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Black Hole Thermodynamics in AdSd_ d

ยท5 mins

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

Z=Aโˆซ[Dg]eโˆ’I[g].\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 D[g]\mathcal{D}[g], and integrand eโˆ’I[g]e^{-\mathcal{I}[g]} of metrics gg 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 Z\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 I[g]โ‰ซ1\mathcal{I}[g]\gg1, the integrand eI[g]โ‰ช1e^{\mathcal{I}[g]}\ll1. Hence, the most dominant contributions are given by the minima of the function where we can approximate

I[g]=โˆ‘iI[gi]+12(gโˆ’gi)2ฮด2ฮดg2I[g]โˆฃg=gi+โ€ฆ\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

Zโ‰ˆAeโˆ’โˆ‘iI[gi]โˆซ[Dg]eโˆ‘i12(gโˆ’gi)2ฮด2ฮดg2I[g]โˆฃg=gi+โ€ฆ=eโˆ’โˆ‘iI[gi].\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., I[g]=ฮฒF=(1/T)F\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 AdSd_d #

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

I=โˆ’18ฯ€Gd+2ร—[SX+Sโˆ‚X]\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

SX=12โˆซXdd+2xโˆ’g[Rโˆ’2ฮ›],Sโˆ‚X=โˆซโˆ‚Xdd+1xโˆ’h[K+d+1โ„“2+c1(d)R+c2(d)(RabRabโˆ’c3(d)R2)+โ‹ฏโ€‰].\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 ฮ›=โˆ’d(d+1)/2โ„“2\Lambda = -d(d+1)/2\ell^2, R(g)R(g) is the scalar curvature, and K\mathcal{K} and R\mathcal{R} are the extrinsic and scalar curvatures of the boundary metric hh respectively. Note that higher curvature corrections to the boundary action appear in dโ‰ฅ6d\ge 6 and the coefficients ci(d)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

Rabโˆ’12gabR+ฮ›gab=0d2R+(d+2)ฮ›=0\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

SX=12โˆซXdd+2xโˆ’g(4ฮ›d)=2ฮ›dVol(X). \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, I\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 Sโˆ‚X\mathcal{S}_ {\partial\mathcal{X}} regulates this divergence rendering I\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 ฯ„=it\tau = i t to Euclidean space. The general solution, in static coordinates, is given by

ds2=โˆ’f(r)dt2+dr2f(r)+r2โ„“2dฮฃd,k2f(r)=kโˆ’mr(dโˆ’1)+r2โ„“2.\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-AdSd+2_ {d+2} solution, m>0m > 0, and the pure AdSd+2_ {d+2}, m=0m=0, solution. Here, the dd-dimensional metric dฮฃd,k\text{d}\Sigma_{d,k} corresponds to the various boundary geometries which respect the isometries of AdSd+2_ {d+2} spacetime. In particular,

dฮฃd,k2={โ„“2dฮฉd2fork=1,โˆ‘i=1ddxi2fork=0,โ„“2dฮžd2fork=โˆ’1,\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 dฮฉd2\text{d}\Omega_d^2 and dฮžd2\text{d}\Xi_d^2 are the unit metrics on SdS^d and HdH^d. Thus k=+1,0,โˆ’1k=+1,0,-1 represents spherical, planar, and hyperbolic boundaries respectively. We denote the outer horizon by r+r_+ which is located at the largest positive root of f(r+)=0f(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 ฯ„โˆผฯ„+4ฯ€/fโ€™(r+).\tau \sim \tau + {4\pi}/{fโ€™(r_+)}. The temperature may be computed via the standard Euclidean trick

T(r+)=fโ€ฒ(r+)4ฯ€=(dโˆ’1)k4ฯ€r++(d+1)r+4ฯ€โ„“2\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

Iโˆ—=ฮฒฯƒd,k16ฯ€Gd+2[rdโˆ’1(kโˆ’r2โ„“2)]โˆ’Ik0,E=โˆ‚ฮฒI=d,ฯƒd,k16ฯ€Gd+2mโˆ’Ek0,S=A(H)4Gd+2=ฯƒdr+d4Gd+2\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 mm appearing in the metric function is related to the ADM mass of the black hole. Here, ฯƒd\sigma_d is the volume of the unit dd-dimensional boundary. Note that we may solve for r+(T)r_+(T) from T(r+)T(r_+).

For k=+1,0k=+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 AdSd_ d space, i.e., m=0m=0, as the reference ground state in which case r+,k=0r_ {+,k}=0. However, with k=โˆ’1k=-1, the locally AdSd_ d solution contains a bifurcate Killing horizon at rc=โ„“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 Ik0\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

C=dฯƒd,k4Gd+2r+d((d+1)r+2+(dโˆ’1)kโ„“2(d+1)r+2โˆ’(dโˆ’1)kโ„“2).\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 kk, the heat capacity is always non-negative indicating planar, and hyperbolic black holes are thermodynamically stable. For k=โˆ’1k=-1, the heat capacity vanishes precisely when r+,c=(dโˆ’1)/(d+1)โ„“.r_{+,c} = \sqrt{(d-1)/(d+1)}\ell.

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