Abstract
A variant of the ADT method for the determination of gravitational charges as integrals at infinity is applied to ‘Chern–Simons-like’ theories of 3D gravity, and the result is used to find the mass and angular momentum of the BTZ black hole considered as a solution of a variety of massive 3D gravity field equations. The results agree with many obtained previously by other methods, including our own results for ‘Minimal Massive Gravity’, but they disagree with others, including recently reported results for ‘Exotic Massive Gravity’. We also find the central charges of the asymptotic conformal symmetry algebra for the generic 3D gravity model with AdS vacuum and discuss implications for black hole thermodynamics.
Highlights
In general relativity (GR), conserved ‘charges’ such as mass and angular momentum are generically expressible only as integrals at spatial infinity; the prototype is the Arnowitt– Deser–Misner (ADM) mass formula for an asymptotically-flat spacetime [1]
We find the central charges of the asymptotic conformal symmetry algebra for the generic 3D gravity model with anti-de Sitter (AdS) vacuum and discuss implications for black hole thermodynamics
We recover the results for Minimal Massive Gravity’ (MMG) reported in [19], and we find the mass and angular momentum of the BTZ black hole solution of exotic massive gravity’ (EMG)
Summary
In general relativity (GR), conserved ‘charges’ such as mass and angular momentum are generically expressible only as integrals at spatial infinity; the prototype is the Arnowitt– Deser–Misner (ADM) mass formula for an asymptotically-flat spacetime [1]. Our main result is a simple and general formula for conserved charges as line integrals at spatial infinity We use this formula to determine the mass and angular momentum of the BTZ black hole as a solution of a variety of 3D gravity theories. One advantage of the method used here to determine the asymptotic charges carried by the BTZ black hole is that we start from the most general possible linear coupling of the one-form fields (which include auxiliary fields for massive 3D gravity) to a generic 2-form source consistent with Noether identities This source plays no role in the final formula, as is the case for ADT but we do not encounter the problem of an inconsistent initial assumption. Our results are both internally consistent and consistent with the discussion in [12] for exotic 3D GR, but they disagree with some other recent results [34, 35]
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