Introduction
The quasinormal-mode (QNM) ringdown of a perturbed black hole encodes the parameters of the remnant and, through the no-hair theorems, provides a stringent test of general relativity [1 , 2 , 3 ] . With the detection of gravitational waves from binary mergers [4 ] , the ringdown phase has become an observational probe of strong-field gravity, and the program of black-hole spectroscopy aims to test the Kerr hypothesis by measuring multiple QNM frequencies [5 , 6 , 7 , 8 ] . Any consistent modification of general relativity generically alters the QNM spectrum, so precise predictions of these deviations are essential to interpreting present and future ringdown observations.
Among the best-motivated extensions is dynamical Chern–Simons (dCS) gravity [9 , 10 , 11 ] , a theory with parity-violating phenomenology in which a dynamical pseudoscalar couples to the gravitational Pontryagin density. It arises in string theory through the Green–Schwarz anomaly-cancellation mechanism [12 , 13 ] , in loop quantum gravity when the Barbero–Immirzi parameter is promoted to a field [14 ] , and in effective field theories of inflation [15 ] , and is the leading parity-odd curvature-squared correction to the Einstein–Hilbert action. Solar-System tests constrain its nondynamical version [16 ] , and the dynamical theory is constrained by multimessenger neutron-star observations and by gravitational waves from binary mergers [17 , 18 ] , and binary-black-hole coalescences have been simulated in it [19 , 20 ] .
Since the Pontryagin density vanishes on any spherically symmetric spacetime, the Schwarzschild and Reissner–Nordström solutions persist as exact backgrounds in dCS gravity; the theory nonetheless modifies their perturbations, and the QNM spectrum of Schwarzschild black holes in dCS gravity has been studied in detail [21 , 22 , 23 ] . The rotating case has attracted recent attention: the quasinormal modes of slowly rotating black holes in dCS gravity have been computed by direct integration and, more recently, through modified-Teukolsky and spectral methods [24 , 25 , 26 , 27 ] .
The odd parity of the pseudoscalar has a sharp consequence for the spectrum. In general relativity the axial (odd-parity) and polar (even-parity) perturbations of a black hole are isospectral : they share identical QNM frequencies, a degeneracy guaranteed by the Chandrasekhar–Detweiler transformation [28 ] . An odd-parity field that couples to the axial sector alone lifts this degeneracy. In dCS gravity the pseudoscalar couples only to the axial sector, shifting the axial gravitational modes while leaving the polar sector untouched, so that the axial and polar spectra no longer coincide [21 , 22 ] . This breaking of isospectrality is a characteristic ringdown signature of the dCS coupling, arising because the pseudoscalar perturbation has odd parity, distinct from a mere numerical shift of individual frequencies. Isospectrality breaking has been identified as a discriminator in other extensions of general relativity, including dilatonic, higher-curvature and f ( R ) f(R) f ( R ) [29 , 30 ] , and is built into parametrized descriptions of the ringdown [31 , 32 ] ; here it is driven by the odd-parity Chern–Simons pseudoscalar and controlled by the black-hole charge.
While the Schwarzschild case is well studied, the role of black-hole charge has received little attention. Astrophysical black holes are expected to be nearly neutral, but charge is a useful theoretical control parameter, minicharged and dark-sector models allow sizeable charge-to-mass ratios [33 ] , and gravitational-wave observations already constrain it [34 , 35 ] . Charge also enters the dCS problem in a nontrivial way. It couples the electromagnetic and gravitational perturbations already in general relativity [28 , 36 , 37 , 38 , 39 , 40 ] , and it deforms the very curvature invariants through which the pseudoscalar sources the metric. Its rotating (Kerr–Newman) generalization already has a rich gravito-electromagnetic spectrum in general relativity, including eigenvalue repulsion between mode families [41 , 42 , 43 ] . The present problem should also be contrasted with our earlier study of stationary axial perturbations of the same background [44 ] , where time-reversal symmetry forces the Chern–Simons scalar to be constant and no observable effect survives at first order in the coupling. The quasinormal modes considered here carry complex frequencies and therefore break time-reversal symmetry; the coupling is dynamical, entering at O ( ζ ) O(\zeta) O ( ζ )
In this work we derive the dCS couplings of the axial sector of the RN black hole and compute the resulting isospectrality breaking. Part of the calculation carries over from the Schwarzschild case: by direct reduction of the linearized field equations, the coupling of the pseudoscalar to the axial gravitational field and its reciprocal are both built from the kernel ( M r − Q 2 ) Δ / r 8 (Mr-Q^2)\Delta/r^8 ( M r − Q 2 ) Δ/ r 8 M f / r 5 Mf/r^5 M f / r 5 [22 ] at Q = 0 Q=0 Q = 0
An electromagnetic-led branch that is also shifted. At finite charge the reduction produces a new channel: the pseudoscalar couples, in both directions, to Ψ + χ Ψ e \Psi+\chi\Psi_e Ψ + χ Ψ e χ = 2 Q / [ r ( l − 1 ) ( l + 2 ) ] \chi=2Q/[r\sqrt{(l-1)(l+2)}] χ = 2 Q / [ r ( l − 1 ) ( l + 2 ) ] Q c = 2 M l ( l + 1 ) + 1 / [ l ( l + 1 ) + 2 ] Q_c=2M\sqrt{l(l+1)+1/[l(l+1)+2]} Q c = 2 M l ( l + 1 ) + 1/ [ l ( l + 1 ) + 2 ] O ( ζ ) O(\sqrt{\zeta}) O ( ζ ) [43 ] .
Charge controls both branches. The Einstein–Maxwell sector decouples under a constant, Q Q Q Q Q Q l 3 l^3 l 3 Q = 0 Q=0 Q = 0 l 2 / Q l^2/Q l 2 / Q Q > 0 Q>0 Q > 0 Q l ≫ M Ql\gg M Ql ≫ M Q 2 Q^2 Q 2
The extremal limit. The kernel ( M r − Q 2 ) (Mr-Q^2) ( M r − Q 2 ) [45 , 46 , 47 ] , which exist only near extremality, as 1 − Q 2 / M 2 1-Q^2/M^2 1 − Q 2 / M 2
We compute the shifts with an exterior-complex-scaling spectral method for the photon-sphere modes and a Frobenius–shooting method for the zero-damping family, and we state throughout which results are exact, which are numerical, and which are asymptotic estimates (Table V).
The remainder of this paper is organized as follows. In Sec. II we set up dynamical Chern–Simons gravity on the static Reissner–Nordström background and establish the perturbation scheme, and in Sec. III we show that the odd parity of the pseudoscalar confines the Chern–Simons effect to the axial sector, breaking isospectrality. Section IV records the general-relativistic axial backbone—the coupled electromagnetic–gravitational Regge–Wheeler system of the charged black hole. In Secs. V and VI we derive the axial Cotton source and reduce it to the closed-form coupling C d C S C_{\mathrm{dCS}} C dCS
We work in geometrized units G = c = 1 G=c=1 G = c = 1 ( − , + , + , + ) (-,+,+,+) ( − , + , + , + )
Theoretical Framework
Action and Field Equations
We work in dynamical Chern–Simons (dCS) gravity minimally coupled to a Maxwell field, in the gravitational formulation in which the pseudoscalar ϑ \vartheta ϑ
S = ∫ d 4 x − g [ 1 2 κ R + α 4 ϑ ∗ R R − β 2 g μ ν ∇ μ ϑ ∇ ν ϑ − 1 2 κ F μ ν F μ ν ] , κ ≡ 8 π G (1) S=\int d^4x\sqrt{-g}\left[\frac{1}{2\kappa}R+\frac{\alpha}{4}\vartheta\,{}^\ast RR-\frac{\beta}{2}g^{\mu\nu}\nabla_\mu\vartheta\nabla_\nu\vartheta-\frac{1}{2\kappa}F_{\mu\nu}F^{\mu\nu}\right],\qquad\kappa\equiv8\pi G \tag*{(1)} S = ∫ d 4 x − g [ 2 κ 1 R + 4 α ϑ ∗ R R − 2 β g μν ∇ μ ϑ ∇ ν ϑ − 2 κ 1 F μν F μν ] , κ ≡ 8 π G (1) where F μ ν = ∂ μ A ν − ∂ ν A μ F_{\mu\nu} = \partial_\mu A_\nu- \partial_\nu A_\mu F μν = ∂ μ A ν − ∂ ν A μ
∗ R R = ∗ R μ ρ σ ν R ρ μ ρ σ , ∗ R μ ν ρ σ = 1 2 ε ρ σ τ η R μ ν τ η . (2) {}^*RR = {}^*R^\mu{}_{\rho\sigma}{}^\nu R^\rho{}_{\mu\rho\sigma}, \qquad{}^*R^\mu{}_\nu{}^{\rho\sigma} = \frac{1}{2}\varepsilon^{\rho\sigma\tau\eta}R^\mu{}_{\nu\tau\eta}. \tag*{(2)} ∗ R R = ∗ R μ ρ σ ν R ρ μ ρ σ , ∗ R μ ν ρ σ = 2 1 ε ρ σ τ η R μ ν τ η . (2) with ε ρ σ τ η \varepsilon^{\rho\sigma\tau\eta} ε ρ σ τ η
Varying (1) with respect to the metric, the pseudoscalar, and the gauge field gives, respectively,
R μ ν + 2 κ α C μ ν = κ ( T μ ν − 1 2 g μ ν T ) , (3) R_{\mu\nu} + 2\kappa\alpha C_{\mu\nu} = \kappa\left(T_{\mu\nu} - \frac{1}{2}g_{\mu\nu}T\right), \tag*{(3)} R μν + 2 κ α C μν = κ ( T μν − 2 1 g μν T ) , (3) β □ ϑ = − α 4 ∗ R R , (4) \beta\Box\vartheta= -\frac{\alpha}{4}{}^*RR, \tag*{(4)} β □ ϑ = − 4 α ∗ R R , (4) ∇ μ F μ ν = 0 , (5) \nabla_\mu F^{\mu\nu} = 0, \tag*{(5)} ∇ μ F μν = 0 , (5) where T μ ν = T μ ν E M + T μ ν ϑ T_{\mu\nu}=T_{\mu\nu}^{\rm EM}+T_{\mu\nu}^{\vartheta} T μν = T μν EM + T μν ϑ C C C
C μ ν = ( ∇ ρ ϑ ) ε ρ σ τ ( μ ∇ τ R ν ) σ + ( ∇ ρ ∇ σ ϑ ) ∗ R σ ( μ ν ) ρ . (6) C^{\mu\nu} = (\nabla_\rho\vartheta)\varepsilon^{\rho\sigma\tau(\mu}\nabla_\tau R^{\nu)}{}_\sigma+ (\nabla_\rho\nabla_\sigma\vartheta){}^*R^{\sigma(\mu\nu)\rho}. \tag*{(6)} C μν = ( ∇ ρ ϑ ) ε ρ σ τ ( μ ∇ τ R ν ) σ + ( ∇ ρ ∇ σ ϑ ) ∗ R σ ( μν ) ρ . (6) Round brackets denote symmetrization with weight 1 / 2 1/2 1/2
T μ ν E M = 2 κ ( F μ α F ν α − 1 4 g μ ν F α β F α β ) , T μ ν ϑ = β ( ∇ μ ϑ ∇ ν ϑ − 1 2 g μ ν ∇ α ϑ ∇ α ϑ ) . (7) T_{\mu\nu}^{\rm EM} = \frac{2}{\kappa}\left(F_{\mu\alpha}F_\nu{}^\alpha-\frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right), \qquad
T_{\mu\nu}^{\vartheta} = \beta\left(\nabla_\mu\vartheta\nabla_\nu\vartheta-\frac{1}{2}g_{\mu\nu}\nabla_\alpha\vartheta\nabla^\alpha\vartheta\right). \tag*{(7)} T μν EM = κ 2 ( F μα F ν α − 4 1 g μν F α β F α β ) , T μν ϑ = β ( ∇ μ ϑ ∇ ν ϑ − 2 1 g μν ∇ α ϑ ∇ α ϑ ) . (7) a. Conventions.
Orientation. The Levi-Civita tensor is ε μ ν ρ σ = − g [ μ ν ρ σ ] \varepsilon_{\mu\nu\rho\sigma}=\sqrt{-g}\,[\mu\nu\rho\sigma] ε μν ρ σ = − g [ μν ρ σ ] [ t r θ ϕ ] = + 1 [tr\theta\phi]=+1 [ t r θ ϕ ] = + 1 ε t r θ ϕ = − 1 / − g \varepsilon^{tr\theta\phi}=-1/\sqrt{-g} ε t r θ ϕ = − 1/ − g (2) . Reversing the orientation flips the sign of every off-diagonal Chern–Simons coupling below simultaneously (C d C S C_{\rm dCS} C dCS C e Θ C_{e\Theta} C e Θ C Θ g C_{\Theta g} C Θ g C Θ e C_{\Theta e} C Θ e Θ → − Θ \Theta\to-\Theta Θ → − Θ
Harmonics. We work with axisymmetric modes, Y l ≡ Y l 0 ∝ P l ( cos θ ) Y_l\equiv Y_{l0}\propto P_l(\cos\theta) Y l ≡ Y l 0 ∝ P l ( cos θ ) m m m ( 2 l + 1 ) (2l+1) ( 2 l + 1 ) W ( θ ) = sin θ ∂ θ Y l W(\theta)=\sin\theta\,\partial_\theta Y_l W ( θ ) = sin θ ∂ θ Y l S ( θ ) S(\theta) S ( θ ) (12) ]; it obeys W ′ ′ = cot θ W ′ − l ( l + 1 ) W W''=\cot\theta\,W'-l(l+1)W W ′′ = cot θ W ′ − l ( l + 1 ) W
The Maxwell equation (5) is unmodified: there is no direct ϑ F F ~ \vartheta F\widetilde F ϑF F § 1.5 .
Background and expansion scheme
The static Reissner–Nordström background metric and gauge field are
d s 2 = − f d t 2 + d r 2 f + r 2 d Ω 2 , A μ d x μ = − Q r d t , (8) ds^2=-f\,dt^2+\frac{dr^2}{f}+r^2d\Omega^2,\qquad A_\mu dx^\mu=-\frac{Q}{r}\,dt, \tag*{(8)} d s 2 = − f d t 2 + f d r 2 + r 2 d Ω 2 , A μ d x μ = − r Q d t , (8) with f ( r ) = 1 − 2 M / r + Q 2 / r 2 = Δ / r 2 f(r)=1-2M/r+Q^2/r^2=\Delta/r^2 f ( r ) = 1 − 2 M / r + Q 2 / r 2 = Δ/ r 2 Δ = ( r − r + ) ( r − r − ) \Delta=(r-r_+)(r-r_-) Δ = ( r − r + ) ( r − r − ) r ± = M ± M 2 − Q 2 r_\pm=M\pm\sqrt{M^2-Q^2} r ± = M ± M 2 − Q 2 − F μ ν F μ ν / ( 2 κ ) -F_{\mu\nu}F^{\mu\nu}/(2\kappa) − F μν F μν / ( 2 κ ) (1) , the geometrized RN charge parameter Q Q Q f f f Q 2 / r 2 Q^2/r^2 Q 2 / r 2 Q Q Q
We expand about the RN background in two small quantities: the amplitude ϵ \epsilon ϵ
g μ ν = g μ ν R N + ϵ ( h μ ν G R + α h μ ν ( 1 ) + ⋯ ) , g_{\mu\nu}=g_{\mu\nu}^{\rm RN}+\epsilon\left(h_{\mu\nu}^{\rm GR}+\alpha h_{\mu\nu}^{(1)}+\cdots\right), g μν = g μν RN + ϵ ( h μν GR + α h μν ( 1 ) + ⋯ ) , A μ = A μ R N + ϵ ( δ A μ G R + α δ A μ ( 1 ) + ⋯ ) , (9) A_\mu=A_\mu^{\rm RN}+\epsilon\left(\delta A_\mu^{\rm GR}+\alpha\delta A_\mu^{(1)}+\cdots\right), \tag*{(9)} A μ = A μ RN + ϵ ( δ A μ GR + α δ A μ ( 1 ) + ⋯ ) , (9) ϑ = ϵ ( ϑ ( 0 ) + α β ϑ ( 1 ) + ⋯ ) . \vartheta=\epsilon\left(\vartheta^{(0)}+\frac{\alpha}{\beta}\vartheta^{(1)}+\cdots\right). ϑ = ϵ ( ϑ ( 0 ) + β α ϑ ( 1 ) + ⋯ ) . Here ϑ ( 0 ) \vartheta^{(0)} ϑ ( 0 ) ϑ ( 0 ) = 0 \vartheta^{(0)}=0 ϑ ( 0 ) = 0 O ( α / β ) \mathcal{O}(\alpha/\beta) O ( α / β ) ∗ R R ∣ R N = 0 {}^*RR|_{\rm RN}=0 ∗ R R ∣ RN = 0 ϵ \epsilon ϵ
the pseudoscalar drives the axial metric with a strength ∝ α \propto\alpha ∝ α 2 κ α C μ ν 2\kappa\alpha C_{\mu\nu} 2 κ α C μν
the metric drives the pseudoscalar with a strength ∝ α / β \propto\alpha/\beta ∝ α / β − α ∗ R R / 4 β -\alpha{}^*RR/4\beta − α ∗ R R /4 β
A shift of a gravitational quasinormal frequency requires both, so it is controlled by their product ∝ α 2 / β \propto\alpha^2/\beta ∝ α 2 / β
ζ ≡ α 2 β κ M 4 . (10) \zeta\equiv\frac{\alpha^2}{\beta\kappa M^4}. \tag*{(10)} ζ ≡ β κ M 4 α 2 . (10) For numerical values we use units M = α = 1 M=\alpha=1 M = α = 1 κ = 8 π \kappa=8\pi κ = 8 π ζ = 1 / ( 8 π β ) \zeta=1/(8\pi\beta) ζ = 1/ ( 8 π β ) β δ ω \beta\,\delta\omega β δ ω M δ ω = 8 π ( β δ ω ) ζ M\delta\omega=8\pi(\beta\,\delta\omega)\zeta M δ ω = 8 π ( β δ ω ) ζ
The orders that appear below are:
Off resonance (gravitational and pseudoscalar frequencies well separated): δ ω = O ( ζ ) \delta\omega=\mathcal{O}(\zeta) δ ω = O ( ζ )
Near a degeneracy with a pseudoscalar mode: an avoided crossing gives δ ω = O ( ζ ) \delta\omega=\mathcal{O}(\sqrt{\zeta}) δ ω = O ( ζ ) § 1.10 ).
Strictly stationary perturbations: the direct source of the Regge–Wheeler channel vanishes at ω = 0 \omega=0 ω = 0 § 1.8 ).
Table 1. Notation and conventions. The last column gives n n n n ^n n G = c = 1 G=c=1 G = c = 1 n = 0 n=0 n = 0
Pseudoscalar stress-energy: T μ ν ϑ T_{\mu\nu}^{\vartheta} T μν ϑ ϑ = O ( ϵ ) \vartheta=\mathcal{O}(\epsilon) ϑ = O ( ϵ ) O ( ϵ 2 ) \mathcal{O}(\epsilon^2) O ( ϵ 2 ) α \alpha α O ( ζ ) \mathcal{O}(\zeta) O ( ζ )
The linear metric and Maxwell perturbations are therefore sourced only through the C C C [6 ] , evaluated on the dynamical pseudoscalar and the RN background. The O ( α 2 / β ) \mathcal{O}(\alpha^2/\beta) O ( α 2 / β ) δ V Θ \delta V_\Theta δ V Θ O ( ζ 2 ) \mathcal{O}(\zeta^2) O ( ζ 2 )
Parity structure of the coupled perturbations
The coupling α 4 ϑ ∗ R R \frac{\alpha}{4}\vartheta\,{}^\ast RR 4 α ϑ ∗ R R [1 ] is parity invariant, and the linearized equations split into multiplets of definite parity. A pseudoscalar perturbation transforms as ( − 1 ) l + 1 (-1)^{l+1} ( − 1 ) l + 1 δ ϑ \delta\vartheta δ ϑ
polar: { δ Ψ e v e n , δ ϕ e v e n } , axial: { δ Ψ o d d , δ ϕ o d d , δ ϑ } . (11) \text{polar: }\{\delta\Psi^{\rm even},\delta\phi^{\rm even}\},\qquad
\text{axial: }\{\delta\Psi^{\rm odd},\delta\phi^{\rm odd},\delta\vartheta\}. \tag*{(11)} polar: { δ Ψ even , δ ϕ even } , axial: { δ Ψ odd , δ ϕ odd , δ ϑ } . (11) Maxwell theory is parity even, so charge adds an electromagnetic member to each sector without mixing them, and the minimal theory has no ϑ F F ~ \vartheta F\widetilde F ϑF F δ ( ∗ R R ) ∣ p o l a r = 0 \delta({}^\ast RR)|_{\rm polar}=0 δ ( ∗ R R ) ∣ polar = 0
The general-relativistic axial backbone
Before introducing the Chern–Simons source, we record the general-relativistic axial sector of the charged black hole: the coupled electromagnetic–gravitational Regge–Wheeler system of the Reissner–Nordström (RN) spacetime. This is the backbone onto which the dynamical Chern–Simons coupling is layered in the following sections.
