Applied Mathematics

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Dynamics of radiating stars in the strong gravity regime.
(2017) Brassel, Byron Perry.; Maharaj, Sunil Dutt.; Goswami, Rituparno.
We model the dynamics of a spherically symmetric radiating dynamical star, emitting outgoing null radiation, with three spacetime regions. The local internal atmosphere is a two-component system consisting of standard pressure-free, null radiation and an additional string fluid with energy density and nonzero pressure obeying all physically realistic energy conditions. The middle region is purely radiative which matches to a third region which is the Schwarzschild exterior. A large family of solutions to the field equations are presented for various realistic equations of state. A comparison of our solutions with earlier well known results is undertaken and we show that all these solutions, including those of Husain, are contained in our family. We then generalise our class of solutions to higher dimensions and consider the effects of diffusive transport We also study the gravitational collapse in the context of the cosmic censorship conjecture. We outline the general mathematical framework to study the conditions on the mass function so that future directed nonspacelike geodesics can terminate at the singularity in the past. Mass functions for several equations of state are analysed using this framework and it is shown that the collapse in each case terminates at a locally naked central singularity. These singularities are strong curvature singularities which implies that no extension of spacetime through them is possible. These results are then extended to modified gravity. We establish the result that the standard Boulware-Deser spacetime can radiate. This allows us to model the dynamics of a spherically symmetric radiating dynamical star in ve-dimensional Einstein-Gauss-Bonnet gravity with three spacetime regions. Finally, the junction conditions are derived entirely in five dimensional Einstein-Gauss-Bonnet gravity via the matching of two spacetime region leading to a model for a radiating star in higher order gravity.
New models in general relativity and Einstein-Gauss-Bonnet gravity.
(2021) Naicker, Shavani.; Maharaj, Sunil Dutt.; Brassel, Byron Perry.
We generate the Einstein-Gauss-Bonnet field equations in five dimensions for a spherically symmetric static spacetime. The matter distributions considered are both neutral and charged. The introduction of a coordinate transformation brings the condition of isotropic pressure to a single master ordinary differential equation that is an Abel equation of the second kind. We demonstrate that the master equation can be reduced to a first order nonlinear canonical differential equation. Firstly, we consider uncharged gravitating matter. Several new classes of exact solutions are found in explicit and implicit forms. One of the potentials is determined completely. The second potential satisfies a constraint equation. Secondly, we study charged gravitating matter with Maxwell’s equations. We find new classes of exact charged solutions in explicit and implicit forms using two approaches. In the first approach, we can find new exact models without integration. In the second approach the Abelian pressure isotropy equation has to be integrated, which we demonstrate is possible in a number of cases. The inclusion of the electromagnetic field provides an extra degree of freedom that leads to viable exact solutions. An interesting feature characterising the new models is that a general relativity limit does not exist. Our new solutions exist only in Einstein-Gauss-Bonnet gravity. In addition, we have considered the dynamics of a shear-free fluid in Einstein gravity in higher dimensions with nonvanishing heat flux in a spherically symmetric manifold. This endeavour generates new exact models, being a generalisation of models developed in earlier treatments.
Shear-free models for relativistic fluids with heat flow and pressure isotropy.
(2014) Brassel, Byron Perry.; Govender, Gabriel.; Maharaj, Sunil Dutt.
We model the interior dynamics of a relativistic radiating fuid in a nonstatic spher- ically symmetric spacetime. The matter distribution takes the form of an imperfect fuid with a nonvanishing radially directed heat flux. The fluid pressure is isotropic and the spherically symmetric spacetime manifold is described by a shear-free line el- ement. In our investigation, the isotropy of pressure is a consistency condition which realises a second order nonlinear ordinary differential equation with variable coefficients in the gravitational potentials. We examine this governing equation by imposing vari- ous forms for these potentials and review classes of physically acceptable models that are applicable in relativistic astrophysics. Several new classes of new exact solutions to the condition of pressure isotropy are also found. A comparison of our solutions with earlier well known results is undertaken. A physical analysis of two of the new models is performed where the spatial and temporal evolution of the matter and grav- itational variables are probed. We demonstrate that the fluid pressure, energy density and heat flux are regular and well behaved for both models throughout the interior, and our results indicate that one of the models is consistent with the well established core-envelope framework for compact stellar scenarios. We also analyse the energy conditions for the radiating fluid and demonstrate consistent behaviour, with only the dominant condition being violated. Finally, an analysis of the relativistic thermody- namics of two solutions is undertaken in the Israel-Stewart theory and the temperature profiles for both the noncausal and causal cases are presented.

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Browsing Applied Mathematics by Author "Brassel, Byron Perry."