Einstein static universe

Example

The Einstein static universe is a Friedmann–Lemaître–Robertson–Walker metric with positive spatial curvature. Its scale factor is a constant $a(t) = a_0 > 0$: $$ds^2 = -dt^2 + a_0^2 \left( d\chi^2 + \sin^2\chi \, d\Omega^2 \right), \qquad 0 \le \chi \le \pi,$$ on $\R \times S^3$, in units with $c = 1$. Here $d\Omega^2$ is the round metric on the unit sphere, and the curvature constant is $k = 1$. It is filled with pressureless dust of constant density $\rho > 0$ and has a cosmological constant $\Lambda$.

Check: with $\dot a = \ddot a = 0$ and $p = 0$, the second Friedmann equation gives $0 = -\tfrac{4\pi G}{3}\rho + \tfrac{\Lambda}{3}$, so $\Lambda = 4\pi G \rho$. The first gives $0 = \tfrac{8\pi G}{3}\rho - \tfrac{1}{a_0^2} + \tfrac{\Lambda}{3}$, so $\tfrac{1}{a_0^2} = \tfrac{8\pi G}{3}\rho + \tfrac{4\pi G}{3}\rho = 4\pi G \rho = \Lambda$. So the metric solves the Einstein equations with dust exactly when $\Lambda = 4\pi G\rho = 1/a_0^2$.