Introduction
Topological methods provide a way to classify black-hole properties that is less sensitive to the details of a particular solution. The approach adopted here is based on Duan’s ϕ \phi ϕ [1 ] : a two-dimensional vector field is constructed from a scalar function, and its zero points carry integer winding numbers. Their sum, the total topological charge (TTC), remains unchanged under smooth deformations, provided the relevant boundary conditions are preserved, and can therefore be used to distinguish topological classes. This framework has been applied to light rings and photon spheres [58 , 59 , 60 , 61 ] and to black-hole thermodynamics [72 , 73 ] , and has since been explored in a wide range of gravitational models [2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 ] .
For photon spheres, Cunha and collaborators established topological results for light rings around ultracompact objects and black holes, including the existence and stability properties of the relevant circular null orbits [58 , 59 , 60 ] . We subsequently related these results to a topological charge and extended the analysis asymptotically anti-de Sitter and de Sitter spacetimes [61 ] . In our previous work, this correspondence was used to classify the spacetime around black holes and to organize parameter ranges by their horizon and orbit structure [62 , 63 , 64 , 65 , 66 , 67 , 68 ] . We apply this construction to the circular null geodesics of the background metric and to the extraordinary electromagnetic characteristics. The distinction is relevant in nonlinear electrodynamics, where the propagation cone depends on derivatives of the electromagnetic Lagrangian [99 , 100 ] .
A complementary line of research applies topological methods to black-hole thermodynamics. In the conventional approach, one studies the temperature as a function of the horizon radius and examines the Gibbs free energy to identify phase transitions; a crossing of the locally stable branches of a free-energy swallowtail, for example, identifies a first-order small–large black-hole transition [69 , 70 , 71 ] . Wei, Liu and Mann showed that this information can also be encoded in the zeros of a vector field constructed from a generalized free energy, with the associated topological charges distinguishing stable and unstable branches [72 , 73 ] . Subsequent studies have applied this method to a variety of black-hole solutions and thermodynamic ensembles. The present analysis uses the topological classification alongside the temperature and free-energy curves, so that the topological results can be checked against the conventional thermodynamic description.
The holographic description of black-hole thermodynamics provides a further connection between bulk gravitational physics and the thermodynamics of a dual conformal field theory (CFT). In extended black-hole thermodynamics, variations of the cosmological constant and Newton’s constant can be translated into variations of boundary thermodynamic quantities, including the CFT central charge and its conjugate chemical potential [92 , 94 ] . This dictionary has been used to study ensemble-dependent phase transitions and critical behavior for charged and rotating AdS black holes [93 , 95 ] . Related work has examined holographic phase transitions in charged Gauss–Bonnet gravity [71 ] , the relation between CFT thermodynamics and the weak gravity and weak cosmic censorship conjectures [98 ] , and classical and topological CFT thermodynamics in Lifshitz and hyperscaling-violating geometries [96 ] . Recent applications to Einstein–Maxwell–power–Yang–Mills AdS black holes further illustrate how nonlinear gauge sectors can influence the phase structure of the boundary theory [97 ] . Together, these studies provide the context for examining the holographic thermodynamics of the magnetically charged nonlinear-electrodynamics black hole considered here.
The black hole studied in this work is the magnetically charged solution of Einstein gravity with a negative cosmological constant coupled to a two-parameter nonlinear electrodynamics, introduced by Kruglov [89 ] . That work presented the metric and mass functions, the extended first law, the Smarr relation, the heat capacity and the Gibbs free energy, and estimated a critical pressure of approximately 0.003 for q = 1 q=1 q = 1 β = 0.1 \beta=0.1 β = 0.1 σ = 0.75 \sigma=0.75 σ = 0.75 β \beta β σ \sigma σ q q q 2 β B 2\beta\mathcal{B} 2 β B [89 ] , and extend the holographic dictionary of Refs. [92 , 93 , 94 ] to this model.
The paper is organized as follows. Section II summarizes the topological construction for photon spheres. Section III introduces the model, discusses the relation between the integration constant in the metric and the total mass, and presents the background classification and the effective-optical comparison. Section IV gives the extended first law and Smarr relation and examines the thermodynamic topology and free energy. Section V constructs the holographic dictionary, checks the first law and Euler relation of the dual CFT, and studies the relevant ensembles, critical point and critical exponents. Section VI presents our conclusions.
Topological Description of Circular Null Orbits
Photon spheres are linked to gravitational lensing and to the shadow of a black hole [4 , 74 , 75 , 76 , 77 , 78 , 79 , 80 ] , and circular null orbits, their stability and the associated photon surfaces have been studied extensively [81 , 82 , 83 , 84 , 85 , 86 , 87 , 88 ] . The construction used here is described in detail in Refs. [58 , 59 , 60 , 61 , 62 , 63 , 64 ] ; only the relations needed below are recalled. The construction applies to a static metric on a nondegenerate domain. In the background analysis below, “photon sphere” denotes a circular null orbit of the spacetime metric; the effective electromagnetic orbits are treated in § 1.2 H. A static, spherically symmetric metric is written as
d s 2 = − f ( r ) d t 2 + d r 2 g ( r ) + h ( r ) ( d θ 2 + sin 2 θ d φ 2 ) . (1) ds^2=-f(r)dt^2+\frac{dr^2}{g(r)}+h(r)\left(d\theta^2+\sin^2\theta\,d\varphi^2\right). \tag*{(1)} d s 2 = − f ( r ) d t 2 + g ( r ) d r 2 + h ( r ) ( d θ 2 + sin 2 θ d φ 2 ) . (1) For null geodesics in the equatorial plane θ = π / 2 \theta=\pi/2 θ = π /2 L = 1 2 g μ ν x ˙ μ x ˙ ν = 0 \mathcal{L}=\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=0 L = 2 1 g μν x ˙ μ x ˙ ν = 0 E = f t ˙ E= f\dot{t} E = f t ˙ L = h ϕ ˙ L=h\dot{\phi} L = h ϕ ˙ L = 0 \mathcal{L}=0 L = 0 r ˙ 2 + V e f f ( r ) = 0 \dot{r}^{2}+V_{\mathrm{eff}}(r)=0 r ˙ 2 + V eff ( r ) = 0
V e f f ( r ) = g ( r ) ( L 2 h ( r ) − E 2 f ( r ) ) . (2) V_{\mathrm{eff}}(r)=g(r)\left(\frac{L^2}{h(r)}-\frac{E^2}{f(r)}\right). \tag*{(2)} V eff ( r ) = g ( r ) ( h ( r ) L 2 − f ( r ) E 2 ) . (2) A circular null orbit satisfies V e f f = V e f f ′ = 0 V_{\mathrm{eff}}=V_{\mathrm{eff}}'=0 V eff = V eff ′ = 0 V e f f V_{\mathrm{eff}} V eff E E E L L L
H ( r , θ ) = − g t t g φ φ = 1 sin θ f ( r ) h ( r ) (3) \mathbf{H}(r,\theta)=\sqrt{\frac{-g_{tt}}{g_{\varphi\varphi}}}=\frac{1}{\sin\theta}\sqrt{\frac{f(r)}{h(r)}} \tag*{(3)} H ( r , θ ) = g φφ − g tt = sin θ 1 h ( r ) f ( r ) (3) is used instead, since it is built from the metric alone. Its extrema in r r r θ = π / 2 \theta=\pi/2 θ = π /2 θ \theta θ H \mathbf{H} H θ = π / 2 \theta=\pi/2 θ = π /2
ϕ = ( ϕ r , ϕ θ ) = ( g ∂ r H , ∂ θ H h ) (4) \boldsymbol{\phi}=(\phi^r,\phi^\theta)=\left(\sqrt{g}\,\partial_r\mathbf{H},\frac{\partial_\theta\mathbf{H}}{\sqrt{h}}\right) \tag*{(4)} ϕ = ( ϕ r , ϕ θ ) = ( g ∂ r H , h ∂ θ H ) (4) vanishes precisely at these orbits. With ϕ = ∥ ϕ ∥ e i Ω \boldsymbol{\phi}=\|\boldsymbol{\phi}\|e^{i\Omega} ϕ = ∥ ϕ ∥ e i Ω z i z_i z i
ω i = 1 2 π ∮ C i d Ω , Ω = arctan ϕ θ ϕ r , (5) \omega_i=\frac{1}{2\pi}\oint_{C_i}d\Omega,\qquad\Omega=\arctan\frac{\phi^\theta}{\phi^r}, \tag*{(5)} ω i = 2 π 1 ∮ C i d Ω , Ω = arctan ϕ r ϕ θ , (5) where C i C_i C i ( r , θ ) (r,\theta) ( r , θ ) z i z_i z i Ω \Omega Ω C i C_i C i ω i = κ i ζ i \omega_i=\kappa_i\zeta_i ω i = κ i ζ i κ i ≥ 1 \kappa_i\geq1 κ i ≥ 1 n = ϕ / ∥ ϕ ∥ \boldsymbol{n}=\boldsymbol{\phi}/\|\boldsymbol{\phi}\| n = ϕ /∥ ϕ ∥ z i z_i z i ζ i = ± 1 \zeta_i=\pm1 ζ i = ± 1 ϕ \boldsymbol{\phi} ϕ j μ = ( 2 π ) − 1 ϵ μ ν ρ ϵ a b ∂ ν ϕ a ∂ ρ ϕ b j^\mu=(2\pi)^{-1}\epsilon^{\mu\nu\rho}\epsilon_{ab}\partial_\nu\phi^a\partial_\rho\phi^b j μ = ( 2 π ) − 1 ϵ μν ρ ϵ ab ∂ ν ϕ a ∂ ρ ϕ b ( t , r , θ ) (t,r,\theta) ( t , r , θ ) ∂ μ j μ = 0 \partial_\mu j^\mu=0 ∂ μ j μ = 0 ϕ \boldsymbol{\phi} ϕ
Q = ∑ i ω i . (6) Q=\sum_i\omega_i. \tag*{(6)} Q = i ∑ ω i . (6) Near a maximum of H \mathbf{H} H r r r ( ϕ r , ϕ θ ) ∝ ( − x , y ) (\phi^r,\phi^\theta)\propto(-x,y) ( ϕ r , ϕ θ ) ∝ ( − x , y ) ω = − 1 \omega=-1 ω = − 1 ( x , y ) (x,y) ( x , y ) ω = + 1 \omega=+1 ω = + 1 [58 , 59 , 60 , 61 ] , the background charge is Q = − 1 Q=-1 Q = − 1 Q = 0 Q=0 Q = 0 + 1 − 1 +1-1 + 1 − 1 [102 ] .
The Four-Dimensional AdS Magnetic Black Hole
Solution and mass parameters
The action is
I = ∫ d 4 x − g [ R − 2 Λ 16 π G + L ( F ) ] . (7) I=\int d^4x\sqrt{-g}\left[\frac{R-2\Lambda}{16\pi G}+\mathcal{L}(\mathcal{F})\right]. \tag*{(7)} I = ∫ d 4 x − g [ 16 π G R − 2Λ + L ( F ) ] . (7) with Λ = − 3 / l 2 \Lambda=-3/l^{2} Λ = − 3/ l 2 [89 ]
L ( F ) = − F 4 π ( 1 + 2 β F ) σ , F = F μ ν F μ ν 4 = B 2 − E 2 2 . (8) \mathcal{L}(\mathcal{F})=-\frac{\mathcal{F}}{4\pi(1+2\beta\mathcal{F})^{\sigma}},\qquad\mathcal{F}=\frac{F_{\mu\nu}F^{\mu\nu}}{4}=\frac{B^{2}-E^{2}}{2}. \tag*{(8)} L ( F ) = − 4 π ( 1 + 2 β F ) σ F , F = 4 F μν F μν = 2 B 2 − E 2 . (8) The coupling β > 0 \beta>0 β > 0 [ L 4 ] [L^{4}] [ L 4 ] c = ℏ = 1 c=\hbar=1 c = ℏ = 1 [ L 2 ] [L^{2}] [ L 2 ] G = c = 1 G=c=1 G = c = 1 q q q σ > 0 \sigma>0 σ > 0 B = q / r 2 B=q/r^{2} B = q / r 2 G = 1 G=1 G = 1
f ( r ) = 1 − 2 m 0 r + r 2 l 2 − I ( r ) r , (9) f(r)=1-\frac{2m_{0}}{r}+\frac{r^{2}}{l^{2}}-\frac{\mathcal{I}(r)}{r}, \tag*{(9)} f ( r ) = 1 − r 2 m 0 + l 2 r 2 − r I ( r ) , (9) I ( r ) = q 2 r 4 σ − 1 ( 4 σ − 1 ) ( β q 2 ) σ 2 F 1 ( σ , σ − 1 4 ; σ + 3 4 ; − r 4 β q 2 ) , (10) \mathcal{I}(r)=\frac{q^{2}r^{4\sigma-1}}{(4\sigma-1)(\beta q^{2})^{\sigma}}\,{}_2F_{1}\left(\sigma,\sigma-\frac{1}{4};\sigma+\frac{3}{4};-\frac{r^{4}}{\beta q^{2}}\right), \tag*{(10)} I ( r ) = ( 4 σ − 1 ) ( β q 2 ) σ q 2 r 4 σ − 1 2 F 1 ( σ , σ − 4 1 ; σ + 4 3 ; − β q 2 r 4 ) , (10) where m 0 m_{0} m 0 2 F 1 ( a , b ; c ; z ) {}_2F_{1}(a,b;c;z) 2 F 1 ( a , b ; c ; z ) I \mathcal{I} I
I ′ ( r ) = q 2 r 4 σ − 2 ( r 4 + β q 2 ) σ . (11) \mathcal{I}'(r)=\frac{q^{2}r^{4\sigma-2}}{(r^{4}+\beta q^{2})^{\sigma}}. \tag*{(11)} I ′ ( r ) = ( r 4 + β q 2 ) σ q 2 r 4 σ − 2 . (11) For σ > 1 / 4 \sigma>1/4 σ > 1/4 m M m_{M} m M M = m 0 + m M M=m_{0}+m_{M} M = m 0 + m M [89 ] , with
m M = q 3 / 2 Γ ( σ − 1 4 ) Γ ( 5 4 ) 2 β 1 / 4 Γ ( σ ) . (12) m_{M}=\frac{q^{3/2}\Gamma\left(\sigma-\frac{1}{4}\right)\Gamma\left(\frac{5}{4}\right)}{2\beta^{1/4}\Gamma(\sigma)}. \tag*{(12)} m M = 2 β 1/4 Γ ( σ ) q 3/2 Γ ( σ − 4 1 ) Γ ( 4 5 ) . (12) The metric function then takes the form
f ( r ) = 1 − 2 M r + q 2 r 2 2 F 1 ( σ , 1 4 ; 5 4 ; − β q 2 r 4 ) + r 2 l 2 , (13) f(r)=1-\frac{2M}{r}+\frac{q^{2}}{r^{2}}\,{}_2F_{1}\left(\sigma,\frac{1}{4};\frac{5}{4};-\frac{\beta q^{2}}{r^{4}}\right)+\frac{r^{2}}{l^{2}}, \tag*{(13)} f ( r ) = 1 − r 2 M + r 2 q 2 2 F 1 ( σ , 4 1 ; 4 5 ; − r 4 β q 2 ) + l 2 r 2 , (13) in which M M M σ ≤ 1 / 4 \sigma\leq1/4 σ ≤ 1/4 m 0 m_{0} m 0 m M m_{M} m M σ = 1 / 4 \sigma=1/4 σ = 1/4 Γ ( σ − 1 / 4 ) \Gamma(\sigma-1/4) Γ ( σ − 1/4 ) I \mathcal{I} I σ = 1 / 4 \sigma=1/4 σ = 1/4 m 0 m_{0} m 0 σ ≠ 1 / 4 \sigma\neq1/4 σ = 1/4 M = m 0 + m M M=m_{0}+m_{M} M = m 0 + m M m M m_{M} m M σ < 1 / 4 ) , i s i d e n t i c a l t o E q . ( 9 ) ; t h i s i d e n t i t y w a s v e r i f i e d n u m e r i c a l l y t o $ 10 − 15 \sigma<1/4), is identical to Eq. (9); this identity was verified numerically to \$10^{-15} σ < 1/4 ) , i s i d e n t i c a l t o E q . ( 9 ) ; t hi s i d e n t i t y w a s v er i f i e d n u m er i c a l l y t o $1 0 − 15 σ = 0.12 \sigma=0.12 σ = 0.12 0.3 0.3 0.3 0.9 0.9 0.9 [62 , 63 , 64 , 65 , 66 , 67 , 68 ] , the constant m 0 m_{0} m 0 m 0 m_{0} m 0 M M M M M M
Background circular-null-orbit functions
With g = f g=f g = f h = r 2 h=r^{2} h = r 2 (3) and (4) give
H ( r , θ ) = f ( r ) r sin θ . \mathbf{H}(r,\theta)=\frac{\sqrt{f(r)}}{r\sin\theta}. H ( r , θ ) = r sin θ f ( r ) . ϕ r = r f ′ ( r ) − 2 f ( r ) 2 r 2 sin θ , (14) \phi^{r}=\frac{rf'(r)-2f(r)}{2r^{2}\sin\theta}, \tag*{(14)} ϕ r = 2 r 2 sin θ r f ′ ( r ) − 2 f ( r ) , (14) ϕ θ = − f ( r ) cos θ r 2 sin 2 θ , \phi^{\theta}=-\frac{\sqrt{f(r)}\cos\theta}{r^{2}\sin^{2}\theta}, ϕ θ = − r 2 sin 2 θ f ( r ) cos θ , and, from (9) and (11),
r f ′ − 2 f = 6 m 0 r − 2 + 3 I ( r ) r − I ′ ( r ) . (15) rf'-2f=\frac{6m_{0}}{r}-2+\frac{3\mathcal{I}(r)}{r}-\mathcal{I}'(r). \tag*{(15)} r f ′ − 2 f = r 6 m 0 − 2 + r 3 I ( r ) − I ′ ( r ) . (15) The radial component of ϕ \phi ϕ σ + 1 \sigma+1 σ + 1 σ + 3 / 4 \sigma+3/4 σ + 3/4 H \mathbf{H} H σ \sigma σ ϕ r ( r , π / 2 ) = 0 \phi^{r}(r,\pi/2)=0 ϕ r ( r , π /2 ) = 0 l l l r 2 / l 2 r^{2}/l^{2} r 2 / l 2 (15) . The positive roots of f ( r ) = 0 f(r)=0 f ( r ) = 0
All numerical values were obtained with arbitrary-precision evaluation of the hypergeometric functions. Roots were bracketed on a logarithmic grid and refined by Brent’s method, and threshold values of σ \sigma σ f = f ′ = 0 f=f'=0 f = f ′ = 0 ϕ r = ∂ r ϕ r = 0 \phi^{r}=\partial_{r}\phi^{r}=0 ϕ r = ∂ r ϕ r = 0 σ \sigma σ (5) by numerical integration along small closed counterclockwise curves around each zero point, and agree with the stated values. We set m 0 = 1 m_{0}=1 m 0 = 1 l = 1 l=1 l = 1 q = 0.5 q=0.5 q = 0.5 q < m 0 q<m_{0} q < m 0 q = 1.2 q=1.2 q = 1.2 q > m 0 q>m_{0} q > m 0
Case q < m 0 q<m_{0} q < m 0
For q = 0.5 q=0.5 q = 0.5 σ \sigma σ β \beta β Table 1 . The geometry has two horizons for σ < σ 1 \sigma<\sigma_{1} σ < σ 1 σ 1 < σ < 1 / 4 \sigma_{1}<\sigma<1/4 σ 1 < σ < 1/4 σ > 1 / 4 \sigma>1/4 σ > 1/4 σ 2 \sigma_{2} σ 2 Table 1 is the value at which the pair of photon spheres of the horizonless geometry disappears. Because the thresholds move with β \beta β β = 0.1 \beta=0.1 β = 0.1 Figure 1 .
