# Symmetry and Equivalence in Szekeres Models

## TL;DR

This paper classifies all Szekeres cosmological models with a single Killing vector, revealing symmetry types based on the curvature parameter and conditions like shell crossings, and explores their physical equivalence.

## Contribution

It provides a complete classification of Szekeres models with one symmetry and extends understanding of their equivalence transformations.

## Key findings

- Translational symmetries occur only in shell crossing quasi hyperboloidal models.
- Quasi planar models have no or full planar symmetry.
- Rotational symmetries are the only single symmetries in quasi spherical models.

## Abstract

We solve for all Szekeres metrics that have a single Killing vector. For quasi hyperboloidal ($\epsilon = -1$) metrics, we find that translational symmetries are possible, but only in metrics that have shell crossings somewhere, while metrics that can be made free of shell crossings only permit rotations. The quasi planar metrics ($\epsilon = 0$) either have no Killing vectors or they admit full planar symmetry. Single symmetries in quasi spherical metrics ($\epsilon = +1$) are all rotations. The rotations correspond to a known family of axially symmetric metrics, which for each $\epsilon$ value, are equivalent to each other. We consider Szekeres metrics in which the line of dipole extrema is required to be geodesic in the 3-space, and show the same set of families emerges. We investigate when two Szekeres metrics are physically equivalent, and complete a previous list of transformations of the arbitrary functions.