Hawking–Page Transition in AdS Gravity

• Hawking–Page transition is a gravitational phase change between thermal AdS space and AdS black holes, marked by a discontinuous shift in free energy.
• It exhibits a sudden jump in geometry, where thermal AdS is favored at low temperatures and a large black hole emerges above a critical temperature (e.g., T_HP = 1/(πL) in 4D).
• This transition provides a holographic link to confinement/deconfinement in gauge theories and adapts to extensions like massive gravity and higher-curvature corrections.

The Hawking–Page transition is a first-order gravitational phase transition between thermal anti-de Sitter (AdS) space and AdS black holes, with profound implications for quantum gravity, gauge/gravity duality, and high-energy physics. It exemplifies black hole ensemble thermodynamics and encodes dual confinement/deconfinement transitions in holographic field theories. The structure and realization of the transition depend on spacetime topology, boundary conditions, matter content, and possible modifications to gravity.

1. Fundamental Definition and Geometry

The original Hawking–Page transition was established for asymptotically AdS spacetimes with compact boundary topology, specifically for Schwarzschild–AdS black holes in global coordinates (boundary $S^{d-1}\times S^1$) (Braga et al., 16 Sep 2025, Eune et al., 2013). The system admits two leading Euclidean saddle points at fixed temperature $T$:

• Thermal AdS (no horizon): A regular, horizonless background, representing radiation at low $T$.
• Black hole (with horizon): A static black hole solution characterized by horizon radius $r_+$.

The canonical (fixed-temperature) ensemble free energy is

$F = M - T S$

where $M$ is the black hole mass and $S$ the Bekenstein–Hawking entropy. For $T < T_\mathrm{HP}$ thermal AdS is favored ($F_\mathrm{BH}>0$). Above $T_\mathrm{HP}$ the large black hole becomes the global minimum ($F_\mathrm{BH}<0$), corresponding to a discontinuous jump in the preferred geometry and the order parameter (horizon area).

The transition is first order: the entropy and energy exhibit discontinuities, and the difference in free energies at $T=T_\mathrm{HP}$ sets the latent heat. For example, in four spacetime dimensions ($d=4$), the critical temperature is $T_\mathrm{HP}=1/(\pi L)$, where $L$ is the AdS radius (Braga et al., 16 Sep 2025).

2. Mechanism and Physical Interpretation

The Hawking–Page transition fundamentally arises from the interplay of gravitational thermodynamics, negative cosmological constant (stabilizing large AdS black holes), and boundary topology:

• IR scale and boundary topology: In AdS with compact ($S^{d-1}$) boundary, the spatial volume provides an infrared scale required for the transition. With planar or noncompact boundaries ($\mathbb{R}^n \times S^1$), ordinary AdS gravity lacks a Hawking–Page transition due to absence of such a scale; $F_\mathrm{BH}$ is always negative (Braga et al., 16 Sep 2025).
• Gauge/gravity duality: The transition is dual to the confinement/deconfinement transition in the boundary large-$N$ gauge theory, via AdS/CFT. The thermal AdS phase maps to a confined, low-temperature state; the black hole phase to a deconfined plasma (Braga et al., 6 Apr 2024).

In the presence of modifications—e.g., massive gravity terms (Adams et al., 2014, Braga et al., 16 Sep 2025), higher-order curvature corrections (Wang et al., 2020, Cui et al., 2021), or alternative boundary conditions (reflecting wall, finite cavity) (Zhao et al., 2020)—the character and criteria for the transition change accordingly.

3. Mathematical Structure in Massive Gravity and Beyond

Including a graviton mass term or other IR scalar(s) introduces an explicit infrared scale and breaks diffeomorphism invariance in boundary directions, modifying the phase structure (Braga et al., 16 Sep 2025, Adams et al., 2014, Yerra et al., 2021):

• General action (massive gravity):

$$\mathcal{I} = -\frac{1}{2\kappa^2} \int d^{n+2}x \sqrt{g}\, (\mathcal{R} - \Lambda) -\frac{1}{2\kappa^2} \int d^{n+2}x\sqrt{g}\, (\lambda_1 \mathcal{U}_1 + \lambda_2 \mathcal{U}_2)$$

with $\lambda_{1,2} \sim m^2$.