Axial ansatz and the electromagnetic–gravitational coupling
For an odd-parity mode of harmonic index l l l ω \omega ω
h t φ = h 0 ( r ) e − i ω t S ( θ ) , h r φ = h 1 ( r ) e − i ω t S ( θ ) , h_{t\varphi}=h_0(r)e^{-i\omega t}S(\theta),\qquad
h_{r\varphi}=h_1(r)e^{-i\omega t}S(\theta), h tφ = h 0 ( r ) e − iω t S ( θ ) , h r φ = h 1 ( r ) e − iω t S ( θ ) , δ A φ = u ( r ) e − i ω t S ( θ ) , S ( θ ) ≡ sin θ ∂ θ P l ( cos θ ) . (12) \delta A_\varphi=u(r)e^{-i\omega t}S(\theta),\qquad
S(\theta)\equiv\sin\theta\,\partial_\theta P_l(\cos\theta). \tag*{(12)} δ A φ = u ( r ) e − iω t S ( θ ) , S ( θ ) ≡ sin θ ∂ θ P l ( cos θ ) . (12) with all remaining components fixed to zero by the gauge choice.
The charge couples the electromagnetic and gravitational perturbations already at the linear level. The background Coulomb field F t r = − Q / r 2 F_{tr}=-Q/r^2 F t r = − Q / r 2 δ g φ t = − h t φ / ( f r 2 sin 2 θ ) \delta g^{\varphi t}=-h_{t\varphi}/(fr^2\sin^2\theta) δ g φt = − h tφ / ( f r 2 sin 2 θ ) δ g φ r = − f h r φ / ( r 2 sin 2 θ ) \delta g^{\varphi r}=-fh_{r\varphi}/(r^2\sin^2\theta) δ g φ r = − f h r φ / ( r 2 sin 2 θ )
F c o u p t φ = − Q h r φ r 4 sin 2 θ , F c o u p r φ = + Q h t φ r 4 sin 2 θ . (13) F_{\rm coup}^{t\varphi}=-\frac{Qh_{r\varphi}}{r^4\sin^2\theta},\qquad
F_{\rm coup}^{r\varphi}=+\frac{Qh_{t\varphi}}{r^4\sin^2\theta}. \tag*{(13)} F coup tφ = − r 4 sin 2 θ Q h r φ , F coup r φ = + r 4 sin 2 θ Q h tφ . (13) Substituting into ∇ μ F μ φ = ( 1 / − g ) ∂ μ ( − g F μ φ ) \nabla_\mu F^{\mu\varphi}=(1/\sqrt{-g})\,\partial_\mu(\sqrt{-g}\,F^{\mu\varphi}) ∇ μ F μ φ = ( 1/ − g ) ∂ μ ( − g F μ φ ) − g = r 2 sin θ \sqrt{-g}=r^2\sin\theta − g = r 2 sin θ
[ ∇ μ F μ φ ] c o u p = Q r 5 [ r ( h 0 ′ + i ω h 1 ) − 2 h 0 ] e − i ω t S ( θ ) sin 2 θ , (14) \left[\nabla_\mu F^{\mu\varphi}\right]_{\rm coup}
=\frac{Q}{r^5}\left[r(h_0'+i\omega h_1)-2h_0\right]e^{-i\omega t}\frac{S(\theta)}{\sin^2\theta}, \tag*{(14)} [ ∇ μ F μ φ ] coup = r 5 Q [ r ( h 0 ′ + iω h 1 ) − 2 h 0 ] e − iω t sin 2 θ S ( θ ) , (14) which we have verified by direct computation. The common factor g φ φ = 1 / ( r 2 sin 2 θ ) g^{\varphi\varphi}=1/(r^2\sin^2\theta) g φφ = 1/ ( r 2 sin 2 θ ) ν = φ \nu=\varphi ν = φ Q Q Q u u u h 0 , h 1 h_0,h_1 h 0 , h 1 δ T a x i a l E M \delta T^{\mathrm{EM}}_{\mathrm{axial}} δ T axial EM Q Q Q (14) by the self-adjointness of the linearised Einstein–Maxwell system.
After separation with tensor spherical harmonics and reduction to master variables, the axial sector reduces to the coupled RN system of Chandrasekhar, which we take in the form given by Berti and Kokkotas [40 ] . With Δ = r 2 − 2 M r + Q 2 \Delta=r^2-2Mr+Q^2 Δ = r 2 − 2 M r + Q 2 d r ∗ / d r = r 2 / Δ \mathrm{d}r_*/\mathrm{d}r=r^2/\Delta d r ∗ / d r = r 2 /Δ H 2 ( − ) H_2^{(-)} H 2 ( − ) H 1 ( − ) H_1^{(-)} H 1 ( − )
( d 2 d r ∗ 2 + ω 2 ) H 2 ( − ) = Δ r 5 { [ l ( l + 1 ) r − 6 M + 4 Q 2 r ] H 2 ( − ) + 2 Q ( l − 1 ) ( l + 2 ) H 1 ( − ) } (15) \left(\frac{\mathrm{d}^2}{\mathrm{d}r_*^2}+\omega^2\right)H_2^{(-)}=\frac{\Delta}{r^5}\left\{\left[l(l+1)r-6M+\frac{4Q^2}{r}\right]H_2^{(-)}+2Q\sqrt{(l-1)(l+2)}H_1^{(-)}\right\} \tag*{(15)} ( d r ∗ 2 d 2 + ω 2 ) H 2 ( − ) = r 5 Δ { [ l ( l + 1 ) r − 6 M + r 4 Q 2 ] H 2 ( − ) + 2 Q ( l − 1 ) ( l + 2 ) H 1 ( − ) } (15) ( d 2 d r ∗ 2 + ω 2 ) H 1 ( − ) = Δ r 5 { [ l ( l + 1 ) r + 4 Q 2 r ] H 1 ( − ) + 2 Q ( l − 1 ) ( l + 2 ) H 2 ( − ) } (16) \left(\frac{\mathrm{d}^2}{\mathrm{d}r_*^2}+\omega^2\right)H_1^{(-)}=\frac{\Delta}{r^5}\left\{\left[l(l+1)r+\frac{4Q^2}{r}\right]H_1^{(-)}+2Q\sqrt{(l-1)(l+2)}H_2^{(-)}\right\} \tag*{(16)} ( d r ∗ 2 d 2 + ω 2 ) H 1 ( − ) = r 5 Δ { [ l ( l + 1 ) r + r 4 Q 2 ] H 1 ( − ) + 2 Q ( l − 1 ) ( l + 2 ) H 2 ( − ) } (16) The gravitational equation carries the Regge–Wheeler term − 6 M Δ / r 5 = f ( − 6 M / r 3 ) -6M\Delta/r^5=f(-6M/r^3) − 6 M Δ/ r 5 = f ( − 6 M / r 3 ) 4 Q 2 / r 4Q^2/r 4 Q 2 / r
C Q ( r ) = Δ r 5 2 Q ( l − 1 ) ( l + 2 ) , (17) C_Q(r)=\frac{\Delta}{r^5}2Q\sqrt{(l-1)(l+2)}, \tag*{(17)} C Q ( r ) = r 5 Δ 2 Q ( l − 1 ) ( l + 2 ) , (17) of order Q Q Q l = 1 l=1 l = 1 l ≥ 2 l\ge2 l ≥ 2 C Q ∝ Q / r 5 C_Q\propto Q/r^5 C Q ∝ Q / r 5 (17) matches that of the component-level coupling (14) , providing a consistency check between the master-variable system and the direct calculation.2 Ψ e = − 2 u / μ \Psi_e=-2u/\mu Ψ e = − 2 u / μ δ A φ = − ( μ / 2 ) Ψ e e − i ω t S ( θ ) \delta A_\varphi=-(\mu/2)\Psi_e e^{-i\omega t}S(\theta) δ A φ = − ( μ /2 ) Ψ e e − iω t S ( θ ) Ψ R W = f h 1 / r \Psi_{\mathrm{RW}}=fh_1/r Ψ RW = f h 1 / r
Chandrasekhar decoupling
A constant rotation in field space decouples (15) –(16) into two Schrödinger-type equations
( d 2 d r ∗ 2 + ω 2 ) Z i ( − ) = V i ( − ) Z i ( − ) , (18) \left(\frac{\mathrm{d}^2}{\mathrm{d}r_*^2}+\omega^2\right)Z_i^{(-)}=V_i^{(-)}Z_i^{(-)}, \tag*{(18)} ( d r ∗ 2 d 2 + ω 2 ) Z i ( − ) = V i ( − ) Z i ( − ) , (18) V i ( − ) = Δ r 5 [ l ( l + 1 ) r − q j + 4 Q 2 r ] , ( i ≠ j ) , V_i^{(-)}=\frac{\Delta}{r^5}\left[l(l+1)r-q_j+\frac{4Q^2}{r}\right],\qquad(i\ne j), V i ( − ) = r 5 Δ [ l ( l + 1 ) r − q j + r 4 Q 2 ] , ( i = j ) , with
q 1 , 2 = 3 M ± 9 M 2 + 4 Q 2 ( l − 1 ) ( l + 2 ) . (19) q_{1,2}=3M\pm\sqrt{9M^2+4Q^2(l-1)(l+2)}. \tag*{(19)} q 1 , 2 = 3 M ± 9 M 2 + 4 Q 2 ( l − 1 ) ( l + 2 ) . (19) The rotation is exact and independent of r r r (15) –(16) as ( Δ / r 5 ) [ ( l ( l + 1 ) r + 4 Q 2 / r ) I + A ] (\Delta/r^5)[(l(l+1)r+4Q^2/r)I+A] ( Δ/ r 5 ) [( l ( l + 1 ) r + 4 Q 2 / r ) I + A ] I I I 2 × 2 2\times2 2 × 2
A = ( − 6 M 2 Q μ 2 Q μ 0 ) (20) A=\begin{pmatrix}-6M&2Q\mu\\2Q\mu&0\end{pmatrix} \tag*{(20)} A = ( − 6 M 2 Q μ 2 Q μ 0 ) (20) has constant entries, with eigenvalues − q 1 , 2 -q_{1,2} − q 1 , 2
s = 9 M 2 + 4 Q 2 μ 2 , tan φ = s − 3 M 2 Q μ . (21) s=\sqrt{9M^2+4Q^2\mu^2},\qquad\tan\varphi=\frac{s-3M}{2Q\mu}. \tag*{(21)} s = 9 M 2 + 4 Q 2 μ 2 , tan φ = 2 Q μ s − 3 M . (21) Then
Z 2 ( − ) = cos φ Ψ − sin φ Ψ e , Z 1 ( − ) = sin φ Ψ + cos φ Ψ e . (22) Z_2^{(-)}=\cos\varphi\,\Psi-\sin\varphi\,\Psi_e,\qquad Z_1^{(-)}=\sin\varphi\,\Psi+\cos\varphi\,\Psi_e. \tag*{(22)} Z 2 ( − ) = cos φ Ψ − sin φ Ψ e , Z 1 ( − ) = sin φ Ψ + cos φ Ψ e . (22) Z 2 ( − ) Z_2^{(-)} Z 2 ( − ) q 1 q_1 q 1 (18) . It is the gravitational-led branch, continuous with the Schwarzschild Regge–Wheeler mode; we also write it Z g Z_g Z g
Z 1 ( − ) Z_1^{(-)} Z 1 ( − ) q 2 q_2 q 2 electromagnetic-led branch, Z e Z_e Z e
The mixing angle grows from φ = 0 \varphi=0 φ = 0 Q = 0 Q=0 Q = 0 φ → π / 4 \varphi\to\pi/4 φ → π /4 Q μ ≫ M Q\mu\gg M Q μ ≫ M l l l Ψ \Psi Ψ Ψ e \Psi_e Ψ e
With the i ≠ j i\ne j i = j (18) , Z g ≡ Z 2 ( − ) Z_g\equiv Z_2^{(-)} Z g ≡ Z 2 ( − ) q 1 q_1 q 1 Z e ≡ Z 1 ( − ) Z_e\equiv Z_1^{(-)} Z e ≡ Z 1 ( − ) q 2 q_2 q 2 A ( cos φ , − sin φ ) T = − q 1 ( cos φ , − sin φ ) T A(\cos\varphi,-\sin\varphi)^T=-q_1(\cos\varphi,-\sin\varphi)^T A ( cos φ , − sin φ ) T = − q 1 ( cos φ , − sin φ ) T A ( sin φ , cos φ ) T = − q 2 ( sin φ , cos φ ) T A(\sin\varphi,\cos\varphi)^T=-q_2(\sin\varphi,\cos\varphi)^T A ( sin φ , cos φ ) T = − q 2 ( sin φ , cos φ ) T Q → 0 Q\to0 Q → 0 φ → 0 \varphi\to0 φ → 0 q 1 → 6 M q_1\to6M q 1 → 6 M Z g → Ψ Z_g\to\Psi Z g → Ψ Z e → Ψ e Z_e\to\Psi_e Z e → Ψ e Q = M Q=M Q = M q 1 = 2 M ( l + 2 ) q_1=2M(l+2) q 1 = 2 M ( l + 2 ) q 2 = − 2 M ( l − 1 ) q_2=-2M(l-1) q 2 = − 2 M ( l − 1 ) h g = l h_g=l h g = l h e = l + 2 h_e=l+2 h e = l + 2 (18) give the pure RN spectrum. Their fundamental frequencies are the baseline against which the Chern–Simons shift is measured (Sec. XIII). The pseudoscalar couples to a fixed, r r r
The axial Chern–Simons source on static Reissner–Nordström
In dynamical Chern–Simons (dCS) gravity the pseudoscalar ϑ \vartheta ϑ M f / r 5 → ( M r − Q 2 ) Δ / r 8 Mf/r^5 \to(Mr-Q^2)\Delta/r^8 M f / r 5 → ( M r − Q 2 ) Δ/ r 8 (40) . The genuinely new structure, the electromagnetic channel, appears in § 1.5 .
Perturbation and its covariant derivatives
We take the dynamical scalar perturbation on the static RN background (a = 0 a=0 a = 0
δ ϑ = ∑ l R l ( r ) Y l ( θ ) e − i ω t , m = 0 , (23) \delta\vartheta= \sum_l R_l(r)Y_l(\theta)e^{-i\omega t}, \qquad m=0, \tag*{(23)} δ ϑ = l ∑ R l ( r ) Y l ( θ ) e − iω t , m = 0 , (23) where the radial profile R l ( r ) = Θ ( r ) / r R_l(r)=\Theta(r)/r R l ( r ) = Θ ( r ) / r δ ϑ \delta\vartheta δ ϑ R l R_l R l R l R_l R l § 1.5 ). Writing v μ ≡ ∇ μ δ ϑ = ∂ μ δ ϑ v_\mu\equiv\nabla_\mu\delta\vartheta=\partial_\mu\delta\vartheta v μ ≡ ∇ μ δ ϑ = ∂ μ δ ϑ v μ ν ≡ ∇ μ v ν = ∂ μ v ν − Γ μ ν λ v λ v_{\mu\nu}\equiv\nabla_\mu v_\nu=\partial_\mu v_\nu-\Gamma^\lambda_{\mu\nu}v_\lambda v μν ≡ ∇ μ v ν = ∂ μ v ν − Γ μν λ v λ e − i ω t e^{-i\omega t} e − iω t ∂ t → − i ω \partial_t\to-i\omega ∂ t → − iω
v t = − i ω R Y , v r = R ′ Y , v θ = R Y ′ , v ϕ = 0 , (24) v_t=-i\omega RY,\qquad v_r=R'Y,\qquad v_\theta=RY',\qquad v_\phi=0, \tag*{(24)} v t = − iω R Y , v r = R ′ Y , v θ = R Y ′ , v ϕ = 0 , (24) and, using the RN Christoffel symbols,
v t t = − ω 2 R Y − 1 2 f f ′ R ′ Y , v t r = v r t = − i ω R ′ Y + i ω f ′ 2 f R Y , v t θ = − i ω R Y ′ , v r r = R ′ ′ Y + f ′ 2 f R ′ Y , v r θ = R ′ Y ′ − 1 r R Y ′ , v θ θ = R Y ′ ′ + r f R ′ Y , v ϕ ϕ = r f sin 2 θ R ′ Y + sin θ cos θ R Y ′ , v t ϕ = v r ϕ = v θ ϕ = 0. (25) \begin{aligned}
v_{tt}&=-\omega^2RY-\frac{1}{2}ff'R'Y,\\
v_{tr}=v_{rt}&=-i\omega R'Y+\frac{i\omega f'}{2f}RY,\\
v_{t\theta}&=-i\omega RY',\qquad v_{rr}=R''Y+\frac{f'}{2f}R'Y,\\
v_{r\theta}&=R'Y'-\frac{1}{r}RY',\qquad v_{\theta\theta}=RY''+rfR'Y,\\
v_{\phi\phi}&=rf\sin^2\theta R'Y+\sin\theta\cos\theta RY',\\
v_{t\phi}&=v_{r\phi}=v_{\theta\phi}=0. \tag*{(25)}
\end{aligned} v tt v t r = v r t v tθ v r θ v ϕϕ v tϕ = − ω 2 R Y − 2 1 f f ′ R ′ Y , = − iω R ′ Y + 2 f iω f ′ R Y , = − iω R Y ′ , v r r = R ′′ Y + 2 f f ′ R ′ Y , = R ′ Y ′ − r 1 R Y ′ , v θ θ = R Y ′′ + r f R ′ Y , = r f sin 2 θ R ′ Y + sin θ cos θ R Y ′ , = v r ϕ = v θ ϕ = 0. (25) All components in (24)–(25) are verified symbolically.
C-tensor
We split Eq. (6) as C μ ν = C ( 1 ) μ ν + C ( 2 ) μ ν C^{\mu\nu}=C^{\mu\nu}_{(1)}+C^{\mu\nu}_{(2)} C μν = C ( 1 ) μν + C ( 2 ) μν (39) of Ref. [44 ] ):
C ( 1 ) μ ν = 1 2 v ρ ( ϵ ρ σ τ μ ∇ τ R σ ν + ϵ ρ σ τ ν ∇ τ R σ μ ) , (26) C^{\mu\nu}_{(1)}=\frac{1}{2}v_\rho\left(\epsilon^{\rho\sigma\tau\mu}\nabla_\tau R^\nu_{\ \sigma}+\epsilon^{\rho\sigma\tau\nu}\nabla_\tau R^\mu_{\ \sigma}\right), \tag*{(26)} C ( 1 ) μν = 2 1 v ρ ( ϵ ρ σ τ μ ∇ τ R σ ν + ϵ ρ σ τ ν ∇ τ R σ μ ) , (26) C ( 2 ) μ ν = 1 2 v ρ σ ( ∗ R σ μ ν ρ + ∗ R σ ν μ ρ ) , (27) C^{\mu\nu}_{(2)}=\frac{1}{2}v_{\rho\sigma}\left({}^{*}R^{\sigma\mu\nu\rho}+{}^{*}R^{\sigma\nu\mu\rho}\right), \tag*{(27)} C ( 2 ) μν = 2 1 v ρ σ ( ∗ R σ μν ρ + ∗ R σ ν μ ρ ) , (27) where C ( 1 ) C_{(1)} C ( 1 ) C ( 2 ) C_{(2)} C ( 2 )
R t t = R r r = − Q 2 r 4 , R θ θ = R ϕ ϕ = + Q 2 r 4 , (28) R^t_{\ t}=R^r_{\ r}=-\frac{Q^2}{r^4},\qquad R^\theta_{\ \theta}=R^\phi_{\ \phi}=+\frac{Q^2}{r^4}, \tag*{(28)} R t t = R r r = − r 4 Q 2 , R θ θ = R ϕ ϕ = + r 4 Q 2 , (28) and the RN Riemann tensor is standard [28 ] . Only the axial components C t ϕ C^{t\phi} C tϕ C r ϕ C^{r\phi} C r ϕ
Axial components
a. Dual-Riemann part C ( 2 ) C_{(2)} C ( 2 ) Contracting (27) with v σ τ v_{\sigma\tau} v σ τ ( σ , τ ) = ( r , θ ) (\sigma,\tau)=(r,\theta) ( σ , τ ) = ( r , θ ) ( θ , r ) (\theta,r) ( θ , r ) C ( 2 ) t ϕ C^{t\phi}_{(2)} C ( 2 ) tϕ ( t , θ ) (t,\theta) ( t , θ ) ( θ , t ) (\theta,t) ( θ , t ) C ( 2 ) r ϕ C^{r\phi}_{(2)} C ( 2 ) r ϕ Δ \Delta Δ
C ( 2 ) t ϕ = − 3 ( M r − Q 2 ) r 6 sin θ v r θ , C ( 2 ) r ϕ = 3 ( M r − Q 2 ) r 6 sin θ v t θ . (29) C^{t\phi}_{(2)}=-\frac{3(Mr-Q^2)}{r^6\sin\theta}v_{r\theta},\qquad C^{r\phi}_{(2)}=\frac{3(Mr-Q^2)}{r^6\sin\theta}v_{t\theta}. \tag*{(29)} C ( 2 ) tϕ = − r 6 sin θ 3 ( M r − Q 2 ) v r θ , C ( 2 ) r ϕ = r 6 sin θ 3 ( M r − Q 2 ) v tθ . (29) b. Ricci part C ( 1 ) C_{(1)} C ( 1 ) Because the mixed Ricci difference R t t − R ϕ ϕ R^t_{\ t}-R^\phi_{\ \phi} R t t − R ϕ ϕ (28) is a function of r r r ∇ θ \nabla_\theta ∇ θ ∇ t \nabla_t ∇ t C ( 1 ) t ϕ C^{t\phi}_{(1)} C ( 1 ) tϕ C ( 1 ) r ϕ C^{r\phi}_{(1)} C ( 1 ) r ϕ
C ( 1 ) t ϕ = − 3 Q 2 r 7 R l S ϕ , C ( 1 ) r ϕ = 0 , S ϕ ≡ ∂ θ Y l sin θ . (30) C^{t\phi}_{(1)}=-\frac{3Q^2}{r^7}R_lS^\phi,\qquad C^{r\phi}_{(1)}=0,\qquad S^\phi\equiv\frac{\partial_\theta Y_l}{\sin\theta}. \tag*{(30)} C ( 1 ) tϕ = − r 7 3 Q 2 R l S ϕ , C ( 1 ) r ϕ = 0 , S ϕ ≡ sin θ ∂ θ Y l . (30) Consistently, C ( 1 ) t ϕ → 0 C^{t\phi}_{(1)}\to0 C ( 1 ) tϕ → 0 Q → 0 Q\to0 Q → 0
c. Combined source. Inserting v r θ = ( R l ′ − R l / r ) ∂ θ Y l v_{r\theta}=(R_l'-R_l/r)\partial_\theta Y_l v r θ = ( R l ′ − R l / r ) ∂ θ Y l v t θ = − i ω R l ∂ θ Y l v_{t\theta}=-i\omega R_l\partial_\theta Y_l v tθ = − iω R l ∂ θ Y l (25) into Eqs. (29) –(30) , both axial components carry the common angular factor S ϕ S^\phi S ϕ
C t ϕ = [ − 3 ( M r − Q 2 ) r 6 R l ′ + 3 ( M r − 2 Q 2 ) r 7 R l ] S ϕ , (31) C^{t\phi}=\left[-\frac{3(Mr-Q^2)}{r^6}R_l'+\frac{3(Mr-2Q^2)}{r^7}R_l\right]S^\phi, \tag*{(31)} C tϕ = [ − r 6 3 ( M r − Q 2 ) R l ′ + r 7 3 ( M r − 2 Q 2 ) R l ] S ϕ , (31) C r ϕ = − 3 i ω ( M r − Q 2 ) r 6 R l S ϕ . (32) C^{r\phi}=-\frac{3i\omega(Mr-Q^2)}{r^6}R_lS^\phi. \tag*{(32)} C r ϕ = − r 6 3 iω ( M r − Q 2 ) R l S ϕ . (32) Thus C t ϕ C^{t\phi} C tϕ { R l ′ , R l } \{R_l',R_l\} { R l ′ , R l } C r ϕ C^{r\phi} C r ϕ i ω i\omega iω e − i ω t e^{-i\omega t} e − iω t ω \omega ω C ( 2 ) C_{(2)} C ( 2 ) (31) –(32) are the input to the master-variable reduction of Sec. VI.
