Interior volume of the charged BTZ black holes

In Schwarzschild spacetime, Reinhart (1973) has shown the hypersurface $r_{ss}=3M/2$ (the subscript stands for “steady-state”) to be the maximal hypersurface. This steady-state radius $r_{ss}$ plays a crucial role in defining and evaluating the interior volume of a black hole. In this article, we investigate various methods to compute the maximal interior volume of a charged BTZ black hole. We find that the presence of charge Q in a black hole introduces a “log” term in the metric as a result of which, an analytical solution for the volume does not exist. So we first compute the volume of the black hole for the limiting case when the charge Q is very small (i.e., $Q\ll 1$: Q is a dimensionless parameter in (2+1) dimensions) 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 v. We further investigate the functional behavior 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 from the horizon entropy.