Schwarzschild-AdS Black Hole

Updated 12 December 2025

Schwarzschild–AdS black holes are static, spherically symmetric solutions to Einstein’s equations with a negative cosmological constant, central to gravitational thermodynamics and gauge/gravity duality.
They exhibit distinct thermodynamic behaviors, including phase transitions from unstable small black holes to stable large ones, as characterized by the Hawking temperature and heat capacity.
Extended analyses treat the mass as enthalpy and incorporate quantum corrections, offering insights into microstate counting and potential resolutions of the information paradox.

A Schwarzschild–Anti-de Sitter (Schwarzschild–AdS) black hole is a static, spherically symmetric solution to the Einstein equations with a negative cosmological constant, describing an uncharged black hole embedded in an asymptotically Anti-de Sitter spacetime. It forms a central paradigm for gravitational thermodynamics, phase transitions, and gauge/gravity duality in asymptotically AdS backgrounds.

1. Metric Structure and Horizon Properties

In four dimensions, the Schwarzschild–AdS solution in static, spherically symmetric coordinates $(t, r, \theta, \phi)$ is given by

$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2(d\theta^2 + \sin^2\theta\,d\phi^2)$

where the AdS lapse function is

$f(r) = 1 - \frac{2M}{r} + \frac{r^2}{\ell^2}$

or equivalently $f(r) = 1 - 2M/r - (\Lambda/3)\,r^2$ with %%%%2%%%% and $\ell^2 = -3/\Lambda$ (Lin et al., 2016, 1711.02744).

The event horizon $r_h$ is defined as the largest real positive root of $f(r_h) = 0$ , leading to

$r_h^3 + \ell^2 r_h - 2 M \ell^2 = 0$

or, inverted,

$M = \frac{r_h}{2} \left(1 + \frac{r_h^2}{\ell^2}\right)$

This provides a direct relationship between the ADM mass $M$ and the horizon radius $r_h$ .

2. Thermodynamics: Hawking Temperature and Entropy

The surface gravity at the horizon is $\kappa = \frac12 f'(r_h)$ , so the Hawking temperature follows as

$T_H = \frac{\kappa}{2\pi} = \frac{1}{4\pi} \left(\frac{1}{r_h} + \frac{3 r_h}{\ell^2}\right)$

This matches the formula derived by multiple methods: Euclidean regularity, topological arguments, and tunneling calculations (Lin et al., 2016, 1711.02744, Rahman et al., 2012, Robson et al., 2019).

The Bekenstein–Hawking entropy is given by one quarter of the horizon area: $S = \frac{A}{4} = \pi r_h^2$ in units $G = \hbar = c = k_B = 1$ .

For fixed mass, the AdS curvature reduces $r_h$ compared to the asymptotically flat Schwarzschild case, leading to lower entropy and higher temperature for the same $M$ (Alexanian, 2019).

3. Thermodynamic Phase Structure and Stability

Heat Capacity and Phases

The heat capacity at fixed AdS scale is

$C = 2\pi r_h^2\, \frac{1 + 3\, r_h^2 / \ell^2}{3 r_h^2 / \ell^2 - 1}$

This exhibits a divergence at $r_h = \ell/\sqrt{3}$ , corresponding to a minimum in the temperature curve. For $r_h < \ell/\sqrt{3}$ (small black holes), $C < 0$ and the solution is thermodynamically unstable; for $r_h > \ell/\sqrt{3}$ (large black holes), $C > 0$ implying stability (1711.02744).

Hawking–Page Phase Transition

The free energy of the Schwarzschild–AdS black hole,

$F_{\rm BH} = M - T_H S = \frac{r_h}{4} \left(1 - \frac{r_h^2}{\ell^2}\right)$

becomes negative for $r_h > \ell$ , marking a first-order Hawking–Page transition from thermal AdS spacetime (no black hole, $F_{\rm AdS} = 0$ ) to a dominant large black hole saddle for $T > T_{\rm HP} = 1/(\pi \ell)$ . This transition is a key signature of black hole thermodynamics in AdS and enters prominently in holographic QFT duals (1711.02744, Angsachon et al., 2022, Robson et al., 2019).