The contravariant components are proportional to S ϕ S^\phi S ϕ C μ ϕ = g μ ϕ C μ ϕ C_{\mu\phi}=g_{\mu\phi}C^{\mu\phi} C μ ϕ = g μ ϕ C μ ϕ W W W l l l
The axial master equation and the dynamical Chern–Simons coupling
We now reduce the axial sector to a single Schrödinger-type master equation and extract the coupling coefficient C d C S C_{\mathrm{dCS}} C dCS
Axial field equations
In Regge–Wheeler gauge the odd-parity metric perturbation is carried by two radial functions h 0 ( r ) = h t ϕ h_0(r)=h_{t\phi} h 0 ( r ) = h tϕ h 1 ( r ) = h r ϕ h_1(r)=h_{r\phi} h 1 ( r ) = h r ϕ W ( θ ) = sin θ ∂ θ Y l W(\theta)=\sin\theta\,\partial_\theta Y_l W ( θ ) = sin θ ∂ θ Y l e − i ω t e^{-i\omega t} e − iω t W ′ ′ = cot θ W ′ − L W W''=\cot\theta\,W'-LW W ′′ = cot θ W ′ − L W W ′ W' W ′
δ G t ϕ = W [ − Δ 2 r 2 h 0 ′ ′ − i ω Δ 2 r 2 h 1 ′ + L r 2 − 4 M r + 4 Q 2 2 r 4 h 0 − i ω Δ r 3 h 1 ] , (33) \delta G_{t\phi}=W\left[-\frac{\Delta}{2r^2}h_0''-\frac{i\omega\Delta}{2r^2}h_1'+\frac{Lr^2-4Mr+4Q^2}{2r^4}h_0-\frac{i\omega\Delta}{r^3}h_1\right], \tag*{(33)} δ G tϕ = W [ − 2 r 2 Δ h 0 ′′ − 2 r 2 iω Δ h 1 ′ + 2 r 4 L r 2 − 4 M r + 4 Q 2 h 0 − r 3 iω Δ h 1 ] , (33) δ G r ϕ = W [ i ω r 2 2 Δ h 0 ′ − i ω r Δ h 0 + N ( r ) 2 r 4 Δ h 1 ] . (34) \delta G_{r\phi}=W\left[\frac{i\omega r^2}{2\Delta}h_0'-\frac{i\omega r}{\Delta}h_0+\frac{\mathcal{N}(r)}{2r^4\Delta}h_1\right]. \tag*{(34)} δ G r ϕ = W [ 2Δ iω r 2 h 0 ′ − Δ iω r h 0 + 2 r 4 Δ N ( r ) h 1 ] . (34) with
N ( r ) = L r 2 Δ + 4 M r ( r 2 − Q 2 ) − 2 ( r 2 − Q 2 ) ( r 2 + Q 2 ) − ω 2 r 6 . (35) \mathcal{N}(r)=Lr^2\Delta+4Mr(r^2-Q^2)-2(r^2-Q^2)(r^2+Q^2)-\omega^2r^6. \tag*{(35)} N ( r ) = L r 2 Δ + 4 M r ( r 2 − Q 2 ) − 2 ( r 2 − Q 2 ) ( r 2 + Q 2 ) − ω 2 r 6 . (35) The field equation (3) turns these into sourced equations, δ G μ ϕ = − 2 κ α C μ ϕ + κ δ T μ ϕ E M \delta G_{\mu\phi}=-2\kappa\alpha\,C_{\mu\phi}+\kappa\,\delta T^{\mathrm{EM}}_{\mu\phi} δ G μ ϕ = − 2 κ α C μ ϕ + κ δ T μ ϕ EM 2 κ α = 16 π 2\kappa\alpha=16\pi 2 κ α = 16 π α = 1 \alpha=1 α = 1 R = 0 R=0 R = 0 δ R = 0 \delta R=0 δ R = 0 T ϑ T^{\vartheta} T ϑ O ( ϵ 2 ) O(\epsilon^2) O ( ϵ 2 ) C Q C_Q C Q
Angular projection: diagonal coupling
A key point fixes the whole calculation. The contravariant Cotton components of Sec. V are proportional to S ϕ = ∂ θ Y l / sin θ S^\phi=\partial_\theta Y_l/\sin\theta S ϕ = ∂ θ Y l / sin θ C μ ϕ = g μ μ g ϕ ϕ C μ ϕ ∝ W C_{\mu\phi}=g_{\mu\mu}g_{\phi\phi}C^{\mu\phi}\propto W C μ ϕ = g μμ g ϕϕ C μ ϕ ∝ W (33) –(34) . Because the background is spherically symmetric, the map δ ϑ ↦ C μ ν [ δ ϑ ] \delta\vartheta\mapsto C_{\mu\nu}[\delta\vartheta] δ ϑ ↦ C μν [ δ ϑ ] S O ( 3 ) SO(3) S O ( 3 ) W W W ∫ S ˉ A S A d Ω = l ( l + 1 ) \int\bar S^A S_A\,d\Omega=l(l+1) ∫ S ˉ A S A d Ω = l ( l + 1 ) l l l l ( l + 1 ) l(l+1) l ( l + 1 ) l → l ± 2 l\to l\pm2 l → l ± 2 W W W
− 16 π C t ϕ ∣ W = − 48 π f r 5 [ r ( M r − Q 2 ) R l ′ − ( M r − 2 Q 2 ) R l ] , (36) -16\pi C_{t\phi}\big|_W=-\frac{48\pi f}{r^5}\left[r(Mr-Q^2)R_l'-(Mr-2Q^2)R_l\right], \tag*{(36)} − 16 π C tϕ W = − r 5 48 π f [ r ( M r − Q 2 ) R l ′ − ( M r − 2 Q 2 ) R l ] , (36) − 16 π C r ϕ ∣ W = 48 π i ω ( M r − Q 2 ) f r 4 R l . (37) -16\pi C_{r\phi}\big|_W=\frac{48\pi i\omega(Mr-Q^2)}{fr^4}R_l. \tag*{(37)} − 16 π C r ϕ W = f r 4 48 π iω ( M r − Q 2 ) R l . (37) Elimination, master variable, and calibration
Equation (34) is first order in h 0 h_0 h 0 (33) renders h 0 h_0 h 0 h 1 h_1 h 1
Ψ R W ≡ Δ r 3 h 1 , (38) \Psi_{\mathrm{RW}}\equiv\frac{\Delta}{r^3}h_1, \tag*{(38)} Ψ RW ≡ r 3 Δ h 1 , (38) this equation takes the form of a Schrödinger type master equation. In the neutral limit the reduction was verified to reproduce the Schwarzschild Regge–Wheeler potential f [ l ( l + 1 ) / r 2 − 6 M / r 3 ] f[l(l+1)/r^2-6M/r^3] f [ l ( l + 1 ) / r 2 − 6 M / r 3 ] V R W = ( Δ / r 5 ) [ l ( l + 1 ) r − 6 M + 4 Q 2 / r ] V_{\mathrm{RW}}=(\Delta/r^5)[l(l+1)r-6M+4Q^2/r] V RW = ( Δ/ r 5 ) [ l ( l + 1 ) r − 6 M + 4 Q 2 / r ] C Q C_Q C Q Ψ e \Psi_e Ψ e
Following the treatment of the Schwarzschild case, we adopt the axial master function Ψ ≡ i Ψ R W / ω \Psi\equiv i\Psi_{\mathrm{RW}}/\omega Ψ ≡ i Ψ RW / ω i / ω i/\omega i / ω ω \omega ω (37) , converting what would otherwise be a first-derivative coupling into a purely algebraic one. The master equation becomes
[ d 2 d r ∗ 2 + ω 2 − V R W ] Ψ = C Q Ψ e + C d C S Θ , (39) \left[\frac{d^2}{dr_*^2}+\omega^2-V_{\mathrm{RW}}\right]\Psi=C_Q\Psi_e+C_{\mathrm{dCS}}\Theta, \tag*{(39)} [ d r ∗ 2 d 2 + ω 2 − V RW ] Ψ = C Q Ψ e + C dCS Θ , (39) where the scalar radial function R l R_l R l (23) enters through Θ = r R l \Theta=rR_l Θ = r R l Θ \Theta Θ [ d 2 / d r ∗ 2 + ω 2 − V Θ ] Θ = 0 [d^2/dr_*^2+\omega^2-V_\Theta]\Theta=0 [ d 2 / d r ∗ 2 + ω 2 − V Θ ] Θ = 0 V Θ = ( Δ / r 5 ) [ l ( l + 1 ) r + 2 M − 2 Q 2 / r ] V_\Theta=(\Delta/r^5)[l(l+1)r+2M-2Q^2/r] V Θ = ( Δ/ r 5 ) [ l ( l + 1 ) r + 2 M − 2 Q 2 / r ] Θ ′ ′ \Theta'' Θ ′′
The coupling coefficient
Carrying out the reduction, the first-derivative ( Θ ′ ) (\Theta') ( Θ ′ ) Q Q Q
C d C S ( r ) = 96 π α ( M r − Q 2 ) Δ r 8 = 96 π α ( M r − Q 2 ) f r 6 . (40) C_{\mathrm{dCS}}(r)=\frac{96\pi\alpha(Mr-Q^2)\Delta}{r^8}=\frac{96\pi\alpha(Mr-Q^2)f}{r^6}. \tag*{(40)} C dCS ( r ) = r 8 96 π α ( M r − Q 2 ) Δ = r 6 96 π α ( M r − Q 2 ) f . (40) The coupling is built from the same combination ( M r − Q 2 ) (Mr-Q^2) ( M r − Q 2 ) Δ \Delta Δ l l l (40) .
Multipole structure. As in the neutral case, the angular momentum factors ( l − 1 ) l ( l + 1 ) ( l + 2 ) \sqrt{(l-1)l(l+1)(l+2)} ( l − 1 ) l ( l + 1 ) ( l + 2 ) l = 1 l=1 l = 1
The Schwarzschild comparison can be made complete. Setting Q → 0 Q \to0 Q → 0 (58) reduces to
[ d 2 d r ∗ 2 + ω 2 − f ( l ( l + 1 ) r 2 − 6 M r 3 ) ] Ψ = 96 π M f r 5 α Θ (41) \left[\frac{d^2}{dr_*^2}+\omega^2-f\left(\frac{l(l+1)}{r^2}-\frac{6M}{r^3}\right)\right]\Psi=\frac{96\pi M f}{r^5}\alpha\Theta \tag*{(41)} [ d r ∗ 2 d 2 + ω 2 − f ( r 2 l ( l + 1 ) − r 3 6 M ) ] Ψ = r 5 96 π M f α Θ (41) [ d 2 d r ∗ 2 + ω 2 − f ( l ( l + 1 ) r 2 + 2 M r 3 ) ] Θ = 6 M α β ( l + 2 ) ! ( l − 2 ) ! f r 5 Ψ (42) \left[\frac{d^2}{dr_*^2}+\omega^2-f\left(\frac{l(l+1)}{r^2}+\frac{2M}{r^3}\right)\right]\Theta=\frac{6M\alpha}{\beta}\frac{(l+2)!}{(l-2)!}\frac{f}{r^5}\Psi \tag*{(42)} [ d r ∗ 2 d 2 + ω 2 − f ( r 2 l ( l + 1 ) + r 3 2 M ) ] Θ = β 6 M α ( l − 2 )! ( l + 2 )! r 5 f Ψ (42) which coincides term by term with Eqs. (2.8)–(2.9) of Ref. [22 ] once the O ( α 2 / β ) \mathcal{O}(\alpha^2/\beta) O ( α 2 / β ) δ V Θ \delta V_\Theta δ V Θ 96 π α M f / r 5 96\pi\alpha M f/r^5 96 π α M f / r 5 Ψ = i Ψ R W / ω \Psi=i\Psi_{\mathrm{RW}}/\omega Ψ = i Ψ RW / ω ( M r − Q 2 ) Δ / r 8 (Mr-Q^2)\Delta/r^8 ( M r − Q 2 ) Δ/ r 8
The same elimination also feeds the pseudoscalar into the Maxwell equation, because the master equation for Ψ e \Psi_e Ψ e h 0 h_0 h 0 C Q C_Q C Q { Ψ , Ψ e , Θ } \{\Psi,\Psi_e,\Theta\} { Ψ , Ψ e , Θ }
The scalar-sector source, the reciprocal coupling, and the electromagnetic channel
Consistency of the coupled system requires the reciprocal channel, in which the axial perturbation sources the pseudoscalar through
□ ϑ = − α 4 β ∗ R R (43) \Box\vartheta=-\frac{\alpha}{4\beta}\,{}^\ast RR \tag*{(43)} □ ϑ = − 4 β α ∗ R R (43) On the static RN background ∗ R R = 0 {}^\ast RR=0 ∗ R R = 0 δ ( ∗ R R ) \delta({}^\ast RR) δ ( ∗ R R ) h 0 , h 1 h_0,h_1 h 0 , h 1 Ψ e \Psi_e Ψ e Ψ \Psi Ψ
The linearized Pontryagin density
For the Regge–Wheeler-gauge perturbation (12) , a direct computation gives
δ ( ∗ R R ) = − 24 L ( M r − Q 2 ) r 7 [ r h 0 ′ − 2 h 0 + i ω r h 1 ] Y l (44) \delta({}^\ast RR)=-\frac{24L(Mr-Q^2)}{r^7}\left[rh'_0-2h_0+i\omega r h_1\right]Y_l \tag*{(44)} δ ( ∗ R R ) = − r 7 24 L ( M r − Q 2 ) [ r h 0 ′ − 2 h 0 + iω r h 1 ] Y l (44) exactly, at all Q Q Q ∂ θ Y l \partial_\theta Y_l ∂ θ Y l ( M r − Q 2 ) (Mr-Q^2) ( M r − Q 2 )
∗ R μ ν ρ σ R μ ν ρ σ ′ = ∗ C μ ν ρ σ C μ ν ρ σ ′ (45) {}^\ast R^{\mu\nu\rho\sigma}R'_{\mu\nu\rho\sigma}={}^\ast C^{\mu\nu\rho\sigma}C'_{\mu\nu\rho\sigma} \tag*{(45)} ∗ R μν ρ σ R μν ρ σ ′ = ∗ C μν ρ σ C μν ρ σ ′ (45) because in four dimensions every term containing the Ricci tensor or scalar vanishes in that parity-odd contraction by the symmetries of ε \varepsilon ε [11 ] . The RN background is of Petrov type D, with Ψ 2 = − ( M r − Q 2 ) / r 4 \Psi_2=-(Mr-Q^2)/r^4 Ψ 2 = − ( M r − Q 2 ) / r 4 δ ( ∗ R R ) ∝ Ψ 2 Im δ Ψ 2 \delta({}^\ast RR)\propto\Psi_2\operatorname{Im}\delta\Psi_2 δ ( ∗ R R ) ∝ Ψ 2 Im δ Ψ 2 Im δ Ψ 2 \operatorname{Im}\delta\Psi_2 Im δ Ψ 2
Reduction to master variables
At the order required, h 0 h_0 h 0 h 1 h_1 h 1 r ϕ r\phi r ϕ
h 0 ′ − 2 h 0 r + i ω h 1 = L − 2 r [ Ψ + χ Ψ e ] , χ ( r ) ≡ 2 Q r μ (46) h'_0-\frac{2h_0}{r}+i\omega h_1=\frac{L-2}{r}\left[\Psi+\chi\Psi_e\right],\qquad\chi(r)\equiv\frac{2Q}{r\mu} \tag*{(46)} h 0 ′ − r 2 h 0 + iω h 1 = r L − 2 [ Ψ + χ Ψ e ] , χ ( r ) ≡ r μ 2 Q (46) with Ψ = i Ψ R W / ω \Psi=i\Psi_{\mathrm{RW}}/\omega Ψ = i Ψ RW / ω Ψ e \Psi_e Ψ e (15) –(16) . The Ψ e \Psi_e Ψ e δ T r ϕ E M ∝ Q ω δ A ϕ \delta T^{\mathrm{EM}}_{r\phi}\propto Q\omega\delta A_\phi δ T r ϕ EM ∝ Q ω δ A ϕ (44) ,
δ ( ∗ R R ) = − 24 ( l − 1 ) l ( l + 1 ) ( l + 2 ) ( M r − Q 2 ) r 7 [ Ψ + χ Ψ e ] Y l (47) \delta({}^\ast RR)=-\frac{24(l-1)l(l+1)(l+2)(Mr-Q^2)}{r^7}\left[\Psi+\chi\Psi_e\right]Y_l \tag*{(47)} δ ( ∗ R R ) = − r 7 24 ( l − 1 ) l ( l + 1 ) ( l + 2 ) ( M r − Q 2 ) [ Ψ + χ Ψ e ] Y l (47) The tensor weight ( l − 1 ) l ( l + 1 ) ( l + 2 ) = L ( L − 2 ) (l-1)l(l+1)(l+2)=L(L-2) ( l − 1 ) l ( l + 1 ) ( l + 2 ) = L ( L − 2 ) L L L L − 2 = μ 2 L-2=\mu^2 L − 2 = μ 2 δ ϑ = ( Θ / r ) Y l e − i ω t \delta\vartheta=(\Theta/r)Y_l e^{-i\omega t} δ ϑ = ( Θ/ r ) Y l e − iω t (43) becomes
[ d 2 d r ∗ 2 + ω 2 − V Θ ] Θ = C Θ g Ψ + C Θ e Ψ e (48) \left[\frac{d^2}{dr_*^2}+\omega^2-V_\Theta\right]\Theta=C_{\Theta g}\Psi+C_{\Theta e}\Psi_e \tag*{(48)} [ d r ∗ 2 d 2 + ω 2 − V Θ ] Θ = C Θ g Ψ + C Θ e Ψ e (48) with
C Θ g = 6 α ( M r − Q 2 ) ( l − 1 ) l ( l + 1 ) ( l + 2 ) Δ β r 8 (49) C_{\Theta g}=\frac{6\alpha(Mr-Q^2)(l-1)l(l+1)(l+2)\Delta}{\beta r^8} \tag*{(49)} C Θ g = β r 8 6 α ( M r − Q 2 ) ( l − 1 ) l ( l + 1 ) ( l + 2 ) Δ (49) C Θ e = χ C Θ g . C_{\Theta e}=\chi C_{\Theta g}. C Θ e = χ C Θ g . The electromagnetic channel on the gravity side
The Maxwell equation is unmodified, but it involves h 0 h_0 h 0 h 0 h_0 h 0 sourced Einstein equations carries the C C C
[ d 2 d r ∗ 2 + ω 2 − V E M ] Ψ e = C Q Ψ + C e Θ Θ , C e Θ = χ C d C S . (50) \left[\frac{d^2}{dr_*^2}+\omega^2-V_{\mathrm{EM}}\right]\Psi_e=C_Q\Psi+C_{e\Theta}\Theta,\qquad C_{e\Theta}=\chi C_{\mathrm{dCS}}. \tag*{(50)} [ d r ∗ 2 d 2 + ω 2 − V EM ] Ψ e = C Q Ψ + C e Θ Θ , C e Θ = χ C dCS . (50) Thus the pseudoscalar couples to the single combination Ψ + χ Ψ e \Psi+\chi\Psi_e Ψ + χ Ψ e both directions, and
C Θ g C d C S = C e Θ C e Θ = ( l − 1 ) l ( l + 1 ) ( l + 2 ) 16 π β . (51) \frac{C_{\Theta g}}{C_{\mathrm{dCS}}}=\frac{C_{e\Theta}}{C_{e\Theta}}=\frac{(l-1)l(l+1)(l+2)}{16\pi\beta}. \tag*{(51)} C dCS C Θ g = C e Θ C e Θ = 16 π β ( l − 1 ) l ( l + 1 ) ( l + 2 ) . (51) This is the reciprocity required by the single bilinear interaction α 4 ∫ − g δ ϑ δ ( ∗ R R ) \frac{\alpha}{4}\int\sqrt{-g}\,\delta\vartheta\,\delta({}^{*}RR) 4 α ∫ − g δ ϑ δ ( ∗ R R )
