# New exact solutions for neutral and charged shear-free relativistic fluids.

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## Date

2022

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## Abstract

We study shear-free gravitating fluids in general relativity. We first analyse the integrability of the
Emden-Fowler equation that governs the behaviour of shear-free neutral perfect fluid distributions.
We find a new exact solution and generate a new first integral. The first integral is subject to an
integrability condition which can be expressed as a third order differential equation whose solution
can be expressed in terms of elementary functions and elliptic integrals. We extend this approach
to include the effect of the electromagnetic charge. The Einstein-Maxwell system for a charged
shear-free matter can be reduced to a generalized Emden-Fowler equation. We integrate this
equation and find a new first integral. For this solution to exist two integral equations arise as
integrability conditions. The integrability conditions can be solved to find new solutions. In both
cases the first integrals are given parametrically. Our investigations suggest that complexity of a
self-gravitating fluid is related to the existence of a first integral. For both neutral and charged
fluids the general form of the parametric solution depends on a cubic and quartic polynomial
respectively. The special case of repeated roots leads to simplification and this regains earlier
results. We also study relativistic charged shear-free gravitating fluids in higher dimensions. Two
classes of exact solutions to the Einstein-Maxwell equations are found. We obtain these solutions
by reducing the Einstein-Maxwell equations to a single second order nonlinear partial differential
equation containing two arbitrary functions. This generalizes the condition of pressure isotropy
to higher dimensions; the new condition is functionally different from four dimensions. The new
exact solutions obtained in higher dimensions reduce to known results in four dimensions. The
presence of higher dimensions affects the dynamics of relativistic fluids in general relativity.

## Description

Doctoral Degree. University of KwaZulu-Natal, Durban.