ε 2 τ ( u , v Λ ( u ) ) − 3 + δ + ∫ v v Λ ( u ) ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v ′ |\tilde m(u,v)| \lesssim A\varepsilon^2\tau(u,v^\Lambda(u))^{-3+\delta} + \int_v^{v^\Lambda(u)} (|\psi_+|^2 + |\psi_-|^2 + |\partial_v\psi_+|^2)\,dv' ∣ m ~ ( u , v ) ∣ ≲ A ε 2 τ ( u , v Λ ( u ) ) − 3 + δ + ∫ v v Λ ( u ) ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v ′ ≲ A ε 2 τ ( u , v Λ ( u ) ) − 3 + δ + F ( u , τ ( u , v ) ) ≲ A ε 2 τ ( u , v ) − 3 + δ . (87) \lesssim A\varepsilon^2\tau(u,v^\Lambda(u))^{-3+\delta} + \mathcal{F}(u,\tau(u,v)) \lesssim A\varepsilon^2\tau(u,v)^{-3+\delta}. \tag*{(87)} ≲ A ε 2 τ ( u , v Λ ( u ) ) − 3 + δ + F ( u , τ ( u , v )) ≲ A ε 2 τ ( u , v ) − 3 + δ . (87) Combining (86) and (87) , (84) finally follows. □
We now put all of the estimates proved in this section together and improve the bootstrap assumptions for the geometry, (42) –(44) .
Proof of Proposition 4.1. Summarizing Propositions 4.11 and 4.13, we have found on D τ f D_{\tau_f} D τ f
∣ r ~ ∣ ≲ A ε 3 / 2 τ − 2 + δ , |\tilde r| \lesssim A\varepsilon^{3/2}\tau^{-2+\delta}, ∣ r ~ ∣ ≲ A ε 3/2 τ − 2 + δ , ∣ ν ν ˉ − 1 ∣ ≲ A 2 ε 3 / 2 τ − 1 + δ , \left|\frac{\nu}{\bar\nu}-1\right| \lesssim A^2\varepsilon^{3/2}\tau^{-1+\delta}, ν ˉ ν − 1 ≲ A 2 ε 3/2 τ − 1 + δ , where the implicit constant does not depend on A A A A A A
∣ r ~ ∣ ≤ 1 2 A ε 3 / 2 τ − 2 + δ , |\tilde r| \le\frac{1}{2}A\varepsilon^{3/2}\tau^{-2+\delta}, ∣ r ~ ∣ ≤ 2 1 A ε 3/2 τ − 2 + δ , ∣ ν ν ˉ − 1 ∣ ≤ 1 2 A 3 ε 3 / 2 τ − 1 + δ . \left|\frac{\nu}{\bar\nu}-1\right| \le\frac{1}{2}A^3\varepsilon^{3/2}\tau^{-1+\delta}. ν ˉ ν − 1 ≤ 2 1 A 3 ε 3/2 τ − 1 + δ . Once A A A
∣ m ~ ∣ ≲ A ε 2 τ − 3 + δ , |\tilde m| \lesssim A\varepsilon^2\tau^{-3+\delta}, ∣ m ~ ∣ ≲ A ε 2 τ − 3 + δ , which then implies, for ε \varepsilon ε
∣ m ~ ∣ ≤ 1 2 A ε 3 / 2 τ − 3 + δ |\tilde m| \le\frac{1}{2}A\varepsilon^{3/2}\tau^{-3+\delta} ∣ m ~ ∣ ≤ 2 1 A ε 3/2 τ − 3 + δ in D τ f D_{\tau_f} D τ f
Preliminary estimates for the coupled system
The coupling between the Dirac field and the geometry presents additional difficulties compared to the case of the scalar wave equation. For instance, as discussed in Remark 2.6, exploiting the symmetric structure of the Dirac system on a Schwarzschild background is essential to prove energy estimates. Thus, from (22) –(23) , we infer that suitable bounds for ∂ u log Ω \partial_u\log\Omega ∂ u log Ω ∂ v log Ω \partial_v\log\Omega ∂ v log Ω log Ω \log\Omega log Ω Lemma 5.2 . Next, in Lemma 5.6 we derive pointwise estimates for the derivatives of the Dirac field. Lastly, we employ these bounds to control in Lemma 5.8 the derivatives of the geometric quantities (such as κ \kappa κ γ \gamma γ m m m § 8 . We conclude with a final lemma, which follows as a direct consequence of these bounds.
To this end, we adapt the strategy used in [4 Section 7.2] , where analogous pointwise decay estimates are obtained for the scalar field on a perturbed extremal Reissner–Nordström black hole.
Lemma 5.1. For any A ≥ 1 A \ge1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_f \in\mathcal{B} τ f ∈ B
r ˉ 1 / 2 ∣ ψ + ∣ ≲ A 1 / 2 ε τ − 3 / 2 + δ / 2 , (88) \bar r^{1/2}|\psi_+| \lesssim A^{1/2}\varepsilon\tau^{-3/2+\delta/2}, \tag*{(88)} r ˉ 1/2 ∣ ψ + ∣ ≲ A 1/2 ε τ − 3/2 + δ /2 , (88) r ˉ ∣ ψ + ∣ ≲ A 1 / 2 ε τ − 1 + δ / 2 , (89) \bar r|\psi_+| \lesssim A^{1/2}\varepsilon\tau^{-1+\delta/2}, \tag*{(89)} r ˉ ∣ ψ + ∣ ≲ A 1/2 ε τ − 1 + δ /2 , (89) r ˉ ∣ ψ − ∣ ≲ A 1 / 2 ε τ − 3 / 2 + δ / 2 , (90) \bar r|\psi_-| \lesssim A^{1/2}\varepsilon\tau^{-3/2+\delta/2}, \tag*{(90)} r ˉ ∣ ψ − ∣ ≲ A 1/2 ε τ − 3/2 + δ /2 , (90) r ˉ 7 / 2 ∣ T u v ν ∣ ≲ A ε 2 τ − 3 + δ , (91) \bar r^{7/2}\left|\frac{T_{uv}}{\nu}\right| \lesssim A\varepsilon^{2}\tau^{-3+\delta}, \tag*{(91)} r ˉ 7/2 ν T uv ≲ A ε 2 τ − 3 + δ , (91) on D τ f D_{\tau_f} D τ f
Proof. Proof of (88) –(90) for r ≥ Λ r \ge\Lambda r ≥ Λ χ 1 = χ 1 ( r ) \chi_1=\chi_1(r) χ 1 = χ 1 ( r ) χ 1 ( r ) = 1 \chi_1(r)=1 χ 1 ( r ) = 1 r ≥ 2 Λ r \ge2\Lambda r ≥ 2Λ χ 1 ( r ) = 0 \chi_1(r)=0 χ 1 ( r ) = 0 r ≤ Λ r \le\Lambda r ≤ Λ Lemma 4.8 , the set { r = 2 Λ } \{r=2\Lambda\} { r = 2Λ } D τ f D_{\tau_f} D τ f ( u , v ) ∈ D τ f ∩ { r ≥ 2 Λ } (u,v)\in D_{\tau_f}\cap\{r\ge2\Lambda\} ( u , v ) ∈ D τ f ∩ { r ≥ 2Λ } { u } × [ v Λ ( u ) , v ] \{u\}\times[v^\Lambda(u),v] { u } × [ v Λ ( u ) , v ] [ u , u Λ ( v ) ] × { v } [u,u^\Lambda(v)]\times\{v\} [ u , u Λ ( v )] × { v } D τ f D_{\tau_f} D τ f β ∈ { 1 , 2 } \beta\in\{1,2\} β ∈ { 1 , 2 } ( u , v ) ∈ D τ f ∩ { r ≥ 2 Λ } (u,v)\in D_{\tau_f}\cap\{r\ge2\Lambda\} ( u , v ) ∈ D τ f ∩ { r ≥ 2Λ }
r β ∣ ψ + ∣ 2 = r β − 2 ∣ r ψ + ∣ 2 = ∣ ∫ v Λ ( u ) v ( χ 1 ′ λ r β ∣ ψ + ∣ 2 + χ 1 ( β − 2 ) λ r β − 3 ∣ r ψ + ∣ 2 + 2 χ 1 r β − 2 ℜ [ r ψ + ‾ ∂ v ( r ψ + ) ] ) d v ′ ∣ ≲ ∫ v Λ ( u ) v ( ∣ β − 2 ∣ r β − 1 ∣ ψ + ∣ 2 + r β − 1 ∣ ψ + ∣ ∣ ∂ v ( r ψ + ) ∣ ) d v ′ ≲ ( E 0 ( τ ( u , v ) ) ) 1 / 2 ( E 2 β − 2 ( τ ( u , v ) ) ) 1 / 2 ≲ A ε 2 τ − 4 + β + δ ( u , v ) , \begin{aligned}
r^\beta|\psi_+|^2&=r^{\beta-2}|r\psi_+|^2\\
&=\left|\int_{v^\Lambda(u)}^v\left(\chi_1'\lambda r^\beta|\psi_+|^2+\chi_1(\beta-2)\lambda r^{\beta-3}|r\psi_+|^2+2\chi_1r^{\beta-2}\Re[r\overline{\psi_+}\partial_v(r\psi_+)]\right)\,dv'\right|\\
&\lesssim\int_{v^\Lambda(u)}^v\left(|\beta-2|r^{\beta-1}|\psi_+|^2+r^{\beta-1}|\psi_+||\partial_v(r\psi_+)|\right)\,dv'\\
&\lesssim\bigl(\mathcal{E}_0(\tau(u,v))\bigr)^{1/2}\bigl(\mathcal{E}_{2\beta-2}(\tau(u,v))\bigr)^{1/2}\\
&\lesssim A\varepsilon^2\tau^{-4+\beta+\delta}(u,v),
\end{aligned} r β ∣ ψ + ∣ 2 = r β − 2 ∣ r ψ + ∣ 2 = ∫ v Λ ( u ) v ( χ 1 ′ λ r β ∣ ψ + ∣ 2 + χ 1 ( β − 2 ) λ r β − 3 ∣ r ψ + ∣ 2 + 2 χ 1 r β − 2 ℜ [ r ψ + ∂ v ( r ψ + )] ) d v ′ ≲ ∫ v Λ ( u ) v ( ∣ β − 2∣ r β − 1 ∣ ψ + ∣ 2 + r β − 1 ∣ ψ + ∣∣ ∂ v ( r ψ + ) ∣ ) d v ′ ≲ ( E 0 ( τ ( u , v )) ) 1/2 ( E 2 β − 2 ( τ ( u , v )) ) 1/2 ≲ A ε 2 τ − 4 + β + δ ( u , v ) , r 2 ∣ ψ − ∣ 2 = ∣ ∫ u u Λ ( v ) ( χ 1 ′ ν r 2 ∣ ψ − ∣ 2 + 2 χ 1 r ν ∣ ψ − ∣ 2 + 2 χ 1 r 2 ℜ [ ψ − ‾ ∂ u ψ − ] ) d u ′ ∣ ≲ ∫ u u Λ ( v ) ( r 2 ∣ ψ − ∣ 2 + r 2 ∣ ∂ u ψ − ∣ 2 ) d u ′ ≲ F ( τ ( u , v ) ) ≲ A ε 2 τ − 3 + δ ( u , v ) , \begin{aligned}
r^2|\psi_-|^2&=\left|\int_u^{u^\Lambda(v)}\left(\chi_1'\nu r^2|\psi_-|^2+2\chi_1r\nu|\psi_-|^2+2\chi_1r^2\Re[\overline{\psi_-}\partial_u\psi_-]\right)\,du'\right|\\
&\lesssim\int_u^{u^\Lambda(v)}\left(r^2|\psi_-|^2+r^2|\partial_u\psi_-|^2\right)\,du'\\
&\lesssim\mathcal{F}(\tau(u,v))\lesssim A\varepsilon^2\tau^{-3+\delta}(u,v),
\end{aligned} r 2 ∣ ψ − ∣ 2 = ∫ u u Λ ( v ) ( χ 1 ′ ν r 2 ∣ ψ − ∣ 2 + 2 χ 1 r ν ∣ ψ − ∣ 2 + 2 χ 1 r 2 ℜ [ ψ − ∂ u ψ − ] ) d u ′ ≲ ∫ u u Λ ( v ) ( r 2 ∣ ψ − ∣ 2 + r 2 ∣ ∂ u ψ − ∣ 2 ) d u ′ ≲ F ( τ ( u , v )) ≲ A ε 2 τ − 3 + δ ( u , v ) , proving (88) –(90) for r ≥ 2 Λ r\ge2\Lambda r ≥ 2Λ ( u , v ) ∈ { Λ ≤ r ≤ 2 Λ } ∩ D τ f (u,v)\in\{\Lambda\le r\le2\Lambda\}\cap D_{\tau_f} ( u , v ) ∈ { Λ ≤ r ≤ 2Λ } ∩ D τ f
u ∗ ( v ) : = { u 2 Λ ( v ) if r ( 0 , v ) ≥ 2 Λ , 0 otherwise . u^*(v):=
\begin{cases}
u^{2\Lambda}(v) & \text{if } r(0,v)\ge2\Lambda,\\
0 & \text{otherwise}.
\end{cases} u ∗ ( v ) := { u 2Λ ( v ) 0 if r ( 0 , v ) ≥ 2Λ , otherwise . Noting that [ u ∗ ( v ) , u ] × { v } ⊂ D τ f [u^*(v),u]\times\{v\}\subset D_{\tau_f} [ u ∗ ( v ) , u ] × { v } ⊂ D τ f 0 ≤ u − u ∗ ( v ) ≲ 1 0\le u-u^*(v)\lesssim1 0 ≤ u − u ∗ ( v ) ≲ 1 τ ( u ∗ ( v ) , v ) ∼ τ ( u , v ) \tau(u^*(v),v)\sim\tau(u,v) τ ( u ∗ ( v ) , v ) ∼ τ ( u , v )
∣ ψ − ( u , v ) ∣ ≤ ∣ ψ − ( u ∗ ( v ) , v ) ∣ + ∫ u ∗ ( v ) u ∣ ∂ u ψ − ∣ d u ′ ≲ A 1 / 2 ε τ ( u ∗ ( v ) , v ) − 3 / 2 + δ / 2 + ( F ( u , τ ( u ∗ ( v ) , v ) ) ) 1 / 2 ≲ A 1 / 2 ε τ − 3 / 2 + δ / 2 ( u , v ) , \begin{aligned}
|\psi_-(u,v)|&\le|\psi_-(u^*(v),v)|+\int_{u^*(v)}^u|\partial_u\psi_-|\,du'\\
&\lesssim A^{1/2}\varepsilon\tau(u^*(v),v)^{-3/2+\delta/2}+\bigl(\mathcal{F}(u,\tau(u^*(v),v))\bigr)^{1/2}\\
&\lesssim A^{1/2}\varepsilon\tau^{-3/2+\delta/2}(u,v),
\end{aligned} ∣ ψ − ( u , v ) ∣ ≤ ∣ ψ − ( u ∗ ( v ) , v ) ∣ + ∫ u ∗ ( v ) u ∣ ∂ u ψ − ∣ d u ′ ≲ A 1/2 ε τ ( u ∗ ( v ) , v ) − 3/2 + δ /2 + ( F ( u , τ ( u ∗ ( v ) , v )) ) 1/2 ≲ A 1/2 ε τ − 3/2 + δ /2 ( u , v ) , ∣ ψ + ( u , v ) ∣ ≲ ∣ Φ + ( u , v ) ∣ ≤ ∣ Φ + ( u ∗ ( v ) , v ) ∣ + ∫ u ∗ ( v ) u ∣ ∂ u Φ + ∣ d u ′ ≲ ∣ ψ + ( u ∗ ( v ) , v ) ∣ + ∫ u ∗ ( v ) u ∣ ψ − ∣ d u ′ ≲ A 1 / 2 ε τ ( u ∗ ( v ) , v ) − 3 / 2 + δ / 2 + ( F ( u , τ ( u ∗ ( v ) , v ) ) ) 1 / 2 ≲ A 1 / 2 ε τ − 3 / 2 + δ / 2 ( u , v ) , \begin{aligned}
|\psi_+(u,v)|&\lesssim|\Phi_+(u,v)|\le|\Phi_+(u^*(v),v)|+\int_{u^*(v)}^u|\partial_u\Phi_+|\,du'\\
&\lesssim|\psi_+(u^*(v),v)|+\int_{u^*(v)}^u|\psi_-|\,du'\\
&\lesssim A^{1/2}\varepsilon\tau(u^*(v),v)^{-3/2+\delta/2}+\bigl(\mathcal{F}(u,\tau(u^*(v),v))\bigr)^{1/2}\\
&\lesssim A^{1/2}\varepsilon\tau^{-3/2+\delta/2}(u,v),
\end{aligned} ∣ ψ + ( u , v ) ∣ ≲ ∣ Φ + ( u , v ) ∣ ≤ ∣ Φ + ( u ∗ ( v ) , v ) ∣ + ∫ u ∗ ( v ) u ∣ ∂ u Φ + ∣ d u ′ ≲ ∣ ψ + ( u ∗ ( v ) , v ) ∣ + ∫ u ∗ ( v ) u ∣ ψ − ∣ d u ′ ≲ A 1/2 ε τ ( u ∗ ( v ) , v ) − 3/2 + δ /2 + ( F ( u , τ ( u ∗ ( v ) , v )) ) 1/2 ≲ A 1/2 ε τ − 3/2 + δ /2 ( u , v ) , where we used ψ + = κ − 1 / 2 Φ + ∼ Φ + \psi_{+}=\kappa^{-1/2}\Phi_{+}\sim\Phi_{+} ψ + = κ − 1/2 Φ + ∼ Φ + ∂ u Φ + = κ 1 / 2 ν ψ − ∼ ν ψ − \partial_{u}\Phi_{+}=\kappa^{1/2}\nu\psi_{-}\sim\nu\psi_{-} ∂ u Φ + = κ 1/2 ν ψ − ∼ ν ψ − { Λ ≤ r ≤ 2 Λ } \{\Lambda\le r\le2\Lambda\} { Λ ≤ r ≤ 2Λ } (52) and (54) . This proves (88) –(90) in the whole region { r ≥ Λ } \{r\ge\Lambda\} { r ≥ Λ }
Proof of (5.1)–(5.3) for r ≤ Λ r\le\Lambda r ≤ Λ Let χ 2 = χ 2 ( r ) \chi_{2}=\chi_{2}(r) χ 2 = χ 2 ( r ) χ 2 ( r ) = 1 \chi_{2}(r)=1 χ 2 ( r ) = 1 r ≤ Λ / 2 r\le\Lambda/2 r ≤ Λ/2 χ 2 ( r ) = 0 \chi_{2}(r)=0 χ 2 ( r ) = 0 r ≥ Λ r\ge\Lambda r ≥ Λ { r = Λ / 2 } \{r=\Lambda/2\} { r = Λ/2 } D τ f D_{\tau_{f}} D τ f ( u , v ) ∈ D τ f (u,v)\in D_{\tau_{f}} ( u , v ) ∈ D τ f r ( u , v ) ≤ Λ / 2 r(u,v)\le\Lambda/2 r ( u , v ) ≤ Λ/2 [ u Λ ( v ) , u ] × { v } [u^{\Lambda}(v),u]\times\{v\} [ u Λ ( v ) , u ] × { v } D τ f D_{\tau_{f}} D τ f
∣ ψ + ∣ 2 ( u , v ) ≲ ∣ Φ + ∣ 2 ( u , v ) ≲ ∣ ∫ u Λ ( v ) u ( χ 2 ′ ν ∣ Φ + ∣ 2 + 2 χ 2 ℜ [ Φ + ‾ ∂ u Φ + ] ) d u ′ ∣ |\psi_{+}|^{2}(u,v)\lesssim|\Phi_{+}|^{2}(u,v)\lesssim\left|\int_{u^{\Lambda}(v)}^{u}\left(\chi_{2}'\nu|\Phi_{+}|^{2}+2\chi_{2}\Re[\overline{\Phi_{+}}\partial_{u}\Phi_{+}]\right)\,du'\right| ∣ ψ + ∣ 2 ( u , v ) ≲ ∣ Φ + ∣ 2 ( u , v ) ≲ ∫ u Λ ( v ) u ( χ 2 ′ ν ∣ Φ + ∣ 2 + 2 χ 2 ℜ [ Φ + ∂ u Φ + ] ) d u ′ ≲ ∫ u Λ ( v ) u ( − ν ˉ ) ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 ) d u ′ ≲ E ( τ ( u , v ) ) ≲ A ε 2 τ − 3 + δ ( u , v ) , \lesssim\int_{u^{\Lambda}(v)}^{u}(-\bar{\nu})\left(|\psi_{+}|^{2}+|\psi_{-}|^{2}\right)\,du'\lesssim\mathcal{E}(\tau(u,v))\lesssim A\varepsilon^{2}\tau^{-3+\delta}(u,v), ≲ ∫ u Λ ( v ) u ( − ν ˉ ) ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 ) d u ′ ≲ E ( τ ( u , v )) ≲ A ε 2 τ − 3 + δ ( u , v ) , ∣ ψ − ∣ 2 ( u , v ) = ∣ ∫ u Λ ( v ) u ( χ 2 ′ ν ∣ ψ − ∣ 2 + 2 χ 2 ℜ [ ψ − ‾ ∂ u ψ − ] ) d u ′ ∣ |\psi_{-}|^{2}(u,v)=\left|\int_{u^{\Lambda}(v)}^{u}\left(\chi_{2}'\nu|\psi_{-}|^{2}+2\chi_{2}\Re[\overline{\psi_{-}}\partial_{u}\psi_{-}]\right)\,du'\right| ∣ ψ − ∣ 2 ( u , v ) = ∫ u Λ ( v ) u ( χ 2 ′ ν ∣ ψ − ∣ 2 + 2 χ 2 ℜ [ ψ − ∂ u ψ − ] ) d u ′ ≲ ∫ u Λ ( v ) u ( − ν ˉ ∣ ψ − ∣ 2 + ∣ ∂ u ψ − ∣ 2 − ν ˉ ) d u ′ ≲ E ( τ ( u , v ) ) ≲ A ε 2 τ − 3 + δ ( u , v ) , \lesssim\int_{u^{\Lambda}(v)}^{u}\left(-\bar{\nu}|\psi_{-}|^{2}+\frac{|\partial_{u}\psi_{-}|^{2}}{-\bar{\nu}}\right)\,du'\lesssim\mathcal{E}(\tau(u,v))\lesssim A\varepsilon^{2}\tau^{-3+\delta}(u,v), ≲ ∫ u Λ ( v ) u ( − ν ˉ ∣ ψ − ∣ 2 + − ν ˉ ∣ ∂ u ψ − ∣ 2 ) d u ′ ≲ E ( τ ( u , v )) ≲ A ε 2 τ − 3 + δ ( u , v ) , where in the first line we used again ψ + ∼ Φ + \psi_{+}\sim\Phi_{+} ψ + ∼ Φ + ∂ u Φ + = κ 1 / 2 ν ψ − ∼ ν ψ − \partial_{u}\Phi_{+}=\kappa^{1/2}\nu\psi_{-}\sim\nu\psi_{-} ∂ u Φ + = κ 1/2 ν ψ − ∼ ν ψ − (88) –(90) for r ≤ Λ / 2 r\le\Lambda/2 r ≤ Λ/2 Λ / 2 ≤ r ≤ Λ \Lambda/2\le r\le\Lambda Λ/2 ≤ r ≤ Λ u u u Λ ≤ r ≤ 2 Λ \Lambda\le r\le2\Lambda Λ ≤ r ≤ 2Λ
Proof of (5.4): This follows immediately by (14) and the previous estimates. □
Let us now proceed with establishing estimates on the derivatives of the null lapse on D τ f D_{\tau_{f}} D τ f
Lemma 5.2. For any A ≥ 1 A\ge1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_{f}\in\mathcal{B} τ f ∈ B
∣ ∂ v log Ω − M r ˉ − 2 ∣ ≲ A 2 ε 3 / 2 τ − 2 + δ (92) |\partial_{v}\log\Omega-M\bar{r}^{-2}|\lesssim A^{2}\varepsilon^{3/2}\tau^{-2+\delta} \tag*{(92)} ∣ ∂ v log Ω − M r ˉ − 2 ∣ ≲ A 2 ε 3/2 τ − 2 + δ (92) r ˉ ∣ ∂ v log Ω − M r ˉ − 2 ∣ ≲ A 3 ε 3 / 2 τ − 1 + δ (93) \bar{r}|\partial_{v}\log\Omega-M\bar{r}^{-2}|\lesssim A^{3}\varepsilon^{3/2}\tau^{-1+\delta} \tag*{(93)} r ˉ ∣ ∂ v log Ω − M r ˉ − 2 ∣ ≲ A 3 ε 3/2 τ − 1 + δ (93) r ˉ 2 ∣ ∂ u log Ω + M r ˉ − 2 ∣ ≲ A 3 ε 3 / 2 τ − 3 / 4 + δ , (94) \bar{r}^{2}|\partial_{u}\log\Omega+M\bar{r}^{-2}|\lesssim A^{3}\varepsilon^{3/2}\tau^{-3/4+\delta}, \tag*{(94)} r ˉ 2 ∣ ∂ u log Ω + M r ˉ − 2 ∣ ≲ A 3 ε 3/2 τ − 3/4 + δ , (94) in D τ f D_{\tau_{f}} D τ f
Proof. Proof of (5.5) and (5.6): We start by noting that, since κ = 1 \kappa=1 κ = 1 [ Γ u ( τ f ) ] × [ 0 , Γ v ( τ f ) ] [\Gamma^{u}(\tau_{f})]\times[0,\Gamma^{v}(\tau_{f})] [ Γ u ( τ f )] × [ 0 , Γ v ( τ f )]
0 = ∂ v κ = ∂ u ( ( − ν ) − 1 Ω 2 ) = ( − ν ) − 1 ( ∂ v Ω 2 + ∂ v ν ) = 2 ∂ v log Ω + ( − ν ) − 1 ∂ v ν , 0=\partial_{v}\kappa=\partial_{u}((-\nu)^{-1}\Omega^{2})=(-\nu)^{-1}(\partial_{v}\Omega^{2}+\partial_{v}\nu)=2\partial_{v}\log\Omega+(-\nu)^{-1}\partial_{v}\nu, 0 = ∂ v κ = ∂ u (( − ν ) − 1 Ω 2 ) = ( − ν ) − 1 ( ∂ v Ω 2 + ∂ v ν ) = 2 ∂ v log Ω + ( − ν ) − 1 ∂ v ν , yielding
∂ v log Ω = m r 2 + 2 π r T u v ν . \partial_{v}\log\Omega=\frac{m}{r^{2}}+2\pi r\frac{T_{uv}}{\nu}. ∂ v log Ω = r 2 m + 2 π r ν T uv . Therefore, we readily deduce
∣ ∂ v log Ω − M r ˉ 2 ∣ ( Γ u ( τ f ) , v ) ≲ ∣ m ~ ∣ r 2 + ∣ r ~ ∣ r 3 + r ∣ T u v − ν ∣ ≲ A 2 ε 3 / 2 τ − 2 + δ , (95) \left|\partial_{v}\log\Omega-\frac{M}{\bar{r}^{2}}\right|(\Gamma^{u}(\tau_{f}),v)\lesssim\frac{|\widetilde{m}|}{r^{2}}+\frac{|\widetilde{r}|}{r^{3}}+r\left|\frac{T_{uv}}{-\nu}\right|\lesssim A^{2}\varepsilon^{3/2}\tau^{-2+\delta}, \tag*{(95)} ∂ v log Ω − r ˉ 2 M ( Γ u ( τ f ) , v ) ≲ r 2 ∣ m ∣ + r 3 ∣ r ∣ + r − ν T uv ≲ A 2 ε 3/2 τ − 2 + δ , (95) for v ∈ [ 0 , Γ v ( τ f ) ] v\in[0,\Gamma^{v}(\tau_{f})] v ∈ [ 0 , Γ v ( τ f )] (91) and the geometric estimates established in the previous section. Next, using (11) and (15) , we compute
∂ u ( ∂ v log Ω − κ M r − 2 ) = 2 Ω 2 r − 3 m ~ − 4 Ω 2 r − 2 ℑ [ ψ − ‾ ψ + ] + 2 M r − 1 ℑ [ ψ − ‾ ∂ u ψ − ] . \partial_{u}(\partial_{v}\log\Omega-\kappa M r^{-2})=2\Omega^{2}r^{-3}\widetilde{m}-4\Omega^{2}r^{-2}\Im[\overline{\psi_{-}}\psi_{+}]+2Mr^{-1}\Im[\overline{\psi_{-}}\partial_{u}\psi_{-}]. ∂ u ( ∂ v log Ω − κ M r − 2 ) = 2 Ω 2 r − 3 m − 4 Ω 2 r − 2 ℑ [ ψ − ψ + ] + 2 M r − 1 ℑ [ ψ − ∂ u ψ − ] . Therefore, integrating backwards in u u u { r ≤ Λ } \{r\le\Lambda\} { r ≤ Λ }
∣ ∂ v log Ω − κ M r − 2 ∣ ( u , v ) ≲ ∣ ∂ v log Ω − κ M r − 2 ∣ ( Γ u ( τ f ) , v ) + ∫ u Γ u ( τ f ) ( 2 Ω 2 r − 3 ∣ m ~ ∣ + Ω 2 ∣ ψ − ∣ ∣ ψ + ∣ + ∣ ψ − ∣ ∣ ∂ u ψ − ∣ ) d u ′ . \left|\partial_{v}\log\Omega-\kappa Mr^{-2}\right|(u,v)\lesssim\left|\partial_{v}\log\Omega-\kappa Mr^{-2}\right|(\Gamma^{u}(\tau_{f}),v)
+\int_{u}^{\Gamma^{u}(\tau_{f})}\left(2\Omega^{2}r^{-3}|\widetilde{m}|+\Omega^{2}|\psi_{-}||\psi_{+}|+|\psi_{-}||\partial_{u}\psi_{-}|\right)\,du'. ∂ v log Ω − κ M r − 2 ( u , v ) ≲ ∂ v log Ω − κ M r − 2 ( Γ u ( τ f ) , v ) + ∫ u Γ u ( τ f ) ( 2 Ω 2 r − 3 ∣ m ∣ + Ω 2 ∣ ψ − ∣∣ ψ + ∣ + ∣ ψ − ∣∣ ∂ u ψ − ∣ ) d u ′ . After noting that Ω 2 = κ ( − ν ) ∼ ( − ν ˉ ) \Omega^{2}=\kappa(-\nu)\sim(-\bar{\nu}) Ω 2 = κ ( − ν ) ∼ ( − ν ˉ ) (50) and (54) , we find that
∫ u Γ u ( τ f ) 2 Ω 2 r − 3 ∣ m ~ ∣ d u ′ ≲ A ε 2 τ − 3 + δ ∫ u Γ u ( τ f ) ( − ν ˉ ) d u ′ ≲ A ε 2 τ − 3 + δ ∫ 2 M Λ d r ˉ ≲ A ε 2 τ − 3 + δ , \int_{u}^{\Gamma^{u}(\tau_{f})}2\Omega^{2}r^{-3}|\widetilde{m}|\,du'\lesssim A\varepsilon^{2}\tau^{-3+\delta}\int_{u}^{\Gamma^{u}(\tau_{f})}(-\bar{\nu})\,du'\lesssim A\varepsilon^{2}\tau^{-3+\delta}\int_{2M}^{\Lambda}d\bar{r}\lesssim A\varepsilon^{2}\tau^{-3+\delta}, ∫ u Γ u ( τ f ) 2 Ω 2 r − 3 ∣ m ∣ d u ′ ≲ A ε 2 τ − 3 + δ ∫ u Γ u ( τ f ) ( − ν ˉ ) d u ′ ≲ A ε 2 τ − 3 + δ ∫ 2 M Λ d r ˉ ≲ A ε 2 τ − 3 + δ , yielding, from (95) ,
∣ ∂ v log Ω − κ M r − 2 ∣ ( u , v ) ≲ A 2 ε 3 / 2 τ − 2 + δ + A ε 2 τ − 3 + δ + E ( τ ) ≲ A 2 ε 3 / 2 τ − 2 + δ , (96) \left|\partial_v \log\Omega-\kappa M r^{-2}\right|(u,v)\lesssim A^2\varepsilon^{3/2}\tau^{-2+\delta}+A\varepsilon^2\tau^{-3+\delta}+\mathcal{E}(\tau)\lesssim A^2\varepsilon^{3/2}\tau^{-2+\delta}, \tag*{(96)} ∂ v log Ω − κ M r − 2 ( u , v ) ≲ A 2 ε 3/2 τ − 2 + δ + A ε 2 τ − 3 + δ + E ( τ ) ≲ A 2 ε 3/2 τ − 2 + δ , (96) on D τ f ∩ { r ≤ Λ } D_{\tau_f}\cap\{r\leq\Lambda\} D τ f ∩ { r ≤ Λ } ( u , v ) ∈ { r ≥ Λ } (u,v)\in\{r\geq\Lambda\} ( u , v ) ∈ { r ≥ Λ } u u u
∣ ∂ v log Ω − κ M r − 2 ∣ ( u , v ) ≲ ∣ ∂ v log Ω − κ M r − 2 ∣ ( u Λ ( v ) , v ) + ∫ u u Λ ( v ) ( r − 3 ∣ m ~ ∣ + r − 2 ∣ ψ − ∣ ∣ ψ + ∣ + r − 1 ∣ ψ − ∣ ∣ ∂ u ψ − ∣ ) d u ′ ≲ A 2 ε 3 / 2 τ − 2 + δ + F ( v , τ ) ≲ A 2 ε 3 / 2 τ − 2 + δ , (97) \begin{aligned}
\left|\partial_v\log\Omega-\kappa M r^{-2}\right|(u,v)&\lesssim\left|\partial_v\log\Omega-\kappa M r^{-2}\right|(u^\Lambda(v),v)\\
&\quad+\int_u^{u^\Lambda(v)}\left(r^{-3}|\widetilde m|+r^{-2}|\psi_-||\psi_+|+r^{-1}|\psi_-||\partial_u\psi_-|\right)\,du'\\
&\lesssim A^2\varepsilon^{3/2}\tau^{-2+\delta}+\mathcal{F}(v,\tau)\lesssim A^2\varepsilon^{3/2}\tau^{-2+\delta}, \tag*{(97)}
\end{aligned} ∂ v log Ω − κ M r − 2 ( u , v ) ≲ ∂ v log Ω − κ M r − 2 ( u Λ ( v ) , v ) + ∫ u u Λ ( v ) ( r − 3 ∣ m ∣ + r − 2 ∣ ψ − ∣∣ ψ + ∣ + r − 1 ∣ ψ − ∣∣ ∂ u ψ − ∣ ) d u ′ ≲ A 2 ε 3/2 τ − 2 + δ + F ( v , τ ) ≲ A 2 ε 3/2 τ − 2 + δ , (97) where we used (47) , (70) and (84) . By combining (96) and (5.10) together with the previous geometric estimates on $\kappa$ and $r$, we obtain (5.5)` as desired,
∣ ∂ v log Ω − M r ‾ − 2 ∣ ≲ A 2 ε 3 / 2 τ − 2 + δ . \left|\partial_v\log\Omega-M\overline r^{-2}\right|\lesssim A^2\varepsilon^{3/2}\tau^{-2+\delta}. ∂ v log Ω − M r − 2 ≲ A 2 ε 3/2 τ − 2 + δ . To prove (93) , we proceed as in the previous lemma and consider a smooth cutoff χ 1 = χ 1 ( r ) \chi_1=\chi_1(r) χ 1 = χ 1 ( r ) χ 1 ( r ) = 1 \chi_1(r)=1 χ 1 ( r ) = 1 r ≥ 2 Λ r\geq2\Lambda r ≥ 2Λ χ 1 ( r ) = 0 \chi_1(r)=0 χ 1 ( r ) = 0 r ≤ Λ r\leq\Lambda r ≤ Λ ( u , v ) ∈ D τ f ∩ { r ≥ 2 Λ } (u,v)\in D_{\tau_f}\cap\{r\geq2\Lambda\} ( u , v ) ∈ D τ f ∩ { r ≥ 2Λ }
r ‾ ∣ ∂ v log Ω − κ M r − 2 ∣ ( u , v ) ≲ ∫ u u Λ ( v ) ∣ χ 1 ′ ν ‾ r ‾ ( ∂ v log Ω − κ M r − 2 ) ∣ + ∣ χ 1 ν ‾ ( ∂ v log Ω − κ M r − 2 ) ∣ + r ‾ ( r − 3 ∣ m ~ ∣ + r − 2 ∣ ψ − ∣ ∣ ψ + ∣ + r − 1 ∣ ψ − ∣ ∣ ∂ u ψ − ∣ ) d u ′ ≲ ∫ u u Λ ( v ) ( ∣ ∂ v log Ω − κ M r − 2 ∣ + r − 2 ∣ m ~ ∣ + r − 1 ∣ ψ − ∣ ∣ ψ + ∣ + ∣ ψ − ∣ ∣ ∂ u ψ − ∣ ) d u ′ ≲ A 2 ε 3 / 2 τ − 1 + δ + F ( v , τ ) ≲ A 2 ε 3 / 2 τ − 1 + δ , \begin{aligned}
\overline r\left|\partial_v\log\Omega-\kappa M r^{-2}\right|(u,v)&\lesssim\int_u^{u^\Lambda(v)}\left|\chi_1'\overline\nu\overline r\left(\partial_v\log\Omega-\kappa M r^{-2}\right)\right|+\left|\chi_1\overline\nu\left(\partial_v\log\Omega-\kappa M r^{-2}\right)\right|\\
&\quad+\overline r\left(r^{-3}|\widetilde m|+r^{-2}|\psi_-||\psi_+|+r^{-1}|\psi_-||\partial_u\psi_-|\right)\,du'\\
&\lesssim\int_u^{u^\Lambda(v)}\left(\left|\partial_v\log\Omega-\kappa M r^{-2}\right|+r^{-2}|\widetilde m|+r^{-1}|\psi_-||\psi_+|+|\psi_-||\partial_u\psi_-|\right)\,du'\\
&\lesssim A^2\varepsilon^{3/2}\tau^{-1+\delta}+\mathcal{F}(v,\tau)\lesssim A^2\varepsilon^{3/2}\tau^{-1+\delta},
\end{aligned} r ∂ v log Ω − κ M r − 2 ( u , v ) ≲ ∫ u u Λ ( v ) χ 1 ′ ν r ( ∂ v log Ω − κ M r − 2 ) + χ 1 ν ( ∂ v log Ω − κ M r − 2 ) + r ( r − 3 ∣ m ∣ + r − 2 ∣ ψ − ∣∣ ψ + ∣ + r − 1 ∣ ψ − ∣∣ ∂ u ψ − ∣ ) d u ′ ≲ ∫ u u Λ ( v ) ( ∂ v log Ω − κ M r − 2 + r − 2 ∣ m ∣ + r − 1 ∣ ψ − ∣∣ ψ + ∣ + ∣ ψ − ∣∣ ∂ u ψ − ∣ ) d u ′ ≲ A 2 ε 3/2 τ − 1 + δ + F ( v , τ ) ≲ A 2 ε 3/2 τ − 1 + δ , where in particular we used (47) , (70) and (92) . Using the previous geometric estimates on κ \kappa κ r r r (93) .
Remark 5.3. Before turning to the estimate for ∂ u log Ω \partial_u\log\Omega ∂ u log Ω ∂ v log Ω \partial_v\log\Omega ∂ v log Ω M r ‾ − 2 M\overline r^{-2} M r − 2 Ω ~ \widetilde\Omega Ω r ~ \widetilde r r τ \tau τ u u u r ≥ Λ r\geq\Lambda r ≥ Λ (93) directly with an r ‾ 2 \overline r^2 r 2 ∫ u u Λ ( v ) r ∣ ∂ v log Ω − κ M r − 2 ∣ d u ′ \int_u^{u^\Lambda(v)}r|\partial_v\log\Omega-\kappa M r^{-2}|\,du' ∫ u u Λ ( v ) r ∣ ∂ v log Ω − κ M r − 2 ∣ d u ′ (92) . For this reason, we first establish the unweighted estimate (92) and subsequently recover the weaker weighted estimate (93) .
Proof of (94) : Proceeding similarly, we start by noting that, since γ = 1 \gamma=1 γ = 1 [ 0 , Γ u ( τ f ) ] × { Γ v ( τ f ) } [0,\Gamma^u(\tau_f)]\times\{\Gamma^v(\tau_f)\} [ 0 , Γ u ( τ f )] × { Γ v ( τ f )}
0 = ∂ u γ = ∂ u ( λ − 1 Ω 2 ) = λ − 1 ( ∂ u Ω 2 − ∂ u λ ) = 2 ∂ u log Ω − λ − 1 ∂ u λ , 0=\partial_u\gamma=\partial_u(\lambda^{-1}\Omega^2)=\lambda^{-1}(\partial_u\Omega^2-\partial_u\lambda)=2\partial_u\log\Omega-\lambda^{-1}\partial_u\lambda, 0 = ∂ u γ = ∂ u ( λ − 1 Ω 2 ) = λ − 1 ( ∂ u Ω 2 − ∂ u λ ) = 2 ∂ u log Ω − λ − 1 ∂ u λ , yielding
∂ u log Ω = − m r 2 + 2 π r T u v λ = − m r 2 + 2 π r T u v Ω 2 , \partial_u\log\Omega=-\frac{m}{r^2}+2\pi r\frac{T_{uv}}{\lambda}=-\frac{m}{r^2}+2\pi r\frac{T_{uv}}{\Omega^2}, ∂ u log Ω = − r 2 m + 2 π r λ T uv = − r 2 m + 2 π r Ω 2 T uv , where we again used γ = λ − 1 Ω 2 = 1 \gamma=\lambda^{-1}\Omega^2=1 γ = λ − 1 Ω 2 = 1
∣ ∂ u log Ω + M r 2 ∣ ( u , Γ v ( τ f ) ) ≲ ∣ m ~ ∣ r 2 + ∣ r ~ ∣ r 3 + r ∣ T u v − ν ∣ ≲ A 2 ε 3 / 2 r ‾ − 2 τ − 2 + δ , \left|\partial_u\log\Omega+\frac{M}{r^2}\right|(u,\Gamma^v(\tau_f))\lesssim\frac{|\widetilde m|}{r^2}+\frac{|\widetilde r|}{r^3}+r\left|\frac{T_{uv}}{-\nu}\right|\lesssim A^2\varepsilon^{3/2}\overline r^{-2}\tau^{-2+\delta}, ∂ u log Ω + r 2 M ( u , Γ v ( τ f )) ≲ r 2 ∣ m ∣ + r 3 ∣ r ∣ + r − ν T uv ≲ A 2 ε 3/2 r − 2 τ − 2 + δ , for u ∈ [ 0 , Γ u ( τ f ) ] u\in[0,\Gamma^u(\tau_f)] u ∈ [ 0 , Γ u ( τ f )] (91) and the geometric estimates established in the previous section. Next, using (11) and (16) , we compute
∂ v ( ∂ u log Ω + γ m r − 2 ) = − 3 κ 1 / 2 ( − ν ) r 2 ℑ [ ψ − ψ + ‾ ] + 1 r 2 λ ( 2 γ m r − ν ) ℑ [ ψ + ∂ v ψ + ] . \partial_v(\partial_u\log\Omega+\gamma m r^{-2})=-\frac{3\kappa^{1/2}(-\nu)}{r^2}\Im[\psi_-\overline{\psi_+}]+\frac{1}{r^2\lambda}\left(\frac{2\gamma m}{r}-\nu\right)\Im[\psi_+\partial_v\psi_+]. ∂ v ( ∂ u log Ω + γ m r − 2 ) = − r 2 3 κ 1/2 ( − ν ) ℑ [ ψ − ψ + ] + r 2 λ 1 ( r 2 γ m − ν ) ℑ [ ψ + ∂ v ψ + ] . Therefore, first integrating backwards in v v v { r ≥ Λ } \{r \ge\Lambda\} { r ≥ Λ } r 2 r^2 r 2
r 2 ∣ ∂ u log Ω + γ m r − 2 ∣ ( u , v ) ≲ r 2 ( u , v ) ∣ ∂ u log Ω + γ m r − 2 ∣ ( u , Γ v ( τ f ) ) + r 2 ( u , v ) ∫ v Γ v ( τ f ) ( r − 2 ∣ ψ − ∣ ∣ ψ + ∣ + r − 2 ∣ ψ + ∣ ∣ ∂ v ψ + ∣ ) d v ′ ≲ A 2 ε 3 / 2 τ − 2 + δ + ∫ v Γ v ( τ f ) ( ∣ ψ − ∣ ∣ ψ + ∣ + ∣ ψ + ∣ ∣ ∂ v ψ + ∣ ) d v ′ ≲ A 2 ε 3 / 2 τ − 2 + δ + E 0 ( τ ) ≲ A 2 ε 3 / 2 τ − 2 + δ . (98) \begin{aligned}
r^2\lvert\partial_u \log\Omega+\gamma mr^{-2}\rvert(u,v)
&\lesssim r^2(u,v)\lvert\partial_u\log\Omega+\gamma mr^{-2}\rvert(u,\Gamma^v(\tau_f)) \\
&\quad+r^2(u,v)\int_v^{\Gamma^v(\tau_f)}\left(r^{-2}\lvert\psi_-\rvert\lvert\psi_+\rvert+r^{-2}\lvert\psi_+\rvert\lvert\partial_v\psi_+\rvert\right)\,dv' \\
&\lesssim A^2\varepsilon^{3/2}\tau^{-2+\delta}+\int_v^{\Gamma^v(\tau_f)}\left(\lvert\psi_-\rvert\lvert\psi_+\rvert+\lvert\psi_+\rvert\lvert\partial_v\psi_+\rvert\right)\,dv' \\
&\lesssim A^2\varepsilon^{3/2}\tau^{-2+\delta}+\mathcal{E}_0(\tau) \\
&\lesssim A^2\varepsilon^{3/2}\tau^{-2+\delta}. \tag*{(98)}
\end{aligned} r 2 ∣ ∂ u log Ω + γ m r − 2 ∣ ( u , v ) ≲ r 2 ( u , v ) ∣ ∂ u log Ω + γ m r − 2 ∣ ( u , Γ v ( τ f )) + r 2 ( u , v ) ∫ v Γ v ( τ f ) ( r − 2 ∣ ψ − ∣ ∣ ψ + ∣ + r − 2 ∣ ψ + ∣ ∣ ∂ v ψ + ∣ ) d v ′ ≲ A 2 ε 3/2 τ − 2 + δ + ∫ v Γ v ( τ f ) ( ∣ ψ − ∣ ∣ ψ + ∣ + ∣ ψ + ∣ ∣ ∂ v ψ + ∣ ) d v ′ ≲ A 2 ε 3/2 τ − 2 + δ + E 0 ( τ ) ≲ A 2 ε 3/2 τ − 2 + δ . (98) where we used (56) to bound r 2 ( u , v ) ≤ r 2 ( u , Γ v ( τ f ) ) r^2(u,v)\le r^2(u,\Gamma^v(\tau_f)) r 2 ( u , v ) ≤ r 2 ( u , Γ v ( τ f )) r r r γ \gamma γ r r r m m m (94) in D τ f ∩ { r ≥ Λ } D_{\tau_f}\cap\{r\ge\Lambda\} D τ f ∩ { r ≥ Λ }
For { r ≤ Λ } \{r\le\Lambda\} { r ≤ Λ }
ξ : = e r ˉ ( ∂ u log Ω + M r ˉ − 2 ) , \xi:=e^{\bar r}(\partial_u\log\Omega+M\bar r^{-2}), ξ := e r ˉ ( ∂ u log Ω + M r ˉ − 2 ) , which satisfies
∂ v ξ − λ ˉ ξ = e r ˉ F , F : = ( 2 Ω 2 m r 3 − 8 π T u v − 2 Ω 2 ‾ M r ˉ 3 ) , \partial_v\xi-\bar\lambda\xi=e^{\bar r}F,\qquad F:=\left(\frac{2\Omega^2m}{r^3}-8\pi T_{uv}-\frac{2\overline{\Omega^2}M}{\bar r^3}\right), ∂ v ξ − λ ˉ ξ = e r ˉ F , F := ( r 3 2 Ω 2 m − 8 π T uv − r ˉ 3 2 Ω 2 M ) , indeed
∂ v ξ = ∂ v ( e r ˉ ( ∂ u log Ω + M r ˉ − 2 ) ) = λ ˉ e r ˉ ( ∂ u log Ω + M r ˉ − 2 ) + e r ˉ ( ∂ v ∂ u log Ω − 2 M λ ˉ r ˉ − 3 ) = λ ˉ ξ + e r ˉ ( 2 Ω 2 m r − 3 − 8 π T u v − 2 Ω 2 ‾ M r ˉ − 3 ) . \begin{aligned}
\partial_v\xi
&=\partial_v\left(e^{\bar r}(\partial_u\log\Omega+M\bar r^{-2})\right) \\
&=\bar\lambda e^{\bar r}(\partial_u\log\Omega+M\bar r^{-2})+e^{\bar r}(\partial_v\partial_u\log\Omega-2M\bar\lambda\bar r^{-3}) \\
&=\bar\lambda\xi+e^{\bar r}(2\Omega^2mr^{-3}-8\pi T_{uv}-2\overline{\Omega^2}M\bar r^{-3}).
\end{aligned} ∂ v ξ = ∂ v ( e r ˉ ( ∂ u log Ω + M r ˉ − 2 ) ) = λ ˉ e r ˉ ( ∂ u log Ω + M r ˉ − 2 ) + e r ˉ ( ∂ v ∂ u log Ω − 2 M λ ˉ r ˉ − 3 ) = λ ˉ ξ + e r ˉ ( 2 Ω 2 m r − 3 − 8 π T uv − 2 Ω 2 M r ˉ − 3 ) . Multiplying by ξ 3 \xi^3 ξ 3 v v v Γ \Gamma Γ
1 4 ξ 4 ( u , ν Λ ( u ) ) − 1 4 ξ 4 ( u , v ) − ∫ v ν Λ ( u ) λ ˉ ξ 4 d v ′ = ∫ v ν Λ ( u ) e r ˉ F ξ 3 d v ′ , \frac{1}{4}\xi^4(u,\nu^\Lambda(u))-\frac{1}{4}\xi^4(u,v)-\int_v^{\nu^\Lambda(u)}\bar\lambda\xi^4\,dv'=\int_v^{\nu^\Lambda(u)}e^{\bar r}F\xi^3\,dv', 4 1 ξ 4 ( u , ν Λ ( u )) − 4 1 ξ 4 ( u , v ) − ∫ v ν Λ ( u ) λ ˉ ξ 4 d v ′ = ∫ v ν Λ ( u ) e r ˉ F ξ 3 d v ′ , hence
1 4 ξ 4 ( u , v ) + ∫ v ν Λ ( u ) λ ˉ ξ 4 d v ′ ≤ 1 4 ξ 4 ( u , ν Λ ) + ∫ v ν Λ ( u ) e r ˉ ∣ F ∣ ∣ ξ ∣ 3 d v ′ ≤ 1 4 ξ 4 ( u , ν Λ ) + ∫ v ν Λ ( u ) ( e 4 r ˉ λ ˉ − 3 ∣ F ∣ 4 + λ ˉ 2 ∣ ξ ∣ 4 ) d v ′ , \frac{1}{4}\xi^4(u,v)+\int_v^{\nu^\Lambda(u)}\bar\lambda\xi^4\,dv'\le\frac{1}{4}\xi^4(u,\nu^\Lambda)+\int_v^{\nu^\Lambda(u)}e^{\bar r}|F||\xi|^3\,dv'\le\frac{1}{4}\xi^4(u,\nu^\Lambda)+\int_v^{\nu^\Lambda(u)}\left(e^{4\bar r}\bar\lambda^{-3}|F|^4+\frac{\bar\lambda}{2}|\xi|^4\right)\,dv', 4 1 ξ 4 ( u , v ) + ∫ v ν Λ ( u ) λ ˉ ξ 4 d v ′ ≤ 4 1 ξ 4 ( u , ν Λ ) + ∫ v ν Λ ( u ) e r ˉ ∣ F ∣∣ ξ ∣ 3 d v ′ ≤ 4 1 ξ 4 ( u , ν Λ ) + ∫ v ν Λ ( u ) ( e 4 r ˉ λ ˉ − 3 ∣ F ∣ 4 + 2 λ ˉ ∣ ξ ∣ 4 ) d v ′ , yielding
∣ ∂ u log Ω + M r ˉ − 2 ∣ 4 ( u , v ) ≲ 1 4 ξ 4 ( u , v ) + ∫ v ν Λ ( u ) λ ˉ 2 ξ 4 d v ′ ≤ 1 4 ξ 4 ( u , ν Λ ( u ) ) + ∫ v ν Λ ( u ) λ ˉ − 3 ∣ F ∣ 4 d v ′ . (99) |\partial_u\log\Omega+M\bar r^{-2}|^4(u,v)\lesssim\frac{1}{4}\xi^4(u,v)+\int_v^{\nu^\Lambda(u)}\frac{\bar\lambda}{2}\xi^4\,dv'\le\frac{1}{4}\xi^4(u,\nu^\Lambda(u))+\int_v^{\nu^\Lambda(u)}\bar\lambda^{-3}|F|^4\,dv'. \tag*{(99)} ∣ ∂ u log Ω + M r ˉ − 2 ∣ 4 ( u , v ) ≲ 4 1 ξ 4 ( u , v ) + ∫ v ν Λ ( u ) 2 λ ˉ ξ 4 d v ′ ≤ 4 1 ξ 4 ( u , ν Λ ( u )) + ∫ v ν Λ ( u ) λ ˉ − 3 ∣ F ∣ 4 d v ′ . (99) where we used the fact that e 4 r ˉ e^{4\bar r} e 4 r ˉ { r ≤ Λ } \{r\le\Lambda\} { r ≤ Λ } ∣ Ω 2 ~ ∣ ≲ ( − ν ˉ ) A 3 ε 3 / 2 τ − 1 + δ |\widetilde{\Omega^2}|\lesssim(-\bar\nu)A^3\varepsilon^{3/2}\tau^{-1+\delta} ∣ Ω 2 ∣ ≲ ( − ν ˉ ) A 3 ε 3/2 τ − 1 + δ (57) ),
∣ F ∣ ≲ ∣ Ω 2 ~ ∣ + Ω 2 ‾ ∣ m ~ ∣ + Ω 2 ‾ ∣ r ~ ∣ + Ω 2 ∣ ψ − ∣ ∣ ψ + ∣ ≲ Ω 2 ‾ A 3 ε 3 / 2 τ − 1 + δ |F|\lesssim|\widetilde{\Omega^2}|+\overline{\Omega^2}|\widetilde m|+\overline{\Omega^2}|\widetilde r|+\Omega^2|\psi_-||\psi_+|\lesssim\overline{\Omega^2}A^3\varepsilon^{3/2}\tau^{-1+\delta} ∣ F ∣ ≲ ∣ Ω 2 ∣ + Ω 2 ∣ m ∣ + Ω 2 ∣ r ∣ + Ω 2 ∣ ψ − ∣∣ ψ + ∣ ≲ Ω 2 A 3 ε 3/2 τ − 1 + δ on { r ≤ Λ } \{r\le\Lambda\} { r ≤ Λ } (99) by
∫ v ν Λ ( u ) λ ˉ − 3 ∣ F ∣ 4 d v ′ ≲ ( A 3 ε 3 / 2 ) 4 ∫ v ν Λ ( u ) Ω 2 ‾ τ − 4 + 4 δ d v ′ ≲ ( A 3 ε 3 / 2 ) 4 τ − 3 + 4 δ . \int_v^{\nu^\Lambda(u)}\bar\lambda^{-3}|F|^4\,dv'\lesssim(A^3\varepsilon^{3/2})^4\int_v^{\nu^\Lambda(u)}\overline{\Omega^2}\tau^{-4+4\delta}\,dv'\lesssim(A^3\varepsilon^{3/2})^4\tau^{-3+4\delta}. ∫ v ν Λ ( u ) λ ˉ − 3 ∣ F ∣ 4 d v ′ ≲ ( A 3 ε 3/2 ) 4 ∫ v ν Λ ( u ) Ω 2 τ − 4 + 4 δ d v ′ ≲ ( A 3 ε 3/2 ) 4 τ − 3 + 4 δ . Therefore, using (98) , we finally obtain
∣ ∂ u log Ω + M r ˉ − 2 ∣ ( u , v ) ≲ A 3 ε 3 / 2 τ − 3 / 4 + δ , |\partial_u\log\Omega+M\bar r^{-2}|(u,v)\lesssim A^3\varepsilon^{3/2}\tau^{-3/4+\delta}, ∣ ∂ u log Ω + M r ˉ − 2 ∣ ( u , v ) ≲ A 3 ε 3/2 τ − 3/4 + δ , concluding the proof.
Remark 5.4. For ∂ u log Ω \partial_u\log\Omega ∂ u log Ω r ≤ Λ r\le\Lambda r ≤ Λ γ M r − 2 \gamma Mr^{-2} γ M r − 2 ∂ v γ ∼ ( 1 − μ ) − 1 \partial_v\gamma\sim(1-\mu)^{-1} ∂ v γ ∼ ( 1 − μ ) − 1 ξ \xi ξ λ ˉ ξ \bar\lambda\xi λ ˉ ξ v v v (99) , allowing us to absorb part of the source term by Young’s inequality, in the spirit of [21 Lemma 15.1.1] . This argument allows us to recover a sufficient decay in τ \tau τ ∂ v γ \partial_v\gamma ∂ v γ
In order to bound the derivatives of the Dirac field and thus the components T u u T_{uu} T uu T v v T_{vv} T v v D \mathcal{D} D (25) . The proof follows [4 Lemma 7.3] , with the difference that our seed data norm controls only one spinor component on each initial null hypersurface. We exploit the transport structure of the Dirac equations (20) –(21) to recover a control of the missing components and thereby bound the energies at the initial time τ = 1 \tau= 1 τ = 1
Lemma 5.5. For any A ≥ 1 A \ge1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_f \in\mathcal{B} τ f ∈ B
E ( 1 ) + E p ( 1 ) ≲ ( D [ S 0 ] ) 2 . (100) \mathcal{E}(1) + \mathcal{E}_p(1) \lesssim\bigl(\mathcal{D}[S_0]\bigr)^2 . \tag*{(100)} E ( 1 ) + E p ( 1 ) ≲ ( D [ S 0 ] ) 2 . (100) Proof. Let us start with C ^ i n \widehat{C}_{\mathrm{in}} C in ν ∼ ν ˉ \nu\sim\bar{\nu} ν ∼ ν ˉ D [ S 0 ] \mathcal{D}[S_0] D [ S 0 ] (25) , we have on C ^ i n \widehat{C}_{\mathrm{in}} C in
∣ ψ − ∣ + ∣ ∂ u ψ − ∣ − ν ˉ ≲ D [ S 0 ] . (101) |\psi_-| + \frac{|\partial_u\psi_-|}{-\bar{\nu}} \lesssim\mathcal{D}[S_0]. \tag*{(101)} ∣ ψ − ∣ + − ν ˉ ∣ ∂ u ψ − ∣ ≲ D [ S 0 ] . (101) Thus, it follows
∫ C ^ i n ( − ν ˉ ∣ ψ − ∣ 2 + ∣ ∂ u ψ − ∣ 2 − ν ˉ ) d u ≲ ( D [ S 0 ] ) 2 ∫ C ^ i n ( − ν ˉ ) d u ≲ ( D [ S 0 ] ) 2 ∫ 2 M Λ d r ˉ ≲ ( D [ S 0 ] ) 2 . \int_{\widehat{C}_{\mathrm{in}}} \left( -\bar{\nu}|\psi_-|^2 + \frac{|\partial_u\psi_-|^2}{-\bar{\nu}} \right)\,du \lesssim\bigl(\mathcal{D}[S_0]\bigr)^2 \int_{\widehat{C}_{\mathrm{in}}}(-\bar{\nu})\,du \lesssim\bigl(\mathcal{D}[S_0]\bigr)^2 \int_{2M}^{\Lambda}d\bar{r} \lesssim\bigl(\mathcal{D}[S_0]\bigr)^2. ∫ C in ( − ν ˉ ∣ ψ − ∣ 2 + − ν ˉ ∣ ∂ u ψ − ∣ 2 ) d u ≲ ( D [ S 0 ] ) 2 ∫ C in ( − ν ˉ ) d u ≲ ( D [ S 0 ] ) 2 ∫ 2 M Λ d r ˉ ≲ ( D [ S 0 ] ) 2 . Furthermore, recalling that κ ∼ 1 \kappa\sim1 κ ∼ 1 (20) that
∣ ψ + ∣ ( u , 0 ) ≲ ∣ Φ + ∣ ( u , 0 ) ≲ ∣ Φ + ∣ ( 0 , 0 ) + ∫ 0 u ∣ Ω r Φ − ∣ d u ′ ≲ ∣ ψ + ∣ ( 0 , 0 ) + ∫ C ^ i n ∣ ψ − ∣ ( − ν ˉ ) d u ′ ≲ D [ S 0 ] , |\psi_+|(u,0) \lesssim|\Phi_+|(u,0) \lesssim|\Phi_+|(0,0) + \int_0^u \left|\frac{\Omega}{r}\Phi_-\right|\,du' \lesssim|\psi_+|(0,0) + \int_{\widehat{C}_{\mathrm{in}}}|\psi_-|(-\bar{\nu})\,du' \lesssim\mathcal{D}[S_0], ∣ ψ + ∣ ( u , 0 ) ≲ ∣ Φ + ∣ ( u , 0 ) ≲ ∣ Φ + ∣ ( 0 , 0 ) + ∫ 0 u r Ω Φ − d u ′ ≲ ∣ ψ + ∣ ( 0 , 0 ) + ∫ C in ∣ ψ − ∣ ( − ν ˉ ) d u ′ ≲ D [ S 0 ] , and consequently
∫ C ^ i n − ν ˉ ∣ ψ + ∣ 2 d u ≲ ( D [ S 0 ] ) 2 ∫ C ^ i n ( − ν ˉ ) d u ≲ ( D [ S 0 ] ) 2 ∫ 2 M Λ d r ˉ ≲ ( D [ S 0 ] ) 2 , \int_{\widehat{C}_{\mathrm{in}}}-\bar{\nu}|\psi_+|^2\,du \lesssim\bigl(\mathcal{D}[S_0]\bigr)^2\int_{\widehat{C}_{\mathrm{in}}}(-\bar{\nu})\,du \lesssim\bigl(\mathcal{D}[S_0]\bigr)^2\int_{2M}^{\Lambda}d\bar{r} \lesssim\bigl(\mathcal{D}[S_0]\bigr)^2, ∫ C in − ν ˉ ∣ ψ + ∣ 2 d u ≲ ( D [ S 0 ] ) 2 ∫ C in ( − ν ˉ ) d u ≲ ( D [ S 0 ] ) 2 ∫ 2 M Λ d r ˉ ≲ ( D [ S 0 ] ) 2 , proving E ( 1 ) ≲ ( D [ S 0 ] ) 2 \mathcal{E}(1) \lesssim\bigl(\mathcal{D}[S_0]\bigr)^2 E ( 1 ) ≲ ( D [ S 0 ] ) 2 C ^ o u t \widehat{C}_{\mathrm{out}} C out λ ∼ 1 \lambda\sim1 λ ∼ 1
r ∣ ψ + ∣ + r 2 ∣ ∂ v ψ + ∣ + r 2 ∣ ∂ v ( r ψ + ) ∣ ≲ D [ S 0 ] (102) r|\psi_+| + r^2|\partial_v\psi_+| + r^2|\partial_v(r\psi_+)| \lesssim\mathcal{D}[S_0] \tag*{(102)} r ∣ ψ + ∣ + r 2 ∣ ∂ v ψ + ∣ + r 2 ∣ ∂ v ( r ψ + ) ∣ ≲ D [ S 0 ] (102) on C ^ o u t \widehat{C}_{\mathrm{out}} C out
∫ C ^ o u t ( r T f p ∣ ∂ v ( r ψ + ) ∣ 2 + r T f p ∗ + 2 ∣ ∂ v ψ + ∣ 2 + r T f p ∗ ∣ ψ + ∣ 2 ) d v ≲ ( D [ S 0 ] ) 2 \int_{\widehat{C}_{\mathrm{out}}}\left(r_{T_f}^{p}|\partial_v(r\psi_+)|^2 + r_{T_f}^{p_*+2}|\partial_v\psi_+|^2 + r_{T_f}^{p_*}|\psi_+|^2\right)\,dv \lesssim\bigl(\mathcal{D}[S_0]\bigr)^2 ∫ C out ( r T f p ∣ ∂ v ( r ψ + ) ∣ 2 + r T f p ∗ + 2 ∣ ∂ v ψ + ∣ 2 + r T f p ∗ ∣ ψ + ∣ 2 ) d v ≲ ( D [ S 0 ] ) 2 as p ∈ [ 0 , 3 − δ ] p \in[0,3-\delta] p ∈ [ 0 , 3 − δ ] − ν ∼ 1 -\nu\sim1 − ν ∼ 1 C ^ o u t \widehat{C}_{\mathrm{out}} C out (21) that
r ∣ ψ − ∣ ( 0 , v ) ≲ ∣ Φ − ( 0 , v ) ∣ ≲ ∣ Φ − ( 0 , 0 ) ∣ + ∫ 0 v ∣ Ω r Φ + ∣ d v ′ ≲ ∣ ψ − ( 0 , 0 ) ∣ + ∫ C ^ o u t ( r ∣ ψ + ∣ ) 1 r 2 d v ′ ≲ D [ S 0 ] , r|\psi_-|(0,v) \lesssim|\Phi_-(0,v)| \lesssim|\Phi_-(0,0)| + \int_0^v \left|\frac{\Omega}{r}\Phi_+\right|\,dv' \lesssim|\psi_-(0,0)| + \int_{\widehat{C}_{\mathrm{out}}}(r|\psi_+|)\frac{1}{r^2}\,dv' \lesssim\mathcal{D}[S_0], r ∣ ψ − ∣ ( 0 , v ) ≲ ∣ Φ − ( 0 , v ) ∣ ≲ ∣ Φ − ( 0 , 0 ) ∣ + ∫ 0 v r Ω Φ + d v ′ ≲ ∣ ψ − ( 0 , 0 ) ∣ + ∫ C out ( r ∣ ψ + ∣ ) r 2 1 d v ′ ≲ D [ S 0 ] , allowing us to bound the remaining term in E p ( 1 ) \mathcal{E}_p(1) E p ( 1 )
We are now ready to establish bounds on the derivatives of the Dirac components.
