Heat Capacity and Local Thermodynamic Stability

Local thermodynamic stability asks whether the appropriate potential has the right curvature under the fluctuations allowed by a specified ensemble. A negative heat capacity is therefore decisive in a canonical energy-exchange direction but not in a microcanonical problem where energy is fixed. A cavity or anti-de Sitter boundary can change the response of the same local horizon family, and a locally stable saddle can still lose globally to another phase.

Required background. Gravitational Actions and Boundary Terms supplies the ensemble-dependent boundary variation and thermodynamic potential.

Helpful background. Quantum Fields in Curved Spacetime supplies equilibrium fluctuations; Einstein Equations supplies the stationary branches; and Black-Hole Geometry and Mechanics supplies horizon charges and the first law.

Response functions and the allowed fluctuation space

For a canonical system at fixed additional extensive variables $X$, the heat capacity is

$$C_X=\left(\frac{\partial E}{\partial T}\right)_X =T\left(\frac{\partial S}{\partial T}\right)_X.$$

Using $1/T=(\partial S/\partial E)_X$ gives

$$\left(\frac{\partial^2S}{\partial E^2}\right)_X =-\frac{1}{T^2C_X}.$$

The canonical weight near an equilibrium energy is $\exp[S(E)-\beta E]$. Its Gaussian energy integral converges only when the entropy curvature is negative, hence $C_X\gt0$. With angular momentum or charge exchange, this single condition is replaced by positivity of the appropriate susceptibility matrix, or equivalently by definiteness of the Hessian of the chosen thermodynamic potential. Holding $J$ fixed and holding $\Omega$ fixed test different matrices.

This criterion can also be derived by putting the black hole in contact with a finite environment. If a small energy $\delta E$ moves from the environment to the hole, expansion of the total entropy gives

$$\delta^2S_{\mathrm{tot}} =-\frac{(\delta E)^2}{2T^2} \left(\frac1{C_{\mathrm{BH}}}+\frac1{C_{\mathrm{env}}}\right).$$

A maximum requires the parenthesis to be positive. An infinite reservoir has $1/C_{\mathrm{env}}\to0$ and cannot stabilize $C_{\mathrm{BH}}\lt0$, whereas a finite reservoir can do so when its response is sufficiently small. This derivation explains why gravitational boundary conditions, which control the available radiation and boundary response, are part of a stability claim. For $C_{\mathrm{env}}\gt0$ and $C_{\mathrm{BH}}\lt0$, the condition is $C_{\mathrm{env}}\lt\lvert C_{\mathrm{BH}}\rvert$. This finite-reservoir argument and cavity stabilization are distinct mechanisms: in the cavity calculation below, even an ideal wall thermostat can be stable because the energy and temperature conjugate to the wall clock have a different response.

Schwarzschild’s negative heat capacity

For four-dimensional asymptotically flat Schwarzschild,

$$T_{\mathrm H}=\frac{1}{8\pi GM}, \qquad S_{\mathrm{BH}}=4\pi GM^2,$$

so

$$C=\frac{dM}{dT_{\mathrm H}}=-8\pi GM^2\lt0.$$

A black hole slightly hotter than an infinite heat bath loses energy, becomes hotter still, and runs away; one slightly colder absorbs energy and cools further. This is a canonical instability. It does not make the fixed-energy microcanonical entropy meaningless, and it is not by itself a statement about Lorentzian metric perturbations.

The boundary conditions are essential. In asymptotically flat space an infinite bath also contains arbitrarily large radiation volume, so the naive canonical partition function is not a controlled comparison. A reflecting cavity turns the question into a finite boundary-value problem.

