Physics Reports Chern–Simons modified general relativity

S. Alexander, N. Yunes / Physics Reports 480 (2009) 1–55 5where here Ɣ is the Christoffel connection. One can now integrate S CS by parts to obtain∫S CS = α ( ϑ K a) | ∂V − α 2Vd 4 x √ −g (∇ a ϑ) K a , (10)where the first term is usually discarded since it is evaluated on the boundary of the manifold. 3 The second term clearlydepends on the covariant derivative of ϑ, which vanishes if ϑ = const. and, in that case, CS modified gravity reduces to GR.For any finite, yet arbitrarily small ∇ a ϑ, CS modified gravity becomes substantially different from GR. The quantity ∇ a ϑcan be thought of as an embedding coordinate, because it embeds a generalization of the standard 3-dimensional CS theoryinto a 4-dimensional spacetime. In this sense, ∇ a ϑ and ∇ a ∇ b ϑ act as deformation parameters in the phase space of alltheories. One can then picture GR as a stable fixed point in this phase space. Away from this ‘‘saddle point’’, CS modifiedgravity induces corrections to the Einstein equations that are proportional to the steepness of the ϑ deformation parameter.The equations of motion of CS modified gravity can be obtained by variation of the action in Eq. (1). Exploiting the wellknownrelationsandδR b acd = ∇ c δƔ b ad − ∇ dδƔ b ac(11)δƔ b ac = 1 2 g bd (∇ a δg dc + ∇ c δg ad − ∇ d δg ac ) , (12)one finds∫δS = κ√ (d 4 x −g R ab − 1V2 g abR + α κ C ab − 1 )2κ T ab δg ab∫+√ [ ]αd 4 ∗ dVx −g R R + β□ϑ − β δϑ + Σ EH + Σ CS + Σ ϑ (13)4dϑVwhere □ := g ab ∇ a ∇ b is the D’Alembertian operator and T ab is the total stress–energy tensor, defined viaT ab = −√ 2 ( )δLmat+ δLϑ, (14)−g δg ab δg abwhere L ϑ is the Lagrangian density of the scalar field action, i.e. the integrand equation (4) divided by √ −g, such thatS ϑ = ∫ V L ϑ d 4 x. Thus, the total stress–energy tensor can be split into external matter contributions T abmatand a scalar fieldcontribution, which is explicitly given byT ϑ = β ab[(∇ a ϑ) (∇ b ϑ) − 1 2 g ab (∇ a ϑ) ( ∇ a ϑ ) ]− g ab V(ϑ) . (15)The tensor C ab that appears in Eq. (13) is a 4-dimensional generalization of the 3-dimensional Cotton–York tensor, whichin order to distinguish it from the latter we shall call the C-tensor. 4 This quantity is given bywhereC ab := v c ɛ cde(a ∇ e R b) d + v cd∗ R d(ab)c , (16)v a := ∇ a ϑ, v ab := ∇ a ∇ b ϑ = ∇ (a ∇ b) ϑ (17)are the velocity and covariant acceleration of ϑ, respectively. The last line of Eq. (13) represents surface terms that arise dueto repeated integrations by parts. Such terms play an interesting role for the thermodynamics of black hole solutions, whichwe shall review in Section 2.5.The vanishing of Eq. (13) leads to the equations of motion of CS modified gravity. The equations of motion for the metricdegrees of freedom (the modified field equations) are simplyG ab + α κ C ab = 12κ T ab, (18)where G ab = R ab − 1 g 2 abR is the Einstein tensor. The trace-reversed form of the modified field equationsR ab + α κ C ab = 1 (T ab − 1 )2κ 2 g abT , (19)3 The implications of discarding this boundary term will be discussed in Section 2.5.4 In the original work of [22], the C-tensor was incorrectly called ‘‘Cotton tensor’’, but the concept of a higher-dimensional Cotton–York tensor alreadyexists [38] and differs from the definition of Eq. (16).