Three checks support these results. First, the unused t ϕ t\phi tϕ C C C Q → 0 Q\to0 Q → 0 [22 ] : χ = 0 \chi=0 χ = 0 C Θ g → ( 6 M α / β ) [ ( l + 2 ) ! / ( l − 2 ) ! ] f / r 5 C_{\Theta g}\to(6M\alpha/\beta)[(l+2)!/(l-2)!]f/r^5 C Θ g → ( 6 M α / β ) [( l + 2 )! / ( l − 2 )!] f / r 5 C Θ g C_{\Theta g} C Θ g l = 1 l=1 l = 1 χ ∝ Q / ( r μ ) \chi\propto Q/(r\mu) χ ∝ Q / ( r μ ) 1 / l 1/l 1/ l
The coupled axial system
Collecting the gravitational (Sec. VI), electromagnetic (Sec. IV) and scalar (§ 1.5 ) sectors, the axial perturbations of the charged black hole in dCS gravity obey the three coupled Schrödinger-type equations
[ d 2 d r ∗ 2 + ω 2 − V R W ] Ψ = C Q Ψ e + C d C S Θ , (52) \left[\frac{d^2}{dr_*^2}+\omega^2-V_{\mathrm{RW}}\right]\Psi=C_Q\Psi_e+C_{\mathrm{dCS}}\Theta, \tag*{(52)} [ d r ∗ 2 d 2 + ω 2 − V RW ] Ψ = C Q Ψ e + C dCS Θ , (52) [ d 2 d r ∗ 2 + ω 2 − V E M ] Ψ e = C Q Ψ + χ C d C S Θ , (53) \left[\frac{d^2}{dr_*^2}+\omega^2-V_{\mathrm{EM}}\right]\Psi_e=C_Q\Psi+\chi C_{\mathrm{dCS}}\Theta, \tag*{(53)} [ d r ∗ 2 d 2 + ω 2 − V EM ] Ψ e = C Q Ψ + χ C dCS Θ , (53) [ d 2 d r ∗ 2 + ω 2 − V Θ ] Θ = C Θ g ( Ψ + χ Ψ e ) , (54) \left[\frac{d^2}{dr_*^2}+\omega^2-V_\Theta\right]\Theta=C_{\Theta g}(\Psi+\chi\Psi_e), \tag*{(54)} [ d r ∗ 2 d 2 + ω 2 − V Θ ] Θ = C Θ g ( Ψ + χ Ψ e ) , (54) where Ψ = i Ψ R W / ω \Psi=i\Psi_{\mathrm{RW}}/\omega Ψ = i Ψ RW / ω Ψ e \Psi_e Ψ e Θ \Theta Θ χ = 2 Q / ( r μ ) \chi=2Q/(r\mu) χ = 2 Q / ( r μ )
V R W = Δ r 5 [ l ( l + 1 ) r − 6 M + 4 Q 2 r ] , (55) V_{\mathrm{RW}}=\frac{\Delta}{r^5}\left[l(l+1)r-6M+\frac{4Q^2}{r}\right], \tag*{(55)} V RW = r 5 Δ [ l ( l + 1 ) r − 6 M + r 4 Q 2 ] , (55) V E M = Δ r 5 [ l ( l + 1 ) r + 4 Q 2 r ] , (56) V_{\mathrm{EM}}=\frac{\Delta}{r^5}\left[l(l+1)r+\frac{4Q^2}{r}\right], \tag*{(56)} V EM = r 5 Δ [ l ( l + 1 ) r + r 4 Q 2 ] , (56) V Θ = Δ r 5 [ l ( l + 1 ) r + 2 M − 2 Q 2 r ] + δ V Θ , (57) V_\Theta=\frac{\Delta}{r^5}\left[l(l+1)r+2M-\frac{2Q^2}{r}\right]+\delta V_\Theta, \tag*{(57)} V Θ = r 5 Δ [ l ( l + 1 ) r + 2 M − r 2 Q 2 ] + δ V Θ , (57) and the couplings, in units α = 1 \alpha=1 α = 1 [22 ] , α \alpha α β → α 2 β \beta\to\alpha^2\beta β → α 2 β Θ → Θ / α \Theta\to\Theta/\alpha Θ → Θ/ α ζ \zeta ζ C d C S ∝ α C_{\mathrm{dCS}}\propto\alpha C dCS ∝ α C Θ g ∝ α / β C_{\Theta g}\propto\alpha/\beta C Θ g ∝ α / β
C Q = 2 Q μ Δ r 5 , C_Q=\frac{2Q\mu\Delta}{r^5}, C Q = r 5 2 Q μ Δ , C d C S = 96 π ( M r − Q 2 ) Δ r 8 , C_{\mathrm{dCS}}=\frac{96\pi(Mr-Q^2)\Delta}{r^8}, C dCS = r 8 96 π ( M r − Q 2 ) Δ , C Θ g = 6 ( M r − Q 2 ) ( l − 1 ) l ( l + 1 ) ( l + 2 ) Δ β r 8 . (58) C_{\Theta g}=\frac{6(Mr-Q^2)(l-1)l(l+1)(l+2)\Delta}{\beta r^8}. \tag*{(58)} C Θ g = β r 8 6 ( M r − Q 2 ) ( l − 1 ) l ( l + 1 ) ( l + 2 ) Δ . (58) The term δ V Θ = O ( α 2 / β ) \delta V_\Theta=O(\alpha^2/\beta) δ V Θ = O ( α 2 / β ) δ V Θ = 576 π l ( l + 1 ) α 2 ( M r − Q 2 ) Δ / ( β r 12 ) \delta V_\Theta=576\pi l(l+1)\alpha^2(Mr-Q^2)\Delta/(\beta r^{12}) δ V Θ = 576 π l ( l + 1 ) α 2 ( M r − Q 2 ) Δ/ ( β r 12 ) [22 ] at Q = 0 Q=0 Q = 0 O ( ζ 2 ) O(\zeta^2) O ( ζ 2 )
Branch structure
In the branch basis ((22) ) the general-relativistic block is diagonal. The pseudoscalar couples to Z g ≡ Z 2 ( − ) Z_g\equiv Z_2^{(-)} Z g ≡ Z 2 ( − ) Z e ≡ Z 1 ( − ) Z_e\equiv Z_1^{(-)} Z e ≡ Z 1 ( − )
Ψ + χ Ψ e = w g ( r ) Z g + w e ( r ) Z e , \Psi+\chi\Psi_e=w_g(r)Z_g+w_e(r)Z_e, Ψ + χ Ψ e = w g ( r ) Z g + w e ( r ) Z e , w g = cos ϕ − χ sin ϕ , w e = sin ϕ + χ cos ϕ . (59) w_g=\cos\phi-\chi\sin\phi,\qquad w_e=\sin\phi+\chi\cos\phi. \tag*{(59)} w g = cos ϕ − χ sin ϕ , w e = sin ϕ + χ cos ϕ . (59) Away from degeneracies, and after this branch projection, the two branches communicate only through Θ \Theta Θ a ∈ { g , e } a\in\{g,e\} a ∈ { g , e } O ( ζ ) O(\zeta) O ( ζ )
[ d 2 d r ∗ 2 + ω 2 − V a ] Z a = w a C d C S Θ , \left[\frac{d^2}{dr_*^2}+\omega^2-V_a\right]Z_a=w_a C_{\mathrm{dCS}}\Theta, [ d r ∗ 2 d 2 + ω 2 − V a ] Z a = w a C dCS Θ , [ d 2 d r ∗ 2 + ω 2 − V Θ ] Θ = w a C Θ g Z a . (60) \left[\frac{d^2}{dr_*^2}+\omega^2-V_\Theta\right]\Theta=w_a C_{\Theta g}Z_a. \tag*{(60)} [ d r ∗ 2 d 2 + ω 2 − V Θ ] Θ = w a C Θ g Z a . (60) with V g V_g V g V e V_e V e (18) carrying q 1 q_1 q 1 q 2 q_2 q 2 (60) reproduces the O ( ζ ) O(\zeta) O ( ζ ) 10 − 7 10^{-7} 1 0 − 7 w a ( r ) w_a(r) w a ( r )
Δ ω i s o ( a ) = ω a a x i a l ( α , Q ) − ω a p o l a r ( 0 , Q ) = ω a ( α , Q ) − ω a ( 0 , Q ) , (61) \Delta\omega_{\mathrm{iso}}^{(a)}=\omega_a^{\mathrm{axial}}(\alpha,Q)-\omega_a^{\mathrm{polar}}(0,Q)=\omega_a(\alpha,Q)-\omega_a(0,Q), \tag*{(61)} Δ ω iso ( a ) = ω a axial ( α , Q ) − ω a polar ( 0 , Q ) = ω a ( α , Q ) − ω a ( 0 , Q ) , (61) where each branch is continued from its general-relativistic (α → 0 \alpha\to0 α → 0 and the electromagnetic-led branches are shifted.
Eikonal limit: the isospectrality-breaking shift
Integrating out Θ \Theta Θ (60) in the local approximation, appropriate at large l l l
V ± ( a ) = V a + V Θ 2 ± D a 2 + ω a 2 P , V_{\pm}^{(a)}=\frac{V_a+V_{\Theta}}{2}\pm\sqrt{D_a^2+\omega_a^2P}, V ± ( a ) = 2 V a + V Θ ± D a 2 + ω a 2 P , D a = V a − V Θ 2 = − Δ 2 r 5 [ 2 M − 6 Q 2 r + q ( a ) ] , D_a=\frac{V_a-V_{\Theta}}{2}=-\frac{\Delta}{2r^5}\left[2M-\frac{6Q^2}{r}+q_{(a)}\right], D a = 2 V a − V Θ = − 2 r 5 Δ [ 2 M − r 6 Q 2 + q ( a ) ] , P ≡ C d C S C Θ g = 576 π ( l − 1 ) l ( l + 1 ) ( l + 2 ) ( M r − Q 2 ) 2 Δ 2 β r 16 , (62) P\equiv C_{dCS}C_{\Theta g}=\frac{576\pi(l-1)l(l+1)(l+2)(Mr-Q^2)^2\Delta^2}{\beta r^{16}}, \tag*{(62)} P ≡ C d C S C Θ g = β r 16 576 π ( l − 1 ) l ( l + 1 ) ( l + 2 ) ( M r − Q 2 ) 2 Δ 2 , (62) with q ( g ) = q 1 q_{(g)}=q_1 q ( g ) = q 1 q ( e ) = q 2 q_{(e)}=q_2 q ( e ) = q 2 Q = 0 Q=0 Q = 0 D g = − 4 M Δ / r 5 D_g=-4M\Delta/r^5 D g = − 4 M Δ/ r 5 (62) –(69) provide a local, eikonal interpretation of the branch shifts. The finite-l l l
Off-resonance regime
For ω a 2 P ≪ D a 2 \omega_a^2P\ll D_a^2 ω a 2 P ≪ D a 2
δ ( ω a 2 ) ≃ ω a 2 P 2 D a ∣ r p e a k , δ ω a = δ ( ω a 2 ) 2 ω a , (63) \delta(\omega_a^2)\simeq\left.\frac{\omega_a^2P}{2D_a}\right|_{r_{\mathrm{peak}}},\qquad\delta\omega_a=\frac{\delta(\omega_a^2)}{2\omega_a}, \tag*{(63)} δ ( ω a 2 ) ≃ 2 D a ω a 2 P r peak , δ ω a = 2 ω a δ ( ω a 2 ) , (63) which is O ( ζ ) O(\zeta) O ( ζ ) ≃ 11 % \simeq11\% ≃ 11% l = 2 l=2 l = 2 ≤ 2 % \leq2\% ≤ 2% l = 4 l=4 l = 4 ≤ 1 % \leq1\% ≤ 1% l = 12 l=12 l = 12 Q / M = 0 Q/M=0 Q / M = 0 0.3 0.3 0.3 0.6 0.6 0.6
a . a. a . Neutral hole. At Q = 0 Q=0 Q = 0 D g D_g D g l l l P ∝ l 4 P\propto l^4 P ∝ l 4 r p h = 3 M r_{\mathrm{ph}}=3M r ph = 3 M ω g ≃ l / ( 3 3 M ) \omega_g\simeq l/(3\sqrt{3}M) ω g ≃ l / ( 3 3 M )
M δ ω g ∣ Q = 0 ≃ − 32 3 π 2 243 ζ l 3 ≈ − 2.25 ζ l 3 . (64) M\left.\delta\omega_g\right|_{Q=0}\simeq-\frac{32\sqrt{3}\pi^2}{243}\zeta l^3\approx-2.25\,\zeta l^3. \tag*{(64)} M δ ω g ∣ Q = 0 ≃ − 243 32 3 π 2 ζ l 3 ≈ − 2.25 ζ l 3 . (64) b . b. b . Charged hole. For Q > 0 Q>0 Q > 0 l l l Q μ ≳ 2 3 M Q\mu\gtrsim\frac{2}{3}M Q μ ≳ 3 2 M V Θ − V g ≃ 2 Q μ Δ / r 5 V_{\Theta}-V_g\simeq2Q\mu\Delta/r^5 V Θ − V g ≃ 2 Q μ Δ/ r 5 ω g 2 → 1 2 \omega_g^2\to\frac{1}{2} ω g 2 → 2 1
δ ω g ≃ − 72 π l 2 β Q K 2 r 6 f Δ ∣ r p h ( Q ) ∝ l 2 Q , K ≡ ( M r − Q 2 ) Δ r 8 , (65) \delta\omega_g\simeq-\left.\frac{72\pi l^2}{\beta Q}\frac{K^2r^6}{\sqrt{f}\Delta}\right|_{r_{\mathrm{ph}}(Q)}\propto\frac{l^2}{Q},\qquad K\equiv\frac{(Mr-Q^2)\Delta}{r^8}, \tag*{(65)} δ ω g ≃ − β Q 72 π l 2 f Δ K 2 r 6 r ph ( Q ) ∝ Q l 2 , K ≡ r 8 ( M r − Q 2 ) Δ , (65) in units α = 1 \alpha=1 α = 1 r p h r_{\mathrm{ph}} r ph
r p h = 3 M + 9 M 2 − 8 Q 2 2 . (66) r_{\mathrm{ph}}=\frac{3M+\sqrt{9M^2-8Q^2}}{2}. \tag*{(66)} r ph = 2 3 M + 9 M 2 − 8 Q 2 . (66) The limits Q → 0 Q\to0 Q → 0 l → ∞ l\to\infty l → ∞ l = 11 l=11 l = 11 12 12 12 d ln ∣ Re δ ω g ∣ / d ln l \mathrm{d}\ln|\operatorname{Re}\delta\omega_g|/\mathrm{d}\ln l d ln ∣ Re δ ω g ∣/ d ln l 2.91 2.91 2.91 Q = 0 Q=0 Q = 0 2.20 2.20 2.20 Q / M = 0.3 Q/M=0.3 Q / M = 0.3 2.10 2.10 2.10 Q / M = 0.6 Q/M=0.6 Q / M = 0.6 l ≥ 3 l\geq3 l ≥ 3
Extending the reduced spectral calculation to l = 30 l=30 l = 30 Figure 2 ). The ratio of Re δ ω g \operatorname{Re}\delta\omega_g Re δ ω g (64) behaves as 1 + 1.24 / l 1+1.24/l 1 + 1.24/ l Q = 0 Q=0 Q = 0 (65) behaves as 1 − 0.6 M / ( Q l ) 1-0.6M/(Ql) 1 − 0.6 M / ( Ql ) Q = 0.3 M Q=0.3M Q = 0.3 M 0.6 M 0.6M 0.6 M 1 / l 1/l 1/ l l = 12 l=12 l = 12 30 30 30 1.004 1.004 1.004 0.987 0.987 0.987 0.999 0.999 0.999 l 2 / Q l^2/Q l 2 / Q Q l ≫ M Ql\gg M Ql ≫ M l → ∞ l\to\infty l → ∞ Q > 0 Q>0 Q > 0 Q → 0 Q\to0 Q → 0 l l l (64) is recovered only for Q l ≪ M Ql\ll M Ql ≪ M
Figure 2. Ratio of the spectral gravitational-led coefficient Re δ ω g \operatorname{Re}\delta\omega_g Re δ ω g M δ ω ≃ − 2.25 ζ l 3 M\delta\omega\simeq-2.25\,\zeta l^3 M δ ω ≃ − 2.25 ζ l 3 Q = 0 Q=0 Q = 0 (64) ] and the l 2 / Q l^2/Q l 2 / Q (65) at Q = 0.3 M , 0.6 M Q=0.3M,0.6M Q = 0.3 M , 0.6 M l = 2 l=2 l = 2 30 30 30 1 / l 1/l 1/ l l ≥ 12 l\geq12 l ≥ 12 1.004 1.004 1.004 0.987 0.987 0.987 0.999 0.999 0.999
Strong-mixing regime and the validity of the effective theory
The off-resonance regime requires ω a 2 P ≪ D a 2 \omega_a^2P\ll D_a^2 ω a 2 P ≪ D a 2 ζ l 4 ≪ 1 \zeta l^4\ll1 ζ l 4 ≪ 1 Q = 0 Q=0 Q = 0 ζ l 2 ≪ ( Q / M ) 2 \zeta l^2\ll(Q/M)^2 ζ l 2 ≪ ( Q / M ) 2 Q > 0 Q>0 Q > 0 Q l ≫ M Ql\gg M Ql ≫ M δ ( ω a 2 ) ≃ − ∣ ω a ∣ P \delta(\omega_a^2)\simeq-|\omega_a|\sqrt{P} δ ( ω a 2 ) ≃ − ∣ ω a ∣ P δ ω a ∝ l ζ \delta\omega_a\propto l\sqrt{\zeta} δ ω a ∝ l ζ O ( ζ ) O(\sqrt{\zeta}) O ( ζ ) l l l Q = 0 Q=0 Q = 0 α = 1 \alpha=1 α = 1
δ ω g ∣ Q = 0 ≃ − 4 3 π 81 l M 3 β . (67) \left.\delta\omega_g\right|_{Q=0}\simeq-\frac{4\sqrt{3\pi}}{81}\frac{l}{M^3\sqrt{\beta}}. \tag*{(67)} δ ω g ∣ Q = 0 ≃ − 81 4 3 π M 3 β l . (67) At fixed small ζ \zeta ζ ζ → 0 \zeta\to0 ζ → 0 l l l l l l
The electromagnetic-led branch: resonance and sign reversal
For the electromagnetic-led branch, q 2 = 3 M − s q_2=3M-s q 2 = 3 M − s Q Q Q D e D_e D e D e ( r p h ) = 0 D_e(r_{\mathrm{ph}})=0 D e ( r ph ) = 0
2 M − 6 Q 2 r p h + 3 M − 9 M 2 + 4 Q 2 μ 2 = 0 , (68) 2M-\frac{6Q^2}{r_{\mathrm{ph}}}+3M-\sqrt{9M^2+4Q^2\mu^2}=0, \tag*{(68)} 2 M − r ph 6 Q 2 + 3 M − 9 M 2 + 4 Q 2 μ 2 = 0 , (68) locates it to within 0.5 % 0.5\% 0.5% l = 2 l=2 l = 2
Q c = 2 M L + 1 L + 2 → l ≫ 1 2 M l , (69) Q_c=\frac{2M\sqrt{L+1}}{L+2}\xrightarrow[l\gg1]{}\frac{2M}{l}, \tag*{(69)} Q c = L + 2 2 M L + 1 l ≫ 1 l 2 M , (69) one has s = M ( 5 L + 2 ) / ( L + 2 ) s=M(5L+2)/(L+2) s = M ( 5 L + 2 ) / ( L + 2 ) V Θ = V e ( + ) V_{\Theta}=V_e^{(+)} V Θ = V e ( + ) V e ( ± ) = ± q 2 d F / d r + q 2 2 F 2 + μ 2 ( μ 2 + 2 ) F V_e^{(\pm)}=\pm q_2\,\mathrm{d}F/\mathrm{d}r+q_2^2F^2+\mu^2(\mu^2+2)F V e ( ± ) = ± q 2 d F / d r + q 2 2 F 2 + μ 2 ( μ 2 + 2 ) F F = Δ / [ r 3 ( μ 2 r + q 2 ) ] F=\Delta/[r^3(\mu^2r+q_2)] F = Δ/ [ r 3 ( μ 2 r + q 2 )] [28 ] . By the Chandrasekhar isospectrality of V e ( ± ) V_e^{(\pm)} V e ( ± ) ω Θ = ω e \omega_{\Theta}=\omega_e ω Θ = ω e Q c Q_c Q c Q c / M = 0.66144 Q_c/M=0.66144 Q c / M = 0.66144 0.51508 0.51508 0.51508 0.15861 0.15861 0.15861 l = 2 , 3 , 12 l=2,3,12 l = 2 , 3 , 12 ∣ ω e − ω Θ ∣ < 10 − 10 |\omega_e-\omega_{\Theta}|<10^{-10} ∣ ω e − ω Θ ∣ < 1 0 − 10 Q c Q_c Q c V e < V Θ V_e<V_{\Theta} V e < V Θ Q c Q_c Q c Re δ ω e > 0 \operatorname{Re}\delta\omega_e>0 Re δ ω e > 0 Im δ ω e > 0 \operatorname{Im}\delta\omega_e>0 Im δ ω e > 0 longer-lived [Fig. 1(c)].