Lemma 5.6. For any A ≥ 1 A \ge1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_f \in\mathcal{B} τ f ∈ B 1 ≤ τ 1 ≤ τ 2 ≤ τ f 1 \le\tau_1 \le\tau_2 \le\tau_f 1 ≤ τ 1 ≤ τ 2 ≤ τ f
r ˉ 2 ∣ ∂ v ψ + ∣ ≲ A 1 / 2 ε , (103) \bar{r}^{2}|\partial_v\psi_+| \lesssim A^{1/2}\varepsilon, \tag*{(103)} r ˉ 2 ∣ ∂ v ψ + ∣ ≲ A 1/2 ε , (103) r ˉ 2 ∣ ∂ v ( r ψ + ) ∣ ≲ A 1 / 2 ε , (104) \bar{r}^{2}|\partial_v(r\psi_+)| \lesssim A^{1/2}\varepsilon, \tag*{(104)} r ˉ 2 ∣ ∂ v ( r ψ + ) ∣ ≲ A 1/2 ε , (104) r ˉ − ν ∣ ∂ u ψ − ∣ ≲ A 1 / 2 ε , (105) \frac{\bar{r}}{-\nu}|\partial_u\psi_-| \lesssim A^{1/2}\varepsilon, \tag*{(105)} − ν r ˉ ∣ ∂ u ψ − ∣ ≲ A 1/2 ε , (105) r ˉ 2 ∣ T u u ∣ ν 2 ≲ A ε 2 τ − 3 / 2 + δ / 2 , (106) \bar{r}^{2}\frac{|T_{uu}|}{\nu^2} \lesssim A\varepsilon^2\tau^{-3/2+\delta/2}, \tag*{(106)} r ˉ 2 ν 2 ∣ T uu ∣ ≲ A ε 2 τ − 3/2 + δ /2 , (106) r ˉ 7 / 2 ∣ T v v ∣ ≲ A ε 2 τ − 3 / 2 + δ / 2 , (107) \bar{r}^{7/2}|T_{vv}| \lesssim A\varepsilon^2\tau^{-3/2+\delta/2}, \tag*{(107)} r ˉ 7/2 ∣ T v v ∣ ≲ A ε 2 τ − 3/2 + δ /2 , (107) on D τ f D_{\tau_f} D τ f
Proof. Proof of (103) : We consider again the two regions { r ≤ Λ } \{r \le\Lambda\} { r ≤ Λ } { r ≥ Λ } \{r \ge\Lambda\} { r ≥ Λ } Φ ~ + : = r ( − ν ) − 1 / 2 ψ + \widetilde{\Phi}_{+} := r(-\nu)^{-1/2}\psi_{+} Φ + := r ( − ν ) − 1/2 ψ +
∂ u ∂ v Φ ~ + + ( ∂ u log Ω − ν r ) ∂ v Φ ~ + + ( 6 Ω 2 m r 3 − 12 π T u v ) Φ ~ + = 0. \partial_{u}\partial_{v}\widetilde{\Phi}_{+}+\left(\partial_{u}\log\Omega-\frac{\nu}{r}\right)\partial_{v}\widetilde{\Phi}_{+}+\left(\frac{6\Omega^{2}m}{r^{3}}-12\pi T_{uv}\right)\widetilde{\Phi}_{+}=0. ∂ u ∂ v Φ + + ( ∂ u log Ω − r ν ) ∂ v Φ + + ( r 3 6 Ω 2 m − 12 π T uv ) Φ + = 0. Hence, from
∂ u ( r 2 ∂ v Φ ~ + ) = ( 3 ν r − ∂ u log Ω ) ( r 2 ∂ v Φ ~ + ) − r 2 ( 6 Ω 2 m r 3 − 12 π T u v ) Φ ~ + , \partial_{u}\left(r^{2}\partial_{v}\widetilde{\Phi}_{+}\right)=\left(\frac{3\nu}{r}-\partial_{u}\log\Omega\right)\left(r^{2}\partial_{v}\widetilde{\Phi}_{+}\right)-r^{2}\left(\frac{6\Omega^{2}m}{r^{3}}-12\pi T_{uv}\right)\widetilde{\Phi}_{+}, ∂ u ( r 2 ∂ v Φ + ) = ( r 3 ν − ∂ u log Ω ) ( r 2 ∂ v Φ + ) − r 2 ( r 3 6 Ω 2 m − 12 π T uv ) Φ + , we compute for ( u , v ) ∈ D τ f ∩ { r ≥ Λ } (u,v)\in D_{\tau_f}\cap\{r\ge\Lambda\} ( u , v ) ∈ D τ f ∩ { r ≥ Λ }
( r 2 ∂ v Φ ~ + ) ( u , v ) = exp ( ∫ 0 u ( 3 ν r − ∂ u log Ω ) d u ′ ) ( r 2 ∂ v Φ ~ + ) ( 0 , v ) − ∫ 0 u exp ( ∫ u ′ u ( 3 ν r − ∂ u log Ω ) d u ′ ′ ) r 2 ( 6 Ω 2 m r 3 − 12 π T u v ) Φ ~ + d u ′ . (108) \left(r^{2}\partial_{v}\widetilde{\Phi}_{+}\right)(u,v)=\exp\left(\int_{0}^{u}\left(\frac{3\nu}{r}-\partial_{u}\log\Omega\right)\,du'\right)\left(r^{2}\partial_{v}\widetilde{\Phi}_{+}\right)(0,v)
-\int_{0}^{u}\exp\left(\int_{u'}^{u}\left(\frac{3\nu}{r}-\partial_{u}\log\Omega\right)\,du''\right)r^{2}\left(\frac{6\Omega^{2}m}{r^{3}}-12\pi T_{uv}\right)\widetilde{\Phi}_{+}\,du'. \tag*{(108)} ( r 2 ∂ v Φ + ) ( u , v ) = exp ( ∫ 0 u ( r 3 ν − ∂ u log Ω ) d u ′ ) ( r 2 ∂ v Φ + ) ( 0 , v ) − ∫ 0 u exp ( ∫ u ′ u ( r 3 ν − ∂ u log Ω ) d u ′′ ) r 2 ( r 3 6 Ω 2 m − 12 π T uv ) Φ + d u ′ . (108) Using (52) and (94) , it holds
exp ( ∫ u 2 u 1 ( 3 ν r − ∂ u log Ω ) d u ′ ) ≲ exp ( ∫ u 2 u 1 r − 2 ( M + A 3 ε 3 / 2 τ − 3 / 4 + δ ) d u ′ ) ≲ 1 , \exp\left(\int_{u_{2}}^{u_{1}}\left(\frac{3\nu}{r}-\partial_{u}\log\Omega\right)\,du'\right)\lesssim\exp\left(\int_{u_{2}}^{u_{1}}r^{-2}\left(M+A^{3}\varepsilon^{3/2}\tau^{-3/4+\delta}\right)\,du'\right)\lesssim1, exp ( ∫ u 2 u 1 ( r 3 ν − ∂ u log Ω ) d u ′ ) ≲ exp ( ∫ u 2 u 1 r − 2 ( M + A 3 ε 3/2 τ − 3/4 + δ ) d u ′ ) ≲ 1 , on D τ f ∩ { r ≥ Λ } D_{\tau_f}\cap\{r\ge\Lambda\} D τ f ∩ { r ≥ Λ } (70) and (88) , we bound
∫ 0 u r 2 ( 6 Ω 2 m r 3 − 12 π T u v ) Φ ~ + d u ′ ≲ ∫ 0 u ∣ ψ + ∣ d u ′ ≲ A 1 / 2 ε ∫ 0 u τ − 3 / 2 + δ / 2 d u ′ ≲ A 1 / 2 ε . \int_{0}^{u}r^{2}\left(\frac{6\Omega^{2}m}{r^{3}}-12\pi T_{uv}\right)\widetilde{\Phi}_{+}\,du'\lesssim\int_{0}^{u}|\psi_{+}|\,du'\lesssim A^{1/2}\varepsilon\int_{0}^{u}\tau^{-3/2+\delta/2}\,du'\lesssim A^{1/2}\varepsilon. ∫ 0 u r 2 ( r 3 6 Ω 2 m − 12 π T uv ) Φ + d u ′ ≲ ∫ 0 u ∣ ψ + ∣ d u ′ ≲ A 1/2 ε ∫ 0 u τ − 3/2 + δ /2 d u ′ ≲ A 1/2 ε . Therefore, (108) , together with
r 2 ∣ ∂ v Φ ~ + ∣ ( 0 , v ) ≲ r 2 ( ∣ ∂ v ( r ψ + ) ∣ + r ∣ ψ + ∣ ∣ ∂ v ν ∣ ) ( 0 , v ) ≲ ( r 2 ∣ ∂ v ( r ψ + ) ∣ + r ∣ ψ + ∣ ) ≲ D [ S 0 ] ≲ ε , r^{2}|\partial_{v}\widetilde{\Phi}_{+}|(0,v)\lesssim r^{2}\left(|\partial_{v}(r\psi_{+})|+r|\psi_{+}||\partial_{v}\nu|\right)(0,v)\lesssim\left(r^{2}|\partial_{v}(r\psi_{+})|+r|\psi_{+}|\right)\lesssim\mathcal{D}[S_{0}]\lesssim\varepsilon, r 2 ∣ ∂ v Φ + ∣ ( 0 , v ) ≲ r 2 ( ∣ ∂ v ( r ψ + ) ∣ + r ∣ ψ + ∣∣ ∂ v ν ∣ ) ( 0 , v ) ≲ ( r 2 ∣ ∂ v ( r ψ + ) ∣ + r ∣ ψ + ∣ ) ≲ D [ S 0 ] ≲ ε , which controls the boundary term, yields
∣ r 2 ∂ v Φ ~ + ∣ ≲ A 1 / 2 ε , |r^{2}\partial_{v}\widetilde{\Phi}_{+}|\lesssim A^{1/2}\varepsilon, ∣ r 2 ∂ v Φ + ∣ ≲ A 1/2 ε , and therefore
r 2 ∣ ∂ v ψ + ∣ ≲ r ∣ ψ + ∣ + r ∣ ∂ v Φ ~ + ∣ ≲ A 1 / 2 ε , r 2 ∣ ∂ v ( r ψ + ) ∣ ≲ r ∣ ψ + ∣ + r 2 ∣ ∂ v Φ ~ + ∣ ≲ A 1 / 2 ε , r^{2}|\partial_{v}\psi_{+}|\lesssim r|\psi_{+}|+r|\partial_{v}\widetilde{\Phi}_{+}|\lesssim A^{1/2}\varepsilon,\qquad r^{2}|\partial_{v}(r\psi_{+})|\lesssim r|\psi_{+}|+r^{2}|\partial_{v}\widetilde{\Phi}_{+}|\lesssim A^{1/2}\varepsilon, r 2 ∣ ∂ v ψ + ∣ ≲ r ∣ ψ + ∣ + r ∣ ∂ v Φ + ∣ ≲ A 1/2 ε , r 2 ∣ ∂ v ( r ψ + ) ∣ ≲ r ∣ ψ + ∣ + r 2 ∣ ∂ v Φ + ∣ ≲ A 1/2 ε , on D τ f ∩ { r ≥ Λ } D_{\tau_f}\cap\{r\ge\Lambda\} D τ f ∩ { r ≥ Λ } { r ≤ Λ } \{r\le\Lambda\} { r ≤ Λ } Φ + \Phi_{+} Φ +
∂ u ∂ v Φ + = ν κ 1 / 2 ( ( ∂ v log Ω − λ r − 1 ) ψ − + κ r − 2 ψ + ) . \partial_{u}\partial_{v}\Phi_{+}=\nu\kappa^{1/2}\left((\partial_{v}\log\Omega-\lambda r^{-1})\psi_{-}+\kappa r^{-2}\psi_{+}\right). ∂ u ∂ v Φ + = ν κ 1/2 ( ( ∂ v log Ω − λ r − 1 ) ψ − + κ r − 2 ψ + ) . Therefore, for ( u , v ) ∈ D τ f ∩ { r ≤ Λ } (u,v)\in D_{\tau_f}\cap\{r\le\Lambda\} ( u , v ) ∈ D τ f ∩ { r ≤ Λ } (88) and (90) ,
∣ ∂ v Φ + ∣ ( u , v ) ≲ ∣ ∂ v Φ + ∣ ( u Λ ( v ) , v ) + ∫ u Λ ( v ) u ν κ 1 / 2 ( ( ∂ v log Ω − λ r − 1 ) ψ − + κ r − 2 ψ + ) d u ′ |\partial_{v}\Phi_{+}|(u,v)\lesssim|\partial_{v}\Phi_{+}|(u^{\Lambda}(v),v)+\int_{u^{\Lambda}(v)}^{u}\nu\kappa^{1/2}\left((\partial_{v}\log\Omega-\lambda r^{-1})\psi_{-}+\kappa r^{-2}\psi_{+}\right)\,du' ∣ ∂ v Φ + ∣ ( u , v ) ≲ ∣ ∂ v Φ + ∣ ( u Λ ( v ) , v ) + ∫ u Λ ( v ) u ν κ 1/2 ( ( ∂ v log Ω − λ r − 1 ) ψ − + κ r − 2 ψ + ) d u ′ ≲ A 1 / 2 ε + A 1 / 2 ε ∫ u Λ ( v ) u ( − ν ) d u ′ ≲ A 1 / 2 ε + A 1 / 2 ε ∫ 2 M Λ d r ˉ ≲ A 1 / 2 ε . (109) \lesssim A^{1/2}\varepsilon+A^{1/2}\varepsilon\int_{u^{\Lambda}(v)}^{u}(-\nu)\,du'\lesssim A^{1/2}\varepsilon+A^{1/2}\varepsilon\int_{2M}^{\Lambda}d\bar r\lesssim A^{1/2}\varepsilon. \tag*{(109)} ≲ A 1/2 ε + A 1/2 ε ∫ u Λ ( v ) u ( − ν ) d u ′ ≲ A 1/2 ε + A 1/2 ε ∫ 2 M Λ d r ˉ ≲ A 1/2 ε . (109) where we controlled the boundary term by noting that
∣ ∂ v Φ + ∣ ( u Λ ( v ) , v ) ≲ ( ( ∂ v log Ω + r − 1 ) ∣ ψ + ∣ + r − 1 ∣ ∂ v Φ ~ + ∣ ) ( u Λ ( v ) , v ) ≲ ( ∣ ψ + ∣ + ∣ ∂ v Φ ~ + ∣ ) ( u Λ ( v ) , v ) ≲ A 1 / 2 ε , |\partial_{v}\Phi_{+}|(u^{\Lambda}(v),v)\lesssim\left((\partial_{v}\log\Omega+r^{-1})|\psi_{+}|+r^{-1}|\partial_{v}\widetilde{\Phi}_{+}|\right)(u^{\Lambda}(v),v)\lesssim\left(|\psi_{+}|+|\partial_{v}\widetilde{\Phi}_{+}|\right)(u^{\Lambda}(v),v)\lesssim A^{1/2}\varepsilon, ∣ ∂ v Φ + ∣ ( u Λ ( v ) , v ) ≲ ( ( ∂ v log Ω + r − 1 ) ∣ ψ + ∣ + r − 1 ∣ ∂ v Φ + ∣ ) ( u Λ ( v ) , v ) ≲ ( ∣ ψ + ∣ + ∣ ∂ v Φ + ∣ ) ( u Λ ( v ) , v ) ≲ A 1/2 ε , by (57) , (75) and (92) . Hence, we conclude on D τ f ∩ { r ≤ Λ } D_{\tau_f}\cap\{r\le\Lambda\} D τ f ∩ { r ≤ Λ }
∣ ∂ v ψ + ∣ ≲ ∣ ∂ v Φ + ∣ + ∣ ψ + ∣ ∣ ∂ v κ ∣ ≲ A 1 / 2 ε , |\partial_{v}\psi_{+}|\lesssim|\partial_{v}\Phi_{+}|+|\psi_{+}||\partial_{v}\kappa|\lesssim A^{1/2}\varepsilon, ∣ ∂ v ψ + ∣ ≲ ∣ ∂ v Φ + ∣ + ∣ ψ + ∣∣ ∂ v κ ∣ ≲ A 1/2 ε , where we used (88) , (109) and the following fact:
∣ ∂ v κ ∣ = ∣ 2 κ ( ∂ v log Ω − κ m r − 2 + 2 π r T u v − ν ) ∣ ≲ ∣ ∂ v log Ω − M r ‾ − 2 ∣ + ∣ κ ~ ∣ r ‾ − 2 + ∣ m ~ ∣ r ‾ − 2 + ∣ r ~ ∣ r ‾ − 3 + r ∣ T u v − ν ∣ ≲ A 3 ε 3 / 2 , |\partial_v\kappa|=\left|2\kappa\left(\partial_v\log\Omega-\kappa mr^{-2}+2\pi r\frac{T_{uv}}{-\nu}\right)\right|\lesssim|\partial_v\log\Omega-M\overline{r}^{-2}|+|\widetilde{\kappa}| \overline{r}^{-2}+|\widetilde{m}|\overline{r}^{-2}+|\widetilde{r}|\overline{r}^{-3}+r\left|\frac{T_{uv}}{-\nu}\right|
\lesssim A^{3}\varepsilon^{3/2}, ∣ ∂ v κ ∣ = 2 κ ( ∂ v log Ω − κm r − 2 + 2 π r − ν T uv ) ≲ ∣ ∂ v log Ω − M r − 2 ∣ + ∣ κ ∣ r − 2 + ∣ m ∣ r − 2 + ∣ r ∣ r − 3 + r − ν T uv ≲ A 3 ε 3/2 , again by the geometric estimates, (91) and (92) . Moreover, we immediately obtain
∣ ∂ v ( r ψ + ) ∣ ≲ λ ∣ ψ + ∣ + r ∣ ∂ v ψ + ∣ ≲ ∣ ψ + ∣ + ∣ ∂ v ψ + ∣ ≲ A 1 / 2 ε . |\partial_v(r\psi_+)|\lesssim\lambda|\psi_+|+r|\partial_v\psi_+|\lesssim|\psi_+|+|\partial_v\psi_+|\lesssim A^{1/2}\varepsilon. ∣ ∂ v ( r ψ + ) ∣ ≲ λ ∣ ψ + ∣ + r ∣ ∂ v ψ + ∣ ≲ ∣ ψ + ∣ + ∣ ∂ v ψ + ∣ ≲ A 1/2 ε . Proof of (105) : Let us consider Φ ~ − : = r Ω Φ − = r 2 κ − 1 / 2 ψ − \widetilde{\Phi}_-:=\frac{r}{\Omega}\Phi_-=r^2\kappa^{-1/2}\psi_- Φ − := Ω r Φ − = r 2 κ − 1/2 ψ −
∂ u ∂ v Φ ~ − = − ( ∂ v log Ω − λ r ) ∂ u Φ ~ − − ( 6 Ω 2 m r 3 − 12 π T u v ) Φ ~ − . \partial_u\partial_v\widetilde{\Phi}_-=-\left(\partial_v\log\Omega-\frac{\lambda}{r}\right)\partial_u\widetilde{\Phi}_--\left(\frac{6\Omega^2m}{r^3}-12\pi T_{uv}\right)\widetilde{\Phi}_-. ∂ u ∂ v Φ − = − ( ∂ v log Ω − r λ ) ∂ u Φ − − ( r 3 6 Ω 2 m − 12 π T uv ) Φ − . Thus, we compute
∂ v ( 1 − ν r ∂ u Φ ~ − ) = ( 1 − ν r ∂ u Φ ~ − ) ( ∂ v ν − ν − ∂ v log Ω ) + 1 ν r ( 6 Ω 2 m r 3 − 12 π T u v ) Φ ~ − , \partial_v\left(\frac{1}{-\nu r}\partial_u\widetilde{\Phi}_-\right)=\left(\frac{1}{-\nu r}\partial_u\widetilde{\Phi}_-\right)\left(\frac{\partial_v\nu}{-\nu}-\partial_v\log\Omega\right)+\frac{1}{\nu r}\left(\frac{6\Omega^2m}{r^3}-12\pi T_{uv}\right)\widetilde{\Phi}_-, ∂ v ( − ν r 1 ∂ u Φ − ) = ( − ν r 1 ∂ u Φ − ) ( − ν ∂ v ν − ∂ v log Ω ) + ν r 1 ( r 3 6 Ω 2 m − 12 π T uv ) Φ − , which can be solved for
( ∂ u Φ ~ − − ν r ) ( u , v ) = exp ( ∫ 0 v ( ∂ v ν − ν − ∂ v log Ω ) d v ′ ) ( ∂ u Φ ~ − − ν r ) ( u , 0 ) + ∫ 0 v exp ( ∫ v ′ v ( ∂ v ν − ν − ∂ v log Ω ) d v ′ ′ ) 1 ν r ( 6 Ω 2 m r 3 − 12 π T u v ) Φ ~ − d v ′ . (110) \begin{aligned}
\left(\frac{\partial_u\widetilde{\Phi}_-}{-\nu r}\right)(u,v)=\exp\left(\int_0^v\left(\frac{\partial_v\nu}{-\nu}-\partial_v\log\Omega\right)dv'\right)\left(\frac{\partial_u\widetilde{\Phi}_-}{-\nu r}\right)(u,0) \\
&+\int_0^v\exp\left(\int_{v'}^v\left(\frac{\partial_v\nu}{-\nu}-\partial_v\log\Omega\right)dv''\right)\frac{1}{\nu r}\left(\frac{6\Omega^2m}{r^3}-12\pi T_{uv}\right)\widetilde{\Phi}_-\,dv'. \tag*{(110)}
\end{aligned} ( − ν r ∂ u Φ − ) ( u , v ) = exp ( ∫ 0 v ( − ν ∂ v ν − ∂ v log Ω ) d v ′ ) ( − ν r ∂ u Φ − ) ( u , 0 ) + ∫ 0 v exp ( ∫ v ′ v ( − ν ∂ v ν − ∂ v log Ω ) d v ′′ ) ν r 1 ( r 3 6 Ω 2 m − 12 π T uv ) Φ − d v ′ . (110) Let us start by bounding the integrating factor. If ( u , v 1 ) , ( u , v 2 ) ∈ D τ f ∩ { r ≤ Λ } (u,v_1),(u,v_2)\in D_{\tau_f}\cap\{r\leq\Lambda\} ( u , v 1 ) , ( u , v 2 ) ∈ D τ f ∩ { r ≤ Λ } (91) and (92) ,
exp ( ∫ v 1 v 2 ( ∂ v ν − ν − ∂ v log Ω ) d v ′ ) = exp ( ∫ v 1 v 2 ( − 2 κ m r 2 − ∂ v log Ω + 4 π r T u v − ν ) d v ′ ) \exp\left(\int_{v_1}^{v_2}\left(\frac{\partial_v\nu}{-\nu}-\partial_v\log\Omega\right)dv'\right)=\exp\left(\int_{v_1}^{v_2}\left(-\frac{2\kappa m}{r^2}-\partial_v\log\Omega+4\pi r\frac{T_{uv}}{-\nu}\right)dv'\right) exp ( ∫ v 1 v 2 ( − ν ∂ v ν − ∂ v log Ω ) d v ′ ) = exp ( ∫ v 1 v 2 ( − r 2 2 κm − ∂ v log Ω + 4 π r − ν T uv ) d v ′ ) ≲ exp ( ∫ v 1 v 2 ( − 3 M r 2 + A 2 ε 3 / 2 τ − 2 + δ + A ε 2 r ‾ − 5 / 2 τ − 3 + δ ) d v ′ ) ≲ 1 + A 2 ε 3 / 2 v 1 − 1 + δ ≲ 1 ; \lesssim\exp\left(\int_{v_1}^{v_2}\left(-\frac{3M}{r^2}+A^2\varepsilon^{3/2}\tau^{-2+\delta}+A\varepsilon^2\overline{r}^{-5/2}\tau^{-3+\delta}\right)dv'\right)\lesssim1+A^2\varepsilon^{3/2}v_1^{-1+\delta}\lesssim1; ≲ exp ( ∫ v 1 v 2 ( − r 2 3 M + A 2 ε 3/2 τ − 2 + δ + A ε 2 r − 5/2 τ − 3 + δ ) d v ′ ) ≲ 1 + A 2 ε 3/2 v 1 − 1 + δ ≲ 1 ; on the other hand, when ( u , v 1 ) , ( u , v 2 ) ∈ D τ f ∩ { r ≥ Λ } (u,v_1),(u,v_2)\in D_{\tau_f}\cap\{r\geq\Lambda\} ( u , v 1 ) , ( u , v 2 ) ∈ D τ f ∩ { r ≥ Λ }
exp ( ∫ v 1 v 2 ( ∂ v ν − ν − ∂ v log Ω ) d v ′ ) = exp ( ∫ v 1 v 2 − ∂ v log ( ( − ν ) Ω ) d v ′ ) = ( − ν ) Ω ( u , v 1 ) ( − ν ) Ω ( u , v 2 ) ≲ 1 , \exp\left(\int_{v_1}^{v_2}\left(\frac{\partial_v\nu}{-\nu}-\partial_v\log\Omega\right)dv'\right)=\exp\left(\int_{v_1}^{v_2}-\partial_v\log\left((-\nu)\Omega\right)dv'\right)=\frac{(-\nu)\Omega(u,v_1)}{(-\nu)\Omega(u,v_2)}\lesssim1, exp ( ∫ v 1 v 2 ( − ν ∂ v ν − ∂ v log Ω ) d v ′ ) = exp ( ∫ v 1 v 2 − ∂ v log ( ( − ν ) Ω ) d v ′ ) = ( − ν ) Ω ( u , v 2 ) ( − ν ) Ω ( u , v 1 ) ≲ 1 , by (52) and (57) . The case when ( u , v 1 ) , ( u , v 2 ) (u,v_1),(u,v_2) ( u , v 1 ) , ( u , v 2 ) Γ \Gamma Γ v Γ ( u ) v^\Gamma(u) v Γ ( u )
∫ 0 v 1 r ( 6 κ m r 3 + 12 π ∣ T u v − ν ∣ ) ∣ Φ ~ − ∣ d v ′ ≲ ∫ 0 v ( m r 3 + ∣ T u v − ν ∣ ) ∣ r ψ − ∣ d v ′ \int_0^v\frac{1}{r}\left(\frac{6\kappa m}{r^3}+12\pi\left|\frac{T_{uv}}{-\nu}\right|\right)|\widetilde{\Phi}_-|\,dv'\lesssim\int_0^v\left(\frac{m}{r^3}+\left|\frac{T_{uv}}{-\nu}\right|\right)|r\psi_-|\,dv' ∫ 0 v r 1 ( r 3 6 κm + 12 π − ν T uv ) ∣ Φ − ∣ d v ′ ≲ ∫ 0 v ( r 3 m + − ν T uv ) ∣ r ψ − ∣ d v ′ ≲ A 1 / 2 ε ∫ 0 v ( r − 3 τ − 3 / 2 + δ / 2 + A 2 ε 2 r ‾ − 7 / 2 τ − 9 / 2 + 3 δ / 2 ) d v ′ ≲ A 1 / 2 ε , \lesssim A^{1/2}\varepsilon\int_0^v\left(r^{-3}\tau^{-3/2+\delta/2}+A^2\varepsilon^2\overline{r}^{-7/2}\tau^{-9/2+3\delta/2}\right)dv'\lesssim A^{1/2}\varepsilon, ≲ A 1/2 ε ∫ 0 v ( r − 3 τ − 3/2 + δ /2 + A 2 ε 2 r − 7/2 τ − 9/2 + 3 δ /2 ) d v ′ ≲ A 1/2 ε , using (49) , (90) and (91) . Therefore, from (110) and the fact that
∣ ∂ u Φ ~ − − ν r ∣ ( u , 0 ) ≲ ( ∣ ψ − ∣ ( 1 + ∣ ∂ u κ − ν ∣ + ∣ ∂ u ψ − − ν ∣ ) ) ( u , 0 ) ≲ ( ∣ ψ − ∣ ( 1 + ∣ ψ − ∣ ∣ ∂ u ψ − − ν ∣ + ∣ ∂ u ψ − − ν ∣ ) ) ( u , 0 ) ≲ ε \left|\frac{\partial_u\widetilde{\Phi}_-}{-\nu r}\right|(u,0)\lesssim\left(|\psi_-|\left(1+\left|\frac{\partial_u\kappa}{-\nu}\right|+\left|\frac{\partial_u\psi_-}{-\nu}\right|\right)\right)(u,0)\lesssim\left(|\psi_-|\left(1+|\psi_-|\left|\frac{\partial_u\psi_-}{-\nu}\right|+\left|\frac{\partial_u\psi_-}{-\nu}\right|\right)\right)(u,0)\lesssim\varepsilon − ν r ∂ u Φ − ( u , 0 ) ≲ ( ∣ ψ − ∣ ( 1 + − ν ∂ u κ + − ν ∂ u ψ − ) ) ( u , 0 ) ≲ ( ∣ ψ − ∣ ( 1 + ∣ ψ − ∣ − ν ∂ u ψ − + − ν ∂ u ψ − ) ) ( u , 0 ) ≲ ε by (101) , we deduce
∣ ∂ u Φ ~ − − ν r ∣ ≲ A 1 / 2 ε . (111) \left|\frac{\partial_u\widetilde{\Phi}_-}{-\nu r}\right|\lesssim A^{1/2}\varepsilon. \tag*{(111)} − ν r ∂ u Φ − ≲ A 1/2 ε . (111) Then, (105) follows by ψ − = κ 1 / 2 r − 2 Φ ~ − \psi_-=\kappa^{1/2}r^{-2}\widetilde{\Phi}_- ψ − = κ 1/2 r − 2 Φ − (90) and (111) , indeed
r ∣ ∂ u ψ − ∣ − ν ≲ ( 1 + r ∣ ∂ u κ ∣ − ν ) ∣ ψ − ∣ + ∣ ∂ u Φ ~ − ∣ − ν r ≲ ( 1 + ∣ ∂ u Φ ~ − ∣ − ν r ) ∣ r ψ − ∣ + ∣ ∂ u Φ ~ − ∣ − ν r ≲ A 1 / 2 ε , r\frac{|\partial_u\psi_-|}{-\nu}\lesssim\left(1+r\frac{|\partial_u\kappa|}{-\nu}\right)|\psi_-|+\frac{|\partial_u\widetilde{\Phi}_-|}{-\nu r}\lesssim\left(1+\frac{|\partial_u\widetilde{\Phi}_-|}{-\nu r}\right)|r\psi_-|+\frac{|\partial_u\widetilde{\Phi}_-|}{-\nu r}\lesssim A^{1/2}\varepsilon, r − ν ∣ ∂ u ψ − ∣ ≲ ( 1 + r − ν ∣ ∂ u κ ∣ ) ∣ ψ − ∣ + − ν r ∣ ∂ u Φ − ∣ ≲ ( 1 + − ν r ∣ ∂ u Φ − ∣ ) ∣ r ψ − ∣ + − ν r ∣ ∂ u Φ − ∣ ≲ A 1/2 ε , where we rewrote (15) as ∂ u κ = − 2 r S [ ψ ~ − ∂ u ψ − ] = − 2 r − 1 κ 3 / 2 S [ ψ ~ − ∂ u Φ ~ − ] \partial_u\kappa=-2r\mathcal{S}[\widetilde{\psi}_-\partial_u\psi_-]=-2r^{-1}\kappa^{3/2}\mathcal{S}[\widetilde{\psi}_-\partial_u\widetilde{\Phi}_-] ∂ u κ = − 2 r S [ ψ − ∂ u ψ − ] = − 2 r − 1 κ 3/2 S [ ψ − ∂ u Φ − ]
Proof of (106) and (107) : These follow immediately by (14) and the previous estimates. □
Remark 5.7. The meaning and role of the reweightings Φ ~ ± \widetilde{\Phi}_\pm Φ ± Φ ~ − \widetilde{\Phi}_- Φ − Φ ~ + \widetilde{\Phi}_+ Φ + r p r^p r p
As a consequence, we finally establish pointwise boundedness estimates for the Einstein equations.
Lemma 5.8. For any A ≥ 1 A\ge1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_f\in\mathbb{B} τ f ∈ B 1 ≤ τ 1 ≤ τ 2 ≤ τ f 1\le\tau_1\le\tau_2\le\tau_f 1 ≤ τ 1 ≤ τ 2 ≤ τ f
∣ ∂ u ∂ v r − ν + 2 M r 2 ∣ ≲ A 2 ε 3 / 2 r − 2 τ − 2 + δ , (112) \left|\frac{\partial_u\partial_v r}{-\nu}+\frac{2M}{r^2}\right|\lesssim A^2\varepsilon^{3/2}r^{-2}\tau^{-2+\delta}, \tag*{(112)} − ν ∂ u ∂ v r + r 2 2 M ≲ A 2 ε 3/2 r − 2 τ − 2 + δ , (112) ∣ ∂ u ∂ v log Ω − ν − 2 M r 3 ∣ ≲ A 2 ε 3 / 2 r − 2 τ − 2 + δ , (113) \left|\frac{\partial_u\partial_v\log\Omega}{-\nu}-\frac{2M}{r^3}\right|\lesssim A^2\varepsilon^{3/2}r^{-2}\tau^{-2+\delta}, \tag*{(113)} − ν ∂ u ∂ v log Ω − r 3 2 M ≲ A 2 ε 3/2 r − 2 τ − 2 + δ , (113) ∣ ∂ u m − ν ∣ ≲ A ε 2 τ − 3 / 2 + δ / 2 , (114) \left|\frac{\partial_u m}{-\nu}\right|\lesssim A\varepsilon^2\tau^{-3/2+\delta/2}, \tag*{(114)} − ν ∂ u m ≲ A ε 2 τ − 3/2 + δ /2 , (114) ∣ ∂ v m ∣ ≲ A ε 2 r − 3 / 2 τ − 3 / 2 + δ / 2 , (115) |\partial_v m|\lesssim A\varepsilon^2 r^{-3/2}\tau^{-3/2+\delta/2}, \tag*{(115)} ∣ ∂ v m ∣ ≲ A ε 2 r − 3/2 τ − 3/2 + δ /2 , (115) ∣ ∂ u κ − ν ∣ ≲ A ε 2 r − 1 τ − 3 / 2 + δ / 2 (116) \left|\frac{\partial_u\kappa}{-\nu}\right|\lesssim A\varepsilon^2r^{-1}\tau^{-3/2+\delta/2} \tag*{(116)} − ν ∂ u κ ≲ A ε 2 r − 1 τ − 3/2 + δ /2 (116) in D τ f \mathcal{D}_{\tau_f} D τ f
∣ ∂ v γ ∣ ≲ A ε 2 r − 7 / 2 τ − 3 / 2 + δ / 2 (117) |\partial_v\gamma|\lesssim A\varepsilon^2r^{-7/2}\tau^{-3/2+\delta/2} \tag*{(117)} ∣ ∂ v γ ∣ ≲ A ε 2 r − 7/2 τ − 3/2 + δ /2 (117) in D τ f ∩ { r ≥ Λ } \mathcal{D}_{\tau_f}\cap\{r\ge\Lambda\} D τ f ∩ { r ≥ Λ }
Proof. The bounds follow readily by (10) , (11) and (15) –(18) , together with Lemmas 5.1 and 5.6. □
We conclude this section with an argument that has been adapted from [53 Proposition 5.12] , which relies on the previous lemma. This result will be crucial in controlling the gauge map Φ τ f \Phi_{\tau_f} Φ τ f
Lemma 5.9. For any A ≥ 1 A\ge1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_f\in\mathbb{B} τ f ∈ B
− ν ∼ e − ( u − v ) (118) -\nu\sim e^{-(u-v)} \tag*{(118)} − ν ∼ e − ( u − v ) (118) on D τ f ∩ { r ≤ Λ } \mathcal{D}_{\tau_f}\cap\{r\le\Lambda\} D τ f ∩ { r ≤ Λ }
Proof. Step 1. We start by proving
sup Γ ∩ D τ f ∣ u − v ∣ ≲ A ε 2 . (119) \sup_{\Gamma\cap\mathcal{D}_{\tau_f}}|u-v|\lesssim A\varepsilon^2. \tag*{(119)} Γ ∩ D τ f sup ∣ u − v ∣ ≲ A ε 2 . (119) Recalling Definition 3.11 and denoting by T : = 1 κ ∂ v + 1 γ ∂ u T:=\frac{1}{\kappa}\partial_v+\frac{1}{\gamma}\partial_u T := κ 1 ∂ v + γ 1 ∂ u [42 ] , we note that
( u − v ) ( Γ ( 1 ) ) = ( u − v ) ( 0 , 0 ) = 0 , Γ ˙ ( τ ) = 1 2 1 − μ T ∣ Γ ( τ ) . (u-v)(\Gamma(1))=(u-v)(0,0)=0,\qquad\dot{\Gamma}(\tau)=\frac{1}{2\sqrt{1-\mu}}T|_{\Gamma(\tau)}. ( u − v ) ( Γ ( 1 )) = ( u − v ) ( 0 , 0 ) = 0 , Γ ˙ ( τ ) = 2 1 − μ 1 T ∣ Γ ( τ ) . Thus, on Γ ( τ ) \Gamma(\tau) Γ ( τ ) τ ∈ [ 1 , τ f ] \tau\in[1,\tau_f] τ ∈ [ 1 , τ f ]
∣ d d τ ( u − v ) ∣ = 1 2 1 − μ ∣ T ( u − v ) ∣ = 1 2 1 − μ ∣ 1 γ − 1 κ ∣ ≲ 1 2 1 − μ ( ∣ γ ~ ∣ + ∣ κ ~ ∣ ) , \left|\frac{d}{d\tau}(u-v)\right|=\frac{1}{2\sqrt{1-\mu}}|T(u-v)|=\frac{1}{2\sqrt{1-\mu}}\left|\frac{1}{\gamma}-\frac{1}{\kappa}\right|\lesssim\frac{1}{2\sqrt{1-\mu}}(|\widetilde{\gamma}|+|\widetilde{\kappa}|), d τ d ( u − v ) = 2 1 − μ 1 ∣ T ( u − v ) ∣ = 2 1 − μ 1 γ 1 − κ 1 ≲ 2 1 − μ 1 ( ∣ γ ∣ + ∣ κ ∣ ) , where we used that κ \kappa κ γ \gamma γ Γ \Gamma Γ (53) and (58) . Hence,
( u − v ) ( Γ ( τ ) ) ≲ ∫ 1 τ 1 2 1 − μ ( ∣ γ ~ ∣ + ∣ κ ~ ∣ ) d τ ′ (u-v)(\Gamma(\tau)) \lesssim\int_1^\tau\frac{1}{2\sqrt{1-\mu}} \left(|\widetilde{\gamma}|+|\widetilde{\kappa}|\right)\,d\tau' ( u − v ) ( Γ ( τ )) ≲ ∫ 1 τ 2 1 − μ 1 ( ∣ γ ∣ + ∣ κ ∣ ) d τ ′ ≲ ( sup Γ ∣ γ ∣ ) ∫ 0 u ( τ ) ∣ γ ~ ( u ′ , v Λ ( u ′ ) ) ∣ d u ′ + ( sup Γ ∣ κ ∣ ) ∫ 0 v ( τ ) ∣ κ ~ ( u Λ ( v ′ ) , v ′ ) ∣ d v ′ . \lesssim\left(\sup_\Gamma|\gamma|\right)\int_0^{u(\tau)}|\widetilde{\gamma}(u',v^\Lambda(u'))|\,du' + \left(\sup_\Gamma|\kappa|\right)\int_0^{v(\tau)}|\widetilde{\kappa}(u^\Lambda(v'),v')|\,dv'. ≲ ( Γ sup ∣ γ ∣ ) ∫ 0 u ( τ ) ∣ γ ( u ′ , v Λ ( u ′ )) ∣ d u ′ + ( Γ sup ∣ κ ∣ ) ∫ 0 v ( τ ) ∣ κ ( u Λ ( v ′ ) , v ′ ) ∣ d v ′ . Using again the uniform bounds for κ \kappa κ γ \gamma γ Γ \Gamma Γ (16) and (117) to write
∫ 0 u ( τ f ) ∣ γ ~ ( u ′ , v Λ ( u ′ ) ) ∣ d u ′ ≲ ∫ 0 u ( τ f ) ∫ v Λ ( u ′ ) v ( τ f ) A ε 2 r − 7 / 2 τ − 3 / 2 + δ / 2 d v ′ d u ′ \int_0^{u(\tau_f)}|\widetilde{\gamma}(u',v^\Lambda(u'))|\,du' \lesssim\int_0^{u(\tau_f)}\int_{v^\Lambda(u')}^{v(\tau_f)} A\varepsilon^2 r^{-7/2}\tau^{-3/2+\delta/2}\,dv'\,du' ∫ 0 u ( τ f ) ∣ γ ( u ′ , v Λ ( u ′ )) ∣ d u ′ ≲ ∫ 0 u ( τ f ) ∫ v Λ ( u ′ ) v ( τ f ) A ε 2 r − 7/2 τ − 3/2 + δ /2 d v ′ d u ′ ≲ A ε 2 ∫ 0 u ( τ f ) ( τ ′ ) − 3 / 2 + δ / 2 d u ′ ∫ v Λ ( u ′ ) v ( τ f ) r − 7 / 2 d v ′ d u ′ ≲ A ε 2 , \lesssim A\varepsilon^2\int_0^{u(\tau_f)}(\tau')^{-3/2+\delta/2}\,du'\int_{v^\Lambda(u')}^{v(\tau_f)}r^{-7/2}\,dv'\,du' \lesssim A\varepsilon^2, ≲ A ε 2 ∫ 0 u ( τ f ) ( τ ′ ) − 3/2 + δ /2 d u ′ ∫ v Λ ( u ′ ) v ( τ f ) r − 7/2 d v ′ d u ′ ≲ A ε 2 , where in the last line we used the definition of τ \tau τ (31) , together with the integrable decay in both τ \tau τ r r r A ε 2 A\varepsilon^2 A ε 2 (15) and (116) ,
∫ 0 v ( τ f ) ∣ κ ~ ( u Λ ( v ′ ) , v ′ ) ∣ d v ′ ≲ ∫ 0 v ( τ f ) ∫ u Λ ( v ′ ) u ( τ f ) A ε 2 ( − ν ) r ˉ − 1 τ − 3 / 2 + δ / 2 d u ′ d v ′ \int_0^{v(\tau_f)}|\widetilde{\kappa}(u^\Lambda(v'),v')|\,dv' \lesssim\int_0^{v(\tau_f)}\int_{u^\Lambda(v')}^{u(\tau_f)} A\varepsilon^2(-\nu)\bar r^{-1}\tau^{-3/2+\delta/2}\,du'\,dv' ∫ 0 v ( τ f ) ∣ κ ( u Λ ( v ′ ) , v ′ ) ∣ d v ′ ≲ ∫ 0 v ( τ f ) ∫ u Λ ( v ′ ) u ( τ f ) A ε 2 ( − ν ) r ˉ − 1 τ − 3/2 + δ /2 d u ′ d v ′ ≲ A ε 2 ∫ 0 v ( τ f ) ( τ ′ ) − 3 / 2 + δ / 2 d v ′ ∫ u Λ ( v ′ ) u ( τ f ) ( − ν ) d v ′ d u ′ ≲ A ε 2 , \lesssim A\varepsilon^2\int_0^{v(\tau_f)}(\tau')^{-3/2+\delta/2}\,dv'\int_{u^\Lambda(v')}^{u(\tau_f)}(-\nu)\,dv'\,du' \lesssim A\varepsilon^2, ≲ A ε 2 ∫ 0 v ( τ f ) ( τ ′ ) − 3/2 + δ /2 d v ′ ∫ u Λ ( v ′ ) u ( τ f ) ( − ν ) d v ′ d u ′ ≲ A ε 2 , where in the last line we used again (31) , together with the integrable decay in τ \tau τ − ν -\nu − ν u u u A ε 2 A\varepsilon^2 A ε 2
Step 2. Let us now prove (118) on D τ f ∩ { r ≤ Λ } D_{\tau_f}\cap\{r\leq\Lambda\} D τ f ∩ { r ≤ Λ } (10) , we readily deduce
∂ v log ( − ν ) = 2 κ m r 2 + 4 π r ν T u v . \partial_v\log(-\nu)=\frac{2\kappa m}{r^2}+\frac{4\pi r}{\nu}T_{uv}. ∂ v log ( − ν ) = r 2 2 κm + ν 4 π r T uv . Noting by (91) and the previous geometric estimates established in § 6 that
0 < C ≤ ∣ 2 κ m r 2 + 4 π r ν T u v ∣ ≤ C ′ , 0<C\leq\left|\frac{2\kappa m}{r^2}+\frac{4\pi r}{\nu}T_{uv}\right|\leq C', 0 < C ≤ r 2 2 κm + ν 4 π r T uv ≤ C ′ , for some strictly positive constants C , C ′ C,C' C , C ′ v v v ( u , v ) ∈ D τ f ∩ { r ≤ Λ } (u,v)\in D_{\tau_f}\cap\{r\leq\Lambda\} ( u , v ) ∈ D τ f ∩ { r ≤ Λ }
( − ν ) ( u , v ) = ( − ν ) ( u , v Λ ( u ) ) exp ( − ∫ v v Λ ( u ) ( 2 κ m r 2 + 4 π r ν T u v ) d v ′ ) ∼ exp ( − ( v Λ ( u ) − u ) ) ∼ exp ( − ( u − v ) ) (-\nu)(u,v)=(-\nu)(u,v^\Lambda(u))\exp\left(-\int_v^{v^\Lambda(u)}\left(\frac{2\kappa m}{r^2}+\frac{4\pi r}{\nu}T_{uv}\right)\,dv'\right)\sim\exp\left(-(v^\Lambda(u)-u)\right)\sim\exp(-(u-v)) ( − ν ) ( u , v ) = ( − ν ) ( u , v Λ ( u )) exp ( − ∫ v v Λ ( u ) ( r 2 2 κm + ν 4 π r T uv ) d v ′ ) ∼ exp ( − ( v Λ ( u ) − u ) ) ∼ exp ( − ( u − v )) where we used (52) to control the boundary term and (119) to compare v Λ ( u ) − u v^\Lambda(u)-u v Λ ( u ) − u u − v u-v u − v
Energy estimates for the Dirac field
The aim of this section is to improve the bootstrap assumptions on the Dirac field, namely (45) –(48) , by employing the vector-field method. In particular, in § 8.1 we establish an integrated local energy decay estimate by combining a suitable Morawetz multiplier and the current conservation for the Dirac field. Since this estimate degenerates towards the horizon and does not have optimal decay in the far region, we establish red-shift estimates in § 8.2 and r p r^p r p § 8.5 we prove the following result via a pigeonhole argument.
Proposition 6.1 (Improving the bootstrap assumptions for the Dirac field). There exist positive constants A 0 A_0 A 0 ε 0 \varepsilon_0 ε 0 M 0 M_0 M 0 δ \delta δ A = A 0 A=A_0 A = A 0 ε ≤ ε 0 \varepsilon\leq\varepsilon_0 ε ≤ ε 0 τ f ∈ B ( S , ε , A ) \tau_f\in\mathcal{B}(S,\varepsilon,A) τ f ∈ B ( S , ε , A )
E τ f ( τ ) ≤ 1 2 A ε 2 τ − 3 + δ , (120) \mathcal{E}^{\tau_f}(\tau)\leq\frac{1}{2}A\varepsilon^2\tau^{-3+\delta}, \tag*{(120)} E τ f ( τ ) ≤ 2 1 A ε 2 τ − 3 + δ , (120) E p τ f ( τ ) ≤ 1 2 A ε 2 τ − 3 + p + δ , (121) \mathcal{E}^{\tau_f}_p(\tau)\leq\frac{1}{2}A\varepsilon^2\tau^{-3+p+\delta}, \tag*{(121)} E p τ f ( τ ) ≤ 2 1 A ε 2 τ − 3 + p + δ , (121) F τ f ( v , τ ) ≤ 1 2 A ε 2 τ − 3 + δ , (122) \mathcal{F}^{\tau_f}(v,\tau)\leq\frac{1}{2}A\varepsilon^2\tau^{-3+\delta}, \tag*{(122)} F τ f ( v , τ ) ≤ 2 1 A ε 2 τ − 3 + δ , (122) F τ f ( u , τ ) ≤ 1 2 A ε 2 τ − 3 + δ . (123) \mathcal{F}^{\tau_f}(u,\tau)\leq\frac{1}{2}A\varepsilon^2\tau^{-3+\delta}. \tag*{(123)} F τ f ( u , τ ) ≤ 2 1 A ε 2 τ − 3 + δ . (123) hold for every τ ∈ [ 1 , τ f ] \tau\in[1,\tau_f] τ ∈ [ 1 , τ f ] ( u , v ) ∈ D τ f (u,v) \in\mathcal{D}_{\tau_f} ( u , v ) ∈ D τ f p ∈ [ δ , 3 − δ ] p \in[\delta,3-\delta] p ∈ [ δ , 3 − δ ]
The energy estimates in this section (except in Lemma 6.4) will be carried out in a region R ⊂ D τ f \mathcal{R} \subset\mathcal{D}_{\tau_f} R ⊂ D τ f τ f ∈ B \tau_f \in\mathcal{B} τ f ∈ B ( u 1 , v 1 ) , ( u 2 , v 2 ) ∈ Γ ∩ D τ f (u_1,v_1),(u_2,v_2) \in\Gamma\cap\mathcal{D}_{\tau_f} ( u 1 , v 1 ) , ( u 2 , v 2 ) ∈ Γ ∩ D τ f ( u 2 , v 2 ) (u_2,v_2) ( u 2 , v 2 ) ( u 1 , v 1 ) (u_1,v_1) ( u 1 , v 1 ) u 2 ′ > u 2 u'_2 > u_2 u 2 ′ > u 2 v 2 ′ > v 2 v'_2 > v_2 v 2 ′ > v 2
R : = ( [ u 1 , u 2 ] × [ v 1 , v 2 ′ ] ) ∪ ( [ u 1 , u 2 ′ ] × [ v 1 , v 2 ] ) , R ≤ Λ : = R ∩ { r ≤ Λ } , R ≥ Λ : = R ∩ { r ≥ Λ } . \mathcal{R} := ([u_1,u_2] \times[v_1,v'_2]) \cup([u_1,u'_2] \times[v_1,v_2]), \qquad\mathcal{R}_{\leq\Lambda} := \mathcal{R} \cap\{r \leq\Lambda\}, \qquad\mathcal{R}_{\geq\Lambda} := \mathcal{R} \cap\{r \geq\Lambda\}. R := ([ u 1 , u 2 ] × [ v 1 , v 2 ′ ]) ∪ ([ u 1 , u 2 ′ ] × [ v 1 , v 2 ]) , R ≤ Λ := R ∩ { r ≤ Λ } , R ≥ Λ := R ∩ { r ≥ Λ } . The null segments constituting ∂ R \partial\mathcal{R} ∂ R Figure 7 . We define τ 1 \tau_1 τ 1 τ \tau τ ∪ \cup ∪ τ 2 \tau_2 τ 2 τ \tau τ ∪ \cup ∪
Figure 7. A Penrose diagram of the region R \mathcal{R} R
Integrated local energy decay (ILED) estimate
We begin with the first estimate for the Dirac field. The argument combines the Dirac energy current, which provides control of the boundary terms, with a Morawetz multiplier yielding the required coercivity in the bulk.
Proposition 6.2. For any A ≥ 1 A \geq1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_f \in\mathcal{B} τ f ∈ B ( u 1 , v 1 ) , ( u 2 , v 2 ) ∈ Γ ∩ D τ f (u_1,v_1),(u_2,v_2) \in\Gamma\cap\mathcal{D}_{\tau_f} ( u 1 , v 1 ) , ( u 2 , v 2 ) ∈ Γ ∩ D τ f u 2 ′ > u 2 u'_2 > u_2 u 2 ′ > u 2 v 2 ′ > v 2 v'_2 > v_2 v 2 ′ > v 2 ( u 2 , v 2 ) (u_2,v_2) ( u 2 , v 2 ) ( u 1 , v 1 ) (u_1,v_1) ( u 1 , v 1 )
∫ I I I ( r 2 ∣ ψ − ∣ 2 + r − 2 ∣ ψ + ∣ 2 + r 2 ∣ ∂ u ψ − ∣ 2 ) d u + ∫ I V ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v + ∫ V − ν ˉ ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 + ∣ ∂ u ψ − ∣ 2 ) d u + ∫ V I ( ∣ ψ + ∣ 2 + ν ˉ 2 ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v + ∬ R − ν ˉ r ˉ − 4 ( ∣ ψ + ∣ 2 + r 2 ∣ ∂ v ψ + ∣ 2 ) + ν ˉ 2 r ˉ − 2 ( ∣ ψ − ∣ 2 + r 2 ∣ ∂ u ψ − ∣ 2 ) d u d v ≲ ∫ I − ν ˉ ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 + ∣ ∂ u ψ − ∣ 2 ) d u + ∫ I I ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v + ε 3 τ 1 − 7 / 2 + 3 δ . (124) \begin{aligned}
&\int_{\mathrm{III}} \left(r^2|\psi_-|^2+r^{-2}|\psi_+|^2+r^2|\partial_u\psi_-|^2\right)\,du
+\int_{\mathrm{IV}}\left(|\psi_+|^2+|\psi_-|^2+|\partial_v\psi_+|^2\right)\,dv\\
&\quad+\int_{\mathrm{V}}-\bar{\nu}\left(|\psi_-|^2+|\psi_+|^2+|\partial_u\psi_-|^2\right)\,du
+\int_{\mathrm{VI}}\left(|\psi_+|^2+\bar{\nu}^2|\psi_-|^2+|\partial_v\psi_+|^2\right)\,dv\\
&\quad+\iint_{\mathcal{R}}-\bar{\nu}\bar{r}^{-4}\left(|\psi_+|^2+r^2|\partial_v\psi_+|^2\right)+\bar{\nu}^2\bar{r}^{-2}\left(|\psi_-|^2+r^2|\partial_u\psi_-|^2\right)\,du\,dv\\
&\lesssim\int_{\mathrm{I}}-\bar{\nu}\left(|\psi_-|^2+|\psi_+|^2+|\partial_u\psi_-|^2\right)\,du
+\int_{\mathrm{II}}\left(|\psi_+|^2+|\psi_-|^2+|\partial_v\psi_+|^2\right)\,dv
+\varepsilon^3\tau_1^{-7/2+3\delta}. \tag*{(124)}
\end{aligned} ∫ III ( r 2 ∣ ψ − ∣ 2 + r − 2 ∣ ψ + ∣ 2 + r 2 ∣ ∂ u ψ − ∣ 2 ) d u + ∫ IV ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v + ∫ V − ν ˉ ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 + ∣ ∂ u ψ − ∣ 2 ) d u + ∫ VI ( ∣ ψ + ∣ 2 + ν ˉ 2 ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v + ∬ R − ν ˉ r ˉ − 4 ( ∣ ψ + ∣ 2 + r 2 ∣ ∂ v ψ + ∣ 2 ) + ν ˉ 2 r ˉ − 2 ( ∣ ψ − ∣ 2 + r 2 ∣ ∂ u ψ − ∣ 2 ) d u d v ≲ ∫ I − ν ˉ ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 + ∣ ∂ u ψ − ∣ 2 ) d u + ∫ II ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v + ε 3 τ 1 − 7/2 + 3 δ . (124) At this stage, before turning to the proof, we highlight the degeneracy of certain coefficients near the horizon, such as ∂ u ψ − \partial_u\psi_- ∂ u ψ − ψ − \psi_- ψ − r r r
Proof of the ILED estimate
Let us begin with the following lemma, which yields a control on the boundary terms.
Lemma 6.3 (Degenerate energy estimate – I). With hypotheses as in Proposition 6.2, it holds that
∫ I I I ( r − 2 ∣ ψ + ∣ 2 + r 2 ∣ ψ − ∣ 2 + r 2 ∣ ∂ u ψ − ∣ 2 ) d u + ∫ I V ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v + ∫ V − ν ˉ ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ∣ ∂ u ψ − ∣ 2 ) d u + ∫ V I ( ∣ ψ + ∣ 2 + ν ˉ 2 ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v ≲ ∫ I − ν ˉ ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ∣ ∂ u ψ − ∣ 2 ν ˉ 2 ) d u + ∫ I I ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v + ε r r T + ε 3 τ 1 − 7 / 2 + 3 δ , (125) \begin{aligned}
&\int_{\mathrm{III}}\left(r^{-2}|\psi_+|^2+r^2|\psi_-|^2+r^2|\partial_u\psi_-|^2\right)\,du
+\int_{\mathrm{IV}}\left(|\psi_+|^2+|\psi_-|^2+|\partial_v\psi_+|^2\right)\,dv\\
&\quad+\int_{\mathrm{V}}-\bar{\nu}\left(|\psi_+|^2+|\psi_-|^2+|\partial_u\psi_-|^2\right)\,du
+\int_{\mathrm{VI}}\left(|\psi_+|^2+\bar{\nu}^2|\psi_-|^2+|\partial_v\psi_+|^2\right)\,dv\\
&\lesssim\int_{\mathrm{I}}-\bar{\nu}\left(|\psi_+|^2+|\psi_-|^2+\frac{|\partial_u\psi_-|^2}{\bar{\nu}^2}\right)\,du
+\int_{\mathrm{II}}\left(|\psi_+|^2+|\psi_-|^2+|\partial_v\psi_+|^2\right)\,dv+\varepsilon rr^T+\varepsilon^3\tau_1^{-7/2+3\delta}, \tag*{(125)}
\end{aligned} ∫ III ( r − 2 ∣ ψ + ∣ 2 + r 2 ∣ ψ − ∣ 2 + r 2 ∣ ∂ u ψ − ∣ 2 ) d u + ∫ IV ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v + ∫ V − ν ˉ ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ∣ ∂ u ψ − ∣ 2 ) d u + ∫ VI ( ∣ ψ + ∣ 2 + ν ˉ 2 ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v ≲ ∫ I − ν ˉ ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ν ˉ 2 ∣ ∂ u ψ − ∣ 2 ) d u + ∫ II ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v + ε r r T + ε 3 τ 1 − 7/2 + 3 δ , (125) where
E r r T : = A 3 ε 0 2 ∬ R Ω ‾ 2 r ‾ − 4 ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 + r ‾ 2 ∣ ∂ v Φ + ∣ 2 + r ‾ 2 ∣ ∂ u Φ − ∣ 2 ) d u d v . (126) \mathcal{E}_{rr}^{T}:=A^{3}\varepsilon_{0}^{2}\iint_{\mathcal{R}}\overline{\Omega}^{2}\overline{r}^{-4}\left(|\Phi_{+}|^{2}+|\Phi_{-}|^{2}+\overline{r}^{2}|\partial_{v}\Phi_{+}|^{2}+\overline{r}^{2}|\partial_{u}\Phi_{-}|^{2}\right)\,du\,dv. \tag*{(126)} E r r T := A 3 ε 0 2 ∬ R Ω 2 r − 4 ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 + r 2 ∣ ∂ v Φ + ∣ 2 + r 2 ∣ ∂ u Φ − ∣ 2 ) d u d v . (126) In the proof, we follow the strategy in [61 Proposition 3.9] , which crucially relies on the structure of the Dirac equations in the form (20) –(21) . To control the first-order derivatives, we commute the equations with the vector field T ‾ : = ∂ v + ∂ u \overline{T}:=\partial_{v}+\partial_{u} T := ∂ v + ∂ u T ‾ \overline{T} T Kodama vector field [42 ]
K : = ( 1 κ ∂ v + 1 γ ∂ u ) . K:=\left(\frac{1}{\kappa}\partial_{v}+\frac{1}{\gamma}\partial_{u}\right). K := ( κ 1 ∂ v + γ 1 ∂ u ) . and treat the resulting error terms in an analogous manner. Indeed, error terms such as ∂ u κ \partial_{u}\kappa ∂ u κ ∂ v γ \partial_{v}\gamma ∂ v γ Lemma 5.8 , while ∂ u γ \partial_{u}\gamma ∂ u γ ∂ v κ \partial_{v}\kappa ∂ v κ Lemma 5.2 . We do not pursue this approach here in order to avoid lengthy computations. Finally, to obtain full control of the null hypersurface terms, we follow [4 Proposition 6.1] and introduce an additional "null Lagrangian" current [16 ] . This allows us to recover the missing control of Φ − \Phi_{-} Φ − u u u Φ + \Phi_{+} Φ + v v v
Proof of Lemma 6.3. We begin by multiplying (20) by Φ ‾ + \overline{\Phi}_{+} Φ + (21) by Φ ‾ − \overline{\Phi}_{-} Φ −
∂ u ( ∣ Φ + ∣ 2 ) + ∂ v ( ∣ Φ − ∣ 2 ) = 0. (127) \partial_{u}(|\Phi_{+}|^{2})+\partial_{v}(|\Phi_{-}|^{2})=0. \tag*{(127)} ∂ u ( ∣ Φ + ∣ 2 ) + ∂ v ( ∣ Φ − ∣ 2 ) = 0. (127) Let us now commute the Dirac equations (20) –(21) with the vector field T ‾ = ∂ v + ∂ u \overline{T}=\partial_{v}+\partial_{u} T = ∂ v + ∂ u
∂ u ( T ‾ Φ + ) = − Ω r ( ∂ v log Ω + ∂ u log Ω − ( λ + ν ) r − 1 ) Φ − − Ω 2 r 2 Φ + − Ω r ∂ u Φ − , \partial_{u}(\overline{T}\Phi_{+})=-\frac{\Omega}{r}\left(\partial_{v}\log\Omega+\partial_{u}\log\Omega-(\lambda+\nu)r^{-1}\right)\Phi_{-}-\frac{\Omega^{2}}{r^{2}}\Phi_{+}-\frac{\Omega}{r}\partial_{u}\Phi_{-}, ∂ u ( T Φ + ) = − r Ω ( ∂ v log Ω + ∂ u log Ω − ( λ + ν ) r − 1 ) Φ − − r 2 Ω 2 Φ + − r Ω ∂ u Φ − , ∂ v ( T ‾ Φ − ) = Ω r ( ∂ v log Ω + ∂ u log Ω − ( λ + ν ) r − 1 ) Φ + − Ω 2 r 2 Φ − + Ω r ∂ v Φ + . \partial_{v}(\overline{T}\Phi_{-})=\frac{\Omega}{r}\left(\partial_{v}\log\Omega+\partial_{u}\log\Omega-(\lambda+\nu)r^{-1}\right)\Phi_{+}-\frac{\Omega^{2}}{r^{2}}\Phi_{-}+\frac{\Omega}{r}\partial_{v}\Phi_{+}. ∂ v ( T Φ − ) = r Ω ( ∂ v log Ω + ∂ u log Ω − ( λ + ν ) r − 1 ) Φ + − r 2 Ω 2 Φ − + r Ω ∂ v Φ + . Similarly as before, we multiply by T ‾ Φ ‾ ± \overline{T}\overline{\Phi}_{\pm} T Φ ± T ‾ Φ + = ∂ v Φ + − ( Ω / r ) Φ − \overline{T}\Phi_{+}=\partial_{v}\Phi_{+}-(\Omega/r)\Phi_{-} T Φ + = ∂ v Φ + − ( Ω/ r ) Φ − T ‾ Φ − = ∂ u Φ − + ( Ω / r ) Φ + \overline{T}\Phi_{-}=\partial_{u}\Phi_{-}+(\Omega/r)\Phi_{+} T Φ − = ∂ u Φ − + ( Ω/ r ) Φ +
1 2 ( ∂ u ( ∣ T ‾ Φ + ∣ 2 ) + ∂ v ( ∣ T ‾ Φ − ∣ 2 ) ) = A ( ℜ [ Φ ‾ + ∂ u Φ − ] − ℜ [ Φ ‾ − ∂ v Φ + ] + Ω r ( ∣ Φ − ∣ 2 + ∣ Φ + ∣ 2 ) ) . (128) \frac{1}{2}\left(\partial_{u}(|\overline{T}\Phi_{+}|^{2})+\partial_{v}(|\overline{T}\Phi_{-}|^{2})\right)=\mathcal{A}\left(\mathop{\Re}[\overline{\Phi}_{+}\partial_{u}\Phi_{-}]-\mathop{\Re}[\overline{\Phi}_{-}\partial_{v}\Phi_{+}]+\frac{\Omega}{r}(|\Phi_{-}|^{2}+|\Phi_{+}|^{2})\right). \tag*{(128)} 2 1 ( ∂ u ( ∣ T Φ + ∣ 2 ) + ∂ v ( ∣ T Φ − ∣ 2 ) ) = A ( ℜ [ Φ + ∂ u Φ − ] − ℜ [ Φ − ∂ v Φ + ] + r Ω ( ∣ Φ − ∣ 2 + ∣ Φ + ∣ 2 ) ) . (128) where
A : = Ω r ( ∂ v log Ω + ∂ u log Ω − ( λ + ν ) r − 1 ) . \mathcal{A}:=\frac{\Omega}{r}\left(\partial_{v}\log\Omega+\partial_{u}\log\Omega-(\lambda+\nu)r^{-1}\right). A := r Ω ( ∂ v log Ω + ∂ u log Ω − ( λ + ν ) r − 1 ) . Recalling (57) , (93) , (94) , and the fact that
∣ λ + ν ∣ = ∣ ( 1 − μ ‾ ) ( κ − ν / ν ‾ ) + ( 1 − μ ~ ) κ ∣ ≲ A 3 ε 0 3 / 2 τ − 1 + δ , |\lambda+\nu|=|(1-\overline{\mu})(\kappa-\nu/\overline{\nu})+(1-\widetilde{\mu})\kappa|\lesssim A^{3}\varepsilon_{0}^{3/2}\tau^{-1+\delta}, ∣ λ + ν ∣ = ∣ ( 1 − μ ) ( κ − ν / ν ) + ( 1 − μ ) κ ∣ ≲ A 3 ε 0 3/2 τ − 1 + δ , we have
∣ A ∣ ≲ A 3 ε 0 3 / 2 ( − ν ‾ ) 1 / 2 r ‾ − 2 τ − 3 / 4 + δ . |\mathcal{A}|\lesssim A^{3}\varepsilon_{0}^{3/2}(-\overline{\nu})^{1/2}\overline{r}^{-2}\tau^{-3/4+\delta}. ∣ A ∣ ≲ A 3 ε 0 3/2 ( − ν ) 1/2 r − 2 τ − 3/4 + δ . Thus, we integrate (128) over R \mathcal{R} R
∬ R ∣ A ∣ ∣ Φ + ∣ ∣ ∂ u Φ − ∣ d u d v ≲ A 3 ε 3 / 2 ( ∬ R ≤ Λ ( − ν ˉ ∣ Φ + ∣ 2 + τ − 3 / 2 + 2 δ ∣ ∂ u Φ − ∣ 2 ) d u d v + ∬ R ≥ Λ ( r ˉ − 4 ∣ ∂ u Φ − ∣ 2 + τ − 3 / 2 + 2 δ ∣ Φ + ∣ 2 ) d u d v ) ≲ A 3 ε 3 / 2 ( ∬ R ≤ Λ ( − ν ˉ ( ∣ Φ + ∣ 2 + τ − 3 / 2 + 2 δ ( ∣ ψ − ∣ 2 + ∣ ∂ u ψ − ∣ 2 ) ) ) d u d v + ∬ R ≥ Λ ( r ˉ − 4 ∣ ∂ u Φ − ∣ 2 + τ − 3 / 2 + 2 δ ∣ ψ + ∣ 2 ) d u d v ) ≲ E r r T + A 3 ε 3 / 2 ( ∫ τ 1 τ 2 τ − 3 / 2 + 2 δ E ( τ ) d τ + ∫ τ 1 τ 2 τ − 3 / 2 + 2 δ E 0 ( τ ) d τ ) ≲ E r r T + ε 3 τ 1 − 7 / 2 + 3 δ . \begin{aligned}
\iint_{\mathcal{R}} |\mathcal{A}||\Phi_+||\partial_u\Phi_-|\,du\,dv
&\lesssim A^3\varepsilon^{3/2}\left(\iint_{\mathcal{R}_{\leq\Lambda}}\left(-\bar{\nu}|\Phi_+|^2+\tau^{-3/2+2\delta}|\partial_u\Phi_-|^2\right)\,du\,dv\right.\\
&\qquad\left.+\iint_{\mathcal{R}_{\geq\Lambda}}\left(\bar{r}^{-4}|\partial_u\Phi_-|^2+\tau^{-3/2+2\delta}|\Phi_+|^2\right)\,du\,dv\right)\\
&\lesssim A^3\varepsilon^{3/2}\left(\iint_{\mathcal{R}_{\leq\Lambda}}\left(-\bar{\nu}\left(|\Phi_+|^2+\tau^{-3/2+2\delta}(|\psi_-|^2+|\partial_u\psi_-|^2)\right)\right)\,du\,dv\right.\\
&\qquad\left.+\iint_{\mathcal{R}_{\geq\Lambda}}\left(\bar{r}^{-4}|\partial_u\Phi_-|^2+\tau^{-3/2+2\delta}|\psi_+|^2\right)\,du\,dv\right)\\
&\lesssim\mathop{\mathrm{Err}}^T+A^3\varepsilon^{3/2}\left(\int_{\tau_1}^{\tau_2}\tau^{-3/2+2\delta}\mathcal{E}(\tau)\,d\tau+\int_{\tau_1}^{\tau_2}\tau^{-3/2+2\delta}\mathcal{E}_0(\tau)\,d\tau\right)\\
&\lesssim\mathop{\mathrm{Err}}^T+\varepsilon^3\tau_1^{-7/2+3\delta}.
