Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 38A (General Relativity)

$G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }$. The contracted Bianchi identity ${\nabla }^{\mu }{G}_{\mu \nu }=0$ is required to match ${\nabla }^{\mu }{T}_{\mu \nu }=0$.

Computing $R=6(\ddot{a}/a+({\dot{a}}^2+k)/a^2)$ givesThus $\alpha =8\pi \rho /3$, $\beta =-k$, and $\gamma =-4\pi (\rho +3p)/3$ in the stated notation.

Staticity requires ${\rho }_0+3p_0=0$, so ${\rho }_m=\Lambda /(4\pi )$. The first Friedmann equation then requires $k=1$ and $a_0={\Lambda }^{-1/2}$.

Linearizing the constraint gives $\delta \rho =-2{\rho }_0\delta a/a_0$. Since $\delta p=\delta \rho /3$, the acceleration equation gives $\delta \ddot{a}=2\Lambda \delta a$. Hence perturbations contain $e^{\sqrt{2\Lambda }t}$: the Einstein static universe is unstable.