Black hole mass and angular momentum in topologically massive gravity

TL;DR

This paper generalizes the Abbott-Deser-Tekin (ADT) construction to compute Killing charges in topologically massive gravity (TMG) around arbitrary backgrounds, enabling consistent definitions of mass and angular momentum for black holes with non-constant-curvature asymptotics. It shows how the ADT currents decompose into Einstein and Cotton contributions and provides a coordinate-invariant charge formula Q(ξ) = (1/κ) ∮ √|g| [F_E^{0i}(ξ) + (1/μ) F_C^{0i}(ξ)] dS_i. Applying the formalism to stationary rotationally symmetric spacetimes via a SL(2,R) reduction yields explicit expressions for mass and angular momentum in terms of perturbations of a superangular-momentum vector, and demonstrates that the ADT angular momentum matches the SAM result while the ADT mass receives specific corrections. For the ACL black hole of cosmological TMG, the authors obtain M_ADT = (8π μ)/(9 κ) β^2(1−β^2) ω and J = (4π μ)/(9 κ) β^2[(1−β^2) ω^2 − ((1+β^2)/(1−β^2)) ρ0^2], and show that the total entropy S = (8π^2)/(3 κ √(1−β^2))[(1+β^2)ρ0 + (1−β^2)ω], together with the Hawking temperature and horizon angular velocity, satisfies the first law dM = T_H dS + Ω_h dJ. The work thus validates the ADT approach for non-constant-curvature backgrounds and highlights its compatibility with black-hole thermodynamics, while suggesting extensions to other TMG-related black holes and theories of gravity.

Abstract

We extend the Abbott-Deser-Tekin approach to the computation of the Killing charge for a solution of topologically massive gravity (TMG) linearized around an arbitrary background. This is then applied to evaluate the mass and angular momentum of black hole solutions of TMG with non-constant curvature asymptotics. The resulting values, together with the appropriate black hole entropy, fit nicely into the first law of black hole thermodynamics.