\end{aligned} ∬ R ∣ A ∣∣ Φ + ∣∣ ∂ u Φ − ∣ d u d v ≲ A 3 ε 3/2 ( ∬ R ≤ Λ ( − ν ˉ ∣ Φ + ∣ 2 + τ − 3/2 + 2 δ ∣ ∂ u Φ − ∣ 2 ) d u d v + ∬ R ≥ Λ ( r ˉ − 4 ∣ ∂ u Φ − ∣ 2 + τ − 3/2 + 2 δ ∣ Φ + ∣ 2 ) d u d v ) ≲ A 3 ε 3/2 ( ∬ R ≤ Λ ( − ν ˉ ( ∣ Φ + ∣ 2 + τ − 3/2 + 2 δ ( ∣ ψ − ∣ 2 + ∣ ∂ u ψ − ∣ 2 ) ) ) d u d v + ∬ R ≥ Λ ( r ˉ − 4 ∣ ∂ u Φ − ∣ 2 + τ − 3/2 + 2 δ ∣ ψ + ∣ 2 ) d u d v ) ≲ Err T + A 3 ε 3/2 ( ∫ τ 1 τ 2 τ − 3/2 + 2 δ E ( τ ) d τ + ∫ τ 1 τ 2 τ − 3/2 + 2 δ E 0 ( τ ) d τ ) ≲ Err T + ε 3 τ 1 − 7/2 + 3 δ . by the bootstrap assumptions (3.21), (3.22) and the definition of E r r T \mathop{\mathrm{Err}}^T Err T (126) . In this manner, we obtain
∣ ∬ R ( R H S o f x r e f ‘ ( 6.9 ) ‘ ) d u d v ∣ ≲ A 3 ε 3 / 2 ( ∬ R − ν ˉ r ˉ − 4 ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 + r 2 ∣ ∂ v Φ + ∣ 2 + r 2 ∣ ∂ u Φ − ∣ 2 ) d u d v + ∬ R ≤ Λ τ − 3 / 2 + 2 δ ( − ν ˉ ) ( ∣ ψ − ∣ 2 + ∣ ∂ u ψ − ∣ 2 + ∣ ψ + ∣ 2 ) d u d v + ∬ R ≥ Λ τ − 3 / 2 + 2 δ ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 ) d u d v ) ≲ E r r T + ε 3 τ 1 − 7 / 2 + 3 δ . \begin{aligned}
\left|\iint_{\mathcal{R}}(\mathrm{RHS\ of\ }{xref}`(6.9)`) \,du\,dv\right|
&\lesssim A^3\varepsilon^{3/2}\left(\iint_{\mathcal{R}}-\bar{\nu}\bar{r}^{-4}\left(|\Phi_+|^2+|\Phi_-|^2+r^2|\partial_v\Phi_+|^2+r^2|\partial_u\Phi_-|^2\right)\,du\,dv\right.\\
&\qquad+\iint_{\mathcal{R}_{\leq\Lambda}}\tau^{-3/2+2\delta}(-\bar{\nu})\left(|\psi_-|^2+|\partial_u\psi_-|^2+|\psi_+|^2\right)\,du\,dv\\
&\qquad\left.+\iint_{\mathcal{R}_{\geq\Lambda}}\tau^{-3/2+2\delta}\left(|\psi_+|^2+|\psi_-|^2\right)\,du\,dv\right)\\
&\lesssim\mathop{\mathrm{Err}}^T+\varepsilon^3\tau_1^{-7/2+3\delta}.
\end{aligned} ∬ R ( RHS of x r e f ‘ ( 6.9 ) ‘ ) d u d v ≲ A 3 ε 3/2 ( ∬ R − ν ˉ r ˉ − 4 ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 + r 2 ∣ ∂ v Φ + ∣ 2 + r 2 ∣ ∂ u Φ − ∣ 2 ) d u d v + ∬ R ≤ Λ τ − 3/2 + 2 δ ( − ν ˉ ) ( ∣ ψ − ∣ 2 + ∣ ∂ u ψ − ∣ 2 + ∣ ψ + ∣ 2 ) d u d v + ∬ R ≥ Λ τ − 3/2 + 2 δ ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 ) d u d v ) ≲ Err T + ε 3 τ 1 − 7/2 + 3 δ . Next, we consider the trivial identity ∂ u ∂ v ( σ ˉ ∣ Φ ± ∣ 2 ) − ∂ v ∂ u ( σ ˉ ∣ Φ ± ∣ 2 ) = 0 \partial_u\partial_v(\bar{\sigma}|\Phi_\pm|^2)-\partial_v\partial_u(\bar{\sigma}|\Phi_\pm|^2)=0 ∂ u ∂ v ( σ ˉ ∣ Φ ± ∣ 2 ) − ∂ v ∂ u ( σ ˉ ∣ Φ ± ∣ 2 ) = 0 σ ˉ : = Ω ‾ 2 = 1 − 2 M / r ˉ \bar{\sigma}:=\overline{\Omega}^2=1-2M/\bar{r} σ ˉ := Ω 2 = 1 − 2 M / r ˉ
∂ u ( 2 M λ ˉ r ˉ − 2 ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 ) + 2 Ω ‾ 2 ( ℜ [ Φ ‾ + ∂ v Φ + ] + Ω r ˉ − 1 ℜ [ Φ ‾ − Φ + ] ) ) − ∂ v ( 2 M ν ˉ r ˉ − 2 ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 ) + 2 Ω ‾ 2 ( ℜ [ Φ ‾ − ∂ u Φ − ] − Ω r ˉ − 1 ℜ [ Φ ‾ − Φ + ] ) ) = 0. (129) \begin{aligned}
\partial_u\Bigl(2M\bar{\lambda}\bar{r}^{-2}(|\Phi_+|^2+|\Phi_-|^2)+2\overline{\Omega}^2\bigl(\Re[\overline{\Phi}_+\partial_v\Phi_+]+\Omega\bar{r}^{-1}\Re[\overline{\Phi}_-\Phi_+]\bigr)\Bigr)\\
-\partial_v\Bigl(2M\bar{\nu}\bar{r}^{-2}(|\Phi_+|^2+|\Phi_-|^2)+2\overline{\Omega}^2\bigl(\Re[\overline{\Phi}_-\partial_u\Phi_-]-\Omega\bar{r}^{-1}\Re[\overline{\Phi}_-\Phi_+]\bigr)\Bigr)=0. \tag*{(129)}
\end{aligned} ∂ u ( 2 M λ ˉ r ˉ − 2 ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 ) + 2 Ω 2 ( ℜ [ Φ + ∂ v Φ + ] + Ω r ˉ − 1 ℜ [ Φ − Φ + ] ) ) − ∂ v ( 2 M ν ˉ r ˉ − 2 ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 ) + 2 Ω 2 ( ℜ [ Φ − ∂ u Φ − ] − Ω r ˉ − 1 ℜ [ Φ − Φ + ] ) ) = 0. (129) Finally, adding (127) –(128) –(129) yields
∬ R ∂ u ( ∣ Φ + ∣ 2 + Ω ‾ 2 r ˉ − 2 ∣ Φ − ∣ 2 + ∣ ∂ v Φ + ∣ 2 ) + ∂ v ( Ω ‾ 2 r ˉ − 2 ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 + ∣ ∂ u Φ − ∣ 2 ) ≲ E r r T + ε 3 τ 1 − 7 / 2 + 3 δ . \iint_{\mathcal{R}}\partial_u\left(|\Phi_+|^2+\overline{\Omega}^2\bar{r}^{-2}|\Phi_-|^2+|\partial_v\Phi_+|^2\right)+\partial_v\left(\overline{\Omega}^2\bar{r}^{-2}|\Phi_+|^2+|\Phi_-|^2+|\partial_u\Phi_-|^2\right)\lesssim\mathop{\mathrm{Err}}^T+\varepsilon^3\tau_1^{-7/2+3\delta}. ∬ R ∂ u ( ∣ Φ + ∣ 2 + Ω 2 r ˉ − 2 ∣ Φ − ∣ 2 + ∣ ∂ v Φ + ∣ 2 ) + ∂ v ( Ω 2 r ˉ − 2 ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 + ∣ ∂ u Φ − ∣ 2 ) ≲ Err T + ε 3 τ 1 − 7/2 + 3 δ . Recalling the definition (19) relating Φ ± \Phi_\pm Φ ± ψ ± \psi_\pm ψ ±
With the same argument, one also proves the following.
Lemma 6.4 (Degenerate energy estimate – II). With hypotheses as in Proposition 6.2, it holds that
∫ i i i ( r ˉ − 2 ∣ ψ + ∣ 2 + r ˉ 2 ∣ ψ − ∣ 2 + r ˉ 2 ∣ ∂ u ψ − ∣ 2 ) d u + ∫ i v ( ∣ ψ + ∣ 2 + ν ˉ 2 ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v ≲ ∫ i − ν ˉ ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ∣ ∂ u ψ − ∣ 2 ν ˉ 2 ) d u + ∫ i i ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v + E r r T + ε 3 τ 1 − 7 / 2 + 3 δ . (130) \begin{aligned}
\int_{\mathrm{iii}}\left(\bar{r}^{-2}|\psi_+|^2+\bar{r}^2|\psi_-|^2+\bar{r}^2|\partial_u\psi_-|^2\right)\,du
+\int_{\mathrm{iv}}\left(|\psi_+|^2+\bar{\nu}^2|\psi_-|^2+|\partial_v\psi_+|^2\right)\,dv\\
\lesssim\int_{\mathrm{i}}-\bar{\nu}\left(|\psi_+|^2+|\psi_-|^2+\frac{|\partial_u\psi_-|^2}{\bar{\nu}^2}\right)\,du
+\int_{\mathrm{ii}}\left(|\psi_+|^2+|\psi_-|^2+|\partial_v\psi_+|^2\right)\,dv+\mathop{\mathrm{Err}}^T+\varepsilon^3\tau_1^{-7/2+3\delta}. \tag*{(130)}
\end{aligned} ∫ iii ( r ˉ − 2 ∣ ψ + ∣ 2 + r ˉ 2 ∣ ψ − ∣ 2 + r ˉ 2 ∣ ∂ u ψ − ∣ 2 ) d u + ∫ iv ( ∣ ψ + ∣ 2 + ν ˉ 2 ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v ≲ ∫ i − ν ˉ ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ν ˉ 2 ∣ ∂ u ψ − ∣ 2 ) d u + ∫ ii ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v + Err T + ε 3 τ 1 − 7/2 + 3 δ . (130) where E r r T \mathop{\mathrm{Err}}^T Err T Figure 8 .
Figure 8. A Penrose diagram of the region considered in Lemma 6.4.
As a second step, we now prove the estimate that allows us to recover a globally coercive bulk term.
Lemma 6.5 (Morawetz estimate). With hypotheses as in Proposition 6.2, it holds that
∬ R − ν ˉ r ˉ − 4 ( ∣ ψ + ∣ 2 + r ˉ 2 ∣ ∂ v ψ + ∣ 2 ) + ν ˉ 2 r ˉ − 2 ( ∣ ψ − ∣ 2 + r ˉ 2 ∣ ∂ u ψ − ∣ 2 ) d u d v ≲ ( L H S o f x r e f ‘ ( 6.6 ) ‘ ) + ε 3 τ 1 − 7 / 2 + 3 δ . (131) \iint_{\mathcal{R}}-\bar{\nu}\bar{r}^{-4}\left(|\psi_+|^2+\bar{r}^2|\partial_v\psi_+|^2\right)+\bar{\nu}^2\bar{r}^{-2}\left(|\psi_-|^2+\bar{r}^2|\partial_u\psi_-|^2\right)\,du\,dv\lesssim\left(\mathrm{LHS\ of\ }{xref}`(6.6)`\right)+\varepsilon^3\tau_1^{-7/2+3\delta}. \tag*{(131)} ∬ R − ν ˉ r ˉ − 4 ( ∣ ψ + ∣ 2 + r ˉ 2 ∣ ∂ v ψ + ∣ 2 ) + ν ˉ 2 r ˉ − 2 ( ∣ ψ − ∣ 2 + r ˉ 2 ∣ ∂ u ψ − ∣ 2 ) d u d v ≲ ( LHS of x r e f ‘ ( 6.6 ) ‘ ) + ε 3 τ 1 − 7/2 + 3 δ . (131) As noted in Remark 2.6, the symmetry between the equations for the two Dirac components is crucial for establishing the Morawetz estimate in the linear case. Our strategy is therefore to regard the equations for Φ ± \Phi_{\pm} Φ ± [61 Theorem 4.1] to our double-null gauge and carry out an analogous analysis of the bulk terms.
Proof of Lemma 6.5. Thanks to (4.9), (5.6), (5.7) and the geometric estimates established in § 6 , we can linearize equations (22) –(23) as follows
∂ u ∂ v Φ ± + ( Ω ‾ 2 r ‾ 2 + E r r ± ± ) Φ ± + ( Ω ‾ ( 3 M − r ‾ ) r ‾ 3 + E r r ∓ ± ) Φ ∓ = 0 , (132) \partial_u\partial_v\Phi_{\pm}+\left(\frac{\overline{\Omega}^{2}}{\overline{r}^{2}}+\mathrm{Err}^{\pm\pm}\right)\Phi_{\pm}+\left(\frac{\overline{\Omega}(3M-\overline{r})}{\overline{r}^{3}}+\mathrm{Err}^{\mp\pm}\right)\Phi_{\mp}=0, \tag*{(132)} ∂ u ∂ v Φ ± + ( r 2 Ω 2 + Err ±± ) Φ ± + ( r 3 Ω ( 3 M − r ) + Err ∓± ) Φ ∓ = 0 , (132) where
∣ E r r ± ± ∣ ≲ A 3 ε 3 / 2 Ω ‾ 2 r ‾ − 2 τ − 1 + δ , ∣ E r r ∓ ± ∣ ≲ A 3 ε 3 / 2 Ω ‾ r ‾ − 2 τ − 3 / 4 + δ . (133) \left|\mathrm{Err}^{\pm\pm}\right|\lesssim A^{3}\varepsilon^{3/2}\overline{\Omega}^{2}\overline{r}^{-2}\tau^{-1+\delta},\qquad\left|\mathrm{Err}^{\mp\pm}\right|\lesssim A^{3}\varepsilon^{3/2}\overline{\Omega}\overline{r}^{-2}\tau^{-3/4+\delta}. \tag*{(133)} Err ±± ≲ A 3 ε 3/2 Ω 2 r − 2 τ − 1 + δ , Err ∓± ≲ A 3 ε 3/2 Ω r − 2 τ − 3/4 + δ . (133) Let us apply to (132) the following Morawetz multiplier
X Φ ± : = f ( ∂ v − ∂ u ) Φ ± + q Φ ± , X\Phi_{\pm}:=f(\partial_v-\partial_u)\Phi_{\pm}+q\Phi_{\pm}, X Φ ± := f ( ∂ v − ∂ u ) Φ ± + q Φ ± , where f = f ( r ‾ ) f=f(\overline{r}) f = f ( r ) q = q ( r ‾ ) q=q(\overline{r}) q = q ( r )
ℜ [ X Φ + ∂ u ∂ v Φ + ] = ∂ u ( f 2 ∣ ∂ v Φ + ∣ 2 + q ℜ [ Φ + ∂ v Φ + ] ) − ∂ v ( f Ω ‾ 2 2 r ‾ 2 ∣ Φ − ∣ 2 − Ω ‾ 2 q ′ 2 ∣ Φ + ∣ 2 ) + Ω ‾ 2 f ′ 2 ( ∣ ∂ v Φ + ∣ 2 + Ω 2 r ‾ 2 ∣ Φ − ∣ 2 ) − Ω ‾ 2 ( Ω ‾ 2 q ′ ) ′ 2 ∣ Φ + ∣ 2 + q Ω r ‾ ℜ [ Φ ‾ − ∂ v Φ + ] , \begin{aligned}
\Re[X\Phi_{+}\partial_u\partial_v\Phi_{+}]={}&\partial_u\left(\frac{f}{2}|\partial_v\Phi_{+}|^2+q\Re[\Phi_{+}\partial_v\Phi_{+}]\right)-\partial_v\left(\frac{f\overline{\Omega}^{2}}{2\overline{r}^{2}}|\Phi_{-}|^2-\frac{\overline{\Omega}^{2}q'}{2}|\Phi_{+}|^2\right)\\
&+\frac{\overline{\Omega}^{2}f'}{2}\left(|\partial_v\Phi_{+}|^2+\frac{\Omega^{2}}{\overline{r}^{2}}|\Phi_{-}|^2\right)-\frac{\overline{\Omega}^{2}(\overline{\Omega}^{2}q')'}{2}|\Phi_{+}|^2+\frac{q\Omega}{\overline{r}}\Re[\overline{\Phi}_{-}\partial_v\Phi_{+}],
\end{aligned} ℜ [ X Φ + ∂ u ∂ v Φ + ] = ∂ u ( 2 f ∣ ∂ v Φ + ∣ 2 + q ℜ [ Φ + ∂ v Φ + ] ) − ∂ v ( 2 r 2 f Ω 2 ∣ Φ − ∣ 2 − 2 Ω 2 q ′ ∣ Φ + ∣ 2 ) + 2 Ω 2 f ′ ( ∣ ∂ v Φ + ∣ 2 + r 2 Ω 2 ∣ Φ − ∣ 2 ) − 2 Ω 2 ( Ω 2 q ′ ) ′ ∣ Φ + ∣ 2 + r q Ω ℜ [ Φ − ∂ v Φ + ] , ℜ [ X Φ − ∂ u ∂ v Φ − ] = ∂ u ( f Ω 2 2 r ‾ 2 ∣ Φ + ∣ 2 − Ω ‾ 2 q ′ 2 ∣ Φ − ∣ 2 ) − ∂ v ( f 2 ∣ ∂ u Φ − ∣ 2 − q ℜ [ Φ ‾ − ∂ u Φ − ] ) + Ω 2 f ′ 2 ( ∣ ∂ u Φ − ∣ 2 + Ω ‾ 2 r ‾ 2 ∣ Φ + ∣ 2 ) − Ω ‾ 2 ( Ω ‾ 2 q ′ ) ′ 2 ∣ Φ − ∣ 2 − q Ω r ‾ ℜ [ Φ ‾ + ∂ u Φ − ] . (134) \begin{aligned}
\Re[X\Phi_{-}\partial_u\partial_v\Phi_{-}]={}&\partial_u\left(\frac{f\Omega^{2}}{2\overline{r}^{2}}|\Phi_{+}|^2-\frac{\overline{\Omega}^{2}q'}{2}|\Phi_{-}|^2\right)-\partial_v\left(\frac{f}{2}|\partial_u\Phi_{-}|^2-q\Re[\overline{\Phi}_{-}\partial_u\Phi_{-}]\right)\\
&+\frac{\Omega^{2}f'}{2}\left(|\partial_u\Phi_{-}|^2+\frac{\overline{\Omega}^{2}}{\overline{r}^{2}}|\Phi_{+}|^2\right)-\frac{\overline{\Omega}^{2}(\overline{\Omega}^{2}q')'}{2}|\Phi_{-}|^2-\frac{q\Omega}{\overline{r}}\Re[\overline{\Phi}_{+}\partial_u\Phi_{-}]. \tag*{(134)}
\end{aligned} ℜ [ X Φ − ∂ u ∂ v Φ − ] = ∂ u ( 2 r 2 f Ω 2 ∣ Φ + ∣ 2 − 2 Ω 2 q ′ ∣ Φ − ∣ 2 ) − ∂ v ( 2 f ∣ ∂ u Φ − ∣ 2 − q ℜ [ Φ − ∂ u Φ − ] ) + 2 Ω 2 f ′ ( ∣ ∂ u Φ − ∣ 2 + r 2 Ω 2 ∣ Φ + ∣ 2 ) − 2 Ω 2 ( Ω 2 q ′ ) ′ ∣ Φ − ∣ 2 − r q Ω ℜ [ Φ + ∂ u Φ − ] . (134) Next,
∑ ± ( Ω ‾ 2 r ‾ 2 + E r r ± ± ) ℜ [ X Φ ± Φ ± ] = ∑ ± ∂ v ( Ω ‾ 2 f 2 r ‾ 2 ∣ Φ ± ∣ 2 ) − ∂ u ( Ω ‾ 2 f 2 r ‾ 2 ∣ Φ ± ∣ 2 ) + ( − Ω ‾ 2 ( Ω ‾ 2 f r ‾ 2 ) ′ + Ω ‾ 2 q r ‾ 2 ) ∣ Φ ± ∣ 2 + E r r ± ± ℜ [ X Φ ± Φ ± ] . (135) \sum_{\pm}\left(\frac{\overline{\Omega}^{2}}{\overline{r}^{2}}+\mathrm{Err}^{\pm\pm}\right)\Re[X\Phi_{\pm}\Phi_{\pm}]=\sum_{\pm}\partial_v\left(\frac{\overline{\Omega}^{2}f}{2\overline{r}^{2}}|\Phi_{\pm}|^2\right)-\partial_u\left(\frac{\overline{\Omega}^{2}f}{2\overline{r}^{2}}|\Phi_{\pm}|^2\right)+\left(-\overline{\Omega}^{2}\left(\frac{\overline{\Omega}^{2}f}{\overline{r}^{2}}\right)'+\frac{\overline{\Omega}^{2}q}{\overline{r}^{2}}\right)|\Phi_{\pm}|^2+\mathrm{Err}^{\pm\pm}\Re[X\Phi_{\pm}\Phi_{\pm}]. \tag*{(135)} ± ∑ ( r 2 Ω 2 + Err ±± ) ℜ [ X Φ ± Φ ± ] = ± ∑ ∂ v ( 2 r 2 Ω 2 f ∣ Φ ± ∣ 2 ) − ∂ u ( 2 r 2 Ω 2 f ∣ Φ ± ∣ 2 ) + ( − Ω 2 ( r 2 Ω 2 f ) ′ + r 2 Ω 2 q ) ∣ Φ ± ∣ 2 + Err ±± ℜ [ X Φ ± Φ ± ] . (135) Regarding the last term, we compute
∑ ± ( Ω ‾ ( 3 M − r ˉ ) r ˉ 3 + E r r ∓ ± ) ℜ [ X ‾ Φ ± Φ ∓ ] = Ω ‾ ( 3 M − r ˉ ) r ˉ 3 [ f ℜ [ ∂ v Φ ‾ + Φ − − ∂ u Φ ‾ − ‾ Φ + ] + Ω r ˉ ( ∣ Φ − ∣ 2 + ∣ Φ + ∣ 2 ) + 2 q ℜ [ Φ ‾ − Φ + ] ] + E r r − + ℜ [ X ‾ Φ + Φ − ] + E r r − + ℜ [ X ‾ Φ − Φ + ] = ∂ v ( f Ω ‾ ( 3 M − r ˉ ) r ˉ 3 ℜ [ Φ ‾ + Φ − ] ) − ∂ u ( f Ω ‾ ( 3 M − r ˉ ) r ˉ 3 ℜ [ Φ ‾ + Φ − ] ) + 2 ( − Ω ‾ 2 ( f Ω ‾ ( 3 M − r ˉ ) r ˉ 3 ) ′ + q Ω ‾ ( 3 M − r ˉ ) r ˉ 3 ) ℜ [ Φ ‾ + Φ − ] + E r r − + ℜ [ X ‾ Φ + Φ − ] + E r r − + ℜ [ X ‾ Φ − Φ + ] , (136) \begin{aligned}
\sum_{\pm}\left(\frac{\overline{\Omega}(3M-\bar r)}{\bar r^3}+\mathcal{E}rr^{\mp\pm}\right)\Re[\overline{\mathsf{X}}_{\Phi_\pm}\Phi_{\mp}]
&= \frac{\overline{\Omega}(3M-\bar r)}{\bar r^3}\left[f\Re[\partial_v\overline{\Phi}_+\Phi_- - \overline{\partial_u\overline{\Phi}_-}\Phi_+] + \frac{\Omega}{\bar r}\left(|\Phi_-|^2+|\Phi_+|^2\right)+2q\Re[\overline{\Phi}_-\Phi_+]\right] \\
&\quad+ \mathcal{E}rr^{-+}\Re[\overline{\mathsf{X}}_{\Phi_+}\Phi_-]+\mathcal{E}rr^{-+}\Re[\overline{\mathsf{X}}_{\Phi_-}\Phi_+] \\
&= \partial_v\left(\frac{f\overline{\Omega}(3M-\bar r)}{\bar r^3}\Re[\overline{\Phi}_+\Phi_-]\right)-\partial_u\left(\frac{f\overline{\Omega}(3M-\bar r)}{\bar r^3}\Re[\overline{\Phi}_+\Phi_-]\right) \\
&\quad+2\left(-\overline{\Omega}^2\left(\frac{f\overline{\Omega}(3M-\bar r)}{\bar r^3}\right)' + \frac{q\overline{\Omega}(3M-\bar r)}{\bar r^3}\right)\Re[\overline{\Phi}_+\Phi_-] \\
&\quad+\mathcal{E}rr^{-+}\Re[\overline{\mathsf{X}}_{\Phi_+}\Phi_-]+\mathcal{E}rr^{-+}\Re[\overline{\mathsf{X}}_{\Phi_-}\Phi_+], \tag*{(136)}
\end{aligned} ± ∑ ( r ˉ 3 Ω ( 3 M − r ˉ ) + E r r ∓± ) ℜ [ X Φ ± Φ ∓ ] = r ˉ 3 Ω ( 3 M − r ˉ ) [ f ℜ [ ∂ v Φ + Φ − − ∂ u Φ − Φ + ] + r ˉ Ω ( ∣ Φ − ∣ 2 + ∣ Φ + ∣ 2 ) + 2 q ℜ [ Φ − Φ + ] ] + E r r −+ ℜ [ X Φ + Φ − ] + E r r −+ ℜ [ X Φ − Φ + ] = ∂ v ( r ˉ 3 f Ω ( 3 M − r ˉ ) ℜ [ Φ + Φ − ] ) − ∂ u ( r ˉ 3 f Ω ( 3 M − r ˉ ) ℜ [ Φ + Φ − ] ) + 2 ( − Ω 2 ( r ˉ 3 f Ω ( 3 M − r ˉ ) ) ′ + r ˉ 3 q Ω ( 3 M − r ˉ ) ) ℜ [ Φ + Φ − ] + E r r −+ ℜ [ X Φ + Φ − ] + E r r −+ ℜ [ X Φ − Φ + ] , (136) where we used the Dirac equations (20) –(21) to reduce the bulk to total derivatives terms plus one cross contribution ℜ [ Φ ‾ + Φ − ] \Re[\overline{\Phi}_+\Phi_-] ℜ [ Φ + Φ − ]
F : = 2 ( − Ω ‾ 2 ( f Ω ‾ ( 3 M − r ˉ ) r ˉ 3 ) ′ + q Ω ‾ ( 3 M − r ˉ ) r ˉ 3 ) , \mathcal{F}:=2\left(-\overline{\Omega}^2\left(\frac{f\overline{\Omega}(3M-\bar r)}{\bar r^3}\right)'+\frac{q\overline{\Omega}(3M-\bar r)}{\bar r^3}\right), F := 2 ( − Ω 2 ( r ˉ 3 f Ω ( 3 M − r ˉ ) ) ′ + r ˉ 3 q Ω ( 3 M − r ˉ ) ) , and writing
1 = r ˉ Ω ‾ Ω r + E r r , ∣ E r r ∣ ≲ A 3 E 3 / 2 τ − 1 + δ , 1=\frac{\bar r}{\overline{\Omega}}\frac{\Omega}{r}+\mathcal{E}rr,\qquad|\mathcal{E}rr|\lesssim A^3\mathcal{E}^{3/2}\tau^{-1+\delta}, 1 = Ω r ˉ r Ω + E r r , ∣ E r r ∣ ≲ A 3 E 3/2 τ − 1 + δ , we further rewrite the term ℜ [ Φ ‾ + Φ − ] \Re[\overline{\Phi}_+\Phi_-] ℜ [ Φ + Φ − ] (136) as
F ℜ [ Φ ‾ + Φ − ] = F ( r ˉ Ω ‾ Ω r + E r r ) ℜ [ Φ ‾ + Φ − ] = F r ˉ 2 Ω ‾ ( ℜ [ ∂ v Φ ‾ − Φ − ] − ℜ [ Φ ‾ + ∂ u Φ + ] ) + F E r r ℜ [ Φ ‾ + Φ − ] = ∂ v ( F r ˉ 4 Ω ‾ ∣ Φ − ∣ 2 ) − ∂ u ( F r ˉ 4 Ω ‾ ∣ Φ + ∣ 2 ) − ∂ v ( F r ˉ 4 Ω ‾ ) ∣ Φ − ∣ 2 + ∂ u ( F r ˉ 4 Ω ‾ ) ∣ Φ + ∣ 2 + F E r r ℜ [ Φ ‾ + Φ − ] = ∂ v ( F r ˉ 4 Ω ‾ ∣ Φ − ∣ 2 ) − ∂ u ( F r ˉ 4 Ω ‾ ∣ Φ + ∣ 2 ) − Ω ‾ 2 ( F r ˉ 4 Ω ‾ ) ′ ( ∣ Φ − ∣ 2 + ∣ Φ + ∣ 2 ) + F E r r ℜ [ Φ ‾ + Φ − ] . \begin{aligned}
\mathcal{F}\Re[\overline{\Phi}_+\Phi_-]
&=\mathcal{F}\left(\frac{\bar r}{\overline{\Omega}}\frac{\Omega}{r}+\mathcal{E}rr\right)\Re[\overline{\Phi}_+\Phi_-] \\
&=\frac{\mathcal{F}\bar r}{2\overline{\Omega}}\left(\Re[\overline{\partial_v\Phi}_-\Phi_-]-\Re[\overline{\Phi}_+\partial_u\Phi_+]\right)+\mathcal{F}\mathcal{E}rr\Re[\overline{\Phi}_+\Phi_-] \\
&=\partial_v\left(\frac{\mathcal{F}\bar r}{4\overline{\Omega}}|\Phi_-|^2\right)-\partial_u\left(\frac{\mathcal{F}\bar r}{4\overline{\Omega}}|\Phi_+|^2\right)-\partial_v\left(\frac{\mathcal{F}\bar r}{4\overline{\Omega}}\right)|\Phi_-|^2+\partial_u\left(\frac{\mathcal{F}\bar r}{4\overline{\Omega}}\right)|\Phi_+|^2+\mathcal{F}\mathcal{E}rr\Re[\overline{\Phi}_+\Phi_-] \\
&=\partial_v\left(\frac{\mathcal{F}\bar r}{4\overline{\Omega}}|\Phi_-|^2\right)-\partial_u\left(\frac{\mathcal{F}\bar r}{4\overline{\Omega}}|\Phi_+|^2\right)-\overline{\Omega}^2\left(\frac{\mathcal{F}\bar r}{4\overline{\Omega}}\right)'(|\Phi_-|^2+|\Phi_+|^2)+\mathcal{F}\mathcal{E}rr\Re[\overline{\Phi}_+\Phi_-].
\end{aligned} F ℜ [ Φ + Φ − ] = F ( Ω r ˉ r Ω + E r r ) ℜ [ Φ + Φ − ] = 2 Ω F r ˉ ( ℜ [ ∂ v Φ − Φ − ] − ℜ [ Φ + ∂ u Φ + ] ) + FE r r ℜ [ Φ + Φ − ] = ∂ v ( 4 Ω F r ˉ ∣ Φ − ∣ 2 ) − ∂ u ( 4 Ω F r ˉ ∣ Φ + ∣ 2 ) − ∂ v ( 4 Ω F r ˉ ) ∣ Φ − ∣ 2 + ∂ u ( 4 Ω F r ˉ ) ∣ Φ + ∣ 2 + FE r r ℜ [ Φ + Φ − ] = ∂ v ( 4 Ω F r ˉ ∣ Φ − ∣ 2 ) − ∂ u ( 4 Ω F r ˉ ∣ Φ + ∣ 2 ) − Ω 2 ( 4 Ω F r ˉ ) ′ ( ∣ Φ − ∣ 2 + ∣ Φ + ∣ 2 ) + FE r r ℜ [ Φ + Φ − ] . In this manner, we are able to exploit, also in the perturbed setting, Ma and Zhang’s trick to transform the mixed term ℜ [ X ‾ Φ ± Φ ∓ ] \Re[\overline{\mathsf{X}}_{\Phi_\pm}\Phi_{\mp}] ℜ [ X Φ ± Φ ∓ ] ∣ Φ ± ∣ 2 |\Phi_\pm|^2 ∣ Φ ± ∣ 2 (134) , (135) and (136) yields
Ω ‾ 2 f ′ 2 [ ( ∂ v Φ + ∣ 2 + Ω 2 r 2 ∣ Φ − ∣ 2 + 2 Ω r q Ω ‾ 2 f ′ ℜ [ Φ ‾ − ∂ v Φ + ] ) + ( ∣ ∂ u Φ − ∣ 2 + Ω 2 r 2 ∣ Φ + ∣ 2 − 2 Ω r q Ω ‾ 2 f ′ ℜ [ Φ ‾ + ∂ u Φ − ] ) ] + Ω ‾ 2 ( − ( Ω ‾ 2 q ′ ) ′ 2 − ( Ω ‾ 2 f r 2 ) ′ + q r 2 − ( F r ˉ 4 Ω ‾ ) ′ ) ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 ) + E r r s = − ∂ u F u − ∂ v F v , (137) \begin{aligned}
\frac{\overline{\Omega}^2f'}{2}\Bigg[\left(\partial_v\Phi_+|^2+\frac{\Omega^2}{r^2}|\Phi_-|^2+\frac{2\Omega}{r}\frac{q}{\overline{\Omega}^2f'}\Re[\overline{\Phi}_-\partial_v\Phi_+]\right)
+\left(|\partial_u\Phi_-|^2+\frac{\Omega^2}{r^2}|\Phi_+|^2-\frac{2\Omega}{r}\frac{q}{\overline{\Omega}^2f'}\Re[\overline{\Phi}_+\partial_u\Phi_-]\right)\Bigg] \\
+\overline{\Omega}^2\left(-\frac{(\overline{\Omega}^2q')'}{2}-\left(\frac{\overline{\Omega}^2f}{r^2}\right)'+\frac{q}{r^2}-\left(\frac{\mathcal{F}\bar r}{4\overline{\Omega}}\right)'\right)(|\Phi_+|^2+|\Phi_-|^2)+\mathcal{E}rr\mathord{\mathrm{s}}=-\partial_uF_u-\partial_vF_v, \tag*{(137)}
\end{aligned} 2 Ω 2 f ′ [ ( ∂ v Φ + ∣ 2 + r 2 Ω 2 ∣ Φ − ∣ 2 + r 2Ω Ω 2 f ′ q ℜ [ Φ − ∂ v Φ + ] ) + ( ∣ ∂ u Φ − ∣ 2 + r 2 Ω 2 ∣ Φ + ∣ 2 − r 2Ω Ω 2 f ′ q ℜ [ Φ + ∂ u Φ − ] ) ] + Ω 2 ( − 2 ( Ω 2 q ′ ) ′ − ( r 2 Ω 2 f ) ′ + r 2 q − ( 4 Ω F r ˉ ) ′ ) ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 ) + E r r s = − ∂ u F u − ∂ v F v , (137) where
F u : = f 2 Ω ‾ 2 ∣ ∂ v Φ + ∣ 2 + [ f 2 ( Ω 2 r ˉ 2 − Ω ‾ 2 r ˉ 2 ) + r ˉ Ω ‾ 2 ( f Ω ‾ ( 3 M − r ˉ ) r ˉ 3 ) ′ − q ( 3 M − r ˉ ) 2 r ˉ 2 ] ∣ Φ + ∣ 2 − Ω ‾ 2 2 ( q ′ + f r − 2 ) ∣ Φ − ∣ 2 + q ℜ [ Φ ‾ + ∂ v Φ + ] − f Ω ‾ ( 3 M − r ˉ ) r ˉ 3 ℜ [ Φ ‾ + Φ − ] , \begin{aligned}
F_u&:=\frac{f}{2\overline{\Omega}^2}|\partial_v\Phi_+|^2+\left[\frac{f}{2}\left(\frac{\Omega^2}{\bar r^2}-\frac{\overline{\Omega}^2}{\bar r^2}\right)+\frac{\bar r\overline{\Omega}}{2}\left(\frac{f\overline{\Omega}(3M-\bar r)}{\bar r^3}\right)'-\frac{q(3M-\bar r)}{2\bar r^2}\right]|\Phi_+|^2 \\
&\quad-\frac{\overline{\Omega}^2}{2}(q'+fr^{-2})|\Phi_-|^2+q\Re[\overline{\Phi}_+\partial_v\Phi_+]-\frac{f\overline{\Omega}(3M-\bar r)}{\bar r^3}\Re[\overline{\Phi}_+\Phi_-],
\end{aligned} F u := 2 Ω 2 f ∣ ∂ v Φ + ∣ 2 + [ 2 f ( r ˉ 2 Ω 2 − r ˉ 2 Ω 2 ) + 2 r ˉ Ω ( r ˉ 3 f Ω ( 3 M − r ˉ ) ) ′ − 2 r ˉ 2 q ( 3 M − r ˉ ) ] ∣ Φ + ∣ 2 − 2 Ω 2 ( q ′ + f r − 2 ) ∣ Φ − ∣ 2 + q ℜ [ Φ + ∂ v Φ + ] − r ˉ 3 f Ω ( 3 M − r ˉ ) ℜ [ Φ + Φ − ] , F v : = − f 2 ∣ ∂ u Φ − ∣ 2 + [ f 2 ( Ω ‾ 2 r ˉ 2 − Ω 2 r 2 ) − r ˉ Ω ‾ 2 ( f Ω ‾ ( 3 M − r ˉ ) r ˉ 3 ) ′ + q ( 3 M − r ˉ ) 2 r ˉ 2 ] ∣ Φ − ∣ 2 + Ω ‾ 2 2 ( q ′ + f r − 2 ) ∣ Φ + ∣ 2 + q ℜ [ Φ ‾ − ∂ u Φ − ] + f Ω ‾ ( 3 M − r ˉ ) r ˉ 3 ℜ [ Φ ‾ + Φ − ] , \begin{aligned}
F_v&:=-\frac{f}{2}|\partial_u\Phi_-|^2+\left[\frac{f}{2}\left(\frac{\overline{\Omega}^2}{\bar r^2}-\frac{\Omega^2}{r^2}\right)-\frac{\bar r\overline{\Omega}}{2}\left(\frac{f\overline{\Omega}(3M-\bar r)}{\bar r^3}\right)'+\frac{q(3M-\bar r)}{2\bar r^2}\right]|\Phi_-|^2 \\
&\quad+\frac{\overline{\Omega}^2}{2}(q'+fr^{-2})|\Phi_+|^2+q\Re[\overline{\Phi}_-\partial_u\Phi_-]+\frac{f\overline{\Omega}(3M-\bar r)}{\bar r^3}\Re[\overline{\Phi}_+\Phi_-],
\end{aligned} F v := − 2 f ∣ ∂ u Φ − ∣ 2 + [ 2 f ( r ˉ 2 Ω 2 − r 2 Ω 2 ) − 2 r ˉ Ω ( r ˉ 3 f Ω ( 3 M − r ˉ ) ) ′ + 2 r ˉ 2 q ( 3 M − r ˉ ) ] ∣ Φ − ∣ 2 + 2 Ω 2 ( q ′ + f r − 2 ) ∣ Φ + ∣ 2 + q ℜ [ Φ − ∂ u Φ − ] + r ˉ 3 f Ω ( 3 M − r ˉ ) ℜ [ Φ + Φ − ] , E r r o r s : = ∑ ± , ± ′ E r r ± ± ′ ℜ [ X ‾ Φ ± Φ ± ′ ] + F E r r ℜ [ Φ ‾ + Φ − ] . \mathcal{E}rrors:=\sum_{\pm,\pm'}\mathcal{E}rr^{\pm\pm'}\Re[\overline{\mathsf{X}}_{\Phi_\pm}\Phi_{\pm'}]+\mathcal{F}\mathcal{E}rr\Re[\overline{\Phi}_+\Phi_-]. E r r or s := ± , ± ′ ∑ E r r ± ± ′ ℜ [ X Φ ± Φ ± ′ ] + FE r r ℜ [ Φ + Φ − ] . By integrating over R \mathbb{R} R
f : = 1 − 3 M r ˉ , q : = Ω ‾ 2 f ′ = 3 M Ω ‾ 2 r ˉ 2 , f := 1-\frac{3M}{\bar{r}}, \qquad q := \overline{\Omega}^{2}f'=\frac{3M\overline{\Omega}^{2}}{\bar{r}^{2}}, f := 1 − r ˉ 3 M , q := Ω 2 f ′ = r ˉ 2 3 M Ω 2 , the LHS of (137) becomes
∬ R 3 M Ω ‾ 2 2 r ˉ 2 [ ∣ ∂ v Φ + − ∂ u Φ + ∣ 2 + ∣ ∂ v Φ − − ∂ u Φ − ∣ 2 ] + M Ω ‾ 2 r ˉ 6 ( 9 r ˉ 2 − 48 M r ˉ + 135 2 M 2 ) ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 ) d u d v , (138) \iint_{\mathcal{R}}\frac{3M\overline{\Omega}^{2}}{2\bar{r}^{2}}\left[|\partial_{v}\Phi_{+}-\partial_{u}\Phi_{+}|^{2}+|\partial_{v}\Phi_{-}-\partial_{u}\Phi_{-}|^{2}\right]+\frac{M\overline{\Omega}^{2}}{\bar{r}^{6}}\left(9\bar{r}^{2}-48M\bar{r}+\frac{135}{2}M^{2}\right)(|\Phi_{+}|^{2}+|\Phi_{-}|^{2})\,du\,dv, \tag*{(138)} ∬ R 2 r ˉ 2 3 M Ω 2 [ ∣ ∂ v Φ + − ∂ u Φ + ∣ 2 + ∣ ∂ v Φ − − ∂ u Φ − ∣ 2 ] + r ˉ 6 M Ω 2 ( 9 r ˉ 2 − 48 M r ˉ + 2 135 M 2 ) ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 ) d u d v , (138) which is strictly positive on D τ f D_{\tau_{f}} D τ f
∣ F u ∣ ≲ r − 2 ∣ Φ + ∣ 2 − ν r − 2 ∣ Φ − ∣ 2 + ∣ ∂ v Φ + ∣ 2 ≲ r − 2 ∣ ψ + ∣ 2 + ν 2 ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 , |F_{u}|\lesssim r^{-2}|\Phi_{+}|^{2}-\nu r^{-2}|\Phi_{-}|^{2}+|\partial_{v}\Phi_{+}|^{2}\lesssim r^{-2}|\psi_{+}|^{2}+\nu^{2}|\psi_{-}|^{2}+|\partial_{v}\psi_{+}|^{2}, ∣ F u ∣ ≲ r − 2 ∣ Φ + ∣ 2 − ν r − 2 ∣ Φ − ∣ 2 + ∣ ∂ v Φ + ∣ 2 ≲ r − 2 ∣ ψ + ∣ 2 + ν 2 ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 , ∣ F v ∣ ≲ Ω 2 r − 2 ∣ Φ + ∣ 2 + r − 2 ∣ Φ − ∣ 2 + ∣ ∂ u Φ − ∣ 2 ≲ − ν ˉ ( r − 2 ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + r 2 ∣ ∂ u ψ − ∣ 2 ) , |F_{v}|\lesssim\Omega^{2}r^{-2}|\Phi_{+}|^{2}+r^{-2}|\Phi_{-}|^{2}+|\partial_{u}\Phi_{-}|^{2}\lesssim-\bar{\nu}(r^{-2}|\psi_{+}|^{2}+|\psi_{-}|^{2}+r^{2}|\partial_{u}\psi_{-}|^{2}), ∣ F v ∣ ≲ Ω 2 r − 2 ∣ Φ + ∣ 2 + r − 2 ∣ Φ − ∣ 2 + ∣ ∂ u Φ − ∣ 2 ≲ − ν ˉ ( r − 2 ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + r 2 ∣ ∂ u ψ − ∣ 2 ) , and thus are bounded by the LHS of (125) .
On the other hand, by choosing
f : = 1 − 3 M r ˉ , q : = − Ω ‾ 2 f ′ = − 3 M Ω ‾ 2 r ˉ 2 , f := 1-\frac{3M}{\bar{r}}, \qquad q := -\overline{\Omega}^{2}f'=-\frac{3M\overline{\Omega}^{2}}{\bar{r}^{2}}, f := 1 − r ˉ 3 M , q := − Ω 2 f ′ = − r ˉ 2 3 M Ω 2 , the LHS of (137) becomes
∬ R 3 M Ω ‾ 2 2 r ˉ 2 [ ∣ ∂ v Φ + + ∂ u Φ + ∣ 2 + ∣ ∂ v Φ − + ∂ u Φ − ∣ 2 ] + C ( r ˉ ) ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 ) d u d v , (139) \iint_{\mathcal{R}}\frac{3M\overline{\Omega}^{2}}{2\bar{r}^{2}}\left[|\partial_{v}\Phi_{+}+\partial_{u}\Phi_{+}|^{2}+|\partial_{v}\Phi_{-}+\partial_{u}\Phi_{-}|^{2}\right]+C(\bar{r})(|\Phi_{+}|^{2}+|\Phi_{-}|^{2})\,du\,dv, \tag*{(139)} ∬ R 2 r ˉ 2 3 M Ω 2 [ ∣ ∂ v Φ + + ∂ u Φ + ∣ 2 + ∣ ∂ v Φ − + ∂ u Φ − ∣ 2 ] + C ( r ˉ ) ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 ) d u d v , (139) where
∣ C ( r ˉ ) ∣ = ∣ 3 M Ω ‾ 2 2 r ˉ 6 ( 20 r ˉ 2 − 144 M r ˉ + 225 M 2 ) ∣ ≲ Ω ‾ 2 r ˉ − 4 , |C(\bar{r})|=\left|\frac{3M\overline{\Omega}^{2}}{2\bar{r}^{6}}(20\bar{r}^{2}-144M\bar{r}+225M^{2})\right|\lesssim\overline{\Omega}^{2}\bar{r}^{-4}, ∣ C ( r ˉ ) ∣ = 2 r ˉ 6 3 M Ω 2 ( 20 r ˉ 2 − 144 M r ˉ + 225 M 2 ) ≲ Ω 2 r ˉ − 4 , with boundary terms analogous to those above. Thus, combining (138) and (139) , we obtain
∬ R Ω ‾ 2 r ˉ − 4 ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 + r ˉ 2 ∣ ∂ v Φ + ∣ 2 + r ˉ 2 ∣ ∂ u Φ − ∣ 2 ) d u d v + E r r o r s ≲ (LHS of (6.6)) . \iint_{\mathcal{R}}\overline{\Omega}^{2}\bar{r}^{-4}\left(|\Phi_{+}|^{2}+|\Phi_{-}|^{2}+\bar{r}^{2}|\partial_{v}\Phi_{+}|^{2}+\bar{r}^{2}|\partial_{u}\Phi_{-}|^{2}\right)\,du\,dv+\mathit{Errors}\lesssim\text{(LHS of (6.6))}. ∬ R Ω 2 r ˉ − 4 ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 + r ˉ 2 ∣ ∂ v Φ + ∣ 2 + r ˉ 2 ∣ ∂ u Φ − ∣ 2 ) d u d v + Errors ≲ (LHS of (6.6)) . We conclude the proof by briefly discussing the error terms E r r o r s \mathit{Errors} Errors E r r − + \mathit{Err}^{-+} Err −+
∣ E r r − + ℜ [ X ‾ Φ − Φ + ] ∣ ≲ ∣ E r r − + ∣ ( Ω r − 1 ∣ Φ + ∣ 2 + ∣ ∂ u Φ − ∣ ∣ Φ + ∣ + Ω 2 r − 2 ∣ Φ − ∣ ∣ Φ + ∣ ) |\mathit{Err}^{-+}\Re[\overline{X}\Phi_{-}\Phi_{+}]|\lesssim|\mathit{Err}^{-+}|(\Omega r^{-1}|\Phi_{+}|^{2}+|\partial_{u}\Phi_{-}||\Phi_{+}|+\Omega^{2}r^{-2}|\Phi_{-}||\Phi_{+}|) ∣ Err −+ ℜ [ X Φ − Φ + ] ∣ ≲ ∣ Err −+ ∣ ( Ω r − 1 ∣ Φ + ∣ 2 + ∣ ∂ u Φ − ∣∣ Φ + ∣ + Ω 2 r − 2 ∣ Φ − ∣∣ Φ + ∣ ) ≲ A 3 ε 3 / 2 ( 1 r ≤ Λ ( Ω 2 ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 ) + τ − 3 / 2 + 2 δ ∣ ∂ u Φ − ∣ 2 ) + 1 r ≥ Λ ( r − 4 ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 + r 2 ∣ ∂ u Φ − ∣ 2 ) + τ − 2 + 2 δ ∣ Φ + ∣ 2 ) ) . \lesssim A^{3}\varepsilon^{3/2}\left(\mathbb{1}_{r\leq\Lambda}\left(\Omega^{2}(|\Phi_{+}|^{2}+|\Phi_{-}|^{2})+\tau^{-3/2+2\delta}|\partial_{u}\Phi_{-}|^{2}\right)
+\mathbb{1}_{r\geq\Lambda}\left(r^{-4}(|\Phi_{+}|^{2}+|\Phi_{-}|^{2}+r^{2}|\partial_{u}\Phi_{-}|^{2})+\tau^{-2+2\delta}|\Phi_{+}|^{2}\right)\right). ≲ A 3 ε 3/2 ( 1 r ≤ Λ ( Ω 2 ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 ) + τ − 3/2 + 2 δ ∣ ∂ u Φ − ∣ 2 ) + 1 r ≥ Λ ( r − 4 ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 + r 2 ∣ ∂ u Φ − ∣ 2 ) + τ − 2 + 2 δ ∣ Φ + ∣ 2 ) ) . Noting that, with the choices f ∼ 1 f\sim1 f ∼ 1 q ∼ Ω ‾ 2 r ˉ − 2 q\sim\overline{\Omega}^{2}\bar{r}^{-2} q ∼ Ω 2 r ˉ − 2 F \mathcal{F} F E r r \mathit{Err} Err F ∼ Ω ‾ r ˉ − 3 \mathcal{F}\sim\overline{\Omega}\bar{r}^{-3} F ∼ Ω r ˉ − 3
∬ R E r r o r s d u d v ≲ A 3 ε 3 / 2 ∬ R ( Ω ‾ 2 r ˉ − 4 ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 + r 2 ∣ ∂ v Φ + ∣ 2 + r 2 ∣ ∂ u Φ − ∣ 2 ) + 1 r ≤ Λ τ − 3 / 2 + 2 δ ( ∣ ∂ u Φ − ∣ 2 + ∣ Φ − ∣ 2 ) + 1 r ≥ Λ τ − 2 + 2 δ ( ∣ Φ + ∣ 2 + ∣ ∂ v Φ + ∣ 2 + r − 2 ∣ Φ − ∣ 2 ) ) d u d v , \iint_{\mathcal{R}}\mathit{Errors}\,du\,dv\lesssim A^{3}\varepsilon^{3/2}\iint_{\mathcal{R}}\left(\overline{\Omega}^{2}\bar{r}^{-4}(|\Phi_{+}|^{2}+|\Phi_{-}|^{2}+r^{2}|\partial_{v}\Phi_{+}|^{2}+r^{2}|\partial_{u}\Phi_{-}|^{2})
+\mathbb{1}_{r\leq\Lambda}\tau^{-3/2+2\delta}(|\partial_{u}\Phi_{-}|^{2}+|\Phi_{-}|^{2})+\mathbb{1}_{r\geq\Lambda}\tau^{-2+2\delta}(|\Phi_{+}|^{2}+|\partial_{v}\Phi_{+}|^{2}+r^{-2}|\Phi_{-}|^{2})\right)\,du\,dv, ∬ R Errors d u d v ≲ A 3 ε 3/2 ∬ R ( Ω 2 r ˉ − 4 ( ∣ Φ + ∣ 2 + ∣ Φ − ∣ 2 + r 2 ∣ ∂ v Φ + ∣ 2 + r 2 ∣ ∂ u Φ − ∣ 2 ) + 1 r ≤ Λ τ − 3/2 + 2 δ ( ∣ ∂ u Φ − ∣ 2 + ∣ Φ − ∣ 2 ) + 1 r ≥ Λ τ − 2 + 2 δ ( ∣ Φ + ∣ 2 + ∣ ∂ v Φ + ∣ 2 + r − 2 ∣ Φ − ∣ 2 ) ) d u d v , where the first term can be absorbed into the main coercive bulk term for ε \varepsilon ε (45) –(46) :
A 3 ε 3 / 2 ∬ R ( 1 r ≤ Λ τ − 3 / 2 + 2 δ ( ∣ ∂ u Φ − ∣ 2 + ∣ Φ − ∣ 2 ) + 1 r ≥ Λ τ − 2 + 2 δ ( ∣ Φ + ∣ 2 + ∣ ∂ v Φ + ∣ 2 + r − 2 ∣ Φ − ∣ 2 ) ) d u d v A^{3}\varepsilon^{3/2}\iint_{\mathcal{R}}\left(\mathbb{1}_{r\leq\Lambda}\tau^{-3/2+2\delta}(|\partial_{u}\Phi_{-}|^{2}+|\Phi_{-}|^{2})+\mathbb{1}_{r\geq\Lambda}\tau^{-2+2\delta}(|\Phi_{+}|^{2}+|\partial_{v}\Phi_{+}|^{2}+r^{-2}|\Phi_{-}|^{2})\right)\,du\,dv A 3 ε 3/2 ∬ R ( 1 r ≤ Λ τ − 3/2 + 2 δ ( ∣ ∂ u Φ − ∣ 2 + ∣ Φ − ∣ 2 ) + 1 r ≥ Λ τ − 2 + 2 δ ( ∣ Φ + ∣ 2 + ∣ ∂ v Φ + ∣ 2 + r − 2 ∣ Φ − ∣ 2 ) ) d u d v ≲ A 3 ε 3 / 2 ( ∫ τ 1 τ 2 τ − 3 / 2 + 2 δ E ( τ ) d τ + ∫ τ 1 τ 2 τ − 2 + 2 δ E 0 ( τ ) d τ ) ≲ ε 3 τ 1 − 7 / 2 + 3 δ . \lesssim A^{3}\varepsilon^{3/2}\left(\int_{\tau_{1}}^{\tau_{2}}\tau^{-3/2+2\delta}\mathcal{E}(\tau)\,d\tau+\int_{\tau_{1}}^{\tau_{2}}\tau^{-2+2\delta}\mathcal{E}_{0}(\tau)\,d\tau\right)\lesssim\varepsilon^{3}\tau_{1}^{-7/2+3\delta}. ≲ A 3 ε 3/2 ( ∫ τ 1 τ 2 τ − 3/2 + 2 δ E ( τ ) d τ + ∫ τ 1 τ 2 τ − 2 + 2 δ E 0 ( τ ) d τ ) ≲ ε 3 τ 1 − 7/2 + 3 δ . Proof of Proposition 6.2. The result follows by adding (125) and (131) together. Indeed, for ε \varepsilon ε E r r T \mathit{Err}_{T} Err T (125) into the strictly coercive LHS of (131) .
Red-shift estimate near the horizon
As illustrated in the previous proposition, the ILED estimate (124) degenerates towards the event horizon. The following red-shift estimate allows us to remove this degeneracy.
Proposition 6.6. For any A ≥ 1 A \ge1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_f \in\mathcal{B} τ f ∈ B ( u 1 , v 1 ) , ( u 2 , v 2 ) ∈ Γ ∩ D τ f (u_1,v_1),(u_2,v_2) \in\Gamma\cap D_{\tau_f} ( u 1 , v 1 ) , ( u 2 , v 2 ) ∈ Γ ∩ D τ f u 2 ′ > u 2 u_2' > u_2 u 2 ′ > u 2 v 2 ′ > v 2 v_2' > v_2 v 2 ′ > v 2 ( u 2 , v 2 ) (u_2,v_2) ( u 2 , v 2 ) ( u 1 , v 1 ) (u_1,v_1) ( u 1 , v 1 ) r 1 : = 2.5 M r_1 := 2.5M r 1 := 2.5 M r 2 : = 2.7 M r_2 := 2.7M r 2 := 2.7 M
∫ V I ≤ r 1 ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v + ∫ V ≤ r 1 − ν ˉ ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 + ∣ ∂ u ψ − ∣ 2 ν ˉ 2 ) d u + ∫ R ≤ r 1 − ν ˉ ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 + ∣ ∂ v ψ + ∣ 2 + ∣ ∂ u ψ − ∣ 2 ν ˉ 2 ) d u d v \begin{aligned}
\int_{V_{I\le r_1}}\left(|\psi_-|^2+|\psi_+|^2+|\partial_v\psi_+|^2\right)\,dv+\int_{V_{\le r_1}}-\bar{\nu}\left(|\psi_-|^2+|\psi_+|^2+\frac{|\partial_u\psi_-|^2}{\bar{\nu}^2}\right)\,du \\
&+\int_{R_{\le r_1}}-\bar{\nu}\left(|\psi_-|^2+|\psi_+|^2+|\partial_v\psi_+|^2+\frac{|\partial_u\psi_-|^2}{\bar{\nu}^2}\right)\,du\,dv
\end{aligned} ∫ V I ≤ r 1 ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v + ∫ V ≤ r 1 − ν ˉ ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 + ν ˉ 2 ∣ ∂ u ψ − ∣ 2 ) d u + ∫ R ≤ r 1 − ν ˉ ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 + ∣ ∂ v ψ + ∣ 2 + ν ˉ 2 ∣ ∂ u ψ − ∣ 2 ) d u d v ≲ ∫ I − ν ˉ ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 + ∣ ∂ u ψ − ∣ 2 ν ˉ 2 ) d u + ∫ I I ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v + ε 3 τ 1 − 7 / 2 + 3 δ . (140) \lesssim\int_I-\bar{\nu}\left(|\psi_-|^2+|\psi_+|^2+\frac{|\partial_u\psi_-|^2}{\bar{\nu}^2}\right)\,du+\int_{II}\left(|\psi_+|^2+|\psi_-|^2+|\partial_v\psi_+|^2\right)\,dv+\varepsilon^3\tau_1^{-7/2+3\delta}. \tag*{(140)} ≲ ∫ I − ν ˉ ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 + ν ˉ 2 ∣ ∂ u ψ − ∣ 2 ) d u + ∫ I I ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v + ε 3 τ 1 − 7/2 + 3 δ . (140) The proof of this estimate will be given in the next section after a series of lemmas.
Proof of the red-shift estimate
To improve the ILED estimate, we exploit once again the coupled system satisfied by the spinorial components Φ ± \Phi_\pm Φ ± Φ ~ − \widetilde{\Phi}_- Φ − Φ − \Phi_- Φ − Φ ~ − \widetilde{\Phi}_- Φ − Φ + \Phi_+ Φ + Φ ~ − \widetilde{\Phi}_- Φ − Φ + \Phi_+ Φ + (141) below, we start by proving the following red-shift identities.
Lemma 6.7 (Near-horizon identities). Let O \mathcal{O} O O ‾ ⊆ D τ f \overline{\mathcal{O}} \subseteq D_{\tau_f} O ⊆ D τ f φ ∈ C 2 ( O ‾ ) \varphi\in C^2(\overline{\mathcal{O}}) φ ∈ C 2 ( O )
∂ u ∂ v φ + b u ∂ u φ + b 0 φ = 0 , (141) \partial_u\partial_v\varphi+b_u\partial_u\varphi+b_0\varphi=0, \tag*{(141)} ∂ u ∂ v φ + b u ∂ u φ + b 0 φ = 0 , (141) where the coefficients b u ∈ C 0 ( O ) b_u \in C^0(\mathcal{O}) b u ∈ C 0 ( O ) b 0 ∈ C 1 ( O ) b_0 \in C^1(\mathcal{O}) b 0 ∈ C 1 ( O ) F F F χ \chi χ C 1 C^1 C 1 ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) χ = χ ( r ) \chi=\chi(r) χ = χ ( r ) χ ′ = d d r χ ( r ) \chi'=\frac{d}{dr}\chi(r) χ ′ = d r d χ ( r )
∂ u ( χ F b 0 2 ( − ν ) ∣ φ ∣ 2 ) + ∂ v ( χ F 2 ( − ν ) ∣ ∂ u φ ∣ 2 ) + χ ∣ φ ∣ 2 2 ν [ b 0 ∂ u F + F ( ∂ u b 0 − b 0 ∂ u ν ν ) ] + [ F ( b u + ∂ v ν 2 ν ) − ∂ v F 2 ] χ ∣ ∂ u φ ∣ 2 − ν = − χ ′ F 2 ( b 0 ∣ φ ∣ 2 + λ ν ∣ ∂ u φ ∣ 2 ) , (142) \begin{aligned}
\partial_u\left(\frac{\chi F b_0}{2(-\nu)}|\varphi|^2\right)+\partial_v\left(\frac{\chi F}{2(-\nu)}|\partial_u\varphi|^2\right)+\frac{\chi|\varphi|^2}{2\nu}\left[b_0\partial_uF+F\left(\partial_ub_0-\frac{b_0\partial_u\nu}{\nu}\right)\right] \\
&+\left[F\left(b_u+\frac{\partial_v\nu}{2\nu}\right)-\frac{\partial_vF}{2}\right]\frac{\chi|\partial_u\varphi|^2}{-\nu}
=-\frac{\chi'F}{2}\left(b_0|\varphi|^2+\frac{\lambda}{\nu}|\partial_u\varphi|^2\right), \tag*{(142)}
\end{aligned} ∂ u ( 2 ( − ν ) χ F b 0 ∣ φ ∣ 2 ) + ∂ v ( 2 ( − ν ) χ F ∣ ∂ u φ ∣ 2 ) + 2 ν χ ∣ φ ∣ 2 [ b 0 ∂ u F + F ( ∂ u b 0 − ν b 0 ∂ u ν ) ] + [ F ( b u + 2 ν ∂ v ν ) − 2 ∂ v F ] − ν χ ∣ ∂ u φ ∣ 2 = − 2 χ ′ F ( b 0 ∣ φ ∣ 2 + ν λ ∣ ∂ u φ ∣ 2 ) , (142) and
∂ u ( χ r 2 2 ∣ ∂ v φ ∣ 2 ) + ∂ v ( χ r 2 b 0 2 ∣ φ ∣ 2 ) + χ ( − ν ) r ∣ ∂ v φ ∣ 2 + χ r 2 b u ℜ [ ∂ v φ ‾ ∂ u φ ] = − ∂ u ( χ r 2 b 0 2 ∣ φ ∣ 2 ) − ∂ v ( χ r 2 2 ∣ ∂ u φ ∣ 2 ) + ( ∂ v ( χ r 2 2 ) − χ r 2 b u ) ∣ ∂ u φ ∣ 2 + χ ′ ν r 2 2 ∣ ∂ v φ ∣ 2 + ( ∂ u + ∂ v ) ( χ r 2 b 0 2 ) ∣ φ ∣ 2 . (143) \begin{aligned}
\partial_u\left(\frac{\chi r^2}{2}|\partial_v\varphi|^2\right)+\partial_v\left(\frac{\chi r^2b_0}{2}|\varphi|^2\right)+\chi(-\nu)r|\partial_v\varphi|^2+\chi r^2b_u\Re[\overline{\partial_v\varphi}\partial_u\varphi] \\
&=-\partial_u\left(\frac{\chi r^2b_0}{2}|\varphi|^2\right)-\partial_v\left(\frac{\chi r^2}{2}|\partial_u\varphi|^2\right)+\left(\partial_v\left(\frac{\chi r^2}{2}\right)-\chi r^2b_u\right)|\partial_u\varphi|^2 \\
&+\chi'\frac{\nu r^2}{2}|\partial_v\varphi|^2+(\partial_u+\partial_v)\left(\frac{\chi r^2b_0}{2}\right)|\varphi|^2. \tag*{(143)}
\end{aligned} ∂ u ( 2 χ r 2 ∣ ∂ v φ ∣ 2 ) + ∂ v ( 2 χ r 2 b 0 ∣ φ ∣ 2 ) + χ ( − ν ) r ∣ ∂ v φ ∣ 2 + χ r 2 b u ℜ [ ∂ v φ ∂ u φ ] = − ∂ u ( 2 χ r 2 b 0 ∣ φ ∣ 2 ) − ∂ v ( 2 χ r 2 ∣ ∂ u φ ∣ 2 ) + ( ∂ v ( 2 χ r 2 ) − χ r 2 b u ) ∣ ∂ u φ ∣ 2 + χ ′ 2 ν r 2 ∣ ∂ v φ ∣ 2 + ( ∂ u + ∂ v ) ( 2 χ r 2 b 0 ) ∣ φ ∣ 2 . (143) Proof. We start by multiplying (141) by χ F − ν ∂ u φ ‾ \frac{\chi F}{-\nu}\overline{\partial_u\varphi} − ν χ F ∂ u φ
χ F − ν ℜ [ ∂ u φ ‾ ∂ v ∂ u φ ] = ∂ v ( χ F 2 ( − ν ) ∣ ∂ u φ ∣ 2 ) − ∂ v ( χ F 2 ( − ν ) ) ∣ ∂ u φ ∣ 2 , \frac{\chi F}{-\nu}\Re[\overline{\partial_u\varphi}\partial_v\partial_u\varphi]=\partial_v\left(\frac{\chi F}{2(-\nu)}|\partial_u\varphi|^2\right)-\partial_v\left(\frac{\chi F}{2(-\nu)}\right)|\partial_u\varphi|^2, − ν χ F ℜ [ ∂ u φ ∂ v ∂ u φ ] = ∂ v ( 2 ( − ν ) χ F ∣ ∂ u φ ∣ 2 ) − ∂ v ( 2 ( − ν ) χ F ) ∣ ∂ u φ ∣ 2 , χ F − ν ℜ [ ∂ u φ ‾ b u ∂ u φ ] = χ F b u − ν ∣ ∂ u φ ∣ 2 , \frac{\chi F}{-\nu}\Re[\overline{\partial_u\varphi}b_u\partial_u\varphi]=\frac{\chi F b_u}{-\nu}|\partial_u\varphi|^2, − ν χ F ℜ [ ∂ u φ b u ∂ u φ ] = − ν χ F b u ∣ ∂ u φ ∣ 2 , χ F − ν ℜ [ ∂ u φ ‾ b 0 φ ] = ∂ u ( χ F b 0 2 ( − ν ) ∣ φ ∣ 2 ) − ∂ u ( χ F b 0 2 ( − ν ) ) ∣ φ ∣ 2 . \frac{\chi F}{-\nu}\Re[\overline{\partial_u\varphi}b_0\varphi]=\partial_u\left(\frac{\chi F b_0}{2(-\nu)}|\varphi|^2\right)-\partial_u\left(\frac{\chi F b_0}{2(-\nu)}\right)|\varphi|^2. − ν χ F ℜ [ ∂ u φ b 0 φ ] = ∂ u ( 2 ( − ν ) χ F b 0 ∣ φ ∣ 2 ) − ∂ u ( 2 ( − ν ) χ F b 0 ) ∣ φ ∣ 2 . Expanding the last term of the first and third lines by the product rule and rearranging terms, we arrive at (142) .
