Black Hole Chemistry & Extended Thermodynamics

• Black hole chemistry is an extended framework of thermodynamics where the cosmological constant is treated as pressure and the black hole mass is interpreted as enthalpy.
• It analogizes black hole phase behavior to that of Van der Waals fluids, exhibiting critical phenomena, phase transitions, and universal scaling laws.
• The approach bridges gravitational physics with holography and quantum criticality, offering insights into higher-dimensional and modified gravity systems.

Black hole chemistry is the extended framework of black hole thermodynamics in which the cosmological constant is promoted to a thermodynamic variable identified as pressure, and the black hole mass is reinterpreted as enthalpy. This paradigm generates a mathematical and physical analogy between the phase structure of black holes in Anti–de Sitter (AdS) spacetimes and chemical systems, such as Van der Waals fluids, and exposes a rich spectrum of critical phenomena, phase transitions, and universal scaling behaviors. The development has led to deep connections with holography, higher-dimensional gravity, quantum criticality, and geometric approaches to thermodynamic phase space.

1. Fundamental Principles: Extended Phase Space Thermodynamics

In black hole chemistry, the negative cosmological constant $\Lambda$ is identified with a positive thermodynamic pressure,

$P = -\frac{\Lambda}{8\pi G_N}$

where $G_N$ is Newton’s constant. In AdS$_d$ with curvature radius $L$, this gives

$P = \frac{(d-1)(d-2)}{16\pi G_N L^2}$

The conjugate variable is the thermodynamic volume,

$$V = \left( \frac{\partial M}{\partial P} \right)_{S,Q,J,\dots}$$

with $M$ the ADM mass, which is reinterpreted as the enthalpy $H$ of the spacetime,

$M = H(S,P,Q,J,\dots)$

The extended first law reads

$$dM = T\,dS + V\,dP + \sum_i \Omega_i\,dJ_i + \sum_j \Phi_j\,dQ_j + \cdots$$

where $T$ is the Hawking temperature, $S$ the Bekenstein–Hawking entropy, $\Omega_i$ angular velocities, $J_i$ angular momenta, $\Phi_j$ electrostatic potentials, and $Q_j$ charges.

A scaling (Euler) argument yields the generalized Smarr relation in $d$ spacetime dimensions

$$(d-3)M = (d-2)TS - 2PV + (d-3)\sum_j \Phi_j Q_j + \sum_i (d-2)[\Omega_i-\Omega_{\infty,i}]J_i$$

For static, charged, nonrotating black holes, this simplifies to

$(d-3)M = (d-2)TS - 2PV + (d-3)\Phi Q$

These relations are preserved for nontrivial couplings and extended matter content via additional work terms (Karch et al., 2015, Meessen et al., 2022, Mann, 3 Aug 2025).

2. Thermodynamic Volume, Reverse Isoperimetric Inequality, and Generalizations

For spherically symmetric AdS black holes, the thermodynamic volume agrees with the naive geometric volume, $V=\tfrac{4}{3}\pi r_+^3$, with $r_+$ the horizon radius. In rotating (Kerr–AdS) cases, extra spin contributions modify $V$: $$V = \frac{2\pi (r_+^2 + a^2) r_+}{3 \Xi}, \quad \Xi=1-a^2/L^2$$ These contributions reflect horizon deformation due to rotation.

The reverse isoperimetric inequality conjectures that for AdS black holes

$$\mathcal{R} \equiv \left(\frac{(d-1)V}{\omega_{d-2}}\right)^{1/(d-1)} \left(\frac{\omega_{d-2}}{A}\right)^{1/(d-2)} \geq 1$$

where $A$ is the horizon area and $\omega_{d-2}$ is the volume of the unit sphere. Equality holds for Schwarzschild–AdS, providing an upper bound on black hole entropy at fixed volume.