Near Q c Q_c Q c Δ 0 ≡ ω e − ω Θ ≃ ( Q − Q c ) Δ ′ \Delta_0\equiv\omega_e-\omega_{\Theta}\simeq(Q-Q_c)\Delta' Δ 0 ≡ ω e − ω Θ ≃ ( Q − Q c ) Δ ′ M 2 Δ ′ = 0.0961 − 0.0052 i M^2\Delta'=0.0961-0.0052i M 2 Δ ′ = 0.0961 − 0.0052 i l = l= l = 2 ) 2) 2 ) ω ± = ω ˉ ± 1 2 Δ 0 2 + ( 2 g ) 2 \omega_\pm=\bar{\omega}\pm\frac{1}{2}\sqrt{\Delta_0^2+(2g)^2} ω ± = ω ˉ ± 2 1 Δ 0 2 + ( 2 g ) 2 Q = Q c Q=Q_c Q = Q c M ∣ ω + − ω − ∣ = 0.4800 / β = 2.41 ζ M|\omega_+-\omega_-|=0.4800/\sqrt{\beta}=2.41\sqrt{\zeta} M ∣ ω + − ω − ∣ = 0.4800/ β = 2.41 ζ l = 2 l=2 l = 2 0.5762 / β = 2.89 ζ 0.5762/\sqrt{\beta}=2.89\sqrt{\zeta} 0.5762/ β = 2.89 ζ l = 3 l=3 l = 3 10 3 ≤ β ≤ 10 10 10^3\leq\beta\leq10^{10} 1 0 3 ≤ β ≤ 1 0 10 M = α = 1 M=\alpha=1 M = α = 1 Figure 3 ). Isospectrality breaking is therefore O ( ζ ) \mathcal{O}(\sqrt{\zeta}) O ( ζ ) Q c Q_c Q c O ( ζ ) \mathcal{O}(\zeta) O ( ζ ) (63) applies only for ∣ Q − Q c ∣ ≫ 2 g / ∣ Δ ′ ∣ ≃ 25 ζ M |Q-Q_c|\gg2g/|\Delta'|\simeq25\sqrt{\zeta}M ∣ Q − Q c ∣ ≫ 2 g /∣ Δ ′ ∣ ≃ 25 ζ M l = 2 l=2 l = 2 Figure 3 b).
Figure 3. Avoided crossing of the electromagnetic-led and pseudoscalar modes (l = 2 l=2 l = 2 M = α = 1 M=\alpha=1 M = α = 1 Re ω \operatorname{Re}\omega Re ω Q c Q_c Q c (69) , vertical line] for β = 10 4 , 10 5 , 10 6 \beta=10^4,10^5,10^6 β = 1 0 4 , 1 0 5 , 1 0 6 β → ∞ \beta\to\infty β → ∞ Q c Q_c Q c Θ \Theta Θ 16 π β / [ L ( L − 2 ) ] \sqrt{16\pi\beta/[L(L-2)]} 16 π β / [ L ( L − 2 )] ∝ β − 1 / 2 \propto\beta^{-1/2} ∝ β − 1/2 Q = Q c Q=Q_c Q = Q c l = 2 , 3 l=2,3 l = 2 , 3 M ∣ ω + − ω − ∣ ∝ β − 1 / 2 M|\omega_+-\omega_-|\propto\beta^{-1/2} M ∣ ω + − ω − ∣ ∝ β − 1/2 O ( ζ ) O(\sqrt{\zeta}) O ( ζ )
Low-frequency limit and the static sector
The opposite regime, ω → 0 \omega\to0 ω → 0 [44 ] , while isolating the dynamical origin of the present coupling.
The source is linear in frequency
The normalisation Ψ = i Ψ R W / ω \Psi=i\Psi_{\rm RW}/\omega Ψ = i Ψ RW / ω ω = 0 \omega=0 ω = 0 Ψ R W \Psi_{\rm RW} Ψ RW
[ d 2 d r ∗ 2 + ω 2 − V R W ] Ψ R W = S R W , \left[\frac{d^2}{dr_*^2}+\omega^2-V_{\rm RW}\right]\Psi_{\rm RW}=S_{\rm RW}, [ d r ∗ 2 d 2 + ω 2 − V RW ] Ψ RW = S RW , S R W = − i ω C d C S Θ = − i ω 96 π ( M r − Q 2 ) Δ r 8 Θ . (70) S_{\rm RW}=-i\omega C_{\rm dCS}\Theta=-i\omega\frac{96\pi(Mr-Q^2)\Delta}{r^8}\Theta. \tag*{(70)} S RW = − iω C dCS Θ = − iω r 8 96 π ( M r − Q 2 ) Δ Θ. (70) The source is exactly linear in ω \omega ω (50) , the Maxwell equation in physical variables also acquires a Θ \Theta Θ Ψ = i Ψ R W / ω \Psi=i\Psi_{\rm RW}/\omega Ψ = i Ψ RW / ω C d C S C_{\rm dCS} C dCS ω \omega ω ω \omega ω ω \omega ω C r ϕ ∝ i ω R ℓ C^{r\phi}\propto i\omega R_{\ell} C r ϕ ∝ iω R ℓ C t ϕ C^{t\phi} C tϕ C r ϕ C^{r\phi} C r ϕ Q Q Q ω \omega ω θ φ \theta\varphi θ φ h 0 = ( i f / ω ) ( f h 1 ) ′ h_0=(if/\omega)(fh_1)' h 0 = ( i f / ω ) ( f h 1 ) ′ h 1 = O ( ω h 0 ) h_1=O(\omega h_0) h 1 = O ( ω h 0 ) Ψ = i f h 1 / ( ω r ) \Psi=ifh_1/(\omega r) Ψ = i f h 1 / ( ω r ) ω → 0 \omega\to0 ω → 0 ω \omega ω (70) reflects the vanishing of Ψ R W \Psi_{\mathrm{RW}} Ψ RW
Static limit of the Regge–Wheeler channel
Two consequences follow from Eq. (70) :
Regge–Wheeler channel. S R W → 0 S_{\mathrm{RW}}\to0 S RW → 0 ω → 0 \omega\to0 ω → 0 [44 ] , now expressed as a frequency scaling. The statement does not extend to the static tidal response. The static axial sector is carried by h 0 h_0 h 0 ω \omega ω C t ϕ C_{t\phi} C tϕ δ ( ∗ R R ) \delta({}^*RR) δ ( ∗ R R ) h 0 h_0 h 0 (44) ]. Static axial (magnetic-type) Love numbers are therefore generically modified at O ( ζ ) O(\zeta) O ( ζ ) [48 ] ; we do not compute them here.
Order counting
Eliminating Θ \Theta Θ
( − i ω C d C S ) G Θ ( i ω C Θ g ) = C d C S G Θ C Θ g (-i\omega C_{d\mathrm{CS}})G_\Theta\left(\frac{i}{\omega}C_{\Theta g}\right)=C_{d\mathrm{CS}}G_\Theta C_{\Theta g} ( − iω C d CS ) G Θ ( ω i C Θ g ) = C d CS G Θ C Θ g with G Θ = ( □ + ω 2 − V Θ ) − 1 G_\Theta=(\Box+\omega^2-V_\Theta)^{-1} G Θ = ( □ + ω 2 − V Θ ) − 1 ω \omega ω O ( α ) O(\alpha) O ( α ) O ( ζ ) O(\zeta) O ( ζ ) ω ∼ M − 1 \omega\sim M^{-1} ω ∼ M − 1
Near-extremal limit
The opposite corner of parameter space is the near-extremal black hole, Q → M Q\to M Q → M r + → r − → M r_+\to r_-\to M r + → r − → M T H → 0 T_H\to0 T H → 0 A d S 2 × S 2 \mathrm{AdS}_2\times S^2 AdS 2 × S 2 suppressed and vanishes at extremality, while photon-sphere modes are not suppressed.
The source develops a node at the degenerate horizon
Writing b ≡ M 2 − Q 2 b\equiv\sqrt{M^2-Q^2} b ≡ M 2 − Q 2 r ± = M ± b r_\pm=M\pm b r ± = M ± b b → 0 b\to0 b → 0 ( M r − Q 2 ) (Mr-Q^2) ( M r − Q 2 ) C d C S C_{d\mathrm{CS}} C d CS C Θ g C_{\Theta g} C Θ g
( M r − Q 2 ) ∣ r = r + = b r + = M 2 − Q 2 r + , (71) (Mr-Q^2)\big|_{r=r_+}=br_+=\sqrt{M^2-Q^2}\,r_+, \tag*{(71)} ( M r − Q 2 ) r = r + = b r + = M 2 − Q 2 r + , (71) which vanishes linearly in b b b
Introducing the near-horizon coordinate x x x r − r + = b x r-r_+=bx r − r + = b x Δ = b 2 x ( x + 2 ) \Delta=b^2x(x+2) Δ = b 2 x ( x + 2 ) ( M r − Q 2 ) ≃ b M ( 1 + x ) (Mr-Q^2)\simeq bM(1+x) ( M r − Q 2 ) ≃ b M ( 1 + x ) b b b b 2 b^2 b 2
C d C S ≃ 96 π x ( x + 1 ) ( x + 2 ) M 7 b 3 , C Θ g ∝ b 3 . (72) C_{\mathrm{dCS}} \simeq\frac{96\pi x(x+1)(x+2)}{M^7}b^3,\qquad C_{\Theta g}\propto b^3. \tag*{(72)} C dCS ≃ M 7 96 π x ( x + 1 ) ( x + 2 ) b 3 , C Θ g ∝ b 3 . (72) Each field acquires an A d S 2 \mathrm{AdS}_2 AdS 2 h h h h ( h − 1 ) = U ( r + ) / r + h(h-1)=U(r_+)/r_+ h ( h − 1 ) = U ( r + ) / r + V = ( Δ / r 5 ) U V=(\Delta/r^5)U V = ( Δ/ r 5 ) U ( 9 M 2 + 4 M 2 μ 2 ) = M 2 ( 2 l + 1 ) 2 (9M^2+4M^2\mu^2)=M^2(2l+1)^2 ( 9 M 2 + 4 M 2 μ 2 ) = M 2 ( 2 l + 1 ) 2 q 1 = 2 M ( l + 2 ) q_1=2M(l+2) q 1 = 2 M ( l + 2 ) q 2 = − 2 M ( l − 1 ) q_2=-2M(l-1) q 2 = − 2 M ( l − 1 )
h g = l , h Θ = l + 1 , h e = l + 2. (73) h_g=l,\qquad h_\Theta=l+1,\qquad h_e=l+2. \tag*{(73)} h g = l , h Θ = l + 1 , h e = l + 2. (73) The corresponding zero-damping modes (ZDMs) are
ω ≃ − i κ + ( h + n ) , κ + = r + − r − 2 r + 2 , (74) \omega\simeq-i\kappa_+(h+n),\qquad\kappa_+=\frac{r_+-r_-}{2r_+^2}, \tag*{(74)} ω ≃ − i κ + ( h + n ) , κ + = 2 r + 2 r + − r − , (74) in agreement with Zimmerman and Mark [47 ] , who established the existence of gravito-electromagnetic ZDMs of near-extremal RN. In their notation ∣ δ j ∣ + 1 2 = h |\delta_j|+\frac{1}{2}=h ∣ δ j ∣ + 2 1 = h (73) implies near-degeneracies: the gravitational-led overtone n = 1 n=1 n = 1 h + n = l + 1 h+n=l+1 h + n = l + 1 fundamental gravitational-led ZDM (h + n = l h+n=l h + n = l
The shift of the zero-damping modes
In the throat both couplings carry one power of b b b (72) . For a non-degenerate ZDM, second-order perturbation theory therefore gives δ ω ∼ κ + ζ b 2 \delta\omega\sim\kappa_+\zeta b^2 δ ω ∼ κ + ζ b 2 δ ω / ω ∼ ζ ( M 2 − Q 2 ) / M 2 \delta\omega/\omega\sim\zeta(M^2-Q^2)/M^2 δ ω / ω ∼ ζ ( M 2 − Q 2 ) / M 2 (60) applies. We have computed its shift with that system, using a Frobenius–shooting solver (Appendix E) validated against the ZDM frequencies of Ref. [47 ] . For the fundamental gravitational-led ZDM with l = 2 l=2 l = 2
The shift is purely imaginary, β δ ω = − i C κ + b 2 \beta\,\delta\omega=-iC\kappa_+b^2 β δ ω = − i C κ + b 2
C = 477.2 , 474.8 , 471.4 , 468.2 , 462.5 , 457.4 , 448.9 , 442.1 C=477.2,\ 474.8,\ 471.4,\ 468.2,\ 462.5,\ 457.4,\ 448.9,\ 442.1 C = 477.2 , 474.8 , 471.4 , 468.2 , 462.5 , 457.4 , 448.9 , 442.1 σ ≡ ( r + − r − ) / r + = 0.004 \sigma\equiv(r_+-r_-)/r_+=0.004 σ ≡ ( r + − r − ) / r + = 0.004 0.048 0.048 0.048 C 0 = 481.7 ± 0.6 C_0=481.7\pm0.6 C 0 = 481.7 ± 0.6 σ \sigma σ M = α = 1 M=\alpha=1 M = α = 1
A fit with free exponent, δ ω / ω 0 ∝ b p \delta\omega/\omega_0\propto b^p δ ω / ω 0 ∝ b p p = 1.98 p=1.98 p = 1.98 σ = 0.005 \sigma=0.005 σ = 0.005 p = 2.00 p=2.00 p = 2.00 σ → 0 \sigma\to0 σ → 0 Figure 4 ); the small deviation is the O ( σ ) O(\sigma) O ( σ ) C ( σ ) C(\sigma) C ( σ ) Re ω = 0 \operatorname{Re}\omega=0 Re ω = 0 σ \sigma σ ω / κ + → 2 \omega/\kappa_+\to2 ω / κ + → 2
Figure 4. Fundamental gravitational-led ZDM, l = 2 l=2 l = 2 M = α = 1 M=\alpha=1 M = α = 1 β δ ω / ω 0 \beta\,\delta\omega/\omega_0 β δ ω / ω 0 b 2 = 1 − Q 2 / M 2 b^2=1-Q^2/M^2 b 2 = 1 − Q 2 / M 2 C ( σ ) C(\sigma) C ( σ ) C 0 = 481.7 C_0=481.7 C 0 = 481.7
Hence
δ ω ω 0 = 4 π C 0 ζ ( 1 − Q 2 M 2 ) ≃ 6.0 × 10 3 ζ ( 1 − Q 2 M 2 ) . (75) \frac{\delta\omega}{\omega_0}=4\pi C_0\zeta\left(1-\frac{Q^2}{M^2}\right)\simeq6.0\times10^3\zeta\left(1-\frac{Q^2}{M^2}\right). \tag*{(75)} ω 0 δ ω = 4 π C 0 ζ ( 1 − M 2 Q 2 ) ≃ 6.0 × 1 0 3 ζ ( 1 − M 2 Q 2 ) . (75) real and positive: the mode becomes more damped, and the breaking vanishes at extremality. The large coefficient restricts Eq. (75) to ζ ≪ 2 × 10 − 4 \zeta\ll2\times10^{-4} ζ ≪ 2 × 1 0 − 4
For the near-degenerate overtones the suppression is weaker. The gravitational-led n = 1 n=1 n = 1 Δ ω / κ + ≃ 2 × 10 − 4 \Delta\omega/\kappa_+\simeq2\times10^{-4} Δ ω / κ + ≃ 2 × 1 0 − 4 3 × 10 − 3 3\times10^{-3} 3 × 1 0 − 3 σ = 0.012 \sigma=0.012 σ = 0.012 0.048 0.048 0.048 σ 2 \sigma^2 σ 2 β δ ω ≃ + ( 33 \beta\,\delta\omega\simeq+(33 β δ ω ≃ + ( 33 38 ) i κ + 38)i\kappa_+ 38 ) i κ + b 2 b^2 b 2 δ ω / ω = O ( ζ ( M 2 − Q 2 ) / M ) \delta\omega/\omega=O(\sqrt{\zeta(M^2-Q^2)/M}) δ ω / ω = O ( ζ ( M 2 − Q 2 ) / M )
The two families together
The node of the Cotton source at the degenerate horizon suppresses only the modes that live there. Photonsphere modes remain relevant to the oscillatory ringdown; their relative excitation compared with ZDMs is not computed here. These modes are not suppressed: their shift stays finite and grows as Q → M Q \to M Q → M β δ ω = − 0.97 − 0.35 i \beta\,\delta\omega=-0.97-0.35i β δ ω = − 0.97 − 0.35 i Q / M = 0.98 Q/M=0.98 Q / M = 0.98 l = 2 l=2 l = 2 Table IV ). Table II summarizes the limits.
Table II. Limiting behavior of the dCS isospectrality-breaking shift.
Table IV. Isospectrality-breaking shift of the gravitational-led fundamental (l = 2 l=2 l = 2 n = 0 n=0 n = 0 M = α = 1 M=\alpha=1 M = α = 1 β \beta β χ \chi χ
Weak-coupling limit and validity of the perturbative expansion
The results of the preceding sections rest on treating the Chern–Simons coupling perturbatively. We now make this expansion explicit, identify its dimensionless control parameter, and delimit its regime of validity.
The expansion parameter
Restoring the coupling α \alpha α C d C S ∝ α C_{\mathrm{dCS}}\propto\alpha C dCS ∝ α C Θ g ∝ α / β C_{\Theta g}\propto\alpha/\beta C Θ g ∝ α / β C d C S C Θ g ∝ α 2 / β C_{\mathrm{dCS}}C_{\Theta g}\propto\alpha^2/\beta C dCS C Θ g ∝ α 2 / β a a a
ϵ a ≡ ω a 2 C d C S C Θ g D a 2 = 2304 π α 2 ( l − 1 ) l ( l + 1 ) ( l + 2 ) ( M r − Q 2 ) 2 ω a 2 β r 6 [ 2 M − 6 Q 2 / r + q ( a ) ] 2 . (76) \epsilon_a \equiv\frac{\omega_a^2 C_{\mathrm{dCS}}C_{\Theta g}}{D_a^2}
= \frac{2304\pi\alpha^2(l-1)l(l+1)(l+2)(Mr-Q^2)^2\omega_a^2}{\beta r^6\left[2M-6Q^2/r+q_{(a)}\right]^2}. \tag*{(76)} ϵ a ≡ D a 2 ω a 2 C dCS C Θ g = β r 6 [ 2 M − 6 Q 2 / r + q ( a ) ] 2 2304 π α 2 ( l − 1 ) l ( l + 1 ) ( l + 2 ) ( M r − Q 2 ) 2 ω a 2 . (76) The horizon factor cancels, so ϵ a \epsilon_a ϵ a r r r ϵ g ∼ ζ l 4 \epsilon_g\sim\zeta l^4 ϵ g ∼ ζ l 4 Q = 0 Q=0 Q = 0 ϵ g ∼ ζ l 2 ( M / Q ) 2 \epsilon_g\sim\zeta l^2(M/Q)^2 ϵ g ∼ ζ l 2 ( M / Q ) 2 Q > 0 Q>0 Q > 0 Q l ≫ M Ql\gg M Ql ≫ M ϵ e ( r ) \epsilon_e(r) ϵ e ( r ) D e ( r ) D_e(r) D e ( r ) (69) , where the scalar potential coincides with the polar partner of the axial electromagnetic-led potential. Near this resonance, perturbative validity must be assessed using the complex mode separation.
Regime of validity
The O ( ζ ) O(\zeta) O ( ζ ) a a a ϵ a ≪ 1 \epsilon_a\ll1 ϵ a ≪ 1
ζ ≪ 1 and { l ≪ ζ − 1 / 4 , Q = 0 , l ≪ ( Q / M ) ζ − 1 / 2 , Q > 0 , Q l ≫ M . (77) \zeta\ll1\quad\text{and}\quad
\begin{cases}
l\ll\zeta^{-1/4}, & Q=0,\\
l\ll(Q/M)\zeta^{-1/2}, & Q>0,\ Ql\gg M.
