Davies Phase Transition in Black Holes

• Davies Phase Transition Point is a critical locus in black hole thermodynamics where the heat capacity diverges, indicating a continuous, second-order phase transition.
• It is analyzed using analytic, dynamical, and topological diagnostics that reveal distinct signatures in quasinormal modes and photon-sphere dynamics.
• Topological methods assign a universal winding number of -1 to the transition, highlighting deep connections between black hole geometry and thermodynamic stability.

The Davies Phase Transition Point designates a critical locus in the parameter space of certain black hole solutions, where the canonical–ensemble heat capacity diverges and a continuous (second-order) thermodynamic phase transition occurs. This transition is marked by the sign change in the heat capacity, separating branches of local thermodynamic stability and instability. It is a universal feature of black hole solutions possessing additional conserved charges (e.g., electric charge, angular momentum) and is manifested through analytic, dynamical, and topological diagnostics.

1. Thermodynamic Definition and Conditions

In black hole thermodynamics, for a system with extensive parameters $X_i$ (such as charge $Q$ or angular momentum $J$), the first law reads

$dM = T dS + \sum_i Y_i dX_i\,,$

where $T$ is the Hawking temperature, $S$ the Bekenstein–Hawking entropy, and $Y_i$ the conjugate intensive variables. The key thermodynamic response function is the heat capacity at fixed $X_i$: $$C_{X_i} = T\left(\frac{\partial S}{\partial T}\right)_{X_i}\,.$$ The Davies point occurs at those values of $X_i$ where $C_{X_i}$ diverges. Analytically, this divergence corresponds to

$\partial_S\left(\frac{1}{T}\right)_{X_i} = 0\,,$

which, for given black hole solutions, becomes an explicit algebraic relation among the parameters. For Reissner–Nordström black holes,

$$T(r_+,Q) = \frac{1}{4\pi r_+}\left(1-\frac{Q^2}{r_+^2}\right)\,,$$

the divergence of $C_Q$ occurs at $r_+ = \sqrt{3} Q$, or equivalently, at $Q_D = (\sqrt{3}/2) M$ (Wei et al., 2019, Bhattacharya et al., 2024, Fernandes et al., 2024). In general, the Davies point manifests as a continuous (second-order) phase transition: the free energy and its first derivatives remain continuous, but the heat capacity diverges and changes sign.

2. Microphysical and Tunneling Interpretation

Within quantum descriptions, the Davies point is physically interpreted as the transition where nonthermal back-reaction corrections to Hawking emission, arising via the Parikh–Wilczek tunneling framework, vanish. The emission rate of a quantum of energy $\omega$ is of the form

$$\Gamma \sim e^{-\beta_H \omega - \frac{\omega^2}{2} \frac{\beta_H^2}{M c_Q} + \mathcal{O}(\omega^3)}\,,$$

with $c_Q$ the specific heat. At the Davies point, $c_Q \to \infty$, eliminating the nonthermal $\omega^2$ correction: the emission becomes exactly thermal (La, 2010). This demarcates two regimes:

• For $c_Q < 0$, the nonthermal correction enhances the radiative rate.
• For $c_Q > 0$, the correction suppresses the rate. The Davies locus thus separates black hole phases distinguished by the qualitative character of quantum emission.

3. Dynamical and Geodesic Signatures

A precise correspondence has been found between the Davies point and features in the dynamical response of black holes, particularly their eikonal-limit quasinormal modes (QNMs). In spherically symmetric spacetimes, the frequencies of QNMs in the large angular momentum ($\ell \gg 1$) regime are determined by photon-sphere dynamics: $$\omega_Q \approx \ell \Omega - i (n + 1/2) \lambda\,,$$ where $\Omega$ is the angular velocity and $\lambda$ is the Lyapunov exponent at the photon sphere. The Davies point coincides exactly with the local maximum of the Hawking temperature when expressed as a function of either $\Omega$ or $\lambda$: $$\left.\frac{dT}{d\Omega}\right|_{Q_D} = 0\,, \quad \left.\frac{dT}{d\lambda}\right|_{Q_D} = 0\,.$$ This exact matching persists for both asymptotically flat and de Sitter Reissner–Nordström spacetimes, and also in higher dimensions where the spiral structure seen in QNM parameter planes disappears, but the extremal property remains. Thus, the Davies point is encoded in dynamical null-geodesic observables (Wei et al., 2019).