The proof of (143) follows similarly by considering χ r 2 T φ ‾ = χ r 2 ( ∂ v φ ‾ + ∂ u φ ‾ ) \chi r^2\overline{T\varphi}=\chi r^2(\overline{\partial_v\varphi}+\overline{\partial_u\varphi}) χ r 2 T φ = χ r 2 ( ∂ v φ + ∂ u φ )
χ r 2 ℜ [ T φ ∂ v ∂ u φ ] = ∂ v ( χ r 2 2 ∣ ∂ u φ ∣ 2 ) + ∂ u ( χ r 2 2 ∣ ∂ v φ ∣ 2 ) − ∂ v ( χ r 2 2 ) ∣ ∂ u φ ∣ 2 − ∂ u ( χ r 2 2 ) ∣ ∂ v φ ∣ 2 , \chi r^2\Re[T\varphi\partial_v\partial_u\varphi]=\partial_v\left(\frac{\chi r^2}{2}|\partial_u\varphi|^2\right)+\partial_u\left(\frac{\chi r^2}{2}|\partial_v\varphi|^2\right)-\partial_v\left(\frac{\chi r^2}{2}\right)|\partial_u\varphi|^2-\partial_u\left(\frac{\chi r^2}{2}\right)|\partial_v\varphi|^2, χ r 2 ℜ [ T φ ∂ v ∂ u φ ] = ∂ v ( 2 χ r 2 ∣ ∂ u φ ∣ 2 ) + ∂ u ( 2 χ r 2 ∣ ∂ v φ ∣ 2 ) − ∂ v ( 2 χ r 2 ) ∣ ∂ u φ ∣ 2 − ∂ u ( 2 χ r 2 ) ∣ ∂ v φ ∣ 2 , χ r 2 ℜ [ T b u ‾ ∂ u φ ] = χ r 2 b u ( ∣ ∂ u φ ∣ 2 + ℜ [ ∂ v φ ‾ ∂ u φ ] ) , \chi r^2\Re[T\overline{b_u}\partial_u\varphi]=\chi r^2b_u\left(|\partial_u\varphi|^2+\Re[\overline{\partial_v\varphi}\partial_u\varphi]\right), χ r 2 ℜ [ T b u ∂ u φ ] = χ r 2 b u ( ∣ ∂ u φ ∣ 2 + ℜ [ ∂ v φ ∂ u φ ] ) , χ r 2 ℜ [ ∂ u φ ‾ b 0 φ ] = ∂ u ( χ r 2 b 0 2 ∣ φ ∣ 2 ) + ∂ v ( χ r 2 b 0 2 ∣ φ ∣ 2 ) − ( ∂ u + ∂ v ) ( χ r 2 b 0 2 ) ∣ φ ∣ 2 . \chi r^2\Re[\overline{\partial_u\varphi}b_0\varphi]=\partial_u\left(\frac{\chi r^2b_0}{2}|\varphi|^2\right)+\partial_v\left(\frac{\chi r^2b_0}{2}|\varphi|^2\right)-(\partial_u+\partial_v)\left(\frac{\chi r^2b_0}{2}\right)|\varphi|^2. χ r 2 ℜ [ ∂ u φ b 0 φ ] = ∂ u ( 2 χ r 2 b 0 ∣ φ ∣ 2 ) + ∂ v ( 2 χ r 2 b 0 ∣ φ ∣ 2 ) − ( ∂ u + ∂ v ) ( 2 χ r 2 b 0 ) ∣ φ ∣ 2 . Rearranging the terms yields (143) .
We are now in position to state the main ingredient for the proof of Proposition 6.6.
Lemma 6.8. With hypotheses as in Proposition 6.6, it holds that
∫ V I ≤ r 1 ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 ) d v + ∫ V ≤ r 1 ( − ν ‾ ∣ ψ − ∣ 2 + ∣ ∂ u ψ − ∣ 2 − ν ‾ ) d u \int_{\mathrm{VI}_{\le r_1}}(|\psi_-|^2+|\psi_+|^2)\,dv+\int_{\mathrm{V}_{\le r_1}}\left(-\overline{\nu}|\psi_-|^2+\frac{|\partial_u\psi_-|^2}{-\overline{\nu}}\right)\,du ∫ VI ≤ r 1 ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 ) d v + ∫ V ≤ r 1 ( − ν ∣ ψ − ∣ 2 + − ν ∣ ∂ u ψ − ∣ 2 ) d u + ∬ R ≤ r 1 ( − ν ‾ ∣ ψ − ∣ 2 − ν ‾ ∣ ψ + ∣ 2 + ∣ ∂ u ψ − ∣ 2 − ν ‾ ) d u d v (144) +\iint_{\mathrm{R}_{\le r_1}}\left(-\overline{\nu}|\psi_-|^2-\overline{\nu}|\psi_+|^2+\frac{|\partial_u\psi_-|^2}{-\overline{\nu}}\right)\,du\,dv \tag*{(144)} + ∬ R ≤ r 1 ( − ν ∣ ψ − ∣ 2 − ν ∣ ψ + ∣ 2 + − ν ∣ ∂ u ψ − ∣ 2 ) d u d v (144) ≲ ∫ I ( − ν ‾ ∣ ψ − ∣ 2 + ∣ ∂ u ψ − ∣ 2 − ν ‾ ) d u + ∬ R r 1 , r 2 ( − ν ‾ ∣ ψ − ∣ 2 − ν ‾ ∣ ψ + ∣ 2 + ∣ ∂ u ψ − ∣ 2 − ν ‾ ) d u d v + ε τ 1 − 7 / 2 + 3 δ . \lesssim\int_{\mathrm{I}}\left(-\overline{\nu}|\psi_-|^2+\frac{|\partial_u\psi_-|^2}{-\overline{\nu}}\right)\,du+\iint_{\mathrm{R}_{r_1,r_2}}\left(-\overline{\nu}|\psi_-|^2-\overline{\nu}|\psi_+|^2+\frac{|\partial_u\psi_-|^2}{-\overline{\nu}}\right)\,du\,dv+\varepsilon\tau_1^{-7/2+3\delta}. ≲ ∫ I ( − ν ∣ ψ − ∣ 2 + − ν ∣ ∂ u ψ − ∣ 2 ) d u + ∬ R r 1 , r 2 ( − ν ∣ ψ − ∣ 2 − ν ∣ ψ + ∣ 2 + − ν ∣ ∂ u ψ − ∣ 2 ) d u d v + ε τ 1 − 7/2 + 3 δ . Proof. Step 1. Let χ H + \chi_{\mathcal{H}^+} χ H + 0 ≤ χ ≤ 1 0\le\chi\le1 0 ≤ χ ≤ 1 1 1 1 { r ≤ r 1 } \{r\le r_1\} { r ≤ r 1 } 0 0 0 { r ≥ r 2 } \{r\ge r_2\} { r ≥ r 2 } { r = r 1 } \{r=r_1\} { r = r 1 } { r = r 2 } \{r=r_2\} { r = r 2 } D τ f D_{\tau_f} D τ f Lemma 4.8 , and noting that λ ∣ { r = r i } ∼ λ ‾ ∣ { r = r i } ∼ 1 \lambda|_{\{r=r_i\}}\sim\overline{\lambda}|_{\{r=r_i\}}\sim1 λ ∣ { r = r i } ∼ λ ∣ { r = r i } ∼ 1 (75) . Let us consider
Φ ~ − : = r Ω Φ − , \widetilde{\Phi}_-:=\frac{r}{\Omega}\Phi_-, Φ − := Ω r Φ − , which solves
∂ u ∂ v Φ ~ − + b u − ∂ u Φ ~ − + b 0 − Φ ~ − = 0 , \partial_u\partial_v\widetilde{\Phi}_-+b_u^-\partial_u\widetilde{\Phi}_-+b_0^-\widetilde{\Phi}_-=0, ∂ u ∂ v Φ − + b u − ∂ u Φ − + b 0 − Φ − = 0 , where
b u − : = ∂ v log Ω − λ r , b 0 − : = 6 Ω 2 m r 3 − 12 π T u v . b_u^-:=\partial_v\log\Omega-\frac{\lambda}{r},\qquad b_0^-:=\frac{6\Omega^2m}{r^3}-12\pi T_{uv}. b u − := ∂ v log Ω − r λ , b 0 − := r 3 6 Ω 2 m − 12 π T uv . Choosing χ = χ H + \chi=\chi_{\mathcal{H}^+} χ = χ H +
F : = 1 + C 0 ( 1 − μ ) , F:=1+C_0(1-\mu), F := 1 + C 0 ( 1 − μ ) , as in [61 Proposition 3.15] , with C 0 C_0 C 0 (142) and obtain
∂ u ( χ H + ( 1 + C 0 ( 1 − μ ) ) b 0 − 2 ( − ν ) ∣ Φ ~ − ∣ 2 ) + ∂ v ( χ H + ( 1 + C 0 ( 1 − μ ) ) 2 ( − ν ) ∣ ∂ u Φ ~ − ∣ 2 ) − χ H + ν ( B 0 − ∣ Φ ~ − ∣ 2 + B u − ∣ ∂ u Φ ~ − ∣ 2 ν 2 ) = − χ H + ′ ( 1 + C 0 ( 1 − μ ) ) 2 ( b 0 − ∣ Φ ~ − ∣ 2 + λ ν ∣ ∂ u Φ ~ − ∣ 2 ) , \begin{aligned}
\partial_u\left(\frac{\chi_{\mathcal{H}^+}(1+C_0(1-\mu))b_0^-}{2(-\nu)}|\widetilde{\Phi}_-|^2\right)+\partial_v\left(\frac{\chi_{\mathcal{H}^+}(1+C_0(1-\mu))}{2(-\nu)}|\partial_u\widetilde{\Phi}_-|^2\right)-\chi_{\mathcal{H}^+}\nu\left(\mathcal{B}_0^-|\widetilde{\Phi}_-|^2+\mathcal{B}_u^-\frac{|\partial_u\widetilde{\Phi}_-|^2}{\nu^2}\right) \\
&=-\frac{\chi_{\mathcal{H}^+}'(1+C_0(1-\mu))}{2}\left(b_0^-|\widetilde{\Phi}_-|^2+\frac{\lambda}{\nu}|\partial_u\widetilde{\Phi}_-|^2\right),
\end{aligned} ∂ u ( 2 ( − ν ) χ H + ( 1 + C 0 ( 1 − μ )) b 0 − ∣ Φ − ∣ 2 ) + ∂ v ( 2 ( − ν ) χ H + ( 1 + C 0 ( 1 − μ )) ∣ ∂ u Φ − ∣ 2 ) − χ H + ν ( B 0 − ∣ Φ − ∣ 2 + B u − ν 2 ∣ ∂ u Φ − ∣ 2 ) = − 2 χ H + ′ ( 1 + C 0 ( 1 − μ )) ( b 0 − ∣ Φ − ∣ 2 + ν λ ∣ ∂ u Φ − ∣ 2 ) , where
B 0 − : = B 0 , I − − B 0 , I I − : = C 0 b 0 − r ( − ν ) ( m r + ∂ u m − ν ) − 1 + C 0 ( 1 − μ ) 2 ν 2 ( ∂ u b 0 − − b 0 − ∂ v ν ν ) , \mathcal{B}_0^-:=\mathcal{B}_{0,\mathrm{I}}^--\mathcal{B}_{0,\mathrm{II}}^-:=\frac{C_0b_0^-}{r(-\nu)}\left(\frac{m}{r}+\frac{\partial_um}{-\nu}\right)-\frac{1+C_0(1-\mu)}{2\nu^2}\left(\partial_ub_0^--\frac{b_0^-\partial_v\nu}{\nu}\right), B 0 − := B 0 , I − − B 0 , II − := r ( − ν ) C 0 b 0 − ( r m + − ν ∂ u m ) − 2 ν 2 1 + C 0 ( 1 − μ ) ( ∂ u b 0 − − ν b 0 − ∂ v ν ) , B u − : = ( 1 + C 0 ( 1 − μ ) ) ( b u − + ∂ v ν 2 ν ) − C 0 r ( m λ r − ∂ v m ) . \mathcal{B}_u^-:=(1+C_0(1-\mu))\left(b_u^-+\frac{\partial_v\nu}{2\nu}\right)-\frac{C_0}{r}\left(\frac{m\lambda}{r}-\partial_vm\right). B u − := ( 1 + C 0 ( 1 − μ )) ( b u − + 2 ν ∂ v ν ) − r C 0 ( r mλ − ∂ v m ) . Let us start with the analysis of the boundary terms. We readily compute by (53) , (60) , (91) and the other geometric estimates,
χ H + ( 1 + C 0 ( 1 − μ ) ) b 0 − 2 ( − ν ) = χ H + ( 3 M r ˉ 3 ( 1 + C 0 ( 1 − μ ˉ ) ) + C 0 A 2 ε 3 / 2 τ − 2 + δ ) , \frac{\chi_{\mathcal{H}^{+}}(1+C_{0}(1-\mu))b_{0}^{-}}{2(-\nu)}
=\chi_{\mathcal{H}^{+}}\left(\frac{3M}{\bar{r}^{3}}(1+C_{0}(1-\bar{\mu}))+C_{0}A^{2}\varepsilon^{3/2}\tau^{-2+\delta}\right), 2 ( − ν ) χ H + ( 1 + C 0 ( 1 − μ )) b 0 − = χ H + ( r ˉ 3 3 M ( 1 + C 0 ( 1 − μ ˉ )) + C 0 A 2 ε 3/2 τ − 2 + δ ) , χ H + ( 1 + C 0 ( 1 − μ ) ) 2 ( − ν ) = χ H + − ν ( 1 2 ( 1 + C 0 ( 1 − μ ˉ ) ) + C 0 A 2 ε 3 / 2 τ − 2 + δ ) , (145) \frac{\chi_{\mathcal{H}^{+}}(1+C_{0}(1-\mu))}{2(-\nu)}
=\frac{\chi_{\mathcal{H}^{+}}}{-\nu}\left(\frac{1}{2}(1+C_{0}(1-\bar{\mu}))+C_{0}A^{2}\varepsilon^{3/2}\tau^{-2+\delta}\right), \tag*{(145)} 2 ( − ν ) χ H + ( 1 + C 0 ( 1 − μ )) = − ν χ H + ( 2 1 ( 1 + C 0 ( 1 − μ ˉ )) + C 0 A 2 ε 3/2 τ − 2 + δ ) , (145) which are both strictly coercive for ε \varepsilon ε
We now focus on the bulk coefficients. Regarding B u − \mathcal{B}_{u}^{-} B u − (92) , (112) , (115) , we have
B u − ∼ ( 1 + C 0 ( 1 − μ ˉ ) ) ( 4 M − r ˉ r ˉ 2 ) + C 0 A 2 ε 3 / 2 τ − 2 + δ − C 0 M λ ˉ r ˉ 2 + C 0 A ε 3 / 2 r ˉ − 3 / 2 + δ / 2 = 4 M − r ˉ r ˉ 2 + C 0 ( 1 − 2 M r ˉ ) 3 M − r ˉ r ˉ 2 + C 0 A 2 ε 3 / 2 τ − 3 / 2 + δ / 2 , (146) \begin{aligned}
\mathcal{B}_{u}^{-}\sim(1+C_{0}(1-\bar{\mu}))\left(\frac{4M-\bar{r}}{\bar{r}^{2}}\right)+C_{0}A^{2}\varepsilon^{3/2}\tau^{-2+\delta}-\frac{C_{0}M\bar{\lambda}}{\bar{r}^{2}}+C_{0}A\varepsilon^{3/2}\bar{r}^{-3/2+\delta/2} \\
&=\frac{4M-\bar{r}}{\bar{r}^{2}}+C_{0}\left(1-\frac{2M}{\bar{r}}\right)\frac{3M-\bar{r}}{\bar{r}^{2}}+C_{0}A^{2}\varepsilon^{3/2}\tau^{-3/2+\delta/2}, \tag*{(146)}
\end{aligned} B u − ∼ ( 1 + C 0 ( 1 − μ ˉ )) ( r ˉ 2 4 M − r ˉ ) + C 0 A 2 ε 3/2 τ − 2 + δ − r ˉ 2 C 0 M λ ˉ + C 0 A ε 3/2 r ˉ − 3/2 + δ /2 = r ˉ 2 4 M − r ˉ + C 0 ( 1 − r ˉ 2 M ) r ˉ 2 3 M − r ˉ + C 0 A 2 ε 3/2 τ − 3/2 + δ /2 , (146) which is strictly positive for r ≤ r 1 r\le r_{1} r ≤ r 1 ε \varepsilon ε B 0 − \mathcal{B}_{0}^{-} B 0 −
B 0 , I − ∼ 6 C 0 M 2 r ˉ 5 + C 0 A 2 ε 3 / 2 τ − 3 / 2 + δ / 2 , \mathcal{B}_{0,\mathrm{I}}^{-}\sim\frac{6C_{0}M^{2}}{\bar{r}^{5}}+C_{0}A^{2}\varepsilon^{3/2}\tau^{-3/2+\delta/2}, B 0 , I − ∼ r ˉ 5 6 C 0 M 2 + C 0 A 2 ε 3/2 τ − 3/2 + δ /2 , by (114) and the estimates established in the previous section; on the other hand,
B 0 , I I − = ( 1 + C 0 ( 1 − μ ) ) ( 9 κ m r ˉ 4 − 6 π ν 2 ( ∂ u T u v − ∂ u log Ω 2 T u v ) + 3 κ r ˉ 3 ∂ u m − ν − 12 π κ m T u v r ˉ 2 + 24 π 2 r T u v − ν T u v ν 2 ) ∼ ( 1 + C 0 ( 1 − μ ˉ ) ) 9 M r ˉ 4 + C 0 A 2 ε 3 / 2 τ − 3 / 2 + δ / 2 − 6 π ( 1 + C 0 ( 1 − μ ˉ ) ) [ ∂ u T u v − ∂ u log Ω 2 T u v ν 2 ] , \begin{aligned}
\mathcal{B}_{0,\mathrm{II}}^{-}=(1+C_{0}(1-\mu))\left(\frac{9\kappa m}{\bar{r}^{4}}-\frac{6\pi}{\nu^{2}}(\partial_{u}T_{uv}-\partial_{u}\log\Omega^{2}T_{uv})+\frac{3\kappa}{\bar{r}^{3}}\frac{\partial_{u}m}{-\nu}-\frac{12\pi\kappa mT_{uv}}{\bar{r}^{2}}+24\pi^{2}r\frac{T_{uv}}{-\nu}\frac{T_{uv}}{\nu^{2}}\right) \\
&\sim(1+C_{0}(1-\bar{\mu}))\frac{9M}{\bar{r}^{4}}+C_{0}A^{2}\varepsilon^{3/2}\tau^{-3/2+\delta/2}-6\pi(1+C_{0}(1-\bar{\mu}))\left[\frac{\partial_{u}T_{uv}-\partial_{u}\log\Omega^{2}T_{uv}}{\nu^{2}}\right],
\end{aligned} B 0 , II − = ( 1 + C 0 ( 1 − μ )) ( r ˉ 4 9 κm − ν 2 6 π ( ∂ u T uv − ∂ u log Ω 2 T uv ) + r ˉ 3 3 κ − ν ∂ u m − r ˉ 2 12 π κm T uv + 24 π 2 r − ν T uv ν 2 T uv ) ∼ ( 1 + C 0 ( 1 − μ ˉ )) r ˉ 4 9 M + C 0 A 2 ε 3/2 τ − 3/2 + δ /2 − 6 π ( 1 + C 0 ( 1 − μ ˉ )) [ ν 2 ∂ u T uv − ∂ u log Ω 2 T uv ] , by (91) , (106) , (114) . Computing explicitly the last term in square brackets, one finds that it is equal to
κ − ν ∂ u ( 1 2 π r 2 κ 1 / 2 ℑ [ ψ ‾ − ψ + ] ) ∼ ( − 1 + ∂ u κ − ν ) ℑ [ ψ ‾ − ψ + ] + ℑ [ ∂ u ψ ‾ − − ν ψ + ] , \frac{\kappa}{-\nu}\partial_{u}\left(\frac{1}{2\pi r^{2}\kappa^{1/2}}\Im[\overline{\psi}_{-}\psi_{+}]\right)
\sim\left(-1+\frac{\partial_{u}\kappa}{-\nu}\right)\Im[\overline{\psi}_{-}\psi_{+}]+\Im\left[\frac{\partial_{u}\overline{\psi}_{-}}{-\nu}\psi_{+}\right], − ν κ ∂ u ( 2 π r 2 κ 1/2 1 ℑ [ ψ − ψ + ] ) ∼ ( − 1 + − ν ∂ u κ ) ℑ [ ψ − ψ + ] + ℑ [ − ν ∂ u ψ − ψ + ] , which can be estimated thanks to Lemmas 5.1, 5.6 and 5.8 , yielding
B 0 , I I − ∼ ( 1 + C 0 ( 1 − μ ˉ ) ) 9 M r ˉ 4 + C 0 A 2 ε 3 / 2 τ − 3 / 2 + δ / 2 . \mathcal{B}_{0,\mathrm{II}}^{-}\sim(1+C_{0}(1-\bar{\mu}))\frac{9M}{\bar{r}^{4}}+C_{0}A^{2}\varepsilon^{3/2}\tau^{-3/2+\delta/2}. B 0 , II − ∼ ( 1 + C 0 ( 1 − μ ˉ )) r ˉ 4 9 M + C 0 A 2 ε 3/2 τ − 3/2 + δ /2 . Therefore, we find
B 0 − = B 0 , I − − B 0 , I I − ∼ 3 M r ˉ 5 ( C 0 ( 8 M − 3 r ˉ ) − 3 r ˉ ) + C 0 A 2 ε 3 / 2 τ − 3 / 2 + δ / 2 , (147) \mathcal{B}_{0}^{-}=\mathcal{B}_{0,\mathrm{I}}^{-}-\mathcal{B}_{0,\mathrm{II}}^{-}\sim\frac{3M}{\bar{r}^{5}}\left(C_{0}(8M-3\bar{r})-3\bar{r}\right)+C_{0}A^{2}\varepsilon^{3/2}\tau^{-3/2+\delta/2}, \tag*{(147)} B 0 − = B 0 , I − − B 0 , II − ∼ r ˉ 5 3 M ( C 0 ( 8 M − 3 r ˉ ) − 3 r ˉ ) + C 0 A 2 ε 3/2 τ − 3/2 + δ /2 , (147) which is strictly positive in { r ≤ r 1 } \{r\le r_{1}\} { r ≤ r 1 } C 0 C_{0} C 0 C 0 = 20 C_{0}=20 C 0 = 20 ε \varepsilon ε R \mathcal{R} R
∫ V I ≤ r 1 ∣ Φ ~ − ∣ 2 d v + ∫ V ≤ r 1 ∣ ∂ u Φ ~ − ∣ 2 − ν d u + ∬ R ≤ r 1 ( − ν ∣ Φ ~ − ∣ 2 + ∣ ∂ u Φ ~ − ∣ 2 − ν ) d u d v \int_{V_{\mathrm{I}\le r_{1}}}|\widetilde{\Phi}_{-}|^{2}\,dv+\int_{V_{\le r_{1}}}\frac{|\partial_{u}\widetilde{\Phi}_{-}|^{2}}{-\nu}\,du+\iint_{\mathcal{R}_{\le r_{1}}}\left(-\nu|\widetilde{\Phi}_{-}|^{2}+\frac{|\partial_{u}\widetilde{\Phi}_{-}|^{2}}{-\nu}\right)\,du\,dv ∫ V I ≤ r 1 ∣ Φ − ∣ 2 d v + ∫ V ≤ r 1 − ν ∣ ∂ u Φ − ∣ 2 d u + ∬ R ≤ r 1 ( − ν ∣ Φ − ∣ 2 + − ν ∣ ∂ u Φ − ∣ 2 ) d u d v ≲ ∫ I ∣ ∂ u Φ ~ − ∣ 2 − ν d u + ∬ R r 1 , r 2 ( − ν ∣ Φ ~ − ∣ 2 + ∣ ∂ u Φ ~ − ∣ 2 − ν ) d u d v + C 0 A 2 ε 3 / 2 E r r r e d , (148) \lesssim\int_{I}\frac{|\partial_{u}\widetilde{\Phi}_{-}|^{2}}{-\nu}\,du+\iint_{\mathcal{R}_{r_{1},r_{2}}}\left(-\nu|\widetilde{\Phi}_{-}|^{2}+\frac{|\partial_{u}\widetilde{\Phi}_{-}|^{2}}{-\nu}\right)\,du\,dv+C_{0}A^{2}\varepsilon^{3/2}\mathcal{E}rr^{\mathrm{red}}, \tag*{(148)} ≲ ∫ I − ν ∣ ∂ u Φ − ∣ 2 d u + ∬ R r 1 , r 2 ( − ν ∣ Φ − ∣ 2 + − ν ∣ ∂ u Φ − ∣ 2 ) d u d v + C 0 A 2 ε 3/2 E r r red , (148) where the error appearing in (145) –(146) –(147) is explicitly given by
E r r r e d : = ∬ R χ H + τ − 3 / 2 + δ / 2 ( − ν ∣ Φ ~ − ∣ 2 + ∣ ∂ u Φ ~ − ∣ 2 − ν ) d u d v + ∫ V I χ H + τ − 2 + δ ∣ Φ ~ − ∣ 2 d v + ∫ V χ H + τ − 2 + δ ∣ ∂ u Φ ~ − ∣ 2 d u + ∫ I χ H + τ − 2 + δ ∣ ∂ u Φ ~ − ∣ 2 d u , \begin{aligned}
\mathcal{E}rr^{\mathrm{red}}:=\iint_{\mathcal{R}}\chi_{\mathcal{H}^{+}}\tau^{-3/2+\delta/2}\left(-\nu|\widetilde{\Phi}_{-}|^{2}+\frac{|\partial_{u}\widetilde{\Phi}_{-}|^{2}}{-\nu}\right)\,du\,dv+\int_{V_{\mathrm{I}}}\chi_{\mathcal{H}^{+}}\tau^{-2+\delta}|\widetilde{\Phi}_{-}|^{2}\,dv \\
&+\int_{V}\chi_{\mathcal{H}^{+}}\tau^{-2+\delta}|\partial_{u}\widetilde{\Phi}_{-}|^{2}\,du+\int_{I}\chi_{\mathcal{H}^{+}}\tau^{-2+\delta}|\partial_{u}\widetilde{\Phi}_{-}|^{2}\,du,
\end{aligned} E r r red := ∬ R χ H + τ − 3/2 + δ /2 ( − ν ∣ Φ − ∣ 2 + − ν ∣ ∂ u Φ − ∣ 2 ) d u d v + ∫ V I χ H + τ − 2 + δ ∣ Φ − ∣ 2 d v + ∫ V χ H + τ − 2 + δ ∣ ∂ u Φ − ∣ 2 d u + ∫ I χ H + τ − 2 + δ ∣ ∂ u Φ − ∣ 2 d u , while the cutoff terms in χ ′ \chi' χ ′ { r 1 ≤ r ≤ r 2 } \{r_1 \le r \le r_2\} { r 1 ≤ r ≤ r 2 } (45) and (48) , we bound
∣ E r r e d ∣ ≲ ∬ R r ≤ r 2 τ − 3 / 2 + δ / 2 ( − ν ∣ ψ − ∣ 2 + ∣ ∂ u ψ − ∣ 2 − ν ) d u d v + ∫ V I ≤ r 2 τ − 2 + δ ∣ ψ − ∣ 2 d v + ∫ V I ≤ r 2 τ − 2 + δ ( − ν ∣ ψ − ∣ 2 + ∣ ∂ u ψ − ∣ 2 − ν ) d u + ∫ I r ≤ r 2 τ − 2 + δ ( − ν ∣ ψ − ∣ 2 + ∣ ∂ u ψ − ∣ 2 − ν ) d u ≲ ∫ τ 1 τ 2 τ − 3 / 2 + δ / 2 E ( τ ) d τ + τ 2 − 2 + δ F ( τ 2 ) + τ 2 − 2 + δ E ( τ 2 ) + τ 1 − 2 + δ E ( τ 1 ) ≲ A ε 2 τ 1 − 7 / 2 + 3 δ / 2 . \begin{aligned}
|\mathcal{E}^{\mathrm{rred}}| \lesssim{}& \iint_{R_{r\le r_2}} \tau^{-3/2+\delta/2}\left(-\nu|\psi_-|^2+\frac{|\partial_u\psi_-|^2}{-\nu}\right)\,du\,dv+\int_{V_{I\le r_2}}\tau^{-2+\delta}|\psi_-|^2\,dv\\
&+\int_{V_{I\le r_2}}\tau^{-2+\delta}\left(-\nu|\psi_-|^2+\frac{|\partial_u\psi_-|^2}{-\nu}\right)\,du+\int_{I_{r\le r_2}}\tau^{-2+\delta}\left(-\nu|\psi_-|^2+\frac{|\partial_u\psi_-|^2}{-\nu}\right)\,du\\
&\lesssim\int_{\tau_1}^{\tau_2}\tau^{-3/2+\delta/2}\mathcal{E}(\tau)\,d\tau+\tau_2^{-2+\delta}\mathcal{F}(\tau_2)+\tau_2^{-2+\delta}\mathcal{E}(\tau_2)+\tau_1^{-2+\delta}\mathcal{E}(\tau_1)\lesssim A\varepsilon^2\tau_1^{-7/2+3\delta/2}.
\end{aligned} ∣ E rred ∣ ≲ ∬ R r ≤ r 2 τ − 3/2 + δ /2 ( − ν ∣ ψ − ∣ 2 + − ν ∣ ∂ u ψ − ∣ 2 ) d u d v + ∫ V I ≤ r 2 τ − 2 + δ ∣ ψ − ∣ 2 d v + ∫ V I ≤ r 2 τ − 2 + δ ( − ν ∣ ψ − ∣ 2 + − ν ∣ ∂ u ψ − ∣ 2 ) d u + ∫ I r ≤ r 2 τ − 2 + δ ( − ν ∣ ψ − ∣ 2 + − ν ∣ ∂ u ψ − ∣ 2 ) d u ≲ ∫ τ 1 τ 2 τ − 3/2 + δ /2 E ( τ ) d τ + τ 2 − 2 + δ F ( τ 2 ) + τ 2 − 2 + δ E ( τ 2 ) + τ 1 − 2 + δ E ( τ 1 ) ≲ A ε 2 τ 1 − 7/2 + 3 δ /2 . Step 2. Let us now look at the component Φ + \Phi_+ Φ +
∂ u ∂ v Φ + + b u + ∂ u Φ + + b 0 + Φ + = 0 , (149) \partial_u\partial_v\Phi_+ + b_u^+\partial_u\Phi_+ + b_0^+\Phi_+ = 0, \tag*{(149)} ∂ u ∂ v Φ + + b u + ∂ u Φ + + b 0 + Φ + = 0 , (149) where
b u + : = − ∂ v log Ω + λ r , b 0 + : = Ω 2 r 2 . b_u^+ := -\partial_v\log\Omega+\frac{\lambda}{r},\qquad b_0^+ := \frac{\Omega^2}{r^2}. b u + := − ∂ v log Ω + r λ , b 0 + := r 2 Ω 2 . Relying again on (142) with χ = χ H + \chi=\chi_{\mathcal{H}+} χ = χ H + F : = 1 + C 0 ( 1 − μ ) F:=1+C_0(1-\mu) F := 1 + C 0 ( 1 − μ )
∂ u ( χ H + ( 1 + C 0 ( 1 − μ ) ) b 0 + 2 ( − ν ) ∣ Φ + ∣ 2 ) + ∂ v ( χ H + ( 1 + C 0 ( 1 − μ ) ) 2 ( − ν ) ∣ ∂ u Φ + ∣ 2 ) + χ H + ( − ν ) ( B 0 + ∣ Φ + ∣ 2 + B u + ∣ ∂ u Φ + ∣ 2 ν 2 ) = − χ H + ′ ( 1 + C 0 ( 1 − μ ) ) 2 ( b 0 + ∣ Φ + ∣ 2 + λ ν ∣ ∂ u Φ + ∣ 2 ) , \begin{aligned}
\partial_u\left(\frac{\chi_{\mathcal{H}+}(1+C_0(1-\mu))b_0^+}{2(-\nu)}|\Phi_+|^2\right)&+\partial_v\left(\frac{\chi_{\mathcal{H}+}(1+C_0(1-\mu))}{2(-\nu)}|\partial_u\Phi_+|^2\right)\\
&+\chi_{\mathcal{H}+}(-\nu)\left(B_0^+|\Phi_+|^2+B_u^+\frac{|\partial_u\Phi_+|^2}{\nu^2}\right)\\
&=-\frac{\chi_{\mathcal{H}+}'(1+C_0(1-\mu))}{2}\left(b_0^+|\Phi_+|^2+\frac{\lambda}{\nu}|\partial_u\Phi_+|^2\right),
\end{aligned} ∂ u ( 2 ( − ν ) χ H + ( 1 + C 0 ( 1 − μ )) b 0 + ∣ Φ + ∣ 2 ) + ∂ v ( 2 ( − ν ) χ H + ( 1 + C 0 ( 1 − μ )) ∣ ∂ u Φ + ∣ 2 ) + χ H + ( − ν ) ( B 0 + ∣ Φ + ∣ 2 + B u + ν 2 ∣ ∂ u Φ + ∣ 2 ) = − 2 χ H + ′ ( 1 + C 0 ( 1 − μ )) ( b 0 + ∣ Φ + ∣ 2 + ν λ ∣ ∂ u Φ + ∣ 2 ) , where
B 0 + : = B 0 , I + − B 0 , I I + : = C 0 b 0 + r ( − ν ) ( m r + ∂ u m − ν ) − 1 + C 0 ( 1 − μ ) 2 ν 2 ( ∂ u b 0 + + b 0 + ∂ u ν − ν ) , B_0^+ := B_{0,I}^+-B_{0,II}^+ := \frac{C_0b_0^+}{r(-\nu)}\left(\frac{m}{r}+\frac{\partial_um}{-\nu}\right)-\frac{1+C_0(1-\mu)}{2\nu^2}\left(\partial_ub_0^++\frac{b_0^+\partial_u\nu}{-\nu}\right), B 0 + := B 0 , I + − B 0 , I I + := r ( − ν ) C 0 b 0 + ( r m + − ν ∂ u m ) − 2 ν 2 1 + C 0 ( 1 − μ ) ( ∂ u b 0 + + − ν b 0 + ∂ u ν ) , B u + : = ( 1 + C 0 ( 1 − μ ) ) ( b u + + ∂ v ν 2 ν ) − C 0 r ( m λ r − ∂ v m ) . B_u^+ := (1+C_0(1-\mu))\left(b_u^++\frac{\partial_v\nu}{2\nu}\right)-\frac{C_0}{r}\left(\frac{m\lambda}{r}-\partial_vm\right). B u + := ( 1 + C 0 ( 1 − μ )) ( b u + + 2 ν ∂ v ν ) − r C 0 ( r mλ − ∂ v m ) . As before, the boundary terms rewrite
χ H + ( 1 + C 0 ( 1 − μ ) ) b 0 + 2 ( − ν ) = χ H + ( 1 + C 0 ( 1 − μ ˉ ) 2 r ˉ 2 + C 0 A 2 ε 3 / 2 τ − 2 + δ ) , \frac{\chi_{\mathcal{H}+}(1+C_0(1-\mu))b_0^+}{2(-\nu)}
=\chi_{\mathcal{H}+}\left(\frac{1+C_0(1-\bar\mu)}{2\bar r^2}+C_0A^2\varepsilon^{3/2}\tau^{-2+\delta}\right), 2 ( − ν ) χ H + ( 1 + C 0 ( 1 − μ )) b 0 + = χ H + ( 2 r ˉ 2 1 + C 0 ( 1 − μ ˉ ) + C 0 A 2 ε 3/2 τ − 2 + δ ) , χ H + ( 1 + C 0 ( 1 − μ ) ) 2 ( − ν ) = χ H + − ν ( 1 2 ( 1 + C 0 ( 1 − μ ˉ ) ) + C 0 A 2 ε 3 / 2 τ − 2 + δ ) . \frac{\chi_{\mathcal{H}+}(1+C_0(1-\mu))}{2(-\nu)}
=\frac{\chi_{\mathcal{H}+}}{-\nu}\left(\frac{1}{2}(1+C_0(1-\bar\mu))+C_0A^2\varepsilon^{3/2}\tau^{-2+\delta}\right). 2 ( − ν ) χ H + ( 1 + C 0 ( 1 − μ )) = − ν χ H + ( 2 1 ( 1 + C 0 ( 1 − μ ˉ )) + C 0 A 2 ε 3/2 τ − 2 + δ ) . Proceeding with the bulk terms, we have that
B u + ∼ ( 1 + C 0 ( 1 − μ ˉ ) ) ( r ˉ − 2 M r ˉ 2 ) + C 0 A 2 ε 3 / 2 τ − 2 + δ − C 0 M λ ˉ r ˉ 2 + C 0 A ε 3 / 2 r ˉ − 2 τ − 3 / 2 + δ / 2 = r ˉ − 2 M r ˉ 3 ( r ˉ + C 0 ( r ˉ − 3 M ) ) + C 0 A 2 ε 3 / 2 τ − 3 / 2 + δ / 2 , \begin{aligned}
B_u^+&\sim(1+C_0(1-\bar\mu))\left(\frac{\bar r-2M}{\bar r^2}\right)+C_0A^2\varepsilon^{3/2}\tau^{-2+\delta}-\frac{C_0M\bar\lambda}{\bar r^2}+C_0A\varepsilon^{3/2}\bar r^{-2}\tau^{-3/2+\delta/2}\\
&=\frac{\bar r-2M}{\bar r^3}\left(\bar r+C_0(\bar r-3M)\right)+C_0A^2\varepsilon^{3/2}\tau^{-3/2+\delta/2},
\end{aligned} B u + ∼ ( 1 + C 0 ( 1 − μ ˉ )) ( r ˉ 2 r ˉ − 2 M ) + C 0 A 2 ε 3/2 τ − 2 + δ − r ˉ 2 C 0 M λ ˉ + C 0 A ε 3/2 r ˉ − 2 τ − 3/2 + δ /2 = r ˉ 3 r ˉ − 2 M ( r ˉ + C 0 ( r ˉ − 3 M ) ) + C 0 A 2 ε 3/2 τ − 3/2 + δ /2 , which is degenerate towards the horizon. Therefore, we use the Dirac equation to rewrite ( − ν ) − 1 ∣ ∂ u Φ + ∣ 2 ∼ ( − ν ) ∣ ψ − ∣ 2 (-\nu)^{-1}|\partial_u\Phi_+|^2\sim(-\nu)|\psi_-|^2 ( − ν ) − 1 ∣ ∂ u Φ + ∣ 2 ∼ ( − ν ) ∣ ψ − ∣ 2 (148) . Regarding B 0 + B_0^+ B 0 +
B 0 , I + ∼ C 0 M r ˉ 4 + C 0 A 2 ε 3 / 2 τ − 3 / 2 + δ / 2 ; B_{0,I}^+\sim\frac{C_0M}{\bar r^4}+C_0A^2\varepsilon^{3/2}\tau^{-3/2+\delta/2}; B 0 , I + ∼ r ˉ 4 C 0 M + C 0 A 2 ε 3/2 τ − 3/2 + δ /2 ; B 0 , I I + = ( 1 + C 0 ( 1 − μ ) ) ( κ r ˉ 3 − 2 π κ T u u ν 2 ) ∼ 1 + C 0 ( 1 − μ ˉ ) r ˉ 3 + C 0 A 2 ε 3 / 2 τ − 3 / 2 + δ / 2 . B_{0,II}^+=(1+C_0(1-\mu))\left(\frac{\kappa}{\bar r^3}-\frac{2\pi\kappa T_{uu}}{\nu^2}\right)\sim\frac{1+C_0(1-\bar\mu)}{\bar r^3}+C_0A^2\varepsilon^{3/2}\tau^{-3/2+\delta/2}. B 0 , I I + = ( 1 + C 0 ( 1 − μ )) ( r ˉ 3 κ − ν 2 2 π κ T uu ) ∼ r ˉ 3 1 + C 0 ( 1 − μ ˉ ) + C 0 A 2 ε 3/2 τ − 3/2 + δ /2 . Thus, we find
B 0 + = B 0 , I + − B 0 , I I + ∼ 1 r ˉ 4 ( C 0 ( 3 M − r ˉ ) − r ˉ ) + C 0 A 2 ε 3 / 2 τ − 3 / 2 + δ / 2 . B_0^+=B_{0,I}^+-B_{0,II}^+\sim\frac{1}{\bar r^4}\left(C_0(3M-\bar r)-\bar r\right)+C_0A^2\varepsilon^{3/2}\tau^{-3/2+\delta/2}. B 0 + = B 0 , I + − B 0 , I I + ∼ r ˉ 4 1 ( C 0 ( 3 M − r ˉ ) − r ˉ ) + C 0 A 2 ε 3/2 τ − 3/2 + δ /2 . which is strictly positive in { r ≤ r 1 } \{r \le r_{1}\} { r ≤ r 1 } C 0 C_{0} C 0 C 0 = 20 C_{0}=20 C 0 = 20 ε \varepsilon ε R \mathcal{R} R
∫ V I ≤ r 1 ∣ Φ + ∣ 2 d v + ∫ V ≤ r 1 − ν ∣ Φ ~ − ∣ 2 d u + ∬ R ≤ r 1 − ν ∣ Φ + ∣ 2 d u d v ≲ ∬ R r 1 , r 2 − ν ∣ Φ + ∣ 2 d u d v + ( R H S o f x r e f ‘ ( 6.29 ) ‘ ) + ε 3 τ 1 − 7 / 2 + 3 / 2 δ . (150) \int_{V_{I}\le r_{1}}|\Phi_{+}|^{2}\,dv+\int_{V\le r_{1}}-\nu|\widetilde{\Phi}_{-}|^{2}\,du+\iint_{\mathcal{R}\le r_{1}}-\nu|\Phi_{+}|^{2}\,dudv
\lesssim\iint_{\mathcal{R}_{r_{1},r_{2}}}-\nu|\Phi_{+}|^{2}\,dudv+(\mathrm{RHS\ of\ }{xref}`(6.29)`)+\varepsilon^{3}\tau_{1}^{-7/2+3/2\delta}. \tag*{(150)} ∫ V I ≤ r 1 ∣ Φ + ∣ 2 d v + ∫ V ≤ r 1 − ν ∣ Φ − ∣ 2 d u + ∬ R ≤ r 1 − ν ∣ Φ + ∣ 2 d u d v ≲ ∬ R r 1 , r 2 − ν ∣ Φ + ∣ 2 d u d v + ( RHS of x r e f ‘ ( 6.29 ) ‘ ) + ε 3 τ 1 − 7/2 + 3/2 δ . (150) where the error terms are treated as in the previous step.
Finally, adding (148) and (150) , the proof is completed. Indeed, recalling that Φ ~ − = r 2 κ − 1 / 2 ψ − \widetilde{\Phi}_{-}=r^{2}\kappa^{-1/2}\psi_{-} Φ − = r 2 κ − 1/2 ψ − Φ + = κ 1 / 2 ψ + \Phi_{+}=\kappa^{1/2}\psi_{+} Φ + = κ 1/2 ψ + R ≤ Λ \mathcal{R}_{\le\Lambda} R ≤ Λ
∣ Φ ~ − ∣ ∼ ∣ ψ − ∣ , ∣ ∂ u Φ ~ − ∣ ∼ ∣ ψ − ∣ + ∣ ∂ u ψ − ∣ , ∣ Φ + ∣ ∼ ∣ ψ + ∣ , |\widetilde{\Phi}_{-}|\sim|\psi_{-}|,\qquad|\partial_{u}\widetilde{\Phi}_{-}|\sim|\psi_{-}|+|\partial_{u}\psi_{-}|,\qquad|\Phi_{+}|\sim|\psi_{+}|, ∣ Φ − ∣ ∼ ∣ ψ − ∣ , ∣ ∂ u Φ − ∣ ∼ ∣ ψ − ∣ + ∣ ∂ u ψ − ∣ , ∣ Φ + ∣ ∼ ∣ ψ + ∣ , thanks to the geometric estimates.
The next lemma provides control of the remaining terms involving ψ + \psi_{+} ψ + (143) , thereby completing the proof of Proposition 6.6.
Lemma 6.9. With hypotheses as in Proposition 6.6, it holds that
∫ V I ≤ r 1 ∣ ∂ v ψ + ∣ 2 d v + ∫ V ≤ r 1 − ν ‾ ∣ ψ + ∣ 2 d u + ∬ R ≤ r 1 − ν ‾ ∣ ∂ v ψ + ∣ 2 d u d v ≲ ∫ I − ν ‾ ∣ ψ + ∣ 2 d u + ∬ R r 1 , r 2 − ν ‾ ∣ ∂ v ψ + ∣ 2 d u d v + ( R H S o f x r e f ‘ ( 6.25 ) ‘ ) . (151) \int_{V_{I}\le r_{1}}|\partial_{v}\psi_{+}|^{2}\,dv+\int_{V\le r_{1}}-\overline{\nu}|\psi_{+}|^{2}\,du+\iint_{\mathcal{R}\le r_{1}}-\overline{\nu}|\partial_{v}\psi_{+}|^{2}\,dudv
\lesssim\int_{I}-\overline{\nu}|\psi_{+}|^{2}\,du+\iint_{\mathcal{R}_{r_{1},r_{2}}}-\overline{\nu}|\partial_{v}\psi_{+}|^{2}\,dudv+(\mathrm{RHS\ of\ }{xref}`(6.25)`). \tag*{(151)} ∫ V I ≤ r 1 ∣ ∂ v ψ + ∣ 2 d v + ∫ V ≤ r 1 − ν ∣ ψ + ∣ 2 d u + ∬ R ≤ r 1 − ν ∣ ∂ v ψ + ∣ 2 d u d v ≲ ∫ I − ν ∣ ψ + ∣ 2 d u + ∬ R r 1 , r 2 − ν ∣ ∂ v ψ + ∣ 2 d u d v + ( RHS of x r e f ‘ ( 6.25 ) ‘ ) . (151) Proof. Since Φ + \Phi_{+} Φ + (149) , we apply identity (143) with χ H + \chi_{\mathcal{H}+} χ H +
∂ u ( χ H + r 2 2 ∣ ∂ v Φ + ∣ 2 ) + ∂ v ( χ H + + r 2 b 0 + 2 ∣ Φ + ∣ 2 ) + χ H + ( − ν ) r ∣ ∂ v Φ + ∣ 2 + χ H + r 2 b u + R [ ∂ v Φ + ∂ u Φ + ] \partial_{u}\left(\frac{\chi_{\mathcal{H}+}r^{2}}{2}|\partial_{v}\Phi_{+}|^{2}\right)+\partial_{v}\left(\frac{\chi_{\mathcal{H}+}+r^{2}b_{0}^{+}}{2}|\Phi_{+}|^{2}\right)+\chi_{\mathcal{H}+}(-\nu)r|\partial_{v}\Phi_{+}|^{2}+\chi_{\mathcal{H}+}r^{2}b_{u}^{+}\mathfrak{R}[\partial_{v}\Phi_{+}\partial_{u}\Phi_{+}] ∂ u ( 2 χ H + r 2 ∣ ∂ v Φ + ∣ 2 ) + ∂ v ( 2 χ H + + r 2 b 0 + ∣ Φ + ∣ 2 ) + χ H + ( − ν ) r ∣ ∂ v Φ + ∣ 2 + χ H + r 2 b u + R [ ∂ v Φ + ∂ u Φ + ] = − ∂ u ( χ H + + r 2 b 0 + 2 ∣ Φ + ∣ 2 ) − ∂ v ( χ H + r 2 2 ∣ ∂ u Φ + ∣ 2 ) + ( ∂ v ( χ H + r 2 2 ) − χ H + r 2 b u + ) ∣ ∂ u Φ + ∣ 2 + χ H + ν r 2 2 ∣ ∂ v Φ + ∣ 2 + ( ∂ u + ∂ v ) ( χ H + + r 2 b 0 + 2 ) ∣ Φ + ∣ 2 . =-\partial_{u}\left(\frac{\chi_{\mathcal{H}+}+r^{2}b_{0}^{+}}{2}|\Phi_{+}|^{2}\right)-\partial_{v}\left(\frac{\chi_{\mathcal{H}+}r^{2}}{2}|\partial_{u}\Phi_{+}|^{2}\right)+\left(\partial_{v}\left(\frac{\chi_{\mathcal{H}+}r^{2}}{2}\right)-\chi_{\mathcal{H}+}r^{2}b_{u}^{+}\right)|\partial_{u}\Phi_{+}|^{2}+\chi_{\mathcal{H}+}\frac{\nu r^{2}}{2}|\partial_{v}\Phi_{+}|^{2}+(\partial_{u}+\partial_{v})\left(\frac{\chi_{\mathcal{H}+}+r^{2}b_{0}^{+}}{2}\right)|\Phi_{+}|^{2}. = − ∂ u ( 2 χ H + + r 2 b 0 + ∣ Φ + ∣ 2 ) − ∂ v ( 2 χ H + r 2 ∣ ∂ u Φ + ∣ 2 ) + ( ∂ v ( 2 χ H + r 2 ) − χ H + r 2 b u + ) ∣ ∂ u Φ + ∣ 2 + χ H + 2 ν r 2 ∣ ∂ v Φ + ∣ 2 + ( ∂ u + ∂ v ) ( 2 χ H + + r 2 b 0 + ) ∣ Φ + ∣ 2 . Let us start with the analysis of the LHS. Combining the flux terms with (144) , we readily obtain
χ H + ( r 2 2 ∣ ∂ v Φ + ∣ 2 + ∣ ψ + ∣ 2 ) ≳ χ H + ∣ ∂ v ψ + ∣ 2 , χ H + + r 2 b 0 + 2 ∣ Φ + ∣ 2 ∼ χ H + ( − ν ‾ ) ∣ ψ + ∣ 2 . \chi_{\mathcal{H}+}\left(\frac{r^{2}}{2}|\partial_{v}\Phi_{+}|^{2}+|\psi_{+}|^{2}\right)\gtrsim\chi_{\mathcal{H}+}|\partial_{v}\psi_{+}|^{2},\qquad\frac{\chi_{\mathcal{H}+}+r^{2}b_{0}^{+}}{2}|\Phi_{+}|^{2}\sim\chi_{\mathcal{H}+}(-\overline{\nu})|\psi_{+}|^{2}. χ H + ( 2 r 2 ∣ ∂ v Φ + ∣ 2 + ∣ ψ + ∣ 2 ) ≳ χ H + ∣ ∂ v ψ + ∣ 2 , 2 χ H + + r 2 b 0 + ∣ Φ + ∣ 2 ∼ χ H + ( − ν ) ∣ ψ + ∣ 2 . Proceeding analogously with the bulk term ∣ ∂ v Φ + ∣ 2 |\partial_{v}\Phi_{+}|^{2} ∣ ∂ v Φ + ∣ 2 ∣ ψ + ∣ |\psi_{+}| ∣ ψ + ∣ (144) , to estimate
χ H + ( − ν ) ( ∣ ∂ v Φ + ∣ 2 + ∣ ψ + ∣ 2 ) ≳ χ H + ( − ν ‾ ) ∣ ∂ v ψ + ∣ 2 . \chi_{\mathcal{H}+}(-\nu)(|\partial_{v}\Phi_{+}|^{2}+|\psi_{+}|^{2})\gtrsim\chi_{\mathcal{H}+}(-\overline{\nu})|\partial_{v}\psi_{+}|^{2}. χ H + ( − ν ) ( ∣ ∂ v Φ + ∣ 2 + ∣ ψ + ∣ 2 ) ≳ χ H + ( − ν ) ∣ ∂ v ψ + ∣ 2 . On the other hand, we split the cross term R [ ∂ v Φ + ∂ u Φ + ] = − Ω r − 1 R [ ∂ v Φ + Φ − ] \mathfrak{R}[\partial_{v}\Phi_{+}\partial_{u}\Phi_{+}]=-\Omega r^{-1}\mathfrak{R}[\partial_{v}\Phi_{+}\Phi_{-}] R [ ∂ v Φ + ∂ u Φ + ] = − Ω r − 1 R [ ∂ v Φ + Φ − ]
∣ χ H + r Ω ( ∂ v log Ω − λ r ) R [ ∂ v Φ + Φ − ] ∣ ≲ χ H + ( − ν ‾ ) ( ε 2 ∣ ∂ v ψ + ∣ 2 + r 4 2 ε ∣ ∂ v log Ω − λ r ∣ 2 ∣ ψ − ∣ 2 ) \left|\chi_{\mathcal{H}+}r\Omega\left(\partial_{v}\log\Omega-\frac{\lambda}{r}\right)\mathfrak{R}[\partial_{v}\Phi_{+}\Phi_{-}]\right|\lesssim\chi_{\mathcal{H}+}(-\overline{\nu})\left(\frac{\varepsilon}{2}|\partial_{v}\psi_{+}|^{2}+\frac{r^{4}}{2\varepsilon}|\partial_{v}\log\Omega-\frac{\lambda}{r}|^{2}|\psi_{-}|^{2}\right) χ H + r Ω ( ∂ v log Ω − r λ ) R [ ∂ v Φ + Φ − ] ≲ χ H + ( − ν ) ( 2 ε ∣ ∂ v ψ + ∣ 2 + 2 ε r 4 ∣ ∂ v log Ω − r λ ∣ 2 ∣ ψ − ∣ 2 ) ≲ χ H + ( − ν ‾ ) ( ε 2 ∣ ∂ v ψ + ∣ 2 + 1 2 ε ∣ ψ − ∣ 2 ) , \lesssim\chi_{\mathcal{H}+}(-\overline{\nu})\left(\frac{\varepsilon}{2}|\partial_{v}\psi_{+}|^{2}+\frac{1}{2\varepsilon}|\psi_{-}|^{2}\right), ≲ χ H + ( − ν ) ( 2 ε ∣ ∂ v ψ + ∣ 2 + 2 ε 1 ∣ ψ − ∣ 2 ) , so that the first summand can be absorbed on the LHS, while the second can be controlled by the RHS of (144) . Regarding the right-hand side, the bulk ∣ ∂ u Φ + ∣ ∼ ( − ν ) ∣ ψ − ∣ |\partial_{u}\Phi_{+}|\sim(-\nu)|\psi_{-}| ∣ ∂ u Φ + ∣ ∼ ( − ν ) ∣ ψ − ∣
Proof of Proposition 6.6. Combining (144) and (151) , one obtains
∫ V I ≤ r 1 ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v + ∫ V ≤ r 1 ( − ν ˉ ∣ ψ − ∣ 2 − ν ˉ ∣ ψ + ∣ 2 + ∣ ∂ u ψ − ∣ 2 − ν ˉ ) d u + ∬ R ≤ r 1 ( − ν ˉ ∣ ψ − ∣ 2 − ν ˉ ∣ ψ + ∣ 2 − ν ˉ ∣ ∂ v ψ + ∣ 2 + ∣ ∂ u ψ − ∣ 2 − ν ˉ ) d u d v \begin{aligned}
\int_{\mathrm{VI}\le r_1}\left(|\psi_-|^2+|\psi_+|^2+|\partial_v\psi_+|^2\right)\,dv+\int_{\mathrm{V}\le r_1}\left(-\bar{\nu}|\psi_-|^2-\bar{\nu}|\psi_+|^2+\frac{|\partial_u\psi_-|^2}{-\bar{\nu}}\right)\,du \\
&+\iint_{\mathcal{R}\le r_1}\left(-\bar{\nu}|\psi_-|^2-\bar{\nu}|\psi_+|^2-\bar{\nu}|\partial_v\psi_+|^2+\frac{|\partial_u\psi_-|^2}{-\bar{\nu}}\right)\,du\,dv
\end{aligned} ∫ VI ≤ r 1 ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 + ∣ ∂ v ψ + ∣ 2 ) d v + ∫ V ≤ r 1 ( − ν ˉ ∣ ψ − ∣ 2 − ν ˉ ∣ ψ + ∣ 2 + − ν ˉ ∣ ∂ u ψ − ∣ 2 ) d u + ∬ R ≤ r 1 ( − ν ˉ ∣ ψ − ∣ 2 − ν ˉ ∣ ψ + ∣ 2 − ν ˉ ∣ ∂ v ψ + ∣ 2 + − ν ˉ ∣ ∂ u ψ − ∣ 2 ) d u d v ≲ ∫ I ( − ν ˉ ∣ ψ − ∣ 2 − ν ˉ ∣ ψ + ∣ 2 + ∣ ∂ u ψ − ∣ 2 − ν ˉ ) d u + ∬ R r 1 , r 2 ( − ν ˉ ∣ ψ − ∣ 2 − ν ˉ ∣ ψ + ∣ 2 + ∣ ∂ u ψ − ∣ 2 − ν ˉ − ν ˉ ∣ ∂ v ψ + ∣ 2 ) d u d v + ε 3 τ 1 − 7 / 2 + 3 δ . \begin{aligned}
\lesssim\int_I\left(-\bar{\nu}|\psi_-|^2-\bar{\nu}|\psi_+|^2+\frac{|\partial_u\psi_-|^2}{-\bar{\nu}}\right)\,du \\
&+\iint_{\mathcal{R}_{r_1,r_2}}\left(-\bar{\nu}|\psi_-|^2-\bar{\nu}|\psi_+|^2+\frac{|\partial_u\psi_-|^2}{-\bar{\nu}}-\bar{\nu}|\partial_v\psi_+|^2\right)\,du\,dv+\varepsilon^3\tau_1^{-7/2+3\delta}.
\end{aligned} ≲ ∫ I ( − ν ˉ ∣ ψ − ∣ 2 − ν ˉ ∣ ψ + ∣ 2 + − ν ˉ ∣ ∂ u ψ − ∣ 2 ) d u + ∬ R r 1 , r 2 ( − ν ˉ ∣ ψ − ∣ 2 − ν ˉ ∣ ψ + ∣ 2 + − ν ˉ ∣ ∂ u ψ − ∣ 2 − ν ˉ ∣ ∂ v ψ + ∣ 2 ) d u d v + ε 3 τ 1 − 7/2 + 3 δ . The estimate (140) then follows by bounding the spacetime term on the right-hand side by the LHS of (124) .
r p r^p r p This section is devoted to upgrading (124) in the far region by obtaining additional decay estimates for the energy. In particular, we establish the celebrated Dafermos–Rodnianski hierarchy of r p r^p r p
Proposition 6.10. For any p ∈ [ 0 , 3 − δ ] p\in[0,3-\delta] p ∈ [ 0 , 3 − δ ] A ≥ 1 A\ge1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_f\in\mathcal{B} τ f ∈ B ( u 1 , v 1 ) , ( u 2 , v 2 ) ∈ Γ ∩ D τ f (u_1,v_1),(u_2,v_2)\in\Gamma\cap\mathcal{D}_{\tau_f} ( u 1 , v 1 ) , ( u 2 , v 2 ) ∈ Γ ∩ D τ f u 2 ′ > u 2 u_2'>u_2 u 2 ′ > u 2 v 2 ′ > v 2 v_2'>v_2 v 2 ′ > v 2 ( u 2 , v 2 ) (u_2,v_2) ( u 2 , v 2 ) ( u 1 , v 1 ) (u_1,v_1) ( u 1 , v 1 )
∫ I V ( r p ∣ ∂ v ( r ψ + ) ∣ 2 + c p ∗ r p ∗ + 2 ∣ ∂ v ψ + ∣ 2 + c p ∗ r p ∗ ∣ ψ + ∣ 2 ) d v + ∬ R ≥ Λ ( r p − 1 ∣ ∂ v Φ ~ + ∣ 2 + c ( p − 1 ) ∗ r ( p − 1 ) ∗ + 2 ∣ ∂ v ψ + ∣ 2 + c ( p − 1 ) ∗ r ( p − 1 ) ∗ ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 ) d u d v \begin{aligned}
\int_{\mathrm{IV}}\left(r^p|\partial_v(r\psi_+)|^2+c_{p_*}r^{p_*+2}|\partial_v\psi_+|^2+c_{p_*}r^{p_*}|\psi_+|^2\right)\,dv \\
&+\iint_{\mathcal{R}\ge\Lambda}\left(r^{p-1}|\partial_v\widetilde{\Phi}_+|^2+c_{(p-1)_*}r^{(p-1)_*+2}|\partial_v\psi_+|^2+c_{(p-1)_*}r^{(p-1)_*}|\psi_+|^2+|\psi_-|^2\right)\,du\,dv
\end{aligned} ∫ IV ( r p ∣ ∂ v ( r ψ + ) ∣ 2 + c p ∗ r p ∗ + 2 ∣ ∂ v ψ + ∣ 2 + c p ∗ r p ∗ ∣ ψ + ∣ 2 ) d v + ∬ R ≥ Λ ( r p − 1 ∣ ∂ v Φ + ∣ 2 + c ( p − 1 ) ∗ r ( p − 1 ) ∗ + 2 ∣ ∂ v ψ + ∣ 2 + c ( p − 1 ) ∗ r ( p − 1 ) ∗ ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 ) d u d v ≲ ∫ I − ν ˉ ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 + ∣ ∂ u ψ − ∣ 2 ν ˉ 2 ) d u + ∫ I I ( r p ∣ ∂ v Φ ~ + ∣ 2 + r 2 ∣ ∂ v ψ + ∣ 2 + ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 ) d v + ε 3 τ 1 − 7 / 2 + 3 δ , (152) \lesssim\int_I-\bar{\nu}\left(|\psi_-|^2+|\psi_+|^2+\frac{|\partial_u\psi_-|^2}{\bar{\nu}^2}\right)\,du+\int_{\mathrm{II}}\left(r^p|\partial_v\widetilde{\Phi}_+|^2+r^2|\partial_v\psi_+|^2+|\psi_+|^2+|\psi_-|^2\right)\,dv+\varepsilon^3\tau_1^{-7/2+3\delta}, \tag*{(152)} ≲ ∫ I − ν ˉ ( ∣ ψ − ∣ 2 + ∣ ψ + ∣ 2 + ν ˉ 2 ∣ ∂ u ψ − ∣ 2 ) d u + ∫ II ( r p ∣ ∂ v Φ + ∣ 2 + r 2 ∣ ∂ v ψ + ∣ 2 + ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 ) d v + ε 3 τ 1 − 7/2 + 3 δ , (152) where we recall from Definition 3.7 that p ∗ : = p p_*:=p p ∗ := p p ∈ [ 0 , 1 ) p\in[0,1) p ∈ [ 0 , 1 ) c p ∗ : = ( p ∗ − 1 ) 2 c_{p_*}:=(p_*-1)^2 c p ∗ := ( p ∗ − 1 ) 2
The proof of this estimate will be given in the next section after a series of lemmas.
