Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 4 / d / Solution

In the orthonormal frame, the comoving perfect fluid in general relativity has $T_{00}=\rho$ and $T_{ij}=p{\delta }_{ij}$. The Einstein field equations with cosmological constant therefore giveThusChoosinggives $p=0$ and $\rho =1/(16\pi )>0$. The metric is then the Einstein static universe supported by pressureless matter and positive vacuum energy.