In a spatially flat ( ), FRW spacetime with metric in comoving Cartesian coordinates

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Einstein's field equations · Einstein's field equations

The Geometry of Gravity, Ch FRW cosmology as an exact solution

Problem

In a spatially flat (K=0K=0), Λ=0\Lambda=0 FRW spacetime with metric ds2=−c2dt2+a(t)2δijdxidxjds^2=-c^2dt^2+a(t)^2\delta_{ij}dx^idx^j in comoving Cartesian coordinates, a comoving perfect fluid has energy density ρ(t)\rho(t), pressure p(t)p(t), four-velocity uμ=(1,0,0,0)u^\mu=(1,0,0,0) and Tμν=(ρ+p)uμuν/c2+pgμνT^{\mu\nu}=(\rho+p)u^\mu u^\nu/c^2+pg^{\mu\nu}. ((a)) Using the connection components Γ0ij=(aa˙/c2)δij\Gamma^0{}_{ij}=(a\dot a/c^2)\delta_{ij}, Γi0j=Hδji\Gamma^i{}_{0j}=H\delta^i_j (all others zero, H≡a˙/aH\equiv\dot a/a) and the fluid components T00=ρ/c2T^{00}=\rho/c^2, Tij=pδij/a2T^{ij}=p\delta^{ij}/a^2, T0i=0T^{0i}=0, expand ∇μTμ0=∂μTμ0+ΓμμλTλ0+Γ0μλTμλ=0\nabla_\mu T^{\mu0}=\partial_\mu T^{\mu0}+\Gamma^\mu{}_{\mu\lambda}T^{\lambda0}+\Gamma^0{}_{\mu\lambda}T^{\mu\lambda}=0 to obtain the fluid equation ρ˙+3H(ρ+p)=0\dot\rho+3H(\rho+p)=0. ((b)) Taking as given the two Friedmann equations for this universe, H2=8πGρ3c2H^2=\dfrac{8\pi G\rho}{3c^2} and a¨a=−4πG3c2(ρ+3p)\dfrac{\ddot a}{a}=-\dfrac{4\pi G}{3c^2}(\rho+3p), differentiate the first with respect to tt, use the second to eliminate H˙\dot H, and show the result is the same fluid equation as in part (a), so it carries no information beyond the two Friedmann equations. ((c)) For a fluid with constant equation-of-state parameter w≡p/ρ=1w\equiv p/\rho=1 (a 'stiff' fluid), integrate the fluid equation to find the power of aa by which ρ\rho scales.

Browse the problem library