Effects of charge on the interior volume of BTZ black holes

TL;DR

This paper extends the variational technique for estimating the interior volume of charged BTZ black holes, compares methods, and analyzes how charge affects volume growth and entropy behavior.

Contribution

It introduces an extension of the maximal volume estimation technique to charged BTZ black holes and compares analytical and numerical results for different charge values.

Findings

Volume grows monotonically with time.

Charge influences the volume estimation, especially at small Q.

Volume entropy differs from horizon entropy in functional form.

Abstract

In this article we extend the variational technique for maximal volume estimation of a black hole developed by Christodoulou and Rovelli (CR) to the case of a charged BTZ black hole in 2+1 dimensions. The technique involves a study of the equation of motion of a hypothetical particle moving in an auxiliary manifold defined by spacetime variables. We then compare this estimation with the volume computed using maximization method and extrinsic curvature method. The charge Q of the black hole appears as a log term in the metric and hence an analytical solution for the volume does not exist. So first we compute the steady state radius and the volume for limiting case when the charge Q is very small i.e. Q << 1 and then carry out a numerical analysis to solve for the volume for more generic values of the charge. We find that the volume grows monotonically with the advance time. We further investigate the functional behaviour of the entropy of a massless scalar field living on the maximal hypersurface of a near extremal black hole. We show that this volume entropy exhibits a very different functional form compared to the horizon entropy.