Extended AdS Black-Hole Mechanics

Ordinary black-hole mechanics compares nearby solutions at fixed cosmological constant. Extended AdS mechanics enlarges that solution space by varying $\Lambda$ and adding its conjugate work term. With Newton’s constant fixed, define $P=-\Lambda/(8\pi G)$; for AdS, $P$ is positive. The mass then varies as an enthalpy-like state function in this enlarged phase space. This is a classical variational statement. It neither requires Hawking radiation nor identifies the geometric quantities with microscopic thermodynamic degrees of freedom.

Helpful background. Exact Solutions of Einstein’s Equations supplies the parameterized AdS solution families; Causality and Global Structure supplies the choice of exterior and asymptotic component; The Einstein Equation supplies the role of $\Lambda$ in the constraints; The Gibbons-Hawking-York Boundary Term supplies the Hamiltonian boundary terms defining mass; and Black-Hole Geometry and Mechanics supplies the fixed-$\Lambda$ first law and Smarr scaling.

Cosmological pressure and its conjugate

In $D$ spacetime dimensions, AdS radius $L$ means

$$\Lambda=-\frac{(D-1)(D-2)}{2L^2}, \qquad P=-\frac{\Lambda}{8\pi G} =\frac{(D-1)(D-2)}{16\pi G L^2}.$$

Consider a differentiable family of stationary black-hole solutions with the same asymptotic topology and matter theory, but allow $L$ to vary. Hold $G$, dimensionless quotient identifications, and any unlisted couplings fixed. For a connected Killing horizon, the extended first law is

$$\delta M= \frac{\kappa}{8\pi G}\,\delta A +\sum_i\Omega_i\,\delta J_i +\sum_a\Phi_a\,\delta Q_a +V\,\delta P.$$

Every potential is a horizon-to-infinity difference in a compatible frame and gauge. The new conjugate is defined within this phase space by

$$V=\left(\frac{\partial M}{\partial P}\right)_{A,J_i,Q_a}.$$

The variable-$\Lambda$ law, this derivative definition, and the need to use angular velocities relative to the chosen asymptotic frame are summarized in Kubizňák, Mann, and Teo 2017, §2.2, equations (2.14)–(2.18), Open PDF, pp. 6–8. A Hamiltonian derivation integrates the perturbed constraint with a Killing co-potential between the horizon and infinity Kubizňák, Mann, and Teo 2017, Appendix A, equations (A.1)–(A.12), Open PDF, pp. 67–69.

That derivation also explains why $V$ is not defined as the proper volume behind a chosen spatial horizon cut. A Killing co-potential has a gauge freedom, and its separate surface integrals at the horizon and infinity need not be invariant; the regulated difference entering the Hamiltonian identity is the meaningful quantity. The conjugate $V$ is therefore determined by the entire solution and its asymptotic normalization. Only in special symmetric families must it coincide with an elementary Euclidean volume.

Setting $\delta P=0$ recovers the ordinary fixed-$\Lambda$ law. For four-dimensional Kerr–AdS without gauge charge this is exactly

$$\delta M=\frac{\kappa}{8\pi G}\,\delta A+\Omega\,\delta J,$$

with $M$, $J$, and $\Omega$ defined relative to nonrotating infinity Gibbons, Perry, and Pope 2005, §2.1, equations (2.11)–(2.15), Open PDF, pp. 8–9. Thus the extended law adds a direction in solution space; it does not replace or contradict the fixed-$\Lambda$ relation.

Conversely, a process carried out within one theory at fixed $\Lambda$ cannot be assigned a $V\delta P$ contribution merely because neighboring theories can be compared. The allowed variation, not the algebraic availability of $P$, decides which work terms are present.

Smarr scaling

The Smarr relation follows independently from length scaling. Under $r\mapsto\lambda r$ with $G$ fixed, the relevant homogeneity weights are

$$M,Q_a\mapsto\lambda^{D-3}(M,Q_a), \qquad A,J_i\mapsto\lambda^{D-2}(A,J_i), \qquad P\mapsto\lambda^{-2}P.$$

Euler’s theorem applied to $M(A,J_i,Q_a,P)$ gives

$$(D-3)M= (D-2)\frac{\kappa A}{8\pi G} +(D-2)\sum_i\Omega_iJ_i +(D-3)\sum_a\Phi_aQ_a -2PV.$$

The coefficient $-2$ is forced by the scaling weight of $P$. Omitting $PV$ generally makes the AdS Smarr formula fail even when the differential first law at fixed $P$ remains correct. The general scaling derivation appears in Kubizňák, Mann, and Teo 2017, §2.2, equations (2.19)–(2.21), Open PDF, pp. 8–9.