• Planar boundary ($\mathbb{R}^n \times S^1$): Graviton mass terms create a finite IR scale even for noncompact boundaries, yielding a critical point for the Hawking–Page transition at

$$T_c = \frac{n}{2\pi} \sqrt{\lambda_2} + \frac{\lambda_1 L}{4\pi}$$

The order parameter $\phi=1/z_h$ (inverse horizon coordinate) jumps discontinuously at $T_c$, and the Landau-Ginzburg potential $F(\phi)$ shows characteristic first-order behavior (Braga et al., 16 Sep 2025).

• Ruppeiner geometry: The Ruppeiner curvature scalar $R_N$ at the transition point indicates attractive microstructure interactions for AdS black holes. In $d$-dimensional Schwarzschild AdS, $R_N$ is a universal negative constant at $T_\mathrm{HP}$. For massive gravity, $R_N$ acquires explicit $m$-dependence, breaking universality except when the critical temperature is tuned to zero where universality is restored (Yerra et al., 2021).

4. Extensions, Variants, and Generalizations

The Hawking–Page transition is realized and modified in diverse circumstances:

• Asymptotically de Sitter (dS) Space: The presence of both black hole and cosmological horizons generates a closed coexistence loop in the $(P_\mathrm{eff},T_\mathrm{eff})$ phase plane, with two HP branches and a maximal temperature/pressure—a qualitative departure from AdS physics (Du et al., 2021).
• Finite Cavity (Reflecting Wall): Enclosing a black hole in a finite cavity within asymptotically flat spacetime introduces an effective volume and pressure and permits HP-type transitions, subject to bounds related to the wall potential (Zhao et al., 2020).
• Higher-curvature Theories: Gauss–Bonnet (GB) or Einstein–Gauss–Bonnet (4EGB) terms reduce the HP coexistence domain, lower the transition temperature, and introduce new bounds and phase structures. In 4EGB gravity, the HP line occurs only within a finite pressure range determined by the GB coupling (Wang et al., 2020); for hyperbolic horizons in GB gravity, the phase diagram exhibits reentrant transitions and a triple point (Cui et al., 2021).
• Non-equilibrium and Kinetic Realizations: Free-energy landscape methods model the transition as stochastic dynamics, governed by a Fokker–Planck equation. Both additive and multiplicative noise shift the critical temperature and can alter the stability of phases (Ho, 29 Sep 2025). Nonequilibrium kinetic analyses incorporating evaporation model the transition as a reaction-diffusion process exhibiting a kinetic turnover (competition between nucleation and evaporation time scales) (Li et al., 2021). Non-Markovian effects (memory) influence escape rates and can enhance or hinder the transition depending on the bath correlation structure (Li et al., 2022).
• Other Spacetime and Field Theories:
  • In lower-dimensional dilaton gravity (e.g., Jackiw–Teitelboim), the transition occurs between AdS$_2$–Lifshitz$_2$ vacuum and charged 2D black holes, reflecting first-order free-energy jumps even in two dimensions (Lala et al., 2020).
  • Holographic models of QCD at finite density realize the Hawking–Page transition as confinement/deconfinement, with critical temperature decreasing as chemical potential increases, aligning with phenomenological QCD phase diagrams (Braga et al., 6 Apr 2024).
  • The transition can be mapped into quantum simulators, e.g., spin chains and free-fermion models, exhibiting the same entropy jump and nonanalyticity in appropriate free energies (Pérez-García et al., 25 Jan 2024).

5. Novel and Continuous Transitions

• Continuous (Higher-Order) Hawking–Page Transitions: By coupling gravity to a scalar field with specially chosen asymptotic potential—typically exponential in the IR—a variety of continuous transitions, including second order, higher order, and BKT-type infinite order, can be engineered (Gursoy, 2010). These correspond holographically to normal/superfluid transitions and are characterized by vanishing latent heat and continuous order parameters.

