Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 38D (General Relativity)

For $k=-1$, the Friedmann-Lemaitre-Robertson-Walker metric isThe empty first Friedmann equation, with $\Lambda =0$, givesChoosing the expanding branch and shifting the origin of proper time yields

For an FLRW spacetime,With $a=t$ and $k=-1$, both terms vanish, soIndeed the vacuum equations give a vanishing Ricci tensor, and conformal flatness of an FLRW metric then makes the whole Riemann tensor vanish. The apparent singularity at $t=0$ is therefore only a coordinate singularity.

IntroduceThenwhich is the Minkowski metric. Since $t>0$,so these coordinates cover exactly the interior of the future light cone of its vertex $p$. The $t=\text{constant}$ slices are future hyperboloids. Their spatial isometries are precisely the proper orthochronous Lorentz transformations of Minkowski spacetime that fix $p$.