Past exam of the general relativity and cosmology course of the University of Oxford / 2013 / 1 / c / Solution

The Einstein tensor is $G^{\mu }{}_{\nu }=R^{\mu }{}_{\nu }-\frac{1}{2}{\delta }_{\nu }^{\mu }R$. Thus $G^0{}_0=-R_{00}-R/2$, where the time raising introduces the minus sign. Substitution cancels the $\Phi$ and time-derivative terms in that component and givesThe right sides follow from the Einstein field equations $G^{\mu }{}_{\nu }=(8\pi G/c^4)T^{\mu }{}_{\nu }$ with zero cosmological constant. The full equations also contain the momentum constraint $G^0{}_i=-2{\partial }_0{\partial }_i\Psi /c^2=0$ for dust at rest to this order. Specifying arbitrary time-dependent potentials without that constraint would not give a solution of all the Einstein field equations.