Notes on Spinning AdS_3 Black Hole Solution

TL;DR

This paper derives the rotating AdS_3 black hole solution from the nonrotating BTZ solution using Newman’s method, and analyzes the physical meaning of different coordinate systems, revealing the nature of the BTZ time coordinate.

Contribution

It introduces a new derivation of the rotating BTZ black hole solution and clarifies the physical interpretation of its coordinate systems.

Findings

The rotating BTZ solution can be obtained from the nonrotating one via Newman’s method.

Different coordinate systems have distinct physical meanings related to observer rotation.

The BTZ time coordinate corresponds to an observer rotating opposite to the black hole’s spin.

Abstract

By applying Newman's method, the AdS_3 rotating black hole solution is "derived" from the nonrotating black hole solution of Banados, Teitelboim and Zanelli (BTZ). The rotating BTZ solution derived in this fashion is given in "Boyer-Lindquist-type" coordinates whereas the form of the solution originally given by BTZ is given in a kind of an "unfamiliar" coordinates which are related to each other by a transformation of time coordinate alone. The relative physical meaning between these two coordinates is carefully studied by evaluating angular momentum per unit mass, angular velocity, surface gravity and area of the event horizon in two alternative coordinates respectively. The result of this study leads us to the conclusion that the BTZ time coordinate must be the time coordinate of an observer who rotates around the axis of the spinning hole in opposite direction to that of the hole outside its static limit.