Volume of a rotating black hole in 2+1 dimensions

In this article we apply the technique for maximal volume estimation of a black hole developed by Christodoulou and Rovelli [1] for Schwarzschild black hole and by Zhang et al. [3] 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 hyper-surface for the rotating BTZ black hole using the Lagrangian formulation demonstrated in [1]. Further we use maximization technique illustrated earlier by Bengtsson et al. [4] for Kerr black hole to arrive at the similar result for our case. We argue that the maximum contribution to the volume of the hyper-surface 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 derived using independent considerations of vanishing extrinsic curvature. We show that the volume of this segment of the maximal hyper-surface, 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 hyper-surface for a near extremal black hole and show that it is proportional to the horizon entropy of the black hole. (C) 2022 The Author(s). Published by Elsevier B.V.