Extremal black holes, gravitational entropy and nonstationary metric fields

TL;DR

This paper argues extremal black holes have zero entropy due to their time-independence, while non-extremal black holes' entropy arises from nonstationary interior regions where classical microstates are hidden.

Contribution

It explicitly characterizes the phase space of interior black hole regions and links entropy to ignorance of microstates in nonstationary areas, contrasting extremal and non-extremal cases.

Findings

Extremal black holes are time-independent with zero entropy.

Non-extremal black holes have a nonstationary interior phase space contributing to entropy.

Numerical simulations support that entropy originates from nonstationary regions near the horizon.

Abstract

We show that extremal black holes have zero entropy by pointing out a simple fact: they are time-independent throughout the spacetime and correspond to a single classical microstate. We show that non-extremal black holes, including the Schwarzschild black hole, contain a region hidden behind the event horizon where all their Killing vectors are spacelike. This region is nonstationary and the time $t$ labels a continuous set of classical microstates, the phase space $[\,h_{ab}(t), P^{ab}(t)\,]$, where $h_{ab}$ is a three-metric induced on a spacelike hypersurface $\Sigma_t$ and $P^{ab}$ is its momentum conjugate. We determine explicitly the phase space in the interior region of the Schwarzschild black hole. We identify its entropy as a measure of an outside observer's ignorance of the classical microstates in the interior since the parameter $t$ which labels the states lies anywhere between 0 and 2M. We provide numerical evidence from recent simulations of gravitational collapse in isotropic coordinates that the entropy of the Schwarzschild black hole stems from the region inside and near the event horizon where the metric fields are nonstationary; the rest of the spacetime, which is static, makes no contribution. Extremal black holes have an event horizon but in contrast to non-extremal black holes, their extended spacetimes do not possess a bifurcate Killing horizon. This is consistent with the fact that extremal black holes are time-independent and therefore have no distinct time-reverse.