Spatially inhomogeneous and irrotational geometries admitting intrinsic conformal symmetries

Abstract

"Diagonal" spatially inhomogeneous (SI) models are introduced under the assumption of the existence of (proper) intrinsic symmetries and can be seen, in some sense, complementary to the Szekeres models. The structure of this class of spacetimes can be regarded as a generalization of the (twist-free) Locally Rotationally Symmetric (LRS) geometries without any global isometry containing, however, these models as special cases. We consider geometries where a six-dimensional algebra $\mathcal{IC}$ of Intrinsic Conformal Vector Fields (ICVFs) exists acting on a $2-$dimensional (pseudo)-Riemannian manifold. Its members $\mathbf{X}_{\alpha }$, constituted of 3 Intrinsic Killing Vector Fields (IKVFs) and 3 \emph{proper} and \emph{gradient} ICVFs, as well as the specific form of the gravitational field are given explicitly. An interesting consequence, in contrast with the Szekeres models, is the immediate existence of \emph{conserved quantities along null geodesics}. We check computationally that the magnetic part $H_{ab}$ of the Weyl tensor vanishes whereas the shear $\sigma_{ab}$ and the electric part $E_{ab}$ share a common eigenframe irrespective of the fluid interpretation of the models. A side result is the fact that the spacetimes are foliated by a set of \emph{conformally flat } $3-$dimensional \emph{timelike} slices when the anisotropy of the \emph{flux-free} fluid is described only in terms of the 3 \emph{principal inhomogeneous "pressures"} $p_{\alpha}$ or equivalently when the Ricci tensor shares the same basis of eigenvectors with $\sigma_{ab}$ and $E_{ab}$. The conformal flatness also indicates that a \emph{10-dimensional algebra} of ICVFs $\mathbf{\Xi}$ acting on the $3-$dimensional timelike slices is highly possible to exist enriching in that way the set of conserved quantities admitted by the SI models found in the present paper.