Cavity stabilization

Place a Schwarzschild black hole inside a spherical wall at areal radius $r_B\gt2GM$. The proper wall temperature is the Tolman-redshifted value

$$T_B(M)=\frac{1}{8\pi GM\sqrt{1-2GM/r_B}}.$$

Set $x=2GM/r_B$. Apart from a positive constant, $T_B\propto[x\sqrt{1-x}]^{-1}$, and therefore

$$\frac{d\ln T_B}{dx} =-\frac1x+\frac{1}{2(1-x)} =\frac{3x-2}{2x(1-x)}.$$

The temperature has a minimum at $x=2/3$. The quasilocal wall energy may be written

$$E_B=\frac{r_B}{G}\left(1-\sqrt{1-x}\right),$$

and $dE_B/dM\gt0$. It follows that the small branch $x\lt2/3$ has $dT_B/dM\lt0$ and negative canonical heat capacity, while the large branch $x\gt2/3$ has positive heat capacity. This is the explicit cavity stabilization derived from the Euclidean action and density of states by York 1986, §§II–V and VIII. The wall radius, proper wall temperature, subtraction, and quasilocal energy are all part of the result.

In particular, the wall heat capacity differentiates $E_B$, not $M$:

$$C_B=\left(\frac{\partial E_B}{\partial T_B}\right)_{r_B} =\frac{4\pi r_B^2}{G}\, \frac{x^2(1-x)}{3x-2}.$$

The distinction changes its magnitude even though $dE_B/dM\gt0$ preserves the sign inferred above. It also checks the first law: $dE_B=T_B\,dS$ at fixed wall area, with $S=\pi r_B^2x^2/G$.

Local, global, and dynamical stability

Schwarzschild–AdS makes the distinctions especially sharp. With AdS radius $\ell$,

$$T(r_+)=\frac{1+3r_+^2/\ell^2}{4\pi r_+}.$$

The heat capacity changes sign at $r_+=\ell/\sqrt3$, so larger holes are locally canonical-stable. Their free energy relative to thermal AdS changes sign only at $r_+=\ell$, the Hawking–Page point. Thus the interval $\ell/\sqrt3\lt r_+\lt\ell$ is locally stable but globally subdominant. A local Hessian cannot decide which separated saddle has lower free energy.

Euclidean negative modes add another layer. For the stated canonical boundary problem, Prestidge finds that the Schwarzschild–AdS negative mode disappears where the heat capacity changes sign Prestidge 2000, §§5–7, PDF. That correspondence is powerful but conditional on the perturbation sector, gauge, boundary conditions, and ensemble. A Euclidean negative direction does not automatically equal a Lorentzian growing mode, and absence of that one mode does not prove nonlinear stability.

Rotation and charge make the ensemble dependence more than a one-dimensional sign test. In a grand-canonical calculation, variations of $E$, $J$, and $Q$ mix, and stability requires the full covariance or susceptibility matrix to be positive in the allowed subspace. Checking $C_{J,Q}\gt0$ while ignoring a negative isothermal moment of inertia or electric susceptibility is insufficient. Conversely, a negative direction that changes $Q$ is irrelevant to a strictly fixed-charge ensemble. The boundary term decides which of these directions is present.

As independent checks, the cavity formula recovers negative heat capacity as $r_B\to\infty$, and the AdS formula has the expected $T\sim1/(4\pi r_+)$ for a small hole but $T\sim3r_+/(4\pi\ell^2)$ for a large one. Both sign reversals come from the boundary environment rather than a change in the local area law. At a turning point where $dT/dr_+=0$, the corresponding heat capacity diverges rather than vanishes. That divergence marks a change of local branch stability; it does not by itself imply a singular spacetime or a divergent curvature invariant.

Exercise

For a Schwarzschild black hole in a cavity, use $T_B\propto[x\sqrt{1-x}]^{-1}$ to determine the signs of the heat capacity on the two branches and explain why this does not determine global dominance.

Solution

Because $E_B$ increases with $x$, the sign of $C_B=(\partial E_B/\partial T_B)_{r_B}$ is the sign of $dT_B/dx$. The derivative is negative for $x\lt2/3$, so the small branch has $C_B\lt0$; it is positive for $x\gt2/3$, so the large branch has $C_B\gt0$. This is only a local curvature test. Global dominance requires comparing the complete on-shell actions of all saddles with the same wall geometry and temperature.

References

• Prestidge, T. (2000), “Dynamic and Thermodynamic Stability and Negative Modes in Schwarzschild–Anti-de Sitter,” Physical Review D 61, 084002. Open PDF.
• York, J. W. Jr. (1986), “Black-Hole Thermodynamics and the Euclidean Einstein Action,” Physical Review D 33, 2092–2099. DOI.