The Birkhoff theorem for topologically massive gravity

We derive the general Σ2×S solution of topologically massive gravity in vacuum and in presence of a cosmological constant. The field equations reduce to three-dimensional Einstein equations and the solution has constant Ricci tensor. We briefly discuss the emergence of non-Ricci flat solutions when spin is introduced. Pac(s) numbers: 04.50.+h, 04.20.Cv, 04.20.Jb Keyword(s): Alternative theories of gravity, Fundamental Problems, Exact solutions ∗E-mail: [email protected]; web page: http://www.aei-potsdam.mpg.de/ ̃cavaglia 1 Three-dimensional gravity allows for the construction of a Chern-Simons-like term closely related to the four-dimensional Pontryagin invariant [1]. Adding this Chern-Simons term to the action and varying the latter w.r.t. the metric, one obtains field equations corresponding to Einstein equations plus a term proportional to a symmetric, traceless, covariantly constant tensor of third derivative order (Cotton tensor, Cμν) [1] Gν + 1 m Cν = λδ μ ν , (1) Cν = ǫ ∇σ ( Rνκ − 1 4 gνκR ) , (2) where ǫ is the completely antisymmetric Levi Civita tensor in three-dimensions. The parameter m in Eq. (1) has mass dimension (in Planck units) and we have allowed for the existence of a cosmological constant λ. This new theory is called topologically massive gravity (TMG) and has several surprising properties [1]. In particular, it has no unitarity or ghost problems and appears to be renormalizable. A quite large amount of attention has been devoted to the investigation of the fundamental properties of TMG. In particular, TMG allows for non-trivial (non Ricci flat) solutions in three dimensions [2]. In this context, the study of exact solutions of TMG plays a major role. (Quoting Deser [3]: “[. . . ] no-one has succeeded as yet in finding the simplest possible, ‘Schwarzschild’ solution to the nonlinear model, i.e., a circularly symmetric timeindependent (we don’t know if there’s a Birkhoff theorem) exterior geometry that obeys the [Eq. (1)].”) Having this in mind, the aim of this paper is the investigation of cylindrically symmetric solutions of TMG. Given the • Definition 1: A three-dimensional metric with topology Σ2 × S ds = γμν(x)dx dx − ρ(x)dφ , (3) where φ ∈ [0, 2π[ and γμν (μ, ν = 0, 1) is a generic hyperbolic two-dimensional metric with signature (1,−1), is static when it possesses a Killing vector ξ = ξ(x)∂μ (for the definition of staticity in two-dimensional dilaton gravity see [4]); • Definition 2: A solution of TMG equations is called locally Einstein trivial if the Cotton tensor vanishes identically when is evaluated on the solution; we prove the following • Theorem: The most general solution of Eq. (1) with topology Σ2 × S is static and locally Einstein trivial. 2 • Corollary: When spin is introduced, non-static and non-Einstein trivial solutions do appear. The above theorem can be seen as the Birkhoff theorem for TMG. As a consequence of the theorem, the most general solution of TMG with topology Σ2×S is locally flat for λ = 0 and locally De Sitter/Anti-de Sitter for λ 6= 0. As a result, TMG allows for the existence of BTZ black holes [5]. Conventions: Throughout the paper we use the signature (+,−,−) and Landau-Lifshits [6] conventions for the Ricci tensor. • Proof: Let us consider the metric (3). Since in two-dimensions any metric is locally conformally flat we choose conformal light-cone coordinates u, v on Σ2. Equation (3) is cast in the form ds = f(u, v)dudv− ρ(u, v)dφ , f(u, v) ≥ 0 . (4) Using (4) the Einstein tensor Gν and the Cotton tensor read Gν = 2 