Noether and Lie symmetries for charged perfect fluids

TL;DR

This paper analyzes the symmetries of a key differential equation governing spherically symmetric charged fluids in general relativity, deriving conditions for integrability and unifying previous results.

Contribution

It provides a comprehensive symmetry analysis, including Noether and Lie symmetries, for the charged fluid equation, extending earlier uncharged cases and identifying conditions for reduction to quadratures.

Findings

Derived a general Noether first integral.

Identified conditions for Lie symmetry existence.

Unified previous results for uncharged fluids.

Abstract

We study the underlying nonlinear partial differential equation that governs the behaviour of spherically symmetric charged fluids in general relativity. We investigate the conditions for the equation to admit a first integral or be reduced to quadratures using symmetry methods for differential equations. A general Noether first integral is found. We also undertake a comprehensive group analysis of the underlying equation using Lie point symmetries. The existence of a Lie symmetry is subject to solving an integro-differential equation in general; we investigate the conditions under which it can be reduced to quadratures. Earlier results for uncharged fluids and particular first integrals for charged matter are regained as special cases of our treatment.