Black Hole Thermodynamics in AdS$_ d$

The conceptually cleanest way to understand the thermodynamics of black holes is by studying the Euclidean path integral for gravity. This integral, denoted by $\mathcal{Z}$, is defined as follows

$$\begin{equation}\mathcal{Z} = A \int[\mathcal{D}g] e^{-\mathcal{I}[g]}.\end{equation}$$

This is a functional integral with a normalization A, measure $\mathcal{D}[g]$, and integrand $e^{-\mathcal{I}[g]}$ of metrics $g$ with Euclidean signature. (A primer on functional integration will be released shortly.) Essentially, computing the path integral for any system, subject to certain conditions discussed below, allows one to completely characterize the thermodynamics of the system. In practice, the above path integral is impossible to evaluate directly except in special cases. Rather than evaluating $\mathcal{Z}$ exactly, there are several techniques to approximate its value. One such technique is the saddle point approximation. The idea is that when the action $\mathcal{I}[g]\gg1$, the integrand $e^{\mathcal{I}[g]}\ll1$. Hence, the most dominant contributions are given by the minima of the function where we can approximate

$$\mathcal{I}[g] = \sum_i\mathcal{I}[g_ i] + \frac{1}{2}(g-g_ i)^2\frac{\delta^2}{\delta g^2}\mathcal{I}[g]|_ {g=g_i}+\dots$$

Then the path integral simply reduces to a Gaussian integral which can be normalized as follows

$$\begin{equation}\mathcal{Z} \approx A e^{-\sum_{i}\mathcal{I}[g_ i]} \int[\mathcal{D}g]e^{\sum_ i \frac{1}{2}(g-g_ i)^2\frac{\delta^2}{\delta g^2}\mathcal{I}[g]|_ {g=g_ i}+\dots} = e^{-\sum_{i}\mathcal{I}[g_i]}.\end{equation}$$

Thus the entire contribution of the geometry to the Euclidean path integral is encoded in the minima of the action functional.

In order to make contact with thermodynamics, we invoke the following equivalence: the action in Euclidean signature is proportional to the free energy of the system with the proportionality set by the inverse temperature, i.e., $\mathcal{I}[g] = \beta F = (1/T)F$.

To be concrete, I detail the thermodynamics for Euclidean gravity with a negative cosmological constant, or Einstein-AdS gravity. This theory of gravity admits two minima: a Schwarzschild-AdS black hole, and a thermal gas of photons in AdS spacetime. These two minima admit phase transitions at a critical tempeture which will be studied in a later article.

Gravitational Saddle Points in AdS$_d$ #

We will review the purely gravitational saddle points allowed in AdS$_ {d+2}$ space and describe the Hawking-Page phase transition between a thermal AdS background and an SAdS$_ {d+2}$ black hole. Let $\mathcal{M}$ be a Lorentzian manifold and $\partial\mathcal{M}$ a codimension one boundary. We consider the theory

$$\begin{equation} \mathcal{I} = -\frac{1}{8\pi G_ {d+2}} \times\left[\mathcal{S}_ {\mathcal{X}}+\mathcal{S}_ {\partial\mathcal{X}}\right]\end{equation}$$

where the bulk and the boundary actions are respectively

$$\begin{align} \mathcal{S}_ \mathcal{X} =& \frac{1}{2}\int_ \mathcal{X}\text{d}^{d+2}x\sqrt{-g} \left[R - 2\Lambda\right],\newline \mathcal{S}_ {\partial\mathcal{X}} =& \int_{\partial\mathcal{X}}\text{d}^{d+1}x\sqrt{-h}\left[\mathcal{K}+\frac{d+1}{\ell^2} + c_1(d) \mathcal{R} \right.\newline &+\left.c_2(d)\left(\mathcal{R}_ {ab}\mathcal{R}^{ab} - c_3(d)\mathcal{R}^2\right) +\cdots \right]\nonumber. \end{align}$$

The boundary action is necessary for a well-posed variational principle and to regularize the bulk action. Here $\Lambda = -d(d+1)/2\ell^2$, $R(g)$ is the scalar curvature, and $\mathcal{K}$ and $\mathcal{R}$ are the extrinsic and scalar curvatures of the boundary metric $h$ respectively. Note that higher curvature corrections to the boundary action appear in $d\ge 6$ and the coefficients $c_i(d)$ are sensitive to the dimension and the topology of the horizon. The equations of motion and the trace of the Einstein equations read

$$\begin{align*} R_{ab} - \frac{1}{2}g_{ab}R + \Lambda g_{ab} &= 0\newline \frac{d}{2}R + (d+2)\Lambda &=0 \end{align*}$$

from which the on-shell bulk action then simplifies considerably to a single volume integral

$$ \begin{equation} \mathcal{S}_ {\mathcal{X}} = \frac{1}{2}\int_{\mathcal{X}}\text{d}^{d+2}x\sqrt{-g}\left(\frac{4\Lambda}{d}\right)= \frac{2\Lambda}{d} \text{Vol}(\mathcal{X}). \end{equation}$$

Hence, the entire contribution of the bulk action, $\mathcal{I}$, to the thermodynamics is only dependent on the topological nature of the transverse directions to the horizon. Clearly, the bulk action is divergent since we are integrating over the entire space. The boundary action $\mathcal{S}_ {\partial\mathcal{X}}$ regulates this divergence rendering $\mathcal{I}$ finite.

