---
title: Hawking-Page Transition in AdS Spacetime
url: https://www.emergentmind.com/topics/hawking-page-transition-eb27c289-052a-4e97-bfe6-f04618e2d2b3
type: topic
---

# Hawking-Page Transition in AdS Spacetime

The Hawking-Page transition is a first-order thermodynamic phase transition between thermal radiation and a black hole in asymptotically anti-de Sitter (AdS) spacetime. Originally formulated in the context of AdS$_4$ and subsequently generalized to other gravitational backgrounds and matter content, the Hawking-Page transition is both a central result in semiclassical gravity and a cornerstone for gauge/gravity duality, where it realizes the confinement/deconfinement transition in large-$N$ gauge theories. Its characterization now extends to off-shell free energy landscapes, dynamical kinetics under evaporation, effects of higher curvature and massive gravity, statistical and topological classification, and diverse boundary conditions.

## 1. Foundational Geometry and Critical Point Structure

The prototypical Hawking-Page transition occurs in asymptotically AdS spacetime with compact spatial boundary and negative cosmological constant. In three dimensions, the competition is between the BTZ black hole (mass $M\geq0$) and global AdS$_3$ (“thermal soliton”, $M=-1$) [1301.0395]. The Euclidean metrics are smooth at their respective inverse temperatures and admit on-shell (extremal) free energies:
\[
F^{\mathrm{on}}_\mathrm{BH} = -\frac{M}{8G}\,,\qquad 
F^{\mathrm{on}}_\mathrm{sol} = -\frac{1}{8G}
\]
with corresponding energies and entropies determined by horizon radius and AdS scale. The transition temperature $T_c$ is fixed by the degeneracy of these two free energies, yielding
\[
T_c = \frac{1}{2\pi\ell}
\]
where $\ell$ is the AdS radius.

In higher dimensions, analogous crossings are found between the thermal AdS saddle (zero mass, compact Euclidean time circle) and the Schwarzschild-AdS black hole. The transition is always first-order: the entropy jumps discontinuously and the specific heat may change sign.

## 2. Off-Shell Free Energy Landscape and Tunneling Dynamics

The off-shell generalization constructs a continuous, temperature-parameterized free-energy function $F^{\mathrm{off}}(M;T)$ by considering Euclidean geometries with conical deficits at arbitrary temperatures (i.e., fixed period $\beta$, not necessarily matching the Hawking temperature) [1301.0395, 2509.25039]. For the BTZ case:
\[
F^{\mathrm{off}}_\mathrm{BH}(T,M) = \frac{M}{8G} - T \frac{\pi \ell\sqrt{M}}{2G}
\]
for $M\geq 0$ and
\[
F^{\mathrm{off}}_\mathrm{sol}(M) = \frac{1}{8G}\left[-M-2\sqrt{-M}\,\right]
\]
for $-1<M<0$. The full free energy interpolates smoothly across the “mass gap” between AdS soliton and the smallest black hole, forming a potential barrier with a cusp at $M=0$. The extrema at $M=-1$ (soliton) and $M=\ell^2 (2\pi\ell T)^2$ (BH) mark the true equilibrium phases.

Transitions between these states proceed by quantum tunneling: the tunneling rate is $\Gamma\sim \exp(-\beta\,\Delta F)$, with $\Delta F$ the off-shell barrier height. The off-shell landscape thus enables a semiclassical picture of phase transition intermediates, absent from strict equilibrium treatments [1301.0395, 2509.25039].

## 3. Dynamical Kinetics and Non-Equilibrium Effects

The stochastic and kinetic framework models phase transition dynamics via Langevin or Fokker-Planck equations on the free-energy landscape. For the Schwarzschild-AdS black hole, the order parameter is the horizon radius $r_+$ and the key equation is [2509.25039]:
\[
\frac{dr}{dt} = -\frac{1}{\zeta}\frac{dG_{\text{off}}}{dr} + g(r)\xi(t)
\]
where $\zeta$ is a friction coefficient, $\xi(t)$ is Gaussian noise, and $g(r)$ encodes multiplicative noise. The associated Fokker-Planck equation governs the probability density evolution and, under multiplicative noise, the effective free energy is renormalized:
\[
G_{\mathrm{eff}}(r) = \int^r \frac{G'_{\text{off}}(u)}{g^2(u)}\,du + (2-\lambda)k_BT \ln g(r)
\]
Strong multiplicative noise disfavors the formation of large black holes by effectively raising the free-energy barrier, shifting the critical temperature upward [2509.25039].

Evaporation effects (allowing non-reflecting boundaries) introduce a reaction (sink) term and drive a competition between barrier-limited and evaporation-limited escape rates [2105.00229]. Here, the mean first-passage time from the metastable to the preferred phase interpolates between Arrhenius (thermal activation) and evaporation-controlled regimes, with a kinetic turnover when both time scales become comparable.

Non-Markovian (memory) effects in the bath further accelerate transitions: exponentially decaying and oscillatory memory kernels both generically speed up barrier crossing, especially for long correlation times or high oscillation frequencies [2201.06138].

