Extended AdS Phase Space in Black Holes

• Extended AdS phase space is a framework where the negative cosmological constant is treated as pressure with its conjugate thermodynamic volume, extending black hole thermodynamics.
• It reveals fluid-like phase transitions, including first- and second-order criticality, by using an adapted first law and Smarr relation analogous to van der Waals systems.
• The framework incorporates analytical tools like Ruppeiner geometry and Joule-Thomson expansion, and it explores quantum corrections that impact observable black hole microphysics.

The extended anti-de Sitter (AdS) phase space formalism is a framework in black hole thermodynamics in which the negative cosmological constant $\Lambda$ is promoted to a thermodynamic variable identified as pressure, allowing the construction of a generalized thermodynamic phase space with a first law and Smarr relation analogous to those in classical thermodynamics. This approach underpins a wide set of recent advances collectively referred to as "black hole chemistry," in which AdS black holes exhibit phase behavior (including criticality, first-order transitions, and scaling) directly analogous to classical fluid systems such as the van der Waals gas.

1. Fundamental Structure of the Extended AdS Phase Space

The central @@@@2@@@@ of the extended phase space is the identification

$P = -\frac{\Lambda}{8\pi} \;,$

where $\Lambda$ is the cosmological constant (negative for AdS). The conjugate variable to $P$ is the "thermodynamic volume"

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

where $M$ is the ADM mass of the black hole, now universally interpreted as the enthalpy $H$, rather than internal energy.

This leads to a modified first law for black holes. In the case of the Kerr–Newman–AdS solution, the first law and Smarr relation read

$dM = T\,dS + V\,dP + \Omega \, dJ + \Phi\, dQ$

$M = 2 T S - 2 P V + 2\Omega J + \Phi Q.$

The solution data (mass, angular momentum, charge), the AdS radius $l$ (through $P$), and horizon radius $r_+$ (through entropy $S$ and $V$) are linked via closed-form algebraic relations. For example, the explicit expressions for the mass, entropy, and temperature of the Kerr–Newman–AdS black hole are: $$M = \frac{ (r_+^2 + a^2)(1 + r_+^2/l^2) + Q^2 }{ 2 r_+ \Xi^2 }, \qquad \Xi = 1 - \frac{a^2}{l^2}$$

$S = \frac{ \pi ( r_+^2 + a^2 ) }{ \Xi }$

$$T = \frac{1}{4\pi r_+ (r_+^2 + a^2)} \left[ (r_+^2 + a^2) \left(1 + \frac{3 r_+^2}{l^2}\right) - Q^2 - a^2 \right]$$

$$V = \frac{2\pi r_+ (r_+^2 + a^2)}{3\Xi} \left( 1 + \frac{a^2}{l^2} + \frac{2 a^2}{r_+^2} \right).$$

This structure allows the complete extension of black hole thermodynamics to a phase space admitting $P-V$ work and enables precise analogy to classical non-ideal fluid systems (Zhao et al., 2018, Bairagya et al., 2020, Fontana et al., 2018).

2. Criticality and Phase Structure in the Extended Phase Space

With $P$ and $V$ promoted to thermodynamic variables, black holes exhibit nontrivial critical behavior. The equation of state derived from the Hawking temperature, for Reissner–Nordström–AdS in four dimensions, reads in terms of the specific volume $v = 2 r_+$: $$P = \frac{T}{v} - \frac{1}{2\pi v^2} + \frac{2 Q^2}{\pi v^4}.$$ This is formally analogous to the van der Waals equation for fluids. The critical point is determined by the inflection conditions

$$\left( \frac{\partial P}{\partial v} \right)_T = 0, \qquad \left( \frac{\partial^2P}{\partial v^2} \right)_T = 0.$$

At this point the system undergoes a second-order phase transition with mean-field critical exponents $(\alpha, \beta, \gamma, \delta) = (0, 1/2, 1, 3)$.

Analogous criticality is seen in more general black holes and in higher derivative gravities (e.g., Gauss–Bonnet and quasi-topological cases), as well as in solutions with additional matter such as clouds of strings, noncommutative geometry, Planck-scale corrections, or surrounding dark matter. The critical points, coexistence curves, and associated phase diagrams universally reflect fluid-like behavior, including the presence of first-order small/large black hole transitions and characteristic swallowtail structures in the Gibbs free energy (Zhai et al., 2023, Tzikas, 13 Nov 2025, Lobo et al., 2021, Xu et al., 2016).

3. Microphysical Interpretation and Riemannian Geometry

The extended phase space also supports a geometric interpretation via Ruppeiner geometry. The fluctuation metric on the $(T, V)$ or $(S, Q^2)$ manifold leads to a scalar curvature $R$, whose sign and divergence structure encode dominant microscopic interaction channels and signal second-order criticality, respectively. For charged AdS black holes, the normalized curvature

$R_{\text{norm}} = R\, C_V \propto V$

diverges at spinodal points and changes sign across the phase diagram, corresponding to a transition from attraction-dominated to repulsion-dominated microstructure. The equality of $|R_{\text{norm}}|$ across coexisting phases near the critical point confirms the Widom/Kadanoff conjecture for correlation lengths $\xi$ (Bairagya et al., 2020, Xu et al., 2016).