Extensions exist to include additional work terms, such as conical deficits (cosmic string tensions), where the phase space is

$M(S,P,Q,J,\mu_\pm)$

with new chemical potentials and conjugate tensions (Gregory et al., 2019). In DGP braneworlds, brane tension $\sigma$ replaces $\Lambda$ as a pressure-like variable, with $P_\sigma = -\sigma$ and its conjugate volume $V_\sigma$ matches the geometric horizon volume; the first law and Smarr form persist (Kumar, 25 Aug 2025).

3. Phase Transitions: Van der Waals, Multicritical, and Exotic Phenomena

In the extended phase space, charged AdS black holes exhibit Van der Waals–like small/large black hole phase transitions. For four-dimensional Reissner–Nordström–AdS,

$$P = \frac{T}{v} - \frac{1}{2\pi v^2} + \frac{2Q^2}{\pi v^4}$$

with $v = 2 r_+$ and $Q$ the charge. The critical point solves $\partial_v P = \partial_v^2 P = 0$,

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

and the universal ratio $P_c v_c / T_c = 3/8$ matches the Van der Waals fluid. The standard mean-field exponents $(\alpha=0,\,\beta=1/2,\,\gamma=1,\,\delta=3)$ are realized (Kubiznak et al., 2014, Mann, 3 Aug 2025).

Higher-dimensional and rotating (Kerr–AdS) cases lead to reentrant phase transitions, triple points (three-phase coexistence), and exotic “isolated” (polymeric) critical points with non-mean-field exponents,

$$\beta=1,\; \gamma=K-1,\; \delta=K, \quad K>2\ (\textrm{Lovelock order})$$

(Mann, 3 Aug 2025).

Systems with higher-derivative gravity terms (cubic generalized quasi-topological, Lovelock, etc.), or nonlinear electrodynamics, acquire additional critical points, double swallowtails, anomalous phase structures, and can show superfluid-like $\lambda$-lines (continuous lines of second-order transitions) (Mir et al., 2019, Mann, 3 Aug 2025). Accelerating black holes (C-metric) introduce string tensions as new charges and exhibit phase diagrams with “snapping swallowtails” and modified isoperimetric bounds (Gregory et al., 2019).

4. Lower-Dimensional, Topological, and Microstructured Black Hole Chemistry

For the BTZ (2+1D) and lower-dimensional limits, the extended first law and Smarr relation admit well-defined formulations, but Van der Waals–like criticality and phase coexistence are absent. The thermodynamic volume can become charge-dependent, leading to superentropic BTZ black holes that violate the reverse isoperimetric inequality. Restoration of this inequality requires introducing renormalization-scale work terms (Frassino et al., 2015). In D→2 (1+1D) gravity, a sign-flipped pressure, logarithmic entropy, and a maximum mass constraint appear, but again, no nontrivial phase transitions emerge.

The microstructure of black hole “molecules” can be probed by thermodynamic information geometry, particularly via the Ruppeiner scalar curvature $R$, which diverges at criticality and changes sign corresponding to the dominance of attractive versus repulsive inter-molecular interactions, paralleling ordinary fluids (Ghosh et al., 2023, Mann, 3 Aug 2025). The ensemble dependence (“ensemble non-equivalence”) manifests as different divergences in curvatures computed from different thermodynamic potentials.

Topological approaches, including the assignment of integer winding numbers to critical points using φ-mapping techniques, provide a universal characterization of phase transitions in massive gravity and nonlinear electrodynamics, distinguishing creation/annihilation of phases and anomalous critical points (Zhang et al., 2023).

5. Holographic Black Hole Chemistry and AdS/CFT Correspondence

The holographic duality maps extended black hole thermodynamics in the bulk to thermodynamics of the dual conformal field theory (CFT) on the boundary. In the CFT, the pressure $P$ corresponds to parameters controlling the number of colors $N$ or the central charge (e.g., $L^{d-2}/G_N \sim N^2$). The fundamental scaling observed in the bulk (the universal holographic Smarr relation)

$N^2 \frac{\partial \Omega}{\partial N^2} = \Omega$

arises directly from large $N$ scaling of extensive thermodynamic quantities such as free energy, entropy, and charge (Karch et al., 2015).

