Entropy of an extremal electrically charged thin shell and the extremal black hole

TL;DR

This paper investigates the entropy of extremal black holes using a thin shell model, finding that the entropy depends on the horizon radius and may range from zero to the Bekenstein-Hawking value.

Contribution

It introduces a model using extremally charged thin shells to derive the entropy of extremal black holes as a function of the horizon radius.

Findings

Entropy of extremal black holes is a function of the horizon radius.

Range of extremal black hole entropy is from 0 to A+/4.

Entropy approaches S(r_+) as the shell becomes a black hole.

Abstract

There is a debate as to what is the value of the the entropy $S$ of extremal black holes. There are approaches that yield zero entropy $S=0$, while there are others that yield the Bekenstein-Hawking entropy $S=A_+/4$, in Planck units. There are still other approaches that give that $S$ is proportional to $r_+$ or even that $S$ is a generic well-behaved function of $r_+$. Here $r_+$ is the black hole horizon radius and $A_+=4\pi r_+^2$ is its horizon area. Using a spherically symmetric thin matter shell with extremal electric charge, we find the entropy expression for the extremal thin shell spacetime. When the shell's radius approaches its own gravitational radius, and thus turns into an extremal black hole, we encounter that the entropy is $S=S(r_+)$, i.e., the entropy of an extremal black hole is a function of $r_+$ alone. We speculate that the range of values for an extremal black hole is $0\leq S(r_+) \leq A_+/4$.