Table 1. Background-metric thresholds in σ \sigma σ q = 0.5 q=0.5 q = 0.5 m 0 = l = 1 m_{0}=l=1 m 0 = l = 1 σ < σ 1 \sigma<\sigma_{1} σ < σ 1 σ 1 < σ < 1 / 4 \sigma_{1}<\sigma<1/4 σ 1 < σ < 1/4 σ 1 < σ < σ 2 \sigma_{1}<\sigma<\sigma_{2} σ 1 < σ < σ 2
Figure 1. Background metric function f ( r ) f(r) f ( r ) q = 0.5 q = 0.5 q = 0.5 m 0 = 1 m_0 = 1 m 0 = 1 l = 1 l = 1 l = 1 β = 0.1 \beta= 0.1 β = 0.1 σ = 0.12 \sigma= 0.12 σ = 0.12 0.19 0.19 0.19 0.3 0.3 0.3
For σ = 0.12 \sigma=0.12 σ = 0.12 r = 0.1832 r=0.1832 r = 0.1832 r = 0.6953 r=0.6953 r = 0.6953 ϕ \phi ϕ Figure 2 . It has a single zero point at r = 1.8087 r=1.8087 r = 1.8087 − 1 -1 − 1 Q = − 1 Q = -1 Q = − 1 ϕ r \phi^r ϕ r r = 0.2412 r = 0.2412 r = 0.2412
Figure 2. Normalized topological field n n n ( r , θ ) (r,\theta) ( r , θ ) σ = 0.12 \sigma= 0.12 σ = 0.12 q = 0.5 q = 0.5 q = 0.5 m 0 = 1 m_0 = 1 m 0 = 1 l = 1 l = 1 l = 1 β = 0.1 \beta= 0.1 β = 0.1 r + = 0.6953 r_+ = 0.6953 r + = 0.6953 ( r , θ ) = ( 1.8087 , π / 2 ) (r,\theta) = (1.8087,\pi/2) ( r , θ ) = ( 1.8087 , π /2 ) − 1 -1 − 1
For σ = 0.15 \sigma= 0.15 σ = 0.15 Figure 3 (a) shows a stable photon sphere at r = 0.4479 r = 0.4479 r = 0.4479 ω = + 1 \omega= +1 ω = + 1 r = 1.0606 r = 1.0606 r = 1.0606 ω = − 1 \omega= -1 ω = − 1 Figure 3 (b) shows the corresponding minimum and maximum of H \mathbf{H} H Q = 0 Q = 0 Q = 0 σ 1 < σ < σ 2 \sigma_1 < \sigma< \sigma_2 σ 1 < σ < σ 2 0.1465105 < σ < 0.1548466 0.1465105 < \sigma< 0.1548466 0.1465105 < σ < 0.1548466 β = 0.1 \beta= 0.1 β = 0.1 σ 2 < σ < 1 / 4 \sigma_2 < \sigma< 1/4 σ 2 < σ < 1/4
Figure 3. (a) Normalized topological field n n n σ = 0.15 \sigma= 0.15 σ = 0.15 r = 0.4479 r = 0.4479 r = 0.4479 ω = + 1 \omega= +1 ω = + 1 r = 1.0606 r = 1.0606 r = 1.0606 ω = − 1 \omega= -1 ω = − 1 H ( r , π / 2 ) \mathbf{H}(r,\pi/2) H ( r , π /2 ) q = 0.5 q = 0.5 q = 0.5 m 0 = 1 m_0 = 1 m 0 = 1 l = 1 l = 1 l = 1 β = 0.1 \beta= 0.1 β = 0.1
the field ϕ r \phi^r ϕ r M ( σ ) = m 0 + m M ( σ ) M(\sigma) = m_0 + m_M(\sigma) M ( σ ) = m 0 + m M ( σ ) M p s min M_{\mathrm{ps}}^{\min} M ps m i n
For σ > 1 / 4 \sigma> 1/4 σ > 1/4 σ = 0.3 \sigma= 0.3 σ = 0.3 r + = 1.5831 r_+ = 1.5831 r + = 1.5831 r = 8.5046 r = 8.5046 r = 8.5046 Figure 4 shows the single zero of ϕ r \phi^r ϕ r H \mathbf{H} H σ → 1 / 4 + \sigma\to1/4^+ σ → 1/ 4 + r = 27.42 r = 27.42 r = 27.42 σ = 0.26 \sigma= 0.26 σ = 0.26 r = 239.6 r = 239.6 r = 239.6 σ = 0.251 \sigma= 0.251 σ = 0.251
Figure 4. (a) Background radial component ϕ r ( r , π / 2 ) \phi^r(r,\pi/2) ϕ r ( r , π /2 ) σ = 0.3 \sigma= 0.3 σ = 0.3 q = 0.5 q = 0.5 q = 0.5 m 0 = 1 m_0 = 1 m 0 = 1 l = 1 l = 1 l = 1 β = 0.1 \beta= 0.1 β = 0.1 r = 8.5046 r = 8.5046 r = 8.5046 H ( r , π / 2 ) \mathbf{H}(r,\pi/2) H ( r , π /2 ) r + = 1.5831 r_+ = 1.5831 r + = 1.5831
Case q > m 0 q > m_0 q > m 0
For q = 1.2 q = 1.2 q = 1.2 0 < σ < 1 / 4 0 < \sigma< 1/4 0 < σ < 1/4 β = 0.001 \beta= 0.001 β = 0.001 0.1 0.1 0.1 1.2 1.2 1.2 σ = 0.02 \sigma= 0.02 σ = 0.02 0.08 0.08 0.08 0.15 0.15 0.15 0.2 0.2 0.2 0.24 0.24 0.24 β = 0.1 \beta= 0.1 β = 0.1 0.01 0.01 0.01 σ \sigma σ Figure 5 . The field ϕ r \phi^r ϕ r Figure 6 (a), curve for σ = 0.23 \sigma= 0.23 σ = 0.23 § 1.2 G, and the range is classified as NS0 in the background-metric sector. For σ > 1 / 4 \sigma>1/4 σ > 1/4 σ = 0.9 \sigma=0.9 σ = 0.9 r + = 1.3280 r_{+}=1.3280 r + = 1.3280 r = 6.6879 r=6.6879 r = 6.6879 Figure 6 ), and the photon-sphere radius grows without bound as σ → 1 / 4 + \sigma\to1/4^{+} σ → 1/ 4 + r = 93.82 r=93.82 r = 93.82 σ = 0.26 \sigma=0.26 σ = 0.26 r = 882.8 r=882.8 r = 882.8 σ = 0.251 \sigma=0.251 σ = 0.251
Figure 5. Background metric function f ( r ) f(r) f ( r ) q = 1.2 q=1.2 q = 1.2 m 0 = 1 m_0=1 m 0 = 1 l = 1 l=1 l = 1 β = 0.1 \beta=0.1 β = 0.1 σ = 0.1 \sigma=0.1 σ = 0.1 σ = 0.1 < 1 / 4 \sigma=0.1<1/4 σ = 0.1 < 1/4 σ = 0.5 \sigma=0.5 σ = 0.5
Figure 6. (a) Background radial component ϕ r ( r , π / 2 ) \phi^{r}(r,\pi/2) ϕ r ( r , π /2 ) σ = 0.9 \sigma=0.9 σ = 0.9 r = 6.6879 r=6.6879 r = 6.6879 σ = 0.23 \sigma=0.23 σ = 0.23 q = 1.2 q=1.2 q = 1.2 m 0 = 1 m_0=1 m 0 = 1 l = 1 l=1 l = 1 β = 0.1 \beta=0.1 β = 0.1 H ( r , π / 2 ) H(r,\pi/2) H ( r , π /2 ) r + = 1.3280 r_{+}=1.3280 r + = 1.3280
Table 2. Horizon and background circular-null-orbit classification for m 0 = l = 1 m_0=l=1 m 0 = l = 1 β = 0.1 \beta=0.1 β = 0.1 σ \sigma σ = 0 =0 = 0 m 0 m_0 m 0 σ = 1 / 4 \sigma=1/4 σ = 1/4
Dependence on the magnetic charge
The value q = m 0 q=m_0 q = m 0 σ \sigma σ β \beta β β = 0.1 \beta=0.1 β = 0.1 q = 0.3 q=0.3 q = 0.3 σ \sigma σ q = 0.8 < m 0 q=0.8<m_0 q = 0.8 < m 0 σ = 0.05 \sigma=0.05 σ = 0.05 σ = 0.12 \sigma=0.12 σ = 0.12 q = 1 q=1 q = 1 q = 1.2 q=1.2 q = 1.2
Relation to the ADM mass
The parameter m 0 m_0 m 0 σ ≠ 1 / 4 \sigma\ne1/4 σ = 1/4 M = m 0 + m M ( σ ) M=m_0+m_M(\sigma) M = m 0 + m M ( σ ) m 0 m_0 m 0 σ \sigma σ σ = 1 / 4 \sigma=1/4 σ = 1/4 q = 0.5 q=0.5 q = 0.5 m 0 = 1 m_0=1 m 0 = 1 β = 0.1 \beta=0.1 β = 0.1 M = 0.6950 , 0.5105 , − 0.2804 , − 6.571 , 9.145 , 2.854 , 1.369 M=0.6950, 0.5105, -0.2804, -6.571, 9.145, 2.854, 1.369 M = 0.6950 , 0.5105 , − 0.2804 , − 6.571 , 9.145 , 2.854 , 1.369 σ = 0.12 , 0.15 , 0.20 , 0.24 , 0.26 , 0.30 , 0.90 \sigma=0.12,0.15,0.20,0.24,0.26,0.30,0.90 σ = 0.12 , 0.15 , 0.20 , 0.24 , 0.26 , 0.30 , 0.90 σ \sigma σ 0 < σ < 1 / 4 0<\sigma<1/4 0 < σ < 1/4 σ ≃ 0.189 \sigma\simeq0.189 σ ≃ 0.189 − ∞ -\infty − ∞ σ → 1 / 4 − \sigma\to1/4^{-} σ → 1/ 4 − + ∞ +\infty + ∞ σ → 1 / 4 + \sigma\to1/4^{+} σ → 1/ 4 + σ 2 < σ < 1 / 4 \sigma_{2}<\sigma<1/4 σ 2 < σ < 1/4 σ → 1 / 4 + \sigma\to1/4^{+} σ → 1/ 4 + σ = 1 / 4 \sigma=1/4 σ = 1/4 m 0 m_{0} m 0 M M M (13) , which is analytic across σ = 1 / 4 \sigma=1/4 σ = 1/4 σ ≥ 1 \sigma\geq1 σ ≥ 1 m 0 = 0 m_{0}=0 m 0 = 0 [89 ] . The configurations in the fixed-m 0 = 1 m_{0}=1 m 0 = 1 σ ≤ 1 / 4 \sigma\leq1/4 σ ≤ 1/4 M M M r > 0 r>0 r > 0
Table III. Background critical masses at fixed M M M β = 0.1 \beta=0.1 β = 0.1 l = 1 l=1 l = 1 M p s min M_{\rm ps}^{\min} M ps m i n M p s ( r ) M_{\rm ps}(r) M ps ( r ) M e x t M_{\rm ext} M ext M h ( r ) M_{h}(r) M h ( r ) σ < 1 / 4 \sigma<1/4 σ < 1/4 M < M p s min M<M_{\rm ps}^{\min} M < M ps m i n M p s min < M < M e x t M_{\rm ps}^{\min}<M<M_{\rm ext} M ps m i n < M < M ext
Classification at fixed total mass
The thresholds of Table I refer to the slice m 0 = 1 m_{0}=1 m 0 = 1 M M M (13) , which is analytic at σ = 1 / 4 \sigma=1/4 σ = 1/4 M = M h ( r ) M=M_{h}(r) M = M h ( r ) r f ′ − 2 f = 0 rf^{\prime}-2f=0 r f ′ − 2 f = 0 M = M p s ( r ) M=M_{\rm ps}(r) M = M ps ( r )
M h ( r ) = r 2 [ 1 + r 2 l 2 + q 2 r 2 2 F 1 ( σ , 1 4 ; 5 4 ; − β q 2 r 4 ) ] , M p s ( r ) = r 6 [ 2 + 3 q 2 r 2 2 F 1 ( σ , 1 4 ; 5 4 ; − β q 2 r 4 ) + J ′ ( r ) ] . (16) M_{h}(r)=\frac{r}{2}\left[1+\frac{r^{2}}{l^{2}}+\frac{q^{2}}{r^{2}}\,{}_{2}F_{1}\left(\sigma,\frac{1}{4};\frac{5}{4};-\frac{\beta q^{2}}{r^{4}}\right)\right],
\qquad
M_{\rm ps}(r)=\frac{r}{6}\left[2+\frac{3q^{2}}{r^{2}}\,{}_{2}F_{1}\left(\sigma,\frac{1}{4};\frac{5}{4};-\frac{\beta q^{2}}{r^{4}}\right)+\mathcal{J}'(r)\right]. \tag*{(16)} M h ( r ) = 2 r [ 1 + l 2 r 2 + r 2 q 2 2 F 1 ( σ , 4 1 ; 4 5 ; − r 4 β q 2 ) ] , M ps ( r ) = 6 r [ 2 + r 2 3 q 2 2 F 1 ( σ , 4 1 ; 4 5 ; − r 4 β q 2 ) + J ′ ( r ) ] . (16) The number of horizons and of photon spheres at a given M M M M = c o n s t M=\mathrm{const} M = const M M M M h M_{h} M h M p s M_{\rm ps} M ps ( σ , M ) (\sigma,M) ( σ , M ) β = 0.1 \beta=0.1 β = 0.1 l = 1 l=1 l = 1 M = m 0 + m M ( σ ) M=m_{0}+m_{M}(\sigma) M = m 0 + m M ( σ ) m 0 = 1 m_{0}=1 m 0 = 1 σ ≤ 1 / 4 \sigma\leq1/4 σ ≤ 1/4 M ≥ M p s min M\geq M_{\rm ps}^{\min} M ≥ M ps m i n M ≥ M e x t M\geq M_{\rm ext} M ≥ M ext M h M_{h} M h σ \sigma σ σ = 1 / 4 \sigma=1/4 σ = 1/4 M M M m 0 m_{0} m 0 σ \sigma σ M = m 0 + m M ( σ ) M=m_{0}+m_{M}(\sigma) M = m 0 + m M ( σ ) M e x t ( σ 1 ) M_{\rm ext}(\sigma_{1}) M ext ( σ 1 ) M p s min ( σ 2 ) M_{\rm ps}^{\min}(\sigma_{2}) M ps m i n ( σ 2 ) q = 0.5 q=0.5 q = 0.5 m 0 + m M ( σ 1 ) = M e x t ( σ 1 ) m_{0}+m_{M}(\sigma_{1})=M_{\rm ext}(\sigma_{1}) m 0 + m M ( σ 1 ) = M ext ( σ 1 ) m 0 + m M ( σ 2 ) = M p s min ( σ 2 ) m_{0}+m_{M}(\sigma_{2})=M_{\rm ps}^{\min}(\sigma_{2}) m 0 + m M ( σ 2 ) = M ps m i n ( σ 2 ) β \beta β β = 0.1 \beta=0.1 β = 0.1 (16) . The interval σ 2 < σ < 1 / 4 \sigma_{2}<\sigma<1/4 σ 2 < σ < 1/4 M p s min M_{\rm ps}^{\min} M ps m i n q = 1.2 q=1.2 q = 1.2 m 0 m_{0} m 0 σ < 1 / 4 \sigma<1/4 σ < 1/4 m M < 0 m_{M}<0 m M < 0 M < m 0 = 1 < M p s min = 1.13 M<m_{0}=1<M_{\rm ps}^{\min}=1.13 M < m 0 = 1 < M ps m i n = 1.13 σ > 1 / 4 \sigma>1/4 σ > 1/4 m 0 m_{0} m 0 M = m 0 + m M M=m_{0}+m_{M} M = m 0 + m M M e x t M_{\rm ext} M ext
Effective optical geometry
Electromagnetic disturbances in nonlinear electrodynamics propagate along characteristic cones determined by the field Lagrangian [99 , 100 ] . We consider the geometric-optics limit on the fixed magnetic background and compare its extraordinary circular rays with the background null geodesics.
