Black Holes and Thermodynamics

Abstract

Quantum mechanically, black holes have thermodynamic properties just like ordinary statistical systems. In this chapter, we explain the relation between black holes and thermodynamics using the example of the Schwarzschild black hole.

Appendix: Black Holes and Thermodynamic Relations \({}^\blacklozenge \)

We saw that black holes satisfy the first law. The first law should be always satisfied because the law simply represents the energy conservation. On the other hand, the other thermodynamic relations such as the Euler relation and the Gibbs-Duhem relation do not take standard forms for black holes with compact horizon. The story is different for black branes though: these relations take standard forms like standard thermodynamic systems.

For example, thermodynamic quantities of the Schwarzschild black hole are given by

$$\begin{aligned} T = \frac{1}{4\pi r_0}, \quad S = \frac{\pi r_0^2}{G_4}, \quad E = \frac{r_0}{2G_4}. \end{aligned}$$

(3.61)

So, the Euler relation does not take the form $E=\textit{TS}$, but

$$\begin{aligned} E=2\textit{TS}, \end{aligned}$$

(3.62)

which is known as the Smarr’s formula.

For black holes with compact horizon, thermodynamic relations do not take standard forms because these black holes do not satisfy fundamental postulates of thermodynamics (Sect. 9.2). In particular, thermodynamics requires that the entropy is additive over the subsystems. This requirement is unlikely to hold in the presence of a long-range force such as gravity. Another related problem is that the volume $V$ and its conjugate quantity, pressure $P$, do not appear in those black holes (at least as independent variables.)

The postulate in particular implies that the so-called fundamental relation of a standard thermodynamic system

$$\begin{aligned} S = S(E, V) \end{aligned}$$

(3.63)

is a homogeneous first order function of extensive variables:

$$\begin{aligned} S( \lambda E, \lambda V) = \lambda S(E, V). \end{aligned}$$

(3.64)

Then, the Euler relation and the Gibbs-Duhem relation follow from the fundamental relation. However, if one rewrites Eq. (3.61) in the form of the fundamental relation, one gets

$$\begin{aligned} S_{\text {BH}_4} = 4\pi G_4 E^2, \end{aligned}$$

(3.65)

which is a homogeneous second order function in $E$. Thus, it is no wonder that thermodynamic relations do not take standard forms.

However, the story is different for branes. For branes, one has an additional extensive variable, volume $V$, and the entropy is additive over the subsystems along the brane direction $V=V_A+V_B$. In this sense, the entropy of branes is additive, and branes satisfy the standard postulates of thermodynamics. A simple example is the neutral black string (3.41). In Eq. (3.44), we wrote the entropy of the black string in the form of the fundamental relation:

$$\begin{aligned} S_\text {string} = 4\pi G_5 \frac{E^2}{V_1}. \end{aligned}$$

(3.66)

This is indeed a homogeneous first order function in extensive variables. As a result, one can check that the black string does satisfy the standard thermodynamic relations $E = \textit{TS}-\textit{PV}_1$ and $\textit{SdT}-V_1\textit{dP}=0$. Note that Eqs. (3.65) and (3.66) are actually the same equation since $V_1/G_5=1/G_4$.