Intrinsic Conformal Symmetries in Szekeres models

Abstract

We show that Spatially Inhomogeneous (SI) and Irrotational dust models admit a \emph{6-dimensional algebra } of \emph{Intrinsic Conformal Vector Fields} (ICVFs) $\mathbf{X}_{\alpha }$ satisfying $p_{a}^{c}p_{b}^{d}\mathcal{L}_{\mathbf{X}_{\alpha }}p_{cd}=2\phi (\mathbf{X}_{\alpha })p_{ab}$ where $p_{ab}$ is the associated metric of the 2d distribution $\mathcal{X}$ normal to the fluid velocity $u^{a}$ and the radial unit spacelike vector field $x^{a}$. The Intrinsic Conformal (IC) algebra is determined for each of the curvature value $\epsilon $ that characterizes the structure of the screen space $\mathcal{X}$. In addition the conformal flatness of the hypersurfaces $\mathbf{u}=\mathbf{0}$ indicates the existence of a \emph{% 10-dimensional algebra} of ICVFs of the 3d metric $h_{ab}$. We illustrate this expectation and propose a method to derive them by giving explicitly the \emph{7 proper} ICVFs of the Lema\^{\i}tre-Tolman-Bondi (LTB) model which represents the simplest subclass within the Szekeres family.