Characteristic cones and propagation domain
For F = F μ ν F μ ν / 4 \mathcal{F}=F_{\mu\nu}F^{\mu\nu}/4 F = F μν F μν /4
∇ μ ( L F F μ ν ) = 0 , ∇ [ λ F μ ν ] = 0. (17) \nabla_{\mu}\left(\mathcal{L}_{\mathcal{F}}F^{\mu\nu}\right)=0,\qquad\nabla_{[\lambda}F_{\mu\nu]}=0. \tag*{(17)} ∇ μ ( L F F μν ) = 0 , ∇ [ λ F μν ] = 0. (17) Writing the leading disturbance as b μ ν = k μ a ν − k ν a μ b_{\mu\nu}=k_{\mu}a_{\nu}-k_{\nu}a_{\mu} b μν = k μ a ν − k ν a μ
L F [ k 2 a ν − k ν ( k ⋅ a ) ] + L F F ζ ν ( ζ ⋅ a ) = 0 , ζ ν = k μ F μ ν . (18) \mathcal{L}_{\mathcal{F}}\left[k^{2}a^{\nu}-k^{\nu}(k\cdot a)\right]+\mathcal{L}_{\mathcal{F}\mathcal{F}}\zeta^{\nu}(\zeta\cdot a)=0,\qquad\zeta^{\nu}=k_{\mu}F^{\mu\nu}. \tag*{(18)} L F [ k 2 a ν − k ν ( k ⋅ a ) ] + L FF ζ ν ( ζ ⋅ a ) = 0 , ζ ν = k μ F μν . (18) Away from coefficient degeneracies, the physical characteristic polynomial factors into an ordinary and an extraordinary cone [101 ] :
( g μ ν k μ k ν ) ( γ α β k α k β ) = 0. (19) \left(g^{\mu\nu}k_{\mu}k_{\nu}\right)\left(\gamma^{\alpha\beta}k_{\alpha}k_{\beta}\right)=0. \tag*{(19)} ( g μν k μ k ν ) ( γ α β k α k β ) = 0. (19) The ordinary factor is the null-cone condition of the background metric g μ ν g_{\mu\nu} g μν
γ μ ν = g μ ν − L F F L F F μ α F α ν . (20) \gamma^{\mu\nu}=g^{\mu\nu}-\frac{\mathcal{L}_{\mathcal{F}\mathcal{F}}}{\mathcal{L}_{\mathcal{F}}}F^{\mu}{}_{\alpha}F^{\alpha\nu}. \tag*{(20)} γ μν = g μν − L F L FF F μ α F α ν . (20) The coefficient in Eq. (20) follows the 1 / 4 1/4 1/4 F \mathcal{F} F
For the magnetic field F θ φ = q sin θ F_{\theta\varphi}=q\sin\theta F θ φ = q sin θ
u = β q 2 r 4 , A ( u ) = 1 + ( 1 − σ ) u , (21) u=\frac{\beta q^{2}}{r^{4}},\qquad A(u)=1+(1-\sigma)u, \tag*{(21)} u = r 4 β q 2 , A ( u ) = 1 + ( 1 − σ ) u , (21) N ( u ) = 1 + ( 2 − 5 σ ) u + ( 1 − σ ) ( 1 − 2 σ ) u 2 . (22) N(u)=1+(2-5\sigma)u+(1-\sigma)(1-2\sigma)u^{2}. \tag*{(22)} N ( u ) = 1 + ( 2 − 5 σ ) u + ( 1 − σ ) ( 1 − 2 σ ) u 2 . (22) Figure 7. Background classification in the ( σ , M ) (\sigma,M) ( σ , M ) β = 0.1 \beta= 0.1 β = 0.1 l = 1 l = 1 l = 1 q = 0.5 q = 0.5 q = 0.5 q = 1.2 q = 1.2 q = 1.2 M = m 0 + m M ( σ ) M = m_0 + m_M(\sigma) M = m 0 + m M ( σ ) m 0 = 1 m_0 = 1 m 0 = 1 σ = 1 / 4 \sigma= 1/4 σ = 1/4 0 < M < 3 0 < M < 3 0 < M < 3
For σ > 1 \sigma> 1 σ > 1 A ( u ) A(u) A ( u ) u = 1 / ( σ − 1 ) u = 1/(\sigma- 1) u = 1/ ( σ − 1 ) r = [ β q 2 ( σ − 1 ) ] 1 / 4 r = [\beta q^2(\sigma- 1)]^{1/4} r = [ β q 2 ( σ − 1 ) ] 1/4 (20) . The extraordinary analysis is therefore restricted to the propagation domain in which A ( u ) ≠ 0 A(u) \ne0 A ( u ) = 0 (8) yields
L F = − A ( u ) 4 π ( 1 + u ) σ + 1 , (23) \mathcal{L}_{\mathcal{F}}=-\frac{A(u)}{4\pi(1+u)^{\sigma+1}}, \tag*{(23)} L F = − 4 π ( 1 + u ) σ + 1 A ( u ) , (23) K ≡ L F + 2 F L F F = − N ( u ) 4 π ( 1 + u ) σ + 2 . \mathcal{K}\equiv\mathcal{L}_{\mathcal{F}}+2\mathcal{F}\mathcal{L}_{\mathcal{F}\mathcal{F}}=-\frac{N(u)}{4\pi(1+u)^{\sigma+2}}. K ≡ L F + 2 F L FF = − 4 π ( 1 + u ) σ + 2 N ( u ) . The ratio
η ( r ) = K L F = N ( u ) ( 1 + u ) A ( u ) (24) \eta(r)=\frac{\mathcal{K}}{\mathcal{L}_{\mathcal{F}}}=\frac{N(u)}{(1+u)A(u)} \tag*{(24)} η ( r ) = L F K = ( 1 + u ) A ( u ) N ( u ) (24) defines the extraordinary propagation metric, up to a regular conformal factor:
d s e 2 = − f d t 2 + d r 2 f + r 2 η d Ω 2 . (25) ds_{\mathrm{e}}^2=-f\,dt^2+\frac{dr^2}{f}+\frac{r^2}{\eta}d\Omega^2. \tag*{(25)} d s e 2 = − f d t 2 + f d r 2 + η r 2 d Ω 2 . (25) Its static optical spatial metric is d t 2 = d r 2 / f 2 + r 2 d Ω 2 / ( f η ) dt^2=dr^2/f^2+r^2d\Omega^2/(f\eta) d t 2 = d r 2 / f 2 + r 2 d Ω 2 / ( f η ) η → 1 \eta\to1 η → 1
We use the static component connected to the asymptotic weak-field region, with
f > 0 , A ( u ) > 0 , N ( u ) > 0. (26) f>0,\qquad A(u)>0,\qquad N(u)>0. \tag*{(26)} f > 0 , A ( u ) > 0 , N ( u ) > 0. (26) These conditions retain the Maxwell signs of L F \mathcal{L}_{\mathcal{F}} L F K \mathcal{K} K σ = 0.12 \sigma=0.12 σ = 0.12 0.15 0.15 0.15 0.30 0.30 0.30 A A A N N N u ≥ 0 u\geq0 u ≥ 0 σ = 0.9 \sigma=0.9 σ = 0.9 N = 1 − 2.5 u − 0.08 u 2 N=1-2.5u-0.08u^2 N = 1 − 2.5 u − 0.08 u 2 u ∗ ≃ 0.3950070225 u_\ast\simeq0.3950070225 u ∗ ≃ 0.3950070225 q = 1.2 q=1.2 q = 1.2 β = 0.1 \beta=0.1 β = 0.1 r d e g ≃ 0.777033 r_{\mathrm{deg}}\simeq0.777033 r deg ≃ 0.777033 r + ≃ 1.328025 r_+\simeq1.328025 r + ≃ 1.328025
Circular rays and topological indices
The functions entering the topological construction are
H e = f η r sin θ , U e = H e 2 ( r , π / 2 ) = f η r 2 , (27) H_{\mathrm{e}}=\frac{\sqrt{f\eta}}{r\sin\theta},\qquad U_{\mathrm{e}}=H_{\mathrm{e}}^2(r,\pi/2)=\frac{f\eta}{r^2}, \tag*{(27)} H e = r sin θ f η , U e = H e 2 ( r , π /2 ) = r 2 f η , (27) ϕ e r = η 2 r 2 sin θ ( r f ′ − 2 f + r f η ′ η ) , ϕ e θ = − η f cos θ r 2 sin 2 θ . (28) \phi_{\mathrm{e}}^r=\frac{\sqrt{\eta}}{2r^2\sin\theta}\left(rf'-2f+rf\frac{\eta'}{\eta}\right),\qquad
\phi_{\mathrm{e}}^\theta=-\frac{\eta\sqrt{f}\cos\theta}{r^2\sin^2\theta}. \tag*{(28)} ϕ e r = 2 r 2 sin θ η ( r f ′ − 2 f + r f η η ′ ) , ϕ e θ = − r 2 sin 2 θ η f cos θ . (28) The circular rays lie at θ = π / 2 \theta=\pi/2 θ = π /2
C e ( r ) = η ( r ) ( r f ′ − 2 f ) + r f η ′ = 0 , (29) \mathcal{C}_{\mathrm{e}}(r)=\eta(r)(rf'-2f)+rf\eta'=0, \tag*{(29)} C e ( r ) = η ( r ) ( r f ′ − 2 f ) + r f η ′ = 0 , (29) equivalently U e ′ = 0 U_{\mathrm{e}}'=0 U e ′ = 0 U e U_{\mathrm{e}} U e − 1 -1 − 1 + 1 +1 + 1 (5) .
Writing f = f 0 + r 2 / l 2 f=f_0+r^2/l^2 f = f 0 + r 2 / l 2 f 0 f_0 f 0 l l l
C e = η ( r f 0 ′ − 2 f 0 ) + r f 0 η ′ + r 3 l 2 η ′ . (30) \mathcal{C}_{\mathrm{e}}=\eta(rf_0'-2f_0)+rf_0\eta'+\frac{r^3}{l^2}\eta'. \tag*{(30)} C e = η ( r f 0 ′ − 2 f 0 ) + r f 0 η ′ + l 2 r 3 η ′ . (30) The extraordinary orbit condition therefore depends explicitly on the AdS scale when η ′ ≠ 0 \eta'\ne0 η ′ = 0
Comparison with the background orbits
Table IV compares selected background and extraordinary orbits at m 0 = l = 1 m_0=l=1 m 0 = l = 1 β = 0.1 \beta=0.1 β = 0.1 r = 10 3 r=10^3 r = 1 0 3 U b = f / r 2 U_b=f/r^2 U b = f / r 2 U e U_e U e 10 − 3 10^{-3} 1 0 − 3 10 − 4 10^{-4} 1 0 − 4
Figure 8 resolves the displacement of the two horizonless extrema and displays the effective vector fields near their zeros. The minimum moves inward and the maximum outward, with local indices + 1 +1 + 1 − 1 -1 − 1
This comparison concerns isolated orbits at the stated parameter values. The labels NS and NS0 in the preceding sections refer to the background-metric sector, equivalently to the ordinary polarization mode. The extraordinary merger boundaries require C e = ∂ r C e = 0 \mathcal{C}_e=\partial_r\mathcal{C}_e=0 C e = ∂ r C e = 0
Topological Phase Diagrams
Generalized free energy
The generalized free energy is [72 ]
F = M − S τ , (31) \mathcal{F}=M-\frac{S}{\tau}, \tag*{(31)} F = M − τ S , (31) where τ \tau τ τ = τ h = 1 / T h \tau=\tau_h=1/T_h τ = τ h = 1/ T h P = − Λ / 8 π = 3 / ( 8 π l 2 ) P=-\Lambda/8\pi=3/(8\pi l^2) P = − Λ/8 π = 3/ ( 8 π l 2 ) S = π r h 2 S=\pi r_h^2 S = π r h 2 M M M [89 ] . On shell, F \mathcal{F} F G = M − T S G=M-TS G = M − T S f ( r h ) = 0 f(r_h)=0 f ( r h ) = 0 (13) gives
M = r h 2 + 4 π P r h 3 3 + q 2 2 r h 2 F 1 ( 1 4 , σ ; 5 4 ; − β q 2 r h 4 ) , (32) M=\frac{r_h}{2}+\frac{4\pi P r_h^3}{3}+\frac{q^2}{2r_h}\,{}_2F_1\left(\frac{1}{4},\sigma;\frac{5}{4};-\frac{\beta q^2}{r_h^4}\right), \tag*{(32)} M = 2 r h + 3 4 π P r h 3 + 2 r h q 2 2 F 1 ( 4 1 , σ ; 4 5 ; − r h 4 β q 2 ) , (32) and the Hawking temperature T = ∂ M / ∂ S = M ′ ( r h ) / ( 2 π r h ) T=\partial M/\partial S=M'(r_h)/(2\pi r_h) T = ∂ M / ∂ S = M ′ ( r h ) / ( 2 π r h )
T = 1 4 π r h [ 1 + 8 π P r h 2 − q 2 r h 2 F 1 + 4 σ β q 4 5 r h 6 F 2 ] , (33) T=\frac{1}{4\pi r_h}\left[1+8\pi P r_h^2-\frac{q^2}{r_h^2}\mathcal{F}_1+\frac{4\sigma\beta q^4}{5r_h^6}\mathcal{F}_2\right], \tag*{(33)} T = 4 π r h 1 [ 1 + 8 π P r h 2 − r h 2 q 2 F 1 + 5 r h 6 4 σ β q 4 F 2 ] , (33) with F 1 = 2 2 F 1 ( 1 4 , σ ; 5 4 ; z ) \mathcal{F}_1=2{}_2F_1\left(\frac{1}{4},\sigma;\frac{5}{4};z\right) F 1 = 2 2 F 1 ( 4 1 , σ ; 4 5 ; z ) F 2 = 2 2 F 1 ( 5 4 , σ + 1 ; 9 4 ; z ) \mathcal{F}_2=2{}_2F_1\left(\frac{5}{4},\sigma+1;\frac{9}{4};z\right) F 2 = 2 2 F 1 ( 4 5 , σ + 1 ; 4 9 ; z ) z = − β q 2 / r h 4 z=-\beta q^2/r_h^4 z = − β q 2 / r h 4 [89 ] , and the derivative of Eq. (32) was checked numerically against Eq. (33) . The vector field is
ϕ = ( ∂ F ∂ r h , − cot Θ csc Θ ) , \boldsymbol{\phi}=\left(\frac{\partial\mathcal{F}}{\partial r_h},-\cot\Theta\csc\Theta\right), ϕ = ( ∂ r h ∂ F , − cot Θ csc Θ ) , ∂ F ∂ r h = M ′ ( r h ) − 2 π r h τ , (34) \frac{\partial\mathcal{F}}{\partial r_h}=M'(r_h)-\frac{2\pi r_h}{\tau}, \tag*{(34)} ∂ r h ∂ F = M ′ ( r h ) − τ 2 π r h , (34) with 0 ≤ Θ ≤ π 0\leq\Theta\leq\pi 0 ≤ Θ ≤ π ϕ Θ \phi^\Theta ϕ Θ Θ = 0 , π \Theta=0,\pi Θ = 0 , π Θ = π / 2 \Theta=\pi/2 Θ = π /2
τ = τ h ( r h ) = 2 π r h M ′ ( r h ) = 20 π r h 7 40 π P r h 8 + 4 σ β q 4 F 2 − 5 q 2 r h 5 F 1 + 5 r h 6 . (35) \begin{aligned}
\tau=\tau_h(r_h)=\frac{2\pi r_h}{M'(r_h)} \\
&=\frac{20\pi r_h^7}{40\pi P r_h^8+4\sigma\beta q^4\mathcal{F}_2-5q^2r_h^5\mathcal{F}_1+5r_h^6}. \tag*{(35)}
\end{aligned} τ = τ h ( r h ) = M ′ ( r h ) 2 π r h = 40 π P r h 8 + 4 σ β q 4 F 2 − 5 q 2 r h 5 F 1 + 5 r h 6 20 π r h 7 . (35) A zero point with d τ h / d r h < 0 d\tau_h/dr_h<0 d τ h / d r h < 0 + 1 +1 + 1 d τ h / d r h > 0 d\tau_h/dr_h>0 d τ h / d r h > 0 − 1 -1 − 1 ( q , β , σ , P ) (q,\beta,\sigma,P) ( q , β , σ , P ) m 0 m_0 m 0 l l l M M M r h r_h r h P P P l l l σ \sigma σ q = 0.5 q=0.5 q = 0.5 q = 1.2 q=1.2 q = 1.2 T > 0 T>0 T > 0 T = 0 T=0 T = 0
Extended first law and Smarr relation
In the extended phase space the mass of Eq. (32) is a function of the four variables S = π r h 2 S=\pi r_h^2 S = π r h 2 P P P q q q β \beta β σ \sigma σ
T = ( ∂ M ∂ S ) P , q , β , V = ( ∂ M ∂ P ) S , q , β = 4 π r h 3 3 , T=\left(\frac{\partial M}{\partial S}\right)_{P,q,\beta},\qquad
V=\left(\frac{\partial M}{\partial P}\right)_{S,q,\beta}=\frac{4\pi r_h^3}{3}, T = ( ∂ S ∂ M ) P , q , β , V = ( ∂ P ∂ M ) S , q , β = 3 4 π r h 3 , Φ = ( ∂ M ∂ q ) S , P , β , B = ( ∂ M ∂ β ) S , P , q . (36) \Phi=\left(\frac{\partial M}{\partial q}\right)_{S,P,\beta},\qquad
\mathcal{B}=\left(\frac{\partial M}{\partial\beta}\right)_{S,P,q}. \tag*{(36)} Φ = ( ∂ q ∂ M ) S , P , β , B = ( ∂ β ∂ M ) S , P , q . (36) Differentiating Eq. (32) with the rule d d z 2 F 1 ( a , b ; c ; z ) = a b c 2 F 1 ( a + 1 , b + 1 ; c + 1 ; z ) \frac{d}{dz}{}_2F_1(a,b;c;z)=\frac{ab}{c}{}_2F_1(a+1,b+1;c+1;z) d z d 2 F 1 ( a , b ; c ; z ) = c ab 2 F 1 ( a + 1 , b + 1 ; c + 1 ; z ) (33) together with
Φ = q r h F 1 − σ β q 3 5 r h 5 F 2 , B = − σ q 4 10 r h 5 F 2 , (37) \Phi=\frac{q}{r_h}\mathcal{F}_1-\frac{\sigma\beta q^3}{5r_h^5}\mathcal{F}_2,\qquad
\mathcal{B}=-\frac{\sigma q^4}{10r_h^5}\mathcal{F}_2, \tag*{(37)} Φ = r h q F 1 − 5 r h 5 σ β q 3 F 2 , B = − 10 r h 5 σ q 4 F 2 , (37) where F 1 \mathcal{F}_1 F 1 F 2 \mathcal{F}_2 F 2 (33) . Here Φ \Phi Φ B \mathcal{B} B β \beta β M M M
d M = T d S + V d P + Φ d q + B d β . (38) dM=T\,dS+V\,dP+\Phi\,dq+\mathcal{B}\,d\beta. \tag*{(38)} d M = T d S + V d P + Φ d q + B d β . (38) Equations (37) and (38) coincide with the corresponding results of Ref. [89 ] (Eqs. (25) –(27) of the arXiv version).
