Friedmann–Lemaître–Robertson–Walker metric

Example

Let $I \subseteq \R$ be an open interval, let $a \colon I \to (0, \infty)$ be a smooth function, and let $k \in \R$. Let $(\Sigma, \gamma_k)$ be the complete simply connected three-dimensional space form of constant sectional curvature $k$. The Friedmann–Lemaître–Robertson–Walker (FLRW) metric on $M = I \times \Sigma$ is $$ds^2 = -dt^2 + a(t)^2 \, \gamma_k,$$ in units with $c = 1$. In comoving coordinates $(\chi, \theta, \phi)$ on $\Sigma$ one has $$\gamma_k = d\chi^2 + S_k(\chi)^2 \left( d\theta^2 + \sin^2\theta \, d\phi^2 \right),$$ where $S_k(\chi) = \sin(\sqrt{k}\,\chi)/\sqrt{k}$ if $k > 0$, $S_k(\chi) = \chi$ if $k = 0$, and $S_k(\chi) = \sinh(\sqrt{-k}\,\chi)/\sqrt{-k}$ if $k < 0$.

Check that it is a Lorentzian metric: $\gamma_k$ is positive definite, so $g$ has signature $(-,+,+,+)$ at every point. Check the cosmological principle for $u = \partial_t$ and the slices $\Sigma_t = \set{t} \times \Sigma$: every isometry $\psi$ of the space form gives an isometry $(t, x) \mapsto (t, \psi(x))$ of $g$. The complete simply connected space form has isometries that move any point to any other point. It also has isometries that fix a point and rotate any tangent direction to any other one. So both homogeneity and isotropy hold.