\end{cases} \tag*{(77)} ζ ≪ 1 and { l ≪ ζ − 1/4 , l ≪ ( Q / M ) ζ − 1/2 , Q = 0 , Q > 0 , Ql ≫ M . (77) The perturbative treatment also fails near genuine degeneracies with pseudoscalar modes, where the shift becomes O ( ζ ) O(\sqrt{\zeta}) O ( ζ )
In practice the first-order shift of branch a a a ∣ δ ω a ∣ ≪ ∣ ω a − ω Θ ∣ |\delta\omega_a|\ll|\omega_a-\omega_\Theta| ∣ δ ω a ∣ ≪ ∣ ω a − ω Θ ∣ Q c Q_c Q c ∣ Q − Q c ∣ ≫ 25 ζ M |Q-Q_c|\gg25\sqrt{\zeta}M ∣ Q − Q c ∣ ≫ 25 ζ M l = 2 l=2 l = 2 (75) restricts it to ζ ≪ 2 × 10 − 4 \zeta\ll2\times10^{-4} ζ ≪ 2 × 1 0 − 4 ζ ≪ 1 \zeta\ll1 ζ ≪ 1
Self-consistency
The half-integer power ζ 1 / 2 \zeta^{1/2} ζ 1/2 G Θ G_\Theta G Θ l l l O ( ζ ) O(\zeta) O ( ζ )
δ ω a = 1 2 ω a ∫ d r ∗ Z a ( 0 ) w a C d C S G Θ w a C Θ g Z a ( 0 ) ∫ d r ∗ ( Z a ( 0 ) ) 2 . (78) \delta\omega_a=\frac{1}{2\omega_a}\frac{\int dr_*\,Z_a^{(0)}w_aC_{\mathrm{dCS}}G_\Theta w_aC_{\Theta g}Z_a^{(0)}}{\int dr_*\left(Z_a^{(0)}\right)^2}. \tag*{(78)} δ ω a = 2 ω a 1 ∫ d r ∗ ( Z a ( 0 ) ) 2 ∫ d r ∗ Z a ( 0 ) w a C dCS G Θ w a C Θ g Z a ( 0 ) . (78) the coupled-equation form of the parametrized ringdown formalism [32 ] , with Z a ( 0 ) Z_a^{(0)} Z a ( 0 )
The unperturbed spectrum: baseline and validation
Before evaluating the Chern–Simons shift we fix the unperturbed RN axial spectrum ω 0 ( Q ) \omega_0(Q) ω 0 ( Q ) [40 , 49 , 50 ] .
Neutral anchor
At Q = 0 Q=0 Q = 0 l = 2 l=2 l = 2
ω 0 M ∣ Q = 0 = 0.3736716844 − 0.0889623157 i (79) \left.\omega_0M\right|_{Q=0}=0.3736716844-0.0889623157i \tag*{(79)} ω 0 M ∣ Q = 0 = 0.3736716844 − 0.0889623157 i (79) for ( l = 2 , n = 0 ) (l=2,\ n=0) ( l = 2 , n = 0 )
is reproduced to all quoted digits by the qnm package (Kerr solver at spin a = 0 a=0 a = 0 [50 ] , providing an independent check of the neutral endpoint. In the Schwarzschild limit, V Θ + δ V Θ V_\Theta+\delta V_\Theta V Θ + δ V Θ V 22 V_{22} V 22 [22 ] [their Eq. (3.6)], confirming the O ( α 2 / β ) O(\alpha^2/\beta) O ( α 2 / β )
Charge dependence
For Q > 0 Q > 0 Q > 0 x ≡ ( Q / M ) 2 x \equiv(Q/M)^2 x ≡ ( Q / M ) 2
ω 0 ( Q ) M = ∑ k = 0 ∞ w k ( Q M ) 2 k , (80) \omega_0(Q)M = \sum_{k=0}^{\infty} w_k \left(\frac{Q}{M}\right)^{2k}, \tag*{(80)} ω 0 ( Q ) M = k = 0 ∑ ∞ w k ( M Q ) 2 k , (80) with coefficients (for l = 2 l = 2 l = 2 n = 0 n = 0 n = 0 [49 ]
w 0 = 0.3736716844 − 0.0889623157 i , w 1 = 0.0258176729 − 0.0028240321 i , w 2 = 0.0251877887 + 0.0002053245 i , w 3 = − 0.0047481702 + 0.0025084021 i , w 4 = 0.0155726501 + 0.0004128797 i . (81) \begin{aligned}
w_0 &= 0.3736716844 - 0.0889623157 i,\\
w_1 &= 0.0258176729 - 0.0028240321 i,\\
w_2 &= 0.0251877887 + 0.0002053245 i,\\
w_3 &= -0.0047481702 + 0.0025084021 i,\\
w_4 &= 0.0155726501 + 0.0004128797 i. \tag*{(81)}
\end{aligned} w 0 w 1 w 2 w 3 w 4 = 0.3736716844 − 0.0889623157 i , = 0.0258176729 − 0.0028240321 i , = 0.0251877887 + 0.0002053245 i , = − 0.0047481702 + 0.0025084021 i , = 0.0155726501 + 0.0004128797 i . (81) The series (80) is built on the Chandrasekhar-decoupled gravitational channel [Eq. (18) with q = q 1 q = q_1 q = q 1
Cross-validation
Table III (PDF p. 16) compares the series and the spectral solution with the continued-fraction results of Berti and Kokkotas [40 ] . The spectral baseline agrees with the tabulated continued-fraction results within approximately 10 − 5 10^{-5} 1 0 − 5
Table III (PDF p. 16). Baseline RN axial gravitational-led fundamental (l = 2 l = 2 l = 2 n = 0 n = 0 n = 0 ω 0 M \omega_0 M ω 0 M (80) , the spectral (exterior-complex-scaling) solution, and the continued-fraction results of Berti and Kokkotas. The spectral solution agrees with Berti–Kokkotas at all charges. The series degrades beyond Q / M ≈ 0.6 Q/M \approx0.6 Q / M ≈ 0.6
As shown in Figure 5 , the expansion (80) loses accuracy for Q / M ≳ 0.7 Q/M \gtrsim0.7 Q / M ≳ 0.7 x ≈ 0.64 x \approx0.64 x ≈ 0.64 [49 ] . All shifts below are therefore computed on the spectral baseline. Its agreement with Ref. [40 ] holds at all charges, and with the series to 8 × 10 − 9 8 \times10^{-9} 8 × 1 0 − 9 Q / M ≤ 0.1 Q/M \leq0.1 Q / M ≤ 0.1
Figure 5. Baseline Reissner–Nordström axial gravitational fundamental (l = 2 l = 2 l = 2 n = 0 n = 0 n = 0 M ω 0 M\omega_0 M ω 0 Eq. (78); open circles are the independent continued-fraction results of Berti and Kokkotas [40 ] . The truncated series agrees with the continued-fraction data to five decimal places for Q / M ≲ 0.4 Q/M \lesssim0.4 Q / M ≲ 0.4 Table III (PDF p. 16) ).
Frequency shifts of the coupled system
Method
In the normalization Ψ = i Ψ R W / ω \Psi= i\Psi_{\mathrm{RW}}/\omega Ψ = i Ψ RW / ω ω \omega ω ω 2 \omega^2 ω 2 [51 , 52 , 53 , 54 ] :
the tortoise coordinate is continued to r ∗ = e i θ s r_* = e^{i\theta}s r ∗ = e i θ s
the operator is discretized by Chebyshev collocation in s s s
The linear coefficient β δ ω \beta\,\delta\omega β δ ω 1 / β 1/\beta 1/ β 4 × 10 − 5 4 \times10^{-5} 4 × 1 0 − 5 ≤ 7 × 10 − 6 \leq7 \times10^{-6} ≤ 7 × 1 0 − 6 θ ≥ 0.6 \theta\geq0.6 θ ≥ 0.6 S ≥ 60 S \geq60 S ≥ 60 1.4 × 10 − 6 1.4 \times10^{-6} 1.4 × 1 0 − 6 1 / β 1/\beta 1/ β 4 × 10 − 4 4 \times10^{-4} 4 × 1 0 − 4 l = 2 l = 2 l = 2 Table IV ) and by 0.4 % 0.4\% 0.4% l = 6 l = 6 l = 6 Table VII ).
Table VII. Estimated numerical uncertainty of the quoted results (units M = α = 1 M=\alpha=1 M = α = 1
Neutral limit
For l = 2 l = 2 l = 2 n = 0 n = 0 n = 0
β δ ω ∣ Q = 0 = − 0.7733 − 0.3946 i ( M = α = 1 ) , (82) \left.\beta\,\delta\omega\right|_{Q=0} = -0.7733 - 0.3946 i \qquad(M = \alpha= 1), \tag*{(82)} β δ ω ∣ Q = 0 = − 0.7733 − 0.3946 i ( M = α = 1 ) , (82) i.e. M δ ω = ( − 19.4 − 9.9 i ) ζ M\delta\omega= (-19.4 - 9.9 i)\,\zeta M δ ω = ( − 19.4 − 9.9 i ) ζ [22 ] , − 0.7731 − 0.3945 i -0.7731 - 0.3945 i − 0.7731 − 0.3945 i M 4 β = 100 M^4\beta= 100 M 4 β = 100 0.4 % 0.4\% 0.4% more strongly damped than its polar partner. For higher multipoles at Q = 0 Q = 0 Q = 0
Table 2.
These approach the l 3 l^3 l 3 (64) with O ( 1 / l ) O(1/l) O ( 1/ l ) l = 6 l = 6 l = 6 l = 12 l = 12 l = 12 β δ ω ≃ − 0.45 + 0.15 i \beta\,\delta\omega\simeq-0.45 + 0.15\,i β δ ω ≃ − 0.45 + 0.15 i Q = 0 Q = 0 Q = 0
Charge dependence
Table IV and Fig. 6 give the shift of the gravitational-led fundamental across the full charge range. Three features stand out.
l = 2 l = 2 l = 2 ∣ Re δ ω ∣ |\operatorname{Re}\delta\omega| ∣ Re δ ω ∣ Q / M ≃ 0.6 Q/M \simeq0.6 Q / M ≃ 0.6 Q / M = 0.98 Q/M = 0.98 Q / M = 0.98
Higher multipoles are strongly suppressed at intermediate charge. The shift falls to 0.49 of its neutral value at l = 6 l = 6 l = 6 Q / M ≃ 0.7 Q/M \simeq0.7 Q / M ≃ 0.7 l 3 → l 2 / Q l^3 \to l^2/Q l 3 → l 2 / Q
The electromagnetic channel matters quantitatively. Without it (χ = 0 \chi= 0 χ = 0 l = 2 l = 2 l = 2 ∣ Im δ ω ∣ |\operatorname{Im}\delta\omega| ∣ Im δ ω ∣ Im δ ω \operatorname{Im}\delta\omega Im δ ω
In all cases Im δ ω < 0 \operatorname{Im}\delta\omega< 0 Im δ ω < 0
Conclusions
We have derived the axial couplings of a dynamical Chern–Simons pseudoscalar to a charged black hole and computed the resulting breaking of axial–polar isospectrality in the Reissner–Nordström quasinormal spectrum. Beyond the charge-dependent kernel, three features have no Schwarzschild analog: the shifted electromagnetic-led branch and its exact degeneracy with the pseudoscalar at Q c Q_c Q c
Couplings. By direct reduction of the linearized field equations we obtained, in closed form, the coupling C d C S = 96 π ( M r − Q 2 ) Δ / r 8 C_{\mathrm{dCS}} = 96\pi(Mr - Q^2)\Delta/r^8 C dCS = 96 π ( M r − Q 2 ) Δ/ r 8 C d C S θ C_{\mathrm{dCS}\,\theta} C dCS θ ( M r − Q 2 ) Δ / r 8 (Mr - Q^2)\Delta/r^8 ( M r − Q 2 ) Δ/ r 8 Ψ 2 \Psi_2 Ψ 2 Ψ + χ Ψ e \Psi+\chi\Psi_e Ψ + χ Ψ e χ = 2 Q / [ r ( l − 1 ) ( l + 2 ) ] \chi=2Q/[r\sqrt{(l-1)(l+2)}] χ = 2 Q / [ r ( l − 1 ) ( l + 2 ) ]
Shifts. We computed the shifts with an exterior-complex-scaling spectral method and, for near-extremal zero-damping modes, a Frobenius–shooting method validated against Ref. [47 ] . In the neutral limit β δ ω = − 0.7733 − 0.3946 i \beta\,\delta\omega=-0.7733-0.3946i β δ ω = − 0.7733 − 0.3946 i l = 2 l=2 l = 2 M = α = 1 M=\alpha=1 M = α = 1 [22 ] . Charge changes the off-resonance scaling of the gravitational-led shift from l 3 l^3 l 3 l 2 / Q l^2/Q l 2 / Q Q c = 2 M L / ( L + 2 ) Q_c=2M\sqrt{L}/(L+2) Q c = 2 M L / ( L + 2 ) O ( ζ ) O(\sqrt{\zeta}) O ( ζ ) M ∣ Δ ω ∣ ≃ 2.4 ζ M|\Delta\omega|\simeq2.4\sqrt{\zeta} M ∣Δ ω ∣ ≃ 2.4 ζ l = 2 l=2 l = 2 δ ω / ω 0 ≃ 6.0 × 10 3 ζ ( 1 − Q 2 / M 2 ) \delta\omega/\omega_0\simeq6.0\times10^3\zeta(1-Q^2/M^2) δ ω / ω 0 ≃ 6.0 × 1 0 3 ζ ( 1 − Q 2 / M 2 ) l = 2 l=2 l = 2 ζ ( M 2 − Q 2 ) / M \sqrt{\zeta(M^2-Q^2)}/M ζ ( M 2 − Q 2 ) / M ω → 0 \omega\to0 ω → 0 h 0 h_0 h 0
Scope. All shifts are at leading order in ζ \zeta ζ ζ \sqrt{\zeta} ζ Table V summarizes the status of each result.
Table V. TABLE V. Status of results.
Outlook. A natural extension is the rotating Kerr–Newman case, where rotation induces parity-violating effects already at first order in spin and coexists with the charge-driven effects found here. Whereas the quasinormal modes of uncharged rotating black holes in dCS gravity have been computed by slow-rotation and spectral methods [24 , 25 , 26 , 27 ] , the charged rotating case remains open. As black-hole spectroscopy sharpens, isospectrality breaking offers a characteristic signature of the dCS coupling in the ringdown, distinct from a mere shift of individual frequencies.
Data availability
The data that support the findings of this article are openly available [55 ] . The archive (GitHub reposhttps://github.com/AbhiRout-cosmo/RN-dCS, release v1.1.0, archived on Zenodo with version DOI 10.5281/zenodo.23331188; all versions: 10.5281/zenodo.23283164) contains the following material, released under the MIT license:
The SymPy derivations of the linearized Pontryagin density, the axial Einstein–Maxwell equations, the C C C { Ψ , Ψ e , Θ } \{\Psi,\Psi_e,\Theta\} { Ψ , Ψ e , Θ } t φ t\varphi tφ
The exterior-complex-scaling Chebyshev solver, the branch-reduced { Z a , Θ } \{Z_a,\Theta\} { Z a , Θ }
one script per numerical result, together with the data behind all tables and Figure 1 , Figure 2 , Figure 3 , Figure 4 and Figure 6 .
Figure 1. Branch-resolved isospectrality breaking (units M = α = 1 M=\alpha=1 M = α = 1 β \beta β ∝ l 3 \propto l^3 ∝ l 3 Q = 0 Q=0 Q = 0 ∝ l 2 \propto l^2 ∝ l 2 Q > 0 Q>0 Q > 0 Q l ≫ M Ql\gg M Ql ≫ M (64) , (65) ]. (b) Charge Q c Q_c Q c (67)}; the local criterion {(citep}(66)} (solid); and the large-l l l 2 M / ( l − 1 ) ( l + 2 ) 2M/\sqrt{(l-1)(l+2)} 2 M / ( l − 1 ) ( l + 2 ) Q c Q_c Q c O ( ζ ) \mathcal{O}(\sqrt{\zeta}) O ( ζ )
Figure 6. Isospectrality-breaking shift of the gravitational-led fundamental from the coupled spectral solution (units M = α = 1 M=\alpha=1 M = α = 1 Re δ ω ( Q ) / Re δ ω ( 0 ) \operatorname{Re}\delta\omega(Q)/\operatorname{Re}\delta\omega(0) Re δ ω ( Q ) / Re δ ω ( 0 ) l = 2 l=2 l = 2 6 6 6 l = 2 l=2 l = 2 χ \chi χ
Every numerical value quoted in this article can be re-generated from these scripts.
Axial Cotton tensor components and their symbolic verification
This appendix records the linearized C C C (31) –(32) ], and the two limits used as cross-checks. All tensor algebra was carried out with a computer-algebra implementation (SymPy), which is available with the numerical codes (Appendix E).
Setup and conventions
We perturb the static Reissner–Nordström (RN) background
d s 2 = − f d t 2 + f − 1 d r 2 + r 2 ( d θ 2 + sin 2 θ d φ 2 ) , f = Δ r 2 , Δ = r 2 − 2 M r + Q 2 , (83) ds^2=-f\,dt^2+f^{-1}dr^2+r^2(d\theta^2+\sin^2\theta\,d\varphi^2),
\qquad
f=\frac{\Delta}{r^2},\quad\Delta=r^2-2Mr+Q^2, \tag*{(83)} d s 2 = − f d t 2 + f − 1 d r 2 + r 2 ( d θ 2 + sin 2 θ d φ 2 ) , f = r 2 Δ , Δ = r 2 − 2 M r + Q 2 , (83) by a dynamical pseudoscalar
δ ϑ = R ( r ) Y l ( θ ) e − i ω t , (84) \delta\vartheta=R(r)Y_l(\theta)e^{-i\omega t}, \tag*{(84)} δ ϑ = R ( r ) Y l ( θ ) e − iω t , (84) with vanishing background scalar (the RN background solves the dCS field equations exactly, since ∗ R R ∣ R N = 0 {}^{*}RR|_{\mathrm{RN}}=0 ∗ R R ∣ RN = 0 C C C
C μ ν = ( ∇ ρ ϑ ) ϵ ρ σ τ ( μ ∇ σ R ν ) τ + ( ∇ ρ ∇ σ ϑ ) ∗ R σ ( μ ν ) ρ , (85) C^{\mu\nu}=(\nabla_\rho\vartheta)\epsilon^{\rho\sigma\tau(\mu}\nabla_\sigma R^{\nu)}{}_\tau+(\nabla_\rho\nabla_\sigma\vartheta){}^{*}R^{\sigma(\mu\nu)\rho}, \tag*{(85)} C μν = ( ∇ ρ ϑ ) ϵ ρ σ τ ( μ ∇ σ R ν ) τ + ( ∇ ρ ∇ σ ϑ ) ∗ R σ ( μν ) ρ , (85) with the round brackets denoting symmetrisation with weight 1 / 2 1/2 1/2 96 π 96\pi 96 π 192 π 192\pi 192 π ϵ μ ν ρ σ = − g [ μ ν ρ σ ] \epsilon_{\mu\nu\rho\sigma}=\sqrt{-g}\,[\mu\nu\rho\sigma] ϵ μν ρ σ = − g [ μν ρ σ ] [ t r θ φ ] = + 1 [tr\theta\varphi]=+1 [ t r θ φ ] = + 1 ϵ t r θ φ = − 1 / − g \epsilon^{tr\theta\varphi}=-1/\sqrt{-g} ϵ t r θ φ = − 1/ − g
The axial (odd-parity) sector is generated by the vector harmonic
S φ = ∂ θ Y l sin θ , (86) S^\varphi=\frac{\partial_\theta Y_l}{\sin\theta}, \tag*{(86)} S φ = sin θ ∂ θ Y l , (86) i.e. the contravariant component of X A = ( − ϵ A B ∂ B Y l ) X_A=(-\epsilon_A{}^B\partial_B Y_l) X A = ( − ϵ A B ∂ B Y l ) ϵ A B \epsilon_{AB} ϵ A B S φ S^\varphi S φ 1 / sin θ 1/\sin\theta 1/ sin θ l l l
The linearised components
Evaluating (A3) on (A1)–(A2) gives two nonzero contravariant components in the axial sector; the algebra naturally separates into the two pieces of (A3)`, which we label C ( 1 ) C_{(1)} C ( 1 ) ∇ ϑ ε ∇ R \nabla\vartheta\,\varepsilon\nabla R ∇ ϑ ε ∇ R C ( 2 ) C_{(2)} C ( 2 ) ∇ ∇ ϑ ∗ R \nabla\nabla\vartheta\,{}^{*}R ∇∇ ϑ ∗ R
a. The t φ t\varphi tφ The two pieces are
C ( 2 ) t φ = − 3 ( M r − Q 2 ) r 6 ( R ′ − R r ) S φ , (87) C^{t\varphi}_{(2)}=-\frac{3(Mr-Q^{2})}{r^{6}}\left(R'-\frac{R}{r}\right)S^{\varphi}, \tag*{(87)} C ( 2 ) tφ = − r 6 3 ( M r − Q 2 ) ( R ′ − r R ) S φ , (87) C ( 1 ) t φ = − 3 Q 2 r 7 R S φ . (88) C^{t\varphi}_{(1)}=-\frac{3Q^{2}}{r^{7}}RS^{\varphi}. \tag*{(88)} C ( 1 ) tφ = − r 7 3 Q 2 R S φ . (88) Adding,
C t φ = [ − 3 ( M r − Q 2 ) r 6 R ′ + 3 ( M r − 2 Q 2 ) r 7 R ] S φ , (89) C^{t\varphi}=\left[-\frac{3(Mr-Q^{2})}{r^{6}}R'+\frac{3(Mr-2Q^{2})}{r^{7}}R\right]S^{\varphi}, \tag*{(89)} C tφ = [ − r 6 3 ( M r − Q 2 ) R ′ + r 7 3 ( M r − 2 Q 2 ) R ] S φ , (89) where the shift Q 2 → 2 Q 2 Q^{2}\to2Q^{2} Q 2 → 2 Q 2 R R R R ′ R' R ′ C ( 1 ) C_{(1)} C ( 1 )
b. The r φ r\varphi r φ Here only the C ( 2 ) C_{(2)} C ( 2 ) ω \omega ω
C r φ = C ( 2 ) r φ = − 3 i ω ( M r − Q 2 ) r 6 R S φ , C ( 1 ) r φ = 0. (90) C^{r\varphi}=C^{r\varphi}_{(2)}=-\frac{3i\omega(Mr-Q^{2})}{r^{6}}RS^{\varphi},\qquad C^{r\varphi}_{(1)}=0. \tag*{(90)} C r φ = C ( 2 ) r φ = − r 6 3 iω ( M r − Q 2 ) R S φ , C ( 1 ) r φ = 0. (90) The factor i ω i\omega iω dynamical character of the effect: it vanishes for a stationary (ω = 0 \omega=0 ω = 0 ω \omega ω Ψ = i Ψ R W / ω \Psi=i\Psi_{\rm RW}/\omega Ψ = i Ψ RW / ω § 1.8 ).
c. The θ φ \theta\varphi θ φ
C θ φ = 0. (91) C^{\theta\varphi}=0. \tag*{(91)} C θ φ = 0. (91) The absence of a tensor-harmonic ( θ φ ) (\theta\varphi) ( θ φ )
Internal consistency and symbolic verification
Three internal checks, all confirmed symbolically, fix confidence in (A7)–(A9).