Table 4. Background and extraordinary circular orbits at m 0 = l = 1 m_0=l=1 m 0 = l = 1 β = 0.1 \beta=0.1 β = 0.1 100 ( r e − r b ) / r b 100(r_e-r_b)/r_b 100 ( r e − r b ) / r b η e = η ( r e ) \eta_e=\eta(r_e) η e = η ( r e ) σ = 0.15 \sigma=0.15 σ = 0.15
The conjugate quantities have a direct interpretation in terms of the field energy. With the energy density ρ = − L = q 2 x 4 σ − 4 / [ 8 π ( x 4 + β q 2 ) σ ] \rho=-\mathcal{L}=q^2x^{4\sigma-4}/[8\pi(x^4+\beta q^2)^\sigma] ρ = − L = q 2 x 4 σ − 4 / [ 8 π ( x 4 + β q 2 ) σ ] I ′ = 8 π x 2 ρ \mathcal{I}'=8\pi x^2\rho I ′ = 8 π x 2 ρ (11) , and the horizon condition together with I ( ∞ ) = 2 m M \mathcal{I}(\infty)=2m_M I ( ∞ ) = 2 m M (32) into
M = r h 2 + 4 π P r h 3 3 + 4 π ∫ r h ∞ ρ ( x ) x 2 d x . (39) M=\frac{r_h}{2}+\frac{4\pi P r_h^3}{3}+4\pi\int_{r_h}^{\infty}\rho(x)x^2\,dx. \tag*{(39)} M = 2 r h + 3 4 π P r h 3 + 4 π ∫ r h ∞ ρ ( x ) x 2 d x . (39) The integral converges for every σ > 0 \sigma>0 σ > 0 ρ x 2 ∼ x − 2 \rho x^2\sim x^{-2} ρ x 2 ∼ x − 2 x x x σ < 1 / 4 \sigma<1/4 σ < 1/4 m M m_M m M Φ = 4 π ∫ r h ∞ ∂ q ρ x 2 d x \Phi=4\pi\int_{r_h}^{\infty}\partial_q\rho\,x^2dx Φ = 4 π ∫ r h ∞ ∂ q ρ x 2 d x B = 4 π ∫ r h ∞ ∂ β ρ x 2 d x \mathcal{B}=4\pi\int_{r_h}^{\infty}\partial_\beta\rho\,x^2dx B = 4 π ∫ r h ∞ ∂ β ρ x 2 d x ∂ β ρ < 0 \partial_\beta\rho<0 ∂ β ρ < 0 B < 0 \mathcal{B}<0 B < 0 T = f ′ ( r h ) / 4 π = [ 1 + 8 π P r h 2 ρ ( r h ) ] / ( 4 π r h ) T=f'(r_h)/4\pi=[1+8\pi P r_h^2\rho(r_h)]/(4\pi r_h) T = f ′ ( r h ) /4 π = [ 1 + 8 π P r h 2 ρ ( r h )] / ( 4 π r h ) (33) .
In the units G = c = 1 G = c = 1 G = c = 1 [ M ] = [ r h ] = [ q ] = L [M] = [r_h] = [q] = L [ M ] = [ r h ] = [ q ] = L [ S ] = L 2 [S] = L^2 [ S ] = L 2 [ P ] = L − 2 [P] = L^{-2} [ P ] = L − 2 [ β ] = L 2 [\beta] = L^2 [ β ] = L 2 M ( λ 2 S , λ − 2 P , λ q , λ 2 β ) = λ M ( S , P , q , β ) M(\lambda^2 S,\lambda^{-2}P,\lambda q,\lambda^2\beta) = \lambda M(S,P,q,\beta) M ( λ 2 S , λ − 2 P , λ q , λ 2 β ) = λ M ( S , P , q , β )
M = 2 T S − 2 P V + Φ q + 2 β B . (40) M = 2TS - 2PV + \Phi q + 2\beta\mathcal{B}. \tag*{(40)} M = 2 T S − 2 P V + Φ q + 2 β B . (40) The term 2 β B 2\beta\mathcal{B} 2 β B M M M 2 β B ≠ 0 2\beta\mathcal{B} \ne0 2 β B = 0 B < 0 \mathcal{B} < 0 B < 0 β > 0 \beta> 0 β > 0 σ > 0 \sigma> 0 σ > 0 β → 0 \beta\to0 β → 0 F 1 → 1 \mathcal{F}_1 \to1 F 1 → 1 (40) reduces to the Smarr relation of the Reissner–Nordström–AdS black hole, M = 2 T S − 2 P V + Φ q M = 2TS - 2PV + \Phi q M = 2 T S − 2 P V + Φ q Φ = q / r h \Phi= q/r_h Φ = q / r h
Both relations were verified numerically. For σ = 0.12 \sigma= 0.12 σ = 0.12 0.15 0.15 0.15 0.45 0.45 0.45 0.9 0.9 0.9 (33) and (37) agree with central-difference derivatives of Eq. (32) to 4 × 10 − 23 4 \times10^{-23} 4 × 1 0 − 23 10 − 10 10^{-10} 1 0 − 10 4 × 10 − 36 4 \times10^{-36} 4 × 1 0 − 36 10 − 17 10^{-17} 1 0 − 17 σ = 0.25 \sigma= 0.25 σ = 0.25 σ = 1.2 \sigma= 1.2 σ = 1.2 (40) , is satisfied by the closed forms to 10 − 51 10^{-51} 1 0 − 51 verify_thermo_precision.py and verify_firstlaw_path.py). The first law, Eq. (38) , was integrated along the straight path from ( r h , P , q , β ) = ( 1.5 , 0.001 , 0.5 , 0.1 ) (r_h,P,q,\beta) = (1.5, 0.001, 0.5, 0.1) ( r h , P , q , β ) = ( 1.5 , 0.001 , 0.5 , 0.1 ) ( 2.2 , 0.004 , 0.8 , 0.3 ) (2.2, 0.004, 0.8, 0.3) ( 2.2 , 0.004 , 0.8 , 0.3 ) 3 × 10 − 51 3 \times10^{-51} 3 × 1 0 − 51 σ = 0.12 \sigma= 0.12 σ = 0.12 0.25 0.25 0.25 0.45 0.45 0.45 0.9 0.9 0.9 1.2 1.2 1.2
The comparison of the temperature isobars across the nonlinear exponent is shown in Figure 9 . In both panels we fix q = 0.5 q = 0.5 q = 0.5 β = 0.1 \beta= 0.1 β = 0.1 P / P c = 0.5 , 1 , 1.8 P/P_c = 0.5, 1, 1.8 P / P c = 0.5 , 1 , 1.8 σ \sigma σ r h r_h r h M h ( r h ) M_h(r_h) M h ( r h ) (32) ; the curves therefore describe a family of black-hole solutions, not the radial profile of a single geometry with fixed mass. Only the branches with T > 0 T > 0 T > 0 σ = 0.12 \sigma= 0.12 σ = 0.12 σ = 1.2 \sigma= 1.2 σ = 1.2 r h r_h r h M h ( r h ) M_h(r_h) M h ( r h ) M h ( r h ) M_h(r_h) M h ( r h ) f f f d M h / d r h = 2 π r h T > 0 dM_h/dr_h = 2\pi r_h T > 0 d M h / d r h = 2 π r h T > 0 M h M_h M h T T T σ = 0.12 \sigma= 0.12 σ = 0.12 T = 0 T = 0 T = 0 r h ≃ 0.47939 r_h \simeq0.47939 r h ≃ 0.47939 0.47068 0.47068 0.47068 0.45858 0.45858 0.45858 P / P c = 0.5 P/P_c = 0.5 P / P c = 0.5 1 1 1 1.8 1.8 1.8 σ = 1.2 \sigma= 1.2 σ = 1.2 T ∼ 1 / ( 4 π r h ) T \sim1/(4\pi r_h) T ∼ 1/ ( 4 π r h ) r h → 0 + r_h \to0^+ r h → 0 + r h = 0 r_h = 0 r h = 0
Figure 9. Hawking temperature T ( r h ) T(r_h) T ( r h ) q = 0.5 q=0.5 q = 0.5 β = 0.1 \beta=0.1 β = 0.1 P = 0.5 P c P=0.5P_c P = 0.5 P c P = P c P=P_c P = P c P = 1.8 P c P=1.8P_c P = 1.8 P c σ = 0.12 \sigma=0.12 σ = 0.12 P c = 0.013304283174 P_c=0.013304283174 P c = 0.013304283174 σ = 1.2 \sigma=1.2 σ = 1.2 P c = 0.013727349243 P_c=0.013727349243 P c = 0.013727349243 q q q ∂ T / ∂ r h = ∂ 2 T / ∂ r h 2 = 0 \partial T/\partial r_h=\partial^2T/\partial r_h^2=0 ∂ T / ∂ r h = ∂ 2 T / ∂ r h 2 = 0 ( r c , T c ) (r_c,T_c) ( r c , T c ) T > 0 T>0 T > 0 σ = 1.2 \sigma=1.2 σ = 1.2 r h r_h r h T → + ∞ T\to+\infty T → + ∞ r h → 0 + r_h\to0^+ r h → 0 + r h r_h r h T > 0 T>0 T > 0 M h M_h M h r h r_h r h
Critical points
Writing T = A ( r h ) + 2 P r h T = A(r_h) + 2Pr_h T = A ( r h ) + 2 P r h ∂ r h T = ∂ r h 2 T = 0 \partial_{r_h}T = \partial^2_{r_h}T = 0 ∂ r h T = ∂ r h 2 T = 0 A ′ ′ ( r c ) = 0 A''(r_c) = 0 A ′′ ( r c ) = 0 P c = − A ′ ( r c ) / 2 P_c = -A'(r_c)/2 P c = − A ′ ( r c ) /2 Table V lists physical solutions with P c > 0 P_c > 0 P c > 0 ( q , σ ) = ( 3.249107 , 1.2 ) (q,\sigma) = (3.249107, 1.2) ( q , σ ) = ( 3.249107 , 1.2 ) ( 3.920545 , 1.5 ) (3.920545, 1.5) ( 3.920545 , 1.5 ) m 0 = M − m M = 0 m_0 = M - m_M = 0 m 0 = M − m M = 0 q q q r c r_c r c T c > 0 T_c > 0 T c > 0 f ( r ) = 0 f(r) = 0 f ( r ) = 0 r h = r c r_h = r_c r h = r c r − = 0.752053 r_- = 0.752053 r − = 0.752053 0.982012 0.982012 0.982012 [89 ] , q = 1 q = 1 q = 1 β = 0.1 \beta= 0.1 β = 0.1 σ = 0.75 \sigma= 0.75 σ = 0.75 r c = 2.4374 r_c = 2.4374 r c = 2.4374 P c = 0.003332 P_c = 0.003332 P c = 0.003332 P c ≈ 0.003 P_c \approx0.003 P c ≈ 0.003 β → 0 \beta\to0 β → 0 q q q r c = 6 q r_c = \sqrt{6}q r c = 6 q P c = 1 / ( 96 π q 2 ) P_c = 1/(96\pi q^2) P c = 1/ ( 96 π q 2 ) T c = 6 / ( 18 π q ) T_c = \sqrt{6}/(18\pi q) T c = 6 / ( 18 π q ) P c r c / T c = 3 / 16 P_c r_c/T_c = 3/16 P c r c / T c = 3/16 Table V also lists the ratios of the critical quantities to these values at the same charge. The fixed-charge examples show that the nonlinear electrodynamics shifts r c r_c r c P c P_c P c T c T_c T c m 0 = 0 m_0 = 0 m 0 = 0 q q q σ \sigma σ m 0 = 0 m_0 = 0 m 0 = 0 m 0 = M − m M m_0 = M - m_M m 0 = M − m M M = M h ( r h , P ) M = M_h(r_h,P) M = M h ( r h , P ) q q q β \beta β σ \sigma σ P c P_c P c m 0 ≠ 0 m_0 \ne0 m 0 = 0 β q 2 / r c 4 \beta q^2/r_c^4 β q 2 / r c 4 P > P c P > P_c P > P c σ = 0.12 \sigma= 0.12 σ = 0.12 σ = 0.45 \sigma= 0.45 σ = 0.45 T > 0 T > 0 T > 0 P = P c P = P_c P = P c P < P c P < P_c P < P c § 1.3 E). For σ = 0.9 \sigma= 0.9 σ = 0.9 1.2 1.2 1.2 T → ∞ T \to\infty T → ∞ r h → 0 r_h \to0 r h → 0 σ = 1.2 \sigma= 1.2 σ = 1.2 q = 0.5 q = 0.5 q = 0.5 P = 10 − 3 P = 10^{-3} P = 1 0 − 3 r h = 0.4790 r_h = 0.4790 r h = 0.4790 0.8044 0.8044 0.8044 6.2469 6.2469 6.2469
Table V. Critical point ( r c , P c , T c ) (r_c,P_c,T_c) ( r c , P c , T c ) β = 0.1 \beta=0.1 β = 0.1 q q q r c R N = 6 q r_c^{\mathrm{RN}}=\sqrt{6}q r c RN = 6 q P c R N = 1 / ( 96 π q 2 ) P_c^{\mathrm{RN}}=1/(96\pi q^2) P c RN = 1/ ( 96 π q 2 ) T c R N = 6 / ( 18 π q ) T_c^{\mathrm{RN}}=\sqrt{6}/(18\pi q) T c RN = 6 / ( 18 π q ) P c r c / T c = 3 / 16 = 0.1875 P_cr_c/T_c=3/16=0.1875 P c r c / T c = 3/16 = 0.1875 q = 0.5 q=0.5 q = 0.5 σ = 1.2 , 1.5 \sigma=1.2,1.5 σ = 1.2 , 1.5 m 0 = 0 m_0=0 m 0 = 0 q q q
Topological charges at P = 10 − 3 P = 10^{-3} P = 1 0 − 3
The benchmark cases below have P = 10 − 3 < P c P = 10^{-3} < P_c P = 1 0 − 3 < P c T ( r h ) T(r_h) T ( r h ) Table VI . The extrema are the radii at which a pair of zero points of opposite charge is created or annihilated as τ \tau τ
Table VI. Zero points of ϕ \phi ϕ β = 0.1 \beta=0.1 β = 0.1 P = 10 − 3 P=10^{-3} P = 1 0 − 3 T ( r h ) T(r_h) T ( r h ) r h r_h r h T T T τ \tau τ τ \tau τ
q = 0.5 q = 0.5 q = 0.5 σ = 0.12 \sigma= 0.12 σ = 0.12 Figure 10 shows τ h ( r h ) \tau_h(r_h) τ h ( r h ) r h = 0.6337 r_h = 0.6337 r h = 0.6337 ( + 1 ) (+1) ( + 1 ) [eq] (-1)$ and a large stable black hole at 23.2926 23.2926 23.2926 ( + 1 ) (+1) ( + 1 ) = + 1 = +1 = + 1 T T T r h = 0.8692 r_h = 0.8692 r h = 0.8692 r h = 6.2469 r_h = 6.2469 r h = 6.2469 15.84 < τ < 39.76 15.84 < \tau< 39.76 15.84 < τ < 39.76 τ > 39.76 \tau> 39.76 τ > 39.76 + 1 +1 + 1 τ \tau τ
Figure 10. σ = 0.12 \sigma= 0.12 σ = 0.12 q = 0.5 q = 0.5 q = 0.5 β = 0.1 \beta= 0.1 β = 0.1 P = 10 − 3 P = 10^{-3} P = 1 0 − 3 τ h ( r h ) \tau_h(r_h) τ h ( r h ) τ = 20 \tau= 20 τ = 20 n n n ( r h , Θ ) (r_h,\Theta) ( r h , Θ ) τ = 20 \tau= 20 τ = 20 r h = 0.6337 r_h = 0.6337 r h = 0.6337
q = 0.5 q=0.5 q = 0.5 σ = 0.9 \sigma=0.9 σ = 0.9 T ( r h ) → ∞ T(r_h)\to\infty T ( r h ) → ∞ r h → 0 r_h\to0 r h → 0 σ = 0.12 \sigma=0.12 σ = 0.12 T → − ∞ T\to-\infty T → − ∞ r h = 0.3948 r_h=0.3948 r h = 0.3948 0.8269 0.8269 0.8269 6.2469 6.2469 6.2469 − 1 , + 1 , − 1 , + 1 -1,+1,-1,+1 − 1 , + 1 , − 1 , + 1 τ = 35 \tau=35 τ = 35 − 1 -1 − 1 + 1 +1 + 1 = 0 =0 = 0 15.55 < τ < 31.91 15.55<\tau<31.91 15.55 < τ < 31.91 τ = 25 \tau=25 τ = 25 τ \tau τ + 1 +1 + 1 σ = 0.12 \sigma=0.12 σ = 0.12 P = 10 − 3 < P c = 0.013599 P=10^{-3}<P_c=0.013599 P = 1 0 − 3 < P c = 0.013599 r h ∈ ( 0.3948 , 0.8269 ) r_h\in(0.3948,0.8269) r h ∈ ( 0.3948 , 0.8269 ) r h > 6.2469 r_h>6.2469 r h > 6.2469 0.03134 < T < 0.06432 0.03134<T<0.06432 0.03134 < T < 0.06432 § 1.3 E), so that the small black hole is metastable. A first-order small–large transition exists for σ = 0.9 \sigma= 0.9 σ = 0.9 P ∗ < P < P c P_* < P < P_c P ∗ < P < P c P ∗ = 0.001307 P_* = 0.001307 P ∗ = 0.001307 P > P c P > P_c P > P c r h r_h r h
q = 0.5 q = 0.5 q = 0.5 σ = 1.2 \sigma= 1.2 σ = 1.2 P = 10 − 3 P = 10^{-3} P = 1 0 − 3 τ h ( r h ) = 1 / T ( r h ) \tau_h(r_h) = 1/T(r_h) τ h ( r h ) = 1/ T ( r h ) r h = 0.479013 r_h = 0.479013 r h = 0.479013 τ h \tau_h τ h τ = 17 \tau= 17 τ = 17 T ( r h ) = 1 / τ T(r_h) = 1/\tau T ( r h ) = 1/ τ r h = 0.4155011 r_h = 0.4155011 r h = 0.4155011 τ = 25 \tau= 25 τ = 25 r h = 2.0960916 r_h = 2.0960916 r h = 2.0960916 Q = 0 Q = 0 Q = 0 Figure 12 compares both inverse-temperature levels and the corresponding normalized vector fields.