Proof of the r p r^p r p
To prove the following estimates, we proceed in a manner analogous to the red-shift argument, with the roles of the two spinorial components interchanged. Here we start with the Φ + \Phi_+ Φ +
Φ ~ + : = r Ω Φ + = r ( − ν ) 1 / 2 ψ + . \widetilde{\Phi}_+:=\frac{r}{\Omega}\Phi_+=\frac{r}{(-\nu)^{1/2}}\psi_+. Φ + := Ω r Φ + = ( − ν ) 1/2 r ψ + . which satisfies a decoupled equation up to nonlinear error terms. As in Section 6.2, we first establish a preliminary lemma for the r p r^p r p (153) , which is satisfied both by Φ ~ + \widetilde{\Phi}_+ Φ + Φ − \Phi_- Φ −
Lemma 6.11 (r p r^p r p Let O \mathcal{O} O O ⊆ D τ f \mathcal{O}\subseteq\mathcal{D}_{\tau_f} O ⊆ D τ f φ ∈ C 2 ( O ‾ ) \varphi\in C^2(\overline{\mathcal{O}}) φ ∈ C 2 ( O )
∂ u ∂ v φ + b v ∂ v φ + b 0 φ = 0 , (153) \partial_u\partial_v\varphi+b_v\partial_v\varphi+b_0\varphi=0, \tag*{(153)} ∂ u ∂ v φ + b v ∂ v φ + b 0 φ = 0 , (153) where the coefficients b v ∈ C 0 ( O ) b_v\in C^0(\mathcal{O}) b v ∈ C 0 ( O ) b 0 ∈ C 1 ( O ) b_0\in C^1(\mathcal{O}) b 0 ∈ C 1 ( O ) χ \chi χ C 1 C^1 C 1 ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) χ = χ ( r ) \chi=\chi(r) χ = χ ( r ) χ ′ = d d r χ ( r ) \chi'=\frac{d}{dr}\chi(r) χ ′ = d r d χ ( r )
∂ u ( 1 2 χ r p ∣ ∂ v φ ∣ 2 ) + ∂ v ( 1 2 χ r p b 0 ∣ φ ∣ 2 ) + 1 2 χ r p − 1 ( ( 2 r b v − p ν ) ∣ ∂ v φ ∣ 2 − ( p λ b 0 + r ∂ v b 0 ) ∣ φ ∣ 2 ) = 1 2 χ ′ r p ( ν ∣ ∂ v φ ∣ 2 + λ b 0 ∣ φ ∣ 2 ) . (154) \begin{aligned}
\partial_u\left(\frac{1}{2}\chi r^p|\partial_v\varphi|^2\right)+\partial_v\left(\frac{1}{2}\chi r^p b_0|\varphi|^2\right)+\frac{1}{2}\chi r^{p-1}\left((2rb_v-p\nu)|\partial_v\varphi|^2-(p\lambda b_0+r\partial_vb_0)|\varphi|^2\right) \\
&=\frac{1}{2}\chi' r^p\left(\nu|\partial_v\varphi|^2+\lambda b_0|\varphi|^2\right). \tag*{(154)}
\end{aligned} ∂ u ( 2 1 χ r p ∣ ∂ v φ ∣ 2 ) + ∂ v ( 2 1 χ r p b 0 ∣ φ ∣ 2 ) + 2 1 χ r p − 1 ( ( 2 r b v − p ν ) ∣ ∂ v φ ∣ 2 − ( p λ b 0 + r ∂ v b 0 ) ∣ φ ∣ 2 ) = 2 1 χ ′ r p ( ν ∣ ∂ v φ ∣ 2 + λ b 0 ∣ φ ∣ 2 ) . (154) Proof. We multiply (6.34) by χ r p ∂ v φ ‾ \chi r^p\overline{\partial_v\varphi} χ r p ∂ v φ
χ r p ℜ [ ∂ v φ ‾ ∂ v ∂ u φ ] = ∂ u ( 1 2 χ r p ∣ ∂ v φ ∣ 2 ) − ∂ u ( 1 2 χ r p ) ∣ ∂ v φ ∣ 2 , \chi r^p\Re\left[\overline{\partial_v\varphi}\partial_v\partial_u\varphi\right]
=\partial_u\left(\frac{1}{2}\chi r^p|\partial_v\varphi|^2\right)
-\partial_u\left(\frac{1}{2}\chi r^p\right)|\partial_v\varphi|^2, χ r p ℜ [ ∂ v φ ∂ v ∂ u φ ] = ∂ u ( 2 1 χ r p ∣ ∂ v φ ∣ 2 ) − ∂ u ( 2 1 χ r p ) ∣ ∂ v φ ∣ 2 , χ r p ℜ [ ∂ v φ ‾ b v ∂ v φ ] = χ r p b v ∣ ∂ v φ ∣ 2 , \chi r^p\Re\left[\overline{\partial_v\varphi}b_v\partial_v\varphi\right]
=\chi r^p b_v|\partial_v\varphi|^2, χ r p ℜ [ ∂ v φ b v ∂ v φ ] = χ r p b v ∣ ∂ v φ ∣ 2 , χ r p ℜ [ ∂ v φ ‾ b 0 φ ] = ∂ v ( 1 2 χ r p b 0 ∣ φ ∣ 2 ) − ∂ v ( 1 2 χ r p b 0 ) ∣ φ ∣ 2 . \chi r^p\Re\left[\overline{\partial_v\varphi}b_0\varphi\right]
=\partial_v\left(\frac{1}{2}\chi r^p b_0|\varphi|^2\right)
-\partial_v\left(\frac{1}{2}\chi r^p b_0\right)|\varphi|^2. χ r p ℜ [ ∂ v φ b 0 φ ] = ∂ v ( 2 1 χ r p b 0 ∣ φ ∣ 2 ) − ∂ v ( 2 1 χ r p b 0 ) ∣ φ ∣ 2 . Expanding the last term of the first and third lines by the product rule and rearranging terms, we arrive at (154) .
We are now ready to state the main ingredient for the proof of Proposition 6.10.
Lemma 6.12. With hypotheses as in Proposition 6.10, it holds that
∫ I V ≥ 4 Λ r p ∣ ∂ v Φ ~ + ∣ 2 d v + ∬ R ≥ 4 Λ ( r p − 1 ∣ ∂ v Φ ~ + ∣ 2 + ∣ ψ − ∣ 2 ) d u d v \int_{\mathcal{I}_{V\geq4\Lambda}}r^p|\partial_v\widetilde{\Phi}_+|^2\,dv
+\iint_{\mathcal{R}_{\geq4\Lambda}}\left(r^{p-1}|\partial_v\widetilde{\Phi}_+|^2+|\psi_-|^2\right)\,du\,dv ∫ I V ≥ 4Λ r p ∣ ∂ v Φ + ∣ 2 d v + ∬ R ≥ 4Λ ( r p − 1 ∣ ∂ v Φ + ∣ 2 + ∣ ψ − ∣ 2 ) d u d v ≲ ∬ R Λ , 4 Λ ( ∣ ∂ v Φ ~ + ∣ 2 + ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 ) d u d v + ∫ I I I ( r p ∣ ∂ v Φ ~ + ∣ 2 + ∣ ψ + ∣ 2 ) d v . (155) \lesssim\iint_{\mathcal{R}_{\Lambda,4\Lambda}}\left(|\partial_v\widetilde{\Phi}_+|^2+|\psi_+|^2+|\psi_-|^2\right)\,du\,dv
+\int_{\mathcal{I}_{\mathrm{II}}}\left(r^p|\partial_v\widetilde{\Phi}_+|^2+|\psi_+|^2\right)\,dv. \tag*{(155)} ≲ ∬ R Λ , 4Λ ( ∣ ∂ v Φ + ∣ 2 + ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 ) d u d v + ∫ I II ( r p ∣ ∂ v Φ + ∣ 2 + ∣ ψ + ∣ 2 ) d v . (155) The reweighted field Φ ~ + \widetilde{\Phi}_+ Φ + reweighted radiation field for ψ + \psi_+ ψ + r p r^p r p [1 , 56 ] and adapted to the Dirac field on Schwarzschild in [61 Definitions 5.5 and 5.14] . In the latter setting, they introduced reweighted fields Φ ~ ± 1 / 2 ( i ) \widetilde{\Phi}^{(i)}_{\pm1/2} Φ ± 1/2 ( i ) i i i r p r^p r p Φ ~ + \widetilde{\Phi}_+ Φ + Φ ~ + 1 / 2 ( 1 ) \widetilde{\Phi}^{(1)}_{+1/2} Φ + 1/2 ( 1 )
Proof of Lemma 6.12. Step 1. Let χ \chi χ 0 ≤ χ ≤ 1 0\leq\chi\leq1 0 ≤ χ ≤ 1 { r ≥ 4 Λ } \{r\geq4\Lambda\} { r ≥ 4Λ } { r ≤ Λ } \{r\leq\Lambda\} { r ≤ Λ } { r = 4 Λ } \{r=4\Lambda\} { r = 4Λ } Φ ~ + = Ω r Φ + \widetilde{\Phi}_+=\frac{\Omega}{r}\Phi_+ Φ + = r Ω Φ +
∂ u ∂ v Φ ~ + + b v + ∂ v Φ ~ + + b 0 + Φ ~ + = 0 , \partial_u\partial_v\widetilde{\Phi}_+ + b_v^+\partial_v\widetilde{\Phi}_+ + b_0^+\widetilde{\Phi}_+ = 0, ∂ u ∂ v Φ + + b v + ∂ v Φ + + b 0 + Φ + = 0 , where
b v + : = ∂ u log Ω − ν r , b 0 + : = 6 Ω 2 m r 3 − 12 π T u v . b_v^+ := \partial_u\log\Omega-\frac{\nu}{r},\qquad
b_0^+ := \frac{6\Omega^2m}{r^3}-12\pi T_{uv}. b v + := ∂ u log Ω − r ν , b 0 + := r 3 6 Ω 2 m − 12 π T uv . Therefore, from Lemma 6.11, we obtain
∂ u ( 1 2 χ r p ∣ ∂ v Φ ~ + ∣ 2 ) + ∂ v ( 1 2 χ r p b 0 + ∣ Φ ~ + ∣ 2 ) + 1 2 χ r p − 1 ( ( 2 r b v + − p ν ) ∣ ∂ v Φ ~ + ∣ 2 − ( p λ b 0 + + r ∂ v b 0 + ) ∣ Φ ~ + ∣ 2 ) = 1 2 χ ′ r p ( ν ∣ ∂ v Φ ~ + ∣ 2 + λ b 0 + ∣ Φ ~ + ∣ 2 ) . (156) \begin{aligned}
\partial_u\left(\frac{1}{2}\chi r^p|\partial_v\widetilde{\Phi}_+|^2\right)
+\partial_v\left(\frac{1}{2}\chi r^p b_0^+|\widetilde{\Phi}_+|^2\right)
+\frac{1}{2}\chi r^{p-1}
\left((2rb_v^+-p\nu)|\partial_v\widetilde{\Phi}_+|^2
-(p\lambda b_0^+ + r\partial_v b_0^+)|\widetilde{\Phi}_+|^2\right) \\
&=\frac{1}{2}\chi' r^p\left(\nu|\partial_v\widetilde{\Phi}_+|^2+\lambda b_0^+|\widetilde{\Phi}_+|^2\right). \tag*{(156)}
\end{aligned} ∂ u ( 2 1 χ r p ∣ ∂ v Φ + ∣ 2 ) + ∂ v ( 2 1 χ r p b 0 + ∣ Φ + ∣ 2 ) + 2 1 χ r p − 1 ( ( 2 r b v + − p ν ) ∣ ∂ v Φ + ∣ 2 − ( p λ b 0 + + r ∂ v b 0 + ) ∣ Φ + ∣ 2 ) = 2 1 χ ′ r p ( ν ∣ ∂ v Φ + ∣ 2 + λ b 0 + ∣ Φ + ∣ 2 ) . (156) As the analysis of the boundary terms is straightforward, we directly focus on the bulk terms. Looking at ∣ ∂ v Φ ~ + ∣ 2 |\partial_v\widetilde{\Phi}_+|^2 ∣ ∂ v Φ + ∣ 2
2 r b v + − p ν = − ν ( 2 + p ) + 2 r ∂ u log Ω ∼ p + 2 > 0 , 2rb_v^+-p\nu=-\nu(2+p)+2r\partial_u\log\Omega\sim p+2>0, 2 r b v + − p ν = − ν ( 2 + p ) + 2 r ∂ u log Ω ∼ p + 2 > 0 , for r ≥ 2 Λ r\geq2\Lambda r ≥ 2Λ ε \varepsilon ε ∣ Φ ~ + ∣ 2 |\widetilde{\Phi}_+|^2 ∣ Φ + ∣ 2 (58) , (91) , (93) , (115) ,
− p λ b 0 + − r ∂ v b 0 + = 6 r − 3 ( Λ 2 γ m ( 3 − p ) + r Ω 2 ( ∂ v m − 2 m ∂ v log Ω ) + 2 π r 3 ( p λ T u v − r ∂ v T u v ) ) ∼ M r − 3 ( 3 − p − M r − 1 + A 2 ε 3 / 2 τ − 3 / 2 + δ / 2 ) , \begin{aligned}
-p\lambda b_0^+-r\partial_v b_0^+
=6r^{-3}\left(\Lambda^2\gamma m(3-p)+r\Omega^2(\partial_v m-2m\partial_v\log\Omega)+2\pi r^3(p\lambda T_{uv}-r\partial_vT_{uv})\right) \\
&\sim Mr^{-3}\left(3-p-Mr^{-1}+A^2\varepsilon^{3/2}\tau^{-3/2+\delta/2}\right),
\end{aligned} − p λ b 0 + − r ∂ v b 0 + = 6 r − 3 ( Λ 2 γ m ( 3 − p ) + r Ω 2 ( ∂ v m − 2 m ∂ v log Ω ) + 2 π r 3 ( p λ T uv − r ∂ v T uv ) ) ∼ M r − 3 ( 3 − p − M r − 1 + A 2 ε 3/2 τ − 3/2 + δ /2 ) , which is strictly positive for p ∈ [ 0 , 3 − δ ] p \in[0,3-\delta] p ∈ [ 0 , 3 − δ ] r ≥ 4 Λ r \ge4\Lambda r ≥ 4Λ ε \varepsilon ε
∣ ∂ v T u v ∣ = ∣ ( ∂ v log Ω − 3 λ r ) T u v + κ 1 / 2 ( − ν ) 2 π r 2 ℑ [ ψ ‾ − ∂ v ψ + ] ∣ ≲ A ε 2 r − 9 / 2 τ − 3 + δ + A ε 2 r − 5 τ − 3 / 2 + δ / 2 ≲ A ε 2 r − 9 / 2 τ − 3 / 2 + δ / 2 . \left|\partial_v T_{uv}\right| = \left|\left(\partial_v \log\Omega- 3\frac{\lambda}{r}\right)T_{uv} + \frac{\kappa^{1/2}(-\nu)}{2\pi r^2}\Im[\overline{\psi}_-\partial_v\psi_+]\right| \lesssim A\varepsilon^2 r^{-9/2}\tau^{-3+\delta} + A\varepsilon^2 r^{-5}\tau^{-3/2+\delta/2}
\lesssim A\varepsilon^2 r^{-9/2}\tau^{-3/2+\delta/2}. ∣ ∂ v T uv ∣ = ( ∂ v log Ω − 3 r λ ) T uv + 2 π r 2 κ 1/2 ( − ν ) ℑ [ ψ − ∂ v ψ + ] ≲ A ε 2 r − 9/2 τ − 3 + δ + A ε 2 r − 5 τ − 3/2 + δ /2 ≲ A ε 2 r − 9/2 τ − 3/2 + δ /2 . Noticing that the RHS of (156) is localized in R Λ , 4 Λ \mathcal{R}_{\Lambda,4\Lambda} R Λ , 4Λ
∬ V ≥ 4 Λ r p ∣ ∂ v Φ ~ + ∣ 2 d v + ∬ R ≥ 4 Λ r p − 1 ∣ ∂ v Φ ~ + ∣ 2 d u d v ≲ ∬ R Λ , 4 Λ ( ∣ ∂ v Φ ~ + ∣ 2 + ∣ ψ + ∣ 2 ) d u d v + ∫ I r p ∣ ∂ v Φ ~ + ∣ 2 d v . (157) \iint_{\mathcal{V}_{\ge4\Lambda}} r^p|\partial_v\widetilde{\Phi}_+|^2\,dv + \iint_{\mathcal{R}_{\ge4\Lambda}} r^{p-1}|\partial_v\widetilde{\Phi}_+|^2\,dudv \lesssim\iint_{\mathcal{R}_{\Lambda,4\Lambda}} \left(|\partial_v\widetilde{\Phi}_+|^2 + |\psi_+|^2\right)\,dudv + \int_{\mathcal{I}} r^p|\partial_v\widetilde{\Phi}_+|^2\,dv. \tag*{(157)} ∬ V ≥ 4Λ r p ∣ ∂ v Φ + ∣ 2 d v + ∬ R ≥ 4Λ r p − 1 ∣ ∂ v Φ + ∣ 2 d u d v ≲ ∬ R Λ , 4Λ ( ∣ ∂ v Φ + ∣ 2 + ∣ ψ + ∣ 2 ) d u d v + ∫ I r p ∣ ∂ v Φ + ∣ 2 d v . (157) for p ∈ [ 0 , 3 − δ ] p \in[0,3-\delta] p ∈ [ 0 , 3 − δ ]
Step 2
Let us now recall that Φ − \Phi_- Φ −
∂ u ∂ v Φ − + b v − ∂ v Φ − + b 0 − Φ − = 0 , \partial_u\partial_v\Phi_- + b_v^-\partial_v\Phi_- + b_0^-\Phi_- = 0, ∂ u ∂ v Φ − + b v − ∂ v Φ − + b 0 − Φ − = 0 , where
b v − : = − b v + = − ∂ u log Ω + ν r , b 0 − : = Ω 2 r 2 . b_v^- := -b_v^+ = -\partial_u\log\Omega+ \frac{\nu}{r}, \qquad b_0^- := \frac{\Omega^2}{r^2}. b v − := − b v + = − ∂ u log Ω + r ν , b 0 − := r 2 Ω 2 . Therefore, with χ \chi χ
∂ u ( 1 2 χ r p ∣ ∂ v Φ − ∣ 2 ) + ∂ v ( 1 2 χ r p b 0 − ∣ Φ − ∣ 2 ) + 1 2 χ r p − 1 ( ( 2 r b v − − p ν ) ∣ ∂ v Φ − ∣ 2 − ( p λ b 0 − + r ∂ v b 0 − ) ∣ Φ − ∣ 2 ) = 1 2 χ ′ r p ( ν ∣ ∂ v Φ − ∣ 2 + λ b 0 − ∣ Φ − ∣ 2 ) . \partial_u\left(\frac{1}{2}\chi r^p|\partial_v\Phi_-|^2\right) + \partial_v\left(\frac{1}{2}\chi r^p b_0^-|\Phi_-|^2\right) + \frac{1}{2}\chi r^{p-1}\left((2rb_v^- - p\nu)|\partial_v\Phi_-|^2 - (p\lambda b_0^- + r\partial_v b_0^-)|\Phi_-|^2\right)
= \frac{1}{2}\chi' r^p(\nu|\partial_v\Phi_-|^2 + \lambda b_0^-|\Phi_-|^2). ∂ u ( 2 1 χ r p ∣ ∂ v Φ − ∣ 2 ) + ∂ v ( 2 1 χ r p b 0 − ∣ Φ − ∣ 2 ) + 2 1 χ r p − 1 ( ( 2 r b v − − p ν ) ∣ ∂ v Φ − ∣ 2 − ( p λ b 0 − + r ∂ v b 0 − ) ∣ Φ − ∣ 2 ) = 2 1 χ ′ r p ( ν ∣ ∂ v Φ − ∣ 2 + λ b 0 − ∣ Φ − ∣ 2 ) . Proceeding as before, one finds that the bulk ∣ ∂ v Φ − ∣ 2 |\partial_v\Phi_-|^2 ∣ ∂ v Φ − ∣ 2
2 r b v − − p ν = ( 2 − p ) ν − 2 r ∂ u log Ω ∼ − ( 2 − p ) + M r ˉ − 1 < 0 , 2rb_v^- - p\nu= (2-p)\nu- 2r\partial_u\log\Omega\sim-(2-p) + Mr\bar{}^{-1} < 0, 2 r b v − − p ν = ( 2 − p ) ν − 2 r ∂ u log Ω ∼ − ( 2 − p ) + M r ˉ − 1 < 0 , for r ≥ 4 Λ r \ge4\Lambda r ≥ 4Λ r p − 1 ∣ ∂ v Φ − ∣ 2 ∼ r p − 3 ∣ Φ + ∣ 2 r^{p-1}|\partial_v\Phi_-|^2 \sim r^{p-3}|\Phi_+|^2 r p − 1 ∣ ∂ v Φ − ∣ 2 ∼ r p − 3 ∣ Φ + ∣ 2 ∣ Φ − ∣ 2 |\Phi_-|^2 ∣ Φ − ∣ 2
− p λ b 0 − − r ∂ v b 0 − = r − 2 ( λ 2 γ ( 2 − p ) − 2 r Ω 2 ∂ v log Ω ) ∼ r − 2 ( 2 − p − M r ˉ − 1 + A 2 ε 3 / 2 r − 1 τ − 2 + δ ) , -p\lambda b_0^- - r\partial_v b_0^- = r^{-2}\left(\lambda^2\gamma(2-p) - 2r\Omega^2\partial_v\log\Omega\right) \sim r^{-2}\left(2-p-Mr\bar{}^{-1}+A^2\varepsilon^{3/2}r^{-1}\tau^{-2+\delta}\right), − p λ b 0 − − r ∂ v b 0 − = r − 2 ( λ 2 γ ( 2 − p ) − 2 r Ω 2 ∂ v log Ω ) ∼ r − 2 ( 2 − p − M r ˉ − 1 + A 2 ε 3/2 r − 1 τ − 2 + δ ) , which is strictly positive for p ∈ [ 0 , 2 − δ ] p \in[0,2-\delta] p ∈ [ 0 , 2 − δ ] r ≥ 4 Λ r \ge4\Lambda r ≥ 4Λ ε \varepsilon ε p = 1 p=1 p = 1 ∣ Φ − ∣ 2 ∼ r 2 ∣ ψ − ∣ 2 |\Phi_-|^2 \sim r^2|\psi_-|^2 ∣ Φ − ∣ 2 ∼ r 2 ∣ ψ − ∣ 2
∬ R ≥ 4 Λ ∣ ψ − ∣ 2 d u d v ≲ ∬ R Λ , 4 Λ ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 ) d u d v + ∫ I r − 1 ∣ ψ + ∣ 2 d v . (158) \iint_{\mathcal{R}_{\ge4\Lambda}}|\psi_-|^2\,dudv \lesssim\iint_{\mathcal{R}_{\Lambda,4\Lambda}}\left(|\psi_+|^2 + |\psi_-|^2\right)\,dudv + \int_{\mathcal{I}}r^{-1}|\psi_+|^2\,dv. \tag*{(158)} ∬ R ≥ 4Λ ∣ ψ − ∣ 2 d u d v ≲ ∬ R Λ , 4Λ ( ∣ ψ + ∣ 2 + ∣ ψ − ∣ 2 ) d u d v + ∫ I r − 1 ∣ ψ + ∣ 2 d v . (158) Putting together (157) and (158) , we conclude the proof.
To obtain control over ψ + \psi_+ ψ + [4 Lemma 6.21] , which are a consequence of [53 Lemma 8.30] . These estimates are also the source of the degenerating exponents and constants p ∗ p_* p ∗ c p ∗ c_{p_*} c p ∗
Lemma 6.13 (r p r^p r p α < − 1 \alpha< -1 α < − 1
( α + 1 ) 2 ∬ R ≥ 4 Λ r α + 2 ∣ ψ + ∣ 2 d u d v ≲ ∬ R ≥ 4 Λ r α + 2 ∣ ∂ v Φ ~ + ∣ 2 d u d v + ∬ R Λ , 4 Λ ( ∣ ∂ v Φ ~ + ∣ 2 + ∣ ψ + ∣ 2 ) d u d v , (\alpha+1)^2 \iint_{\mathcal{R}_{\ge4\Lambda}} r^{\alpha+2}|\psi_+|^2\,dudv \lesssim\iint_{\mathcal{R}_{\ge4\Lambda}} r^{\alpha+2}|\partial_v\widetilde{\Phi}_+|^2\,dudv + \iint_{\mathcal{R}_{\Lambda,4\Lambda}}\left(|\partial_v\widetilde{\Phi}_+|^2 + |\psi_+|^2\right)\,dudv, ( α + 1 ) 2 ∬ R ≥ 4Λ r α + 2 ∣ ψ + ∣ 2 d u d v ≲ ∬ R ≥ 4Λ r α + 2 ∣ ∂ v Φ + ∣ 2 d u d v + ∬ R Λ , 4Λ ( ∣ ∂ v Φ + ∣ 2 + ∣ ψ + ∣ 2 ) d u d v , ( α + 1 ) 2 ∫ v Λ v 2 r α + 2 ∣ ψ + ∣ 2 d v ≲ ∫ v Λ v 2 ( r α + 2 ∣ ∂ v Φ ~ + ∣ 2 + ∣ ψ + ∣ 2 ) d v . (\alpha+1)^2 \int_{v_\Lambda}^{v_2} r^{\alpha+2}|\psi_+|^2\,dv \lesssim\int_{v_\Lambda}^{v_2}\left(r^{\alpha+2}|\partial_v\widetilde{\Phi}_+|^2 + |\psi_+|^2\right)\,dv. ( α + 1 ) 2 ∫ v Λ v 2 r α + 2 ∣ ψ + ∣ 2 d v ≲ ∫ v Λ v 2 ( r α + 2 ∣ ∂ v Φ + ∣ 2 + ∣ ψ + ∣ 2 ) d v . Proof. It suffices to redo the proof of [4 Lemma 6.21] with Φ ~ + \widetilde{\Phi}_{+} Φ + Φ ~ + ∼ r ψ + \widetilde{\Phi}_{+} \sim r\psi_{+} Φ + ∼ r ψ + { r ≥ Λ } \{r \geq\Lambda\} { r ≥ Λ } Γ \Gamma Γ
∣ ∂ v ψ + ∣ 2 ≲ r − 2 ( ∣ ∂ v Φ ~ + ∣ 2 + ∣ ψ + ∣ 2 ) , ∣ ∂ v ( r ψ + ) ∣ 2 ≲ ∣ ∂ v Φ ~ + ∣ 2 + r − 2 ∣ ψ + ∣ 2 , \lvert\partial_{v}\psi_{+}\rvert^{2} \lesssim r^{-2}\left(\lvert\partial_{v}\widetilde{\Phi}_{+}\rvert^{2}+\lvert\psi_{+}\rvert^{2}\right), \qquad\lvert\partial_{v}(r\psi_{+})\rvert^{2} \lesssim\lvert\partial_{v}\widetilde{\Phi}_{+}\rvert^{2}+r^{-2}\lvert\psi_{+}\rvert^{2}, ∣ ∂ v ψ + ∣ 2 ≲ r − 2 ( ∣ ∂ v Φ + ∣ 2 + ∣ ψ + ∣ 2 ) , ∣ ∂ v ( r ψ + ) ∣ 2 ≲ ∣ ∂ v Φ + ∣ 2 + r − 2 ∣ ψ + ∣ 2 , in R ≥ Λ \mathcal{R}_{\geq\Lambda} R ≥ Λ
Proof of Proposition 6.10. The proof follows from Lemmas 6.12 and 6.13, Proposition 6.2, and the relation ψ + = r − 1 ( − ν ) 1 / 2 Φ ~ + \psi_{+}=r^{-1}(-\nu)^{1/2}\widetilde{\Phi}_{+} ψ + = r − 1 ( − ν ) 1/2 Φ +
Improving the bootstrap assumptions for the Dirac field
We are now in position to prove Proposition 6.1 using a pigeonhole principle argument. Recalling the definition of the energies E \mathcal{E} E E p \mathcal{E}_{p} E p F \mathcal{F} F F \mathcal{F} F
G p ( τ ) : = E ( τ ) + E p ( τ ) , \mathcal{G}_{p}(\tau) := \mathcal{E}(\tau)+\mathcal{E}_{p}(\tau), G p ( τ ) := E ( τ ) + E p ( τ ) , and summarize the content of Propositions 6.2, 6.6 and 6.10 as follows.
Proposition 6.14. For any A ≥ 1 A \geq1 A ≥ 1 ε \varepsilon ε τ f ∈ B \tau_{f} \in\mathcal{B} τ f ∈ B 1 ≤ τ 1 ≤ τ 2 ≤ τ f 1 \leq\tau_{1} \leq\tau_{2} \leq\tau_{f} 1 ≤ τ 1 ≤ τ 2 ≤ τ f
sup τ ∈ [ τ 1 , τ 2 ] G p ( τ ) ≲ G p ( τ 1 ) + ε 3 τ 1 − 7 / 2 + 3 δ + p , (159) \sup_{\tau\in[\tau_{1},\tau_{2}]} \mathcal{G}_{p}(\tau) \lesssim\mathcal{G}_{p}(\tau_{1})+\varepsilon^{3}\tau_{1}^{-7/2+3\delta+p}, \tag*{(159)} τ ∈ [ τ 1 , τ 2 ] sup G p ( τ ) ≲ G p ( τ 1 ) + ε 3 τ 1 − 7/2 + 3 δ + p , (159) for every p ∈ [ 0 , 3 − δ ] p \in[0,3-\delta] p ∈ [ 0 , 3 − δ ]
∫ τ 1 τ 2 G p − 1 ( τ ) d τ ≲ G p ( τ 1 ) + ε 3 τ 1 − 7 / 2 + 3 δ + p , (160) \int_{\tau_{1}}^{\tau_{2}}\mathcal{G}_{p-1}(\tau)\,\mathrm{d}\tau\lesssim\mathcal{G}_{p}(\tau_{1})+\varepsilon^{3}\tau_{1}^{-7/2+3\delta+p}, \tag*{(160)} ∫ τ 1 τ 2 G p − 1 ( τ ) d τ ≲ G p ( τ 1 ) + ε 3 τ 1 − 7/2 + 3 δ + p , (160) for every p ∈ [ 1 , 3 − δ ] p \in[1,3-\delta] p ∈ [ 1 , 3 − δ ]
Before proceeding with the proof, we need the following interpolation property, which is an immediate consequence of Hölder’s inequality.
Lemma 6.15. Let 0 ≤ p 1 < p < p 2 ≤ 3 − δ 0 \leq p_{1}<p<p_{2}\leq3-\delta 0 ≤ p 1 < p < p 2 ≤ 3 − δ τ ∈ [ 1 , τ f ] \tau\in[1,\tau_{f}] τ ∈ [ 1 , τ f ]
G p ( τ ) ≲ ( G p 1 ( τ ) ) p 2 − p p 2 − p 1 ( G p 2 ( τ ) ) p − p 1 p 2 − p 1 . \mathcal{G}_{p}(\tau) \lesssim\left(\mathcal{G}_{p_{1}}(\tau)\right)^{\frac{p_{2}-p}{p_{2}-p_{1}}}\left(\mathcal{G}_{p_{2}}(\tau)\right)^{\frac{p-p_{1}}{p_{2}-p_{1}}}. G p ( τ ) ≲ ( G p 1 ( τ ) ) p 2 − p 1 p 2 − p ( G p 2 ( τ ) ) p 2 − p 1 p − p 1 . Proof of Proposition 6.1. We start by noting that for any p ∈ [ 0 , 3 − δ ] p \in[0,3-\delta] p ∈ [ 0 , 3 − δ ]
sup τ ∈ [ 1 , τ f ] G p ( τ ) ≲ G p ( 1 ) + ε 3 ≲ ( D [ S 0 ] ) 2 + ε 3 ≲ ε 2 , (161) \sup_{\tau\in[1,\tau_{f}]}\mathcal{G}_{p}(\tau) \lesssim\mathcal{G}_{p}(1)+\varepsilon^{3}\lesssim\left(\mathcal{D}[S_{0}]\right)^{2}+\varepsilon^{3}\lesssim\varepsilon^{2}, \tag*{(161)} τ ∈ [ 1 , τ f ] sup G p ( τ ) ≲ G p ( 1 ) + ε 3 ≲ ( D [ S 0 ] ) 2 + ε 3 ≲ ε 2 , (161) where we combined (159) applied on [ 1 , τ f ] [1,\tau_{f}] [ 1 , τ f ] [ 1 , τ f ] [1,\tau_{f}] [ 1 , τ f ] [ 2 i , 2 i + 1 ] [2^{i},2^{i+1}] [ 2 i , 2 i + 1 ] i ∈ { 0 , … , I τ f − 1 } i \in\{0,\ldots,I_{\tau_{f}}-1\} i ∈ { 0 , … , I τ f − 1 } I τ f I_{\tau_{f}} I τ f 2 I τ f ≤ τ f 2^{I_{\tau_{f}}}\leq\tau_{f} 2 I τ f ≤ τ f I τ f ≥ 1 I_{\tau_{f}}\geq1 I τ f ≥ 1 (161) already suffices. By the pigeonhole principle applied on each sub-interval, there exists τ i ′ ∈ [ 2 i , 2 i + 1 ] \tau'_{i}\in[2^{i},2^{i+1}] τ i ′ ∈ [ 2 i , 2 i + 1 ] i ∈ { 0 , … , I τ f − 1 } i\in\{0,\ldots,I_{\tau_{f}}-1\} i ∈ { 0 , … , I τ f − 1 }
G 2 − δ ( τ i ′ ) ≲ 1 2 i ∫ 2 i 2 i + 1 G 2 − δ ( τ ) d τ ≲ 2 − i ( G 3 − δ ( 2 i ) + ε 3 2 i ( − 1 / 2 + 2 δ ) ) ≲ ε 2 2 − i , (162) \mathcal{G}_{2-\delta}(\tau'_{i})\lesssim\frac{1}{2^{i}}\int_{2^{i}}^{2^{i+1}}\mathcal{G}_{2-\delta}(\tau)\,\mathrm{d}\tau\lesssim2^{-i}\left(\mathcal{G}_{3-\delta}(2^{i})+\varepsilon^{3}2^{i(-1/2+2\delta)}\right)\lesssim\varepsilon^{2}2^{-i}, \tag*{(162)} G 2 − δ ( τ i ′ ) ≲ 2 i 1 ∫ 2 i 2 i + 1 G 2 − δ ( τ ) d τ ≲ 2 − i ( G 3 − δ ( 2 i ) + ε 3 2 i ( − 1/2 + 2 δ ) ) ≲ ε 2 2 − i , (162) where we combined (160) and (161) with p = 3 − δ p=3-\delta p = 3 − δ (159) with p = 2 − δ p=2-\delta p = 2 − δ (162) to
G 2 − δ ( τ ) ≲ ε 2 τ − 1 , (163) \mathcal{G}_{2-\delta}(\tau)\lesssim\varepsilon^{2}\tau^{-1}, \tag*{(163)} G 2 − δ ( τ ) ≲ ε 2 τ − 1 , (163) for every τ ∈ [ 1 , τ f ] \tau\in[1,\tau_{f}] τ ∈ [ 1 , τ f ] (161) for p = 3 − δ p=3-\delta p = 3 − δ (163) , yielding
G 2 ( τ ) ≲ ε 2 τ − 1 + δ , (164) \mathcal{G}_{2}(\tau)\lesssim\varepsilon^{2}\tau^{-1+\delta}, \tag*{(164)} G 2 ( τ ) ≲ ε 2 τ − 1 + δ , (164) for every τ ∈ [ 1 , τ f ] \tau\in[1,\tau_f] τ ∈ [ 1 , τ f ] τ i ′ ′ ∈ [ 2 i , 2 i + 1 ] \tau_i'' \in[2^i,2^{i+1}] τ i ′′ ∈ [ 2 i , 2 i + 1 ] i ∈ { 0 , … , I τ f − 1 } i \in\{0,\ldots,I_{\tau_f}-1\} i ∈ { 0 , … , I τ f − 1 }
G 1 ( τ i ′ ′ ) ≲ 1 2 i ∫ 2 i 2 i + 1 G 1 ( τ ) d τ ≲ 2 − i ( G 2 ( 2 i ) + ε 2 2 3 i ( − 3 / 2 + 3 δ ) ) ≲ ε 2 ( 2 i ) − 2 + δ . \mathcal{G}_1(\tau_i'') \lesssim\frac{1}{2^i}\int_{2^i}^{2^{i+1}}\mathcal{G}_1(\tau)\,\mathrm{d}\tau\lesssim2^{-i}\left(\mathcal{G}_2(2^i)+\varepsilon^2 2^{3i(-3/2+3\delta)}\right)\lesssim\varepsilon^2(2^i)^{-2+\delta}. G 1 ( τ i ′′ ) ≲ 2 i 1 ∫ 2 i 2 i + 1 G 1 ( τ ) d τ ≲ 2 − i ( G 2 ( 2 i ) + ε 2 2 3 i ( − 3/2 + 3 δ ) ) ≲ ε 2 ( 2 i ) − 2 + δ . where we combined (164) and (160) with p = 2 p=2 p = 2
G 1 ( τ ) ≲ ε 2 τ − 2 + δ , \mathcal{G}_1(\tau)\lesssim\varepsilon^2\tau^{-2+\delta}, G 1 ( τ ) ≲ ε 2 τ − 2 + δ , for every τ ∈ [ 1 , τ f ] \tau\in[1,\tau_f] τ ∈ [ 1 , τ f ]
G 0 ( τ ) ≲ ε 2 τ − 3 + δ , \mathcal{G}_0(\tau)\lesssim\varepsilon^2\tau^{-3+\delta}, G 0 ( τ ) ≲ ε 2 τ − 3 + δ , for every τ ∈ [ 1 , τ f ] \tau\in[1,\tau_f] τ ∈ [ 1 , τ f ] (161) for p = 3 − δ p=3-\delta p = 3 − δ
G p ( τ ) ≲ ε 2 τ − 3 + δ + p , (165) \mathcal{G}_p(\tau)\lesssim\varepsilon^2\tau^{-3+\delta+p}, \tag*{(165)} G p ( τ ) ≲ ε 2 τ − 3 + δ + p , (165) for every p ∈ [ 0 , 3 − δ ] p\in[0,3-\delta] p ∈ [ 0 , 3 − δ ] τ ∈ [ 1 , τ f ] \tau\in[1,\tau_f] τ ∈ [ 1 , τ f ]
F ( u , τ ) ≲ ε 2 τ − 3 + δ , F ( v , τ ) ≲ ε 2 τ − 3 + δ , (166) \mathcal{F}(u,\tau)\lesssim\varepsilon^2\tau^{-3+\delta},\qquad\mathcal{F}(v,\tau)\lesssim\varepsilon^2\tau^{-3+\delta}, \tag*{(166)} F ( u , τ ) ≲ ε 2 τ − 3 + δ , F ( v , τ ) ≲ ε 2 τ − 3 + δ , (166) for any ( u , v ) ∈ D τ f (u,v)\in\mathcal{D}_{\tau_f} ( u , v ) ∈ D τ f τ ∈ [ 1 , τ f ] \tau\in[1,\tau_f] τ ∈ [ 1 , τ f ] A = A 0 A=A_0 A = A 0 (165) and (166) imply (120) –(123) .
Proof of Theorem 2
This section is devoted to the proof of Theorem 2 , concerning the stability of Schwarzschild for the spherically symmetric Einstein–Dirac system.
In particular, in § 9.1 we prove by a continuity argument that the bootstrap set satisfies B = [ 1 , + ∞ ) \mathcal{B}=[1,+\infty) B = [ 1 , + ∞ ) § 9.2 , we control the teleological gauges introduced in § 5.1.2 , showing that the sequence of gauges obtained through successive updates converges to an eschatological gauge. Moreover, we show that the corresponding anchored backgrounds, introduced in § 5.1.3 , converge to a Schwarzschild black hole with mass M ∞ M_\infty M ∞ r ˉ ∞ \bar r_\infty r ˉ ∞ § 9.3 , we complete the proof of Theorem 2 .
Since in this part of the paper we will work with gauges and background solutions corresponding to different bootstrap times, we will no longer suppress the τ f \tau_f τ f A = A 0 A=A_0 A = A 0
The continuity argument
We start by showing that the bootstrap set B ( S , ε , A ) \mathcal{B}(\mathcal{S},\varepsilon,A) B ( S , ε , A ) [ 1 , + ∞ ) [1,+\infty) [ 1 , + ∞ )
Proposition 7.1. Let M 0 M_0 M 0 δ 0 \delta_0 δ 0 Theorem 2 . Then, there exists an ε s t a b ( M 0 , δ ) > 0 \varepsilon_{\mathrm{stab}}(M_0,\delta)>0 ε stab ( M 0 , δ ) > 0 ε \varepsilon ε S \mathcal{S} S 0 < ε ≤ ε s t a b 0<\varepsilon\leq\varepsilon_{\mathrm{stab}} 0 < ε ≤ ε stab B ( S , ε , A 0 ) = [ 1 , + ∞ ) \mathcal{B}(\mathcal{S},\varepsilon,A_0)=[1,+\infty) B ( S , ε , A 0 ) = [ 1 , + ∞ )
Before proving the result, we first show that the bootstrap region remains strictly away from the future boundary of the initial data hypersurface C ^ \widehat{\mathcal{C}} C
Lemma 7.2. There exists a constant θ ∈ ( 0 , 1 ) \theta\in(0,1) θ ∈ ( 0 , 1 ) ε \varepsilon ε τ f ∈ B ( S , ε , A 0 ) \tau_f\in\mathcal{B}(\mathcal{S},\varepsilon,A_0) τ f ∈ B ( S , ε , A 0 ) Γ ^ ( τ f ) ≤ θ U ∗ \widehat{\Gamma}(\tau_f)\leq\theta U_\ast Γ ( τ f ) ≤ θ U ∗
Proof. We begin by noting that
∣ M τ f − M 0 ∣ = ∣ M τ f − m τ f ( 0 , 0 ) ∣ = ∣ m ~ τ f ( 0 , 0 ) ∣ ≲ ε 3 / 2 ≲ ε , |M_{\tau_f}-M_0|=|M_{\tau_f}-m_{\tau_f}(0,0)|=|\widetilde m_{\tau_f}(0,0)|\lesssim\varepsilon^{3/2}\lesssim\varepsilon, ∣ M τ f − M 0 ∣ = ∣ M τ f − m τ f ( 0 , 0 ) ∣ = ∣ m τ f ( 0 , 0 ) ∣ ≲ ε 3/2 ≲ ε , by Proposition 3.2 and (44) . Moreover, since ∂ u r ^ = − 1 \partial_u\hat{r}=-1 ∂ u r ^ = − 1 C ^ i n \hat{C}_{\mathrm{in}} C ^ in r ( Γ u ^ ( τ f ) , 0 ) = Λ − Γ u ^ ( τ f ) r(\Gamma^{\hat{u}}(\tau_f),0)=\Lambda-\Gamma^{\hat{u}}(\tau_f) r ( Γ u ^ ( τ f ) , 0 ) = Λ − Γ u ^ ( τ f ) (43) , we bound
Γ u ^ ( τ f ) = Λ − r ( Γ u ^ ( τ f ) , 0 ) = Λ − r ˉ τ f ∘ Φ τ f ( Γ u ^ ( τ f ) , 0 ) − r ~ ∘ Φ τ f ( Γ u ^ ( τ f ) , 0 ) ≤ 100 M 0 − 2 M τ f + O ( ε ) = 98 M 0 + O ( ε ) , \begin{aligned}
\Gamma^{\hat{u}}(\tau_f)=\Lambda-r(\Gamma^{\hat{u}}(\tau_f),0)=\Lambda-\bar{r}_{\tau_f}\circ\Phi_{\tau_f}(\Gamma^{\hat{u}}(\tau_f),0)-\tilde{r}\circ\Phi_{\tau_f}(\Gamma^{\hat{u}}(\tau_f),0)\leq100M_0-2M_{\tau_f}+O(\varepsilon) \\
&=98M_0+O(\varepsilon),
\end{aligned} Γ u ^ ( τ f ) = Λ − r ( Γ u ^ ( τ f ) , 0 ) = Λ − r ˉ τ f ∘ Φ τ f ( Γ u ^ ( τ f ) , 0 ) − r ~ ∘ Φ τ f ( Γ u ^ ( τ f ) , 0 ) ≤ 100 M 0 − 2 M τ f + O ( ε ) = 98 M 0 + O ( ε ) , which is strictly smaller than U ∗ = 98.5 M 0 U_*=98.5M_0 U ∗ = 98.5 M 0 ε \varepsilon ε
Proof of Proposition 7.1. Let us prove that B ( S , ε , A 0 ) \mathcal{B}(\mathcal{S},\varepsilon,A_0) B ( S , ε , A 0 ) ε \varepsilon ε
Non-emptiness: It is a straightforward consequence of local well-posedness and continuous dependence on the initial data. Indeed, a solution arising from a seed data set S \mathcal{S} S C ^ \hat{C} C ^ ( 0 , 0 ) (0,0) ( 0 , 0 ) ε \varepsilon ε τ f \tau_f τ f A = A 0 A=A_0 A = A 0
Openness: Let us now prove it is open, that is, proving that if τ f ∈ B ( S , ε , A 0 ) \tau_f\in\mathcal{B}(\mathcal{S},\varepsilon,A_0) τ f ∈ B ( S , ε , A 0 ) τ f + η ∈ B ( S , ε , A 0 ) \tau_f+\eta\in\mathcal{B}(\mathcal{S},\varepsilon,A_0) τ f + η ∈ B ( S , ε , A 0 ) η > 0 \eta>0 η > 0 τ f ∈ B \tau_f\in\mathcal{B} τ f ∈ B ( r , Ω ^ , ψ + , ψ − ) (r,\hat{\Omega},\psi_+,\psi_-) ( r , Ω ^ , ψ + , ψ − )
D ^ τ f : = [ 0 , Γ u ^ ( τ f ) ] × [ 0 , Γ v ^ ( τ f ) ] ⊂ Q ^ max . \hat{D}_{\tau_f}:=[0,\Gamma^{\hat{u}}(\tau_f)]\times[0,\Gamma^{\hat{v}}(\tau_f)]\subset\hat{Q}_{\max}. D ^ τ f := [ 0 , Γ u ^ ( τ f )] × [ 0 , Γ v ^ ( τ f )] ⊂ Q ^ m a x . By Proposition 2.9 and Lemma 7.2, there exists σ > 0 \sigma>0 σ > 0
D ^ τ f σ : = [ 0 , Γ u ^ ( τ f ) + σ ] × [ 0 , Γ v ^ ( τ f ) + σ ] ⊂ Q ^ max . \hat{D}^{\sigma}_{\tau_f}:=[0,\Gamma^{\hat{u}}(\tau_f)+\sigma]\times[0,\Gamma^{\hat{v}}(\tau_f)+\sigma]\subset\hat{Q}_{\max}. D ^ τ f σ := [ 0 , Γ u ^ ( τ f ) + σ ] × [ 0 , Γ v ^ ( τ f ) + σ ] ⊂ Q ^ m a x . Our aim is to show that there exists η > 0 \eta>0 η > 0
D ^ τ f + η : = [ 0 , Γ u ^ ( τ f + η ) ] × [ 0 , Γ v ^ ( τ f + η ) ] \hat{D}_{\tau_f+\eta}:=[0,\Gamma^{\hat{u}}(\tau_f+\eta)]\times[0,\Gamma^{\hat{v}}(\tau_f+\eta)] D ^ τ f + η := [ 0 , Γ u ^ ( τ f + η )] × [ 0 , Γ v ^ ( τ f + η )] satisfies
D ^ τ f ⊂ D ^ τ f + η ⊆ D ^ τ f σ ⊂ Q ^ max , \hat{D}_{\tau_f}\subset\hat{D}_{\tau_f+\eta}\subseteq\hat{D}^{\sigma}_{\tau_f}\subset\hat{Q}_{\max}, D ^ τ f ⊂ D ^ τ f + η ⊆ D ^ τ f σ ⊂ Q ^ m a x , and, in addition, properties 1.–3. of Definition 3.11 hold on D ^ τ f + η \hat{D}_{\tau_f+\eta} D ^ τ f + η
Since Γ \Gamma Γ Γ ∣ [ 0 , τ f ] \left.\Gamma\right|_{[0,\tau_f]} Γ ∣ [ 0 , τ f ] D ^ τ f σ \hat{D}^{\sigma}_{\tau_f} D ^ τ f σ σ \sigma σ (52) , (51) and (54) imply, by continuity, that κ ^ > 0 \hat{\kappa}>0 κ ^ > 0 D ^ τ f σ \hat{D}^{\sigma}_{\tau_f} D ^ τ f σ γ ^ > 0 \hat{\gamma}>0 γ ^ > 0 D ^ τ f σ ∩ { r ≥ Λ } \hat{D}^{\sigma}_{\tau_f}\cap\{r\geq\Lambda\} D ^ τ f σ ∩ { r ≥ Λ } σ \sigma σ D ^ τ f + η ⊂ Q ^ max \hat{D}_{\tau_f+\eta}\subset\hat{Q}_{\max} D ^ τ f + η ⊂ Q ^ m a x η \eta η Φ τ f + η \Phi_{\tau_f+\eta} Φ τ f + η ( u τ f + η , v τ f + η ) (u_{\tau_f+\eta},v_{\tau_f+\eta}) ( u τ f + η , v τ f + η )
It remains to verify point 3. By Definition 3.4, we note that u τ f + η → u τ f u_{\tau_f+\eta}\to u_{\tau_f} u τ f + η → u τ f [ 0 , Γ u ^ ( τ f ) ] [0,\Gamma^{\hat{u}}(\tau_f)] [ 0 , Γ u ^ ( τ f )] v τ f + η → v τ f v_{\tau_f+\eta}\to v_{\tau_f} v τ f + η → v τ f [ 0 , Γ v ^ ( τ f ) ] [0,\Gamma^{\hat{v}}(\tau_f)] [ 0 , Γ v ^ ( τ f )] η → 0 \eta\to0 η → 0 (32) and denoting by r ˉ τ f + η \bar{r}_{\tau_f+\eta} r ˉ τ f + η r ˉ τ f \bar{r}_{\tau_f} r ˉ τ f r ˉ τ f + η ∘ Φ τ f + η → r ˉ τ f ∘ Φ τ f \bar{r}_{\tau_f+\eta}\circ\Phi_{\tau_f+\eta}\to\bar{r}_{\tau_f}\circ\Phi_{\tau_f} r ˉ τ f + η ∘ Φ τ f + η → r ˉ τ f ∘ Φ τ f D ^ τ f \hat{D}_{\tau_f} D ^ τ f η → 0 \eta\to0 η → 0 (42) –(48) have been strictly improved on D ^ τ f \hat{D}_{\tau_f} D ^ τ f η > 0 \eta>0 η > 0 (42) –(48) continue to hold on D ^ τ f + η \hat{D}_{\tau_f+\eta} D ^ τ f + η τ f + η ∈ B ( S , ε , A 0 ) \tau_f+\eta\in\mathcal{B}(\mathcal{S},\varepsilon,A_0) τ f + η ∈ B ( S , ε , A 0 )
Closedness: We now show that the bootstrap set is closed, by proving that if τ f n → τ f ∞ \tau_f^n\to\tau_f^\infty τ f n → τ f ∞ n → + ∞ n\to+\infty n → + ∞ τ f n \tau_f^n τ f n B ( S , ε , A 0 ) \mathcal{B}(\mathcal{S},\varepsilon,A_0) B ( S , ε , A 0 ) τ f ∞ \tau_f^\infty τ f ∞ τ f ∞ ∈ B ( S , ε , A 0 ) \tau_f^\infty\in\mathcal{B}(\mathcal{S},\varepsilon,A_0) τ f ∞ ∈ B ( S , ε , A 0 )
We start by noting that, as Γ \Gamma Γ Γ u ^ ( τ f n ) \Gamma^{\hat{u}}(\tau_f^n) Γ u ^ ( τ f n ) Γ v ^ ( τ f n ) \Gamma^{\hat{v}}(\tau_f^n) Γ v ^ ( τ f n ) Γ u ^ ( τ f n ) \Gamma^{\hat{u}}(\tau_f^n) Γ u ^ ( τ f n ) u ^ ∗ < U ∗ \hat{u}_*<U_* u ^ ∗ < U ∗ (69) , Γ v ^ ( τ f n ) \Gamma^{\hat{v}}(\tau_f^n) Γ v ^ ( τ f n ) v ^ ∗ \hat{v}_* v ^ ∗ D ^ ∗ : = [ 0 , u ^ ∗ ) × [ 0 , v ^ ∗ ) \hat{D}_*:=[0,\hat{u}_*)\times[0,\hat{v}_*) D ^ ∗ := [ 0 , u ^ ∗ ) × [ 0 , v ^ ∗ ) Q ^ max \hat{Q}_{\max} Q ^ m a x
D ^ τ f ∞ = D ^ ∗ ⊂ Q ^ max , \hat{D}_{\tau_f^\infty}=\hat{D}_*\subset\hat{Q}_{\max}, D ^ τ f ∞ = D ^ ∗ ⊂ Q ^ m a x , that is showing that the solution ( r , Ω ^ , ψ + , ψ − ) (r,\hat{\Omega},\psi_+,\psi_-) ( r , Ω ^ , ψ + , ψ − ) D ^ ∗ \hat{D}_* D ^ ∗ (55) , (56) , (88) , (90) , (103) , we find 1 ≲ r ≲ 1 + τ f ∞ 1\lesssim r\lesssim1+\tau_f^\infty 1 ≲ r ≲ 1 + τ f ∞ ∣ λ ^ ∣ ≲ 1 |\hat{\lambda}|\lesssim1 ∣ λ ^ ∣ ≲ 1 κ ^ ∼ 1 \hat{\kappa}\sim1 κ ^ ∼ 1 ∣ ψ ± ∣ ≲ 1 |\psi_\pm|\lesssim1 ∣ ψ ± ∣ ≲ 1 ∣ ∂ v ψ + ∣ ≲ 1 |\partial_v\psi_+|\lesssim1 ∣ ∂ v ψ + ∣ ≲ 1 D ^ ∗ \hat{D}_* D ^ ∗ (10) yields ∣ log ( − ν ^ ) ∣ ≲ 1 + τ f ∞ |\log(-\hat{\nu})|\lesssim1+\tau_f^\infty ∣ log ( − ν ^ ) ∣ ≲ 1 + τ f ∞ D ^ ∗ \hat{D}_* D ^ ∗ ∣ ∂ u ^ ψ − ∣ ≲ 1 + τ f ∞ |\partial_{\hat{u}}\psi^-| \lesssim1+\tau_f^\infty ∣ ∂ u ^ ψ − ∣ ≲ 1 + τ f ∞ (105) . Recalling Ω ^ 2 = − κ ^ ν ^ \hat{\Omega}^2=-\hat{\kappa}\hat{\nu} Ω ^ 2 = − κ ^ ν ^ 1 ≲ Ω ^ 2 ≲ 1 + τ f ∞ 1\lesssim\hat{\Omega}^2\lesssim1+\tau_f^\infty 1 ≲ Ω ^ 2 ≲ 1 + τ f ∞ ∣ ∂ u ^ log Ω ∣ + ∣ ∂ v ^ log Ω ∣ ≲ 1 + ( τ f ∞ ) 2 |\partial_{\hat{u}}\log\Omega|+|\partial_{\hat{v}}\log\Omega|\lesssim1+(\tau_f^\infty)^2 ∣ ∂ u ^ log Ω∣ + ∣ ∂ v ^ log Ω∣ ≲ 1 + ( τ f ∞ ) 2 D ^ ⋆ \hat{D}_\star D ^ ⋆ D ^ ⋆ \hat{D}_\star D ^ ⋆ C 1 C^1 C 1 C ∞ C^\infty C ∞ τ f ∞ ∈ B ( S , ε , A 0 ) \tau_f^\infty\in\mathcal{B}(\mathcal{S},\varepsilon,A_0) τ f ∞ ∈ B ( S , ε , A 0 )
Remark 7.3. Let us note that ε s t a b \varepsilon_{\mathrm{stab}} ε stab Proposition 7.15 below.
Figure 9. We depict here the Penrose diagram of the region considered in the openness argument of Proposition 7.1 . Within the extended region D ^ τ f σ \hat{D}^{\sigma}_{\tau_f} D ^ τ f σ η > 0 \eta>0 η > 0 D ^ τ f ⊂ D ^ τ f + η ⊂ D ^ τ f σ ⊂ Q ^ max \hat{D}_{\tau_f}\subset\hat{D}_{\tau_f+\eta}\subset\hat{D}^{\sigma}_{\tau_f}\subset\hat{Q}_{\max} D ^ τ f ⊂ D ^ τ f + η ⊂ D ^ τ f σ ⊂ Q ^ m a x D ^ τ f σ \hat{D}^{\sigma}_{\tau_f} D ^ τ f σ
Control of the gauge and anchored backgrounds
Since we proved B ( S , ε , A 0 ) = [ 1 , + ∞ ) \mathcal{B}(\mathcal{S},\varepsilon,A_0)=[1,+\infty) B ( S , ε , A 0 ) = [ 1 , + ∞ )
D ^ ∞ : = ⋃ τ f ≥ 1 D ^ τ f ⊂ Q ^ max . \hat{D}_\infty:=\bigcup_{\tau_f\geq1}\hat{D}_{\tau_f}\subset\hat{Q}_{\max}. D ^ ∞ := τ f ≥ 1 ⋃ D ^ τ f ⊂ Q ^ m a x . The aim of this section is to properly define the eschatological gauge Φ ∞ \Phi_\infty Φ ∞ D ^ ∞ \hat{D}_\infty D ^ ∞ Φ τ f = ( u τ f , v τ f ) \Phi_{\tau_f}=(u_{\tau_f},v_{\tau_f}) Φ τ f = ( u τ f , v τ f )
The arguments and techniques are analogous to those employed in [4 Section 8.2] . In our setting, however, there are two additional points that need to be addressed. First, due to our different choice of teleological gauge, we are required to control the behaviour of the coordinate component v τ f v_{\tau_f} v τ f M τ f M_{\tau_f} M τ f [4 ] . For these reasons, we provide the details of the argument below.
Proposition 7.4. For ( u ^ , v ^ ) ∈ D ^ ∞ (\hat{u},\hat{v})\in\hat{D}_\infty ( u ^ , v ^ ) ∈ D ^ ∞
Φ ∞ ( u ^ , v ^ ) : = lim τ f → + ∞ Φ τ f ( u ^ , v ^ ) \Phi_\infty(\hat{u},\hat{v}):=\lim_{\tau_f\to+\infty}\Phi_{\tau_f}(\hat{u},\hat{v}) Φ ∞ ( u ^ , v ^ ) := τ f → + ∞ lim Φ τ f ( u ^ , v ^ ) exists and defines a C 2 C^2 C 2 Φ ∞ : D ^ ∞ → [ 0 , + ∞ ) × [ 0 , + ∞ ) \Phi_\infty:\hat{D}_\infty\to[0,+\infty)\times[0,+\infty) Φ ∞ : D ^ ∞ → [ 0 , + ∞ ) × [ 0 , + ∞ )
The proof is given after a series of preliminary lemmas. Let us begin with a straightforward consequence of Lemma 7.2.
