Sunil D. Maharaj

For a static and spherically symmetric space-time, we investigate a class of exact solutions that arise when two fundamental geometric constraints are imposed simultaneously: Karmarkar’s condition and vanishing of the Weyl tensor. These conditions restrict the curvature in such a way that the space-time becomes conformally flat and belongs to the family of embedding class-I solutions. Even though the subsequent solutions, namely, the Schwarzschild interior solution and the de Sitter solution are well known, the novelty of our presentation is that these solutions are shown to be a direct consequence of the imposed geometric constraints. The physical matter composition becomes highly constrained by the associated geometry under such conditions. The Schwarzschild interior solution describes space-time of an incompressible fluid sphere, while the de Sitter solution corresponds to a vacuum energy dominated configuration. Interestingly, pressure anisotropy as well as the ‘‘complexity factor’’ vanish identically once Karmarkar’s condition and the conformal flatness conditions are applied simultaneously. As these two geometric constraints alone are sufficient to determine the background space-time uniquely, application of Karmarkar’s condition might not be a suitable method for the development of realistic stellar models in conformally flat space-time unless one invokes other factors into consideration, such as time-dependent metric potentials.

We investigate the behaviour of charged, static, spherically symmetric matter distributions in Einstein–Gauss–Bonnet (EGB) gravity. We impose pressure isotropy on our system and analyse the resulting charged condition of pressure isotropy. This equation is an Abel differential equation of the second kind and exact solutions are found by requiring that it is separable. The condition of separability leads naturally to specific forms for the electric field intensity. Matching at the surface of the star shows that the total charge within the star depends on hypergeometric functions which is different from earlier treatments. The Gauss–Bonnet contribution to the mass of the star depends on spacetime dimension, the radius of the star and the Gauss–Bonnet coupling parameter. This Gauss–Bonnet contribution leads to behaviour different from general relativity; specifically it grows with increasing radius and increases the overall mass.

We investigate a class of radiating star models with nonzero shear in general relativity by specifying a particular form of the gravitational potential associated with the areal 2-sphere. This choice enables the boundary condition equation to be transformed into a Riccati equation, which we integrate to obtain three distinct classes of exact solutions. These solutions represent a category of shearing metrics not previously studied. A comprehensive physical analysis confirms the physical plausibility of the solutions, highlighting their potential applicability to realistic radiating stellar structures.

We investigate the arrow of time problem in the context of gravitational collapse of radiating stars in higher dimensions for both neutral and charged matter. The interior spacetime is described by a shear-free, isotropic spherically symmetric metric filled with a dissipative fluid. The exterior spacetime of the radiating star is taken as the higher dimensional Vaidya metric. We establish that the arrow of time, measured by the epoch function, is opposite to the thermodynamic arrow of time for all dimensions in such spacetimes. The physical consequences of our results are considered. Our results conform with previous studies on shear-free spherical collapse, which suggests avoidance of the naked singularity as the end state results in a wrong arrow of time, indicating a fundamental problem with the local application of the epoch functions to test the Weyl curvature hypothesis, which we have demonstrated in the context of shear-free, pressure-isotropic subclass of radiating spherical collapse for dimension four and beyond.