Figure 12. Thermodynamic topology for q = 0.5 q = 0.5 q = 0.5 σ = 1.2 \sigma= 1.2 σ = 1.2 β = 0.1 \beta= 0.1 β = 0.1 P = 10 − 3 P = 10^{-3} P = 1 0 − 3 τ h ( r h ) \tau_h(r_h) τ h ( r h ) τ = 17 , 25 \tau= 17,25 τ = 17 , 25 n ( r h , Θ ) n(r_h,\Theta) n ( r h , Θ ) τ = 17 , 25 \tau= 17,25 τ = 17 , 25 ω = − 1 \omega= -1 ω = − 1 + 1 +1 + 1 ϕ r h > 0 \phi^{r_h} > 0 ϕ r h > 0 ϕ r h < 0 \phi^{r_h} < 0 ϕ r h < 0 r h r_h r h
q = 1.2 q = 1.2 q = 1.2 σ = 0.45 \sigma= 0.45 σ = 0.45 τ = 35 \tau= 35 τ = 35 Figure 13 ) there are three zero points, at r h = 1.8321 r_h = 1.8321 r h = 1.8321 r h = 2.2125 r_h = 2.2125 r h = 2.2125 33.51 < τ < 40.44 33.51 < \tau< 40.44 33.51 < τ < 40.44
Figure 13. σ = 0.45 \sigma=0.45 σ = 0.45 q = 1.2 q=1.2 q = 1.2 β = 0.1 \beta=0.1 β = 0.1 P = 10 − 3 P=10^{-3} P = 1 0 − 3 τ h ( r h ) \tau_h(r_h) τ h ( r h ) τ = 35 \tau=35 τ = 35 n n n τ = 35 \tau=35 τ = 35 r h = 1.8321 r_h=1.8321 r h = 1.8321 2.8904 2.8904 2.8904 10.5699 10.5699 10.5699
Free energy
The on-shell free energy G ( T ) = M − T S G(T) = M - TS G ( T ) = M − T S Figure 14 . For σ = 0.12 \sigma= 0.12 σ = 0.12 σ = 0.45 \sigma= 0.45 σ = 0.45 T T T Table VI . The crossing of the two stable branches fixes the first-order transition temperature T t T_t T t T ( r s ) = T ( r l ) T(r_s) = T(r_l) T ( r s ) = T ( r l ) G ( r s ) = G ( r l ) G(r_s) = G(r_l) G ( r s ) = G ( r l ) Table VII ). For σ = 0.9 \sigma= 0.9 σ = 0.9 P = 10 − 3 P = 10^{-3} P = 1 0 − 3 σ \sigma σ σ = 0.9 \sigma= 0.9 σ = 0.9 P > P ∗ = 0.001307 P > P_* = 0.001307 P > P ∗ = 0.001307 σ = 1.2 \sigma= 1.2 σ = 1.2 P = 0.005 P = 0.005 P = 0.005 Table VII . The free energy therefore complements the topological assignment: the local indices label the thermal stability of the computed zeros, whereas the free energy selects the equilibrium branch.
Table VII. First-order coexistence of the small (r s r_s r s r l r_l r l T ( r s ) = T ( r l ) T(r_s) = T(r_l) T ( r s ) = T ( r l ) G ( r s ) = G ( r l ) = G t G(r_s) = G(r_l) = G_t G ( r s ) = G ( r l ) = G t β = 0.1 \beta= 0.1 β = 0.1 Δ S = π ( r l 2 − r s 2 ) \Delta S = \pi(r_l^2-r_s^2) Δ S = π ( r l 2 − r s 2 ) σ = 0.9 \sigma= 0.9 σ = 0.9 1.2 1.2 1.2 P = 10 − 3 P = 10^{-3} P = 1 0 − 3
Figure 14. Gibbs free energy G = M − T S G=M-TS G = M − T S P = 10 − 3 P=10^{-3} P = 1 0 − 3 β = 0.1 \beta=0.1 β = 0.1 T T T σ = 0.12 \sigma=0.12 σ = 0.12 q = 0.5 q=0.5 q = 0.5 σ = 0.9 \sigma=0.9 σ = 0.9 q = 0.5 q=0.5 q = 0.5 σ = 0.45 \sigma=0.45 σ = 0.45 q = 1.2 q=1.2 q = 1.2 T ( r h ) T(r_h) T ( r h )
Figure 15 extends the Gibbs-free-energy analysis to the high-σ \sigma σ q = 0.5 q = 0.5 q = 0.5 σ = 1.2 \sigma= 1.2 σ = 1.2 T ( r h ) T(r_h) T ( r h ) C P = 2 π r h T / ( d T / d r h ) C_P = 2\pi r_h T/(dT/dr_h) C P = 2 π r h T / ( d T / d r h ) C P > 0 C_P > 0 C P > 0 C P < 0 C_P < 0 C P < 0 G G G P = 10 − 3 P = 10^{-3} P = 1 0 − 3
Figure 15. Bulk Gibbs free energy G = M − T S G=M-TS G = M − T S T T T P = 10 − 3 P=10^{-3} P = 1 0 − 3 q = 0.5 q=0.5 q = 0.5 β = 0.1 \beta=0.1 β = 0.1 σ = 1.2 \sigma=1.2 σ = 1.2 σ = 1.5 \sigma=1.5 σ = 1.5 C P > 0 C_P>0 C P > 0 C P < 0 C_P<0 C P < 0 T ( r h ) T(r_h) T ( r h ) T > 0 T>0 T > 0
Holographic Thermodynamics
Assuming a holographic interpretation, the black hole is described by a thermodynamic state of a three-dimensional conformal field theory (CFT) on a sphere of radius R R R [92 , 93 , 94 ] , a variation of the cosmological constant is a variation of the central charge C C C R R R V = 4 π R 2 \mathcal{V} = 4\pi R^2 V = 4 π R 2 β \beta β [93 ] : the phase structure in the canonical, grand-canonical and mixed ensembles, the critical point and the critical exponents.
Dictionary
In the units G = 1 G = 1 G = 1
E = l M R , T ~ = l T R , S = π r h 2 , V = 4 π R 2 , E = \frac{lM}{R}, \qquad\widetilde{T} = \frac{lT}{R}, \qquad S = \pi r_h^2, \qquad\mathcal{V} = 4\pi R^2, E = R l M , T = R l T , S = π r h 2 , V = 4 π R 2 , Figure 11. Thermodynamic topology for σ = 0.9 \sigma= 0.9 σ = 0.9 q = 0.5 q = 0.5 q = 0.5 β = 0.1 \beta= 0.1 β = 0.1 P = 10 − 3 P = 10^{-3} P = 1 0 − 3 τ h ( r h ) \tau_h(r_h) τ h ( r h ) τ = 25 , 35 \tau= 25,35 τ = 25 , 35 n ( r h , Θ ) n(r_h,\Theta) n ( r h , Θ ) τ = 25 , 35 \tau= 25,35 τ = 25 , 35 ω = − 1 \omega= -1 ω = − 1 + 1 +1 + 1 ϕ r h > 0 \phi^{r_h} > 0 ϕ r h > 0 ϕ r h < 0 \phi^{r_h} < 0 ϕ r h < 0 r h r_h r h
C = l 2 4 , Q ~ = q l , Φ ~ = Φ R , β ~ = β , (41) C = \frac{l^2}{4}, \qquad\widetilde{Q} = ql, \qquad\widetilde{\Phi} = \frac{\Phi}{R}, \qquad\widetilde{\beta} = \beta, \tag*{(41)} C = 4 l 2 , Q = q l , Φ = R Φ , β = β , (41) where M M M T T T Φ \Phi Φ (32) , (33) and (37) , and l 2 = 3 / ( 8 π P ) l^2 = 3/(8\pi P) l 2 = 3/ ( 8 π P ) [92 , 93 , 94 ] for a conformal factor ω = R / l \omega= R/l ω = R / l q q q C C C [93 ] . The coupling is carried over unchanged as a formal thermodynamic variable, β ~ = β \widetilde{\beta} = \beta β = β [ L 2 ] [L^2] [ L 2 ] β ~ = 0.1 \widetilde{\beta} = 0.1 β = 0.1 R = 1 R = 1 R = 1 Ψ ~ = l L / R \widetilde{\Psi}=l\mathcal{L}/R Ψ = l L / R Q ~ = q l \widetilde{Q}=ql Q = q l Φ ~ = Φ / R \widetilde{\Phi}=\Phi/R Φ = Φ/ R (37) . For Maxwell theory this replacement is the electric–magnetic dual of the electric dictionary; for the present nonlinear Lagrangian it is a definition by the first law, and the ensembles at fixed Φ ~ \widetilde{\Phi} Φ E E E β ~ \widetilde{\beta} β
p = E 2 V , μ = E − T ~ S ~ − Φ ~ Q ~ − Ψ ~ β ~ C . (42) p=\frac{E}{2\mathcal{V}},\qquad\mu=\frac{E-\widetilde{T}\widetilde{S}-\widetilde{\Phi}\widetilde{Q}-\widetilde{\Psi}\widetilde{\beta}}{C}. \tag*{(42)} p = 2 V E , μ = C E − T S − Φ Q − Ψ β . (42) The expression for p p p Ψ ~ β ~ \widetilde{\Psi}\widetilde{\beta} Ψ β μ \mu μ (45) . In terms of the dimensionless variables x = r h / l x=r_h/l x = r h / l y = q / l = Q ~ / ( 4 C ) y=q/l=\widetilde{Q}/(4C) y = q / l = Q / ( 4 C ) β ^ = β / l 2 = β ~ / ( 4 C ) \widehat{\beta}=\beta/l^2=\widetilde{\beta}/(4C) β = β / l 2 = β / ( 4 C ) z = − β ^ y 2 / x 4 z=-\widehat{\beta}y^2/x^4 z = − β y 2 / x 4 R = 1 R=1 R = 1
E R C = 2 x ( 1 + x 2 ) + 2 y 2 F 1 x , S = 4 π C x 2 , T ~ R = 1 4 π x [ 1 + 3 x 2 − y 2 F 1 x 2 + 4 σ β ^ y 4 F 2 5 x 6 ] , Φ ~ R = y x F 1 − σ β ^ y 3 F 2 5 x 5 , μ R = x − x 3 − y 2 F 1 x + 2 σ β ^ y 4 F 2 5 x 5 . (43) \begin{aligned}
\frac{ER}{C}&=2x(1+x^2)+\frac{2y^2F_1}{x},\qquad S=4\pi Cx^2,\\
\widetilde{T}R&=\frac{1}{4\pi x}\left[1+3x^2-\frac{y^2F_1}{x^2}+\frac{4\sigma\widehat{\beta}y^4F_2}{5x^6}\right],\\
\widetilde{\Phi}R&=\frac{y}{x}F_1-\frac{\sigma\widehat{\beta}y^3F_2}{5x^5},\\
\mu R&=x-x^3-\frac{y^2F_1}{x}+\frac{2\sigma\widehat{\beta}y^4F_2}{5x^5}. \tag*{(43)}
\end{aligned} C E R T R Φ R μ R = 2 x ( 1 + x 2 ) + x 2 y 2 F 1 , S = 4 π C x 2 , = 4 π x 1 [ 1 + 3 x 2 − x 2 y 2 F 1 + 5 x 6 4 σ β y 4 F 2 ] , = x y F 1 − 5 x 5 σ β y 3 F 2 , = x − x 3 − x y 2 F 1 + 5 x 5 2 σ β y 4 F 2 . (43) and p = E / ( 8 π R 2 ) p=E/(8\pi R^2) p = E / ( 8 π R 2 ) F 1 = 2 F 1 ( 1 4 , σ ; 5 4 ; z ) F_1={}_2F_1\left(\frac{1}{4},\sigma;\frac{5}{4};z\right) F 1 = 2 F 1 ( 4 1 , σ ; 4 5 ; z ) F 2 = 2 F 1 ( 3 4 , σ + 1 ; 9 4 ; z ) F_2={}_2F_1\left(\frac{3}{4},\sigma+1;\frac{9}{4};z\right) F 2 = 2 F 1 ( 4 3 , σ + 1 ; 4 9 ; z ) β ^ = 0 \widehat{\beta}=0 β = 0 [93 ] . The last term of μ R \mu R μ R − Ψ ~ β ~ / ( C R − 1 ) -\widetilde{\Psi}\widetilde{\beta}/(CR^{-1}) − Ψ β / ( C R − 1 ) T ~ R \widetilde{T}R T R μ R \mu R μ R Φ ~ R \widetilde{\Phi}R Φ R E / ( C R − 1 ) E/(CR^{-1}) E / ( C R − 1 ) S / C S/C S / C ( x , y ) (x,y) ( x , y ) ( σ , β ^ ) (\sigma,\widehat{\beta}) ( σ , β ) ( σ , β ^ , y ) (\sigma,\widehat{\beta},y) ( σ , β , y ) C C C C C C R = 1 R=1 R = 1 β ~ = 0.1 \widetilde{\beta}=0.1 β = 0.1 C = 5 / 6 C=5/6 C = 5/6 β ^ = 0.03 \widehat{\beta}=0.03 β = 0.03 C C C Q ~ \widetilde{Q} Q β ~ \widetilde{\beta} β β ^ \widehat{\beta} β β ^ = κ y \widehat{\beta}=\kappa y β = κ y κ = β ~ / Q ~ \kappa=\widetilde{\beta}/\widetilde{Q} κ = β / Q β ^ \widehat{\beta} β
First law and Euler relation
The first law of the CFT is
d E = T ~ d S ~ − p d V + μ d C + Φ ~ d Q ~ + Ψ ~ d β ~ , Ψ ~ = l L R . (44) dE=\widetilde{T}\,d\widetilde{S}-p\,d\mathcal{V}+\mu\,dC+\widetilde{\Phi}\,d\widetilde{Q}+\widetilde{\Psi}\,d\widetilde{\beta},\qquad\widetilde{\Psi}=\frac{l\mathcal{L}}{R}. \tag*{(44)} d E = T d S − p d V + μ d C + Φ d Q + Ψ d β , Ψ = R l L . (44) It follows from the bulk first law, Eq. (38) , with d P = − 3 d l / ( 4 π l 3 ) dP=-3\,dl/(4\pi l^3) d P = − 3 d l / ( 4 π l 3 ) d q = d Q ~ / l − Q ~ d l / l 2 dq=d\widetilde{Q}/l-\widetilde{Q}\,dl/l^2 d q = d Q / l − Q d l / l 2 d β = d β ~ d\beta=d\widetilde{\beta} d β = d β E ∝ 1 / R E\propto1/R E ∝ 1/ R S S S C C C Q ~ \widetilde{Q} Q β ~ \widetilde{\beta} β p = − ∂ E / ∂ V = E / ( 2 V ) p=-\partial E/\partial\mathcal{V}=E/(2\mathcal{V}) p = − ∂ E / ∂ V = E / ( 2 V ) β ~ \widetilde{\beta} β d β ~ = 0 d\widetilde{\beta}=0 d β = 0 B d β \mathcal{B}\,d\beta B d β (38) . The Euler relation is
E = T ~ S ~ + μ C + Φ ~ Q ~ + Ψ ~ β ~ , (45) E=\widetilde{T}\widetilde{S}+\mu C+\widetilde{\Phi}\widetilde{Q}+\widetilde{\Psi}\widetilde{\beta}, \tag*{(45)} E = T S + μ C + Φ Q + Ψ β , (45) without a term p V p\mathcal{V} p V [92 , 93 , 94 ] . The term Ψ ~ β ~ \widetilde{\Psi}\widetilde{\beta} Ψ β β ~ = β \widetilde{\beta}=\beta β = β λ 2 \lambda^2 λ 2 ( r h , l , q , β ) → ( λ r h , λ l , λ q , λ 2 β ) (r_h,l,q,\beta)\to(\lambda r_h,\lambda l,\lambda q,\lambda^2\beta) ( r h , l , q , β ) → ( λ r h , λ l , λ q , λ 2 β ) § 1.3 B, in the same way as S S S C C C Q ~ \widetilde{Q} Q E R = l M E_R=lM E R = l M E R E_R E R ( S , C , Q ~ , β ~ ) (S,C,\widetilde{Q},\widetilde{\beta}) ( S , C , Q , β ) R R R (45) is Euler’s theorem for it. The relation is the counterpart of the Smarr relation, Eq. (40) , and the two expressions for μ \mu μ (45) and as ∂ E / ∂ C \partial E/\partial C ∂ E / ∂ C ( S , V , Q ~ , β ~ ) (S,\mathcal{V},\widetilde{Q},\widetilde{\beta}) ( S , V , Q , β ) 2 β B 2\beta\mathcal{B} 2 β B
One could instead hold fixed the dimensionless ratio β ^ = β / l 2 \widehat{\beta}=\beta/l^2 β = β / l 2 μ β ^ = μ + Ψ ~ β ~ / C = x − x 3 − y 2 F 1 / x \mu_{\widehat{\beta}}=\mu+\widetilde{\Psi}\widetilde{\beta}/C=x-x^3-y^2F_1/x μ β = μ + Ψ β / C = x − x 3 − y 2 F 1 / x R = 1 R=1 R = 1 Ψ ~ β ^ = l 3 B / R \widetilde{\Psi}_{\widehat{\beta}}=l^3\mathcal{B}/R Ψ β = l 3 B / R β ^ \widehat{\beta} β C C C β ~ = β \widetilde{\beta}=\beta β = β C C C l l l β ^ \widehat{\beta} β C C C Q ~ \widetilde{Q} Q (41) –(44) with partial derivatives of E ( S , V , C , Q ~ , β ~ ) E(S,\mathcal{V},C,\widetilde{Q},\widetilde{\beta}) E ( S , V , C , Q , β ) 10 − 30 10^{-30} 1 0 − 30 σ = 1.2 \sigma=1.2 σ = 1.2 Table XVII reports the difference between μ \mu μ ∂ E / ∂ C \partial E/\partial C ∂ E / ∂ C
Table XVII. CFT quantities for representative bulk states with β = 0.1 \beta=0.1 β = 0.1 R = 1 R=1 R = 1 C = l 2 / 4 C=l^{2}/4 C = l 2 /4 ∣ ∂ E / ∂ C − μ ∣ \lvert\partial E/\partial C-\mu\rvert ∣ ∂ E / ∂ C − μ ∣ ( S , q ~ , Q ~ , β ~ ) (S,\tilde{q},\tilde{Q},\tilde{\beta}) ( S , q ~ , Q ~ , β ~ ) μ \mu μ (42) .
