(ed.). Gravitational waves (IOP, 2001)(422s).

Appendix C. The string cosmology equations 329exterior differential forms, as this may simplify the variational procedure (see, forinstance, [88]). Here, we will follow however the more traditional approach, byvarying the action with respect to g µν ,φ and B µν . We shall take into account thedynamical stress tensor T µν of the matter sources (defined in the usual way), aswell as the scalar source σ representing a possible direct coupling of the dilatonto the matter fields:δ g ( √ −gL m ) = 1 2√ −gTµν δg µν , δ φ ( √ −gL m ) = √ −gσδφ. (16.186)We start performing the variation with respect to the metric, using thestandard, general relativistic results:δ √ −g =− 1 2√ −ggµν δg µν ,δ( √ −gR) = √ −g(G µν δg µν + g µν ∇ 2 δg µν −∇ µ ∇ ν δg µν ), (16.187)where G µν is the usual Einstein tensor. It must be noted, however, that the secondcovariant derivatives of δg µν , when integrated by parts, are no longer equivalentto a divergence (and then to a surface integral), because of the dilaton factorexp(−φ) in front of the Einstein action, which adds dilatonic gradients to thefull variation. By performing a first integration by part, and using the metricitycondition ∇ α g µν = 0, we get in fact:δ g S = 1 2∫d d+1 x √ |g|T µν δg µν − 12λ d−1 s∫d d+1 x √ |g|e −φ× [G µν +∇ α φg µν ∇ α −∇ µ φ∇ ν +∇ µ φ∇ ν φ − 1 2 g µν(∇φ) 2− 1 2 g µνV (φ) + 1 2 g µν 12 1 H αβγ 2 − 12 3 H µαβ H αβ ν ]δg µν− 1 ∫2λ d−1 d d+1 x √ |g|∇ α [e −φ g µν ∇ α δg µν − e −φ ∇ ν δg να ] = 0.s(16.188)A second integration by parts of ∇δg µν cancels the bilinear term ∇ µ φ∇ ν φ, andleads to the field equations:G µν +∇ µ ∇ ν φ + 1 2 g µν[(∇φ) 2 − 2∇ 2 φ − V (φ) + 112 H 2 αβγ ] − 1 4 H µαβ H ναβ= 1 2 eφ T µν . (16.189)We have chosen units such that 2λ d−1s = 1, so that e φ represents the (d + 1)-dimensional gravitational constant (see appendix A). Also, we have implicitlyadded to the action the boundary term12λ d−1 s∫∂√|g|e −φ K α d α , (16.190)