Gravitational entropy of Kerr black holes

Abstract

Classical invariants of General Relativity can be used to approximate the entropy of the gravitational field. In this work, we study two proposed estimators based on scalars constructed out from the Weyl tensor, in Kerr spacetime. In order to evaluate Clifton, Ellis and Tavakol’s proposal, we calculate the gravitational energy density, gravitational temperature, and gravitational entropy of the Kerr spacetime. We find that in the frame we consider, Clifton et al.’s estimator does not reproduce the Bekenstein–Hawking entropy of a Kerr black hole. The results are compared with previous estimates obtained by the authors using the Rudjord–Gr$\varnothing$n–Hervik approach. We conclude that the latter represents better the expected behaviour of the gravitational entropy of black holes.

Appendix

1.1 Definition of improper integral

Let $\Omega \subseteq \mathfrak {R}^{p}$ be a non-compact domain, and let $f: \Omega \rightarrow \mathfrak {R}$ be integrable on each measurable compact domain $D \subset \Omega$. We say that $f$ is improperly integrable on $\Omega$ iff for every increasing sequence of measurable compact domains, $\left( D_{\mathrm{n}}\right) _{n \in \mathbb {N}}$, which is exhausting⁴ $\Omega$, the secuence:

$$\begin{aligned} \sum \limits _\mathrm{n} = \left( \int _{D_\mathrm{n}} f \ d\mu \right) _\mathrm{n \in \mathbb {N}} \end{aligned}$$

is convergent. In such a case we note:

$$\begin{aligned} \lim _{\mathrm{n} \rightarrow \infty } \int _{D_\mathrm{n}} f \ d\mu = \int _{\Omega } f \ d\mu , \end{aligned}$$

and we call it improper integral of $f$ on $D$. Alternatively we say that the integral of $f$ on $\Omega$ is convergent [26].

1.2 Proof of the divergence of the improper integral:

$$\begin{aligned} I = \int \int _{D} \frac{1}{h_{i}(r)} d\theta \; dr = \int \int _{D} \frac{1}{\gamma _{i} \left( r_{*}-r\right) } \ \ \ \ \ \ \ \ \gamma _{i} \in \mathfrak {R}^{+}, \end{aligned}$$

(57)

where

$$\begin{aligned} D = \left\{ \left( r,\theta \right) \in \mathfrak {R}^{2} / \ r_{-} \le r \le r_{*} \ \wedge \ \theta _{*1} \le \theta \le \theta _{*2} \right\} , \end{aligned}$$

(58)

Since $h_{i}(r)$ does not depend on $\theta$, we can simply integrate (57) on this coordinate:

$$\begin{aligned} \int _{R} \frac{\theta _{*2}-\theta _{*1}}{\gamma _{i} \left( r_{*}-r\right) } dr, \end{aligned}$$

(59)

where,

$$\begin{aligned} R = \left\{ r \in \mathfrak {R}/ r_{-} \le r < r_{*}\right\} . \end{aligned}$$

(60)

The function $1/\gamma _{i} \left( r_{*}-r\right)$ diverges at $r = r*$. We define a subsequence of closed subregions $R_\mathrm{n}$ where the latter integral is well-defined:

$$\begin{aligned} R_{n} = \left\{ r \in \mathfrak {R}/ r_{-} \le r < r_{*}-\left( 1/n\right) \right\} , \end{aligned}$$

(61)

with $n \in {\mathbb N}$, such that

$$\begin{aligned} \lim _{n \rightarrow \infty } R_{n} \rightarrow R. \end{aligned}$$

(62)

We integrate the function $1 = 1/h_{i}(r)$ over the domain $R_{n}$ as follows:

$$\begin{aligned} \int _{R_{n}} \frac{dr}{h_{i}(r)}&= \int _{R_{n}} \frac{dr}{\gamma _{i} \left( r_{*}-r\right) } = \frac{-1}{\gamma _{i}} \ln {\left( r_{*}-r\right) } \ \vert _{r_{-}}^{r_{*}-\left( 1/n\right) }\nonumber \\&= \frac{-1}{\gamma _{i}} \left[ \ln \left( r_{*}-r_{*}+(1/n)\right) -\ln \left( r_{*}-r_{-}\right) \right] . \end{aligned}$$

(63)

The limit of the latter equation when $n \rightarrow \infty$ does not exist. Therefore, the integral given by Eq. (57) is divergent.