Ensembles and free energies
Following Ref. [93 ] , the thermodynamic variables ( Q ~ , Φ ~ ) (\widetilde{Q},\widetilde{\Phi}) ( Q , Φ ) ( p , V ) (p,\mathcal{V}) ( p , V ) ( μ , C ) (\mu,C) ( μ , C ) (45) and E = 2 p V E=2p\mathcal{V} E = 2 p V Table XVIII . The potential of the ensemble ( Φ ~ , V , μ ) (\widetilde{\Phi},\mathcal{V},\mu) ( Φ , V , μ ) [93 ] ; here it equals Ψ ~ β ~ \widetilde{\Psi}\widetilde{\beta} Ψ β (45) , which is zero for β = 0 \beta=0 β = 0 β ~ \widetilde{\beta} β V \mathcal{V} V ( Q ~ , V , C ) (\widetilde{Q},\mathcal{V},C) ( Q , V , C ) ( Φ ~ , V , C ) (\widetilde{\Phi},\mathcal{V},C) ( Φ , V , C ) ( Q ~ , V , μ ) (\widetilde{Q},\mathcal{V},\mu) ( Q , V , μ ) p p p β ^ = 0 \widehat{\beta}=0 β = 0 [93 ] finds no transitions in them. For the present black hole the dependence on p p p V \mathcal{V} V
Table XVIII. Ensembles of the CFT with β ~ \tilde{\beta} β ~ E E E
Canonical ensemble (Q, V, C)
At fixed ( C , Q ~ , β ~ ) (C,\widetilde{Q},\widetilde{\beta}) ( C , Q , β ) l l l q q q β \beta β T ~ R = l T ( r h ) \widetilde{T}R=lT(r_h) T R = l T ( r h ) R R R § 1.3 at the pressure P = 3 / ( 8 π l 2 ) = 3 / ( 32 π C ) P=3/(8\pi l^2)=3/(32\pi C) P = 3/ ( 8 π l 2 ) = 3/ ( 32 π C ) y = Q ~ / ( 4 C ) y=\widetilde{Q}/(4C) y = Q / ( 4 C ) ∂ x T ~ = ∂ x 2 T ~ = 0 \partial_x\widetilde{T}=\partial_x^2\widetilde{T}=0 ∂ x T = ∂ x 2 T = 0 y y y Table VIII for β ^ = 0.03 \widehat{\beta}=0.03 β = 0.03 β ^ = 0 \widehat{\beta}=0 β = 0 [93 ] : x c = 1 / 6 x_c=1/\sqrt{6} x c = 1/ 6 y c = 1 / 6 y_c=1/6 y c = 1/6 C / Q ~ c = 3 / 2 C/\widetilde{Q}_c=3/2 C / Q c = 3/2 T ~ c R = 6 / ( 3 π ) \widetilde{T}_cR=\sqrt{6}/(3\pi) T c R = 6 / ( 3 π ) β ^ > 0 \widehat{\beta}>0 β > 0 C C C C / Q ~ c C/\widetilde{Q}_c C / Q c S c / C S_c/C S c / C μ c R \mu_cR μ c R p c p_c p c Φ ~ c R \widetilde{\Phi}_cR Φ c R σ \sigma σ y < y c y<y_c y < y c T ~ ( x ) \widetilde{T}(x) T ( x ) F = E − T ~ S F=E-\widetilde{T}S F = E − T S Figure 16 a). A first-order transition between a small-entropy and a large-entropy phase at a temperature T ~ t ( y ) \widetilde{T}_t(y) T t ( y ) y y y Table IX gives the transition for several values of y y y Q ~ / C \widetilde{Q}/C Q / C Figure 16 c shows the coexistence line in the ( T ~ , Q ~ / C ) (\widetilde{T},\widetilde{Q}/C) ( T , Q / C ) σ = 0.12 \sigma= 0.12 σ = 0.12 β ^ = 0 \hat{\beta} = 0 β ^ = 0 y y y σ = 0.9 \sigma= 0.9 σ = 0.9 y ∗ ≤ y ≤ y c y_* \le y \le y_c y ∗ ≤ y ≤ y c y ∗ = 0.1618 y_* = 0.1618 y ∗ = 0.1618 σ = 1.2 \sigma= 1.2 σ = 1.2 0.172858 ≲ y ≤ y c = 0.175606 0.172858 \lesssim y \le y_c = 0.175606 0.172858 ≲ y ≤ y c = 0.175606 y = 0.1735 y = 0.1735 y = 0.1735 Figure 16 b, where the cusps are resolved in the inset, and the corresponding coexistence line is the short segment ending at Q ~ / C ≃ 0.70 \widetilde{Q}/C \simeq0.70 Q / C ≃ 0.70 Figure 16 c and Figure 16 d. Below the model-dependent lower threshold there is no small–large free-energy crossing between the stable branches. This y y y C C C P ∗ < P < P c P_* < P < P_c P ∗ < P < P c Q ~ \widetilde{Q} Q β ~ \widetilde{\beta} β σ = 0.45 \sigma= 0.45 σ = 0.45 y ≥ 0.0574 y \ge0.0574 y ≥ 0.0574 y ≲ 0.12 y \lesssim0.12 y ≲ 0.12 x s ≲ 0.1 x_s \lesssim0.1 x s ≲ 0.1 0.105 ≲ y ≲ 0.115 0.105 \lesssim y \lesssim0.115 0.105 ≲ y ≲ 0.115 T ~ ( x ) \widetilde{T}(x) T ( x ) T ~ R ≃ 0.31 \widetilde{T}R \simeq0.31 T R ≃ 0.31 0.34 0.34 0.34 T ~ ( x ) \widetilde{T}(x) T ( x ) F F F Table IX . Since Q ~ = 4 C y \widetilde{Q} = 4Cy Q = 4 C y ( Q ~ , T ~ ) (\widetilde{Q},\widetilde{T}) ( Q , T ) C = 5 / 6 C = 5/6 C = 5/6 β ~ = 0.1 \widetilde{\beta} = 0.1 β = 0.1 ( 1 / C , T ~ ) (1/C,\widetilde{T}) ( 1/ C , T ) Q ~ \widetilde{Q} Q C C C Q ~ ↔ P \widetilde{Q}\leftrightarrow P Q ↔ P Φ ~ , μ ↔ V \widetilde{\Phi},\mu\leftrightarrow V Φ , μ ↔ V [93 ] . The alternative identification 1 / C ↔ P 1/C\leftrightarrow P 1/ C ↔ P β ^ \hat{\beta} β ^ C C C β ~ \widetilde{\beta} β Q ~ \widetilde{Q} Q Figure 16 . At fixed C C C σ \sigma σ Table VIII ).
Table VIII. Canonical critical point in units R = 1 R = 1 R = 1 β ~ = 0.1 \widetilde{\beta} = 0.1 β = 0.1 C = 5 / 6 C = 5/6 C = 5/6 β ^ = 0.03 \hat{\beta} = 0.03 β ^ = 0.03 β ~ = 0 \widetilde{\beta} = 0 β = 0 C / Q ~ c = 1 / ( 4 y c ) C/\widetilde{Q}_c = 1/(4y_c) C / Q c = 1/ ( 4 y c ) C C C C C C Q ~ \widetilde{Q} Q S c S_c S c C C C p c p_c p c C / R 3 C/R^3 C / R 3
Figure 16. Free energy and coexistence lines of the fixed-C C C β ~ = 0.1 \tilde{\beta}=0.1 β ~ = 0.1 C = 5 / 6 C=5/6 C = 5/6 β ^ = 0.03 \hat{\beta}=0.03 β ^ = 0.03 F ~ R / C \tilde{F}_{R}/C F ~ R / C T ~ R \tilde{T}R T ~ R σ = 0.12 \sigma=0.12 σ = 0.12 y = 0.12 y=0.12 y = 0.12 y c = 0.16736 y_c=0.16736 y c = 0.16736 0.22 0.22 0.22 σ = 1.2 \sigma=1.2 σ = 1.2 y = 0.1735 y=0.1735 y = 0.1735 y c = 0.17561 y_c=0.17561 y c = 0.17561 0.18 0.18 0.18 y = 0.1735 y=0.1735 y = 0.1735 ( T ~ R , Q ~ / C ) (\tilde{T}R,\tilde{Q}/C) ( T ~ R , Q ~ / C ) σ = 0.12 \sigma=0.12 σ = 0.12 0.9 0.9 0.9 0.45 0.45 0.45 1.2 1.2 1.2 y = 0.01 y=0.01 y = 0.01 σ = 0.12 \sigma=0.12 σ = 0.12 y ∗ = 0.1618 y_*=0.1618 y ∗ = 0.1618 σ = 0.9 \sigma=0.9 σ = 0.9 y = 0.0574 y=0.0574 y = 0.0574 σ = 0.45 \sigma=0.45 σ = 0.45 y ∗ = 0.172858 y_*=0.172858 y ∗ = 0.172858 σ = 1.2 \sigma=1.2 σ = 1.2
Table IX. First-order transition in the canonical ensemble for fixed β ~ = 0.1 \widetilde{\beta}=0.1 β = 0.1 C = 5 / 6 C=5/6 C = 5/6 β ^ = 0.03 \widehat{\beta}=0.03 β = 0.03 R = 1 R=1 R = 1 x s x_s x s x l x_l x l r h / l r_h/l r h / l Δ S / C = 4 π ( x l 2 − x s 2 ) \Delta S/C=4\pi(x_l^2-x_s^2) Δ S / C = 4 π ( x l 2 − x s 2 ) T ~ t Δ S / C \widetilde{T}_t\Delta S/C T t Δ S / C F ~ t = F / C \widetilde{F}_t=F/C F t = F / C β ^ = 0 \widehat{\beta}=0 β = 0 σ = 0.12 \sigma=0.12 σ = 0.12 σ = 0.9 \sigma=0.9 σ = 0.9 σ = 0.45 \sigma=0.45 σ = 0.45 σ = 1.2 \sigma=1.2 σ = 1.2
Critical point at fixed Q ~ \widetilde{Q} Q C C C
In the identification 1 / C ↔ P 1/C\leftrightarrow P 1/ C ↔ P Q ~ \widetilde{Q} Q β ~ \widetilde{\beta} β C C C l l l κ = β ~ / Q ~ = 0.1 \kappa= \widetilde{\beta}/\widetilde{Q} = 0.1 κ = β / Q = 0.1 y = Q ~ / ( 4 C ) y = \widetilde{Q}/(4C) y = Q / ( 4 C ) β ^ = κ y \hat{\beta} = \kappa y β ^ = κ y β ^ \hat{\beta} β ^ T ~ R ( x , y ) \widetilde{T}R(x,y) T R ( x , y ) (43) with β ^ = κ y \hat{\beta} = \kappa y β ^ = κ y y y y x x x ∂ x T ~ = ∂ x 2 T ~ = 0 \partial_x\widetilde{T} = \partial_x^2\widetilde{T} = 0 ∂ x T = ∂ x 2 T = 0 β ^ = κ y \hat{\beta} = \kappa y β ^ = κ y y y y T ~ ( x s ) = T ~ ( x l ) \widetilde{T}(x_s) = \widetilde{T}(x_l) T ( x s ) = T ( x l ) F ~ ( x s ) = F ~ ( x l ) \widetilde{F}(x_s) = \widetilde{F}(x_l) F ( x s ) = F ( x l ) β ^ \hat{\beta} β ^ Figure 17 . The critical temperature changes by about 0.5 % 0.5\% 0.5% σ = 1.5 \sigma= 1.5 σ = 1.5 C / Q ~ c C/\widetilde{Q}_c C / Q c 3 / 2 3/2 3/2 σ \sigma σ C C C Table VIII ), because β ^ c = κ y c \hat{\beta}_c = \kappa y_c β ^ c = κ y c 0.03 0.03 0.03 Q ~ / C \widetilde{Q}/C Q / C σ = 0.9 \sigma= 0.9 σ = 0.9 1.2 1.2 1.2 1.5 1.5 1.5 y y y σ = 0.12 \sigma= 0.12 σ = 0.12 σ = 0.45 \sigma= 0.45 σ = 0.45 C C C
Figure 17. Canonical coexistence lines in the ( T ~ R , Q ~ / C ) (\tilde{T}R,\tilde{Q}/C) ( T ~ R , Q ~ / C ) Q ~ \tilde{Q} Q ~ β ~ = 0.1 \tilde{\beta}=0.1 β ~ = 0.1 κ = 0.1 \kappa=0.1 κ = 0.1 β ^ = κ y \hat{\beta}=\kappa y β ^ = κ y σ = 0.12 \sigma=0.12 σ = 0.12 σ = 0.45 \sigma=0.45 σ = 0.45 σ = 0.9 \sigma=0.9 σ = 0.9 σ = 1.2 \sigma=1.2 σ = 1.2 σ = 1.5 \sigma=1.5 σ = 1.5
Critical exponents
Near the critical point the exponents are those of mean-field theory, α = 0 \alpha= 0 α = 0 β c = 1 / 2 \beta_c = 1/2 β c = 1/2 γ = 1 \gamma= 1 γ = 1 δ = 3 \delta= 3 δ = 3 β ^ = 0 \hat{\beta} = 0 β ^ = 0 [9 ] . The same exponents hold for the present black hole because T ~ ( x , y ) \widetilde{T}(x,y) T ( x , y ) ∂ x T ~ = ∂ x 2 T ~ = 0 \partial_x\widetilde{T} = \partial_x^2\widetilde{T} = 0 ∂ x T = ∂ x 2 T = 0 ∂ x 3 T ~ \partial_x^3\widetilde{T} ∂ x 3 T 11.46 11.46 11.46 11.28 11.28 11.28 10.65 10.65 10.65 9.30 9.30 9.30 7.49 7.49 7.49 ∂ y T ~ \partial_y\widetilde{T} ∂ y T − 0.390 -0.390 − 0.390 − 0.394 -0.394 − 0.394 − 0.406 -0.406 − 0.406 − 0.431 -0.431 − 0.431 − 0.463 -0.463 − 0.463 σ = 0.12 \sigma= 0.12 σ = 0.12 0.45 0.45 0.45 0.9 0.9 0.9 1.2 1.2 1.2 β ^ = 0.03 \hat{\beta} = 0.03 β ^ = 0.03 verify_critical_coeffs.py); both are nonzero, as required for δ = 3 \delta= 3 δ = 3 β c = 1 / 2 \beta_c = 1/2 β c = 1/2 C C C β ~ \widetilde{\beta} β β ^ \hat{\beta} β ^ σ = 1.2 \sigma= 1.2 σ = 1.2 x l − x s x_l-x_s x l − x s ( 1 − y / y c ) β c (1-y/y_c)^{\beta_c} ( 1 − y / y c ) β c 0.5000 0.5000 0.5000 0.5011 0.5011 0.5011 1 − y / y c = 3 × 10 − 3 1-y/y_c = 3\times10^{-3} 1 − y / y c = 3 × 1 0 − 3 3 × 10 − 4 3\times10^{-4} 3 × 1 0 − 4 3 × 10 − 4 3\times10^{-4} 3 × 1 0 − 4 T ~ = T ~ c \widetilde{T} = \widetilde{T}_c T = T c Q ~ − Q ~ c ∝ ( Φ ~ − Φ ~ c ) δ \widetilde{Q}-\widetilde{Q}_c \propto(\widetilde{\Phi}-\widetilde{\Phi}_c)^\delta Q − Q c ∝ ( Φ − Φ c ) δ 3.027 3.027 3.027 3.012 3.012 3.012 3.006 3.006 3.006 r h r_h r h 1 % 1\% 1% 0.5 % 0.5\% 0.5% 0.2 % 0.2\% 0.2% ( ∂ Φ ~ / ∂ Q ~ ) T ~ (\partial\widetilde{\Phi}/\partial\widetilde{Q})_{\widetilde{T}} ( ∂ Φ / ∂ Q ) T Φ ~ = Φ ~ c \widetilde{\Phi}=\widetilde{\Phi}_c Φ = Φ c ( T ~ / T ~ c − 1 ) − 1 (\widetilde{T}/\widetilde{T}_c-1)^{-1} ( T / T c − 1 ) − 1 0.989 0.989 0.989 0.991 0.991 0.991 0.999 0.999 0.999 T ~ / T ~ c − 1 = 10 − 3 \widetilde{T}/\widetilde{T}_c-1 = 10^{-3} T / T c − 1 = 1 0 − 3 10 − 4 10^{-4} 1 0 − 4 10 − 4 10^{-4} 1 0 − 4 10 − 5 10^{-5} 1 0 − 5 γ = 1 \gamma= 1 γ = 1 Table XI . The heat capacity C Φ ~ / C C_{\widetilde{\Phi}}/C C Φ / C
Table XI. Critical exponents of the canonical ensemble at fixed C C C β c \beta_c β c β \beta β t = T ~ / T ~ c − 1 t=\widetilde{T}/\widetilde{T}_c-1 t = T / T c − 1 C Φ ~ ≡ C Φ ~ V C / C C_{\widetilde{\Phi}}\equiv C_{\widetilde{\Phi}V C}/C C Φ ≡ C Φ V C / C
Table X. Critical point at fixed Q ~ \widetilde{Q} Q β ~ = 0.1 \widetilde{\beta}=0.1 β = 0.1 κ = β ~ / Q ~ = 0.1 \kappa=\widetilde{\beta}/\widetilde{Q}=0.1 κ = β / Q = 0.1 R = 1 R=1 R = 1 C C C Q ~ / C \widetilde{Q}/C Q / C σ = 0.12 \sigma=0.12 σ = 0.12 Q ~ / C = 0.04 \widetilde{Q}/C=0.04 Q / C = 0.04
