Hawking-Page Universality in Black Hole Transitions

• Hawking-Page universality class is a family of gravitational and holographic phase transitions characterized by invariant thermodynamic ratios and distinct geometric/topological features.
• These transitions mark the onset of thermodynamically favored black hole phases, exhibiting universal entropy jumps, critical exponents, and scaling behavior across different models.
• Research in this area connects dimensional dualities with holographic correspondence, linking minimal black hole temperatures in higher dimensions to phase transitions in lower-dimensional systems.

The Hawking-Page universality class encompasses a broad family of gravitational and holographic phase transitions, characterized by distinct universal ratios, geometric/topological invariants, and scaling properties. These transitions demarcate the onset of thermodynamically favored black hole phases, typically as parameters such as temperature or geometric radii are tuned. Universality emerges in both thermodynamic properties (e.g., entropy and temperature jumps, critical exponents) and geometric/topological features (e.g., scalar curvature constants, transition charges), providing a unifying framework for various black hole backgrounds, statistical ensembles, and dual field theories.

1. Foundational Thermodynamics and Universal Ratios

At the core of the Hawking-Page universality class is the identification of two key thermodynamic points: the minimal black hole branch, defined by the lowest temperature at which a black hole can exist, and the Hawking-Page (HP) phase transition point, where the free energy of the black hole equals that of thermal AdS or the analogous reference background. For Schwarzschild-AdS and related spacetimes, the universal entropy and temperature ratios at these points are defined as

$$C_S = \frac{S_{HP} - S_{\min}}{S_{\min}}, \quad C_T = \frac{T_{HP} - T_{\min}}{T_{\min}}.$$

For four-dimensional Schwarzschild-AdS and Reissner–Nordström–AdS black holes, one finds $C_S = 2$ and $C_T = (2-\sqrt{3})/\sqrt{3}$, indicating that the entropy jumps by a factor of three at the transition, independent of additional black hole parameters such as charge (with fixed potential) or slow rotation (with fixed angular velocity). In higher dimensions, analogous ratios are pure dimension-dependent functions, and similar results hold when the cosmological constant is promoted to a thermodynamic variable (pressure) (Belhaj et al., 2020).

2. Geometric and Topological Invariants

A defining characteristic of the Hawking-Page universality class is the appearance of parameter-independent geometric and topological invariants at the transition point:

• Ruppeiner Scalar Curvature: At the HP point, the normalized scalar curvature $R$ of the Ruppeiner thermodynamic metric attains the universal constant

$R|_{HP} = -\frac{(d-1)(d-2)}{2},$

depending solely on the spacetime dimension $d$. This value persists regardless of the cosmological constant, horizon radius, or other black hole charges (Wei et al., 2020). The curvature is interpreted as an interaction threshold for black hole microstructure: for $|R| < |(d-1)(d-2)/2|$ the black hole phase is "virtual", while at the HP point the interaction is sufficiently strong to stabilize the large black hole phase.

• Topological Charge: In settings with Rényi statistics, the Hawking-Page transition is marked by a single topological vortex with charge $Q_t = +1$ located at a universal transition radius, with a vanishing total equilibrium phase charge. These results are invariant under charge variations in flat/charged black holes and serve to distinguish the Hawking-Page class from, e.g., Van-der-Waals transitions, which exhibit a pair of transition points with charges $+1$ and $-1$, and a nontrivial equilibrium phase charge (Barzi et al., 2023).

3. Dimensional Dualities and Holographic Correspondence

An exact duality has been identified between the minimum temperature of a $(d+1)$-dimensional Schwarzschild–AdS black hole, $T_0(d+1)$, and the Hawking-Page transition temperature in $d$ dimensions, $T_{HP}(d)$,

$T_{HP}(d) = T_0(d+1).$

This relation holds for all $d \geq 3$ and suggests a form of interdimensional symmetry reminiscent of the AdS/CFT correspondence, mapping bulk ground-state properties to boundary transition points (Wei et al., 2020). The physical implication is a direct correspondence between the ground-state temperature of the higher-dimensional black hole and the first-order phase transition temperature of the lower-dimensional dual description.