(i) Piecewise reconstruction. The separate pieces (A5)–(A6) reproduce the combined component (A7) term by term: the R ′ R' R ′ − 3 ( M r − Q 2 ) / r 6 -3(Mr-Q^{2})/r^{6} − 3 ( M r − Q 2 ) / r 6 C ( 2 ) C_{(2)} C ( 2 ) R R R 3 ( M r − 2 Q 2 ) / r 7 3(Mr-2Q^{2})/r^{7} 3 ( M r − 2 Q 2 ) / r 7 3 ( M r − Q 2 ) / r 7 − 3 Q 2 / r 7 3(Mr-Q^{2})/r^{7}-3Q^{2}/r^{7} 3 ( M r − Q 2 ) / r 7 − 3 Q 2 / r 7 − R / r -R/r − R / r C ( 2 ) C_{(2)} C ( 2 ) C ( 1 ) C_{(1)} C ( 1 )
(ii) Harmonic structure. Every component carries the single angular factor S φ = ∂ θ Y l / sin θ S^{\varphi}=\partial_{\theta}Y_l/\sin\theta S φ = ∂ θ Y l / sin θ Y l Y_l Y l δ ϑ ↦ C μ ν [ δ ϑ ] \delta\vartheta\mapsto C_{\mu\nu}[\delta\vartheta] δ ϑ ↦ C μν [ δ ϑ ] S O ( 3 ) SO(3) S O ( 3 ) l l l
(iii) Curvature-invariant sub-checks. The full set of background Riemann components entering (A3), the dual ∗ R μ ν ρ σ {}^{*}R^{\mu\nu\rho\sigma} ∗ R μν ρ σ ∇ ∇ ϑ \nabla\nabla\vartheta ∇∇ ϑ
Neutral and extremal limits
a. Schwarzschild limit (Q → 0 Q\to0 Q → 0 The components reduce to
C t φ ∣ Q = 0 = [ − 3 M r 5 R ′ + 3 M r 6 R ] S φ , (92) \left.C^{t\varphi}\right|_{Q=0}=\left[-\frac{3M}{r^{5}}R'+\frac{3M}{r^{6}}R\right]S^{\varphi}, \tag*{(92)} C tφ Q = 0 = [ − r 5 3 M R ′ + r 6 3 M R ] S φ , (92) C r φ ∣ Q = 0 = − 3 i ω M r 5 R S φ , \left.C^{r\varphi}\right|_{Q=0}=-\frac{3i\omega M}{r^{5}}RS^{\varphi}, C r φ ∣ Q = 0 = − r 5 3 iω M R S φ , which are the source components that reduce, in the master-variable calculation of Sec. VI, to the Schwarzschild dCS coupling C d C S = 96 π M f / r 5 C_{\rm dCS}=96\pi M f/r^{5} C dCS = 96 π M f / r 5 [22 ] . This is the component-level form of the calibration presented in the main text.
b. Extremal node. The combination ( M r − Q 2 ) (Mr-Q^{2}) ( M r − Q 2 ) r + = M + b r_{+}=M+b r + = M + b b ≡ M 2 − Q 2 b\equiv\sqrt{M^{2}-Q^{2}} b ≡ M 2 − Q 2 Q 2 = M 2 − b 2 Q^{2}=M^{2}-b^{2} Q 2 = M 2 − b 2
( M r − Q 2 ) ∣ r = r + = b ( M + b ) = b r + → b → 0 0. (93) \left.(Mr-Q^{2})\right|_{r=r_{+}}=b(M+b)=br_{+}\xrightarrow{b\to0}0. \tag*{(93)} ( M r − Q 2 ) r = r + = b ( M + b ) = b r + b → 0 0. (93) The Cotton source therefore develops a node at the degenerate horizon as extremality is approached, which is the microscopic origin of the suppression of the zero-damping modes found in Sec. XI.
Reduction to the master equation and the coupling C d C S C_{\rm dCS} C dCS
This appendix carries out, in detail, the reduction of the axial gravitational field equations to a single master equation, and extracts dynamical Chern–Simons coupling C d C S C_{\rm dCS} C dCS h 0 h_0 h 0 Ψ = i Ψ R W / ω \Psi=i\Psi_{\rm RW}/\omega Ψ = i Ψ RW / ω Θ ′ ′ \Theta'' Θ ′′ C d C S C_{\rm dCS} C dCS
The axial field equations with source
In Regge–Wheeler gauge the odd-parity metric perturbation is carried by h 0 ( r ) = h t φ h_0(r)=h_{t\varphi} h 0 ( r ) = h tφ h 1 ( r ) = h r φ h_1(r)=h_{r\varphi} h 1 ( r ) = h r φ W ≡ sin θ ∂ θ Y l W\equiv\sin\theta\,\partial_\theta Y_l W ≡ sin θ ∂ θ Y l e − i ω t e^{-i\omega t} e − iω t W ′ ′ = cot θ W ′ − L W W''=\cot\theta\,W'-LW W ′′ = cot θ W ′ − L W L = l ( l + 1 ) L=l(l+1) L = l ( l + 1 ) W ′ W' W ′
δ G t φ = W [ − Δ 2 r 2 h 0 ′ ′ − i ω Δ 2 r 2 h 1 ′ + L r 2 − 4 M r + 4 Q 2 2 r 4 h 0 − i ω Δ r 3 h 1 ] , (94) \delta G_{t\varphi}=W\left[-\frac{\Delta}{2r^2}h_0''-\frac{i\omega\Delta}{2r^2}h_1'+\frac{Lr^2-4Mr+4Q^2}{2r^4}h_0-\frac{i\omega\Delta}{r^3}h_1\right], \tag*{(94)} δ G tφ = W [ − 2 r 2 Δ h 0 ′′ − 2 r 2 iω Δ h 1 ′ + 2 r 4 L r 2 − 4 M r + 4 Q 2 h 0 − r 3 iω Δ h 1 ] , (94) δ G r φ = W [ i ω r 2 2 Δ h 0 ′ − i ω r Δ h 0 + N ( r ) 2 r 4 Δ h 1 ] . (95) \delta G_{r\varphi}=W\left[\frac{i\omega r^2}{2\Delta}h_0'-\frac{i\omega r}{\Delta}h_0+\frac{\mathcal{N}(r)}{2r^4\Delta}h_1\right]. \tag*{(95)} δ G r φ = W [ 2Δ iω r 2 h 0 ′ − Δ iω r h 0 + 2 r 4 Δ N ( r ) h 1 ] . (95) with
N ( r ) = L r 2 Δ + 4 M r ( r 2 − Q 2 ) − 2 ( r 2 − Q 2 ) ( r 2 + Q 2 ) − ω 2 r 6 . (96) \mathcal{N}(r)=Lr^2\Delta+4Mr(r^2-Q^2)-2(r^2-Q^2)(r^2+Q^2)-\omega^2r^6. \tag*{(96)} N ( r ) = L r 2 Δ + 4 M r ( r 2 − Q 2 ) − 2 ( r 2 − Q 2 ) ( r 2 + Q 2 ) − ω 2 r 6 . (96) From Eq. (3) , the dCS contribution to the axial field equations is δ R a b ⊃ − 2 κ α C a b = − 16 π α C a b \delta R_{ab}\supset-2\kappa\alpha C_{ab}=-16\pi\alpha C_{ab} δ R ab ⊃ − 2 κ α C ab = − 16 π α C ab O ( ϑ 2 ) O(\vartheta^2) O ( ϑ 2 ) δ G μ φ = − 16 π α C μ φ + κ δ T μ φ E M \delta G_{\mu\varphi}=-16\pi\alpha C_{\mu\varphi}+\kappa\delta T^{\rm EM}_{\mu\varphi} δ G μ φ = − 16 π α C μ φ + κ δ T μ φ EM C Q C_Q C Q α = 1 \alpha=1 α = 1 W W W C μ φ = g μ μ g φ φ C μ φ C_{\mu\varphi}=g_{\mu\mu}g_{\varphi\varphi}C^{\mu\varphi} C μ φ = g μμ g φφ C μ φ
− 16 π C t φ ∣ W = − 48 π f r 5 [ r ( M r − Q 2 ) R ′ − ( M r − 2 Q 2 ) R ] ≡ S t r a w . (97) -16\pi C_{t\varphi}\big|_W=-\frac{48\pi f}{r^5}\left[r(Mr-Q^2)R'-(Mr-2Q^2)R\right]\equiv S_t^{\rm raw}. \tag*{(97)} − 16 π C tφ W = − r 5 48 π f [ r ( M r − Q 2 ) R ′ − ( M r − 2 Q 2 ) R ] ≡ S t raw . (97) − 16 π C r φ ∣ W = 48 π i ω ( M r − Q 2 ) f r 4 R ≡ S r r a w . (98) -16\pi C_{r\varphi}\big|_W=\frac{48\pi i\omega(Mr-Q^2)}{fr^4}R\equiv S_r^{\rm raw}. \tag*{(98)} − 16 π C r φ W = f r 4 48 π iω ( M r − Q 2 ) R ≡ S r raw . (98) (These follow from lowering Eqs. (89) –(90) with g t t = − f g_{tt}=-f g tt = − f g r r = f − 1 g_{rr}=f^{-1} g r r = f − 1 g φ φ = r 2 sin 2 θ g_{\varphi\varphi}=r^2\sin^2\theta g φφ = r 2 sin 2 θ 96 π 96\pi 96 π C d C S C_{\rm dCS} C dCS
Elimination of h 0 h_0 h 0
Dividing (B1) by − Δ / ( 2 r 2 ) -\Delta/(2r^2) − Δ/ ( 2 r 2 ) i ω r 2 / ( 2 Δ ) i\omega r^2/(2\Delta) iω r 2 / ( 2Δ )
h 0 ′ ′ + i ω h 1 ′ + A h 0 + B h 1 = S t , (99) h_0''+i\omega h_1'+Ah_0+Bh_1=S_t, \tag*{(99)} h 0 ′′ + iω h 1 ′ + A h 0 + B h 1 = S t , (99) h 0 ′ = 2 r h 0 + C h 1 + S r , (100) h_0'=\frac{2}{r}h_0+Ch_1+S_r, \tag*{(100)} h 0 ′ = r 2 h 0 + C h 1 + S r , (100) with
A = − L r 2 − 4 M r + 4 Q 2 r 2 Δ , B = 2 i ω r , C = i N ω r 6 , (101) A=-\frac{Lr^2-4Mr+4Q^2}{r^2\Delta},\qquad B=\frac{2i\omega}{r},\qquad C=\frac{i\mathcal{N}}{\omega r^6}, \tag*{(101)} A = − r 2 Δ L r 2 − 4 M r + 4 Q 2 , B = r 2 iω , C = ω r 6 i N , (101) and the normalised sources
S t = − 2 r 2 Δ S t r a w , S r = − 2 i Δ ω r 2 S r r a w . (102) S_t=-\frac{2r^2}{\Delta}S_t^{\rm raw},\qquad S_r=-\frac{2i\Delta}{\omega r^2}S_r^{\rm raw}. \tag*{(102)} S t = − Δ 2 r 2 S t raw , S r = − ω r 2 2 i Δ S r raw . (102) Differentiating (B7) and substituting it and (B7) into (B6) eliminates h 0 ′ ′ h_0'' h 0 ′′ h 0 ′ h_0' h 0 ′ h 0 h_0 h 0
h 0 = S t − S r ′ − 2 r S r − ( C + i ω ) h 1 ′ − ( 2 C r + C ′ + B ) h 1 2 / r 2 + A , (103) h_0=\frac{S_t-S_r'-\frac{2}{r}S_r-\left(C+i\omega\right)h_1'-\left(\frac{2C}{r}+C'+B\right)h_1}{2/r^2+A}, \tag*{(103)} h 0 = 2/ r 2 + A S t − S r ′ − r 2 S r − ( C + iω ) h 1 ′ − ( r 2 C + C ′ + B ) h 1 , (103) where 2 / r 2 + A = [ ( 2 − L ) r 2 − 2 Q 2 ] / ( r 2 Δ ) 2/r^2+A=[(2-L)r^2-2Q^2]/(r^2\Delta) 2/ r 2 + A = [( 2 − L ) r 2 − 2 Q 2 ] / ( r 2 Δ ) h 1 h_1 h 1
Ψ R W ≡ Δ r 3 h 1 , (104) \Psi_{\rm RW}\equiv\frac{\Delta}{r^3}h_1, \tag*{(104)} Ψ RW ≡ r 3 Δ h 1 , (104) one recovers the Schrödinger type equations [ d 2 / d r 2 + ω 2 − V R W ] Ψ R W = S [d^2/dr^2+\omega^2-V_{\rm RW}]\Psi_{\rm RW}=S [ d 2 / d r 2 + ω 2 − V RW ] Ψ RW = S f [ l ( l + 1 ) / r 2 − 6 M / r 3 ] f[l(l+1)/r^2-6M/r^3] f [ l ( l + 1 ) / r 2 − 6 M / r 3 ]
Algebraic normalisation and the reduced source
The reduced source S S S Θ \Theta Θ Θ ′ \Theta' Θ ′ ω \omega ω C r φ ∝ i ω C^{r\varphi}\propto i\omega C r φ ∝ iω
Ψ = i Ψ R W ω , (105) \Psi=\frac{i\Psi_{\rm RW}}{\omega}, \tag*{(105)} Ψ = ω i Ψ RW , (105) is not cosmetic: the factor i / ω i/\omega i / ω ω \omega ω r φ r\varphi r φ Ψ \Psi Ψ ω \omega ω Θ \Theta Θ S Ψ R W S_{\Psi_{\rm RW}} S Ψ RW Ψ R W \Psi_{\rm RW} Ψ RW Ψ \Psi Ψ ( i / ω ) S Ψ R W (i/\omega)S_{\Psi_{\rm RW}} ( i / ω ) S Ψ RW ω \omega ω S r r a w S_r^{\rm raw} S r raw § 1.8 .
On-shell reduction and the coupling
The reduced Ψ \Psi Ψ Θ ′ ′ \Theta'' Θ ′′
Δ r 2 d d r ( Δ r 2 d Θ d r ) + ( ω 2 − V Θ ) Θ = 0 , (106) \frac{\Delta}{r^2}\frac{d}{dr}\left(\frac{\Delta}{r^2}\frac{d\Theta}{dr}\right)+(\omega^2-V_\Theta)\Theta=0, \tag*{(106)} r 2 Δ d r d ( r 2 Δ d r d Θ ) + ( ω 2 − V Θ ) Θ = 0 , (106) V Θ = Δ r 5 [ l ( l + 1 ) r + 2 M − 2 Q 2 r ] , V_\Theta=\frac{\Delta}{r^5}\left[l(l+1)r+2M-\frac{2Q^2}{r}\right], V Θ = r 5 Δ [ l ( l + 1 ) r + 2 M − r 2 Q 2 ] , to eliminate Θ ′ ′ \Theta'' Θ ′′ Θ ′ \Theta' Θ ′ Q Q Q
[ d 2 d r ∗ 2 + ω 2 − V R W ] Ψ = C d C S Θ , C d C S = 96 π ( M r − Q 2 ) Δ r 8 = 96 π ( M r − Q 2 ) f r 6 . (107) \left[\frac{d^2}{dr_*^2}+\omega^2-V_{\rm RW}\right]\Psi=C_{\rm dCS}\Theta,
\qquad
C_{\rm dCS}=\frac{96\pi(Mr-Q^2)\Delta}{r^8}=\frac{96\pi(Mr-Q^2)f}{r^6}. \tag*{(107)} [ d r ∗ 2 d 2 + ω 2 − V RW ] Ψ = C dCS Θ , C dCS = r 8 96 π ( M r − Q 2 ) Δ = r 6 96 π ( M r − Q 2 ) f . (107) Two checks confirm (B14). First, the Θ ′ \Theta' Θ ′ Q Q Q
C d C S ∣ Q → 0 = 96 π M f r 5 , (108) \left.C_{\rm dCS}\right|_{Q\to0}=\frac{96\pi M f}{r^5}, \tag*{(108)} C dCS ∣ Q → 0 = r 5 96 π M f , (108) reproducing the Schwarzschild dCS coupling of Molina et al. [22 ] [their Eq. (2.8)]. The coupling is built from the combination ( M r − Q 2 ) (Mr-Q^2) ( M r − Q 2 ) Δ \Delta Δ l l l
a. Maxwell master equation. The Maxwell equation ∇ μ F μ φ = 0 \nabla_\mu F^{\mu\varphi}=0 ∇ μ F μ φ = 0 h 0 h_0 h 0 h 0 ′ h_0' h 0 ′ (14) ]. When $h_0$ is eliminated through the sourced equations (B6)–{xref}(B7), the $C$-tensor source is carried into the Maxwell master equation. With $\Psi_e$ normalized as in {xref}Eqs. (15)–{xref}(16), this gives the term $\chi C_{\rm dCS}\Theta$ of {xref}Eq. (52b)`. The Θ ′ \Theta' Θ ′ ( t φ ) (t\varphi) ( tφ ) ( M , Q , ω , l ) (M,Q,\omega,l) ( M , Q , ω , l )
It is worth recording why the algebraic form (B14) requires the normalisation (B12) rather than the Cunningham–Price–Moncrief (CPM) master function used in the Martel–Poisson formalism. In the CPM normalisation the master function itself carries a factor 2 r / [ ( l − 1 ) ( l + 2 ) ] 2r/[(l-1)(l+2)] 2 r / [( l − 1 ) ( l + 2 )] 1 / [ ( l − 1 ) ( l + 2 ) ] 1/[(l-1)(l+2)] 1/ [( l − 1 ) ( l + 2 )] Θ ′ \Theta' Θ ′ Ψ = i Ψ R W / ω \Psi=i\Psi_{\rm RW}/\omega Ψ = i Ψ RW / ω ω \omega ω r φ r\varphi r φ 1 / [ ( l − 1 ) ( l + 2 ) ] 1/[(l-1)(l+2)] 1/ [( l − 1 ) ( l + 2 )] l = 1 l=1 l = 1 ( l − 1 ) ( l + 2 ) = 0 (l-1)(l+2)=0 ( l − 1 ) ( l + 2 ) = 0
The reciprocal coupling and the electromagnetic channel
Linearized Pontryagin density
The pseudoscalar obeys β □ ϑ = − α 4 ∗ R R \beta\Box\vartheta=-\frac{\alpha}{4}{}^*RR β □ ϑ = − 4 α ∗ R R ∗ R R = 0 {}^*RR=0 ∗ R R = 0 δ ( ∗ R R ) \delta({}^*RR) δ ( ∗ R R ) h t φ = h 0 ( r ) W e − i ω t h_{t\varphi}=h_0(r)We^{-i\omega t} h tφ = h 0 ( r ) W e − iω t h r φ = h 1 ( r ) W e − i ω t h_{r\varphi}=h_1(r)We^{-i\omega t} h r φ = h 1 ( r ) W e − iω t ∗ R R {}^*RR ∗ R R − g \sqrt{-g} − g
δ ( ∗ R R ) = ε ρ σ τ η R ˉ ν μ ρ σ δ R μ ν τ η , (109) \delta({}^*RR)=\varepsilon^{\rho\sigma\tau\eta}\bar{R}^{\nu}{}_{\mu\rho\sigma}\delta R^{\mu}{}_{\nu\tau\eta}, \tag*{(109)} δ ( ∗ R R ) = ε ρ σ τ η R ˉ ν μ ρ σ δ R μ ν τ η , (109) where R ˉ \bar{R} R ˉ
δ ( ∗ R R ) = − 24 L ( M r − Q 2 ) r 7 [ r h 0 ′ − 2 h 0 + i ω r h 1 ] Y l . (110) \delta({}^*RR)=-\frac{24L(Mr-Q^2)}{r^7}\left[rh_0'-2h_0+i\omega r h_1\right]Y_l. \tag*{(110)} δ ( ∗ R R ) = − r 7 24 L ( M r − Q 2 ) [ r h 0 ′ − 2 h 0 + iω r h 1 ] Y l . (110) exactly. All ∂ θ Y l \partial_\theta Y_l ∂ θ Y l W ′ ′ = cot θ W ′ − L W W''=\cot\theta W'-LW W ′′ = cot θ W ′ − L W
δ ( ∗ R R ) ∣ Q = 0 = − L [ 24 M r 5 h 0 ′ − 48 M r 6 h 0 + 24 i ω M r 5 h 1 ] Y l . (111) \left.\delta({}^*RR)\right|_{Q=0}=-L\left[\frac{24M}{r^5}h_0'-\frac{48M}{r^6}h_0+\frac{24i\omega M}{r^5}h_1\right]Y_l. \tag*{(111)} δ ( ∗ R R ) ∣ Q = 0 = − L [ r 5 24 M h 0 ′ − r 6 48 M h 0 + r 5 24 iω M h 1 ] Y l . (111) Reduction
At the order required, h 0 h_0 h 0 h 1 h_1 h 1 δ A φ \delta A_\varphi δ A φ
The θ φ \theta\varphi θ φ i ω h 0 = − ( Δ / r 5 ) [ r Δ h 1 ′ + 2 ( M r − Q 2 ) h 1 ] i\omega h_0=-(\Delta/r^5)[r\Delta h_1'+2(Mr-Q^2)h_1] iω h 0 = − ( Δ/ r 5 ) [ r Δ h 1 ′ + 2 ( M r − Q 2 ) h 1 ]
The r φ r\varphi r φ (46) , where the Ψ e \Psi_e Ψ e δ T φ r ∝ Q ω δ A φ \delta T^\varphi{}_{r}\propto Q\omega\delta A_\varphi δ T φ r ∝ Q ω δ A φ
In the same variables the Einstein–Maxwell system takes the symmetric form (15) –(16) . This fixes the normalization δ A φ ∝ ( μ / 2 ) Ψ e \delta A_\varphi\propto(\mu/2)\Psi_e δ A φ ∝ ( μ /2 ) Ψ e V R W V_{\rm RW} V RW V E M V_{\rm EM} V EM C Q C_Q C Q
Substituting into Eq. (110) gives Eq. (47) . With δ ϑ = ( Θ / r ) Y l e − i ω t \delta\vartheta=(\Theta/r)Y_l e^{-i\omega t} δ ϑ = ( Θ/ r ) Y l e − iω t
□ δ ϑ = 1 r f [ d 2 d r ∗ 2 + ω 2 − V ϑ ] Θ Y l e − i ω t , (112) \Box\delta\vartheta=\frac{1}{rf}\left[\frac{d^2}{dr_*^2}+\omega^2-V_\vartheta\right]\Theta Y_l e^{-i\omega t}, \tag*{(112)} □ δ ϑ = r f 1 [ d r ∗ 2 d 2 + ω 2 − V ϑ ] Θ Y l e − iω t , (112) so
[ d 2 d r ∗ 2 + ω 2 − V ϑ ] Θ = − α Δ 4 β r δ ( ∗ R R ) ∣ Y l = C Θ g ( Ψ + χ Ψ e ) , (113) \left[\frac{d^2}{dr_*^2}+\omega^2-V_\vartheta\right]\Theta
=-\frac{\alpha\Delta}{4\beta r}\left.\delta({}^*RR)\right|_{Y_l}
=C_{\Theta g}(\Psi+\chi\Psi_e), \tag*{(113)} [ d r ∗ 2 d 2 + ω 2 − V ϑ ] Θ = − 4 β r α Δ δ ( ∗ R R ) ∣ Y l = C Θ g ( Ψ + χ Ψ e ) , (113) with C Θ g C_{\Theta g} C Θ g (49) .