4. Analogs of Classical Thermodynamic Processes and Transport

The extended phase space enables the exploration of traditional thermodynamic processes in black holes, most notably the Joule–Thomson (JT) or throttling process. The isenthalpic (constant-$M$) expansion of a Kerr–Newman–AdS black hole exhibits a JT coefficient

$$\mu = \left(\frac{\partial T}{\partial P}\right)_M,$$

whose sign division defines inversion curves $T_i(P)$. The explicit form for $\mu$ in terms of $S, P, J, Q$ reveals that only a minimum inversion temperature exists (no maximum), nearly always $T_i^{\min} \approx 0.5\, T_c$. Isenthalpic curves in the $T$–$P$ plane display rich behavior controlled by two characteristic masses $M_*$ and $\widetilde M$, depending on whether they intersect the inversion curve and whether they cross each other (Zhao et al., 2018).

Classical heat engine cycles (e.g., Carnot, square cycles) and their efficiencies are also defined in this framework, allowing for quantitative analysis of gravitational heat engines and the influence of quantum corrections (such as Planck-scale modifications) on engine performance (Lobo et al., 2021).

5. Generalizations: Matter Fields, Quantum Corrections, and Higher Dimensions

Many studies have considered extensions of the basic AdS phase space to black holes with extra matter, higher-derivative gravity, or quantum-corrected environments:

• Quantum corrections: Planck-scale dispersion modifies the effective equation of state via a temperature-dependent parameter $\omega(T)$, resulting in logarithmic corrections to thermodynamic potentials and shifts in criticality. The sign of the correction (superluminal or subluminal) controls whether critical temperatures and pressures are raised or suppressed (Lobo et al., 2021).
• Cloud of strings: The presence of a "cloud of strings" parameter $a$ in Gauss–Bonnet–AdS black holes shifts the location of the critical point and modifies heat capacity, but preserves universal mean-field exponents and the fluid-like phase structure (Zhai et al., 2023).
• Noncommutative geometry: Promoting the noncommutativity parameter $\theta$ to a thermodynamic variable introduces a new tension term and its conjugate volume. In four dimensions, the system exhibits first-order small/large black hole transitions and ends in a van der Waals critical point; in lower dimensions, either pure stability (BTZ-like) or novel "anti–Hawking–Page" transitions emerge (Tzikas, 13 Nov 2025).
• Perfect fluid dark matter: By extending the phase space to include $Q^2$ and its conjugate $\psi$, black holes surrounded by perfect fluid dark matter show mean-field criticality and modified Ruppeiner geometry (Xu et al., 2016).

A summary table of the main ingredients across formalisms appears below:

Ingredient	Default Extended AdS	Quantum-corrected / Exotic
$P$ variable	$-\Lambda/8\pi$	$-\Lambda/8\pi$, plus e.g. $-\frac{3}{8\pi\theta}$ (noncommutative)
Conjugate $V$	$(\partial M/\partial P)_S$	$(\partial M/\partial P)_S$, $(\partial M/\partial P_\theta)_S$
Additional variables	$J$, $Q$, GB coupling $\alpha$,...	Planck $\alpha_n$, string cloud $a$, dark matter $a$, $\theta$
First law	$dM = T\,dS + V\,dP + ...$	$dM = T\,dS + V\,dP + V_\theta\,dP_\theta + ...$
Smarr	Euler scaling	Modified for extra variables, all derivatives follow
Critical exponents	Mean-field: $0,1/2,1,3$	Same for all studied corrections

6. Physical Implications, Validity Bounds, and Limitations

Consistency conditions inherited from classical thermodynamic geometry impose restrictions on AdS black hole parameters. For example, the Ruppeiner curvature must remain between the specific and total volume, bounding the range of admissible charge, mass, and temperature for which the fluid analogy is sound (Bairagya et al., 2020). Many works that ignore such bounds may reach unphysical conclusions regarding the nature of phase transitions for certain parameter domains.

The interpretation of the cosmological constant as pressure, and black hole mass as enthalpy, is further justified by quasilocal Hamiltonian analysis combined with Brown–York boundary terms, yielding a constraint-based, symplectic structure that is consistent for all thermodynamic potentials (Fontana et al., 2018).

Extensions to dynamical processes, including black hole absorption and weak cosmic censorship, have confirmed that the first law and thermodynamic structures are maintained when $\Lambda$ is included as a pressure, though the second law may be violated in extremal limits (Chen, 2019).

7. Observables and Probes of Extended Phase Space Structure

Recent advances have shown that observables such as the Lyapunov exponent $\lambda$ of null circular photon orbits also encode the phase structure of AdS black holes in the extended phase space. Across first-order phase transitions, $\lambda$ is discontinuous; at the critical point, it behaves as an order parameter with $\Delta\lambda \propto (T_c-T)^{1/2}$, mirroring the behavior of fluid density gaps and providing an independent dynamical probe of thermodynamic criticality (Xie et al., 27 Oct 2025).

This connection suggests that dynamical features visible to external observers (such as quasinormal modes or photon ring characteristics) can serve as physical manifestation of extended phase space thermodynamic phenomena, potentially linking black hole microphysics with observationally accessible signals.