Intrinsic Conformal Symmetries in Szekeres models

TL;DR

This paper explores the intrinsic conformal symmetries of Szekeres models, revealing a 6-dimensional algebra of vector fields and explicitly deriving 7 proper ICVFs for the Lemaître-Tolman-Bondi subclass, enhancing understanding of their geometric structure.

Contribution

It introduces the intrinsic conformal algebra for Szekeres models and explicitly derives proper ICVFs for the LTB subclass, advancing geometric symmetry analysis.

Findings

Identified a 6-dimensional algebra of ICVFs in Szekeres models.

Derived 7 proper ICVFs for the LTB subclass.

Linked conformal flatness to higher-dimensional ICVF algebra.

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.