In four dimensions this reduces to

$$M=\frac{\kappa A}{4\pi G} +2\Omega J+\sum_a\Phi_aQ_a-2PV.$$

This formula provides a stringent normalization check: every term has the same homogeneity weight and the physical units of mass, and the same asymptotic generator must define both $M$ and $\Omega$.

Schwarzschild–AdS as a direct derivation

For a four-dimensional spherical static black hole,

$$M=\frac{r_+}{2G}\left(1+\frac{r_+^2}{L^2}\right), \qquad A=4\pi r_+^2, \qquad P=\frac{3}{8\pi G L^2}.$$

Eliminating $L$ gives a mass function with independent variables $(r_+,P)$,

$$M(r_+,P)=\frac{r_+}{2G}+\frac{4\pi}{3}Pr_+^3.$$

Holding $A$ fixed is equivalent to holding $r_+$ fixed, so

$$V=\left(\frac{\partial M}{\partial P}\right)_A =\frac{4\pi r_+^3}{3}.$$

For this static spherical family, the conjugate equals the Euclidean volume of a ball with areal radius $r_+$. That equality is not a definition and is not universal: rotation and more general geometries can make $V$ differ from a naive interior volume Kubizňák, Mann, and Teo 2017, §§3.1–3.2, equations (3.1)–(3.4), Open PDF, pp. 14–16.

For example, the $D$-dimensional Kerr–AdS expression may be organized as

$$V=\frac{r_+A}{D-1} \left[ 1+\frac{1+r_+^2/L^2}{(D-2)r_+^2} \sum_i\frac{a_i^2}{\Xi_i} \right], \qquad \Xi_i=1-\frac{a_i^2}{L^2}.$$

The second term is positive for the regular standard family and vanishes when every spin vanishes. It explicitly shows that equal horizon area and radius do not imply equal thermodynamic volume once angular momenta differ.

Enthalpy language and additional work terms

Within the extended thermodynamic interpretation, $M$ is called enthalpy and one defines an internal-energy-like quantity $U=M-PV$. This terminology records which variables are natural for the state function; it does not change the conserved charge measured at infinity. If $\Lambda$ is a fixed coupling of the theory rather than a variable state parameter, only the $\delta P=0$ slice has physical variations.

The displayed law is also conditional on what is held fixed. Varying a planar or string compactification length $\mathcal L_i$ introduces a tension term $\mathcal T_i\delta\mathcal L_i$. Varying a conical deficit, matter coupling, or other boundary modulus requires its own conjugate, with a sign fixed by the adopted definition. If those parameters are fixed, such terms are absent. A complete first law is therefore a statement about a declared phase space, not merely about a metric formula.

Worked transfer diagnostic

Starting only from $M(r_+,P)=r_+/(2G)+(4\pi/3)Pr_+^3$, derive $V$, verify the extended first law, and check the four-dimensional Smarr relation.

Solution

Differentiation gives

$$\delta M= \left(\frac{1}{2G}+4\pi Pr_+^2\right)\delta r_+ +\frac{4\pi r_+^3}{3}\,\delta P.$$

Since $\kappa=1/(2r_+)+4\pi GPr_+$ and $\delta A=8\pi r_+\delta r_+$,

$$\frac{\kappa}{8\pi G}\delta A =\left(\frac{1}{2G}+4\pi Pr_+^2\right)\delta r_+.$$

Thus $V=4\pi r_+^3/3$ and $\delta M=(\kappa/8\pi G)\delta A+V\delta P$. Finally,

$$\frac{\kappa A}{4\pi G}-2PV =\frac{r_+}{2G}+\frac{4\pi}{3}Pr_+^3=M,$$

so the Smarr relation holds independently.

Next, Ultraspinning Limits and Classical Phase Structure compares singular limits, perturbative thresholds, and nonlinear branches without conflating their evidence.

References

• Gibbons, G. W., Perry, M. J., and Pope, C. N. “The First Law of Thermodynamics for Kerr–anti-de Sitter Black Holes.” Classical and Quantum Gravity 22 (2005): 1503–1526. doi:10.1088/0264-9381/22/9/002. Open PDF.
• Kubizňák, D., Mann, R. B., and Teo, M. “Black Hole Chemistry: Thermodynamics with Lambda.” Classical and Quantum Gravity 34 (2017): 063001. doi:10.1088/1361-6382/aa5c69. Open PDF.