## 4. Influence of Matter Content, Boundary Conditions, and Gravity Modifications

The Hawking-Page phenomenon persists or is qualitatively modified across diverse extensions:
- **Higher curvature corrections (Gauss-Bonnet, Einstein-Gauss-Bonnet):** These alter both mass and entropy, shift and bound the transition temperature, and introduce reentrance and triple-point behavior. The phase diagram may acquire two disconnected coexistence lines, corresponding to small and large black holes, with a triple point at which multiple phases coexist and an upper critical point where the transition terminates [2106.13942, 2008.01985].
- **Massive gravity and momentum dissipation:** The graviton mass or disorder-like couplings introduce an IR scale in planar AdS, restoring a sharp Hawking-Page-like transition even for noncompact spatial boundaries and breaking the universality of certain microstructure invariants [2509.13454, 1408.6560, 2107.04504].
- **Charge and rotation:** Electric and angular momenta reduce the critical temperature, favor the black hole at lower $T$, and can ultimately eliminate the transition beyond extremality. The coexistence curve in $(T,Q)$ or $(T,J)$ plane acquires a maximal admissible value, above which no Hawking-Page transition occurs [2603.16560].
- **Cavity boundary conditions:** Imposing a finite-radius “box” in otherwise asymptotically flat space generates Hawking-Page-like transitions with modified temperature-pressure scaling, upper bounds on the electric potential, and parameter regions where no transition is possible [2012.13921].
- **De Sitter space:** In the presence of both event and cosmological horizons, an effective equilibrium analysis reveals a closed-coexistence loop in $(T_\text{eff},P_\text{eff})$, in contrast to the unbounded phase curve in AdS [2106.06675]. The Hawking-Page point is dictated by a joint vanishing of the Gibbs free energy of the dS black hole system.

## 5. Topological and Statistical Structure

Classification of the Hawking-Page transition has advanced beyond thermodynamics to include topological invariants. Using the Bragg–Williams off-shell free energy construction, the transition point can be identified as a topological defect with nontrivial winding (topological charge $+1$), stable under non-linear matter couplings such as Born–Infeld electrodynamics [2312.10784]. This invariance extends to the dual gauge theory's confinement-deconfinement transition and is robust across variations in the AdS radius, chemical potential, and interaction strength.

From a statistical mechanics viewpoint, the Hawking-Page transition is encoded in the large-$N$ behavior of matrix models: for example, in $\mathcal{N}=4$ SYM, the partition function reduces to a one-matrix integral that exhibits a first-order transition at a critical value of the coupling, precisely matching the gravity result. This thermodynamic signature can, remarkably, be simulated in quantum many-body systems, such as randomized spin chains, which encode the entropy jump at transition [2401.13963].

## 6. Generalizations and Novel Transitions

The phenomenology of Hawking-Page transitions admits rich generalizations:
- **Continuous (higher-order) transitions:** In Einstein-scalar gravity, nontrivial scalar potentials permit transitions where the entropy difference vanishes at $T_c$. Depending on the IR structure of $V(\phi)$, thermodynamic functions show higher-order scaling or even essential (Berezinskii-Kosterlitz-Thouless) singularities [1007.0500]. Analytic kink solutions interpolate between AdS and a linear-dilaton background.
- **Multiple topologies and exotic phase structure:** For CFTs on products of spheres $S^{d_1}\times S^{d_2}$, the transition can occur between gravitational saddles of different topologies, e.g., between metrics where one or the other sphere shrinks smoothly in the bulk; the transition is first-order in a broad class and accumulates an infinite number of solutions near critical ratios of radii for $d_1+d_2<9$ [1904.07502]. Singular limits and critical scaling arise as both spheres shrink simultaneously.
- **Modified local temperatures:** Adopting local definitions of temperature enforced by stress-tensor anomalies, such as the Hartle–Hawking effective temperature, can alter stability and yield new locally stable “medium-size” black hole phases, along with regions of zeroth-order transitions in the free energy [2205.09938].

## 7. Quantum Information and Holography

In the context of AdS/CFT, the Hawking-Page transition serves as the gravitational dual of the confinement/deconfinement phase transition. As shown in AdS$_3$/CFT$_2$, the transition can release all the black hole's quantum information at the critical temperature, a mechanism complementary to unitary evaporation via the Page curve and “island” formula. The jump in entanglement entropy across the transition saturates a bulk/boundary first-law relation, and the universal character of the entropy release at $T_c$ suggests a broader role for first-order transitions as unitarity-restoring channels in the quantum theory [2107.05218].

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The Hawking-Page transition thus provides a universal organizing principle for black hole thermodynamics, holographic duality, information transfer, and quantum statistical physics. Its theoretical extensions and precise mathematical underpinnings make it a rich subject at the intersection of gravity, field theory, and statistical mechanics. For detailed derivations and further generalizations, see [1301.0395], [2509.25039], [2105.00229], [2201.06138], [2509.13454], [1408.6560], [2106.13942], [1007.0500], [2312.10784], [2401.13963], [2603.16560].

Source: https://www.emergentmind.com/topics/hawking-page-transition-eb27c289-052a-4e97-bfe6-f04618e2d2b3