Volume of a rotating black hole in 2+1 dimensions

TL;DR

This paper extends the maximal volume estimation technique to rotating 2+1 dimensional black holes, deriving the hypersurface equation, identifying the steady state radius, and relating the hypersurface volume to black hole parameters and entropy.

Contribution

It develops a method to compute the maximal hypersurface volume for rotating BTZ black holes, incorporating angular momentum and AdS scale effects, which was not previously done.

Findings

Maximal hypersurface volume depends on mass, AdS scale, and angular momentum.

Steady state radius is determined by vanishing extrinsic curvature.

Scalar field entropy on the hypersurface is proportional to horizon entropy.

Abstract

In this article we apply the technique for maximal volume estimation of a black hole developed by Christodoulou and Rovelli for Schwarzchild blackhole and by Zhang et al for non rotating BTZ black hole, to the case of a rotating black hole in 2+1 dimensions. We derive the equation of the maximal hypersurface for the rotating BTZ blackhole using the Lagrangian formulation demonstrated by Christodoulou and Rovelli . Further we use maximization technique illustrated earlier by Bengtsson et al for Kerr black hole to arrive at the similar result for our case. We argue that the maximum contribution to the volume of the hypersurface comes from what we call the steady state radius, which we show depends on mass M and the AdS length scale. We demonstrate that this steady state radius can be arrived at using independent considerations of vanishing extrinsic curvature. We show that the volume of this segment of the maximal hypersurface, the CR volume, depends on mass, AdS length scale and angular momentum J. We further compute the entropy of a scalar field living on the maximal hypersurface for a near extremal black hole and show that it is proportional to the horizon entropy of the black hole.