Homogeneity of Riemannian space-times of Gödel type

Abstract

The conditions for space-time homogeneity of a Riemannian manifold with a G\"odel-type metric are examined. The Raychaudhuri-Thakurta necessary conditions for space-time homogeneity are shown to be also sufficient and to lead to five linearly independent Killing vectors. These vector fields are exhibited for the most general case and their algebra is examined. The irreducible set of isometrically independent space-time-homogeneous G\"odel-type metrics is shown to be given, in cylindrical coordinates, by $d{s}^{2}={[dt+(\frac{4\ensuremath{\Omega}}{{m}^{2}})sin{\mathrm{h}}^{2}(\frac{\mathrm{mr}}{2})d\ensuremath{\varphi}]}^{2}\ensuremath{-}(\frac{1}{{m}^{2}})sin{\mathrm{h}}^{2}(\mathrm{mr})d{\ensuremath{\varphi}}^{2}\ensuremath{-}d{r}^{2}\ensuremath{-}d{z}^{2}$, where $\ensuremath{\Omega}$ is the vorticity and $\ensuremath{-}\ensuremath{\infty}\ensuremath{\le}{m}^{2}\ensuremath{\le}+\ensuremath{\infty}$, ${m}^{2}=2{\ensuremath{\Omega}}^{2}$ corresponding to the G\"odel metric. Sources of Einstein's equations leading to these metrics as solutions are examined, and it is shown that the inclusion of a scalar field extends the previously known region of solutions $\ensuremath{-}\ensuremath{\infty}\ensuremath{\le}{m}^{2}\ensuremath{\le}2{\ensuremath{\Omega}}^{2}$ to $\ensuremath{-}\ensuremath{\infty}\ensuremath{\le}{m}^{2}\ensuremath{\le}4{\ensuremath{\Omega}}^{2}$. The problem of ambiguity of physical sources of the same metric and that of violation of causality in G\"odel-type space-time-homogeneous universes are examined. In the case ${m}^{2}=4{\ensuremath{\Omega}}^{2}$, we obtain the first exact G\"odel-type solution of Einstein's equations describing a completely causal space-time-homogeneous rotating universe.