Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 309 / 4 / d / Solution

Contraction in the orthonormal frame yieldsHenceFor $u^{\mu }=(1,0,0)$, pressureless matter has $T_{00}=\rho$ and vanishing spatial components. The spatial Einstein equations $G_{ii}+\Lambda {\eta }_{ii}=0$ give $\Lambda =-1/2$, while the time component gives $1=8\pi \rho$. Thus, for $G_N=1$,If the convention is $G_{ab}+\Lambda g_{ab}=T_{ab}$, the corresponding density is simply $\rho =1$.