6. Topological and Statistical Interpretation

• Off-shell Free Energy and Topological Indices: Off-shell constructions using Bragg–Williams–type potentials provide unified order parameters and classify transitions via topological charge in the thermodynamic parameter space. For Born–Infeld AdS black holes, the HP transition point carries unit topological charge ($+1$), matched exactly by the dual (boundary) confinement–deconfinement transition in the gauge theory (Yerra et al., 2023).
• Universality and Microscopic Interpretation: In standard AdS black hole chemistry, the HP transition is interpreted as the statistical nucleation or decay of black holes, with the off-shell free-energy landscape capturing the tunneling process over the entropy barrier (Eune et al., 2013). Ruppeiner curvature analysis provides insights into the microstructure—negative values at $T_\mathrm{HP}$ indicate net attractive interactions among effective degrees of freedom (Yerra et al., 2021).

7. Physical and Holographic Significance

• Gauge/Gravity Duality: The HP transition translates via AdS/CFT into a nonanalytic change in the boundary theory partition function, encoding a confinement (thermal AdS) to deconfinement (large black hole) transition. The associated entropy jump for $N \to \infty$ is order $O(N^2)$ for large $N$ gauge theories (Braga et al., 16 Sep 2025, Pérez-García et al., 25 Jan 2024).
• Black Hole Information: In dynamical contexts, e.g., evaporation or islands in semiclassical gravity, the HP transition can couple to the Page curve—information escape from black holes may occur as a result of the phase transition by converting horizon geometries, independently of Hawking radiation (Sun, 2021).
• Universality and Model Dependence: While the qualitative features—first-order jump, entropy change, and dual confinement physics—are robust, the presence, critical temperature, and detailed structure of the Hawking–Page transition depend crucially on dimension, horizon topology, IR matter content, and boundary conditions.

• "Hawking-Page transition in anti-de Sitter massive gravity with non-compact spatial boundary" (Braga et al., 16 Sep 2025)
• "Hawking-Page transition in holographic massive gravity" (Adams et al., 2014)
• "Novel relations in massive gravity at Hawking-Page transition" (Yerra et al., 2021)
• "Hawking-Page phase transitions in four-dimensional Einstein--Gauss--Bonnet gravity" (Wang et al., 2020)
• "Hawking-Page transition with reentrance and triple point in Gauss-Bonnet gravity" (Cui et al., 2021)
• "Hawking-Page Phase Transition of the four-dimensional de-Sitter Spacetime with non-linear source" (Du et al., 2021)
• "Hawking--Page phase transitions of the black holes in a cavity" (Zhao et al., 2020)
• "Effective Free Energy Landscapes and Hawking-Page Transitions" (Ho, 29 Sep 2025)
• "The kinetics and its turnover of Hawking-Page phase transition under the black hole evaporation" (Li et al., 2021)
• "Kinetics of Hawking-Page phase transition with the non-Markovian effects" (Li et al., 2022)
• "JT gravity and the models of Hawking-Page transition for 2D black holes" (Lala et al., 2020)
• "Hawking-Page transition in holographic QCD at finite density" (Braga et al., 6 Apr 2024)
• "Generalized Hawking-Page transitions" (Aharony et al., 2019)
• "Hawking-Page transition on a spin chain" (Pérez-García et al., 25 Jan 2024)
• "Hawking-Page phase transition, Page curve and islands in black holes" (Sun, 2021)
• "Topology of Hawking-Page transition in Born-Infeld AdS black holes" (Yerra et al., 2023)
• "Hawking-Page phase transition of the Schwarzschild AdS black hole with the effective Tolman temperature" (Eom et al., 2022)
• "Continuous Hawking-Page transitions in Einstein-scalar gravity" (Gursoy, 2010)
• "Hawking-Page phase transition in BTZ black hole revisited" (Eune et al., 2013)