Entropy of extremal black holes: horizon limits through charged thin shells, a unified approach

TL;DR

This paper investigates the entropy of extremal black holes using a unified approach with charged thin shells, revealing different entropy behaviors depending on the limiting process, and clarifying the extremal black hole entropy debate.

Contribution

It introduces a unified method to analyze extremal black hole entropy via charged thin shells, exploring three different horizon limit procedures and their thermodynamic implications.

Findings

Entropy matches Bekenstein-Hawking in cases 1 and 2

Case 3 allows for arbitrary well-behaved entropy functions

Different limiting procedures yield distinct thermodynamic behaviors

Abstract

Using a unified approach we study the entropy of extremal black holes through the entropy of an electrically charged thin shell. We encounter three cases in which a shell can be taken to its own gravitational or horizon radius and become an extremal spacetime. In case 1, we use a non extremal shell, calculate all the thermodynamics quantities including the entropy, take it to the horizon radius, and then take the extremal limit. In case 2, we take the extremal limit and the horizon radius limit simultaneously, i.e., as the shell approaches its horizon radius it also approaches extremality. In case 3, we build first an extremal shell, and then take its horizon radius. We find that the thermodynamic quantities in general have different expressions in the three different cases. The entropy is the Bekenstein-Hawking entropy $ S=A_+/4$ (where $A_+$ is the horizon area) in cases 1 and 2, and in case 3 it can be any well-behaved function of $A_+$. The contributions from the various thermodynamic quantities for the entropy in all three cases are distinct. Indeed, in cases 1 and 2 the limits agree in what concerns the entropy but they disagree in the behavior of all other thermodynamic quantities. Cases 2 and 3 disagree in what concerns the entropy but agree in the behavior of the local temperature and electric potential. Case 2 is in a sense intermediate between cases 1 and 3. Our approach sheds light on the extremal black hole entropy issue.