Lemma 7.5. The limit
u ^ H + : = lim τ → + ∞ Γ u ^ ( τ ) (167) \hat{u}_{H+}:=\lim_{\tau\to+\infty}\Gamma^{\hat{u}}(\tau) \tag*{(167)} u ^ H + := τ → + ∞ lim Γ u ^ ( τ ) (167) exists and satisfies u ^ H + ≤ θ U ∗ \hat{u}_{H+}\leq\theta U_\ast u ^ H + ≤ θ U ∗ θ ∈ ( 0 , 1 ) \theta\in(0,1) θ ∈ ( 0 , 1 )
D ^ ∞ = [ 0 , u ^ H + ) × [ 0 , + ∞ ) . \hat{D}_\infty=[0,\hat{u}_{H+})\times[0,+\infty). D ^ ∞ = [ 0 , u ^ H + ) × [ 0 , + ∞ ) . We now prove that the family { Φ τ f } τ f \{\Phi_{\tau_f}\}_{\tau_f} { Φ τ f } τ f C l o c 2 C^2_{\mathrm{loc}} C loc 2
Lemma 7.6. For any η > 0 \eta> 0 η > 0 ( u ^ 0 , v ^ 0 ) ∈ [ 0 , u ^ H + ) × [ 0 , + ∞ ) (\widehat{u}_0,\widehat{v}_0) \in[0,\widehat{u}_H^+) \times[0,+\infty) ( u 0 , v 0 ) ∈ [ 0 , u H + ) × [ 0 , + ∞ ) τ f , 1 \tau_{f,1} τ f , 1 τ f , 2 > τ f , 1 \tau_{f,2} > \tau_{f,1} τ f , 2 > τ f , 1
∥ Φ τ f , 2 − Φ τ f , 1 ∥ C 2 ( [ 0 , u ^ 0 ] × [ 0 , v ^ 0 ] ) ≲ u ^ 0 , v ^ 0 ε 2 η . (168) \left\lVert\Phi_{\tau_{f,2}}-\Phi_{\tau_{f,1}}\right\rVert_{C^2([0,\widehat{u}_0]\times[0,\widehat{v}_0])} \lesssim_{\widehat{u}_0,\widehat{v}_0} \varepsilon^2\eta. \tag*{(168)} Φ τ f , 2 − Φ τ f , 1 C 2 ([ 0 , u 0 ] × [ 0 , v 0 ]) ≲ u 0 , v 0 ε 2 η . (168) Proof of Lemma 7.6. Let us recall (16) , which, due to gauge invariance, retains the same form
∂ v ^ γ ^ = 2 γ ^ r ( 1 − μ ) ℑ [ ψ + ∂ v ^ ψ + ] . \partial_{\widehat{v}}\widehat{\gamma}=\frac{2\widehat{\gamma}}{r(1-\mu)}\Im[\psi_+\partial_{\widehat{v}}\psi_+]. ∂ v γ = r ( 1 − μ ) 2 γ ℑ [ ψ + ∂ v ψ + ] . Therefore, from (29) we compute
u τ f , 2 ′ ( u ^ ) − u τ f , 1 ′ ( u ^ ) = γ ^ ( u ^ , Γ v ^ ( τ f , 2 ) ) − γ ^ ( u ^ , Γ v ^ ( τ f , 1 ) ) = γ ^ ( u ^ , Γ v ^ ( τ f , 1 ) ) ( exp ( ∫ Γ v ^ ( τ f , 1 ) Γ v ^ ( τ f , 2 ) 2 r ( 1 − μ ) ℑ [ ψ + ∂ v ^ ψ + ] d v ^ ) − 1 ) , \begin{aligned}
u'_{\tau_{f,2}}(\widehat{u})-u'_{\tau_{f,1}}(\widehat{u})=\widehat{\gamma}(\widehat{u},\Gamma^{\widehat{v}}(\tau_{f,2}))-\widehat{\gamma}(\widehat{u},\Gamma^{\widehat{v}}(\tau_{f,1})) \\
&=\widehat{\gamma}(\widehat{u},\Gamma^{\widehat{v}}(\tau_{f,1}))\left(\exp\left(\int_{\Gamma^{\widehat{v}}(\tau_{f,1})}^{\Gamma^{\widehat{v}}(\tau_{f,2})}\frac{2}{r(1-\mu)}\Im[\psi_+\partial_{\widehat{v}}\psi_+]\,d\widehat{v}\right)-1\right),
\end{aligned} u τ f , 2 ′ ( u ) − u τ f , 1 ′ ( u ) = γ ( u , Γ v ( τ f , 2 )) − γ ( u , Γ v ( τ f , 1 )) = γ ( u , Γ v ( τ f , 1 )) ( exp ( ∫ Γ v ( τ f , 1 ) Γ v ( τ f , 2 ) r ( 1 − μ ) 2 ℑ [ ψ + ∂ v ψ + ] d v ) − 1 ) , for u ^ ∈ [ 0 , u ^ 0 ] \widehat{u}\in[0,\widehat{u}_0] u ∈ [ 0 , u 0 ] Figure 10 , by defining
Figure 10. We depict the Penrose diagram of the spacetime region and the null hypersurfaces considered in the argument controlling u τ f , 2 ′ ( u ^ ) − u τ f , 1 ′ ( u ^ ) u_{\tau_{f,2}}'(\hat{u})-u_{\tau_{f,1}}'(\hat{u}) u τ f , 2 ′ ( u ^ ) − u τ f , 1 ′ ( u ^ ) τ f , 1 \tau_{f,1} τ f , 1 Γ u ^ ( τ f , 2 ) > Γ u ^ ( τ f , 1 ) > u ^ 0 \Gamma^{\hat{u}}(\tau_{f,2})>\Gamma^{\hat{u}}(\tau_{f,1})>\hat{u}_0 Γ u ^ ( τ f , 2 ) > Γ u ^ ( τ f , 1 ) > u ^ 0 v ∗ ( u ^ ) v^*(\hat{u}) v ∗ ( u ^ ) u ^ \hat{u} u ^ { r = 3 M τ f } \{r=3M\tau_{f}\} { r = 3 M τ f } { r ≥ 3 M τ f } \{r\geq3M\tau_{f}\} { r ≥ 3 M τ f } (60) to deduce the positivity of 1 − μ 1-\mu 1 − μ ε \varepsilon ε
v ^ ∗ ( u ^ ) : = { v ^ 3 M ( u ^ ) if r ( u ^ , 0 ) ≤ 3 M τ f , 0 otherwise , \widehat{v}^*(\widehat{u}):=
\begin{cases}
\widehat{v}^{3M}(\widehat{u}) & \text{if } r(\widehat{u},0)\leq3M_{\tau_f},\\
0 & \text{otherwise},
\end{cases} v ∗ ( u ) := { v 3 M ( u ) 0 if r ( u , 0 ) ≤ 3 M τ f , otherwise , we may express γ ^ ( u ^ , Γ v ^ ( τ f , 1 ) ) \widehat{\gamma}(\widehat{u},\Gamma^{\widehat{v}}(\tau_{f,1})) γ ( u , Γ v ( τ f , 1 )) γ ^ \widehat{\gamma} γ v ^ \widehat{v} v v ^ ∗ ( u ^ ) \widehat{v}^*(\widehat{u}) v ∗ ( u )
u τ f , 2 ′ ( u ^ ) − u τ f , 1 ′ ( u ^ ) = 1 1 − μ ( u ^ , v ^ ∗ ( u ^ ) ) exp ( ∫ v ^ ∗ ( u ^ ) Γ v ^ ( τ f , 1 ) 2 r ( 1 − μ ) ℑ [ ψ + ∂ v ^ ψ + ] d v ^ ) × ( exp ( ∫ Γ v ^ ( τ f , 1 ) Γ v ^ ( τ f , 2 ) 2 r ( 1 − μ ) ℑ [ ψ + ∂ v ^ ψ + ] d v ^ ) − 1 ) . (169) \begin{aligned}
u'_{\tau_{f,2}}(\widehat{u})-u'_{\tau_{f,1}}(\widehat{u})
={}&\frac{1}{1-\mu(\widehat{u},\widehat{v}^*(\widehat{u}))}
\exp\left(\int_{\widehat{v}^*(\widehat{u})}^{\Gamma^{\widehat{v}}(\tau_{f,1})}\frac{2}{r(1-\mu)}\Im[\psi_+\partial_{\widehat{v}}\psi_+]\,d\widehat{v}\right)\\
&\times\left(\exp\left(\int_{\Gamma^{\widehat{v}}(\tau_{f,1})}^{\Gamma^{\widehat{v}}(\tau_{f,2})}\frac{2}{r(1-\mu)}\Im[\psi_+\partial_{\widehat{v}}\psi_+]\,d\widehat{v}\right)-1\right). \tag*{(169)}
\end{aligned} u τ f , 2 ′ ( u ) − u τ f , 1 ′ ( u ) = 1 − μ ( u , v ∗ ( u )) 1 exp ( ∫ v ∗ ( u ) Γ v ( τ f , 1 ) r ( 1 − μ ) 2 ℑ [ ψ + ∂ v ψ + ] d v ) × ( exp ( ∫ Γ v ( τ f , 1 ) Γ v ( τ f , 2 ) r ( 1 − μ ) 2 ℑ [ ψ + ∂ v ψ + ] d v ) − 1 ) . (169) As the solution exists and is smooth on D ^ τ f , 1 \widehat{D}_{\tau_{f,1}} D τ f , 1
∣ 1 1 − μ ( u ^ , v ^ ∗ ( u ^ ) ) exp ( ∫ v ^ ∗ ( u ^ ) Γ v ^ ( τ f , 1 ) 2 r ( 1 − μ ) ℑ [ ψ + ∂ v ^ ψ + ] d v ^ ) ∣ ≤ C ( u ^ 0 ) . \left|\frac{1}{1-\mu(\widehat{u},\widehat{v}^*(\widehat{u}))}
\exp\left(\int_{\widehat{v}^*(\widehat{u})}^{\Gamma^{\widehat{v}}(\tau_{f,1})}\frac{2}{r(1-\mu)}\Im[\psi_+\partial_{\widehat{v}}\psi_+]\,d\widehat{v}\right)\right|\leq C(\widehat{u}_0). 1 − μ ( u , v ∗ ( u )) 1 exp ( ∫ v ∗ ( u ) Γ v ( τ f , 1 ) r ( 1 − μ ) 2 ℑ [ ψ + ∂ v ψ + ] d v ) ≤ C ( u 0 ) . Indeed, the choice of v ^ ∗ ( u ^ ) \widehat{v}^*(\widehat{u}) v ∗ ( u ) ( u ^ , v ^ ∗ ( u ^ ) ) (\widehat{u},\widehat{v}^*(\widehat{u})) ( u , v ∗ ( u )) { r ≥ 3 M τ f } \{r\geq3M_{\tau_f}\} { r ≥ 3 M τ f } 1 − μ 1-\mu 1 − μ u ^ ∈ [ 0 , u ^ 0 ] \widehat{u}\in[0,\widehat{u}_0] u ∈ [ 0 , u 0 ] ε \varepsilon ε (60) . The integral can then be subdivided into { r ≤ Λ } \{r\leq\Lambda\} { r ≤ Λ } { r ≥ Λ } \{r\geq\Lambda\} { r ≥ Λ } { 0 ≤ u ^ ≤ u ^ 0 } ∩ { r ≤ Λ } \{0\leq\widehat{u}\leq\widehat{u}_0\}\cap\{r\leq\Lambda\} { 0 ≤ u ≤ u 0 } ∩ { r ≤ Λ } (46) . Regarding the second factor in (169) , we obtain by (89) and (103) that
∣ exp ( ∫ Γ v ^ ( τ f , 1 ) Γ v ^ ( τ f , 2 ) 2 r ( 1 − μ ) ℑ [ ψ + ∂ v ^ ψ + ] d v ^ ) − 1 ∣ ≲ ∫ Γ v ^ ( τ f , 1 ) Γ v ^ ( τ f , 2 ) ε 2 r ( u ^ , v ^ ) − 4 d v ^ ≲ u ^ 0 ε 2 τ f , 1 − 3 , (170) \left|\exp\left(\int_{\Gamma^{\widehat{v}}(\tau_{f,1})}^{\Gamma^{\widehat{v}}(\tau_{f,2})}\frac{2}{r(1-\mu)}\Im[\psi_+\partial_{\widehat{v}}\psi_+]\,d\widehat{v}\right)-1\right|
\lesssim\int_{\Gamma^{\widehat{v}}(\tau_{f,1})}^{\Gamma^{\widehat{v}}(\tau_{f,2})}\varepsilon^2r(\widehat{u},\widehat{v})^{-4}\,d\widehat{v}
\lesssim_{\widehat{u}_0}\varepsilon^2\tau_{f,1}^{-3}, \tag*{(170)} exp ( ∫ Γ v ( τ f , 1 ) Γ v ( τ f , 2 ) r ( 1 − μ ) 2 ℑ [ ψ + ∂ v ψ + ] d v ) − 1 ≲ ∫ Γ v ( τ f , 1 ) Γ v ( τ f , 2 ) ε 2 r ( u , v ) − 4 d v ≲ u 0 ε 2 τ f , 1 − 3 , (170) for τ f , 1 \tau_{f,1} τ f , 1 ∣ u τ f , 2 ′ ( u ^ ) − u τ f , 1 ′ ( u ^ ) ∣ ≲ u ^ 0 ε 2 τ f , 1 − 3 |u'_{\tau_{f,2}}(\widehat{u})-u'_{\tau_{f,1}}(\widehat{u})|\lesssim_{\widehat{u}_0}\varepsilon^2\tau_{f,1}^{-3} ∣ u τ f , 2 ′ ( u ) − u τ f , 1 ′ ( u ) ∣ ≲ u 0 ε 2 τ f , 1 − 3 u ^ \widehat{u} u
∣ u τ f , 2 ( u ^ ) − u τ f , 1 ( u ^ ) ∣ ≲ u ^ 0 ε 2 τ f , 1 − 3 ≲ ε 2 η . |u_{\tau_{f,2}}(\widehat{u})-u_{\tau_{f,1}}(\widehat{u})|\lesssim_{\widehat{u}_0}\varepsilon^2\tau_{f,1}^{-3}\lesssim\varepsilon^2\eta. ∣ u τ f , 2 ( u ) − u τ f , 1 ( u ) ∣ ≲ u 0 ε 2 τ f , 1 − 3 ≲ ε 2 η . for τ f , 1 \tau_{f,1} τ f , 1 u ^ ∈ [ 0 , u ^ 0 ] \hat{u}\in[0,\hat{u}_0] u ^ ∈ [ 0 , u ^ 0 ] v v v (15) , which reads
∂ u ^ κ ^ = − 2 r κ ^ ℑ [ ψ − ‾ ∂ u ^ ψ − ] . \partial_{\hat{u}}\hat{\kappa}=-2r\hat{\kappa}\Im[\overline{\psi_-}\partial_{\hat{u}}\psi_-]. ∂ u ^ κ ^ = − 2 r κ ^ ℑ [ ψ − ∂ u ^ ψ − ] . Thus, from (29) we obtain
v τ f , 2 ′ ( v ^ ) − v τ f , 1 ′ ( v ^ ) = κ ^ ( Γ u ^ ( τ f , 2 ) , v ^ ) − κ ^ ( Γ u ^ ( τ f , 1 ) , v ^ ) = κ ^ ( Γ u ^ ( τ f , 1 ) , v ^ ) ( exp ( ∫ Γ u ^ ( τ f , 1 ) Γ u ^ ( τ f , 2 ) − 2 r ℑ [ ψ − ‾ ∂ u ^ ψ − ] d u ^ ) − 1 ) = 1 1 − μ ( 0 , v ^ ) exp ( ∫ 0 Γ u ^ ( τ f , 1 ) − 2 r ℑ [ ψ − ‾ ∂ u ^ ψ − ] d u ^ ) ( exp ( ∫ Γ u ^ ( τ f , 1 ) Γ u ^ ( τ f , 2 ) − 2 r ℑ [ ψ − ‾ ∂ u ^ ψ − ] d u ^ ) − 1 ) , (171) \begin{aligned}
v_{\tau_{f,2}}'(\hat{v})-v_{\tau_{f,1}}'(\hat{v})=\hat{\kappa}(\Gamma^{\hat{u}}(\tau_{f,2}),\hat{v})-\hat{\kappa}(\Gamma^{\hat{u}}(\tau_{f,1}),\hat{v}) \\
&=\hat{\kappa}(\Gamma^{\hat{u}}(\tau_{f,1}),\hat{v})\left(\exp\left(\int_{\Gamma^{\hat{u}}(\tau_{f,1})}^{\Gamma^{\hat{u}}(\tau_{f,2})}-2r\Im[\overline{\psi_-}\partial_{\hat{u}}\psi_-]\,d\hat{u}\right)-1\right) \\
&=\frac{1}{1-\mu(0,\hat{v})}\exp\left(\int_{0}^{\Gamma^{\hat{u}}(\tau_{f,1})}-2r\Im[\overline{\psi_-}\partial_{\hat{u}}\psi_-]\,d\hat{u}\right)\left(\exp\left(\int_{\Gamma^{\hat{u}}(\tau_{f,1})}^{\Gamma^{\hat{u}}(\tau_{f,2})}-2r\Im[\overline{\psi_-}\partial_{\hat{u}}\psi_-]\,d\hat{u}\right)-1\right), \tag*{(171)}
\end{aligned} v τ f , 2 ′ ( v ^ ) − v τ f , 1 ′ ( v ^ ) = κ ^ ( Γ u ^ ( τ f , 2 ) , v ^ ) − κ ^ ( Γ u ^ ( τ f , 1 ) , v ^ ) = κ ^ ( Γ u ^ ( τ f , 1 ) , v ^ ) ( exp ( ∫ Γ u ^ ( τ f , 1 ) Γ u ^ ( τ f , 2 ) − 2 r ℑ [ ψ − ∂ u ^ ψ − ] d u ^ ) − 1 ) = 1 − μ ( 0 , v ^ ) 1 exp ( ∫ 0 Γ u ^ ( τ f , 1 ) − 2 r ℑ [ ψ − ∂ u ^ ψ − ] d u ^ ) ( exp ( ∫ Γ u ^ ( τ f , 1 ) Γ u ^ ( τ f , 2 ) − 2 r ℑ [ ψ − ∂ u ^ ψ − ] d u ^ ) − 1 ) , (171) for v ^ ∈ [ 0 , v ^ 0 ] \hat{v}\in[0,\hat{v}_0] v ^ ∈ [ 0 , v ^ 0 ]
∣ 1 1 − μ ( 0 , v ^ ) exp ( ∫ 0 Γ u ^ ( τ f , 1 ) − 2 r ℑ [ ψ − ‾ ∂ u ^ ψ − ] d u ^ ) ∣ ≲ C ( v ^ 0 ) , \left|\frac{1}{1-\mu(0,\hat{v})}\exp\left(\int_{0}^{\Gamma^{\hat{u}}(\tau_{f,1})}-2r\Im[\overline{\psi_-}\partial_{\hat{u}}\psi_-]\,d\hat{u}\right)\right|\lesssim C(\hat{v}_0), 1 − μ ( 0 , v ^ ) 1 exp ( ∫ 0 Γ u ^ ( τ f , 1 ) − 2 r ℑ [ ψ − ∂ u ^ ψ − ] d u ^ ) ≲ C ( v ^ 0 ) , by treating separately the regions { r ≤ Λ } \{r\leq\Lambda\} { r ≤ Λ } { r ≥ Λ } \{r\geq\Lambda\} { r ≥ Λ } (51) , (45) and (47) . Regarding the last factor, we use (90) , (105) and the fact that { Γ u ^ ( τ f ) } τ f \{\Gamma^{\hat{u}}(\tau_f)\}_{\tau_f} { Γ u ^ ( τ f ) } τ f (167) , to bound
∣ exp ( ∫ Γ u ^ ( τ f , 1 ) Γ u ^ ( τ f , 2 ) − 2 r ℑ [ ψ − ‾ ∂ u ^ ψ − ] d u ^ ) − 1 ∣ ≲ ε 2 ∣ Γ u ^ ( τ f , 2 ) − Γ u ^ ( τ f , 1 ) ∣ ≲ ε 2 η , \left|\exp\left(\int_{\Gamma^{\hat{u}}(\tau_{f,1})}^{\Gamma^{\hat{u}}(\tau_{f,2})}-2r\Im[\overline{\psi_-}\partial_{\hat{u}}\psi_-]\,d\hat{u}\right)-1\right|\lesssim\varepsilon^2|\Gamma^{\hat{u}}(\tau_{f,2})-\Gamma^{\hat{u}}(\tau_{f,1})|\lesssim\varepsilon^2\eta, exp ( ∫ Γ u ^ ( τ f , 1 ) Γ u ^ ( τ f , 2 ) − 2 r ℑ [ ψ − ∂ u ^ ψ − ] d u ^ ) − 1 ≲ ε 2 ∣ Γ u ^ ( τ f , 2 ) − Γ u ^ ( τ f , 1 ) ∣ ≲ ε 2 η , for τ f , 1 \tau_{f,1} τ f , 1 ∣ v τ f , 2 ′ ( v ^ ) − v τ f , 1 ′ ( v ^ ) ∣ ≲ v ^ 0 ε 2 η |v_{\tau_{f,2}}'(\hat{v})-v_{\tau_{f,1}}'(\hat{v})|\lesssim_{\hat{v}_0}\varepsilon^2\eta ∣ v τ f , 2 ′ ( v ^ ) − v τ f , 1 ′ ( v ^ ) ∣ ≲ v ^ 0 ε 2 η v ^ \hat{v} v ^
∣ v τ f , 2 ( v ^ ) − v τ f , 1 ( v ^ ) ∣ ≲ v ^ 0 ε 2 η , |v_{\tau_{f,2}}(\hat{v})-v_{\tau_{f,1}}(\hat{v})|\lesssim_{\hat{v}_0}\varepsilon^2\eta, ∣ v τ f , 2 ( v ^ ) − v τ f , 1 ( v ^ ) ∣ ≲ v ^ 0 ε 2 η , for τ f , 1 \tau_{f,1} τ f , 1 v ^ ∈ [ 0 , v ^ 0 ] \hat{v}\in[0,\hat{v}_0] v ^ ∈ [ 0 , v ^ 0 ]
Let us now focus on the second derivatives. Differentiating (169) yields
u τ f , 2 ′ ′ ( u ^ ) − u τ f , 1 ′ ′ ( u ^ ) = [ ∂ u ^ μ ( u ^ , v ^ ∗ ( u ^ ) ) 1 − μ ( u ^ , v ^ ∗ ( u ^ ) ) + ∫ v ^ ∗ ( u ^ ) Γ u ^ ( τ f , 1 ) ∂ u ^ ( 2 r ( 1 − μ ) I m [ ψ ‾ + ∂ v ^ ψ + ] ) d v ^ ] ( u τ f , 2 ′ ( u ^ ) − u τ f , 1 ′ ( u ^ ) ) + 1 1 − μ ( u ^ , v ^ ∗ ( u ^ ) ) exp ( ∫ 0 Γ u ^ ( τ f , 2 ) 2 r ( 1 − μ ) I m [ ψ ‾ + ∂ v ^ ψ + ] d v ^ ) ∫ Γ u ^ ( τ f , 1 ) Γ u ^ ( τ f , 2 ) ∂ u ^ ( 2 r ( 1 − μ ) I m [ ψ ‾ + ∂ v ^ ψ + ] ) d v ^ . \begin{aligned}
u_{\tau_{f,2}}^{\prime\prime}(\hat{u})-u_{\tau_{f,1}}^{\prime\prime}(\hat{u})=\left[\frac{\partial_{\hat{u}}\mu(\hat{u},\hat{v}^{*}(\hat{u}))}{1-\mu(\hat{u},\hat{v}^{*}(\hat{u}))}+\int_{\hat{v}^{*}(\hat{u})}^{\Gamma^{\hat{u}}(\tau_{f,1})}\partial_{\hat{u}}\left(\frac{2}{r(1-\mu)}\mathfrak{Im}[\overline{\psi}_{+}\partial_{\hat{v}}\psi_{+}]\right)\,d\hat{v}\right]\left(u_{\tau_{f,2}}^{\prime}(\hat{u})-u_{\tau_{f,1}}^{\prime}(\hat{u})\right) \\
&+\frac{1}{1-\mu(\hat{u},\hat{v}^{*}(\hat{u}))}\exp\left(\int_{0}^{\Gamma^{\hat{u}}(\tau_{f,2})}\frac{2}{r(1-\mu)}\mathfrak{Im}[\overline{\psi}_{+}\partial_{\hat{v}}\psi_{+}]\,d\hat{v}\right)\int_{\Gamma^{\hat{u}}(\tau_{f,1})}^{\Gamma^{\hat{u}}(\tau_{f,2})}\partial_{\hat{u}}\left(\frac{2}{r(1-\mu)}\mathfrak{Im}[\overline{\psi}_{+}\partial_{\hat{v}}\psi_{+}]\right)\,d\hat{v}.
\end{aligned} u τ f , 2 ′′ ( u ^ ) − u τ f , 1 ′′ ( u ^ ) = [ 1 − μ ( u ^ , v ^ ∗ ( u ^ )) ∂ u ^ μ ( u ^ , v ^ ∗ ( u ^ )) + ∫ v ^ ∗ ( u ^ ) Γ u ^ ( τ f , 1 ) ∂ u ^ ( r ( 1 − μ ) 2 Im [ ψ + ∂ v ^ ψ + ] ) d v ^ ] ( u τ f , 2 ′ ( u ^ ) − u τ f , 1 ′ ( u ^ ) ) + 1 − μ ( u ^ , v ^ ∗ ( u ^ )) 1 exp ( ∫ 0 Γ u ^ ( τ f , 2 ) r ( 1 − μ ) 2 Im [ ψ + ∂ v ^ ψ + ] d v ^ ) ∫ Γ u ^ ( τ f , 1 ) Γ u ^ ( τ f , 2 ) ∂ u ^ ( r ( 1 − μ ) 2 Im [ ψ + ∂ v ^ ψ + ] ) d v ^ . Using (2.5)–(2.6), we explicitly bound
∣ ∂ u ^ ( 2 r ( 1 − μ ) I m [ ψ ‾ + ∂ v ^ ψ + ] ) ∣ = ∣ 2 r ( 1 − μ ) [ ( γ ^ r + 2 ∂ u ^ m r ( 1 − μ ) − ∂ u ^ κ ^ κ ^ ) I m [ ψ ‾ + ∂ v ^ ψ + ] + ν ^ I m [ ψ ‾ − ∂ v ^ ψ + ] \left|\partial_{\hat{u}}\left(\frac{2}{r(1-\mu)}\mathfrak{Im}[\overline{\psi}_{+}\partial_{\hat{v}}\psi_{+}]\right)\right|=\left|\frac{2}{r(1-\mu)}\left[\left(\frac{\hat{\gamma}}{r}+\frac{2\partial_{\hat{u}}m}{r(1-\mu)}-\frac{\partial_{\hat{u}}\hat{\kappa}}{\hat{\kappa}}\right)\mathfrak{Im}[\overline{\psi}_{+}\partial_{\hat{v}}\psi_{+}]+\hat{\nu}\mathfrak{Im}[\overline{\psi}_{-}\partial_{\hat{v}}\psi_{+}]\right.\right. ∂ u ^ ( r ( 1 − μ ) 2 Im [ ψ + ∂ v ^ ψ + ] ) = r ( 1 − μ ) 2 [ ( r γ ^ + r ( 1 − μ ) 2 ∂ u ^ m − κ ^ ∂ u ^ κ ^ ) Im [ ψ + ∂ v ^ ψ + ] + ν ^ Im [ ψ − ∂ v ^ ψ + ] + ( 1 2 ∂ v ^ ν ^ + Ω ^ 2 ( 1 − μ ) r ) I m [ ψ ‾ + ψ − ] ] ∣ ≲ u ^ 0 ε 2 r − 4 τ − 3 / 2 + δ / 2 ( 1 + τ − 3 / 2 + δ / 2 ) ≲ u ^ 0 ε 2 r − 4 τ − 3 / 2 + δ / 2 . \left.\left.+\left(\frac{1}{2}\partial_{\hat{v}}\hat{\nu}+\frac{\hat{\Omega}^{2}(1-\mu)}{r}\right)\mathfrak{Im}[\overline{\psi}_{+}\psi_{-}]\right]\right|\lesssim_{\hat{u}_{0}}\varepsilon^{2}r^{-4}\tau^{-3/2+\delta/2}(1+\tau^{-3/2+\delta/2})\lesssim_{\hat{u}_{0}}\varepsilon^{2}r^{-4}\tau^{-3/2+\delta/2}. + ( 2 1 ∂ v ^ ν ^ + r Ω ^ 2 ( 1 − μ ) ) Im [ ψ + ψ − ] ] ≲ u ^ 0 ε 2 r − 4 τ − 3/2 + δ /2 ( 1 + τ − 3/2 + δ /2 ) ≲ u ^ 0 ε 2 r − 4 τ − 3/2 + δ /2 . Therefore, with analogous arguments as before, we obtain ∣ u τ f , 2 ′ ′ − u τ f , 1 ′ ′ ∣ ≲ u ^ 0 ε 2 τ f , 1 − 3 ≲ ε 2 η |u_{\tau_{f,2}}^{\prime\prime}-u_{\tau_{f,1}}^{\prime\prime}|\lesssim_{\hat{u}_{0}}\varepsilon^{2}\tau_{f,1}^{-3}\lesssim\varepsilon^{2\eta} ∣ u τ f , 2 ′′ − u τ f , 1 ′′ ∣ ≲ u ^ 0 ε 2 τ f , 1 − 3 ≲ ε 2 η τ f , 1 \tau_{f,1} τ f , 1
Similarly, we differentiate (171) and find
v τ f , 2 ′ ′ ( v ^ ) − v τ f , 1 ′ ′ ( v ^ ) = [ ∂ v ^ μ ( 0 , v ^ ) 1 − μ ( 0 , v ^ ) + ∫ 0 Γ v ^ ( τ f , 1 ) ∂ v ^ ( − 2 r I m [ ψ ‾ − ∂ u ^ ψ − ] ) d u ^ ] ( v τ f , 2 ′ ( v ^ ) − v τ f , 1 ′ ( v ^ ) ) + 1 1 − μ ( 0 , v ^ ) exp ( ∫ 0 Γ v ^ ( τ f , 2 ) ( − 2 r I m [ ψ ‾ − ∂ u ^ ψ − ] ) d u ^ ) ∫ Γ v ^ ( τ f , 1 ) Γ v ^ ( τ f , 2 ) ∂ v ^ ( − 2 r I m [ ψ ‾ − ∂ u ^ ψ − ] ) d u ^ . \begin{aligned}
v_{\tau_{f,2}}^{\prime\prime}(\hat{v})-v_{\tau_{f,1}}^{\prime\prime}(\hat{v})=\left[\frac{\partial_{\hat{v}}\mu(0,\hat{v})}{1-\mu(0,\hat{v})}+\int_{0}^{\Gamma^{\hat{v}}(\tau_{f,1})}\partial_{\hat{v}}\left(-2r\mathfrak{Im}[\overline{\psi}_{-}\partial_{\hat{u}}\psi_{-}]\right)\,d\hat{u}\right]\left(v_{\tau_{f,2}}^{\prime}(\hat{v})-v_{\tau_{f,1}}^{\prime}(\hat{v})\right) \\
&+\frac{1}{1-\mu(0,\hat{v})}\exp\left(\int_{0}^{\Gamma^{\hat{v}}(\tau_{f,2})}\left(-2r\mathfrak{Im}[\overline{\psi}_{-}\partial_{\hat{u}}\psi_{-}]\right)\,d\hat{u}\right)\int_{\Gamma^{\hat{v}}(\tau_{f,1})}^{\Gamma^{\hat{v}}(\tau_{f,2})}\partial_{\hat{v}}\left(-2r\mathfrak{Im}[\overline{\psi}_{-}\partial_{\hat{u}}\psi_{-}]\right)\,d\hat{u}.
\end{aligned} v τ f , 2 ′′ ( v ^ ) − v τ f , 1 ′′ ( v ^ ) = [ 1 − μ ( 0 , v ^ ) ∂ v ^ μ ( 0 , v ^ ) + ∫ 0 Γ v ^ ( τ f , 1 ) ∂ v ^ ( − 2 r Im [ ψ − ∂ u ^ ψ − ] ) d u ^ ] ( v τ f , 2 ′ ( v ^ ) − v τ f , 1 ′ ( v ^ ) ) + 1 − μ ( 0 , v ^ ) 1 exp ( ∫ 0 Γ v ^ ( τ f , 2 ) ( − 2 r Im [ ψ − ∂ u ^ ψ − ] ) d u ^ ) ∫ Γ v ^ ( τ f , 1 ) Γ v ^ ( τ f , 2 ) ∂ v ^ ( − 2 r Im [ ψ − ∂ u ^ ψ − ] ) d u ^ . Again, we substitute (2.5)–(2.6) to compute
∂ v ^ ( − 2 r I m [ ψ ‾ − ∂ u ^ ψ − ] ) = 2 r ( ∂ v ^ ν ^ − ν ^ − λ ^ r ) I m [ ψ ‾ − ∂ u ^ ψ − ] + κ ^ r 2 I m [ ψ ‾ + ∂ u ^ ψ − ] + ( ∂ u ^ κ ^ 2 r 2 − 2 κ ^ ν ^ r 3 ) I m [ ψ ‾ − ψ + ] , \partial_{\hat{v}}\left(-2r\mathfrak{Im}[\overline{\psi}_{-}\partial_{\hat{u}}\psi_{-}]\right)=2r\left(\frac{\partial_{\hat{v}}\hat{\nu}}{-\hat{\nu}}-\frac{\hat{\lambda}}{r}\right)\mathfrak{Im}[\overline{\psi}_{-}\partial_{\hat{u}}\psi_{-}]+\frac{\hat{\kappa}}{r^{2}}\mathfrak{Im}[\overline{\psi}_{+}\partial_{\hat{u}}\psi_{-}]+\left(\frac{\partial_{\hat{u}}\hat{\kappa}}{2r^{2}}-\frac{2\hat{\kappa}\hat{\nu}}{r^{3}}\right)\mathfrak{Im}[\overline{\psi}_{-}\psi_{+}], ∂ v ^ ( − 2 r Im [ ψ − ∂ u ^ ψ − ] ) = 2 r ( − ν ^ ∂ v ^ ν ^ − r λ ^ ) Im [ ψ − ∂ u ^ ψ − ] + r 2 κ ^ Im [ ψ + ∂ u ^ ψ − ] + ( 2 r 2 ∂ u ^ κ ^ − r 3 2 κ ^ ν ^ ) Im [ ψ − ψ + ] , which can be uniformly bounded as before by Lemmas 5.1, 5.6 and 5.8:
∣ ∂ v ^ ( − 2 r I m [ ψ ‾ − ∂ u ^ ψ − ] ) ∣ ≲ ε 2 . \left|\partial_{\hat{v}}\left(-2r\mathfrak{Im}[\overline{\psi}_{-}\partial_{\hat{u}}\psi_{-}]\right)\right|\lesssim\varepsilon^{2}. ∂ v ^ ( − 2 r Im [ ψ − ∂ u ^ ψ − ] ) ≲ ε 2 . Therefore, using again the fact that { Γ u ^ ( τ f ) } τ f \{\Gamma^{\hat{u}}(\tau_{f})\}_{\tau_{f}} { Γ u ^ ( τ f ) } τ f
∣ ∫ Γ u ^ ( τ f , 1 ) Γ u ^ ( τ f , 2 ) ∂ v ^ ( − 2 r I m [ ψ ‾ − ∂ u ^ ψ − ] ) d u ^ ∣ ≲ v ^ 0 ε 2 ∣ Γ u ^ ( τ f , 2 ) − Γ u ^ ( τ f , 1 ) ∣ ≲ v ^ 0 ε 2 η , \left|\int_{\Gamma^{\hat{u}}(\tau_{f,1})}^{\Gamma^{\hat{u}}(\tau_{f,2})}\partial_{\hat{v}}\left(-2r\mathfrak{Im}[\overline{\psi}_{-}\partial_{\hat{u}}\psi_{-}]\right)\,d\hat{u}\right|\lesssim_{\hat{v}_{0}}\varepsilon^{2}|\Gamma^{\hat{u}}(\tau_{f,2})-\Gamma^{\hat{u}}(\tau_{f,1})|\lesssim_{\hat{v}_{0}}\varepsilon^{2}\eta, ∫ Γ u ^ ( τ f , 1 ) Γ u ^ ( τ f , 2 ) ∂ v ^ ( − 2 r Im [ ψ − ∂ u ^ ψ − ] ) d u ^ ≲ v ^ 0 ε 2 ∣ Γ u ^ ( τ f , 2 ) − Γ u ^ ( τ f , 1 ) ∣ ≲ v ^ 0 ε 2 η , for τ f , 1 \tau_{f,1} τ f , 1 ∣ v τ f , 2 ′ ′ − v τ f , 1 ′ ′ ∣ ≲ v ^ 0 ε 2 η |v_{\tau_{f,2}}^{\prime\prime}-v_{\tau_{f,1}}^{\prime\prime}|\lesssim_{\hat{v}_{0}}\varepsilon^{2}\eta ∣ v τ f , 2 ′′ − v τ f , 1 ′′ ∣ ≲ v ^ 0 ε 2 η τ f , 1 \tau_{f,1} τ f , 1
Let us now prove Proposition 7.4.
Proof of Proposition 7.4. Existence and regularity of Φ ∞ = ( u ∞ , v ∞ ) \Phi_{\infty}=(u_{\infty},v_{\infty}) Φ ∞ = ( u ∞ , v ∞ ) u ∞ u_{\infty} u ∞ v ∞ v_{\infty} v ∞
d u ∞ d u ^ = lim τ f → + ∞ γ ^ ( u ^ , Γ u ^ ( τ f ) ) = 1 1 − μ ( u ^ , v ^ ∗ ( u ^ ) ) exp ( ∫ v ^ ∗ ( u ^ ) + ∞ 2 r ( 1 − μ ) I m [ ψ ‾ + ∂ v ^ ψ + ] d v ^ ) > 0 \frac{du_{\infty}}{d\hat{u}}=\lim_{\tau_{f}\to+\infty}\hat{\gamma}(\hat{u},\Gamma^{\hat{u}}(\tau_{f}))=\frac{1}{1-\mu(\hat{u},\hat{v}^{*}(\hat{u}))}\exp\left(\int_{\hat{v}^{*}(\hat{u})}^{+\infty}\frac{2}{r(1-\mu)}\mathfrak{Im}[\overline{\psi}_{+}\partial_{\hat{v}}\psi_{+}]\,d\hat{v}\right)>0 d u ^ d u ∞ = τ f → + ∞ lim γ ^ ( u ^ , Γ u ^ ( τ f )) = 1 − μ ( u ^ , v ^ ∗ ( u ^ )) 1 exp ( ∫ v ^ ∗ ( u ^ ) + ∞ r ( 1 − μ ) 2 Im [ ψ + ∂ v ^ ψ + ] d v ^ ) > 0 by (4.12), and
d v ∞ d v ^ = lim τ f → + ∞ κ ^ ( Γ v ^ ( τ f ) , v ^ ) > 0 , \frac{dv_{\infty}}{d\hat{v}}=\lim_{\tau_{f}\to+\infty}\hat{\kappa}(\Gamma^{\hat{v}}(\tau_{f}),\hat{v})>0, d v ^ d v ∞ = τ f → + ∞ lim κ ^ ( Γ v ^ ( τ f ) , v ^ ) > 0 , by Lemma 4.10. Moreover, u ∞ u_{\infty} u ∞ [ 0 , + ∞ ) [0,+\infty) [ 0 , + ∞ ) u ∞ ( 0 ) = 0 u_{\infty}(0)=0 u ∞ ( 0 ) = 0 u τ f ( Γ u ^ ( τ f ) ) ∼ τ f u_{\tau_{f}}(\Gamma^{\hat{u}}(\tau_{f}))\sim\tau_{f} u τ f ( Γ u ^ ( τ f )) ∼ τ f τ f → + ∞ \tau_{f}\to+\infty τ f → + ∞ v ∞ v_{\infty} v ∞
From now on, we denote the eschatological gauge by ( u ∞ , v ∞ ) : = Φ ∞ ( u ^ , v ^ ) (u_\infty,v_\infty):=\Phi_\infty(\hat{u},\hat{v}) ( u ∞ , v ∞ ) := Φ ∞ ( u ^ , v ^ ) D ∞ \mathcal{D}_\infty D ∞ Φ ∞ \Phi_\infty Φ ∞ C 2 C^2 C 2 ( r , Ω ^ , ψ + , ψ − ) (r,\hat{\Omega},\psi_+,\psi_-) ( r , Ω ^ , ψ + , ψ − ) C 2 C^2 C 2
Since the previous estimate is not uniform up to the horizon, and therefore problematic to uniformly control the anchored backgrounds, we prove the following proposition, which bounds the mapping Φ τ f , 1 ↦ Φ τ f , 2 \Phi_{\tau_{f,1}}\mapsto\Phi_{\tau_{f,2}} Φ τ f , 1 ↦ Φ τ f , 2 τ f , 1 → + ∞ \tau_{f,1}\to+\infty τ f , 1 → + ∞
Lemma 7.9. For 1 ≤ τ f , 1 < τ f , 2 ≤ + ∞ 1\leq\tau_{f,1}<\tau_{f,2}\leq+\infty 1 ≤ τ f , 1 < τ f , 2 ≤ + ∞ τ f , 1 \tau_{f,1} τ f , 1
sup J − ( Γ ( τ f , 1 ) ) ∩ D τ f , 2 ∣ Φ τ f , 1 ∘ Φ ^ τ f , 2 − Id ∣ ≲ ε 2 τ f , 1 − 1 , (172) \sup_{J^-(\Gamma(\tau_{f,1}))\cap\mathcal{D}_{\tau_{f,2}}}\left|\Phi_{\tau_{f,1}}\circ\hat{\Phi}_{\tau_{f,2}}-\operatorname{Id}\right|\lesssim\varepsilon^2\tau_{f,1}^{-1}, \tag*{(172)} J − ( Γ ( τ f , 1 )) ∩ D τ f , 2 sup Φ τ f , 1 ∘ Φ ^ τ f , 2 − Id ≲ ε 2 τ f , 1 − 1 , (172) sup J − ( Γ ( τ f , 1 ) ) ∩ D τ f , 2 ∣ d ( Φ τ f , 1 ∘ Φ ^ τ f , 2 − Id ) ∣ ≲ ε 2 τ f , 1 − 2 . (173) \sup_{J^-(\Gamma(\tau_{f,1}))\cap\mathcal{D}_{\tau_{f,2}}}\left|d(\Phi_{\tau_{f,1}}\circ\hat{\Phi}_{\tau_{f,2}}-\operatorname{Id})\right|\lesssim\varepsilon^2\tau_{f,1}^{-2}. \tag*{(173)} J − ( Γ ( τ f , 1 )) ∩ D τ f , 2 sup d ( Φ τ f , 1 ∘ Φ ^ τ f , 2 − Id ) ≲ ε 2 τ f , 1 − 2 . (173) Proof. From Definition 3.6, we write Φ τ f , 1 ∘ Φ ^ τ f , 2 = g τ f , 1 ; τ f , 2 × h τ f , 1 ; τ f , 2 \Phi_{\tau_{f,1}}\circ\hat{\Phi}_{\tau_{f,2}}=g_{\tau_{f,1};\tau_{f,2}}\times h_{\tau_{f,1};\tau_{f,2}} Φ τ f , 1 ∘ Φ ^ τ f , 2 = g τ f , 1 ; τ f , 2 × h τ f , 1 ; τ f , 2
g τ f , 1 ; τ f , 2 ( u τ f , 2 ) : = ∫ 0 u τ f , 2 γ τ f , 2 ( u τ f , 2 ′ , Γ τ f , 2 v τ f , 2 ( τ f , 1 ) ) d u τ f , 2 ′ = u τ f , 2 + ∫ 0 u τ f , 2 ( γ τ f , 2 ( u τ f , 2 ′ , Γ τ f , 2 v τ f , 2 ( τ f , 1 ) ) − 1 ) d u τ f , 2 ′ , g_{\tau_{f,1};\tau_{f,2}}(u_{\tau_{f,2}}):=\int_0^{u_{\tau_{f,2}}}\gamma_{\tau_{f,2}}\left(u'_{\tau_{f,2}},\Gamma^{v_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})\right)\,\mathrm{d}u'_{\tau_{f,2}}
=u_{\tau_{f,2}}+\int_0^{u_{\tau_{f,2}}}\left(\gamma_{\tau_{f,2}}\left(u'_{\tau_{f,2}},\Gamma^{v_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})\right)-1\right)\,\mathrm{d}u'_{\tau_{f,2}}, g τ f , 1 ; τ f , 2 ( u τ f , 2 ) := ∫ 0 u τ f , 2 γ τ f , 2 ( u τ f , 2 ′ , Γ τ f , 2 v τ f , 2 ( τ f , 1 ) ) d u τ f , 2 ′ = u τ f , 2 + ∫ 0 u τ f , 2 ( γ τ f , 2 ( u τ f , 2 ′ , Γ τ f , 2 v τ f , 2 ( τ f , 1 ) ) − 1 ) d u τ f , 2 ′ , h τ f , 1 ; τ f , 2 ( v τ f , 2 ) : = ∫ 0 v τ f , 2 κ τ f , 2 ( Γ τ f , 2 u τ f , 2 ( τ f , 1 ) , v τ f , 2 ′ ) d v τ f , 2 ′ = v τ f , 2 + ∫ 0 v τ f , 2 ( κ τ f , 2 ( Γ τ f , 2 u τ f , 2 ( τ f , 1 ) , v τ f , 2 ′ ) − 1 ) d v τ f , 2 ′ . h_{\tau_{f,1};\tau_{f,2}}(v_{\tau_{f,2}}):=\int_0^{v_{\tau_{f,2}}}\kappa_{\tau_{f,2}}\left(\Gamma^{u_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1}),v'_{\tau_{f,2}}\right)\,\mathrm{d}v'_{\tau_{f,2}}
=v_{\tau_{f,2}}+\int_0^{v_{\tau_{f,2}}}\left(\kappa_{\tau_{f,2}}\left(\Gamma^{u_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1}),v'_{\tau_{f,2}}\right)-1\right)\,\mathrm{d}v'_{\tau_{f,2}}. h τ f , 1 ; τ f , 2 ( v τ f , 2 ) := ∫ 0 v τ f , 2 κ τ f , 2 ( Γ τ f , 2 u τ f , 2 ( τ f , 1 ) , v τ f , 2 ′ ) d v τ f , 2 ′ = v τ f , 2 + ∫ 0 v τ f , 2 ( κ τ f , 2 ( Γ τ f , 2 u τ f , 2 ( τ f , 1 ) , v τ f , 2 ′ ) − 1 ) d v τ f , 2 ′ . Hence, by (58) , we bound
sup u τ f , 2 ∈ [ 0 , Γ τ f , 2 u τ f , 2 ( τ f , 1 ) ] ∣ g τ f , 1 ; τ f , 2 ′ ( u τ f , 2 ) − 1 ∣ ≲ sup u τ f , 2 ∈ [ 0 , Γ τ f , 2 u τ f , 2 ( τ f , 1 ) ] ∣ γ ~ τ f , 2 ( u τ f , 2 , Γ τ f , 2 v τ f , 2 ( τ f , 1 ) ) ∣ \sup_{u_{\tau_{f,2}}\in[0,\Gamma^{u_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})]}\left|g'_{\tau_{f,1};\tau_{f,2}}(u_{\tau_{f,2}})-1\right|
\lesssim
\sup_{u_{\tau_{f,2}}\in[0,\Gamma^{u_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})]}\left|\widetilde{\gamma}_{\tau_{f,2}}\left(u_{\tau_{f,2}},\Gamma^{v_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})\right)\right| u τ f , 2 ∈ [ 0 , Γ τ f , 2 u τ f , 2 ( τ f , 1 )] sup g τ f , 1 ; τ f , 2 ′ ( u τ f , 2 ) − 1 ≲ u τ f , 2 ∈ [ 0 , Γ τ f , 2 u τ f , 2 ( τ f , 1 )] sup γ τ f , 2 ( u τ f , 2 , Γ τ f , 2 v τ f , 2 ( τ f , 1 ) ) ≲ sup u τ f , 2 ∈ [ 0 , Γ τ f , 2 u τ f , 2 ( τ f , 1 ) ] ε 2 τ f , 2 − 2 τ f , 1 − 3 + δ ( u τ f , 2 , Γ τ f , 2 v τ f , 2 ( τ f , 1 ) ) ≲ ε 2 τ f , 1 − 2 , \lesssim
\sup_{u_{\tau_{f,2}}\in[0,\Gamma^{u_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})]}
\varepsilon^2\tau_{f,2}^{-2}\tau_{f,1}^{-3+\delta}\left(u_{\tau_{f,2}},\Gamma^{v_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})\right)
\lesssim\varepsilon^2\tau_{f,1}^{-2}, ≲ u τ f , 2 ∈ [ 0 , Γ τ f , 2 u τ f , 2 ( τ f , 1 )] sup ε 2 τ f , 2 − 2 τ f , 1 − 3 + δ ( u τ f , 2 , Γ τ f , 2 v τ f , 2 ( τ f , 1 ) ) ≲ ε 2 τ f , 1 − 2 , where we have split [ 0 , Γ τ f , 2 u τ f , 2 ( τ f , 1 ) ] × { Γ τ f , 2 v τ f , 2 ( τ f , 1 ) } [0,\Gamma^{u_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})]\times\{\Gamma^{v_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})\} [ 0 , Γ τ f , 2 u τ f , 2 ( τ f , 1 )] × { Γ τ f , 2 v τ f , 2 ( τ f , 1 )} { τ ≤ 1 2 τ f , 1 } \{\tau\leq\frac{1}{2}\tau_{f,1}\} { τ ≤ 2 1 τ f , 1 } r ≳ τ f , 1 r\gtrsim\tau_{f,1} r ≳ τ f , 1 { τ ≥ 1 2 τ f , 1 } \{\tau\geq\frac{1}{2}\tau_{f,1}\} { τ ≥ 2 1 τ f , 1 } r ≲ τ f , 1 r\lesssim\tau_{f,1} r ≲ τ f , 1
On the other hand, we analogously need to estimate
sup v τ f , 2 ∈ [ 0 , Γ τ f , 2 v τ f , 2 ( τ f , 1 ) ] ∣ h τ f , 1 ; τ f , 2 ′ ( v τ f , 2 ) − 1 ∣ ≲ sup v τ f , 2 ∈ [ 0 , Γ τ f , 2 v τ f , 2 ( τ f , 1 ) ] ∣ κ ~ τ f , 2 ( Γ τ f , 2 u τ f , 2 ( τ f , 1 ) , v τ f , 2 ) ∣ . \sup_{v_{\tau_{f,2}}\in[0,\Gamma^{v_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})]}\left|h'_{\tau_{f,1};\tau_{f,2}}(v_{\tau_{f,2}})-1\right|
\lesssim
\sup_{v_{\tau_{f,2}}\in[0,\Gamma^{v_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})]}
\left|\widetilde{\kappa}_{\tau_{f,2}}\left(\Gamma^{u_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1}),v_{\tau_{f,2}}\right)\right|. v τ f , 2 ∈ [ 0 , Γ τ f , 2 v τ f , 2 ( τ f , 1 )] sup h τ f , 1 ; τ f , 2 ′ ( v τ f , 2 ) − 1 ≲ v τ f , 2 ∈ [ 0 , Γ τ f , 2 v τ f , 2 ( τ f , 1 )] sup κ τ f , 2 ( Γ τ f , 2 u τ f , 2 ( τ f , 1 ) , v τ f , 2 ) . We proceed by subdividing again { Γ τ f , 2 u τ f , 2 ( τ f , 1 ) } × [ 0 , Γ τ f , 2 v τ f , 2 ( τ f , 1 ) ] \{\Gamma^{u_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})\}\times[0,\Gamma^{v_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})] { Γ τ f , 2 u τ f , 2 ( τ f , 1 )} × [ 0 , Γ τ f , 2 v τ f , 2 ( τ f , 1 )] { τ ≤ 1 2 τ f , 1 } \{\tau\leq\frac{1}{2}\tau_{f,1}\} { τ ≤ 2 1 τ f , 1 } { τ ≥ 1 2 τ f , 1 } \{\tau\geq\frac{1}{2}\tau_{f,1}\} { τ ≥ 2 1 τ f , 1 }
Let us begin with the latter: since u ≲ τ f , 1 u\lesssim\tau_{f,1} u ≲ τ f , 1 (53) readily yields
∣ h τ f , 1 ; τ f , 2 ′ ( v τ f , 2 ) − 1 ∣ ≲ ε 2 τ f , 1 − 3 + δ ( Γ τ f , 2 u τ f , 2 ( τ f , 1 ) , v τ f , 2 ) ≲ ε 2 τ f , 1 − 3 + δ , \left|h'_{\tau_{f,1};\tau_{f,2}}(v_{\tau_{f,2}})-1\right|
\lesssim\varepsilon^2\tau_{f,1}^{-3+\delta}\left(\Gamma^{u_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1}),v_{\tau_{f,2}}\right)
\lesssim\varepsilon^2\tau_{f,1}^{-3+\delta}, h τ f , 1 ; τ f , 2 ′ ( v τ f , 2 ) − 1 ≲ ε 2 τ f , 1 − 3 + δ ( Γ τ f , 2 u τ f , 2 ( τ f , 1 ) , v τ f , 2 ) ≲ ε 2 τ f , 1 − 3 + δ , for v τ f , 2 ∈ [ 0 , Γ τ f , 2 v τ f , 2 ( τ f , 1 ) ] ∩ { τ ≥ 1 2 τ f , 1 } v_{\tau_{f,2}}\in[0,\Gamma^{v_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})]\cap\{\tau\geq\frac{1}{2}\tau_{f,1}\} v τ f , 2 ∈ [ 0 , Γ τ f , 2 v τ f , 2 ( τ f , 1 )] ∩ { τ ≥ 2 1 τ f , 1 } { τ ≤ 1 2 τ f , 1 } \{\tau\leq\frac{1}{2}\tau_{f,1}\} { τ ≤ 2 1 τ f , 1 } κ ~ \widetilde{\kappa} κ u ≳ τ f , 1 u\gtrsim\tau_{f,1} u ≳ τ f , 1 (116) with the exponential decay estimate (118) to obtain
∣ κ ~ ( Γ τ f , 2 u τ f , 2 ( τ f , 1 ) , v τ f , 2 ) ∣ ≲ ∫ Γ τ f , 2 u τ f , 2 ( τ f , 1 ) Γ τ f , 2 u τ f , 2 ( τ f , 2 ) ∣ ∂ u κ ~ ∣ d u ′ ≲ ε 2 ∫ Γ τ f , 2 u τ f , 2 ( τ f , 1 ) Γ τ f , 2 u τ f , 2 ( τ f , 2 ) ( − ν ) r ˉ − 1 τ − 3 / 2 + δ / 2 d u ′ \left|\widetilde{\kappa}\left(\Gamma^{u_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1}),v_{\tau_{f,2}}\right)\right|
\lesssim\int_{\Gamma^{u_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})}^{\Gamma^{u_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,2})}\left|\partial_u\widetilde{\kappa}\right|\,\mathrm{d}u'
\lesssim\varepsilon^2\int_{\Gamma^{u_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})}^{\Gamma^{u_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,2})}(-\nu)\bar{r}^{-1}\tau^{-3/2+\delta/2}\,\mathrm{d}u' κ ( Γ τ f , 2 u τ f , 2 ( τ f , 1 ) , v τ f , 2 ) ≲ ∫ Γ τ f , 2 u τ f , 2 ( τ f , 1 ) Γ τ f , 2 u τ f , 2 ( τ f , 2 ) ∣ ∂ u κ ∣ d u ′ ≲ ε 2 ∫ Γ τ f , 2 u τ f , 2 ( τ f , 1 ) Γ τ f , 2 u τ f , 2 ( τ f , 2 ) ( − ν ) r ˉ − 1 τ − 3/2 + δ /2 d u ′ ≲ ε 2 ∫ Γ τ f , 2 u τ f , 2 ( τ f , 1 ) Γ τ f , 2 u τ f , 2 ( τ f , 2 ) ( − ν ) d u ′ ≲ ε 2 e − Γ τ f , 2 u τ f , 2 ( τ f , 1 ) , \lesssim\varepsilon^2\int_{\Gamma^{u_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})}^{\Gamma^{u_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,2})}(-\nu)\,\mathrm{d}u'
\lesssim\varepsilon^2e^{-\Gamma^{u_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})}, ≲ ε 2 ∫ Γ τ f , 2 u τ f , 2 ( τ f , 1 ) Γ τ f , 2 u τ f , 2 ( τ f , 2 ) ( − ν ) d u ′ ≲ ε 2 e − Γ τ f , 2 u τ f , 2 ( τ f , 1 ) , where we used that κ ~ ( Γ τ f , 2 u τ f , 2 ( τ f , 2 ) , v τ f , 2 ) = 0 \widetilde{\kappa}(\Gamma^{u_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,2}),v_{\tau_{f,2}})=0 κ ( Γ τ f , 2 u τ f , 2 ( τ f , 2 ) , v τ f , 2 ) = 0 r ˉ − 1 τ − 3 / 2 + δ / 2 \bar{r}^{-1}\tau^{-3/2+\delta/2} r ˉ − 1 τ − 3/2 + δ /2 v τ f , 2 ∈ [ 0 , Γ τ f , 2 v τ f , 2 ( τ f , 1 ) ] ∩ { τ ≤ 1 2 τ f , 1 } v_{\tau_{f,2}}\in[0,\Gamma^{v_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})]\cap\{\tau\leq\frac{1}{2}\tau_{f,1}\} v τ f , 2 ∈ [ 0 , Γ τ f , 2 v τ f , 2 ( τ f , 1 )] ∩ { τ ≤ 2 1 τ f , 1 } ε 2 τ f , 1 − 3 + δ \varepsilon^2\tau_{f,1}^{-3+\delta} ε 2 τ f , 1 − 3 + δ τ f , 1 \tau_{f,1} τ f , 1
sup u τ f , 2 ∈ [ 0 , Γ τ f , 2 u τ f , 2 ( τ f , 1 ) ] ∣ g τ f , 1 ; τ f , 2 ′ ( u τ f , 2 ) − 1 ∣ ≲ ε 2 τ f , 1 − 2 , sup v τ f , 2 ∈ [ 0 , Γ τ f , 2 v τ f , 2 ( τ f , 1 ) ] ∣ h τ f , 1 ; τ f , 2 ′ ( v τ f , 2 ) − 1 ∣ ≲ ε 2 τ f , 1 − 3 + δ , \sup_{u_{\tau_{f,2}}\in[0,\Gamma^{u_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})]}\left|g'_{\tau_{f,1};\tau_{f,2}}(u_{\tau_{f,2}})-1\right|\lesssim\varepsilon^2\tau_{f,1}^{-2},
\qquad
\sup_{v_{\tau_{f,2}}\in[0,\Gamma^{v_{\tau_{f,2}}}_{\tau_{f,2}}(\tau_{f,1})]}\left|h'_{\tau_{f,1};\tau_{f,2}}(v_{\tau_{f,2}})-1\right|\lesssim\varepsilon^2\tau_{f,1}^{-3+\delta}, u τ f , 2 ∈ [ 0 , Γ τ f , 2 u τ f , 2 ( τ f , 1 )] sup g τ f , 1 ; τ f , 2 ′ ( u τ f , 2 ) − 1 ≲ ε 2 τ f , 1 − 2 , v τ f , 2 ∈ [ 0 , Γ τ f , 2 v τ f , 2 ( τ f , 1 )] sup h τ f , 1 ; τ f , 2 ′ ( v τ f , 2 ) − 1 ≲ ε 2 τ f , 1 − 3 + δ , which readily implies (172) and (173) .
We next turn to determining the final mass of the anchored Schwarzschild background.
Lemma 7.10. The following limit exists
M ∞ : = lim τ f → + ∞ M τ f . (174) M_{\infty} := \lim_{\tau_f \to+\infty} M_{\tau_f}. \tag*{(174)} M ∞ := τ f → + ∞ lim M τ f . (174) Proof. It is a straightforward consequence of (32) and (44) . Indeed, one can note that the sequence { M τ f } τ f \{M_{\tau_f}\}_{\tau_f} { M τ f } τ f
∣ M τ f , 2 − M τ f , 1 ∣ = ∣ M τ f , 2 − m τ f , 2 ( Γ ( τ f , 1 ) ) ∣ ≲ ∣ m ~ τ f , 2 ( Γ ( τ f , 1 ) ) ∣ ≲ ε 3 / 2 τ f , 1 − 3 + δ , (175) \left|M_{\tau_{f,2}}-M_{\tau_{f,1}}\right|=\left|M_{\tau_{f,2}}-m_{\tau_{f,2}}\left(\Gamma(\tau_{f,1})\right)\right|\lesssim\left|\widetilde{m}_{\tau_{f,2}}\left(\Gamma(\tau_{f,1})\right)\right|\lesssim\varepsilon^{3/2}\tau_{f,1}^{-3+\delta}, \tag*{(175)} M τ f , 2 − M τ f , 1 = M τ f , 2 − m τ f , 2 ( Γ ( τ f , 1 ) ) ≲ m τ f , 2 ( Γ ( τ f , 1 ) ) ≲ ε 3/2 τ f , 1 − 3 + δ , (175) and therefore denote by M ∞ M_{\infty} M ∞
Thanks to Lemmas 7.9 and 7.10 , we are now able to compare r ˉ τ f , 1 \bar{r}_{\tau_{f,1}} r ˉ τ f , 1 r ˉ τ f , 2 \bar{r}_{\tau_{f,2}} r ˉ τ f , 2 D ^ ∞ \widehat{\mathcal{D}}_{\infty} D ∞ (168) , we instead pull r ˉ τ f , 1 \bar{r}_{\tau_{f,1}} r ˉ τ f , 1 ( u τ f , 2 , v τ f , 2 ) (u_{\tau_{f,2}},v_{\tau_{f,2}}) ( u τ f , 2 , v τ f , 2 ) (172) .
Lemma 7.11. For 1 ≤ τ f , 1 < τ f , 2 ≤ + ∞ 1\leq\tau_{f,1}<\tau_{f,2}\leq+\infty 1 ≤ τ f , 1 < τ f , 2 ≤ + ∞ τ f , 1 \tau_{f,1} τ f , 1
sup J − ( Γ ( τ f , 1 ) ) ∩ D τ f , 2 ∣ r ˉ τ f , 2 − r ˉ τ f , 1 ∘ ( Φ τ f , 1 ∘ Φ ^ τ f , 2 ) ∣ ≲ ε 3 / 2 τ f , 1 − 1 / 2 . (176) \sup_{J^{-}(\Gamma(\tau_{f,1}))\cap\mathcal{D}_{\tau_{f,2}}}\left|\bar{r}_{\tau_{f,2}}-\bar{r}_{\tau_{f,1}}\circ\left(\Phi_{\tau_{f,1}}\circ\widehat{\Phi}_{\tau_{f,2}}\right)\right|\lesssim\varepsilon^{3/2}\tau_{f,1}^{-1/2}. \tag*{(176)} J − ( Γ ( τ f , 1 )) ∩ D τ f , 2 sup r ˉ τ f , 2 − r ˉ τ f , 1 ∘ ( Φ τ f , 1 ∘ Φ τ f , 2 ) ≲ ε 3/2 τ f , 1 − 1/2 . (176) Proof. Let us define ϱ : = r ˉ τ f , 2 − r ˉ τ f , 1 ∘ ( Φ τ f , 1 ∘ Φ ^ τ f , 2 ) \varrho:=\bar{r}_{\tau_{f,2}}-\bar{r}_{\tau_{f,1}}\circ\left(\Phi_{\tau_{f,1}}\circ\widehat{\Phi}_{\tau_{f,2}}\right) ϱ := r ˉ τ f , 2 − r ˉ τ f , 1 ∘ ( Φ τ f , 1 ∘ Φ τ f , 2 ) u τ f , 2 u_{\tau_{f,2}} u τ f , 2
∂ u τ f , 2 ϱ = ν ˉ τ f , 2 − ( ν ˉ τ f , 1 ∘ ( Φ τ f , 1 ∘ Φ ^ τ f , 2 ) ) α τ f , 1 ; τ f , 2 ′ = ( 1 − 2 M τ f , 1 r ˉ τ f , 1 ∘ ( Φ τ f , 1 ∘ Φ ^ τ f , 2 ) ) α τ f , 1 ; τ f , 2 ′ − ( 1 − 2 M τ f , 2 r ˉ τ f , 2 ) \partial_{u_{\tau_{f,2}}}\varrho=\bar{\nu}_{\tau_{f,2}}-\left(\bar{\nu}_{\tau_{f,1}}\circ\left(\Phi_{\tau_{f,1}}\circ\widehat{\Phi}_{\tau_{f,2}}\right)\right)\alpha'_{\tau_{f,1};\tau_{f,2}}=\left(1-\frac{2M_{\tau_{f,1}}}{\bar{r}_{\tau_{f,1}}\circ\left(\Phi_{\tau_{f,1}}\circ\widehat{\Phi}_{\tau_{f,2}}\right)}\right)\alpha'_{\tau_{f,1};\tau_{f,2}}-\left(1-\frac{2M_{\tau_{f,2}}}{\bar{r}_{\tau_{f,2}}}\right) ∂ u τ f , 2 ϱ = ν ˉ τ f , 2 − ( ν ˉ τ f , 1 ∘ ( Φ τ f , 1 ∘ Φ τ f , 2 ) ) α τ f , 1 ; τ f , 2 ′ = 1 − r ˉ τ f , 1 ∘ ( Φ τ f , 1 ∘ Φ τ f , 2 ) 2 M τ f , 1 α τ f , 1 ; τ f , 2 ′ − ( 1 − r ˉ τ f , 2 2 M τ f , 2 ) = ( 1 − 2 M τ f , 1 r ˉ τ f , 1 ∘ ( Φ τ f , 1 ∘ Φ ^ τ f , 2 ) ) − ( 1 − 2 M τ f , 2 r ˉ τ f , 2 ) + O ( ε 2 τ f , 1 − 2 ) , =\left(1-\frac{2M_{\tau_{f,1}}}{\bar{r}_{\tau_{f,1}}\circ\left(\Phi_{\tau_{f,1}}\circ\widehat{\Phi}_{\tau_{f,2}}\right)}\right)-\left(1-\frac{2M_{\tau_{f,2}}}{\bar{r}_{\tau_{f,2}}}\right)+O\left(\varepsilon^{2}\tau_{f,1}^{-2}\right), = 1 − r ˉ τ f , 1 ∘ ( Φ τ f , 1 ∘ Φ τ f , 2 ) 2 M τ f , 1 − ( 1 − r ˉ τ f , 2 2 M τ f , 2 ) + O ( ε 2 τ f , 1 − 2 ) , where we used (173) in the last equality. We now sum and subtract the term 2 M τ f , 1 / r ˉ τ f , 2 2M_{\tau_{f,1}}/\bar{r}_{\tau_{f,2}} 2 M τ f , 1 / r ˉ τ f , 2 (175) , and deduce
∂ u τ f , 2 ϱ = 2 ( M τ f , 2 − M τ f , 1 ) r ˉ τ f , 2 + 2 M τ f , 1 ( 1 r ˉ τ f , 2 − 1 r ˉ τ f , 1 ∘ ( Φ τ f , 1 ∘ Φ ^ τ f , 2 ) ) + O ( ε 2 τ f , 1 − 2 ) \partial_{u_{\tau_{f,2}}}\varrho=\frac{2(M_{\tau_{f,2}}-M_{\tau_{f,1}})}{\bar{r}_{\tau_{f,2}}}+2M_{\tau_{f,1}}\left(\frac{1}{\bar{r}_{\tau_{f,2}}}-\frac{1}{\bar{r}_{\tau_{f,1}}\circ\left(\Phi_{\tau_{f,1}}\circ\widehat{\Phi}_{\tau_{f,2}}\right)}\right)+O\left(\varepsilon^{2}\tau_{f,1}^{-2}\right) ∂ u τ f , 2 ϱ = r ˉ τ f , 2 2 ( M τ f , 2 − M τ f , 1 ) + 2 M τ f , 1 r ˉ τ f , 2 1 − r ˉ τ f , 1 ∘ ( Φ τ f , 1 ∘ Φ τ f , 2 ) 1 + O ( ε 2 τ f , 1 − 2 ) = 2 M τ f , 1 ( 1 r ˉ τ f , 2 − 1 r ˉ τ f , 1 ∘ ( Φ τ f , 1 ∘ Φ ^ τ f , 2 ) ) + O ( ε 3 / 2 τ f , 1 − 2 ) , =2M_{\tau_{f,1}}\left(\frac{1}{\bar{r}_{\tau_{f,2}}}-\frac{1}{\bar{r}_{\tau_{f,1}}\circ\left(\Phi_{\tau_{f,1}}\circ\widehat{\Phi}_{\tau_{f,2}}\right)}\right)+O\left(\varepsilon^{3/2}\tau_{f,1}^{-2}\right), = 2 M τ f , 1 r ˉ τ f , 2 1 − r ˉ τ f , 1 ∘ ( Φ τ f , 1 ∘ Φ τ f , 2 ) 1 + O ( ε 3/2 τ f , 1 − 2 ) , for τ f , 1 \tau_{f,1} τ f , 1 ∣ ϱ ∣ ≤ ε τ f , 1 − 1 / 3 |\varrho|\leq\varepsilon\tau_{f,1}^{-1/3} ∣ ϱ ∣ ≤ ε τ f , 1 − 1/3 τ f , 1 \tau_{f,1} τ f , 1
∂ u τ f , 2 ϱ = − 2 M τ f , 1 r ˉ τ f , 2 2 ϱ + O ( r ˉ τ f , 2 − 3 ϱ 2 ) + O ( ε 3 / 2 τ f , 1 − 2 ) . (177) \partial_{u_{\tau_{f,2}}}\varrho=-\frac{2M_{\tau_{f,1}}}{\bar{r}_{\tau_{f,2}}^{2}}\varrho+O\left(\bar{r}_{\tau_{f,2}}^{-3}\varrho^{2}\right)+O\left(\varepsilon^{3/2}\tau_{f,1}^{-2}\right). \tag*{(177)} ∂ u τ f , 2 ϱ = − r ˉ τ f , 2 2 2 M τ f , 1 ϱ + O ( r ˉ τ f , 2 − 3 ϱ 2 ) + O ( ε 3/2 τ f , 1 − 2 ) . (177) From (76) , we note that ∣ ϱ ( Γ ( τ f , 1 ) ) ∣ ≲ ε 2 τ f , 1 − 2 + δ |\varrho(\Gamma(\tau_{f,1}))|\lesssim\varepsilon^{2}\tau_{f,1}^{-2+\delta} ∣ ϱ ( Γ ( τ f , 1 )) ∣ ≲ ε 2 τ f , 1 − 2 + δ Γ ( τ f , 1 ) \Gamma(\tau_{f,1}) Γ ( τ f , 1 ) τ f , 1 \tau_{f,1} τ f , 1 (177) backwards in u u u Γ ( τ f , 1 ) \Gamma(\tau_{f,1}) Γ ( τ f , 1 ) Proposition 4.11 ), we bound
∣ ϱ ( u τ f , 2 , Γ v τ f , 2 ( τ f , 1 ) ) ∣ ≲ ε 2 τ f , 1 − 2 + δ + ∫ u τ f , 2 Γ u τ f , 2 ( τ f , 1 ) ( r ˉ τ f , 2 − 3 ϱ 2 + ε 3 / 2 τ f , 1 − 2 ) d u τ f , 2 ′ ≲ ε 2 τ f , 1 − 2 / 3 + ε 3 / 2 τ f , 1 − 1 ≲ ε 2 τ f , 1 − 2 / 3 , (178) \left|\varrho\left(u_{\tau_{f,2}},\Gamma^{v_{\tau_{f,2}}}(\tau_{f,1})\right)\right|\lesssim\varepsilon^{2}\tau_{f,1}^{-2+\delta}+\int_{u_{\tau_{f,2}}}^{\Gamma^{u_{\tau_{f,2}}}(\tau_{f,1})}\left(\bar{r}_{\tau_{f,2}}^{-3}\varrho^{2}+\varepsilon^{3/2}\tau_{f,1}^{-2}\right)\,du'_{\tau_{f,2}}\lesssim\varepsilon^{2}\tau_{f,1}^{-2/3}+\varepsilon^{3/2}\tau_{f,1}^{-1}\lesssim\varepsilon^{2}\tau_{f,1}^{-2/3}, \tag*{(178)} ϱ ( u τ f , 2 , Γ v τ f , 2 ( τ f , 1 ) ) ≲ ε 2 τ f , 1 − 2 + δ + ∫ u τ f , 2 Γ u τ f , 2 ( τ f , 1 ) ( r ˉ τ f , 2 − 3 ϱ 2 + ε 3/2 τ f , 1 − 2 ) d u τ f , 2 ′ ≲ ε 2 τ f , 1 − 2/3 + ε 3/2 τ f , 1 − 1 ≲ ε 2 τ f , 1 − 2/3 , (178) which improves the bootstrap assumption on ϱ \varrho ϱ τ f , 1 \tau_{f,1} τ f , 1 (178) holds for u τ f , 2 ∈ [ 0 , Γ u τ f , 2 ( τ f , 1 ) ] u_{\tau_{f,2}}\in[0,\Gamma^{u_{\tau_{f,2}}}(\tau_{f,1})] u τ f , 2 ∈ [ 0 , Γ u τ f , 2 ( τ f , 1 )] (177) , we may derive in v τ f , 2 v_{\tau_{f,2}} v τ f , 2
∂ v τ f , 2 ϱ = 2 M τ f , 2 r ˉ τ f , 2 2 ϱ + O ( r ˉ τ f , 2 − 3 ϱ 2 ) + O ( ε 3 / 2 τ f , 1 − 2 ) . \partial_{v_{\tau_{f,2}}}\varrho=\frac{2M_{\tau_{f,2}}}{\bar{r}_{\tau_{f,2}}^{2}}\varrho+O\left(\bar{r}_{\tau_{f,2}}^{-3}\varrho^{2}\right)+O\left(\varepsilon^{3/2}\tau_{f,1}^{-2}\right). ∂ v τ f , 2 ϱ = r ˉ τ f , 2 2 2 M τ f , 2 ϱ + O ( r ˉ τ f , 2 − 3 ϱ 2 ) + O ( ε 3/2 τ f , 1 − 2 ) . This can be integrated backwards from [ 0 , Γ u τ f , 2 ( τ f , 1 ) ] × { Γ v ( τ f , 1 ) } [0,\Gamma^{u_{\tau_{f,2}}}(\tau_{f,1})]\times\{\Gamma^{v}(\tau_{f,1})\} [ 0 , Γ u τ f , 2 ( τ f , 1 )] × { Γ v ( τ f , 1 )} (178) , together with the good sign of the coefficient of the linear term on the right-hand side, one finally obtains (176) .