Black Hole Saddles #

Let us apply this procedure to saddle points of the above action. First, we have to identify geometries which are regular after Wick rotation $\tau = i t$ to Euclidean space. The general solution, in static coordinates, is given by

$$\begin{align} \text{d} s^{2}&=-f(r)\text{d} t^{2}+\frac{\text{d} r^{2}}{f(r)}+\frac{r^{2}}{\ell^2}\text{d}\Sigma_{d,k}^{2} \newline f(r)&=k-\frac{m}{r^{(d-1)}}+\frac{r^{2}}{\ell ^{2}}. \end{align}$$

There are two relevant saddles; the Schwarzschild-AdS$_ {d+2}$ solution, $m > 0$, and the pure AdS$_ {d+2}$, $m=0$, solution. Here, the $d$-dimensional metric $\text{d}\Sigma_{d,k}$ corresponds to the various boundary geometries which respect the isometries of AdS$_ {d+2}$ spacetime. In particular,

$$\begin{equation} \text{d}\Sigma_{d,k}^2 = \begin{cases} \ell^2\text{d}\Omega_d^2 &\text{for}\hspace{.1in} k=1,\newline \sum\limits_{i=1}^d\text{d} x_i^2 &\text{for}\hspace{.1in} k=0,\newline \ell^2\text{d}\Xi_d^2 &\text{for}\hspace{.1in} k=-1, \end{cases} \end{equation}$$

where $\text{d}\Omega_d^2$ and $\text{d}\Xi_d^2$ are the unit metrics on $S^d$ and $H^d$. Thus $k=+1,0,-1$ represents spherical, planar, and hyperbolic boundaries respectively. We denote the outer horizon by $r_+$ which is located at the largest positive root of $f(r_+)=0$.

Black Hole Thermodynamics #

To compute the thermodynamics, we perform an analytic continuation of time into the Euclidean section. In order to avoid a conical singularity, we must have $$\tau \sim \tau + {4\pi}/{f’(r_+)}.$$ The temperature may be computed via the standard Euclidean trick

$$\begin{align} T(r_+)&=\frac{f^\prime(r_+)}{4\pi}=\frac{(d-1)k}{4 \pi r_+} +\frac{(d+1) r_+}{4 \pi \ell ^2} \end{align}$$

The remaining thermodynamic quantities are

$$\begin{align} \mathcal{I}^* &= \beta \frac{\sigma_{d,k}}{16\pi G_{d+2}} \left[r^{d-1}\left(k - \frac{r^{2}}{\ell^2}\right)\right] - \mathcal{I}_ {k}^0, \newline E &= \partial_\beta{\mathcal{I}} = \frac{d,\sigma_{d,k}}{16\pi G_{d+2}}m - E_{k}^0,\newline S &= \frac{A(\mathcal{H})}{4G_{d+2}} = \frac{\sigma_{d}r^{d}_ +}{4G_{d+2}} \end{align}$$

Hence, the parameter $m$ appearing in the metric function is related to the ADM mass of the black hole. Here, $\sigma_d$ is the volume of the unit $d$-dimensional boundary. Note that we may solve for $r_+(T)$ from $T(r_+)$.

For $k=+1,0$, the thermodynamics and the on-shell Euclidean action can either be calculated with the usual counterterm method or through a background subtraction technique. The background subtraction identifies the locally AdS$_ d$ space, i.e., $m=0$, as the reference ground state in which case $r_ {+,k}=0$. However, with $k=-1$, the locally AdS$_ d$ solution contains a bifurcate Killing horizon at $$r_c=\ell$$ at a fixed temperature. In this case, performing either a background subtraction or utilizing the counterterm method introduces a conical singularities. These singularities are removed when one subtracts an extremal saddle leading to the corrections $\mathcal{I}_ k^0$.

The classical and thermodynamic stability of these solutions have been rigorously explored in here, here, and here. The heat capacity for these solutions is

$$\begin{equation} C = \frac{d\sigma_{d,k}}{4G_{d+2}}r_+^d\left(\frac{(d+1)r_+^2 + (d-1)k\ell^2}{(d+1)r_+^2 - (d-1) k\ell^2}\right). \end{equation}$$

We see that, for non-positive $k$, the heat capacity is always non-negative indicating planar, and hyperbolic black holes are thermodynamically stable. For $k=-1$, the heat capacity vanishes precisely when $r_{+,c} = \sqrt{(d-1)/(d+1)}\ell.$