ues −16.76, −16.43, −13.66 and −15.42 for β ^ = 0 \hat{\beta}=0 β ^ = 0 σ = 0.12 \sigma=0.12 σ = 0.12 0.9 0.9 0.9 0.45 0.45 0.45 β ^ = 0.03 \hat{\beta}=0.03 β ^ = 0.03 α = 0 \alpha=0 α = 0 ( Q ~ , Φ ~ ) (\widetilde{Q},\widetilde{\Phi}) ( Q , Φ ) C Φ ~ V C C_{\widetilde{\Phi}VC} C Φ V C
Grand-canonical ensemble (Phi, V, C)
In the grand-canonical calculations we again hold C = 5 / 6 C=5/6 C = 5/6 β ~ = 0.1 \widetilde{\beta}=0.1 β = 0.1 β ^ = 0.03 \hat{\beta}=0.03 β ^ = 0.03 Φ ~ \widetilde{\Phi} Φ y ( x ) y(x) y ( x ) Φ ~ R = ϕ \widetilde{\Phi}R=\phi Φ R = ϕ W / C = μ R + Ψ ~ β ~ / ( C R − 1 ) = x − x 3 − y 2 F 1 / x W/C=\mu R+\widetilde{\Psi}\widetilde{\beta}/(CR^{-1})=x-x^3-y^2F_1/x W / C = μ R + Ψ β / ( C R − 1 ) = x − x 3 − y 2 F 1 / x y = ϕ x y=\phi x y = ϕ x T ~ R = ( 1 + 3 x 2 − ϕ 2 ) / ( 4 π x ) \widetilde{T}R=(1+3x^2-\phi^2)/(4\pi x) T R = ( 1 + 3 x 2 − ϕ 2 ) / ( 4 π x ) W = 0 W=0 W = 0 Figure 18 a). Table XII gives the transition temperature. At Φ ~ R = 0.3 \widetilde{\Phi}R=0.3 Φ R = 0.3 Figure 18 (a) shows the curves for RN and for σ = 0.12 \sigma=0.12 σ = 0.12 0.45 0.45 0.45 0.9 0.9 0.9 1.2 1.2 1.2 Table XII . The plotted curves nearly coincide on the main scale, consistent with the very small shifts in the Hawking–Page temperature; the inset resolves the separation of their W = 0 W=0 W = 0 σ = 1.2 \sigma=1.2 σ = 1.2 T ~ H P R = 0.303638 \widetilde{T}_{\mathrm{HPR}}=0.303638 T HPR = 0.303638 1.1 × 10 − 5 1.1\times10^{-5} 1.1 × 1 0 − 5 0.303648 0.303648 0.303648 σ = 1.2 \sigma=1.2 σ = 1.2
Table XII. Hawking–Page temperature T ~ H P R \widetilde{T}_{\mathrm{HPR}} T HPR ( Φ ~ , V , C ) (\widetilde{\Phi},\mathcal{V},C) ( Φ , V , C ) β ~ = 0.1 \widetilde{\beta}=0.1 β = 0.1 C = 5 / 6 C=5/6 C = 5/6 β ^ = 0.03 \hat{\beta}=0.03 β ^ = 0.03 β ~ = 0 \widetilde{\beta}=0 β = 0
Figure 18. (a) Potential W / C W/C W / C ( Φ ~ , V , C ) (\tilde{\Phi},\mathcal{V},C) ( Φ ~ , V , C ) T ~ R \tilde{T}R T ~ R Φ ~ R = 0.3 \tilde{\Phi}R = 0.3 Φ ~ R = 0.3 σ = 0.12 \sigma= 0.12 σ = 0.12 σ = 0.45 \sigma= 0.45 σ = 0.45 σ = 0.9 \sigma= 0.9 σ = 0.9 σ = 1.2 \sigma= 1.2 σ = 1.2 T ~ min R \tilde{T}_{\min}R T ~ m i n R x x x T ~ 1 R \tilde{T}_{1}R T ~ 1 R μ R \mu R μ R β ~ / Q ~ = 0.1 \tilde{\beta}/\tilde{Q} = 0.1 β ~ / Q ~ = 0.1
same order as the exponent. For Φ ~ R < 0.4 \widetilde{\Phi}R<0.4 Φ R < 0.4 σ \sigma σ β ^ = 0 \hat{\beta}=0 β ^ = 0 Φ ~ \widetilde{\Phi} Φ W ( T ~ ) W(\widetilde{T}) W ( T ) Φ ~ R ≤ 0.4 \widetilde{\Phi}R\leq0.4 Φ R ≤ 0.4
Mixed ensemble (Q, V, mu)
In the mixed ensemble we hold Q ~ \widetilde{Q} Q V \mathcal{V} V μ \mu μ β ~ = 0.1 \widetilde{\beta}=0.1 β = 0.1 C C C l l l Q ~ = 1 \widetilde{Q}=1 Q = 1 R = 1 R=1 R = 1 κ = Q ~ / Φ ~ = 0.1 \kappa=\widetilde{Q}/\widetilde{\Phi}=0.1 κ = Q / Φ = 0.1 β ^ = κ y \hat{\beta}=\kappa y β ^ = κ y β ^ \hat{\beta} β ^ 0.03 0.03 0.03 μ R = c o n s t \mu R=\mathrm{const} μ R = const y 2 F 1 / x − 2 κ y 4 F 2 / ( 5 x 5 ) = x − x 3 − μ R y^2F_1/x-2\kappa y^4F_2/(5x^5)=x-x^3-\mu R y 2 F 1 / x − 2 κ y 4 F 2 / ( 5 x 5 ) = x − x 3 − μ R y ( x ) y(x) y ( x ) S = π Q ~ 2 / y S=\pi\widetilde{Q}^2/y S = π Q 2 / y C μ / Q ~ = T ~ d ( π x 2 / y ) / d x / ( d T ~ / d x ) C_\mu/\widetilde{Q}=\widetilde{T}\,d(\pi x^2/y)/dx/(d\widetilde{T}/dx) C μ / Q = T d ( π x 2 / y ) / d x / ( d T / d x ) μ R > 0 \mu R>0 μ R > 0 x x x x − x 3 = μ R x-x^3=\mu R x − x 3 = μ R x x x T ~ 1 \widetilde{T}_1 T 1 x x x T ~ 2 \widetilde{T}_2 T 2 y → 0 y\to0 y → 0 T ~ R = ( 1 + 3 x 2 ) / ( 4 π x ) \widetilde{T}_R=(1+3x^2)/(4\pi x) T R = ( 1 + 3 x 2 ) / ( 4 π x ) T ~ 0 = T ~ min ( μ ) \widetilde{T}_0=\widetilde{T}_{\min}(\mu) T 0 = T m i n ( μ ) T ~ 1 \widetilde{T}_1 T 1 Table XIII lists T ~ min \widetilde{T}_{\min} T m i n x x x T ~ 1 \widetilde{T}_1 T 1 κ = 0.1 \kappa=0.1 κ = 0.1 Figure 18 b shows T ~ min ( μ ) \widetilde{T}_{\min}(\mu) T m i n ( μ ) T ~ min \widetilde{T}_{\min} T m i n σ \sigma σ μ \mu μ Figure 18 b, σ = 1.2 \sigma=1.2 σ = 1.2 μ R = 0.1 \mu R=0.1 μ R = 0.1 μ R = 0.25 \mu R=0.25 μ R = 0.25 μ = 0 \mu=0 μ = 0 T ~ min → 0 \widetilde{T}_{\min}\to0 T m i n → 0 β ^ > 0 \hat{\beta}>0 β ^ > 0
Table XIII. Lower temperature T ~ min R \widetilde{T}_{\min}R T m i n R x x x T ~ 1 R = ( 1 + 3 x ~ + 2 ) / ( 4 π x + ) \widetilde{T}_1R=(1+3\tilde{x}_+^2)/(4\pi x_+) T 1 R = ( 1 + 3 x ~ + 2 ) / ( 4 π x + ) β ~ = 0.1 \widetilde{\beta}=0.1 β = 0.1 x + x_+ x + x − x 3 = μ R x-x^3=\mu R x − x 3 = μ R y = 0 y=0 y = 0 β ~ = 0 \widetilde{\beta}=0 β = 0 μ = 0 \mu=0 μ = 0 T ~ R = x / π \widetilde{T}R=x/\pi T R = x / π T ~ min = 0 \widetilde{T}_{\min}=0 T m i n = 0
Figure 19 presents the corresponding branch structure and numerical μ \mu μ T ~ \tilde{T} T ~ σ = 0.45 \sigma=0.45 σ = 0.45 T 0 = T min T_0=T_{\min} T 0 = T m i n T 1 T_1 T 1 G R = 0 GR=0 GR = 0 G R GR GR C μ C_\mu C μ T i n t T_{\mathrm{int}} T int T 0 T_0 T 0 T 1 T_1 T 1 T 2 T_2 T 2 T i n t T_{\mathrm{int}} T int μ R \mu R μ R C μ C_\mu C μ
Figure 19. Mixed-ensemble free-energy branches and phase boundaries for nonlinear electrodynamics with σ = 0.45 \sigma= 0.45 σ = 0.45 Q ~ = 1 \tilde{Q} = 1 Q ~ = 1 β ~ / Q ~ = 0.1 \tilde{\beta}/\tilde{Q} = 0.1 β ~ / Q ~ = 0.1 R = 1 R = 1 R = 1 G R G R GR T ~ R \tilde{T}R T ~ R μ R \mu R μ R μ R = 0.1 \mu R = 0.1 μ R = 0.1 T 0 T_0 T 0 T 1 T_1 T 1 x x x T 2 T_2 T 2 x x x T i n t T_{\mathrm{int}} T int C μ = 0 C_\mu= 0 C μ = 0 G R G R GR T 1 T_1 T 1 μ \mu μ T ~ \tilde{T} T ~ C μ C_\mu C μ
For σ = 1.2 \sigma=1.2 σ = 1.2 Figure 20 . The boundary curves T 0 T_0 T 0 T 1 T_1 T 1 T 2 T_2 T 2 T i n t T_{\mathrm{int}} T int σ = 0.45 \sigma=0.45 σ = 0.45 μ R = 0.1 \mu R=0.1 μ R = 0.1 T 0 R = 0.17866 T_0R=0.17866 T 0 R = 0.17866 0.18368 0.18368 0.18368 T i n t R T_{\mathrm{int}}R T int R 0.3988 0.3988 0.3988 0.3940 0.3940 0.3940 T 1 R = 0.30991 T_1R=0.30991 T 1 R = 0.30991 T 2 R = 0.81177 T_2R=0.81177 T 2 R = 0.81177 μ \mu μ σ \sigma σ μ \mu μ T i n t T_{\mathrm{int}} T int T 1 T_1 T 1 μ ∗ R ≃ 0.1514 \mu_*R\simeq0.1514 μ ∗ R ≃ 0.1514 0.1500 0.1500 0.1500 μ \mu μ μ R = − 0.1 \mu R=-0.1 μ R = − 0.1 σ = 0.45 \sigma=0.45 σ = 0.45 T → ∞ T\to\infty T → ∞ x → 0 x\to0 x → 0 σ > 1 / 4 \sigma>1/4 σ > 1/4 T ~ R ≃ 0.0995 \tilde{T}R\simeq0.0995 T ~ R ≃ 0.0995 G R ≃ 0.98 GR\simeq0.98 GR ≃ 0.98 T ~ R = 1.7 \tilde{T}R=1.7 T ~ R = 1.7
Figure 20. Mixed-ensemble free-energy branches and phase boundaries for nonlinear electrodynamics with σ = 1.2 \sigma= 1.2 σ = 1.2 Q ~ = 1 \widetilde{Q}=1 Q = 1 β ~ / Q ~ = 0.1 \widetilde{\beta}/\widetilde{Q}=0.1 β / Q = 0.1 R = 1 R=1 R = 1 Figure 19 . (a) The potential G R GR GR T ~ R \widetilde{T}R T R μ R \mu R μ R μ R = 0.1 \mu R=0.1 μ R = 0.1 T 0 T_0 T 0 T 1 T_1 T 1 T 2 T_2 T 2 x x x x x x T ~ i n t R = 0.394 \widetilde{T}_{\mathrm{int}}R=0.394 T int R = 0.394 μ R = − 0.1 \mu R=-0.1 μ R = − 0.1 T ~ R ≤ 1.7 \widetilde{T}R\leq1.7 T R ≤ 1.7 μ \mu μ T ~ \widetilde{T} T μ ∗ R ≃ 0.150 \mu_*R\simeq0.150 μ ∗ R ≃ 0.150
Pressure–volume plane
Ref. [93 ] finds no criticality in the ( p , V ) (p,\mathcal{V}) ( p , V ) ( T ~ , C , Q ~ ) (\tilde{T},C,\tilde{Q}) ( T ~ , C , Q ~ ) R = τ ( x ) / T ~ R=\tau(x)/\tilde{T} R = τ ( x ) / T ~ τ = T ~ R \tau=\tilde{T}R τ = T ~ R V = 4 π R 2 \mathcal{V}=4\pi R^2 V = 4 π R 2 p = C T ~ 3 h ( x ) / ( 2 π ) p=C\tilde{T}^3h(x)/(2\pi) p = C T ~ 3 h ( x ) / ( 2 π ) h ( x ) = ϵ ( x ) / τ ( x ) h(x)=\epsilon(x)/\tau(x) h ( x ) = ϵ ( x ) / τ ( x ) ϵ = M / l \epsilon=M/l ϵ = M / l ∂ x h = ∂ x 2 h = 0 \partial_xh=\partial_x^2h=0 ∂ x h = ∂ x 2 h = 0 ∂ x τ ≠ 0 \partial_x\tau\ne0 ∂ x τ = 0 p ( V ) p(\mathcal{V}) p ( V ) T ~ 3 \tilde{T}^3 T ~ 3 T ~ \tilde{T} T ~ p V 3 / 2 p\mathcal{V}^{3/2} p V 3/2 x x x y y y ( p , V ) (p,\mathcal{V}) ( p , V ) T ~ \tilde{T} T ~ p V 3 / 2 p\mathcal{V}^{3/2} p V 3/2 p ( V ) p(\mathcal{V}) p ( V ) ( T ~ , C ~ , Q ~ ) (\tilde{T},\tilde{C},\tilde{Q}) ( T ~ , C ~ , Q ~ ) h ( x ) h(x) h ( x ) τ > 0 \tau>0 τ > 0 y ∈ [ 0.005 , 0.40 ] y\in[0.005,0.40] y ∈ [ 0.005 , 0.40 ] β ^ = 0 \hat{\beta}=0 β ^ = 0 β ^ = 0.03 \hat{\beta}=0.03 β ^ = 0.03 ∂ x h = ∂ x 2 h = 0 \partial_xh=\partial_x^2h=0 ∂ x h = ∂ x 2 h = 0 Table XIV ). For RN and σ = 0.12 \sigma=0.12 σ = 0.12 y y y [93 ] . For σ = 0.9 \sigma=0.9 σ = 0.9 σ = 1.2 \sigma=1.2 σ = 1.2 σ = 1.5 \sigma=1.5 σ = 1.5 y y y σ = 0.45 \sigma=0.45 σ = 0.45 y = 0.11168 y=0.11168 y = 0.11168 y = 0.11774 y=0.11774 y = 0.11774 Q ~ / C \tilde{Q}/C Q ~ / C τ ( x ) \tau(x) τ ( x ) Y \mathcal{Y} Y x x x p ( Y ) p(\mathcal{Y}) p ( Y ) Table XIV , d τ / d x d\tau/dx d τ / d x ( Y , C , Q ~ ) (\mathcal{Y},C,\tilde{Q}) ( Y , C , Q ~ ) 8 π C x T ~ / τ ′ 8\pi Cx\tilde{T}/\tau' 8 π C x T ~ / τ ′ τ → ∞ \tau\to\infty τ → ∞ y → 1 / 6 y\to1/\sqrt{6} y → 1/ 6
Table XIV. Critical points of the isotherms p ( Y ) p(\mathcal{Y}) p ( Y ) ( T ~ , C , Q ~ ) (\tilde{T},C,\tilde{Q}) ( T ~ , C , Q ~ ) β ^ = 0.03 \hat{\beta}=0.03 β ^ = 0.03 β ^ = 0.1 \hat{\beta}=0.1 β ^ = 0.1 C = 5 / 6 C=5/6 C = 5/6 R = 1 R=1 R = 1 p c Y c 3 / 2 / C = 4 π ϵ c p_c\mathcal{Y}_c^{3/2}/C=4\sqrt{\pi}\epsilon_c p c Y c 3/2 / C = 4 π ϵ c
Heat capacities and stability
At fixed C C C S = 4 π C x 2 S=4\pi Cx^2 S = 4 π C x 2 X X X
C X = T ~ ( ∂ S ∂ T ~ ) X = 8 π C x T ~ ( d T ~ / d x ) X , (46) C_X=\tilde{T}\left(\frac{\partial S}{\partial\tilde{T}}\right)_X=\frac{8\pi Cx\tilde{T}}{(d\tilde{T}/dx)_X}, \tag*{(46)} C X = T ~ ( ∂ T ~ ∂ S ) X = ( d T ~ / d x ) X 8 π C x T ~ , (46) where T ~ ( x ) \tilde{T}(x) T ~ ( x ) X X X y y y ( Q ~ , Y , C ) (\tilde{Q},\mathcal{Y},C) ( Q ~ , Y , C ) y ( x ) y(x) y ( x ) Φ ~ R = const \tilde{\Phi}R=\text{const} Φ ~ R = const ( Φ ~ , Y , C ) (\tilde{\Phi},\mathcal{Y},C) ( Φ ~ , Y , C ) y ( x ) y(x) y ( x ) μ R = const \mu R=\text{const} μ R = const ( Q ~ , Y , μ ) (\tilde{Q},\mathcal{Y},\mu) ( Q ~ , Y , μ ) (46) does not apply to the mixed ensemble, where C C C S = π Q ~ x 2 / y ( x ) S=\pi\tilde{Q}x^2/y(x) S = π Q ~ x 2 / y ( x ) C μ = T ~ ( d S / d x ) / ( d T ~ / d x ) C_\mu=\tilde{T}(dS/dx)/(d\tilde{T}/dx) C μ = T ~ ( d S / d x ) / ( d T ~ / d x ) C X > 0 C_X>0 C X > 0 Figure 21 shows C X / C C_X/C C X / C Table XIX gives the points at which C X C_X C X T ~ ( x ) \tilde{T}(x) T ~ ( x ) y = 0.12 y=0.12 y = 0.12 σ = 0.12 \sigma=0.12 σ = 0.12 0.45 0.45 0.45 C X > 0 C_X>0 C X > 0 x < x 1 x<x_1 x < x 1 C X < 0 C_X<0 C X < 0 x 1 x_1 x 1 x 2 x_2 x 2 C X > 0 C_X>0 C X > 0 x > x 2 x>x_2 x > x 2 Figure 16 a, and the first-order transition of Table IX connects the two stable branches. The large-black-hole divergence x 2 ≃ 0.532 x_2\simeq0.532 x 2 ≃ 0.532 σ \sigma σ x 1 x_1 x 1 T ~ \tilde{T} T ~ σ = 0.45 \sigma=0.45 σ = 0.45 σ = 0.9 \sigma=0.9 σ = 0.9 σ = 1.2 \sigma=1.2 σ = 1.2 x ≳ 0.02 x\gtrsim0.02 x ≳ 0.02 y = 0.12 y=0.12 y = 0.12 x ≃ 0.5508 x\simeq0.5508 x ≃ 0.5508 Φ ~ R = 0.3 \tilde{\Phi}R=0.3 Φ ~ R = 0.3 T ~ \tilde{T} T ~ T ~ min ( μ ) \tilde{T}_{\min}(\mu) T ~ m i n ( μ )
