Feynman Path Integral Formulation

318 9 Scale Dependent Gravitational Couplings( ) 1 {✷Tμν =rr 2 [ρ(t)+p(t)] ȧ(t) 21 − kr 2 − 3ṗ(t)a(t)ȧ(t) − ¨p(t)a(t) 2}(✷Tμν)θθ = r2 (1 − kr 2 ) ( )✷T μνrr(✷Tμν)ϕϕ = r2 (1 − kr 2 ) sin 2 θ ( ✷T μν)rr, (9.68)with the remaining components equal to zero. Note that a non-vanishing pressurecontribution is generated in the effective field equations, even if one assumesinitially a pressureless fluid, p(t) =0. As before, repeated applications of thed’Alembertian ✷ to the above expressions leads to rapidly escalating complexity,which can only be tamed by introducing some further simplifying assumptions.In the following we will therefore assume that T μν has the perfect fluid form appropriatefor non-relativistic matter, with a power law behavior for the density,ρ(t) =ρ 0 t β , and p(t) =0. Thus all components of T μν vanish in the fluid’s restframe, except the tt one, which is simply ρ(t). Setting k = 0 and a(t) =r 0 t α onethen finds(✷Tμν =)tt ( 6α 2 − β 2 − 3αβ+ β ) ρ 0 t β−2( )✷Tμν = 2r 2 rr 0 t 2α α 2 ρ 0 t β−2 , (9.69)which again shows that the tt and rr components get mixed by the action of the ✷operator, and that a non-vanishing rr component gets generated, even though it wasnot originally present.Higher powers of the d’Alembertian ✷ acting on T μν can then be computed aswell. But a comparison with the left hand (gravitational) side of the effective fieldequation, which always behaves like ∼ 1/t 2 for k = 0, shows that in fact a solutioncan only be achieved at order ✷ n provided the exponent β satisfies β = −2 + 2n,orβ = −2 − 1/ν , (9.70)as was found previously from the trace equation, Eqs. (9.47) and (9.62). As a resultone obtains a much simpler set of expressions, which for general n read(✷ n T μν)tt→ c tt (α,ν)ρ 0 t −2 , (9.71)for the tt component, and similarly for the rr component(✷ n T μν)rr → c rr(α,ν)r 2 0 t 2α ρ 0 t −2 . (9.72)But remarkably one finds for the two coefficients the simple identityc rr (α,ν) = 1 3 c tt(α,ν) , (9.73)as well as c θθ = r 2 c rr and c ϕϕ = r 2 sin 2 θ c rr . The identity c rr = 1 3 c tt implies, fromthe consistency of the tt and rr effective field equations at large times,α = 1 2 . (9.74)