Completing the proof
Thanks to the bounds on the gauges and anchored backgrounds established in the previous sections, we are now in a position to rigorously pass to the limit τ f → + ∞ \tau_f \to+\infty τ f → + ∞ D ∞ D_\infty D ∞ Theorem 2 .
We first extract the limiting anchored Schwarzschild background, with area radius function r ˉ ∞ \bar r_\infty r ˉ ∞ M ∞ M_\infty M ∞ (174) .
Proposition 7.12. The limit
r ˉ ⋆ : = lim τ f → + ∞ r ˉ τ f ( 0 , 0 ) (179) \bar r_\star:= \lim_{\tau_f \to+\infty} \bar r_{\tau_f}(0,0) \tag*{(179)} r ˉ ⋆ := τ f → + ∞ lim r ˉ τ f ( 0 , 0 ) (179) exists and satisfies
∣ r ˉ ⋆ − Λ ∣ ≲ ε 2 . (180) \lvert\bar r_\star- \Lambda\rvert\lesssim\varepsilon^2. \tag*{(180)} ∣ r ˉ ⋆ − Λ ∣ ≲ ε 2 . (180) Let us denote by r ˉ ∞ \bar r_\infty r ˉ ∞ ∞ \infty ∞ D ∞ D_\infty D ∞ M ∞ M_\infty M ∞ r ˉ ∞ ( 0 , 0 ) = r ˉ ⋆ \bar r_\infty(0,0)=\bar r_\star r ˉ ∞ ( 0 , 0 ) = r ˉ ⋆
sup J − ( Γ ( τ f ) ) ∩ D ∞ ∣ r ˉ ∞ − r ˉ τ f ∘ ( Φ τ f ∘ Φ ^ ∞ ) ∣ ≲ ε 2 τ f − 1 / 2 . (181) \sup_{J^{-}(\Gamma(\tau_f))\cap D_\infty} \left\lvert\bar r_\infty-\bar r_{\tau_f}\circ(\Phi_{\tau_f}\circ\hat\Phi_\infty)\right\rvert\lesssim\varepsilon^2\tau_f^{-1/2}. \tag*{(181)} J − ( Γ ( τ f )) ∩ D ∞ sup r ˉ ∞ − r ˉ τ f ∘ ( Φ τ f ∘ Φ ^ ∞ ) ≲ ε 2 τ f − 1/2 . (181) Proof. The argument is analogous to [4 Proposition 8.13] . The existence of the limit (179) follows from (176) evaluated at ( 0 , 0 ) (0,0) ( 0 , 0 ) (180) is a consequence of (76) . To prove (181) , we first note that, by (176) , the limit
r ( u ^ , v ^ ) : = lim τ f → + ∞ r ˉ τ f ∘ Φ τ f r(\hat u,\hat v):=\lim_{\tau_f\to+\infty}\bar r_{\tau_f}\circ\Phi_{\tau_f} r ( u ^ , v ^ ) := τ f → + ∞ lim r ˉ τ f ∘ Φ τ f exists on D ^ ∞ \hat D_\infty D ^ ∞
sup J − ( Γ ( τ f ) ) ∩ D ^ ∞ ∣ r − r ˉ τ f ∘ Φ τ f ∣ ≲ ε 3 / 2 τ f − 1 / 2 . (182) \sup_{J^{-}(\Gamma(\tau_f))\cap\hat D_\infty}\left\lvert r-\bar r_{\tau_f}\circ\Phi_{\tau_f}\right\rvert\lesssim\varepsilon^{3/2}\tau_f^{-1/2}. \tag*{(182)} J − ( Γ ( τ f )) ∩ D ^ ∞ sup r − r ˉ τ f ∘ Φ τ f ≲ ε 3/2 τ f − 1/2 . (182) Claiming that r ˉ ∞ = r ∘ Φ ^ ∞ \bar r_\infty=r\circ\hat\Phi_\infty r ˉ ∞ = r ∘ Φ ^ ∞ (181) from (182) . As r ˉ ∞ ( 0 , 0 ) = r ˉ ⋆ = r ∘ Φ ^ ∞ ( 0 , 0 ) \bar r_\infty(0,0)=\bar r_\star=r\circ\hat\Phi_\infty(0,0) r ˉ ∞ ( 0 , 0 ) = r ˉ ⋆ = r ∘ Φ ^ ∞ ( 0 , 0 ) r ∘ Φ ^ ∞ r\circ\hat\Phi_\infty r ∘ Φ ^ ∞ ( u ∞ , v ∞ ) (u_\infty,v_\infty) ( u ∞ , v ∞ )
∂ u ∞ ( r ˉ τ f ∘ ( Φ τ f ∘ Φ ^ ∞ ) ) = − ( 1 − 2 M τ f r ˉ τ f ∘ ( Φ τ f ∘ Φ ^ ∞ ) ) g τ f , ∞ ′ , \partial_{u_\infty}\left(\bar r_{\tau_f}\circ(\Phi_{\tau_f}\circ\hat\Phi_\infty)\right)
=-\left(1-\frac{2M_{\tau_f}}{\bar r_{\tau_f}\circ(\Phi_{\tau_f}\circ\hat\Phi_\infty)}\right)g'_{\tau_f,\infty}, ∂ u ∞ ( r ˉ τ f ∘ ( Φ τ f ∘ Φ ^ ∞ ) ) = − ( 1 − r ˉ τ f ∘ ( Φ τ f ∘ Φ ^ ∞ ) 2 M τ f ) g τ f , ∞ ′ , ∂ v ∞ ( r ˉ τ f ∘ ( Φ τ f ∘ Φ ^ ∞ ) ) = ( 1 − 2 M τ f r ˉ τ f ∘ ( Φ τ f ∘ Φ ^ ∞ ) ) h τ f , ∞ ′ . \partial_{v_\infty}\left(\bar r_{\tau_f}\circ(\Phi_{\tau_f}\circ\hat\Phi_\infty)\right)
=\left(1-\frac{2M_{\tau_f}}{\bar r_{\tau_f}\circ(\Phi_{\tau_f}\circ\hat\Phi_\infty)}\right)h'_{\tau_f,\infty}. ∂ v ∞ ( r ˉ τ f ∘ ( Φ τ f ∘ Φ ^ ∞ ) ) = ( 1 − r ˉ τ f ∘ ( Φ τ f ∘ Φ ^ ∞ ) 2 M τ f ) h τ f , ∞ ′ . Thanks to Lemmas 7.9 and 7.10, we infer that the two quantities above converge uniformly as τ f → + ∞ \tau_f\to+\infty τ f → + ∞ − ( 1 − 2 M ∞ / r ∘ Φ ^ ∞ ) -(1-2M_\infty/r\circ\hat\Phi_\infty) − ( 1 − 2 M ∞ / r ∘ Φ ^ ∞ ) ( 1 − 2 M ∞ / r ∘ Φ ^ ∞ ) (1-2M_\infty/r\circ\hat\Phi_\infty) ( 1 − 2 M ∞ / r ∘ Φ ^ ∞ ) ∂ u ∞ ( r ∘ Φ ^ ∞ ) \partial_{u_\infty}(r\circ\hat\Phi_\infty) ∂ u ∞ ( r ∘ Φ ^ ∞ ) ∂ v ∞ ( r ∘ Φ ^ ∞ ) \partial_{v_\infty}(r\circ\hat\Phi_\infty) ∂ v ∞ ( r ∘ Φ ^ ∞ )
As noted before, we stress that r ˉ ∞ \bar r_\infty r ˉ ∞ D ∞ D_\infty D ∞ ( u ∞ , v ∞ ) (u_\infty,v_\infty) ( u ∞ , v ∞ ) r ˉ ∞ ∘ Φ ∞ \bar r_\infty\circ\Phi_\infty r ˉ ∞ ∘ Φ ∞ C 2 C^2 C 2 D ^ ∞ \hat D_\infty D ^ ∞ ( u ^ , v ^ ) (\hat u,\hat v) ( u ^ , v ^ )
Proposition 7.13. For any τ ∈ [ 1 , + ∞ ) \tau\in[1,+\infty) τ ∈ [ 1 , + ∞ ) p ∈ [ 0 , 3 − δ ] p\in[0,3-\delta] p ∈ [ 0 , 3 − δ ]
lim τ f → + ∞ E τ f ( τ ) = E ∞ ( τ ) , \lim_{\tau_f\to+\infty}\mathcal{E}^{\tau_f}(\tau)=\mathcal{E}^{\infty}(\tau), τ f → + ∞ lim E τ f ( τ ) = E ∞ ( τ ) , lim τ f → + ∞ E p τ f ( τ ) = E p ∞ ( τ ) . \lim_{\tau_f\to+\infty}\mathcal{E}^{\tau_f}_p(\tau)=\mathcal{E}^{\infty}_p(\tau). τ f → + ∞ lim E p τ f ( τ ) = E p ∞ ( τ ) . Proof. The argument relies on the dominated convergence theorem. We refer to [4 Proposition 8.15] for more details on the proof, as the energies defined in this work are very similar to those considered there. □
Let us now prove the first part of Theorem 2 , that is showing the existence of a black hole region and of a regular event horizon in the maximal development of the seed data.
Proposition 7.14. Given an ε \varepsilon ε ε ≤ ε s t a b \varepsilon\le\varepsilon_{\mathrm{stab}} ε ≤ ε stab Q ^ max \widehat{Q}_{\max} Q m a x u ^ H + ∈ ( 0 , U ∗ ) \widehat{u}_{H+} \in(0,U_*) u H + ∈ ( 0 , U ∗ )
∣ u ^ H + − u ^ H + , 0 ∣ ≲ ε , (183) \lvert\widehat{u}_{H+}-\widehat{u}_{H+,0}\rvert\lesssim\varepsilon, \tag*{(183)} ∣ u H + − u H + , 0 ∣ ≲ ε , (183) where u ^ H + , 0 \widehat{u}_{H+,0} u H + , 0 (26) , such that [ 0 , u ^ H + ] × [ 0 , + ∞ ) ⊂ Q ^ max [0,\widehat{u}_{H+}] \times[0,+\infty) \subset\widehat{Q}_{\max} [ 0 , u H + ] × [ 0 , + ∞ ) ⊂ Q m a x
lim v ^ → + ∞ r ( u ^ , v ^ ) = + ∞ (184) \lim_{\widehat{v}\to+\infty} r(\widehat{u},\widehat{v})=+\infty \tag*{(184)} v → + ∞ lim r ( u , v ) = + ∞ (184) for every u ^ ∈ [ 0 , u ^ H + ) \widehat{u} \in[0,\widehat{u}_{H+}) u ∈ [ 0 , u H + )
lim v ^ → + ∞ r ( u ^ H + , v ^ ) = lim v ^ → + ∞ 2 m ( u ^ H + , v ^ ) = 2 M ∞ . (185) \lim_{\widehat{v}\to+\infty} r(\widehat{u}_{H+},\widehat{v})=\lim_{\widehat{v}\to+\infty}2m(\widehat{u}_{H+},\widehat{v})=2M_\infty. \tag*{(185)} v → + ∞ lim r ( u H + , v ) = v → + ∞ lim 2 m ( u H + , v ) = 2 M ∞ . (185) Therefore, [ 0 , u ^ H + ) × { v ^ = + ∞ } [0,\widehat{u}_{H+}) \times\{\widehat{v}=+\infty\} [ 0 , u H + ) × { v = + ∞ } I + \mathcal{I}^{+} I + J − ( I + ) J^{-}(\mathcal{I}^{+}) J − ( I + ) H + : = { u ^ H + } × [ 0 , + ∞ ) \mathcal{H}^{+}:=\{\widehat{u}_{H+}\}\times[0,+\infty) H + := { u H + } × [ 0 , + ∞ )
B H : = Q ^ max ∖ J − ( I + ) = [ u ^ H + , U ∗ ] × [ 0 , + ∞ ) ≠ ∅ , \mathcal{BH}:=\widehat{Q}_{\max}\setminus J^{-}(\mathcal{I}^{+})=[\widehat{u}_{H+},U_*]\times[0,+\infty)\ne\varnothing, B H := Q m a x ∖ J − ( I + ) = [ u H + , U ∗ ] × [ 0 , + ∞ ) = ∅ , and future null infinity is complete in the sense of Christodoulou [15 ] .
Proof. Thanks to the pointwise estimates for the Dirac field in § 7 and the extension principle proved in Proposition B.1, we have D ^ ∞ ⊂ Q ^ max \widehat{D}_{\infty}\subset\widehat{Q}_{\max} D ∞ ⊂ Q m a x ( u ^ , v ^ ) (\widehat{u},\widehat{v}) ( u , v )
D ^ ‾ ∞ ∖ D ^ ∞ = { u ^ H + } × [ 0 , + ∞ ) . \overline{\widehat{D}}_{\infty}\setminus\widehat{D}_{\infty}=\{\widehat{u}_{H+}\}\times[0,+\infty). D ∞ ∖ D ∞ = { u H + } × [ 0 , + ∞ ) . By (43) and (180) , it holds that
∣ r − r ‾ ∞ ∘ Φ ∞ ∣ ≲ ε 3 / 2 τ − 2 + δ (186) \lvert r-\overline{r}_{\infty}\circ\Phi_{\infty}\rvert\lesssim\varepsilon^{3/2}\tau^{-2+\delta} \tag*{(186)} ∣ r − r ∞ ∘ Φ ∞ ∣ ≲ ε 3/2 τ − 2 + δ (186) on D ^ ∞ \widehat{D}_{\infty} D ∞
lim u ^ → u ^ H + r ‾ ∞ ∘ Φ ∞ ( u ^ , v ^ ) = lim u ∞ → + ∞ r ‾ ∞ ( u ∞ , v ∞ ( v ^ ) ) = 2 M ∞ , \lim_{\widehat{u}\to\widehat{u}_{H+}}\overline{r}_{\infty}\circ\Phi_{\infty}(\widehat{u},\widehat{v})=\lim_{u_{\infty}\to+\infty}\overline{r}_{\infty}(u_{\infty},v_{\infty}(\widehat{v}))=2M_{\infty}, u → u H + lim r ∞ ∘ Φ ∞ ( u , v ) = u ∞ → + ∞ lim r ∞ ( u ∞ , v ∞ ( v )) = 2 M ∞ , for v ^ ∈ [ 0 , + ∞ ) \widehat{v}\in[0,+\infty) v ∈ [ 0 , + ∞ )
∣ r ( u ^ H + , v ^ ) − 2 M ∞ ∣ ≲ ε 3 / 2 τ − 2 + δ ( u ^ H + , v ^ ) , (187) \lvert r(\widehat{u}_{H+},\widehat{v})-2M_{\infty}\rvert\lesssim\varepsilon^{3/2}\tau^{-2+\delta}(\widehat{u}_{H+},\widehat{v}), \tag*{(187)} ∣ r ( u H + , v ) − 2 M ∞ ∣ ≲ ε 3/2 τ − 2 + δ ( u H + , v ) , (187) for v ^ ∈ [ 0 , + ∞ ) \widehat{v}\in[0,+\infty) v ∈ [ 0 , + ∞ )
∣ m − M ∞ ∣ ≲ ε 3 / 2 τ − 3 + δ (188) \lvert m-M_{\infty}\rvert\lesssim\varepsilon^{3/2}\tau^{-3+\delta} \tag*{(188)} ∣ m − M ∞ ∣ ≲ ε 3/2 τ − 3 + δ (188) on D ^ ∞ \widehat{D}_{\infty} D ∞ (44) , together imply (185) . Next, (183) follows by (187) evaluated at v ^ = 0 \widehat{v}=0 v = 0 (188) evaluated at τ = 1 \tau=1 τ = 1 ν ^ ( u ^ , 0 ) = − 1 \widehat{\nu}(\widehat{u},0)=-1 ν ( u , 0 ) = − 1
u ^ H + − u ^ H + , 0 = ( Λ − r ( u ^ H + , 0 ) ) − u ^ H + , 0 = ( Λ − 2 M ∞ + O ( ε ) ) − u ^ H + , 0 = 2 ( M 0 − M ∞ ) + O ( ε ) = O ( ε ) . \widehat{u}_{H+}-\widehat{u}_{H+,0}=(\Lambda-r(\widehat{u}_{H+},0))-\widehat{u}_{H+,0}=(\Lambda-2M_{\infty}+O(\varepsilon))-\widehat{u}_{H+,0}=2(M_{0}-M_{\infty})+O(\varepsilon)=O(\varepsilon). u H + − u H + , 0 = ( Λ − r ( u H + , 0 )) − u H + , 0 = ( Λ − 2 M ∞ + O ( ε )) − u H + , 0 = 2 ( M 0 − M ∞ ) + O ( ε ) = O ( ε ) . Moreover, noting that for u ^ ∈ [ 0 , u ^ H + ) \widehat{u}\in[0,\widehat{u}_{H+}) u ∈ [ 0 , u H + )
lim v ^ → + ∞ r ‾ ∞ ∘ Φ ∞ ( u ^ , v ^ ) = lim v ∞ → + ∞ r ‾ ∞ ( u ∞ ( u ^ ) , v ∞ ) = + ∞ , \lim_{\widehat{v}\to+\infty}\overline{r}_{\infty}\circ\Phi_{\infty}(\widehat{u},\widehat{v})=\lim_{v_{\infty}\to+\infty}\overline{r}_{\infty}(u_{\infty}(\widehat{u}),v_{\infty})=+\infty, v → + ∞ lim r ∞ ∘ Φ ∞ ( u , v ) = v ∞ → + ∞ lim r ∞ ( u ∞ ( u ) , v ∞ ) = + ∞ , by the geometry of Schwarzschild, we deduce again by (186) that
lim v ^ → + ∞ r ( u ^ , v ^ ) = + ∞ , \lim_{\widehat{v}\to+\infty}r(\widehat{u},\widehat{v})=+\infty, v → + ∞ lim r ( u , v ) = + ∞ , for u ^ ∈ [ 0 , u ^ H + ) \widehat{u}\in[0,\widehat{u}_{H+}) u ∈ [ 0 , u H + ) r r r H + \mathcal{H}^{+} H + I + \mathcal{I}^{+} I + [19 Section 5.4] . □ \square □
Before proving Theorem 2 , we collect the final geometric estimates in the eschatological gauge ( u ∞ , v ∞ ) (u_{\infty},v_{\infty}) ( u ∞ , v ∞ ) § 6 and § 7 , whereas others carry improved exponents in ε \varepsilon ε
Although the event horizon H + \mathcal{H}^{+} H + ( u ∞ , v ∞ ) (u_{\infty},v_{\infty}) ( u ∞ , v ∞ ) { u ∞ = + ∞ } \{u_{\infty}=+\infty\} { u ∞ = + ∞ } D ∞ ∪ H + \mathcal{D}_{\infty}\cup\mathcal{H}^{+} D ∞ ∪ H +
Proposition 7.15. For ε s t a b \varepsilon_{\mathrm{stab}} ε stab ( u ∞ , v ∞ ) (u_{\infty},v_{\infty}) ( u ∞ , v ∞ )
∣ γ ∞ − 1 ∣ ≲ ε 2 τ − 2 τ − 3 + δ (189) |\gamma_{\infty}-1|\lesssim\varepsilon^{2}\tau^{-2}\tau^{-3+\delta} \tag*{(189)} ∣ γ ∞ − 1∣ ≲ ε 2 τ − 2 τ − 3 + δ (189) on D ∞ ∩ { r ≥ Λ } \mathcal{D}_{\infty}\cap\{r\geq\Lambda\} D ∞ ∩ { r ≥ Λ }
∣ v ∞ v ˉ ∞ − 1 ∣ ≲ ε 2 τ − 1 + δ (190) \left|\frac{v_{\infty}}{\bar{v}_{\infty}}-1\right|\lesssim\varepsilon^{2}\tau^{-1+\delta} \tag*{(190)} v ˉ ∞ v ∞ − 1 ≲ ε 2 τ − 1 + δ (190) on D ∞ \mathcal{D}_{\infty} D ∞
∣ r − r ˉ ∞ ∣ ≲ ε 2 τ − 2 + δ , (191) |r-\bar{r}_{\infty}|\lesssim\varepsilon^{2}\tau^{-2+\delta}, \tag*{(191)} ∣ r − r ˉ ∞ ∣ ≲ ε 2 τ − 2 + δ , (191) ∣ λ ∞ − λ ˉ ∞ ∣ ≲ ε 2 τ − 2 + δ , (192) |\lambda_{\infty}-\bar{\lambda}_{\infty}|\lesssim\varepsilon^{2}\tau^{-2+\delta}, \tag*{(192)} ∣ λ ∞ − λ ˉ ∞ ∣ ≲ ε 2 τ − 2 + δ , (192) ∣ κ ∞ − 1 ∣ ≲ ε 2 τ − 2 + δ , (193) |\kappa_{\infty}-1|\lesssim\varepsilon^{2}\tau^{-2+\delta}, \tag*{(193)} ∣ κ ∞ − 1∣ ≲ ε 2 τ − 2 + δ , (193) ∣ m − M ∞ ∣ ≲ ε 2 τ − 3 + δ , (194) |m-M_{\infty}|\lesssim\varepsilon^{2}\tau^{-3+\delta}, \tag*{(194)} ∣ m − M ∞ ∣ ≲ ε 2 τ − 3 + δ , (194) ∣ ( 1 − μ ) − ( 1 − μ ˉ ∞ ) ∣ ≲ ε 2 τ − 2 + δ , (195) |(1-\mu)-(1-\bar{\mu}_{\infty})|\lesssim\varepsilon^{2}\tau^{-2+\delta}, \tag*{(195)} ∣ ( 1 − μ ) − ( 1 − μ ˉ ∞ ) ∣ ≲ ε 2 τ − 2 + δ , (195) ∣ ∂ v ∞ log Ω ∞ − M ∞ r ˉ ∞ 2 ∣ ≲ ε 2 τ − 2 + δ , (196) \left|\partial_{v_{\infty}}\log\Omega_{\infty}-\frac{M_{\infty}}{\bar{r}_{\infty}^{2}}\right|\lesssim\varepsilon^{2}\tau^{-2+\delta}, \tag*{(196)} ∂ v ∞ log Ω ∞ − r ˉ ∞ 2 M ∞ ≲ ε 2 τ − 2 + δ , (196) ∣ ∂ u ∞ log Ω ∞ + M ∞ r ˉ ∞ 2 ∣ ≲ ε 2 r − 2 τ − 3 / 4 + δ , (197) \left|\partial_{u_{\infty}}\log\Omega_{\infty}+\frac{M_{\infty}}{\bar{r}_{\infty}^{2}}\right|\lesssim\varepsilon^{2}r^{-2}\tau^{-3/4+\delta}, \tag*{(197)} ∂ u ∞ log Ω ∞ + r ˉ ∞ 2 M ∞ ≲ ε 2 r − 2 τ − 3/4 + δ , (197) on D ∞ ∪ H + \mathcal{D}_{\infty}\cup\mathcal{H}^{+} D ∞ ∪ H +
Proof. Estimates (189) and (193) are simply obtained by passing to the limit in (58) and (53) . On the other hand, (44) is upgraded to (194) by passing to the limit in (84) .
Let us now prove the remaining estimates by exploiting the refined estimate (194) via a bootstrap argument in the eschatological gauge on D ∞ \mathcal{D}_{\infty} D ∞ B ′ \mathcal{B}' B ′ τ f ∈ [ 1 , + ∞ ) \tau_{f}\in[1,+\infty) τ f ∈ [ 1 , + ∞ )
∣ r − r ˉ ∞ ∣ ≲ A ′ ε 2 τ − 2 + δ , ∣ v ∞ v ˉ ∞ − 1 ∣ ≲ A ′ 1 / 2 ε 2 τ − 1 + δ , |r-\bar{r}_{\infty}|\lesssim A'\varepsilon^{2}\tau^{-2+\delta},\qquad\left|\frac{v_{\infty}}{\bar{v}_{\infty}}-1\right|\lesssim A'^{1/2}\varepsilon^{2}\tau^{-1+\delta}, ∣ r − r ˉ ∞ ∣ ≲ A ′ ε 2 τ − 2 + δ , v ˉ ∞ v ∞ − 1 ≲ A ′ 1/2 ε 2 τ − 1 + δ , where A ′ ≥ 1 A'\geq1 A ′ ≥ 1
For τ f ∈ B ′ \tau_{f}\in\mathcal{B}' τ f ∈ B ′ (194) to improve the Taylor expansions in Lemma 4.7 to
∂ u ∞ ( r − r ˉ ∞ ) = − 2 M ∞ γ ∞ r ˉ ∞ 2 ( r − r ˉ ∞ ) + O ( ε 2 τ − 1 τ − 3 + δ ) \partial_{u_{\infty}}(r-\bar{r}_{\infty})=-\frac{2M_{\infty}\gamma_{\infty}}{\bar{r}_{\infty}^{2}}(r-\bar{r}_{\infty})+O(\varepsilon^{2}\tau^{-1}\tau^{-3+\delta}) ∂ u ∞ ( r − r ˉ ∞ ) = − r ˉ ∞ 2 2 M ∞ γ ∞ ( r − r ˉ ∞ ) + O ( ε 2 τ − 1 τ − 3 + δ ) in D ∞ ∩ { τ ≤ τ f } ∩ { r ≥ Λ } \mathcal{D}_{\infty}\cap\{\tau\leq\tau_{f}\}\cap\{r\geq\Lambda\} D ∞ ∩ { τ ≤ τ f } ∩ { r ≥ Λ }
∂ v ( r − r ˉ ∞ ) = 2 M κ ∞ r ˉ ∞ 2 ( r − r ˉ ∞ ) + O ( ε 2 τ − 3 + δ ) \partial_{v}(r-\bar{r}_{\infty})=\frac{2M\kappa_{\infty}}{\bar{r}_{\infty}^{2}}(r-\bar{r}_{\infty})+O(\varepsilon^{2}\tau^{-3+\delta}) ∂ v ( r − r ˉ ∞ ) = r ˉ ∞ 2 2 M κ ∞ ( r − r ˉ ∞ ) + O ( ε 2 τ − 3 + δ ) in D ∞ ∩ { τ ≤ τ f } \mathcal{D}_{\infty}\cap\{\tau\leq\tau_{f}\} D ∞ ∩ { τ ≤ τ f }
Therefore, combining these estimates with ∣ ( r − r ˉ ∞ ) Γ ∣ ≲ ε 2 τ − 2 + δ |(r-\bar{r}_{\infty})\Gamma|\lesssim\varepsilon^{2}\tau^{-2+\delta} ∣ ( r − r ˉ ∞ ) Γ∣ ≲ ε 2 τ − 2 + δ (76) , one may repeat the arguments in Propositions 4.11 and 4.13 and improve the above bootstrap assumptions for A ′ A' A ′ B ′ = [ 1 , + ∞ ) \mathcal{B}'=[1,+\infty) B ′ = [ 1 , + ∞ ) (190) and (191) hold. Finally, thanks to these upgraded bounds, one may improve the remaining ones by repeating the arguments in Lemmas 4.6 and 5.2, and Proposition 4.11. □
Let us now finally put everything together and conclude with the proof of Theorem 2 about the nonlinear stability of Schwarzschild as a solution to the Einstein–Dirac system.
Proof of Theorem 2. Existence of a black hole region: It is a consequence of Propositions 7.4 and 7.14. In addition, ∂ u r ∞ → − 1 \partial_{u}r_{\infty}\to-1 ∂ u r ∞ → − 1 (189) .
Orbital stability: (33) is a consequence of the definition of D \mathcal{D} D (188) evaluated at τ = 1 \tau=1 τ = 1 (34) does not immediately follow by Proposition 7.13 with p = 3 − δ p=3-\delta p = 3 − δ ε ≥ D [ S ] \varepsilon\geq\mathcal{D}[S] ε ≥ D [ S ] E ∞ ( 1 ) + E 3 − δ ∞ ( 1 ) \mathcal{E}^{\infty}(1)+\mathcal{E}^{\infty}_{3-\delta}(1) E ∞ ( 1 ) + E 3 − δ ∞ ( 1 ) D ∞ \mathcal{D}_{\infty} D ∞ B ′ \mathcal{B}' B ′ τ f ∈ [ 1 , + ∞ ) \tau_{f}\in[1,+\infty) τ f ∈ [ 1 , + ∞ ) (42) –(48) hold, where A A A A ′ A' A ′ ε \varepsilon ε ε ′ : = ( E ∞ ( 1 ) + E 3 − δ ∞ ( 1 ) ) 1 / 2 \varepsilon':=(\mathcal{E}^{\infty}(1)+\mathcal{E}^{\infty}_{3-\delta}(1))^{1/2} ε ′ := ( E ∞ ( 1 ) + E 3 − δ ∞ ( 1 ) ) 1/2 τ f \tau_f τ f ∞ \infty ∞ ε \varepsilon ε ε ′ \varepsilon' ε ′ B ′ = [ 1 , + ∞ ) \mathcal{B}' = [1,+\infty) B ′ = [ 1 , + ∞ ) τ ∈ [ 1 , + ∞ ) \tau\in[1,+\infty) τ ∈ [ 1 , + ∞ ) p ∈ [ 0 , 3 − δ ] p \in[0,3-\delta] p ∈ [ 0 , 3 − δ ]
E ∞ ( τ ) + E p ∞ ( τ ) ≲ ε ′ 2 τ − 3 + δ + p = ( E ∞ ( 1 ) + E 3 − δ ∞ ( 1 ) ) τ − 3 + δ + p . \mathcal{E}^{\infty}(\tau)+\mathcal{E}^{\infty}_{p}(\tau)\lesssim{\varepsilon'}^{2}\tau^{-3+\delta+p}=\left(\mathcal{E}^{\infty}(1)+\mathcal{E}^{\infty}_{3-\delta}(1)\right)\tau^{-3+\delta+p}. E ∞ ( τ ) + E p ∞ ( τ ) ≲ ε ′ 2 τ − 3 + δ + p = ( E ∞ ( 1 ) + E 3 − δ ∞ ( 1 ) ) τ − 3 + δ + p . from which (34) follows. Regarding (35) , one may revisit the proofs of Lemmas 5.1 and 5.6 and observe that the estimates only depend on the energies and pointwise norms coming from initial data of the solution restricted to J − ( I + ) ∩ C ^ J^{-}(\mathcal{I}^{+})\cap\widehat{C} J − ( I + ) ∩ C (35) .
Asymptotic stability: It follows by Propositions 7.13, 7.15 and Lemma 5.1.
The Dirac equation in spherical symmetry
The construction of the Dirac operator on a Lorentzian manifold is delicate, but fairly classical, and relies on the introduction of a spin structure together with an orthonormal frame. Section A.1 is devoted to recalling the main ingredients needed for the definition of the Dirac operator, while in Section A.2 the reduction to spherical symmetry is shown.
The Dirac equation in curved spacetime
We briefly recall the definition of the Dirac operator on a Lorentzian manifold. We shall refer to [77 ] for an introduction to spinorial formalism, and to [64 Section 5.6] and [9 ] for more details on the geometric construction of the Dirac operator. We stress that in what follows we will commit an abuse of notation by identifying spinor fields with their C 4 \mathbb{C}^{4} C 4
Let ( M , g ) (\mathcal{M},g) ( M , g ) ( − , + , + , + ) (-,+,+,+) ( − , + , + , + ) ( M , g ) (\mathcal{M},g) ( M , g ) { e a } a = 0 , … , 3 \{e_a\}_{a=0,\ldots,3} { e a } a = 0 , … , 3
e a μ g μ ν e b ν = η a b , e_a^{\ \mu}g_{\mu\nu}e_b^{\ \nu}=\eta_{ab}, e a μ g μν e b ν = η ab , where η : = diag ( − 1 , 1 , 1 , 1 ) \eta:=\operatorname{diag}(-1,1,1,1) η := diag ( − 1 , 1 , 1 , 1 ) g g g η \eta η
The Dirac equation provides the relativistic description of spin-1/2 matter and takes the form
i γ μ D μ Ψ = m Ψ , (198) i\gamma^{\mu}D_{\mu}\Psi=m\Psi, \tag*{(198)} i γ μ D μ Ψ = m Ψ , (198) where m ≥ 0 m\geq0 m ≥ 0 Ψ \Psi Ψ γ μ \gamma^{\mu} γ μ D μ D_{\mu} D μ γ μ ∈ M 4 ( C ) \gamma^{\mu}\in M_{4}(\mathbb{C}) γ μ ∈ M 4 ( C )
{ γ μ , γ ν } = − 2 g μ ν Id 4 , \{\gamma^{\mu},\gamma^{\nu}\}=-2g^{\mu\nu}\operatorname{Id}_{4}, { γ μ , γ ν } = − 2 g μν Id 4 , and are obtained from the flat Dirac matrices γ a \gamma^{a} γ a
γ μ = e a μ γ a , \gamma^{\mu}=e_{a}^{\ \mu}\gamma^{a}, γ μ = e a μ γ a , where the flat Dirac matrices satisfy { γ a , γ b } = − 2 η a b Id 4 \{\gamma^{a},\gamma^{b}\}=-2\eta^{ab}\operatorname{Id}_{4} { γ a , γ b } = − 2 η ab Id 4 [17 ]
γ 0 : = ( 0 2 Id 2 Id 2 0 2 ) , γ j : = ( 0 2 σ j − σ j 0 2 ) , \gamma^{0}:=\begin{pmatrix} 0_{2} & \operatorname{Id}_{2} \\ \operatorname{Id}_{2} & 0_{2} \end{pmatrix},\qquad
\gamma^{j}:=\begin{pmatrix} 0_{2} & \sigma_{j} \\ -\sigma_{j} & 0_{2} \end{pmatrix}, γ 0 := ( 0 2 Id 2 Id 2 0 2 ) , γ j := ( 0 2 − σ j σ j 0 2 ) , where the Pauli matrices σ j \sigma_j σ j
σ 1 = ( 0 1 1 0 ) , σ 2 = ( 0 − i i 0 ) , σ 3 = ( 1 0 0 − 1 ) . \sigma_{1}=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
\sigma_{2}=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\qquad
\sigma_{3}=\begin{pmatrix}1&0\\0&-1\end{pmatrix}. σ 1 = ( 0 1 1 0 ) , σ 2 = ( 0 i − i 0 ) , σ 3 = ( 1 0 0 − 1 ) . The spinorial covariant derivative D μ D_\mu D μ
D μ Ψ : = ∂ μ Ψ − 1 8 ω μ a b [ γ a , γ b ] Ψ , D_\mu\Psi:= \partial_\mu\Psi- \frac{1}{8}\omega_\mu{}^{ab}[\gamma_a,\gamma_b]\Psi, D μ Ψ := ∂ μ Ψ − 8 1 ω μ ab [ γ a , γ b ] Ψ , where ω μ a b = − ω μ b a \omega_\mu{}^{ab}=-\omega_\mu{}^{ba} ω μ ab = − ω μ ba spin connection . In terms of the vierbein and the Christoffel symbols, the spin connection is given by
ω μ a b : = e a ν ( Γ ν μ σ e σ b + ∂ μ e ν b ) . \omega_\mu{}^{ab}:=e^a{}_\nu\left(\Gamma^\nu{}_{\mu\sigma}e^{\sigma b}+\partial_\mu e^{\nu b}\right). ω μ ab := e a ν ( Γ ν μ σ e σ b + ∂ μ e ν b ) . Equivalently, its coefficients are characterized by Cartan’s structure equation
d e a + ω a b ∧ e b = 0 , (199) de^a+\omega^a{}_b\wedge e^b=0, \tag*{(199)} d e a + ω a b ∧ e b = 0 , (199) where e a : = e a μ d x μ e^a:=e^a{}_\mu\,dx^\mu e a := e a μ d x μ ω a b : = ω μ a b d x μ \omega^a{}_b:=\omega_\mu{}^a{}_b\,dx^\mu ω a b := ω μ a b d x μ [78 ] ). Strictly speaking, Ψ \Psi Ψ C 4 \mathbb{C}^4 C 4
For later use, we define the adjoint spinor Ψ ‾ : = Ψ ∗ γ 0 \overline{\Psi}:=\Psi^*\gamma^0 Ψ := Ψ ∗ γ 0 adjoint Dirac equation
i D μ Ψ ‾ γ μ = − m Ψ ‾ , iD_\mu\overline{\Psi}\gamma^\mu=-m\overline{\Psi}, i D μ Ψ γ μ = − m Ψ , where the covariant derivative now acts as
D μ Ψ ‾ = ∂ μ Ψ ‾ + 1 8 Ψ ‾ ω μ a b [ γ a , γ b ] . D_\mu\overline{\Psi}=\partial_\mu\overline{\Psi}+\frac{1}{8}\overline{\Psi}\omega_\mu{}^{ab}[\gamma_a,\gamma_b]. D μ Ψ = ∂ μ Ψ + 8 1 Ψ ω μ ab [ γ a , γ b ] . Finally, we conclude this section by giving the formula of the Dirac energy-momentum tensor
T μ ν : = i 4 ( Ψ ‾ γ μ D ν Ψ − D ν Ψ ‾ γ μ Ψ ) + i 4 ( Ψ ‾ γ ν D μ Ψ − D μ Ψ ‾ γ ν Ψ ) . T_{\mu\nu}:=\frac{i}{4}\left(\overline{\Psi}\gamma_\mu D_\nu\Psi-D_\nu\overline{\Psi}\gamma_\mu\Psi\right)+\frac{i}{4}\left(\overline{\Psi}\gamma_\nu D_\mu\Psi-D_\mu\overline{\Psi}\gamma_\nu\Psi\right). T μν := 4 i ( Ψ γ μ D ν Ψ − D ν Ψ γ μ Ψ ) + 4 i ( Ψ γ ν D μ Ψ − D μ Ψ γ ν Ψ ) . Reduction to spherical symmetry
We proceed here with the derivation of the Dirac equation in spherical symmetry. We follow the approach of [11 , 9 ] , which is closer to the perspective of dispersive PDEs. For a derivation more closely related to the Newman–Penrose formalism, we refer to Appendix A of [61 ] .
Analogously to the Einstein equations, for which a coordinate gauge is fixed, the derivation of the Dirac equation requires the choice of a frame gauge, that is, the choice of a tetrad. Since we consider black hole spacetimes which are perturbations of Schwarzschild, we can make such a choice globally in the exterior region. From § 4.1 , let ( M , g ) (\mathcal{M},g) ( M , g )
g = − 4 Ω 2 ( u , v ) d u d v + r 2 ( u , v ) d σ S 2 . g=-4\Omega^2(u,v)\,du\,dv+r^2(u,v)\,d\sigma_{\mathbb{S}^2}. g = − 4 Ω 2 ( u , v ) d u d v + r 2 ( u , v ) d σ S 2 . Recalling the geometric quantities defined in (5) –(7) , we consider the following gauge invariant double-null tetrad
Y : = 1 − ν ∂ u , V : = 1 − μ λ ∂ v , e 1 : = 1 r ∂ ϑ , e 2 : = 1 r sin ϑ ∂ φ , Y:=\frac{1}{-\nu}\partial_u,\qquad V:=\frac{1-\mu}{\lambda}\partial_v,\qquad e_1:=\frac{1}{r}\partial_\vartheta,\qquad e_2:=\frac{1}{r\sin\vartheta}\partial_\varphi, Y := − ν 1 ∂ u , V := λ 1 − μ ∂ v , e 1 := r 1 ∂ ϑ , e 2 := r sin ϑ 1 ∂ φ , which is regular away from { r = 0 } \{r=0\} { r = 0 } e a = e a μ ∂ μ e_a=e_a{}^\mu\partial_\mu e a = e a μ ∂ μ
e 0 = 1 2 ( V + Y ) = 1 2 ( 1 − μ λ ∂ v + 1 − ν ∂ u ) , e 1 , e 2 , e 3 = 1 2 ( V − Y ) = 1 2 ( 1 − μ λ ∂ v − 1 − ν ∂ u ) . e_0=\frac{1}{2}(V+Y)=\frac{1}{2}\left(\frac{1-\mu}{\lambda}\partial_v+\frac{1}{-\nu}\partial_u\right),\qquad e_1,\qquad e_2,\qquad e_3=\frac{1}{2}(V-Y)=\frac{1}{2}\left(\frac{1-\mu}{\lambda}\partial_v-\frac{1}{-\nu}\partial_u\right). e 0 = 2 1 ( V + Y ) = 2 1 ( λ 1 − μ ∂ v + − ν 1 ∂ u ) , e 1 , e 2 , e 3 = 2 1 ( V − Y ) = 2 1 ( λ 1 − μ ∂ v − − ν 1 ∂ u ) . with associated dual frame given by e a = η a b g μ ν e b ν d x μ e^{a}=\eta^{ab}g_{\mu\nu}e_{b}^{\nu}\,dx^{\mu} e a = η ab g μν e b ν d x μ
e 0 = − ν d u + κ d v , e 1 = r d ϑ , e 2 = r sin ϑ d φ , e 3 = ν d u + κ d v . e^{0}=-\nu\,du+\kappa\,dv,\qquad e^{1}=r\,d\vartheta,\qquad e^{2}=r\sin\vartheta\,d\varphi,\qquad e^{3}=\nu\,du+\kappa\,dv. e 0 = − ν d u + κ d v , e 1 = r d ϑ , e 2 = r sin ϑ d φ , e 3 = ν d u + κ d v .
Computing preliminarily
d e 0 = ( ∂ u κ + ∂ v ν ) d u ∧ d v = ∂ u κ + ∂ v ν 2 Ω 2 e 0 ∧ e 3 , de^{0}=(\partial_{u}\kappa+\partial_{v}\nu)\,du\wedge dv=\frac{\partial_{u}\kappa+\partial_{v}\nu}{2\Omega^{2}}\,e^{0}\wedge e^{3}, d e 0 = ( ∂ u κ + ∂ v ν ) d u ∧ d v = 2 Ω 2 ∂ u κ + ∂ v ν e 0 ∧ e 3 , d e 1 = − 1 r ( m r e 0 ∧ e 1 + ( 1 − m r ) e 1 ∧ e 3 ) , de^{1}=-\frac{1}{r}\left(\frac{m}{r}\,e^{0}\wedge e^{1}+\left(1-\frac{m}{r}\right)e^{1}\wedge e^{3}\right), d e 1 = − r 1 ( r m e 0 ∧ e 1 + ( 1 − r m ) e 1 ∧ e 3 ) , d e 2 = − 1 r ( m r e 0 ∧ e 2 + ( 1 − m r ) e 2 ∧ e 3 ) + cot ϑ r e 1 ∧ e 2 , de^{2}=-\frac{1}{r}\left(\frac{m}{r}\,e^{0}\wedge e^{2}+\left(1-\frac{m}{r}\right)e^{2}\wedge e^{3}\right)+\frac{\cot\vartheta}{r}e^{1}\wedge e^{2}, d e 2 = − r 1 ( r m e 0 ∧ e 2 + ( 1 − r m ) e 2 ∧ e 3 ) + r cot ϑ e 1 ∧ e 2 , d e 3 = ( ∂ u κ − ∂ v ν ) d u ∧ d v = ∂ u κ − ∂ v ν 2 Ω 2 e 0 ∧ e 3 , de^{3}=(\partial_{u}\kappa-\partial_{v}\nu)\,du\wedge dv=\frac{\partial_{u}\kappa-\partial_{v}\nu}{2\Omega^{2}}\,e^{0}\wedge e^{3}, d e 3 = ( ∂ u κ − ∂ v ν ) d u ∧ d v = 2 Ω 2 ∂ u κ − ∂ v ν e 0 ∧ e 3 , we obtain by Cartan’s formula (A.2) the spin connection coefficients:
ω v 03 = ∂ v ν ν , \omega_{v}^{\ 03}=\frac{\partial_{v}\nu}{\nu}, ω v 03 = ν ∂ v ν , ω u 03 = − ∂ u κ κ , \omega_{u}^{\ 03}=-\frac{\partial_{u}\kappa}{\kappa}, ω u 03 = − κ ∂ u κ , ω v 01 = − m r , ω v 13 = 1 − m r , \omega_{v}^{\ 01}=-\frac{m}{r},\qquad\omega_{v}^{\ 13}=1-\frac{m}{r}, ω v 01 = − r m , ω v 13 = 1 − r m , ω φ 02 = − m r sin ϑ , ω φ 23 = ( 1 − m r ) sin ϑ , ω φ 12 = − cos ϑ . \omega_{\varphi}^{\ 02}=-\frac{m}{r}\sin\vartheta,\qquad\omega_{\varphi}^{\ 23}=\left(1-\frac{m}{r}\right)\sin\vartheta,\qquad\omega_{\varphi}^{\ 12}=-\cos\vartheta. ω φ 02 = − r m sin ϑ , ω φ 23 = ( 1 − r m ) sin ϑ , ω φ 12 = − cos ϑ . Thus, recalling D μ = ∂ μ − 1 8 ω μ a b [ γ a , γ b ] D_{\mu}=\partial_{\mu}-\frac{1}{8}\omega_{\mu}^{\ ab}[\gamma_{a},\gamma_{b}] D μ = ∂ μ − 8 1 ω μ ab [ γ a , γ b ]
D 0 = D v = ∂ v − ∂ v ν 2 ν γ 0 γ 3 , D_{0}=D_{v}=\partial_{v}-\frac{\partial_{v}\nu}{2\nu}\gamma^{0}\gamma^{3}, D 0 = D v = ∂ v − 2 ν ∂ v ν γ 0 γ 3 , D 1 = D ϑ = ∂ ϑ + 1 2 ( m r γ 0 γ 1 − ( 1 − m r ) γ 1 γ 3 ) , D_{1}=D_{\vartheta}=\partial_{\vartheta}+\frac{1}{2}\left(\frac{m}{r}\gamma^{0}\gamma^{1}-\left(1-\frac{m}{r}\right)\gamma^{1}\gamma^{3}\right), D 1 = D ϑ = ∂ ϑ + 2 1 ( r m γ 0 γ 1 − ( 1 − r m ) γ 1 γ 3 ) , D 2 = D φ = ∂ φ + sin ϑ 2 ( m r γ 0 γ 2 − ( 1 − m r ) γ 2 γ 3 ) + cos ϑ 2 γ 1 γ 2 , D_{2}=D_{\varphi}=\partial_{\varphi}+\frac{\sin\vartheta}{2}\left(\frac{m}{r}\gamma^{0}\gamma^{2}-\left(1-\frac{m}{r}\right)\gamma^{2}\gamma^{3}\right)+\frac{\cos\vartheta}{2}\gamma^{1}\gamma^{2}, D 2 = D φ = ∂ φ + 2 sin ϑ ( r m γ 0 γ 2 − ( 1 − r m ) γ 2 γ 3 ) + 2 cos ϑ γ 1 γ 2 , D 3 = D u = ∂ u + ∂ u κ 2 κ γ 0 γ 3 , D_{3}=D_{u}=\partial_{u}+\frac{\partial_{u}\kappa}{2\kappa}\gamma^{0}\gamma^{3}, D 3 = D u = ∂ u + 2 κ ∂ u κ γ 0 γ 3 , while the curved Gamma matrices γ μ = e a μ γ a \gamma^{\mu}=e_{a}^{\ \mu}\gamma^{a} γ μ = e a μ γ a
γ 0 = γ v = 1 2 κ ( γ 0 + γ 3 ) , γ 1 = γ ϑ = 1 r γ 1 , γ 2 = γ φ = 1 r sin ϑ γ 2 , γ 3 = γ u = 1 − 2 ν ( γ 0 − γ 3 ) . \gamma^{0}=\gamma^{v}=\frac{1}{2\kappa}(\gamma^{0}+\gamma^{3}),\qquad\gamma^{1}=\gamma^{\vartheta}=\frac{1}{r}\gamma^{1},\qquad\gamma^{2}=\gamma^{\varphi}=\frac{1}{r\sin\vartheta}\gamma^{2},\qquad\gamma^{3}=\gamma^{u}=\frac{1}{-2\nu}(\gamma^{0}-\gamma^{3}). γ 0 = γ v = 2 κ 1 ( γ 0 + γ 3 ) , γ 1 = γ ϑ = r 1 γ 1 , γ 2 = γ φ = r sin ϑ 1 γ 2 , γ 3 = γ u = − 2 ν 1 ( γ 0 − γ 3 ) . Therefore, we are ready to explicitly write the Dirac equation (A.1):
i [ ( γ 0 + γ 3 ) ( 1 2 κ ∂ v − ∂ v ν 4 Ω 2 ) + ( γ 0 − γ 3 ) ( − 1 2 ν ∂ u + ∂ u κ 4 Ω 2 ) + 1 r ( 1 − m r ) γ 3 − m r 2 γ 0 + 1 r ( γ 1 ( ∂ ϑ + cot ϑ 2 ) + γ 2 sin ϑ ∂ φ ) ] Ψ = m Ψ . (201) i\left[(\gamma^{0}+\gamma^{3})\left(\frac{1}{2\kappa}\partial_{v}-\frac{\partial_{v}\nu}{4\Omega^{2}}\right)+(\gamma^{0}-\gamma^{3})\left(-\frac{1}{2\nu}\partial_{u}+\frac{\partial_{u}\kappa}{4\Omega^{2}}\right)+\frac{1}{r}\left(1-\frac{m}{r}\right)\gamma^{3}-\frac{m}{r^{2}}\gamma^{0}+\frac{1}{r}\left(\gamma^{1}\left(\partial_{\vartheta}+\frac{\cot\vartheta}{2}\right)+\frac{\gamma^{2}}{\sin\vartheta}\partial_{\varphi}\right)\right]\Psi=m\Psi. \tag*{(201)} i [ ( γ 0 + γ 3 ) ( 2 κ 1 ∂ v − 4 Ω 2 ∂ v ν ) + ( γ 0 − γ 3 ) ( − 2 ν 1 ∂ u + 4 Ω 2 ∂ u κ ) + r 1 ( 1 − r m ) γ 3 − r 2 m γ 0 + r 1 ( γ 1 ( ∂ ϑ + 2 cot ϑ ) + sin ϑ γ 2 ∂ φ ) ] Ψ = m Ψ. (201) Multiplying (A.4) by γ 0 \gamma^{0} γ 0
[ ( Id 4 − γ 0 γ 3 ) ( − 1 2 ν ∂ u + ∂ u κ 4 Ω 2 ) + ( Id 4 + γ 0 γ 3 ) ( 1 2 κ ∂ v − ∂ v ν 4 Ω 2 ) + γ 0 r D S 2 − m r 2 Id 4 + 1 r ( 1 − m r ) γ 0 γ 3 ] Ψ = − i m γ 0 Ψ , \left[(\operatorname{Id}_{4}-\gamma^{0}\gamma^{3})\left(-\frac{1}{2\nu}\partial_{u}+\frac{\partial_{u}\kappa}{4\Omega^{2}}\right)+(\operatorname{Id}_{4}+\gamma^{0}\gamma^{3})\left(\frac{1}{2\kappa}\partial_{v}-\frac{\partial_{v}\nu}{4\Omega^{2}}\right)+\frac{\gamma^{0}}{r}\mathcal{D}_{S^{2}}-\frac{m}{r^{2}}\operatorname{Id}_{4}+\frac{1}{r}\left(1-\frac{m}{r}\right)\gamma^{0}\gamma^{3}\right]\Psi=-im\gamma^{0}\Psi, [ ( Id 4 − γ 0 γ 3 ) ( − 2 ν 1 ∂ u + 4 Ω 2 ∂ u κ ) + ( Id 4 + γ 0 γ 3 ) ( 2 κ 1 ∂ v − 4 Ω 2 ∂ v ν ) + r γ 0 D S 2 − r 2 m Id 4 + r 1 ( 1 − r m ) γ 0 γ 3 ] Ψ = − im γ 0 Ψ , where we denoted
D S 2 : = γ 1 ( ∂ ϑ + cot ϑ 2 ) + γ 2 sin ϑ ∂ φ . \mathcal{D}_{S^{2}}:=\gamma^{1}\left(\partial_{\vartheta}+\frac{\cot\vartheta}{2}\right)+\frac{\gamma^{2}}{\sin\vartheta}\partial_{\varphi}. D S 2 := γ 1 ( ∂ ϑ + 2 cot ϑ ) + sin ϑ γ 2 ∂ φ . To better highlight the block structure of the matrices, we further transform the equation by applying the orthogonal matrix P = ( 1 1 − 1 1 ) P = \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix} P = ( 1 − 1 1 1 )
[ ( − 1 ν ∂ u + ∂ u κ 2 Ω 2 ) ( I d 2 0 2 0 2 0 2 ) + ( 1 κ ∂ v − ∂ v ν 2 Ω 2 ) ( 0 2 0 2 0 2 I d 2 ) + 1 r ( 0 2 ∇ ^ ∇ ^ 0 2 ) + 1 r ( − I d 2 0 2 0 2 ( 1 − μ ) I d 2 ) ] P Ψ = − i m ( 0 2 σ 3 σ 3 0 2 ) P Ψ , (202) \left[\left(-\frac{1}{\nu}\partial_u+\frac{\partial_u\kappa}{2\Omega^2}\right)\begin{pmatrix}\mathrm{Id}_2 & 0_2 \\ 0_2 & 0_2\end{pmatrix}+\left(\frac{1}{\kappa}\partial_v-\frac{\partial_v\nu}{2\Omega^2}\right)\begin{pmatrix}0_2 & 0_2 \\ 0_2 & \mathrm{Id}_2\end{pmatrix}+\frac{1}{r}\begin{pmatrix}0_2 & \widehat{\nabla} \\ \widehat{\nabla} & 0_2\end{pmatrix}+\frac{1}{r}\begin{pmatrix}-\mathrm{Id}_2 & 0_2 \\ 0_2 & (1-\mu)\mathrm{Id}_2\end{pmatrix}\right]P\Psi=-im\begin{pmatrix}0_2 & \sigma_3 \\ \sigma_3 & 0_2\end{pmatrix}P\Psi, \tag*{(202)} [ ( − ν 1 ∂ u + 2 Ω 2 ∂ u κ ) ( Id 2 0 2 0 2 0 2 ) + ( κ 1 ∂ v − 2 Ω 2 ∂ v ν ) ( 0 2 0 2 0 2 Id 2 ) + r 1 ( 0 2 ∇ ∇ 0 2 ) + r 1 ( − Id 2 0 2 0 2 ( 1 − μ ) Id 2 ) ] P Ψ = − im ( 0 2 σ 3 σ 3 0 2 ) P Ψ , (202) where
∇ ^ S 2 : = σ 1 ( ∂ ϑ + cot ϑ 2 ) + σ 2 sin ϑ ∂ φ \widehat{\nabla}_{\mathbb{S}^2}:=\sigma_1\left(\partial_\vartheta+\frac{\cot\vartheta}{2}\right)+\frac{\sigma_2}{\sin\vartheta}\partial_\varphi ∇ S 2 := σ 1 ( ∂ ϑ + 2 cot ϑ ) + sin ϑ σ 2 ∂ φ is the Dirac operator on the sphere.
The operator ∇ ^ S 2 \widehat{\nabla}_{\mathbb{S}^2} ∇ S 2 partial wave decomposition (see [2 , 8 , 75 ] ), allowing us to simplify the angular dependence in terms of spinorial spherical harmonics E j , m j ± \mathcal{E}^{\pm}_{j,m_j} E j , m j ±
∇ ^ S 2 E j , m j ± = ∓ ( 1 2 + j ) E j , m j ± and σ 3 E j , m j ± = E j , m j ∓ , \widehat{\nabla}_{\mathbb{S}^2}\mathcal{E}^{\pm}_{j,m_j}=\mp\left(\frac{1}{2}+j\right)\mathcal{E}^{\pm}_{j,m_j}\qquad\text{and}\qquad\sigma^3\mathcal{E}^{\pm}_{j,m_j}=\mathcal{E}^{\mp}_{j,m_j}, ∇ S 2 E j , m j ± = ∓ ( 2 1 + j ) E j , m j ± and σ 3 E j , m j ± = E j , m j ∓ , where j ∈ 1 2 + N j\in\frac{1}{2}+\mathbb{N} j ∈ 2 1 + N m j ∈ 1 2 + Z m_j\in\frac{1}{2}+\mathbb{Z} m j ∈ 2 1 + Z − j ≤ m j ≤ j -j\leq m_j\leq j − j ≤ m j ≤ j E j , m j ± \mathcal{E}^{\pm}_{j,m_j} E j , m j ± standard spherical harmonics Y j ± , m ± Y_{j_{\pm},m_{\pm}} Y j ± , m ± j ± : = j ± 1 2 j_{\pm}:=j\pm\frac{1}{2} j ± := j ± 2 1 m j ± : = m j ± 1 2 m^{\pm}_{j}:=m_j\pm\frac{1}{2} m j ± := m j ± 2 1
Moreover, these spinorial spherical harmonics can then be combined into C 4 \mathbb{C}^{4} C 4 H ~ j , m j ± \widetilde{\mathcal{H}}^{\pm}_{j,m_j} H j , m j ± [6 Section 5] , the partial wave subspaces are preserved by the nonlinear dynamics provided that the initial datum has an angular part belonging to one of the four spaces H ~ 1 / 2 , m 1 / 2 ± \widetilde{\mathcal{H}}^{\pm}_{1/2,m_{1/2}} H 1/2 , m 1/2 ± H ~ 1 / 2 , 1 / 2 − : = span ( G 1 / 2 , 1 / 2 + , G 1 / 2 , 1 / 2 − ) \widetilde{\mathcal{H}}^{-}_{1/2,1/2}:=\operatorname{span}(\mathcal{G}^{+}_{1/2,1/2},\mathcal{G}^{-}_{1/2,1/2}) H 1/2 , 1/2 − := span ( G 1/2 , 1/2 + , G 1/2 , 1/2 − )
G 1 / 2 , 1 / 2 + : = ( E 1 / 2 , 1 / 2 + 0 ) : = e i φ / 2 2 π ( cos ϑ 2 − sin ϑ 2 0 0 ) , G 1 / 2 , 1 / 2 − : = ( 0 E 1 / 2 , 1 / 2 − ) : = e i φ / 2 2 π ( 0 0 cos ϑ 2 sin ϑ 2 ) . (203) \mathcal{G}^{+}_{1/2,1/2}:=\begin{pmatrix}\mathcal{E}^{+}_{1/2,1/2}\\0\end{pmatrix}:=\frac{e^{i\varphi/2}}{2\sqrt{\pi}}\begin{pmatrix}\cos\frac{\vartheta}{2}\\-\sin\frac{\vartheta}{2}\\0\\0\end{pmatrix},\qquad
\mathcal{G}^{-}_{1/2,1/2}:=\begin{pmatrix}0\\\mathcal{E}^{-}_{1/2,1/2}\end{pmatrix}:=\frac{e^{i\varphi/2}}{2\sqrt{\pi}}\begin{pmatrix}0\\0\\\cos\frac{\vartheta}{2}\\\sin\frac{\vartheta}{2}\end{pmatrix}. \tag*{(203)} G 1/2 , 1/2 + := ( E 1/2 , 1/2 + 0 ) := 2 π e i φ /2 cos 2 ϑ − sin 2 ϑ 0 0 , G 1/2 , 1/2 − := ( 0 E 1/2 , 1/2 − ) := 2 π e i φ /2 0 0 cos 2 ϑ sin 2 ϑ . (203) Therefore, considering the following ansatz
P Ψ : = Ψ + G 1 / 2 , 1 / 2 + + Ψ − G 1 / 2 , 1 / 2 − = ( Ψ + E 1 / 2 , 1 / 2 + Ψ − E 1 / 2 , 1 / 2 − ) , P\Psi:=\Psi_{+}\mathcal{G}^{+}_{1/2,1/2}+\Psi_{-}\mathcal{G}^{-}_{1/2,1/2}=\begin{pmatrix}\Psi_{+}\mathcal{E}^{+}_{1/2,1/2}\\\Psi_{-}\mathcal{E}^{-}_{1/2,1/2}\end{pmatrix}, P Ψ := Ψ + G 1/2 , 1/2 + + Ψ − G 1/2 , 1/2 − = ( Ψ + E 1/2 , 1/2 + Ψ − E 1/2 , 1/2 − ) , and plugging it into (A.5), we obtain two coupled equations
( 1 − ν ∂ u + ∂ u κ 2 Ω 2 − 1 r ) Ψ + + 1 r Ψ − = − i m Ψ − , \left(\frac{1}{-\nu}\partial_u+\frac{\partial_u\kappa}{2\Omega^2}-\frac{1}{r}\right)\Psi_{+}+\frac{1}{r}\Psi_{-}=-im\Psi_{-}, ( − ν 1 ∂ u + 2 Ω 2 ∂ u κ − r 1 ) Ψ + + r 1 Ψ − = − im Ψ − , ( 1 κ ∂ v − ∂ v ν 2 Ω 2 + 1 − μ r ) Ψ − − 1 r Ψ + = − i m Ψ + . \left(\frac{1}{\kappa}\partial_v-\frac{\partial_v\nu}{2\Omega^2}+\frac{1-\mu}{r}\right)\Psi_{-}-\frac{1}{r}\Psi_{+}=-im\Psi_{+}. ( κ 1 ∂ v − 2 Ω 2 ∂ v ν + r 1 − μ ) Ψ − − r 1 Ψ + = − im Ψ + . Performing the gauge invariant reweights
ψ + : = r Ψ + , ψ − : = Ψ − , \psi_{+}:=r\Psi_{+},\qquad\psi_{-}:=\Psi_{-}, ψ + := r Ψ + , ψ − := Ψ − , we finally obtain the reduced Dirac equations
( 1 − ν ∂ u + ∂ u κ 2 Ω 2 ) ψ + = − ( 1 + i m r ) ψ − , \left(\frac{1}{-\nu}\partial_u+\frac{\partial_u\kappa}{2\Omega^2}\right)\psi_{+}=-(1+imr)\psi_{-}, ( − ν 1 ∂ u + 2 Ω 2 ∂ u κ ) ψ + = − ( 1 + im r ) ψ − , ( 1 κ ∂ v − ∂ v ν 2 Ω 2 + 1 − μ r ) ψ − = 1 r 2 ( 1 − i m r ) ψ + . (204) \left(\frac{1}{\kappa}\partial_v-\frac{\partial_v\nu}{2\Omega^2}+\frac{1-\mu}{r}\right)\psi_{-}=\frac{1}{r^2}(1-imr)\psi_{+}. \tag*{(204)} ( κ 1 ∂ v − 2 Ω 2 ∂ v ν + r 1 − μ ) ψ − = r 2 1 ( 1 − im r ) ψ + . (204) Moreover, assuming ν < 0 \nu< 0 ν < 0
Φ + : = κ 1 / 2 ψ + , Φ − : = r ( − ν ) 1 / 2 ψ − , \Phi_{+} := \kappa^{1/2}\psi_{+}, \qquad\Phi_{-} := r(-\nu)^{1/2}\psi_{-}, Φ + := κ 1/2 ψ + , Φ − := r ( − ν ) 1/2 ψ − , and find
∂ u Φ + = − Ω r ( 1 + i m r ) Φ − , \partial_{u}\Phi_{+} = -\frac{\Omega}{r}(1+imr)\Phi_{-}, ∂ u Φ + = − r Ω ( 1 + im r ) Φ − , ∂ v Φ − = Ω r ( 1 − i m r ) Φ + . (205) \partial_{v}\Phi_{-} = \frac{\Omega}{r}(1-imr)\Phi_{+}. \tag*{(205)} ∂ v Φ − = r Ω ( 1 − im r ) Φ + . (205) Let us now proceed to explicitly compute the components of the Dirac energy-momentum tensor T μ ν T_{\mu\nu} T μν
Ψ = P − 1 ( Ψ + E 1 / 2 , 1 / 2 + Ψ − E 1 / 2 , 1 / 2 − ) = e i φ / 2 2 π ( Ψ + cos ϑ / 2 − Ψ − sin ϑ / 2 Ψ − cos ϑ / 2 − Ψ + sin ϑ / 2 ) , \Psi=P^{-1}\begin{pmatrix}\Psi_{+}\mathcal{E}^{+}_{1/2,1/2}\\ \Psi_{-}\mathcal{E}^{-}_{1/2,1/2}\end{pmatrix}
=\frac{e^{i\varphi/2}}{2\sqrt{\pi}}
\begin{pmatrix}
\Psi_{+}\cos\vartheta/2\\
-\Psi_{-}\sin\vartheta/2\\
\Psi_{-}\cos\vartheta/2\\
-\Psi_{+}\sin\vartheta/2
\end{pmatrix}, Ψ = P − 1 ( Ψ + E 1/2 , 1/2 + Ψ − E 1/2 , 1/2 − ) = 2 π e i φ /2 Ψ + cos ϑ /2 − Ψ − sin ϑ /2 Ψ − cos ϑ /2 − Ψ + sin ϑ /2 , one obtains
T u u = i 2 ( Ψ ‾ γ u D u Ψ − D u Ψ ‾ γ u Ψ ) = i 2 ( ( Ψ ∗ γ 0 ) ( g u v γ v ) ( ∂ u − ∂ u κ 2 κ γ 0 γ 3 ) Ψ − ( ∂ u Ψ ∗ γ 0 + ∂ u κ 2 κ Ψ ∗ γ 0 γ 3 ) ( g u v γ v ) Ψ ) = i ν 2 [ Ψ ∗ ( I d 4 + γ 0 γ 3 ) ∂ u Ψ − ∂ u Ψ ∗ ( I d 4 + γ 0 γ 3 ) Ψ ] = − ν 2 π ℑ [ Ψ ‾ − ∂ u Ψ − ] = − ν 2 π ℑ [ ψ ‾ − ∂ u ψ − ] . \begin{aligned}
T_{uu}
&=\frac{i}{2}\left(\overline{\Psi}\gamma_{u}D_{u}\Psi-D_{u}\overline{\Psi}\gamma_{u}\Psi\right)\\
&=\frac{i}{2}\left(\left(\Psi^{*}\gamma^{0}\right)(g_{uv}\gamma^{v})\left(\partial_{u}-\frac{\partial_{u}\kappa}{2\kappa}\gamma^{0}\gamma^{3}\right)\Psi-\left(\partial_{u}\Psi^{*}\gamma^{0}+\frac{\partial_{u}\kappa}{2\kappa}\Psi^{*}\gamma^{0}\gamma^{3}\right)(g_{uv}\gamma^{v})\Psi\right)\\
&=\frac{i\nu}{2}\left[\Psi^{*}(I\!d_{4}+\gamma^{0}\gamma^{3})\partial_{u}\Psi-\partial_{u}\Psi^{*}(I\!d_{4}+\gamma^{0}\gamma^{3})\Psi\right]\\
&=-\frac{\nu}{2\pi}\Im[\overline{\Psi}_{-}\partial_{u}\Psi_{-}]
=-\frac{\nu}{2\pi}\Im[\overline{\psi}_{-}\partial_{u}\psi_{-}].
