Browsing by Author "Gumede, Sfundo Cebolenkosi."

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First integrals for spherically symmetric shear-free perfect fluid distributions.
(2018) Gumede, Sfundo Cebolenkosi.; Maharaj, Sunil Dutt.; Goswami, Rituparno.
In this dissertation we study spherically symmetric shear-free spacetimes. In particular we analyse the integrability of and find exact solutions to the Emden- Fowler equation yxx = f(x)y2; which is the master equation governing the behaviour of shear-free neutral perfect fluid distributions. We first review the study of Maharaj et al (1996) by finding a first integral to this master equation. This first integral is subject to the integrability condition which we use to find restrictions on the function f(x): We show that this first integral is a generalisation of particular solutions obtained by Stephani (1983) and Srivastava (1987). Furthermore, we use a similar method to obtain a new first integral of the master equation. This is achieved by multiplying the Emden-Fowler equation by an integrating factor. We then study the integrability condition, which is an integral equation, related to the new first integral. We find that the integrability condition can be written as a third order differential equation whose solution can be expressed in terms of elementary functions and elliptic integrals. In general the solution of the integrability condition is given parametrically. We believe that this is a new result. A particular form of f(x) is identified which corresponds to repeated roots of a cubic equation giving an explicit solution.
New exact solutions for neutral and charged shear-free relativistic fluids.
(2022) Gumede, Sfundo Cebolenkosi.; Maharaj, Sunil Dutt.; Govinder, Keshlan Sathasiva.
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.