Keshlan S. Govinder

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.

A simple form of the gravitational field for a radiating relativistic star with nonzero shear is studied. The junction conditions at the surface of the star reduce to an autonomous nonlinear first order differential equation of the Riccati type: it admits exact solutions and is amenable to a phase plane analysis. We can determine the temporal evolution and asymptotic behaviour of the radiating star including shear, charge and the cosmological constant. The presence of shear leads to behaviour which is distinct from shear-free models and fold bifurcations arise; the model collapses from an initial static configuration or a superdense cold star forms. The areal distance plays a central role in the asymptotic analysis and there is an interplay with the charge and the cosmological constant.

From the era of Black–Scholes, European options have dwelt on pay-offs that are a linear function of the asset price. In this paper, we will look at a special case of exotic options - power options - whose payoffs are nonlinear functions of the underlying asset price. Exotic options are derivatives which have features that makes them more complex than commonly traded products - thus finding their fair value is not an always easy task. Previous analyses of the power option partial differential equation (PDE) have only obtained closed form solutions either by guessing solutions, similarity methods or the martingale approach [8, 9, 25]. Using Lie symmetry analysis we obtain an optimal system of the Lie point symmetries of the power option PDE and demonstrate an algorithmic method for finding solutions to the equation. In addition, we find a new analytical solution to the asymmetric type of the power option.

The dynamics of a radiating star in general relativity are studied in higher dimensions for a specified shear-free metric. The temporal evolution of the radiating star depends on the spacetime dimension. In particular, we show explicitly that the gravitational potential changes with increasing spacetime dimension. A detailed analysis of the boundary condition is undertaken. We find new exact solutions and first integrals for the boundary condition equation. Known results in four dimensions are regained as special cases. A phase plane analysis indicates that the model asymptotically approaches a static end state or continues to radiate. The physical features are affected by dimension, and we indicate how the luminosity changes with increasing dimension.

We study the dynamics of a charged radiating star in general relativity. The junction conditions at the surface of the star lead to a restriction that connects the radial pressure to the heat flux. The master equation reduces to a nonlinear second order differential equation which determines the temporal evolution. The dynamical behaviour is studied via a phase plane analysis which reveals interesting behaviour. The presence of both the electromagnetic field and the cosmological constant are included in our treatment. They affect the temporal evolution of the gravitating star. We identify the restrictions on the parameters that lead to a stable asymptotic end state of the star.