\end{aligned} T uu = 2 i ( Ψ γ u D u Ψ − D u Ψ γ u Ψ ) = 2 i ( ( Ψ ∗ γ 0 ) ( g uv γ v ) ( ∂ u − 2 κ ∂ u κ γ 0 γ 3 ) Ψ − ( ∂ u Ψ ∗ γ 0 + 2 κ ∂ u κ Ψ ∗ γ 0 γ 3 ) ( g uv γ v ) Ψ ) = 2 i ν [ Ψ ∗ ( I d 4 + γ 0 γ 3 ) ∂ u Ψ − ∂ u Ψ ∗ ( I d 4 + γ 0 γ 3 ) Ψ ] = − 2 π ν ℑ [ Ψ − ∂ u Ψ − ] = − 2 π ν ℑ [ ψ − ∂ u ψ − ] . Similarly, one has
T v v = κ 2 π ℑ [ Ψ ‾ + ∂ v Ψ + ] = κ 2 π r 2 ℑ [ ψ ‾ + ∂ v ψ + ] , T_{vv}=\frac{\kappa}{2\pi}\Im[\overline{\Psi}_{+}\partial_{v}\Psi_{+}]
=\frac{\kappa}{2\pi r^{2}}\Im[\overline{\psi}_{+}\partial_{v}\psi_{+}], T v v = 2 π κ ℑ [ Ψ + ∂ v Ψ + ] = 2 π r 2 κ ℑ [ ψ + ∂ v ψ + ] , T u v = − ν 4 π ℑ [ Ψ ‾ − ∂ v Ψ − ] + κ 4 π ℑ [ Ψ ‾ + ∂ u Ψ + ] = − ν 4 π ℑ [ ψ ‾ − ∂ v ψ − ] + κ 4 π r 2 ℑ [ ψ ‾ + ∂ u ψ + ] , T_{uv}=-\frac{\nu}{4\pi}\Im[\overline{\Psi}_{-}\partial_{v}\Psi_{-}]
+\frac{\kappa}{4\pi}\Im[\overline{\Psi}_{+}\partial_{u}\Psi_{+}]
=-\frac{\nu}{4\pi}\Im[\overline{\psi}_{-}\partial_{v}\psi_{-}]
+\frac{\kappa}{4\pi r^{2}}\Im[\overline{\psi}_{+}\partial_{u}\psi_{+}], T uv = − 4 π ν ℑ [ Ψ − ∂ v Ψ − ] + 4 π κ ℑ [ Ψ + ∂ u Ψ + ] = − 4 π ν ℑ [ ψ − ∂ v ψ − ] + 4 π r 2 κ ℑ [ ψ + ∂ u ψ + ] , T ϑ ϑ = r 4 π ℑ [ Ψ ‾ − Ψ + ] = 1 4 π ℑ [ ψ ‾ − ψ + ] , T_{\vartheta\vartheta}=\frac{r}{4\pi}\Im[\overline{\Psi}_{-}\Psi_{+}]
=\frac{1}{4\pi}\Im[\overline{\psi}_{-}\psi_{+}], T ϑϑ = 4 π r ℑ [ Ψ − Ψ + ] = 4 π 1 ℑ [ ψ − ψ + ] , T φ φ = sin 2 ϑ T ϑ ϑ . T_{\varphi\varphi}=\sin^{2}\vartheta\,T_{\vartheta\vartheta}. T φφ = sin 2 ϑ T ϑϑ . Using (A.7), we compute
T u v = Ω 2 2 π r 2 ( ℑ [ ψ ‾ − ψ + ] − m r ℜ [ ψ ‾ − ψ + ] ) , T_{uv}=\frac{\Omega^{2}}{2\pi r^{2}}\left(\Im[\overline{\psi}_{-}\psi_{+}]-mr\Re[\overline{\psi}_{-}\psi_{+}]\right), T uv = 2 π r 2 Ω 2 ( ℑ [ ψ − ψ + ] − m r ℜ [ ψ − ψ + ] ) , and thus verify Remark A.1:
tr g T = g μ ν T μ ν = m 2 π r ℜ [ ψ ‾ − ψ + ] = m Ψ ‾ Ψ . \operatorname{tr}_{g}T=g^{\mu\nu}T_{\mu\nu}
=\frac{m}{2\pi r}\Re[\overline{\psi}_{-}\psi_{+}]
=m\overline{\Psi}\Psi. tr g T = g μν T μν = 2 π r m ℜ [ ψ − ψ + ] = m Ψ Ψ. Finally, we recall the Einstein equations in spherical symmetry
∂ u ∂ v r = − 2 Ω 2 m r 2 + 4 π r T u v , \partial_{u}\partial_{v}r=-\frac{2\Omega^{2}m}{r^{2}}+4\pi rT_{uv}, ∂ u ∂ v r = − r 2 2 Ω 2 m + 4 π r T uv , 1 2 ∂ u ∂ v log Ω 2 = 2 Ω 2 m r 3 − 4 π T u v − 4 π Ω 2 g A B T A B , \frac{1}{2}\partial_{u}\partial_{v}\log\Omega^{2}
=\frac{2\Omega^{2}m}{r^{3}}-4\pi T_{uv}-4\pi\Omega^{2}g^{AB}T_{AB}, 2 1 ∂ u ∂ v log Ω 2 = r 3 2 Ω 2 m − 4 π T uv − 4 π Ω 2 g A B T A B , ∂ u ( Ω − 2 ∂ u r ) = − 4 π r Ω 2 T u u , \partial_{u}(\Omega^{-2}\partial_{u}r)=-\frac{4\pi r}{\Omega^{2}}T_{uu}, ∂ u ( Ω − 2 ∂ u r ) = − Ω 2 4 π r T uu , ∂ v ( Ω − 2 ∂ v r ) = − 4 π r Ω 2 T v v . (206) \partial_{v}(\Omega^{-2}\partial_{v}r)=-\frac{4\pi r}{\Omega^{2}}T_{vv}. \tag*{(206)} ∂ v ( Ω − 2 ∂ v r ) = − Ω 2 4 π r T v v . (206) where the capital indices denote the spherical variables. Using the relation g A B T A B = tr g T + Ω − 2 T u v g^{AB}T_{AB}=\operatorname{tr}_{g}T+\Omega^{-2}T_{uv} g A B T A B = tr g T + Ω − 2 T uv (10) –(13) (in the case when m = 0 m=0 m = 0
Local well-posedness for the Einstein–Dirac system
In this section, we briefly trace the argument for local well-posedness of the spherically symmetric Einstein–Dirac system in characteristic slabs. For a treatment in the absence of symmetry, we refer to [80 ] . We include the argument here, since the formulation below is more directly suited to our needs.
As already observed in [14 ] , a direct application of the standard local well-posedness theory to the reduced Einstein–Dirac system is complicated by a potential loss of derivatives. Indeed, the spin connection coefficients in the reduced Dirac equations contain derivatives of the metric coefficients. For instance, in the equation for ψ − \psi_{-} ψ − ∂ u κ \partial_{u}\kappa ∂ u κ
∂ u ψ + = N ( g , ∂ g , ψ , ∂ u ψ − ) , \partial_{u}\psi_{+}=\mathcal{N}(g,\partial g,\psi,\partial_{u}\psi_{-}), ∂ u ψ + = N ( g , ∂ g , ψ , ∂ u ψ − ) , which prevents a direct application of the usual quasi-linear theory. To circumvent this issue, a standard approach consists in considering the second-order equations satisfied by ψ ± \psi_{\pm} ψ ± ∂ v \partial_{v} ∂ v ∂ u \partial_{u} ∂ u ψ + \psi_{+} ψ + ψ − \psi_{-} ψ −
∂ 2 ψ = N ( g , ∂ g , ψ , ∂ ψ ) , \partial^{2}\psi=\mathcal{N}(g,\partial g,\psi,\partial\psi), ∂ 2 ψ = N ( g , ∂ g , ψ , ∂ ψ ) , ∂ 2 g = N ( g , ∂ g , ψ , ∂ ψ ) , \partial^{2}g=\mathcal{N}(g,\partial g,\psi,\partial\psi), ∂ 2 g = N ( g , ∂ g , ψ , ∂ ψ ) , for which local well-posedness holds by [28 ] . One then recovers a solution to the original Einstein–Dirac system by propagating the constraint equations and the Dirac equations from the initial bifurcate null hypersurface.
As this procedure requires many substitutions and involves lengthy computations, we rather opt to formulate the well-posedness problem in terms of ( r , Ω , Φ + , Φ − ) (r,\Omega,\Phi_{+},\Phi_{-}) ( r , Ω , Φ + , Φ − ) ν < 0 \nu<0 ν < 0 absence of anti-trapped spheres , which is not restrictive in our setting. Another reason for carrying out this argument here is that, since the dominant energy condition is not satisfied by the Dirac field, we cannot directly rely on the local well-posedness results of [43 ] . Nevertheless, we follow the same general strategy, while replacing the monotonicity arguments relying on the dominant energy condition by arguments adapted to the Einstein–Dirac system.
For simplicity, let us now recall here Proposition 2.9, which constitutes the main result of this section.
Proposition B.1 (Local existence on slabs). For any m ≥ 0 m\geq0 m ≥ 0 and B > 0 B>0 B > 0 there exists a constant ε s l a b > 0 \varepsilon_{\mathrm{slab}}>0 ε slab > 0 with the following property. Let ( r ˚ , Ω ˚ , ψ + o u t , ψ − i n ) (\mathring r,\mathring\Omega,\psi_{+}^{\mathrm{out}},\psi_{-}^{\mathrm{in}}) ( r ˚ , Ω ˚ , ψ + out , ψ − in ) be a characteristic initial data set for the spherically symmetric Einstein–Dirac system on C ( U 0 , U 1 , V 0 , V 1 ) \mathcal{C}(U_{0},U_{1},V_{0},V_{1}) C ( U 0 , U 1 , V 0 , V 1 ) as in Definition 2.7 satisfying
∥ log r ˚ ∥ C 2 ( C ) + ∥ log Ω ˚ ∥ C 2 ( C ) + ∥ ψ + o u t ∥ C 0 ( C o u t ) + ∥ ∂ v ψ + o u t ∥ C 0 ( C o u t ) + ∥ ψ − i n ∥ C 0 ( C i n ) + ∥ ∂ u ψ − i n ∥ C 0 ( C i n ) ≤ B . \lVert\log\mathring r\rVert_{C^{2}(\mathcal{C})}+\lVert\log\mathring\Omega\rVert_{C^{2}(\mathcal{C})}+\lVert\psi_{+}^{\mathrm{out}}\rVert_{C^{0}(\mathcal{C}_{\mathrm{out}})}+\lVert\partial_{v}\psi_{+}^{\mathrm{out}}\rVert_{C^{0}(\mathcal{C}_{\mathrm{out}})}+\lVert\psi_{-}^{\mathrm{in}}\rVert_{C^{0}(\mathcal{C}_{\mathrm{in}})}+\lVert\partial_{u}\psi_{-}^{\mathrm{in}}\rVert_{C^{0}(\mathcal{C}_{\mathrm{in}})}\leq B. ∥ log r ˚ ∥ C 2 ( C ) + ∥ log Ω ˚ ∥ C 2 ( C ) + ∥ ψ + out ∥ C 0 ( C out ) + ∥ ∂ v ψ + out ∥ C 0 ( C out ) + ∥ ψ − in ∥ C 0 ( C in ) + ∥ ∂ u ψ − in ∥ C 0 ( C in ) ≤ B . Then there exists a unique smooth solution ( r , Ω , ψ + , ψ − ) (r,\Omega,\psi_{+},\psi_{-}) ( r , Ω , ψ + , ψ − ) to the spherically symmetric Einstein–Dirac system ((A.8)–(A.9)) on the slab
D : = R ( U 0 , U 0 + min { ε s l a b , U 1 − U 0 } , V 0 , V 1 ) ∪ R ( U 0 , U 1 , V 0 , V 0 + min { ε s l a b , V 1 − V 0 } ) , \mathcal{D}:=\mathcal{R}(U_{0},U_{0}+\min\{\varepsilon_{\mathrm{slab}},U_{1}-U_{0}\},V_{0},V_{1})\cup\mathcal{R}(U_{0},U_{1},V_{0},V_{0}+\min\{\varepsilon_{\mathrm{slab}},V_{1}-V_{0}\}), D := R ( U 0 , U 0 + min { ε slab , U 1 − U 0 } , V 0 , V 1 ) ∪ R ( U 0 , U 1 , V 0 , V 0 + min { ε slab , V 1 − V 0 }) , satisfying ν < 0 \nu<0 ν < 0 on D \mathcal{D} D and which extends the initial data. Moreover, the norms
∥ log r ∥ C k ( D ) , ∥ log Ω ∥ C k ( D ) , ∥ ψ + ∥ C k ( D ) , ∥ ψ − ∥ C k ( D ) \lVert\log r\rVert_{C^{k}(\mathcal{D})},\lVert\log\Omega\rVert_{C^{k}(\mathcal{D})},\lVert\psi_{+}\rVert_{C^{k}(\mathcal{D})},\lVert\psi_{-}\rVert_{C^{k}(\mathcal{D})} ∥ log r ∥ C k ( D ) , ∥ log Ω ∥ C k ( D ) , ∥ ψ + ∥ C k ( D ) , ∥ ψ − ∥ C k ( D ) are finite for any k k k and can be bounded in terms of appropriate higher order initial data norms.
The proof is a consequence of the following two lemmas. We first establish a local existence result.
Lemma B.2. For any m ≥ 0 m\geq0 m ≥ 0 and B > 0 B>0 B > 0 there exists a constant ε 0 > 0 \varepsilon_{0}>0 ε 0 > 0 with the following property. Let ( r ˚ , Ω ˚ , ψ + o u t , ψ − i n ) (\mathring r,\mathring\Omega,\psi_{+}^{\mathrm{out}},\psi_{-}^{\mathrm{in}}) ( r ˚ , Ω ˚ , ψ + out , ψ − in ) be a characteristic initial data set for the spherically symmetric Einstein–Dirac system on C ( U 0 , U 1 , V 0 , V 1 ) \mathcal{C}(U_{0},U_{1},V_{0},V_{1}) C ( U 0 , U 1 , V 0 , V 1 ) as in Definition 2.7 . If U 1 − U 0 < ε 0 U_{1}-U_{0}<\varepsilon_{0} U 1 − U 0 < ε 0 V 1 − V 0 < ε 0 V_{1}-V_{0}<\varepsilon_{0} V 1 − V 0 < ε 0 and
∥ log r ˚ ∥ C 2 ( C ) + ∥ log Ω ˚ ∥ C 2 ( C ) + ∥ ψ + o u t ∥ C 0 ( C o u t ) + ∥ ∂ v ψ + o u t ∥ C 0 ( C o u t ) + ∥ ψ − i n ∥ C 0 ( C i n ) + ∥ ∂ u ψ − i n ∥ C 0 ( C i n ) ≤ B . \lVert\log\mathring r\rVert_{C^{2}(\mathcal{C})}+\lVert\log\mathring\Omega\rVert_{C^{2}(\mathcal{C})}+\lVert\psi_{+}^{\mathrm{out}}\rVert_{C^{0}(\mathcal{C}_{\mathrm{out}})}+\lVert\partial_{v}\psi_{+}^{\mathrm{out}}\rVert_{C^{0}(\mathcal{C}_{\mathrm{out}})}+\lVert\psi_{-}^{\mathrm{in}}\rVert_{C^{0}(\mathcal{C}_{\mathrm{in}})}+\lVert\partial_{u}\psi_{-}^{\mathrm{in}}\rVert_{C^{0}(\mathcal{C}_{\mathrm{in}})}\leq B. ∥ log r ˚ ∥ C 2 ( C ) + ∥ log Ω ˚ ∥ C 2 ( C ) + ∥ ψ + out ∥ C 0 ( C out ) + ∥ ∂ v ψ + out ∥ C 0 ( C out ) + ∥ ψ − in ∥ C 0 ( C in ) + ∥ ∂ u ψ − in ∥ C 0 ( C in ) ≤ B . Then there exists a unique smooth solution ( r , Ω , ψ + , ψ − ) (r,\Omega,\psi_{+},\psi_{-}) ( r , Ω , ψ + , ψ − ) R ( U 0 , U 1 , V 0 , V 1 ) \mathcal{R}(U_{0},U_{1},V_{0},V_{1}) R ( U 0 , U 1 , V 0 , V 1 ) ν < 0 \nu<0 ν < 0 R \mathcal{R} R
∥ log r ∥ C k ( R ) , ∥ log Ω ∥ C k ( R ) , ∥ ψ + ∥ C k ( R ) , ∥ ψ − ∥ C k ( R ) \lVert\log r\rVert_{C^{k}(\mathcal{R})},\lVert\log\Omega\rVert_{C^{k}(\mathcal{R})},\lVert\psi_{+}\rVert_{C^{k}(\mathcal{R})},\lVert\psi_{-}\rVert_{C^{k}(\mathcal{R})} ∥ log r ∥ C k ( R ) , ∥ log Ω ∥ C k ( R ) , ∥ ψ + ∥ C k ( R ) , ∥ ψ − ∥ C k ( R ) are finite for any k k k
Proof. We follow the approach in [23 ] and is based on a standard contraction mapping argument, which requires a nested iteration due to the coupling of the Dirac components.
Since the initial data satisfy ν ˚ < 0 \mathring{\nu}<0 ν ˚ < 0 C \mathcal{C} C
Φ ˚ + o u t : = κ ˚ 1 / 2 ψ ˚ + o u t , Φ ˚ − i n : = r ˚ ( − ν ˚ ) 1 / 2 ψ ˚ − i n , (208) \mathring{\Phi}_{+}^{\mathrm{out}}:=\mathring{\kappa}^{1/2}\mathring{\psi}_{+}^{\mathrm{out}},\qquad\mathring{\Phi}_{-}^{\mathrm{in}}:=\mathring{r}(-\mathring{\nu})^{1/2}\mathring{\psi}_{-}^{\mathrm{in}}, \tag*{(208)} Φ ˚ + out := κ ˚ 1/2 ψ ˚ + out , Φ ˚ − in := r ˚ ( − ν ˚ ) 1/2 ψ ˚ − in , (208) on C o u t \mathcal{C}_{\mathrm{out}} C out C i n \mathcal{C}_{\mathrm{in}} C in
A : = { ( r , Ω ) ∈ C 2 ( R ) × C 1 ( R ) : ∥ log r ∥ C 2 ( R ) + ∥ log Ω ∥ C 1 ( R ) ≤ 2 B } , \mathcal{A}:=\{(r,\Omega)\in C^{2}(\mathcal{R})\times C^{1}(\mathcal{R}):\lVert\log r\rVert_{C^{2}(\mathcal{R})}+\lVert\log\Omega\rVert_{C^{1}(\mathcal{R})}\leq2B\}, A := {( r , Ω ) ∈ C 2 ( R ) × C 1 ( R ) : ∥ log r ∥ C 2 ( R ) + ∥ log Ω ∥ C 1 ( R ) ≤ 2 B } , A ′ : = { ( r , Ω ) ∈ ( C 2 ( R ) ) 2 : ∥ log r ∥ C 2 ( R ) + ∥ log Ω ∥ C 2 ( R ) ≤ 2 B } ⊂ A . \mathcal{A}':=\{(r,\Omega)\in(C^{2}(\mathcal{R}))^{2}:\lVert\log r\rVert_{C^{2}(\mathcal{R})}+\lVert\log\Omega\rVert_{C^{2}(\mathcal{R})}\leq2B\}\subset\mathcal{A}. A ′ := {( r , Ω ) ∈ ( C 2 ( R ) ) 2 : ∥ log r ∥ C 2 ( R ) + ∥ log Ω ∥ C 2 ( R ) ≤ 2 B } ⊂ A . For ( r , Ω ) ∈ A (r,\Omega)\in\mathcal{A} ( r , Ω ) ∈ A Ψ : ( Φ + , Φ − ) → ( Φ ~ + , Φ ~ − ) \Psi:(\Phi_{+},\Phi_{-})\to(\widetilde{\Phi}_{+},\widetilde{\Phi}_{-}) Ψ : ( Φ + , Φ − ) → ( Φ + , Φ − )
B : = { ( Φ + , Φ − ) ∈ C 0 ( R ) 2 : ∥ Φ + ∥ C 0 ( R ) + ∥ Φ − ∥ C 0 ( R ) ≤ 2 B } , \mathcal{B}:=\{(\Phi_{+},\Phi_{-})\in C^{0}(\mathcal{R})^{2}:\lVert\Phi_{+}\rVert_{C^{0}(\mathcal{R})}+\lVert\Phi_{-}\rVert_{C^{0}(\mathcal{R})}\leq2B\}, B := {( Φ + , Φ − ) ∈ C 0 ( R ) 2 : ∥ Φ + ∥ C 0 ( R ) + ∥ Φ − ∥ C 0 ( R ) ≤ 2 B } , where
Φ ~ + ( U , V ) : = Φ ˚ + o u t ( V ) − ∫ U 0 U ( Ω r ( 1 + i m r ) Φ − ) ( U ′ , V ) d U ′ , \widetilde{\Phi}_{+}(U,V):=\mathring{\Phi}_{+}^{\mathrm{out}}(V)-\int_{U_{0}}^{U}\left(\frac{\Omega}{r}(1+imr)\Phi_{-}\right)(U',V)\,dU', Φ + ( U , V ) := Φ ˚ + out ( V ) − ∫ U 0 U ( r Ω ( 1 + im r ) Φ − ) ( U ′ , V ) d U ′ , Φ ~ − ( U , V ) : = Φ ˚ − i n ( U ) + ∫ V 0 V ( Ω r ( 1 − i m r ) Φ + ) ( U , V ′ ) d V ′ . (209) \widetilde{\Phi}_{-}(U,V):=\mathring{\Phi}_{-}^{\mathrm{in}}(U)+\int_{V_{0}}^{V}\left(\frac{\Omega}{r}(1-imr)\Phi_{+}\right)(U,V')\,dV'. \tag*{(209)} Φ − ( U , V ) := Φ ˚ − in ( U ) + ∫ V 0 V ( r Ω ( 1 − im r ) Φ + ) ( U , V ′ ) d V ′ . (209) We start by proving that Ψ \Psi Ψ Ψ : B → B \Psi:\mathcal{B}\to\mathcal{B} Ψ : B → B
∣ Φ ~ ± ∣ ≤ B + ε 0 C B ≤ 2 B , |\widetilde{\Phi}_{\pm}|\leq B+\varepsilon_{0}CB\leq2B, ∣ Φ ± ∣ ≤ B + ε 0 C B ≤ 2 B , for ε 0 \varepsilon_{0} ε 0 C C C B B B ε 0 \varepsilon_{0} ε 0 ( Φ + ( 1 ) , Φ − ( 1 ) ) , ( Φ + ( 2 ) , Φ − ( 2 ) ) ∈ B (\Phi_{+}^{(1)},\Phi_{-}^{(1)}),(\Phi_{+}^{(2)},\Phi_{-}^{(2)})\in\mathcal{B} ( Φ + ( 1 ) , Φ − ( 1 ) ) , ( Φ + ( 2 ) , Φ − ( 2 ) ) ∈ B ε 0 \varepsilon_{0} ε 0
∣ Φ ~ + ( 1 ) − Φ ~ + ( 2 ) ∣ ≤ ∫ U 0 U ∣ Ω r ∣ ∣ Φ − ( 1 ) − Φ − ( 2 ) ∣ d U ′ ≤ ε l o c C B ∥ Φ − ( 1 ) − Φ − ( 2 ) ∥ C 0 ( R ) < ∥ Φ + ( 1 ) − Φ + ( 2 ) ∥ C 0 ( R ) + ∥ Φ − ( 1 ) − Φ − ( 2 ) ∥ C 0 ( R ) , |\widetilde{\Phi}_{+}^{(1)}-\widetilde{\Phi}_{+}^{(2)}|\leq\int_{U_{0}}^{U}\left|\frac{\Omega}{r}\right||\Phi_{-}^{(1)}-\Phi_{-}^{(2)}|\,dU'\leq\varepsilon_{\mathrm{loc}}CB\lVert\Phi_{-}^{(1)}-\Phi_{-}^{(2)}\rVert_{C^{0}(\mathcal{R})}<\lVert\Phi_{+}^{(1)}-\Phi_{+}^{(2)}\rVert_{C^{0}(\mathcal{R})}+\lVert\Phi_{-}^{(1)}-\Phi_{-}^{(2)}\rVert_{C^{0}(\mathcal{R})}, ∣ Φ + ( 1 ) − Φ + ( 2 ) ∣ ≤ ∫ U 0 U r Ω ∣ Φ − ( 1 ) − Φ − ( 2 ) ∣ d U ′ ≤ ε loc C B ∥ Φ − ( 1 ) − Φ − ( 2 ) ∥ C 0 ( R ) < ∥ Φ + ( 1 ) − Φ + ( 2 ) ∥ C 0 ( R ) + ∥ Φ − ( 1 ) − Φ − ( 2 ) ∥ C 0 ( R ) , and analogously for ∣ Φ ~ − ( 1 ) − Φ ~ − ( 2 ) ∣ |\widetilde{\Phi}_{-}^{(1)}-\widetilde{\Phi}_{-}^{(2)}| ∣ Φ − ( 1 ) − Φ − ( 2 ) ∣
∥ Φ ~ + ( 1 ) − Φ ~ + ( 2 ) ∥ C 0 ( R ) + ∥ Φ ~ − ( 1 ) − Φ ~ − ( 2 ) ∥ C 0 ( R ) < ∥ Φ + ( 1 ) − Φ + ( 2 ) ∥ C 0 ( R ) + ∥ Φ − ( 1 ) − Φ − ( 2 ) ∥ C 0 ( R ) . \lVert\widetilde{\Phi}_{+}^{(1)}-\widetilde{\Phi}_{+}^{(2)}\rVert_{C^{0}(\mathcal{R})}+\lVert\widetilde{\Phi}_{-}^{(1)}-\widetilde{\Phi}_{-}^{(2)}\rVert_{C^{0}(\mathcal{R})}<\lVert\Phi_{+}^{(1)}-\Phi_{+}^{(2)}\rVert_{C^{0}(\mathcal{R})}+\lVert\Phi_{-}^{(1)}-\Phi_{-}^{(2)}\rVert_{C^{0}(\mathcal{R})}. ∥ Φ + ( 1 ) − Φ + ( 2 ) ∥ C 0 ( R ) + ∥ Φ − ( 1 ) − Φ − ( 2 ) ∥ C 0 ( R ) < ∥ Φ + ( 1 ) − Φ + ( 2 ) ∥ C 0 ( R ) + ∥ Φ − ( 1 ) − Φ − ( 2 ) ∥ C 0 ( R ) . Therefore, given ( r , Ω ) ∈ A (r,\Omega)\in\mathcal{A} ( r , Ω ) ∈ A ( Φ + , Φ − ) (\Phi_{+},\Phi_{-}) ( Φ + , Φ − )
∂ u Φ + = − Ω r ( 1 + i m r ) Φ − , ∂ v Φ − = Ω r ( 1 − i m r ) Φ + , (210) \partial_{u}\Phi_{+}=-\frac{\Omega}{r}(1+imr)\Phi_{-},\qquad\partial_{v}\Phi_{-}=\frac{\Omega}{r}(1-imr)\Phi_{+}, \tag*{(210)} ∂ u Φ + = − r Ω ( 1 + im r ) Φ − , ∂ v Φ − = r Ω ( 1 − im r ) Φ + , (210) on the metric defined by r r r Ω \Omega Ω ∣ ∂ u Φ + ∣ , ∣ ∂ v Φ − ∣ ≤ C B |\partial_{u}\Phi_{+}|,|\partial_{v}\Phi_{-}|\leq CB ∣ ∂ u Φ + ∣ , ∣ ∂ v Φ − ∣ ≤ C B
∂ v Φ + ( U , V ) = ∂ v Φ ˚ + o u t ( V ) + ∫ U 0 U ( ∂ u ∂ v Φ + ) ( U ′ , V ) d U ′ = ∂ v Φ ˚ + o u t ( V ) + ∫ U 0 U ( Ω r 2 ( − r ∂ v log Ω ( 1 + i m r ) + λ ) Φ − − Ω 2 r 2 ( 1 + m 2 r 2 ) Φ + ) d U ′ , \begin{aligned}
\partial_{v}\Phi_{+}(U,V)=\partial_{v}\mathring{\Phi}_{+}^{\mathrm{out}}(V)+\int_{U_{0}}^{U}(\partial_{u}\partial_{v}\Phi_{+})(U',V)\,dU' \\
&=\partial_{v}\mathring{\Phi}_{+}^{\mathrm{out}}(V)+\int_{U_{0}}^{U}\left(\frac{\Omega}{r^{2}}(-r\partial_{v}\log\Omega(1+imr)+\lambda)\Phi_{-}-\frac{\Omega^{2}}{r^{2}}(1+m^{2}r^{2})\Phi_{+}\right)\,dU',
\end{aligned} ∂ v Φ + ( U , V ) = ∂ v Φ ˚ + out ( V ) + ∫ U 0 U ( ∂ u ∂ v Φ + ) ( U ′ , V ) d U ′ = ∂ v Φ ˚ + out ( V ) + ∫ U 0 U ( r 2 Ω ( − r ∂ v log Ω ( 1 + im r ) + λ ) Φ − − r 2 Ω 2 ( 1 + m 2 r 2 ) Φ + ) d U ′ , ∂ u Φ − ( U , V ) = ∂ u Φ ˚ − i n ( U ) + ∫ V 0 V ( ∂ u ∂ v Φ − ) ( U , V ′ ) d V ′ = ∂ u Φ ˚ − i n ( U ) + ∫ V 0 V ( Ω r 2 ( r ∂ u log Ω ( 1 − i m r ) − ν ) Φ + − Ω 2 r 2 ( 1 + m 2 r 2 ) Φ − ) d V ′ , \begin{aligned}
\partial_{u}\Phi_{-}(U,V)=\partial_{u}\mathring{\Phi}_{-}^{\mathrm{in}}(U)+\int_{V_{0}}^{V}(\partial_{u}\partial_{v}\Phi_{-})(U,V')\,dV' \\
&=\partial_{u}\mathring{\Phi}_{-}^{\mathrm{in}}(U)+\int_{V_{0}}^{V}\left(\frac{\Omega}{r^{2}}(r\partial_{u}\log\Omega(1-imr)-\nu)\Phi_{+}-\frac{\Omega^{2}}{r^{2}}(1+m^{2}r^{2})\Phi_{-}\right)\,dV',
\end{aligned} ∂ u Φ − ( U , V ) = ∂ u Φ ˚ − in ( U ) + ∫ V 0 V ( ∂ u ∂ v Φ − ) ( U , V ′ ) d V ′ = ∂ u Φ ˚ − in ( U ) + ∫ V 0 V ( r 2 Ω ( r ∂ u log Ω ( 1 − im r ) − ν ) Φ + − r 2 Ω 2 ( 1 + m 2 r 2 ) Φ − ) d V ′ , we similarly deduce the bounds
∣ ∂ v Φ + ∣ , ∣ ∂ u Φ − ∣ ≤ 2 B , |\partial_v\Phi_+|,|\partial_u\Phi_-|\le2B, ∣ ∂ v Φ + ∣ , ∣ ∂ u Φ − ∣ ≤ 2 B , for ε 0 \varepsilon_0 ε 0 ( Φ + , Φ − ) (\Phi_+,\Phi_-) ( Φ + , Φ − ) T u u , T u v , T v v , T ϑ ϑ , T φ φ T_{uu},T_{uv},T_{vv},T_{\vartheta\vartheta},T_{\varphi\varphi} T uu , T uv , T v v , T ϑϑ , T φφ
T u u : = 1 2 π r 2 ℑ [ Φ ‾ − ∂ u Φ − ] , T v v : = 1 2 π r 2 ℑ [ Φ ‾ + ∂ v Φ + ] , T u v : = Ω 2 2 π r 3 ( ℑ [ Φ ‾ − Φ + ] − m r ℜ [ Φ ‾ − Φ + ] ) , T_{uu}:=\frac{1}{2\pi r^2}\Im[\overline{\Phi}_-\partial_u\Phi_-],\qquad T_{vv}:=\frac{1}{2\pi r^2}\Im[\overline{\Phi}_+\partial_v\Phi_+],\qquad T_{uv}:=\frac{\Omega^2}{2\pi r^3}\left(\Im[\overline{\Phi}_-\Phi_+]-mr\Re[\overline{\Phi}_-\Phi_+]\right), T uu := 2 π r 2 1 ℑ [ Φ − ∂ u Φ − ] , T v v := 2 π r 2 1 ℑ [ Φ + ∂ v Φ + ] , T uv := 2 π r 3 Ω 2 ( ℑ [ Φ − Φ + ] − m r ℜ [ Φ − Φ + ] ) , T ϑ ϑ : = 1 4 π r Ω 2 ℑ [ Φ ‾ − Φ + ] , T φ φ : = sin 2 ϑ T ϑ ϑ , T_{\vartheta\vartheta}:=\frac{1}{4\pi r\Omega^2}\Im[\overline{\Phi}_-\Phi_+],\qquad T_{\varphi\varphi}:=\sin^2\vartheta\,T_{\vartheta\vartheta}, T ϑϑ := 4 π r Ω 2 1 ℑ [ Φ − Φ + ] , T φφ := sin 2 ϑ T ϑϑ , which can be bounded in view of the previous estimates. Furthermore, thanks to (B.3), one can infer proper bounds on ∂ u T u v , ∂ v T u v , ∂ u T ϑ ϑ , ∂ v T ϑ ϑ , ∂ u T φ φ , ∂ v T φ φ \partial_uT_{uv},\partial_vT_{uv},\partial_uT_{\vartheta\vartheta},\partial_vT_{\vartheta\vartheta},\partial_uT_{\varphi\varphi},\partial_vT_{\varphi\varphi} ∂ u T uv , ∂ v T uv , ∂ u T ϑϑ , ∂ v T ϑϑ , ∂ u T φφ , ∂ v T φφ
Next, we define Φ : A ′ → A \Phi:A'\to A Φ : A ′ → A ( r , Ω ) ↦ ( r ~ , Ω ~ ) (r,\Omega)\mapsto(\widetilde r,\widetilde\Omega) ( r , Ω ) ↦ ( r , Ω )
r ~ ( U , V ) : = r ˚ ( U 0 , V ) + r ˚ ( U , V 0 ) − r ˚ ( U 0 , V 0 ) + ∫ U 0 U ∫ V 0 V ( − Ω 2 r − λ ν r + 4 π r T u v ) ( U ′ , V ′ ) d U ′ d V ′ , \widetilde r(U,V):=\mathring r(U_0,V)+\mathring r(U,V_0)-\mathring r(U_0,V_0)+\int_{U_0}^{U}\int_{V_0}^{V}\left(-\frac{\Omega^2}{r}-\frac{\lambda\nu}{r}+4\pi rT_{uv}\right)(U',V')\,dU'\,dV', r ( U , V ) := r ˚ ( U 0 , V ) + r ˚ ( U , V 0 ) − r ˚ ( U 0 , V 0 ) + ∫ U 0 U ∫ V 0 V ( − r Ω 2 − r λ ν + 4 π r T uv ) ( U ′ , V ′ ) d U ′ d V ′ , log Ω ~ ( U , V ) : = log Ω ˚ ( U 0 , V ) + log Ω ˚ ( U , V 0 ) − log Ω ˚ ( U 0 , V 0 ) + ∫ U 0 U ∫ V 0 V ( Ω 2 r ~ 2 + λ ~ ν ~ r ~ 2 − 4 π T u v − 4 π Ω 2 r ~ 2 g S 2 A B T A B ) ( U ′ , V ′ ) d U ′ d V ′ , \log\widetilde\Omega(U,V):=\log\mathring\Omega(U_0,V)+\log\mathring\Omega(U,V_0)-\log\mathring\Omega(U_0,V_0)
+\int_{U_0}^{U}\int_{V_0}^{V}\left(\frac{\Omega^2}{\widetilde r^2}+\frac{\widetilde\lambda\widetilde\nu}{\widetilde r^2}-4\pi T_{uv}-\frac{4\pi\Omega^2}{\widetilde r^2}g_{\mathbb S^2}^{AB}T_{AB}\right)(U',V')\,dU'\,dV', log Ω ( U , V ) := log Ω ˚ ( U 0 , V ) + log Ω ˚ ( U , V 0 ) − log Ω ˚ ( U 0 , V 0 ) + ∫ U 0 U ∫ V 0 V ( r 2 Ω 2 + r 2 λ ν − 4 π T uv − r 2 4 π Ω 2 g S 2 A B T A B ) ( U ′ , V ′ ) d U ′ d V ′ , where g S 2 g_{\mathbb S^2} g S 2 ε 0 \varepsilon_0 ε 0 r ~ , Ω ~ \widetilde r,\widetilde\Omega r , Ω Φ \Phi Φ A ′ A' A ′ A A A [23 ] for more details). Given its fixed point ( r , Ω ) (r,\Omega) ( r , Ω ) ( Φ + , Φ − ) (\Phi_+,\Phi_-) ( Φ + , Φ − )
∂ v ( r Ω 2 ∂ u ( Ω − 2 ∂ u r ) + 4 π r 2 T u u ) = 0 , \partial_v\left(r\Omega^2\partial_u(\Omega^{-2}\partial_ur)+4\pi r^2T_{uu}\right)=0, ∂ v ( r Ω 2 ∂ u ( Ω − 2 ∂ u r ) + 4 π r 2 T uu ) = 0 , ∂ u ( r Ω 2 ∂ v ( Ω − 2 ∂ v r ) + 4 π r 2 T v v ) = 0 , \partial_u\left(r\Omega^2\partial_v(\Omega^{-2}\partial_vr)+4\pi r^2T_{vv}\right)=0, ∂ u ( r Ω 2 ∂ v ( Ω − 2 ∂ v r ) + 4 π r 2 T v v ) = 0 , hence propagating the constraint equations. Therefore, writing
∂ u ( Ω − 2 ν ) = − 4 π r Ω 2 T u u = − 2 r Ω 2 ℑ [ Φ ‾ − ∂ u Φ − ] , \partial_u(\Omega^{-2}\nu)=-\frac{4\pi r}{\Omega^2}T_{uu}=-\frac{2}{r\Omega^2}\Im[\overline{\Phi}_-\partial_u\Phi_-], ∂ u ( Ω − 2 ν ) = − Ω 2 4 π r T uu = − r Ω 2 2 ℑ [ Φ − ∂ u Φ − ] , we have on R \mathcal R R
Ω − 2 ν ( U , V ) = Ω ˚ − 2 ν ˚ ( U 0 , V ) − ∫ U 0 U ( 2 r Ω 2 ℑ [ Φ ‾ − ∂ u Φ − ] ) ( U ′ , V ) d U ′ ≤ Ω ˚ − 2 ν ˚ ( U 0 , V ) + ε 0 C B < 0 , \Omega^{-2}\nu(U,V)=\mathring\Omega^{-2}\mathring\nu(U_0,V)-\int_{U_0}^{U}\left(\frac{2}{r\Omega^2}\Im[\overline{\Phi}_-\partial_u\Phi_-]\right)(U',V)\,dU'\le\mathring\Omega^{-2}\mathring\nu(U_0,V)+\varepsilon_0CB<0, Ω − 2 ν ( U , V ) = Ω ˚ − 2 ν ˚ ( U 0 , V ) − ∫ U 0 U ( r Ω 2 2 ℑ [ Φ − ∂ u Φ − ] ) ( U ′ , V ) d U ′ ≤ Ω ˚ − 2 ν ˚ ( U 0 , V ) + ε 0 C B < 0 , for ε 0 \varepsilon_0 ε 0 ψ + : = κ − 1 / 2 Φ + \psi_+:=\kappa^{-1/2}\Phi_+ ψ + := κ − 1/2 Φ + ψ − : = r − 1 ( − ν ) − 1 / 2 Φ − \psi_-:=r^{-1}(-\nu)^{-1/2}\Phi_- ψ − := r − 1 ( − ν ) − 1/2 Φ − ∥ ψ ± ∥ C 1 ( R ) \|\psi_\pm\|_{C^1(\mathcal R)} ∥ ψ ± ∥ C 1 ( R ) ( r , Ω , Φ + , Φ − ) (r,\Omega,\Phi_+,\Phi_-) ( r , Ω , Φ + , Φ − ) ( r , Ω , ψ + , ψ − ) (r,\Omega,\psi_+,\psi_-) ( r , Ω , ψ + , ψ − )
The second ingredient we need is the following local spacetime estimate, in the spirit of [43 , 50 ] . We again stress that the additional assumption ν < 0 \nu<0 ν < 0 Φ ± \Phi_\pm Φ ±
Lemma B.3. For any m ≥ 0 m\ge0 m ≥ 0 B > 0 B>0 B > 0 C ∗ < + ∞ C_*<+\infty C ∗ < + ∞ Let ( r , Ω , ψ + , ψ − ) (r,\Omega,\psi_+,\psi_-) ( r , Ω , ψ + , ψ − ) R ′ : = R ( U 0 , U 1 , V 0 , V 1 ) ∖ { ( U , V ) } \mathcal R':=\mathcal R(U_0,U_1,V_0,V_1)\setminus\{(U,V)\} R ′ := R ( U 0 , U 1 , V 0 , V 1 ) ∖ {( U , V )} U 1 − U 0 ≤ C 0 U_1-U_0\le C_0 U 1 − U 0 ≤ C 0 V 1 − V 0 ≤ C 0 V_1-V_0\le C_0 V 1 − V 0 ≤ C 0
∥ log r ∥ C 2 ( C ) + ∥ log Ω ∥ C 2 ( C ) + ∥ ψ + ∥ C 1 ( C ) + ∥ ψ − ∥ C 1 ( C ) ≤ C 0 , (211) \|\log r\|_{C^2(\mathcal C)}+\|\log\Omega\|_{C^2(\mathcal C)}+\|\psi_+\|_{C^1(\mathcal C)}+\|\psi_-\|_{C^1(\mathcal C)}\le C_0, \tag*{(211)} ∥ log r ∥ C 2 ( C ) + ∥ log Ω ∥ C 2 ( C ) + ∥ ψ + ∥ C 1 ( C ) + ∥ ψ − ∥ C 1 ( C ) ≤ C 0 , (211) and the global estimates
∬ R ′ Ω 2 d u d v ≤ C 0 , sup R ′ ∣ log r ∣ ≤ C 0 , ν < 0 on R ′ . (212) \iint_{\mathcal R'}\Omega^2\,du\,dv\le C_0,\qquad\sup_{\mathcal R'}|\log r|\le C_0,\qquad\nu<0\text{ on }\mathcal R'. \tag*{(212)} ∬ R ′ Ω 2 d u d v ≤ C 0 , R ′ sup ∣ log r ∣ ≤ C 0 , ν < 0 on R ′ . (212) Then the following estimate holds
∥ log r ∥ C 2 ( R ′ ) + ∥ log Ω ∥ C 2 ( R ′ ) + ∥ ψ + ∥ C 1 ( R ′ ) + ∥ ψ − ∥ C 1 ( R ′ ) ≤ C ∗ . \|\log r\|_{C^2(\mathcal R')}+\|\log\Omega\|_{C^2(\mathcal R')}+\|\psi_+\|_{C^1(\mathcal R')}+\|\psi_-\|_{C^1(\mathcal R')}\le C_*. ∥ log r ∥ C 2 ( R ′ ) + ∥ log Ω ∥ C 2 ( R ′ ) + ∥ ψ + ∥ C 1 ( R ′ ) + ∥ ψ − ∥ C 1 ( R ′ ) ≤ C ∗ . Proof. In the following, we will use the notation A ≲ 1 A \lesssim1 A ≲ 1 A ≤ C A \le C A ≤ C C C C m m m C 0 C_0 C 0 R \mathcal{R} R R ′ \mathcal{R}' R ′ R ′ \mathcal{R}' R ′
As ν < 0 \nu< 0 ν < 0 R ′ \mathcal{R}' R ′ Φ ± \Phi_{\pm} Φ ±
∥ Φ + ∥ C 1 ( C ) + ∥ Φ − ∥ C 1 ( C ) ≤ C 0 , (213) \lVert\Phi_{+} \rVert_{C^{1}(\mathcal{C})}+\lVert\Phi_{-} \rVert_{C^{1}(\mathcal{C})}\le C_{0}, \tag*{(213)} ∥ Φ + ∥ C 1 ( C ) + ∥ Φ − ∥ C 1 ( C ) ≤ C 0 , (213) by (B.4). Rewriting the first equation in (A.9) as
4 π r 2 T u v = 1 2 ∂ u ∂ v ( r 2 ) + Ω 2 , 4\pi r^{2}T_{uv}=\frac{1}{2}\partial_{u}\partial_{v}\left(r^{2}\right)+\Omega^{2}, 4 π r 2 T uv = 2 1 ∂ u ∂ v ( r 2 ) + Ω 2 , we may integrate it over R ′ \mathcal{R}' R ′
∬ R ∣ T u v ∣ d U d V ≲ ∬ R r 2 ∣ T u v ∣ d U d V ≲ ∬ R ∣ ∂ u ∂ v ( r 2 ) ∣ d U d V + ∬ R Ω 2 d U d V ≲ 1 , (214) \iint_{\mathcal{R}} \lvert T_{uv}\rvert\,dU\,dV\lesssim\iint_{\mathcal{R}}r^{2}\lvert T_{uv}\rvert\,dU\,dV\lesssim\iint_{\mathcal{R}}\left\lvert\partial_{u}\partial_{v}\left(r^{2}\right)\right\rvert\,dU\,dV+\iint_{\mathcal{R}}\Omega^{2}\,dU\,dV\lesssim1, \tag*{(214)} ∬ R ∣ T uv ∣ d U d V ≲ ∬ R r 2 ∣ T uv ∣ d U d V ≲ ∬ R ∂ u ∂ v ( r 2 ) d U d V + ∬ R Ω 2 d U d V ≲ 1 , (214) by (B.4)–(B.5). Moreover, rewriting again the same equation as
∂ u ( r ∂ v r ) = − Ω 2 + 4 π r 2 T u v , \partial_{u}\left(r\partial_{v}r\right)=-\Omega^{2}+4\pi r^{2}T_{uv}, ∂ u ( r ∂ v r ) = − Ω 2 + 4 π r 2 T uv , and integrating in u u u
sup [ U 0 , U 1 ] × { V } ∣ r ∂ v r ∣ ≲ 1 + ∫ U 0 U 1 ( Ω 2 + r 2 ∣ T u v ∣ ) ( U , V ) d U , \sup_{[U_{0},U_{1}]\times\{V\}}\lvert r\partial_{v}r\rvert\lesssim1+\int_{U_{0}}^{U_{1}}\left(\Omega^{2}+r^{2}\lvert T_{uv}\rvert\right)(U,V)\,dU, [ U 0 , U 1 ] × { V } sup ∣ r ∂ v r ∣ ≲ 1 + ∫ U 0 U 1 ( Ω 2 + r 2 ∣ T uv ∣ ) ( U , V ) d U , for all V ∈ [ V 0 , V 1 ] V\in[V_{0},V_{1}] V ∈ [ V 0 , V 1 ]
∫ V 0 V 1 sup [ U 0 , U 1 ] × { V } ∣ λ ∣ d V ≲ ∫ V 0 V 1 sup [ U 0 , U 1 ] × { V } ∣ r λ ∣ d V ≲ 1. \int_{V_{0}}^{V_{1}}\sup_{[U_{0},U_{1}]\times\{V\}}\lvert\lambda\rvert\,dV\lesssim\int_{V_{0}}^{V_{1}}\sup_{[U_{0},U_{1}]\times\{V\}}\lvert r\lambda\rvert\,dV\lesssim1. ∫ V 0 V 1 [ U 0 , U 1 ] × { V } sup ∣ λ ∣ d V ≲ ∫ V 0 V 1 [ U 0 , U 1 ] × { V } sup ∣ r λ ∣ d V ≲ 1. As ∂ u ( r ∂ v r ) = ∂ v ( r ∂ u r ) \partial_{u}(r\partial_{v}r)=\partial_{v}(r\partial_{u}r) ∂ u ( r ∂ v r ) = ∂ v ( r ∂ u r ) u u u v v v
∫ U 0 U 1 sup { U } × [ V 0 , V 1 ] ∣ ν ∣ d U ≲ 1. \int_{U_{0}}^{U_{1}}\sup_{\{U\}\times[V_{0},V_{1]}}\lvert\nu\rvert\,dU\lesssim1. ∫ U 0 U 1 { U } × [ V 0 , V 1 ] sup ∣ ν ∣ d U ≲ 1. Therefore, estimating
∬ R ∣ λ ν ∣ d U d V ≤ ( ∫ V 0 V 1 sup [ U 0 , U 1 ] × { V } ∣ λ ∣ d V ) ( ∫ U 0 U 1 sup { U } × [ V 0 , V 1 ] ∣ ν ∣ d U ) , \iint_{\mathcal{R}}\lvert\lambda\nu\rvert\,dU\,dV\le\left(\int_{V_{0}}^{V_{1}}\sup_{[U_{0},U_{1}]\times\{V\}}\lvert\lambda\rvert\,dV\right)\left(\int_{U_{0}}^{U_{1}}\sup_{\{U\}\times[V_{0},V_{1]}}\lvert\nu\rvert\,dU\right), ∬ R ∣ λ ν ∣ d U d V ≤ ( ∫ V 0 V 1 [ U 0 , U 1 ] × { V } sup ∣ λ ∣ d V ) ( ∫ U 0 U 1 { U } × [ V 0 , V 1 ] sup ∣ ν ∣ d U ) , we infer a bound on the Hawking mass
∬ R Ω 2 ∣ m ∣ d u d v ≲ 1. \iint_{\mathcal{R}}\Omega^{2}\lvert m\rvert\,du\,dv\lesssim1. ∬ R Ω 2 ∣ m ∣ d u d v ≲ 1. Moreover, (B.7) implies
∬ R ∣ Φ − ∣ ∣ Φ + ∣ d U d V ≲ ∬ R Ω r 2 ∣ Φ − ∣ ∣ Φ + ∣ d U d V ≲ ∬ R ∣ T u v ∣ d U d V ≲ 1 , (215) \iint_{\mathcal{R}}\lvert\Phi_{-}\rvert\lvert\Phi_{+}\rvert\,dU\,dV\lesssim\iint_{\mathcal{R}}\frac{\Omega}{r^{2}}\lvert\Phi_{-}\rvert\lvert\Phi_{+}\rvert\,dU\,dV\lesssim\iint_{\mathcal{R}}\lvert T_{uv}\rvert\,dU\,dV\lesssim1, \tag*{(215)} ∬ R ∣ Φ − ∣ ∣ Φ + ∣ d U d V ≲ ∬ R r 2 Ω ∣ Φ − ∣ ∣ Φ + ∣ d U d V ≲ ∬ R ∣ T uv ∣ d U d V ≲ 1 , (215) and therefore, from the definition of T ϑ ϑ T_{\vartheta\vartheta} T ϑϑ T φ φ T_{\varphi\varphi} T φφ
∬ R Ω 2 g A B ∣ T A B ∣ ≲ 1. \iint_{\mathcal{R}}\Omega^{2}g^{AB}\lvert T_{AB}\rvert\lesssim1. ∬ R Ω 2 g A B ∣ T A B ∣ ≲ 1. Thus, from the wave equation in log Ω \log\Omega log Ω
sup R ′ ∣ log Ω ∣ ≲ 1 + ∣ ∬ R ∂ u ∂ v log Ω d U d V ∣ ≲ 1 + ∬ R ( Ω 2 ∣ m ∣ + ∣ T u v ∣ + Ω 2 g A B ∣ T A B ∣ ) d U d V ≲ 1. \sup_{\mathcal{R}'}\lvert\log\Omega\rvert\lesssim1+\left\lvert\iint_{\mathcal{R}}\partial_{u}\partial_{v}\log\Omega\,dU\,dV\right\rvert\lesssim1+\iint_{\mathcal{R}}\left(\Omega^{2}\lvert m\rvert+\lvert T_{uv}\rvert+\Omega^{2}g^{AB}\lvert T_{AB}\rvert\right)\,dU\,dV\lesssim1. R ′ sup ∣ log Ω ∣ ≲ 1 + ∬ R ∂ u ∂ v log Ω d U d V ≲ 1 + ∬ R ( Ω 2 ∣ m ∣ + ∣ T uv ∣ + Ω 2 g A B ∣ T A B ∣ ) d U d V ≲ 1. Now, using (A.8), we compute
∂ u ∣ Φ + ∣ 2 = 2 R [ Φ ‾ + ∂ u Φ + ] = − 2 Ω r R [ ( 1 + i m r ) Φ ‾ + Φ − ] . \partial_u\lvert\Phi_+\rvert^2=2\mathfrak{R}\left[\overline{\Phi}_+\partial_u\Phi_+\right]=-\frac{2\Omega}{r}\mathfrak{R}\left[(1+imr)\overline{\Phi}_+\Phi_-\right]. ∂ u ∣ Φ + ∣ 2 = 2 R [ Φ + ∂ u Φ + ] = − r 2Ω R [ ( 1 + im r ) Φ + Φ − ] . integrate again in u u u
∫ V 0 V 1 sup [ U 0 , U 1 ] × { V } ∣ Φ + ∣ 2 d V ≲ 1 , \int_{V_0}^{V_1}\sup_{[U_0,U_1]\times\{V\}}\lvert\Phi_+\rvert^2\,dV\lesssim1, ∫ V 0 V 1 [ U 0 , U 1 ] × { V } sup ∣ Φ + ∣ 2 d V ≲ 1 , and similarly,
∫ U 0 U 1 sup { U } × [ V 0 , V 1 ] ∣ Φ − ∣ 2 d U ≲ 1. \int_{U_0}^{U_1}\sup_{\{U\}\times[V_0,V_1]}\lvert\Phi_-\rvert^2\,dU\lesssim1. ∫ U 0 U 1 { U } × [ V 0 , V 1 ] sup ∣ Φ − ∣ 2 d U ≲ 1. In the same manner, using the wave equation in log Ω \log\Omega log Ω u u u
sup [ U 0 , U 1 ] × { V } ∣ ∂ v log Ω ∣ ≲ 1 + ∣ ∫ U 0 U 1 ∂ u ∂ v log Ω d U d V ∣ , \sup_{[U_0,U_1]\times\{V\}}\lvert\partial_v\log\Omega\rvert\lesssim1+\left\lvert\int_{U_0}^{U_1}\partial_u\partial_v\log\Omega\,dU\,dV\right\rvert, [ U 0 , U 1 ] × { V } sup ∣ ∂ v log Ω ∣ ≲ 1 + ∫ U 0 U 1 ∂ u ∂ v log Ω d U d V , yielding
∫ V 0 V 1 sup [ U 0 , U 1 ] × { V } ∣ ∂ v log Ω ∣ d V ≲ 1 , \int_{V_0}^{V_1}\sup_{[U_0,U_1]\times\{V\}}\lvert\partial_v\log\Omega\rvert\,dV\lesssim1, ∫ V 0 V 1 [ U 0 , U 1 ] × { V } sup ∣ ∂ v log Ω ∣ d V ≲ 1 , and in the same fashion
∫ U 0 U 1 sup { U } × [ V 0 , V 1 ] ∣ ∂ u log Ω ∣ d U ≲ 1. \int_{U_0}^{U_1}\sup_{\{U\}\times[V_0,V_1]}\lvert\partial_u\log\Omega\rvert\,dU\lesssim1. ∫ U 0 U 1 { U } × [ V 0 , V 1 ] sup ∣ ∂ u log Ω ∣ d U ≲ 1. Therefore, from the initial bounds and (A.8),
sup R ′ ∣ Φ + ∣ ≲ 1 + ∣ ∬ R ∂ u ∂ v Φ + d U d V ∣ = 1 + ∣ ∬ R Ω r 2 ( − r ∂ v log Ω ( 1 + i m r ) + λ ) Φ − − Ω 2 r 2 ( 1 + m 2 r 2 ) Φ + d U d V ∣ ≲ 1 + ( ∫ V 0 V 1 sup [ U 0 , U 1 ] × { V } ( ∣ ∂ v log Ω ∣ + ∣ λ ∣ ) d V ) ( ∫ U 0 U 1 sup { U } × [ V 0 , V 1 ] ∣ Φ − ∣ d U ) + ( ∫ V 0 V 1 sup [ U 0 , U 1 ] × { V } ∣ Φ + ∣ 2 d V ) ( ∫ U 0 U 1 sup { U } × [ V 0 , V 1 ] Ω 2 r 2 d U ) ≲ 1 , \begin{aligned}
\sup_{\mathcal{R}'}\lvert\Phi_+\rvert&\lesssim1+\left\lvert\iint_{\mathcal{R}}\partial_u\partial_v\Phi_+\,dU\,dV\right\rvert\\
&=1+\left\lvert\iint_{\mathcal{R}}\frac{\Omega}{r^2}\left(-r\partial_v\log\Omega(1+imr)+\lambda\right)\Phi_--\frac{\Omega^2}{r^2}(1+m^2r^2)\Phi_+\,dU\,dV\right\rvert\\
&\lesssim1+\left(\int_{V_0}^{V_1}\sup_{[U_0,U_1]\times\{V\}}(\lvert\partial_v\log\Omega\rvert+\lvert\lambda\rvert)\,dV\right)\left(\int_{U_0}^{U_1}\sup_{\{U\}\times[V_0,V_1]}\lvert\Phi_-\rvert\,dU\right)\\
&\quad+\left(\int_{V_0}^{V_1}\sup_{[U_0,U_1]\times\{V\}}\lvert\Phi_+\rvert^2\,dV\right)\left(\int_{U_0}^{U_1}\sup_{\{U\}\times[V_0,V_1]}\frac{\Omega^2}{r^2}\,dU\right)\lesssim1,
\end{aligned} R ′ sup ∣ Φ + ∣ ≲ 1 + ∬ R ∂ u ∂ v Φ + d U d V = 1 + ∬ R r 2 Ω ( − r ∂ v log Ω ( 1 + im r ) + λ ) Φ − − r 2 Ω 2 ( 1 + m 2 r 2 ) Φ + d U d V ≲ 1 + ( ∫ V 0 V 1 [ U 0 , U 1 ] × { V } sup (∣ ∂ v log Ω ∣ + ∣ λ ∣) d V ) ( ∫ U 0 U 1 { U } × [ V 0 , V 1 ] sup ∣ Φ − ∣ d U ) + ( ∫ V 0 V 1 [ U 0 , U 1 ] × { V } sup ∣ Φ + ∣ 2 d V ) ( ∫ U 0 U 1 { U } × [ V 0 , V 1 ] sup r 2 Ω 2 d U ) ≲ 1 , sup R ′ ∣ Φ − ∣ ≲ 1 , \sup_{\mathcal{R}'}\lvert\Phi_-\rvert\lesssim1, R ′ sup ∣ Φ − ∣ ≲ 1 , where the second estimate follows by analogous computations. These bounds imply a uniform pointwise control also on
sup R ′ ( ∣ T u v ∣ + ∣ T v v ∣ + ∣ T φ φ ∣ ) ≲ 1. \sup_{\mathcal{R}'}\left(\lvert T_{uv}\rvert+\lvert T_{vv}\rvert+\lvert T_{\varphi\varphi}\rvert\right)\lesssim1. R ′ sup ( ∣ T uv ∣ + ∣ T v v ∣ + ∣ T φφ ∣ ) ≲ 1. Therefore, by integrating again ∂ u ( r λ ) = ∂ v ( r v ) \partial_u(r\lambda)=\partial_v(rv) ∂ u ( r λ ) = ∂ v ( r v )
sup R ′ ( ∣ ν ∣ + ∣ λ ∣ ) ≲ 1 , \sup_{\mathcal{R}'}(\lvert\nu\rvert+\lvert\lambda\rvert)\lesssim1, R ′ sup (∣ ν ∣ + ∣ λ ∣) ≲ 1 , yielding a pointwise estimate on the Hawking mass m m m
sup R ′ ( ∣ ∂ u log Ω ∣ + ∣ ∂ v log Ω ∣ + ∣ ∂ u ∂ v log Ω ∣ ) ≲ 1. \sup_{\mathcal{R}'}\left(\lvert\partial_u\log\Omega\rvert+\lvert\partial_v\log\Omega\rvert+\lvert\partial_u\partial_v\log\Omega\rvert\right)\lesssim1. R ′ sup ( ∣ ∂ u log Ω ∣ + ∣ ∂ v log Ω ∣ + ∣ ∂ u ∂ v log Ω ∣ ) ≲ 1. Hence, it follows
sup R ′ ∣ ∂ v Φ + ∣ ≲ 1 + ∣ ∫ U 0 U 1 ∂ u ∂ v Φ + d U ∣ ≲ 1 + ∣ ∫ U 0 U 1 Ω r 2 ( − r ∂ v log Ω ( 1 + i m r ) + λ ) Φ − − Ω 2 r 2 ( 1 + m 2 r 2 ) Φ + d U ∣ ≲ 1 , \begin{aligned}
\sup_{\mathcal{R}'}\lvert\partial_v\Phi_+\rvert&\lesssim1+\left\lvert\int_{U_0}^{U_1}\partial_u\partial_v\Phi_+\,dU\right\rvert\\
&\lesssim1+\left\lvert\int_{U_0}^{U_1}\frac{\Omega}{r^2}\left(-r\partial_v\log\Omega(1+imr)+\lambda\right)\Phi_--\frac{\Omega^2}{r^2}(1+m^2r^2)\Phi_+\,dU\right\rvert\lesssim1,
\end{aligned} R ′ sup ∣ ∂ v Φ + ∣ ≲ 1 + ∫ U 0 U 1 ∂ u ∂ v Φ + d U ≲ 1 + ∫ U 0 U 1 r 2 Ω ( − r ∂ v log Ω ( 1 + im r ) + λ ) Φ − − r 2 Ω 2 ( 1 + m 2 r 2 ) Φ + d U ≲ 1 , and similarly
sup R ′ ∣ ∂ u Φ − ∣ ≲ 1 , \sup_{\mathcal{R}'} |\partial_u\Phi_-| \lesssim1, R ′ sup ∣ ∂ u Φ − ∣ ≲ 1 , implying
sup R ′ ( ∣ T u u ∣ + ∣ T v v ∣ + ∣ ∂ v T u v ∣ + ∣ ∂ u T u v ∣ + ∣ ∂ v T ϑ ϑ ∣ + ∣ ∂ u T ϑ ϑ ∣ + ∣ ∂ v T φ φ ∣ + ∣ ∂ u T φ φ ∣ ) ≲ 1. \sup_{\mathcal{R}'}\left(|T_{uu}|+|T_{vv}|+|\partial_vT_{uv}|+|\partial_uT_{uv}|+|\partial_vT_{\vartheta\vartheta}|+|\partial_uT_{\vartheta\vartheta}|+|\partial_vT_{\varphi\varphi}|+|\partial_uT_{\varphi\varphi}|\right)\lesssim1. R ′ sup ( ∣ T uu ∣ + ∣ T v v ∣ + ∣ ∂ v T uv ∣ + ∣ ∂ u T uv ∣ + ∣ ∂ v T ϑϑ ∣ + ∣ ∂ u T ϑϑ ∣ + ∣ ∂ v T φφ ∣ + ∣ ∂ u T φφ ∣ ) ≲ 1. Thus, by (A.9)
sup R ′ ( ∣ ∂ u 2 r ∣ + ∣ ∂ v 2 r ∣ + ∣ ∂ u ∂ v r ∣ ) ≲ 1 , \sup_{\mathcal{R}'}\left(|\partial_u^2r|+|\partial_v^2r|+|\partial_u\partial_vr|\right)\lesssim1, R ′ sup ( ∣ ∂ u 2 r ∣ + ∣ ∂ v 2 r ∣ + ∣ ∂ u ∂ v r ∣ ) ≲ 1 , and by commuting the wave equation in log Ω \log\Omega log Ω
sup R ′ ( ∣ ∂ u 2 log Ω ∣ + ∣ ∂ v 2 log Ω ∣ ) ≲ 1. \sup_{\mathcal{R}'}\left(|\partial_u^2\log\Omega|+|\partial_v^2\log\Omega|\right)\lesssim1. R ′ sup ( ∣ ∂ u 2 log Ω∣ + ∣ ∂ v 2 log Ω∣ ) ≲ 1. Recalling ψ + = κ − 1 / 2 Φ + , ψ − = r − 1 ( − ν ) − 1 / 2 Φ − \psi_+=\kappa^{-1/2}\Phi_+,\psi_-=r^{-1}(-\nu)^{-1/2}\Phi_- ψ + = κ − 1/2 Φ + , ψ − = r − 1 ( − ν ) − 1/2 Φ −
sup R ′ ( ∣ ψ + ∣ + ∣ ∂ v ψ + ∣ + ∣ ∂ u ψ + ∣ + ∣ ψ − ∣ + ∣ ∂ v ψ − ∣ + ∣ ∂ u ψ − ∣ ) ≲ 1 , \sup_{\mathcal{R}'}\left(|\psi_+|+|\partial_v\psi_+|+|\partial_u\psi_+|+|\psi_-|+|\partial_v\psi_-|+|\partial_u\psi_-|\right)\lesssim1, R ′ sup ( ∣ ψ + ∣ + ∣ ∂ v ψ + ∣ + ∣ ∂ u ψ + ∣ + ∣ ψ − ∣ + ∣ ∂ v ψ − ∣ + ∣ ∂ u ψ − ∣ ) ≲ 1 , concluding the proof.
Proof of Proposition B.1. The proof follows the argument of [50 Proposition 3.17] , using Lemmas B.3 and B.2 together with a bootstrap argument.