Figure 21. Heat capacity C X / C C_X/C C X / C arcsinh ( C X / 5 C ) \operatorname{arcsinh}(C_X/5C) arcsinh ( C X /5 C ) x = r h / l x=r_h/l x = r h / l β ~ = 0.1 \widetilde{\beta}=0.1 β = 0.1 C = 5 / 6 C=5/6 C = 5/6 β ^ = 0.03 \widehat{\beta}=0.03 β = 0.03 β ~ = 0.1 \widetilde{\beta}=0.1 β = 0.1 Q ~ = 1 \widetilde{Q}=1 Q = 1 κ = 0.1 \kappa=0.1 κ = 0.1 β ^ = κ y \widehat{\beta}=\kappa y β = κ y C C C σ = 0.12 \sigma=0.12 σ = 0.12 0.45 0.45 0.45 0.9 0.9 0.9 1.2 1.2 1.2 y = 0.12 y=0.12 y = 0.12 Φ ~ R = 0.3 \widetilde{\Phi}R=0.3 Φ R = 0.3 μ R = 0.1 \mu R=0.1 μ R = 0.1 0.5996 ≤ x ≤ 0.6004 0.5996\leq x\leq0.6004 0.5996 ≤ x ≤ 0.6004 0.425 ≤ x ≤ 0.445 0.425\leq x\leq0.445 0.425 ≤ x ≤ 0.445
Table XIX. Divergences of the heat capacity, given by the extrema of T ~ ( x ) \tilde{T}(x) T ~ ( x ) β ~ = 0.1 \tilde{\beta}=0.1 β ~ = 0.1 C = 5 / 6 C=5/6 C = 5/6 β ^ = 0.03 \hat{\beta}=0.03 β ^ = 0.03 y = 0.12 y=0.12 y = 0.12 Φ ~ R = 0.3 \tilde{\Phi}_{R}=0.3 Φ ~ R = 0.3 β ~ = 0.1 \tilde{\beta}=0.1 β ~ = 0.1 Q ~ = 1 \tilde{Q}=1 Q ~ = 1 μ R = 0.1 \mu_{R}=0.1 μ R = 0.1 κ = 0.1 \kappa=0.1 κ = 0.1 β ^ = κ y \hat{\beta}=\kappa y β ^ = κ y x < x 1 x<x_{1} x < x 1 x 1 < x < x 2 x_{1}<x<x_{2} x 1 < x < x 2 x > x 2 x>x_{2} x > x 2
meet continuously. The finite discontinuity in the branch-valued potential instead occurs at the distinct endpoint T ~ 1 \widetilde{T}_1 T 1 x x x Figure 19 ). A zeroth-order equilibrium transition can be inferred only if the competing branch is locally stable there; the sampled endpoint states fail the Hessian test (Table XVI ). The cusp and endpoint temperatures shift with σ \sigma σ x x x C X ≃ 8 π C x 2 = 2 S C_X \simeq8\pi C x^2 = 2S C X ≃ 8 π C x 2 = 2 S μ R > 0 \mu_R > 0 μ R > 0 x − x 3 = μ R x-x^3=\mu_R x − x 3 = μ R x x x
Table XVI. Numerical endpoint test in the mixed ensemble. T 1 T_{1} T 1 G l o w G_{\mathrm{low}} G low Δ ( G R ) = G l o w R \Delta(GR)=G_{\mathrm{low}}R Δ ( GR ) = G low R G h i g h R → 0 G_{\mathrm{high}}R\to0 G high R → 0 det H \det H det H
A positive heat capacity, C X > 0 C_X>0 C X > 0 x x x C μ > 0 C_\mu>0 C μ > 0 E C C < 0 E_{CC}<0 E C C < 0
Conclusions
We have examined the background circular null orbits, effective electromagnetic propagation, thermodynamic topology, and boundary thermodynamic description of a magnetically charged AdS solution in nonlinear electrodynamics.
The background classification is controlled by horizon formation and the creation or annihilation of circular-null-orbit pairs. For m 0 = l = 1 m_0=l=1 m 0 = l = 1 q = 0.5 q=0.5 q = 0.5 β = 0.1 \beta=0.1 β = 0.1 σ = 0.1465105 \sigma=0.1465105 σ = 0.1465105 σ = 0.1548466 \sigma=0.1548466 σ = 0.1548466 m 0 m_0 m 0 σ = 1 / 4 \sigma=1/4 σ = 1/4 m 0 m_0 m 0 M M M σ ≥ 1 \sigma\geq1 σ ≥ 1 m 0 = 0 m_0=0 m 0 = 0
The extraordinary optical geometry introduces the factor η \eta η 5.19 % 5.19\% 5.19% Figure 8 .
Figure 8. Background and extraordinary optical orbit structure at m 0 = l = 1 m_0=l=1 m 0 = l = 1 β = 0.1 \beta=0.1 β = 0.1 U b = f / r 2 U_b=f/r^2 U b = f / r 2 U e = f η / r 2 U_e=f\eta/r^2 U e = f η / r 2 q = 0.5 q=0.5 q = 0.5 σ = 0.15 \sigma=0.15 σ = 0.15 n e = ϕ e / ∥ ϕ e ∥ \mathbf{n}_e=\boldsymbol{\phi}_e/\lVert\boldsymbol{\phi}_e\rVert n e = ϕ e / ∥ ϕ e ∥ + 1 +1 + 1 − 1 -1 − 1 q = 1.2 q=1.2 q = 1.2 σ = 0.9 \sigma=0.9 σ = 0.9 r b r_b r b r e r_e r e (26) .
The thermodynamic phase structure depends on the small-radius temperature and the crossings of stable Gibbs-free-energy branches. At P = 10 − 3 P=10^{-3} P = 1 0 − 3 ( q , σ ) = ( 0.5 , 0.12 ) (q,\sigma)=(0.5,0.12) ( q , σ ) = ( 0.5 , 0.12 ) ( 1.2 , 0.45 ) (1.2,0.45) ( 1.2 , 0.45 ) T T C = + 1 \mathrm{TTC}=+1 TTC = + 1 q = 0.5 q=0.5 q = 0.5 σ = 0.9 \sigma=0.9 σ = 0.9 1.2 1.2 1.2 + ∞ +\infty + ∞ r h → 0 r_h\to0 r h → 0 P = 10 − 3 P=10^{-3} P = 1 0 − 3 σ = 0.9 \sigma=0.9 σ = 0.9 0.001307 < P < P c 0.001307<P<P_c 0.001307 < P < P c P c = 0.013599 P_c=0.013599 P c = 0.013599 σ = 1.2 \sigma=1.2 σ = 1.2 P = 0.005 P=0.005 P = 0.005 T t = 0.0582955 T_t=0.0582955 T t = 0.0582955 σ = 1 \sigma=1 σ = 1 m 0 = 0 m_0=0 m 0 = 0 ( q , σ ) = ( 3.249 , 1.2 ) (q,\sigma)=(3.249,1.2) ( q , σ ) = ( 3.249 , 1.2 ) ( 3.921 , 1.5 ) (3.921,1.5) ( 3.921 , 1.5 ) Table V ). At these points, the ratio P r c / T c P_{rc}/T_c P r c / T c 3 / 16 3/16 3/16 0.03 % 0.03\% 0.03% σ > 0 \sigma>0 σ > 0 σ = 1 / 4 \sigma=1/4 σ = 1/4 r > r h r>r_h r > r h σ > 0 \sigma>0 σ > 0 m 0 m_0 m 0 B = ∂ M / ∂ β \mathcal{B}=\partial M/\partial\beta B = ∂ M / ∂ β β \beta β 2 β B 2\beta\mathcal{B} 2 β B
In the holographic description, the bulk relations map to a first law and a fixed-R R R
E = T S ~ + μ C + Q ~ Φ ~ + Ψ ~ β , E=T\widetilde{S}+\mu C+\widetilde{Q}\widetilde{\Phi}+\widetilde{\Psi}\beta, E = T S + μ C + Q Φ + Ψ β , with no independent p V p\mathcal{V} p V C C C σ \sigma σ Table VIII ). The ratio C / Q ~ c C/\widetilde{Q}_c C / Q c σ \sigma σ 3 / 2 3/2 3/2 β ~ = 0 \widetilde{\beta}=0 β = 0 β c \beta_c β c γ \gamma γ δ \delta δ α = 0 \alpha=0 α = 0 Q ~ / C \widetilde{Q}/C Q / C σ \sigma σ σ = 1.2 \sigma=1.2 σ = 1.2 0.1729 ≲ y ≲ 0.1756 0.1729\lesssim y\lesssim0.1756 0.1729 ≲ y ≲ 0.1756 1 / C 1/C 1/ C Q ~ \widetilde{Q} Q β ~ \widetilde{\beta} β β ^ = β ~ / ( 4 C ) \widehat{\beta}=\widetilde{\beta}/(4C) β = β / ( 4 C ) C C C ( p , γ ~ ) (p,\widetilde{\gamma}) ( p , γ ) σ ≥ 0.45 \sigma\geq0.45 σ ≥ 0.45 σ = 0.12 \sigma=0.12 σ = 0.12
The grand-canonical ensemble displays a Hawking–Page transition, with its temperature changing only in the fifth decimal place across the models considered. In the mixed ensemble, the free-energy branches meet continuously at the cusp T ~ min ( μ ) \widetilde{T}_{\min}(\mu) T m i n ( μ ) T ~ 1 \widetilde{T}_1 T 1 x x x
The orbit radii characterize the geometry and its electromagnetic propagation cones. Relating them to a shadow requires a specified polarization sector, observer and emission model, together with treatment of the AdS boundary [83 , 84 , 99 ] .
Hessian test of local thermodynamic stability
A positive heat capacity tests thermal stability only against entropy fluctuations at fixed ensemble controls. Full local stability requires the Hessian of the CFT energy with respect to the extensive variables allowed to fluctuate in the chosen ensemble to be positive semidefinite. In the canonical ensemble the only fluctuating extensive variable is S S S C Q > 0 C_Q > 0 C Q > 0 ( S , Q ~ ) (S,\widetilde{Q}) ( S , Q ) ( S , C ) (S,C) ( S , C )
H g c = ∂ 2 E ∂ ( S , Q ~ ) 2 , H m i x = ∂ 2 E ∂ ( S , C ) 2 , (47) H_{\mathrm{gc}} = \frac{\partial^2 E}{\partial(S,\widetilde{Q})^2}, \qquad H_{\mathrm{mix}} = \frac{\partial^2 E}{\partial(S,C)^2}, \tag*{(47)} H gc = ∂ ( S , Q ) 2 ∂ 2 E , H mix = ∂ ( S , C ) 2 ∂ 2 E , (47) where the variables held fixed are those specified by each ensemble, and the physical coupling β ~ \widetilde{\beta} β 2 × 2 2 \times2 2 × 2
C Φ = T ~ E Q Q det H g c , C μ = T ~ E C C det H m i x . (48) C_{\Phi} = \frac{\widetilde{T} E_{QQ}}{\det H_{\mathrm{gc}}}, \qquad C_{\mu} = \frac{\widetilde{T} E_{CC}}{\det H_{\mathrm{mix}}}. \tag*{(48)} C Φ = det H gc T E QQ , C μ = det H mix T E C C . (48) Thus C X > 0 C_X > 0 C X > 0
We evaluated the Hessian derivatives from the closed-form holographic energy while holding the physical coupling β ~ \widetilde{\beta} β E Q Q E_{QQ} E QQ x → ∞ x \to\infty x → ∞ C μ > 0 C_{\mu} > 0 C μ > 0 x x x x x x E C C E_{CC} E C C x x x σ = 1.2 \sigma= 1.2 σ = 1.2 C μ = T ~ E C C / det H m i x C_{\mu} = \widetilde{T} E_{CC}/\det H_{\mathrm{mix}} C μ = T E C C / det H mix T ~ > 0 \widetilde{T} > 0 T > 0 det H m i x < 0 \det H_{\mathrm{mix}} < 0 det H mix < 0 Table XV compares the determinant roots with the heat-capacity-divergence locations and reports the corresponding diagonal bound. These numerical results apply to the parameter choices stated in the caption and do not establish stability for arbitrary parameters.
Table XV. Hessian diagonal test for the grand-canonical and mixed ensembles, with R = 1 R=1 R = 1 β ~ = 0.1 \tilde{\beta}=0.1 β ~ = 0.1 x d e t x_{\mathrm{det}} x det det H \det H det H x c a p x_{\mathrm{cap}} x cap Table XIX . In the grand-canonical ensemble, C = 5 / 6 C=5/6 C = 5/6 Φ ~ R = 0.3 \tilde{\Phi}R=0.3 Φ ~ R = 0.3 inf E Q Q = 0 \inf E_{QQ}=0 inf E QQ = 0 x → ∞ x\to\infty x → ∞ Q ~ = 1 \tilde{Q}=1 Q ~ = 1 μ R = 0.1 \mu R=0.1 μ R = 0.1 E C C E_{CC} E C C C μ > 0 C_{\mu}>0 C μ > 0 x ⋆ x_{\star} x ⋆ x x x
To test the proposed zeroth-order interpretation, we also evaluated the lower-entropy solution at the same temperature T 1 T_1 T 1 G R → 0 GR \to0 GR → 0 Table XVI reports the resulting branch-valued jump together with C μ C_{\mu} C μ μ R \mu R μ R σ \sigma σ C μ C_{\mu} C μ det H < 0 \det H < 0 det H < 0 λ min < 0 \lambda_{\min} < 0 λ m i n < 0
The scan covers the representative parameters listed in Table XVI ; this result does not exclude locally stable endpoint states elsewhere in parameter space. The mixed-ensemble results are therefore reported as a finite branch-end discontinuity whose lower-entropy side is locally unstable at the sampled points.
Supplementary tables
The tables below collect numerical material referred to in Secs. V and V J.
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