Via a precise holographic dictionary, bulk enthalpy variations map to energy or chemical potential variations in the boundary theory. The bulk Smarr relation becomes the Euler (Gibbs–Duhem) relation in the CFT

$E = T S + \Omega J + \Phi Q + \mu C$

where $E$ is energy, $S$ entropy, $J$ angular momentum, $Q$ charge, $C$ central charge, and $\mu$ its conjugate chemical potential. Bulk phase transitions, such as the Hawking–Page transition and small/large black hole transitions, correspond to CFT confinement/deconfinement and gas-liquid transitions respectively. Central charge ($N^2$) criticality and multicritical phenomena can be controlled by tuning $C$, $Q$, or other boundary parameters (Mann, 5 Mar 2024, Mann, 3 Aug 2025).

Open questions pertain to subleading $1/N$ corrections, higher-derivative terms, topology, CFT ensemble choice, and the connection to computational complexity (complexity=volume proposals) (Mann, 5 Mar 2024).

6. Generalizations: Massive Gravity, Extra Dimensions, Hair, and Braneworlds

Introducing massive gravitons (dRGT gravity) and nonlinear electrodynamics (Power Maxwell Invariant) leads to enlarged phase diagrams, additional critical points, dimensional “windows” for vdW, reentrant, and triple transitions, and dependence on horizon topology; all horizon topologies can become “effectively spherical” by adjusting massive couplings (Dehghani et al., 2019, Dehghani et al., 2020, Zhang et al., 2023). Conformal invariance in the matter sector modifies the phase structure further, shifting the dimensions where various transitions occur (Dehghani et al., 2020).

Scalar hair induces new critical points and non-universality in compressibility ratios. In hairy black holes, criticality emerges even in the grand canonical ensemble, forbidden in the RN–AdS case, and the number of critical branches increases (Astefanesei et al., 2019). In regular (Bardeen or nonsingular) AdS black holes, the Van der Waals analogy survives the removal of the central singularity, with appropriate corrections to thermodynamic quantities (Tzikas, 2018, Kumar et al., 2021).

Braneworld constructions, such as the DGP model, realize black hole chemistry by treating brane tension as pressure, providing an extended thermodynamics framework for asymptotically flat spacetimes, and mitigating conceptual issues associated with varying $\Lambda$ in the bulk (Kumar, 25 Aug 2025).

Kaluza–Klein reductions illustrate that extra dimensional moduli can play the role of thermodynamic pressure, with the thermodynamic volume corresponding to horizon size in the lower-dimensional theory. The equation of state and critical behavior in extra dimensions are directly governed by the compactification parameter (Kim et al., 18 Feb 2025).

The embedding tensor formalism unifies all dimensionful couplings (cosmological constant, gauge couplings, higher-curvature terms) as thermodynamic variables, each with conjugate potentials, leading to a generalized first law and Smarr formula invariant under the duality groups of supergravity and string theory (Meessen et al., 2022).

7. Outlook, Mathematical Structures, and Future Directions

Black hole chemistry is deeply connected to contact geometry and information geometry. The thermodynamic phase space is structured as a contact manifold, with the equilibrium states forming a Legendre submanifold. Hessian metrics derived from various thermodynamic potentials provide a geometric approach to fluctuation theory, critical phenomena diagnostics, and microstructure inference (Ghosh et al., 2023, Mann, 3 Aug 2025).

Physical implications extend to holographic heat engines, the role of thermodynamic tension in de Sitter (cosmological) horizons, and topological invariants of phase space such as winding numbers of free energy branch points. The program continues to be extended to non-AdS backgrounds, higher-order curvature gravity, string compactifications, and quantum-corrected entropies.

The general theme is that extending black hole thermodynamics with the full panoply of chemical variables reveals a universal landscape of phase transitions, criticality, and microscopic structures, any satisfactory quantum gravity theory must reproduce in the semiclassical limit (Mann, 3 Aug 2025, Mann, 5 Mar 2024).