Energy in Topologically Massive Gravity

We define conserved gravitational charges in -cosmologically extendedtopologically massive gravity , exhibit them in surface integral form about their de-Sitter or flat vacua and verify their correctness in terms of two basic types of solution. e-mail: deser,[email protected] Address after July 15, Physics Department, Middle East Technical University, 06531 Ankara, Turkey. Recently [1], we defined and computed the conserved charges (particularly energy) for generic higher curvature gravity models in surface integral, flux, form about the relevant, asymptotically constant or zero curvature vacua. Our work involved the formalism of [2] about asymptotically (Anti)-de-Sitter (AdS) backgrounds in cosmological Einstein gravity. Here, we define and compute the energy in the rather different context of topologically massive gravity (TMG) [3] extended to include a cosmological term [4] [Energy was defined in the original, Λ = 0, TMG [3], but not in its surface integral form in terms of generic background Killing vectors.] We begin directly with the vacuum equations Rμν − 1 2 gμνR+ Λgμν + 1 μ Cμν = 0, (1) C ≡ 1 √−g ǫ ∇α(R β − 1 4 δ βR). (2) The Cotton tensor C is symmetric, traceless and identically conserved, while the parameter μ is the mass of the linearized TMG excitations at Λ = 0. C , being the D = 3 conformal curvature tensor, vanishes for any Einstein space, including all external solutions of the cosmological Einstein equations, such as AdS, Schwarzschild-dS and BTZ black holes. More characteristic are geometries that obey the full TMG equations but not their Einstein part alone, some of which are given in [5, 6, 7], but these spaces do not correspond to bounded distributions. There is as yet no known“Schwarzschild ” solution, let alone more complicated asymptotically de-Sitter or flat (for Λ = 0 ) ones. What has been solved explicitly is the linearized metric generated by massive spinning interiors in the Λ = 0 sector [8], which will prove useful in testing our asymptotic expressions for the conserved generators. We consider asymptotically AdS metrics 3 gμν ≡ ḡμν + hμν , (3) where hμν is a (finite) deviation about the background ḡμν that obeys R̄μλνβ = Λ(ḡμν ḡλβ − ḡμβ ḡλν), R̄μν = 2Λḡμν , R̄ = 6Λ. (4) Our conventions are: signature (−, +, +), [∇μ,∇ν ]Vλ = Rμνλ Vσ, Rμν ≡ Rμλν . 2 Next, we expand the field equations (1) about ḡμν ; as usual, the nonlinear part is the energy momentum tensor (including matter if present): G μν + 1 μ C μν ≡ κTμν , (5) where the linear cosmological Einstein and Cotton terms read respectively, G μν ≡ R μν − 1 2 ḡμνRL − 2Λhμν , (6) CL μν = 1 √−ḡ ǫ μαβ ḡβσ∇α { RL σν − 2Λh − 1 4 ḡRL } . (7) The linear part of the Ricci tensor is R μν = 1 2 {−2hμν −∇μ∇νh+∇∇νhσμ +∇∇μhσν} . (8) Both (6) and (7) are conserved, symmetric, and ḡC μν = 0 . To every background Killing vector ξ̄ corresponds a (background) conserved charge Q(ξ̄) = ∫ M dx √−ḡT μν ξ̄ν = ∫ M dx √−ḡ { GL μν ξ̄ν + 1 μ CL μν ξ̄ν } (9) ≡ Q E +QμC. Next we express (9) as a 1-dimensional surface integral on the boundary. In the second paper of [1], we gave a detailed account of how this is done for the Einstein part. Here we simply quote that result and move on to the Cotton part. Q(ξ̄) E = 1 8πG ∫