Lie symmetries for equations in conformal geometries

TL;DR

This paper uses Lie symmetry analysis to find new exact solutions to Einstein's field equations in conformally related Petrov type D spacetimes, expanding the set of known solutions with explicit elementary and Bessel function forms.

Contribution

It introduces a novel application of Lie symmetries to generate explicit solutions for conformally related Einstein spacetimes, including cases with elementary and Bessel function solutions.

Findings

Derived new group invariant solutions for Petrov type D spacetimes.

Solved the master field equation explicitly in three cases.

Obtained special solutions involving Bessel functions.

Abstract

We seek exact solutions to the Einstein field equations which arise when two spacetime geometries are conformally related. Whilst this is a simple method to generate new solutions to the field equations, very few such examples have been found in practice. We use the method of Lie analysis of differential equations to obtain new group invariant solutions to conformally related Petrov type D spacetimes. Four cases arise depending on the nature of the Lie symmetry generator. In three cases we are in a position to solve the master field equation in terms of elementary functions. In the fourth case special solutions in terms of Bessel functions are obtained. These solutions contain known models as special cases.