4. Universality in Generalized Holographic and Topological Models

The universality class extends to holographic backgrounds with more complicated topologies, such as products of spheres $S^{d_1}(R_1) \times S^{d_2}(R_2)$ in higher-dimensional AdS spacetimes. Here, Hawking-Page-type transitions separate distinct topological phases, with the phase structure controlled solely by the total spacetime dimension $d = d_1 + d_2$. Universal features include:

• Existence of exactly two dominant smooth bulk saddle points (topologies) exchanging dominance at the transition,
• A singular "double-shrinker" solution at a unique critical ratio of radii,
• The presence of a first-order transition for $d < 9$ (second-order for $d \geq 9$), and
• Infinite solution branching near the critical ratio as $d \rightarrow 9^-$ (Aharony et al., 2019).

This structure applies to large-$N$, strongly coupled holographic CFTs and persists independent of detailed sphere partition or other background specifics.

5. Rényi Statistics, Alternative Ensembles, and Topological Signatures

The Hawking-Page universality class is robust across both Gibbs–Boltzmann and Rényi statistical frameworks. In Rényi thermodynamics, the off-shell Bragg–Williams free energy is constructed with Rényi entropy, and phase transitions are identified via topological mappings:

• $\psi$-mapping: global, temperature-independent transition points (HP transition has a unique ψ-vortex with $Q_t = +1$),
• $\xi$-mapping: ensemble-sensitive, temperature-dependent transitions (e.g., Van-der-Waals),
• $\eta$-mapping: equilibrium phases (HP: total charge zero).

The universality of the Hawking-Page class is encoded by:

• A single global transition (unique vortex of $+1$),
• A universal transition radius independent of electric potential,
• A vanishing total equilibrium phase charge, in contrast to richer structure and multiple defects in Van-der-Waals-like phases (Barzi et al., 2023).

Feature	Hawking–Page class	Van-der-Waals class
Ensemble	Grand canonical	Canonical (fixed Q)
Transition mapping	Single ψ–vortex ($+1$)	Two ψ–defects ($-1$, $+1$)
Total ψ-charge	$+1$	$0$
Equilibrium η–mapping total charge	$0$	$+1$
Critical behavior	First-order jump	Multiphase, spinodals, inflection

6. Continuous Hawking-Page Transitions and Critical Scaling

While the canonical Hawking-Page transition is first order (latent heat, abrupt entropy jump, no divergent correlation length), continuous and higher-order transitions can be engineered in Einstein-scalar gravity. For scalar potentials that asymptote as $V_\infty e^{\xi \phi}$ with subleading corrections, the order and scaling of the transition are dictated by the form of the subleading term:

• For exponentially suppressed subleading behavior: $V_{\rm sub} \sim C e^{-k\phi}$, the transition is $n$-th order with $\Delta F \sim t^n$ for reduced temperature $t$, where $k = \xi(d-1)/(n-1)$.
• For power-law suppression: $V_{\rm sub} \sim C \phi^{-a}$, transitions of Berezinskii-Kosterlitz-Thouless type occur with $\Delta F \sim \exp(-A t^{-1/a})$.

Universal scaling relations for entropy, specific heat, speed of sound, shear and bulk viscosities at and near the critical point have been established. In particular, the ratio of bulk to shear viscosity at the critical temperature is a universal number, $\zeta/\eta = \xi/2$, insensitive to detailed microphysics of $V(\phi)$ (Gursoy, 2010).

7. Open Problems and Outlook

Key open directions include extending the universality and duality relations to rotating, charged, and non-AdS backgrounds; extracting microscopic interpretations of transition invariants within a quantum gravity context; and exploring universality in other gravitational or ensemble settings. The Hawking-Page universality class provides a framework for classifying black hole thermodynamic phase transitions in analogy to critical exponents in statistical mechanics and may offer guidance for finding new universal structures in gravitational and holographic systems (Wei et al., 2020, Belhaj et al., 2020).