Checks
(i) Neutral limit. χ → 0 \chi\to0 χ → 0 C Θ g → ( 6 M α / β ) [ ( l + 2 ) ! / ( l − 2 ) ! ] f / r 5 C_{\Theta g}\to(6M\alpha/\beta)[(l+2)!/(l-2)!]f/r^5 C Θ g → ( 6 M α / β ) [( l + 2 )! / ( l − 2 )!] f / r 5 V 21 V_{21} V 21 [22 ] [their Eq. (2.9)].
(ii) Reciprocity. The gravity-side channels C d C S C_{\rm dCS} C dCS C Θ e = χ C d C S C_{\Theta e}=\chi C_{\rm dCS} C Θ e = χ C dCS C Θ g C_{\Theta g} C Θ g C Θ e = χ C Θ g C_{\Theta e}=\chi C_{\Theta g} C Θ e = χ C Θ g (51) . This is the structure required by the single bilinear α 4 ∫ − g δ ϑ δ ( ∗ R R ) \frac{\alpha}{4}\int\sqrt{-g}\,\delta\vartheta\,\delta({}^*RR) 4 α ∫ − g δ ϑ δ ( ∗ R R )
(iv) Dipole. C Θ g C_{\Theta g} C Θ g C Θ e C_{\Theta e} C Θ e l = 1 l=1 l = 1 l = 1 l=1 l = 1 Ψ e \Psi_e Ψ e μ \mu μ l ≥ 2 l\ge2 l ≥ 2
Local-WKB proxy for the fundamental-mode shift
This appendix records a local-WKB estimate of the shift of the fundamental (l = 2 l=2 l = 2 n = 0 n=0 n = 0 χ \chi χ
Method
To leading WKB order the quasinormal frequency is fixed by the peak of the potential,
ω 0 2 ≃ V R W ( r p e a k ) − i ( n + 1 2 ) − 2 V R W ′ ′ ( r p e a k ) , (114) \omega_0^2 \simeq V_{\rm RW}(r_{\rm peak})-i\left(n+\frac{1}{2}\right)\sqrt{-2V_{\rm RW}^{\prime\prime}(r_{\rm peak})}, \tag*{(114)} ω 0 2 ≃ V RW ( r peak ) − i ( n + 2 1 ) − 2 V RW ′′ ( r peak ) , (114) where the prime denotes d / d r ∗ d/dr_* d / d r ∗ r p e a k r_{\rm peak} r peak V R W V_{\rm RW} V RW δ V e f f \delta V_{\rm eff} δ V eff δ ( ω 2 ) ≃ δ V e f f ( r p e a k ) \delta(\omega^2)\simeq\delta V_{\rm eff}(r_{\rm peak}) δ ( ω 2 ) ≃ δ V eff ( r peak ) V R W ′ ( r p e a k ) = 0 V_{\rm RW}^{\prime}(r_{\rm peak})=0 V RW ′ ( r peak ) = 0
Integrating out the scalar in the coupled system (52)–(58) generates the effective correction δ V e f f = C d C S G Θ C Θ g \delta V_{\rm eff}=C_{\rm dCS}G_{\Theta}C_{\Theta g} δ V eff = C dCS G Θ C Θ g G Θ = ( □ + ω 0 2 − V Θ ) − 1 G_{\Theta}=(\Box+\omega_0^2-V_{\Theta})^{-1} G Θ = ( □ + ω 0 2 − V Θ ) − 1 G Θ ≃ ( ω 0 2 − V Θ ) − 1 G_{\Theta}\simeq(\omega_0^2-V_{\Theta})^{-1} G Θ ≃ ( ω 0 2 − V Θ ) − 1
δ ( ω 2 ) l , n ≃ C d C S C Θ g ω 0 2 − V Θ ∣ r p e a k , δ ω = δ ( ω 2 ) 2 ω 0 . (115) \delta(\omega^2)_{l,n}\simeq\left.\frac{C_{\rm dCS}C_{\Theta g}}{\omega_0^2-V_{\Theta}}\right|_{r_{\rm peak}},\qquad\delta\omega=\frac{\delta(\omega^2)}{2\omega_0}. \tag*{(115)} δ ( ω 2 ) l , n ≃ ω 0 2 − V Θ C dCS C Θ g r peak , δ ω = 2 ω 0 δ ( ω 2 ) . (115) The potential peak
For l = 2 l=2 l = 2 Q = 0 Q=0 Q = 0 V R W = f ( 6 / r 2 − 6 M / r 3 ) V_{\rm RW}=f(6/r^2-6M/r^3) V RW = f ( 6/ r 2 − 6 M / r 3 ) f = 1 − 2 M / r f=1-2M/r f = 1 − 2 M / r V R W = 6 / r 2 − 18 M / r 3 + 12 M 2 / r 4 V_{\rm RW}=6/r^2-18M/r^3+12M^2/r^4 V RW = 6/ r 2 − 18 M / r 3 + 12 M 2 / r 4 d V R W / d r = 0 dV_{\rm RW}/dr=0 d V RW / d r = 0 2 r 2 − 9 M r + 8 M 2 = 0 2r^2-9Mr+8M^2=0 2 r 2 − 9 M r + 8 M 2 = 0
r p e a k ∣ Q = 0 = 9 + 17 4 M ≃ 3.2808 M , (116) \left.r_{\rm peak}\right|_{Q=0}=\frac{9+\sqrt{17}}{4}M\simeq3.2808\,M, \tag*{(116)} r peak ∣ Q = 0 = 4 9 + 17 M ≃ 3.2808 M , (116) lying just outside the photon sphere r p h = 3 M r_{\rm ph}=3M r ph = 3 M Q > 0 Q>0 Q > 0 V R W = ( Δ / r 5 ) [ l ( l + 1 ) r − 6 M + 4 Q 2 / r ] V_{\rm RW}=(\Delta/r^5)[l(l+1)r-6M+4Q^2/r] V RW = ( Δ/ r 5 ) [ l ( l + 1 ) r − 6 M + 4 Q 2 / r ]
Worked example: the neutral case
We display all intermediate quantities at Q = 0 Q=0 Q = 0 M = 1 M=1 M = 1 α = 1 \alpha=1 α = 1 β \beta β r p e a k = 3.28078 r_{\rm peak}=3.28078 r peak = 3.28078 ω 0 = 0.373672 − 0.088962 i \omega_0=0.373672-0.088962\,i ω 0 = 0.373672 − 0.088962 i § 1.11 ), so that ω 0 2 = 0.131716 − 0.066485 i \omega_0^2=0.131716-0.066485\,i ω 0 2 = 0.131716 − 0.066485 i
V Θ ( r p e a k ) = 0.239728 , ω 0 2 − V Θ ( r p e a k ) = − 0.108012 − 0.066485 i , C d C S ( r p e a k ) = 0.309767 , β C Θ g ( r p e a k ) = 0.147903. (117) \begin{aligned}
V_{\Theta}(r_{\rm peak})&=0.239728,\\
\omega_0^2-V_{\Theta}(r_{\rm peak})&=-0.108012-0.066485\,i,\\
C_{\rm dCS}(r_{\rm peak})&=0.309767,\\
\beta C_{\Theta g}(r_{\rm peak})&=0.147903. \tag*{(117)}
\end{aligned} V Θ ( r peak ) ω 0 2 − V Θ ( r peak ) C dCS ( r peak ) β C Θ g ( r peak ) = 0.239728 , = − 0.108012 − 0.066485 i , = 0.309767 , = 0.147903. (117) where C d C S = 96 π ( M r − Q 2 ) Δ / r 8 C_{\rm dCS}=96\pi(Mr-Q^2)\Delta/r^8 C dCS = 96 π ( M r − Q 2 ) Δ/ r 8 C Θ g = 6 ( M r − Q 2 ) ( l − 1 ) l ( l + 1 ) ( l + 2 ) Δ / ( β r 8 ) C_{\Theta g}=6(Mr-Q^2)(l-1)l(l+1)(l+2)\Delta/(\beta r^8) C Θ g = 6 ( M r − Q 2 ) ( l − 1 ) l ( l + 1 ) ( l + 2 ) Δ/ ( β r 8 ) (58) , and for l = 2 l=2 l = 2 ( l − 1 ) l ( l + 1 ) ( l + 2 ) = 24 (l-1)l(l+1)(l+2)=24 ( l − 1 ) l ( l + 1 ) ( l + 2 ) = 24
β δ ω ∣ Q = 0 W K B ≃ − 0.45 + 0.15 i , (118) \left.\beta\,\delta\omega\right|_{Q=0}^{\rm WKB}\simeq-0.45+0.15\,i, \tag*{(118)} β δ ω ∣ Q = 0 WKB ≃ − 0.45 + 0.15 i , (118) to be compared with the exact − 0.773 − 0.3946 i -0.773-0.3946\,i − 0.773 − 0.3946 i
Charge dependence
Repeating the construction across charge, using the validated baseline ω 0 ( Q ) \omega_0(Q) ω 0 ( Q ) § 1.11 at each point, gives the values in Table VI . As Q → M Q\to M Q → M
Table VI. Local-WKB proxy for the fundamental (l = 2 l=2 l = 2 n = 0 n=0 n = 0 M = α = 1 M=\alpha=1 M = α = 1 Im δ ω \operatorname{Im}\delta\omega Im δ ω
well outside the horizon r + → M r_+ \to M r + → M ( M r − Q 2 ) ∣ r p e a k (Mr-Q^2)|_{r_{\mathrm{peak}}} ( M r − Q 2 ) ∣ r peak O ( 1 ) O(1) O ( 1 ) not suppressed; this is the family distinguished from the near-horizon modes in Sec. XI.
Accuracy
Two approximations enter Eq. (115) : leading-order WKB for the fundamental mode, and the local form of G Θ G_\Theta G Θ ω 0 2 − V Θ \omega_0^2-V_\Theta ω 0 2 − V Θ Im δ ω \operatorname{Im}\delta\omega Im δ ω
Numerical methods and convergence
Exterior complex scaling
In the normalization Ψ = i Ψ R W / ω \Psi=i\Psi_{\mathrm{RW}}/\omega Ψ = i Ψ RW / ω H v = ω 2 v \mathbf{H}\,\mathbf{v}=\omega^2\mathbf{v} H v = ω 2 v
H = − ∂ r ∗ 2 + diag ( V R W , V E M , V Θ ) + C , (119) \mathbf{H}=-\partial_{r_*}^2+\operatorname{diag}(V_{\mathrm{RW}},V_{\mathrm{EM}},V_\Theta)+\mathbf{C}, \tag*{(119)} H = − ∂ r ∗ 2 + diag ( V RW , V EM , V Θ ) + C , (119) where C \mathbf{C} C ω \omega ω
Contour. We continue r ∗ = e i θ s r_*=e^{i\theta}s r ∗ = e i θ s s ∈ [ − S , S ] s\in[-S,S] s ∈ [ − S , S ] θ > ∣ arg ω ∣ \theta>|\arg\omega| θ > ∣ arg ω ∣ e ∓ i ω r ∗ e^{\mp i\omega r_*} e ∓ iω r ∗
Map r ( s ) r(s) r ( s ) The map is obtained by integrating d u / d s = e i θ ( r − r − ) / r 2 du/ds=e^{i\theta}(r-r_-)/r^2 d u / d s = e i θ ( r − r − ) / r 2 u = ln ( r − r + ) u=\ln(r-r_+) u = ln ( r − r + ) 10 − 13 10^{-13} 1 0 − 13
Discretization. Chebyshev collocation with N + 1 N+1 N + 1
Shift extraction. δ ω ( β ) = ω ( β ) − ω ( ∞ ) \delta\omega(\beta)=\omega(\beta)-\omega(\infty) δ ω ( β ) = ω ( β ) − ω ( ∞ ) β = 2 × 10 4 \beta=2\times10^4 β = 2 × 1 0 4 4 × 10 4 4\times10^4 4 × 1 0 4 β \beta β ≤ 4 × 10 − 4 \leq4\times10^{-4} ≤ 4 × 1 0 − 4 l = 2 l=2 l = 2 ≤ 0.4 % \leq0.4\% ≤ 0.4% l = 6 l=6 l = 6
Gates.
Schwarzschild l = 2 l=2 l = 2 M ω = 0.3736716777 − 0.0889623110 i M\omega=0.3736716777-0.0889623110i M ω = 0.3736716777 − 0.0889623110 i 8 × 10 − 9 8\times10^{-9} 8 × 1 0 − 9
RN baseline: agrees with the charge series to 8 × 10 − 9 8\times10^{-9} 8 × 1 0 − 9 Q / M = 0.1 Q/M=0.1 Q / M = 0.1 [40 ] at all charges (Table III (PDF p. 16) ).
The neutral shift (82) agrees with the frequency-domain result of Ref. [22 ] .
The branch-reduced system (60) reproduces the full-system coefficients to 5 × 10 − 7 5\times10^{-7} 5 × 1 0 − 7 l = 2 l=2 l = 2 10 − 4 10^{-4} 1 0 − 4 l = 6 l=6 l = 6 β = 2 × 10 4 \beta=2\times10^4 β = 2 × 1 0 4 β − 2 \beta^{-2} β − 2 1.3 × 10 − 4 1.3\times10^{-4} 1.3 × 1 0 − 4 1.4 × 10 − 6 1.4\times10^{-6} 1.4 × 1 0 − 6 1.5 × 10 − 8 1.5\times10^{-8} 1.5 × 1 0 − 8 β = 2 × 10 4 , 5 , 6 \beta=2\times10^{4,5,6} β = 2 × 1 0 4 , 5 , 6 l = 6 l=6 l = 6 Q = 0.2 M Q=0.2M Q = 0.2 M O ( ζ 2 ) O(\zeta^2) O ( ζ 2 )
Independent check: shooting integration of the full { Ψ , Ψ e , Θ } \{\Psi,\Psi_e,\Theta\} { Ψ , Ψ e , Θ } β δ ω \beta\,\delta\omega β δ ω 1.4 × 10 − 6 1.4\times10^{-6} 1.4 × 1 0 − 6 l = 2 l=2 l = 2 Q / M = 0 Q/M=0 Q / M = 0 0.6 0.6 0.6 0.9 0.9 0.9 ω 0 \omega_0 ω 0 8 × 10 − 9 8\times10^{-9} 8 × 1 0 − 9
Linearity in ζ \zeta ζ β = 5 × 10 3 \beta=5\times10^3 β = 5 × 1 0 3 3.2 × 10 5 3.2\times10^5 3.2 × 1 0 5 β δ ω = c 1 + c 2 / β + c 3 / β 2 \beta\,\delta\omega=c_1+c_2/\beta+c_3/\beta^2 β δ ω = c 1 + c 2 / β + c 3 / β 2 ≤ 6 × 10 − 6 \leq6\times10^{-6} ≤ 6 × 1 0 − 6 c 1 c_1 c 1 10 − 6 10^{-6} 1 0 − 6
Convergence. We test the shift itself, not only the two frequencies. Varying θ ∈ [ 0.5 , 0.8 ] \theta\in[0.5,0.8] θ ∈ [ 0.5 , 0.8 ] S ∈ [ 45 , 120 ] S\in[45,120] S ∈ [ 45 , 120 ] N ∈ [ 220 , 460 ] N\in[220,460] N ∈ [ 220 , 460 ] r ∗ = r ∗ c + e i θ ( s − r ∗ c ) r_*=r_*^c+e^{i\theta}(s-r_*^c) r ∗ = r ∗ c + e i θ ( s − r ∗ c ) r ∗ c ∈ [ − 2 , 5 ] M r_*^c\in[-2,5]M r ∗ c ∈ [ − 2 , 5 ] M β δ ω \beta\,\delta\omega β δ ω 4 × 10 − 5 4\times10^{-5} 4 × 1 0 − 5 l = 2 l=2 l = 2 Q = 0 Q=0 Q = 0 3 × 10 − 5 3\times10^{-5} 3 × 1 0 − 5 Q = 0.6 M Q=0.6M Q = 0.6 M 4 × 10 − 9 4\times10^{-9} 4 × 1 0 − 9 Q = 0.95 M Q=0.95M Q = 0.95 M 2 × 10 − 8 2\times10^{-8} 2 × 1 0 − 8 l = 6 l=6 l = 6 7 × 10 − 7 7\times10^{-7} 7 × 1 0 − 7 Q = 0.3 M Q=0.3M Q = 0.3 M θ = 0.4 \theta=0.4 θ = 0.4 Table VII summarizes the numerical uncertainty of each quoted quantity.
Zero-damping modes
Complex scaling cannot reach arg ω ≃ − π / 2 \arg\omega\simeq-\pi/2 arg ω ≃ − π /2 ω / κ + \omega/\kappa_+ ω / κ + ± i ω / 2 κ + \pm i\omega/2\kappa_+ ± iω /2 κ + [39 ] . We therefore use a shooting method in r r r
Horizon side. The ingoing solutions are generated by a matrix Frobenius series to order 60. The Taylor coefficients of the regularized equation are obtained by FFT on the circle ∣ r − r + ∣ = ( r + − r − ) / 2 |r-r_+|=(r_+-r_-)/2 ∣ r − r + ∣ = ( r + − r − ) /2 r − r + = 0.3 ( r + − r − ) r-r_+=0.3(r_+-r_-) r − r + = 0.3 ( r + − r − )
Infinity side. Outgoing solutions are started on the ray r = 8 M + t e 3 i π / 4 r=8M+t e^{3i\pi/4} r = 8 M + t e 3 iπ /4 e i ω r e^{i\omega r} e iω r ω ≃ − i γ \omega\simeq-i\gamma ω ≃ − iγ
Quasinormal condition. The vanishing of the determinant of horizon and outgoing solutions at r = r + + 2 ( r + − r − ) r=r_++2(r_+-r_-) r = r + + 2 ( r + − r − )
Validation.
Photon-sphere modes are reproduced to 10 − 10 10^{-10} 1 0 − 10 l = 2 l=2 l = 2 n = 1 n=1 n = 1
For the scaled ZDM residual M ( n + 1 2 ) − 1 σ − 2 Δ ω I M(n+\frac{1}{2})^{-1}\sigma^{-2}\Delta\omega_I M ( n + 2 1 ) − 1 σ − 2 Δ ω I [47 ] (their Fig. 10) we obtain:
– − 3.02 -3.02 − 3.02 − 1.35 -1.35 − 1.35 j = 1 j=1 j = 1 l = 1 l=1 l = 1 n = 0 , 1 n=0,1 n = 0 , 1 3.0 3.0 3.0 1.35 1.35 1.35
– − 2.01 -2.01 − 2.01 − 1.01 -1.01 − 1.01 j = 2 j=2 j = 2 l = 2 l=2 l = 2 2.0 2.0 2.0 1.0 1.0 1.0
– − 1.69 -1.69 − 1.69 j = 1 j=1 j = 1 l = 2 l=2 l = 2 n = 1 n=1 n = 1 1.6 1.6 1.6
Code availability. All codes and data used here are publicly available; see the Data availability statement [55 ] .