4. Quantum Corrections and Singularity Resolution

Hamilton–Jacobi and Quantum Gravity Corrections

The Hamilton–Jacobi method reveals that black hole radiation deviates from strict thermality when energy conservation and back-reaction are included. The emission probability is directly related to the change in Bekenstein–Hawking entropy,

$\Gamma_{\rm nonthermal} \propto \exp[\Delta S_{BH}] = \exp[S_{BH}(M-\omega) - S_{BH}(M)]$

and reduces to a Boltzmann factor at leading order in small $\omega$ with corrections governed by $\partial S_{BH}/\partial M$ and higher derivatives (Rahman et al., 2012).

Singularity Resolution in Quantum Unimodular Gravity

Recent advances using the Henneaux–Teitelboim unimodular formulation enable a quantization with genuine Schrödinger evolution in unimodular time. Imposing unitarity leads to a family of quantum-corrected metrics,

$f_{qc}(r) = 1 - \frac{2M}{r} - \frac{\Lambda}{3} r^2 + \delta f_{\rm quantum}(r; r_{\min})$

where $\delta f_{\rm quantum}$ is suppressed by $(r_{\min}/r)^6$ with $r_{\min}$ a new minimal scale fixed by the quantum state. The expectation value of the metric never reaches $r = 0$ ; instead, a nonsingular surface at $r_{\min}$ mediates a black–hole to white–hole transition. The position of the event horizon is shifted by $O(r_{\min}^6/r_H^5)$ , and the Hawking temperature and entropy receive $O(r_{\min}^6/r_H^6)$ corrections (Ried, 28 Aug 2025).

5. Topology, Boundary Conditions, and Quantum Fields

The Euclideanized Schwarzschild–AdS geometry exhibits nontrivial topology, with the Euler characteristic $\chi_4$ jumping at the Hawking–Page transition, signaling a breakdown of simple topological derivations of the Hawking temperature in four dimensions. A two-dimensional reduction removes this jump, rendering $\chi_2 = 1$ for all allowed $r_h$ and stabilizing the topological temperature formula. This demonstrates that black hole thermodynamics in AdS is sensitive to the global spacetime topology and boundaries (Robson et al., 2019).

For linear quantum fields (e.g., the massive Klein–Gordon equation) on the hyperboloidal Schwarzschild–AdS background, the presence of the AdS boundary at infinity requires imposed boundary conditions (Dirichlet, Neumann, or Robin). Depending on mass parameters and boundary conditions, energy decay, boundedness, and even linear instability can arise, with the Breitenlohner–Freedman bound determining the range of stability (Fitz-Gibbon, 2019).

6. Extended Thermodynamics and Phase Transitions

"Black hole chemistry" extends thermodynamic variables to include pressure $P = -\Lambda/(8\pi)$ , interpreting the mass $M$ as enthalpy. The first law becomes

$dM = T_H\, dS + V\, dP$

with $V = (\partial M/\partial P)_S$ the thermodynamic volume. In a quasilocal ensemble, boundary terms yield a well-defined internal energy and consistent equations of state. The heat capacity diverges at a critical radius, signalling a small–large black hole transition analogous to van der Waals criticality. Response functions and Legendre structures are regular even when $\Lambda$ is promoted to a variable, supporting studies of critical behavior in finite-volume spacetimes (Fontana et al., 2018, Angsachon et al., 2022).

The Hawking–Page transition persists in variants such as Beltrami coordinates, but with modifications in horizon structure and phase diagrams due to bounded coordinate ranges. Time-dependent boundaries in these frames correspond to cavity-like finite-size effects (Angsachon et al., 2022).

7. Microscopic Models, Holography, and Information

The entropy of a Schwarzschild–AdS black hole can be microscopically modeled as a system of noninteracting, distinguishable quasiparticles with a maximum temperature set by the smallest mass excitation. In AdS, this yields a statistical entropy that is strictly less than the geometric area law, further reduced relative to the asymptotically flat limit. A proposed resolution to the information paradox invokes a horizon-localized Bose–Einstein condensate of zero-mass modes, forming an accumulation point on the event horizon. The entire system forms a nonseparable, unitary Hilbert space ("myriotic field"), with the condensate structure encoding the black hole's microstate information (Alexanian, 2019).

Quantum corrections arising from "rainbow gravity" or other high-energy effects modify the Hawking temperature and entropy, inducing logarithmic and power-law corrections in bulk dimensions. Through the AdS/CFT correspondence, these lead to explicit modifications in the Cardy–Verlinde formula of the dual boundary CFT, which can be absorbed into redefinitions of the Virasoro operator $L_0$ and central charge $c$ (Sefiedgar, 2015).