4. Topological Characterization

Recent developments apply Duan’s φ-mapping topological current theory to provide a topological invariant for the Davies point. One constructs a two-component vector field $\phi^a$ on the space of (entropy, auxiliary angle), with components derived from the inverse temperature and a chosen angular coordinate. The Davies point is a zero of $\phi^a$ at $\theta = \pi/2$, and the associated topological (winding) number is computed to be $w = -1$. This value is universal across distinct black hole backgrounds and distinguishes the Davies transition from, e.g., the Hawking–Page transition (which has winding $+1$) (Bhattacharya et al., 2024, Hazarika et al., 2024). The singular behavior of the topological current at the Davies point confirms the robustness and universality of the transition’s topological nature.

5. Fractional Order and Critical Behavior

While standard Ehrenfest classification places the Davies transition as "second order", refined analysis via fractional derivatives reveals richer critical behavior. For Reissner–Nordström–AdS black holes, there are two categories of Davies points:

• Type I (extremal-temperature points): Where the temperature as a function of horizon radius attains an extremum; at these points, the generalized order of the phase transition is $3/2$.
• Type II (inflection/critical points): Where two extremal points coalesce, yielding an inflection (i.e., $\partial^2 T/\partial r_h^2 = 0$); the fractional order is $4/3$ (Wang et al., 2022).

Table: Orders of Phase Transitions at Davies Points (RN–AdS Black Hole) | Type | Location (in T–r_h diagram) | Fractional Order | |-----------------------------|-----------------------------|------------------| | Extremal-temperature (I) | Max/min of T(r_h) | 3/2 | | Inflection/critical (II) | T', T'' = 0 (coalescence) | 4/3 |

This fractional classification reflects the singularity structure of the free energy near the Davies points and potentially connects to universality classes in statistical mechanics.

6. Path Integral and Canonical Ensemble Perspective

Within the Gibbons–Hawking Euclidean path integral approach, the Davies point directly corresponds to the saddle (critical) point at which two branches of black hole solutions merge. For Reissner–Nordström black holes in the canonical ensemble (fixed charge and temperature at infinity), the potential exhibits two roots in the horizon-radius variable below the Davies temperature, which coalesce at the Davies point. The heat capacity diverges and changes sign; on the "small" black hole branch ($r_+ < \sqrt{3} Q$), $C_Q > 0$ (locally stable), while on the "large" branch it is negative. The global thermodynamic stability, however, is not achieved since the free energy remains positive relative to hot flat space, indicating that globally the configuration is metastable (Fernandes et al., 2024).

7. Generalizations and Universality

The Davies phase transition is not limited to spherically symmetric or asymptotically flat black holes. Its analytic and topological character, as well as its connection to divergences in heat capacity, extends broadly: to higher-dimensional black holes, rotating (Kerr, Kerr–AdS), charged AdS/dS black holes, Gauss–Bonnet black holes, and other modified gravity solutions (Bhattacharya et al., 2024, Wei et al., 2019, Hazarika et al., 2024). The φ-mapping approach provides a universal criterion: wherever the heat capacity diverges, a Davies-type critical point with topological charge −1 is present. This generality suggests deep connections between black hole thermodynamics, dynamical stability, and topological invariants beyond specific spacetime backgrounds.

• Null geodesics, quasinormal modes, and thermodynamic phase transition for charged black holes in asymptotically flat and dS spacetimes (Wei et al., 2019)
• Topological interpretation of extremal and Davies-type phase transitions of black holes (Bhattacharya et al., 2024)
• Fractional phase transitions of RN-AdS black hole at Davies points (Wang et al., 2022)
• Gibbons-Hawking action for electrically charged black holes in the canonical ensemble and Davies' thermodynamic theory of black holes (Fernandes et al., 2024)
• Davies Critical Point and Tunneling (La, 2010)
• Revisiting thermodynamic topology of Hawking-Page and Davies type phase transitions (Hazarika et al., 2024)