# Constructing the CKVs of Bianchi III and V spacetimes

## TL;DR

This paper systematically determines the conformal algebra of Bianchi III and V spacetimes, identifying those admitting proper conformal Killing vectors and analyzing their implications for cosmological fluid dynamics and wave equations.

## Contribution

It introduces a stepwise algorithm to find CKVs in Bianchi III and V spacetimes, revealing only one such spacetime for each type admits a proper CKV.

## Key findings

- Identified unique Bianchi III and V spacetimes with maximum one proper CKV.
- Derived explicit conformal Killing vectors and factors for these spacetimes.
- Applied results to analyze observer kinematics, fluid dynamics, and wave symmetry generators.

## Abstract

We determine the conformal algebra of Bianchi III and Bianchi V spacetimes or, equivalently, we determine all Bianchi III and Bianchi V spacetimes which admit a proper conformal Killing vector. The algorithm that we use has been developed in Class. Quantum. Grav. 15, 2909 (1998) and concerns the computation of the CKVs of decomposable spacetimes. The main point of this method is that a decomposable space admits a CKV if the reduced space admits a gradient homothetic vector the latter being possible only if the reduced space is flat or a space of constant curvature. We apply this method in a stepwise manner starting from the two dimensional spacetime which admits an infinite number of CKVs and we construct step by step the Bianchi III and V spacetimes by assuming that CKVs survive as we increase the dimension of the space. We find that there is only one Bianchi III and one Bianchi V spacetime which admit at maximum one proper CKV. In each case we determine the conformal Killing vector and the corresponding conformal factor. As an application in the spacetimes we found we study the kinematics of the comoving observers and the dynamics of the corresponding cosmological fluid. As a second application we determine in these spacetimes generators of the Lie symmetries of the wave equation.