Thermodynamics of Charged Black Holes

Updated 12 December 2025

Charged black holes are gravitational objects defined by mass, angular momentum, and electric/magnetic charge, exhibiting unique thermodynamic properties.
Key thermodynamic variables such as temperature, entropy, pressure, and electric potential are derived from horizon geometry and gauge fields.
Their equations of state mirror Van der Waals fluids, with critical phenomena and phase transitions providing insights into gravitational and microstructural behaviors.

Charged black holes are gravitational objects characterized not only by mass and angular momentum, but also by a conserved electric or magnetic charge. These charges fundamentally alter the spacetime geometry and result in distinctive thermodynamic properties, phase structures, and critical phenomena. The paper of thermodynamic quantities associated with charged black holes has revealed deep analogies with classical systems, such as the Van der Waals fluid, and provides unique insight into gravitational, quantum field, and statistical behaviors in strongly curved spacetime.

1. Fundamental Thermodynamic Quantities

The thermodynamic variables of a charged black hole system, in their most general form, are derived from the metric coefficients, horizon geometry, and gauge fields. For the canonical class of four-dimensional Reissner–Nordström–AdS (RN–AdS) black holes, the relevant quantities are as follows (Kubiznak et al., 2012):

Pressure: Identified with the (negative) cosmological constant $\Lambda$ ,

$P = -\frac{\Lambda}{8\pi} = \frac{3}{8\pi l^2}$

Enthalpy (Mass): $M$ is interpreted as the enthalpy $H(S,P,Q)$ ,

$H = M(P,S,Q)$

Entropy: Given by the Bekenstein–Hawking area law,

$S = \frac{A}{4} = \pi r_+^2$

where $r_+$ is the horizon radius.

Temperature (Hawking):

$T = \frac{1}{4\pi r_+}\left[1 + 8\pi P r_+^2 - \frac{Q^2}{r_+^2}\right]$

Electric Potential:

$\Phi = \frac{Q}{r_+}$

Thermodynamic Volume: Conjugate to $P$ ,

$V = \left(\frac{\partial M}{\partial P}\right)_{S,Q} = \frac{4}{3}\pi r_+^3$

Gibbs Free Energy:

$G = M - T S$

Similar thermodynamic definitions extend to higher-curvature, non-linear electrodynamics, or modified gravity backgrounds—with corresponding modifications to each quantity, often derived from the generalized area law for entropy (e.g., Wald's formula), or incorporating additional parameters (such as Born–Infeld scale, Euler–Heisenberg parameter, dilaton coupling) in $T$ , $M$ , and $\Phi$ (Magos et al., 2020, Lala, 2012, Panah, 19 Feb 2024, Bhattacharjee et al., 25 Sep 2025, Hendi et al., 2015).

2. Equations of State and Criticality

Charged black holes in AdS admit a thermodynamical equation of state (EOS) linking $P$ , $V$ , and $T$ , which is typically algebraically analogous to the Van der Waals gas. For RN–AdS: $P = \frac{T}{v} - \frac{1}{2\pi v^2} + \frac{2 Q^2}{\pi v^4}$ where the specific volume $v$ is proportional to the horizon radius, $v=2r_+$ (Kubiznak et al., 2012). This EOS admits a critical point, determined by inflection criteria: $\frac{\partial P}{\partial v}\bigg|_{T=T_c} = 0,\quad \frac{\partial^2 P}{\partial v^2}\bigg|_{T=T_c} = 0$ yielding critical values

$v_c = 2\sqrt{6} Q,\quad T_c = \frac{\sqrt{6}}{18\pi Q},\quad P_c = \frac{1}{96\pi Q^2}$

The universal ratio $P_c v_c / T_c = 3/8$ precisely matches the Van der Waals value, highlighting an isomorphism between charged AdS black hole criticality and classical mean-field fluids (Kubiznak et al., 2012). For higher-curvature or non-linear theories, analogous cubic or higher-order characteristic equations arise for the critical points, with the critical exponents and scaling relations often unaffected (see Section 5) (Magos et al., 2020, Lala, 2012).

3. Phase Transitions and Stability

Charged black holes exhibit both first- and second-order phase transitions, with physical signatures in their free energy landscapes and specific heats.

First-Order Transition: Marked by a discontinuous change in the Gibbs free energy $G(T,P;Q)$ ("swallowtail" structure) and latent heat, corresponding to a coexistence line of small/large black holes (SBH/LBH transition) (Kubiznak et al., 2012).
Second-Order Transition: Identified by a continuous but nonanalytic change in $T$ , $S$ , or $C_Q$ , with $C_Q$ (specific heat at fixed $Q$ or $P$ ) exhibiting a divergence.
Critical Exponents: Near criticality, thermodynamic observables follow mean-field scaling laws:

$\begin{aligned} &C_{V} \sim |t|^{-\alpha},\quad \eta \sim |t|^{\beta},\quad \kappa_T \sim |t|^{-\gamma},\quad |P-P_c| \sim |\epsilon|^{\delta} \ &\alpha = 0,\quad \beta = \frac{1}{2},\quad \gamma = 1,\quad \delta = 3 \end{aligned}$

as in the Van der Waals system (Kubiznak et al., 2012, Lala, 2012, Magos et al., 2020).

Thermodynamic stability is encoded in the sign of heat capacities ( $C_Q>0$ ), and globally in the sign of the Helmholtz or Gibbs free energy; negative $F$ indicates global preference for the black hole phase over pure radiation or vacuum (Lala, 2012, Panah, 19 Feb 2024).

4. Extensions: Nonlinear Theories, Modified Gravity, and Hair

Many generalizations of charged black hole backgrounds lead to refined thermodynamic structures:

Euler–Heisenberg/Nonlinear Electrodynamics: Additional parameters (e.g., $a$ in Euler–Heisenberg Lagrangian) introduce higher-order corrections in the EOS, shifting but not destroying Van der Waals–like criticality. The universal ratio $P_c v_c/T_c$ is nearly unchanged, with small corrections $O(10^{-4} a/Q^2)$ , and critical exponents remain mean-field (Magos et al., 2020).
f(R) Gravity/Rainbow Gravity/Modified Theories: The restricted phase space thermodynamics (RPST) formalism, with the central charge $C$ and its conjugate chemical potential $\mu$ , preserves the first law with $M$ as internal energy. Criticality and thermodynamic geometry (via Legendre-invariant scalar curvature) display singularities coincident with specific-heat divergences, illustrating universal features even with higher-derivative or running coupling effects (Bhattacharjee et al., 25 Sep 2025, Hendi et al., 2015).
Black Holes with Scalar, Dilatonic, or Vector Hair: The energy function and specific heats strongly depend on matter hair parameters. Thermodynamic stability and critical entropy shift with the strength of hair or rotation, but the qualitative hallmark of phase transitions—divergences in $C_Q$ , swallowtails in $G(T,P)$ —persists (Panah, 19 Feb 2024, Wu et al., 30 Apr 2024, Sadeghi et al., 2013, Ndongmo et al., 2019).
Higher Dimensions and Topology: Dimensionality and horizon topology affect the location of the critical point but not the universality class. Additions such as quintessence ( $\omega_q$ ), string clouds, or extended electrogravity backgrounds modulate phase boundaries and critical temperatures but preserve Van der Waals–like phenomenology (Chabab et al., 2020, Ma et al., 2016, Dehghani et al., 2023).

5. Thermodynamic Geometry and Fluctuation Theory

The geometric approach to black hole thermodynamics, particularly the Ruppeiner metric and its scalar curvature $R$ , provides a microstructural interpretation (Singh et al., 2023, Panah, 19 Feb 2024). For charged AdS black holes:

At low temperature, $R$ diverges as $1/(\gamma T)$ near extremality, indicating long-range correlations and a transition from attraction- to repulsion-dominated microstructure.
Quantum fluctuations can regularize this divergence, yielding a finite limiting value ( $R=1/3$ ) at $T\to0$ , in both canonical and grand canonical ensembles.
Divergences in scalar curvature calculated from Legendre-invariant thermodynamic metrics coincide precisely with heat capacity singularities and phase transition points, linking geometric criticality with classical thermodynamic instability (Bhattacharjee et al., 25 Sep 2025, Panah, 19 Feb 2024).

6. Special Cases: Lower Dimensions and Exotic Charges

Three-dimensional Charged BTZ Black Holes: The BTZ solution with electric charge (and/or scalar hair) in 2+1 dimensions follows an analogous, albeit simpler, thermodynamic structure. Under particle absorption, both the first and second laws are respected—entropy always increases, and extremal holes are driven away from extremality, not destroyed (Gwak et al., 2015, Upadhyay et al., 2019, Sadeghi et al., 2013).
Nonlinear and Bardeen-type Charges: Regular black holes with nonlinear (magnetic or electric) charge structure, such as the Bardeen black hole, exhibit critical behavior and local stability/instability branches demarcated by heat capacity sign changes and characteristic off-shell free energy profiles (Man et al., 2013, Ndongmo et al., 2019).

7. Universalities and Deviations

Despite the variety of background modifications—higher-curvature corrections, alternative matter sectors, or altered thermodynamic ensembles—the universality of critical exponents and the fundamental analogy with Van der Waals fluids is robust. Quantitative locations of critical points and coexistence lines depend sensitively on model parameters, but the qualitative phase structure and mean-field scaling persist (Kubiznak et al., 2012, Lala, 2012, Magos et al., 2020, Bhattacharjee et al., 25 Sep 2025, Chabab et al., 2020). Notably, in some settings (e.g., massive/dilaton gravity), novel phenomena such as reverse reentrant phase transitions or multi-critical loci (tricritical points) appear, introducing rich new thermodynamic landscapes (Yue et al., 20 Mar 2024).

In summary, the thermodynamic quantities of charged black holes define a rich structure of equations of state, critical phenomena, and phase transitions. While detailed forms of the state variables and critical points depend on spacetime geometry, gauge sector, and matter content, the mean-field universality class and the basic structure of phase behavior remain a robust outcome across gravitational theories and dimensions (Kubiznak et al., 2012, Lala, 2012, Magos et al., 2020, Bhattacharjee et al., 25 Sep 2025, Yue et al., 20 Mar 2024).