Differential equations for relativistic radiating stars.

Abstract

We consider radiating spherical stars in general relativity when they are conformally flat, geodesic with shear, and accelerating, expanding and shearing. We study the junction conditions relating the pressure to the heat flux at the boundary of the star in each case. The boundary conditions are nonlinear partial differential equations in the metric functions. We transform the governing equations to ordinary differential equations using the geometric method of Lie. The Lie symmetry generators that leave the equations invariant are identified, and we generate the optimal system in each case. Each element of the optimal system is used to reduce the partial differential equations to ordinary differential equations which are further analyzed. As a result, particular solutions to the junction conditions are presented for all types of radiating stars. New exact solutions, which are group invariant under the action of Lie point infinitesimal symmetries, are found. Our solutions contain families of traveling wave solutions, self-similar variables, and other forms with different combinations of the spacetime variables. The gravitational potentials are given in terms of elementary functions, and the line elements can be given explicitly in all cases. We show that the Friedmann dust model is regained as a special case in particular solutions. We can connect our results to earlier investigations and we show explicitly that our models are generalizations. Some of our solutions satisfy a linear equation of state. We also regain previously obtained solutions for the Euclidean star as a special case in our accelerating model. Our results highlight the importance of Lie symmetries of differential equations for problems arising in relativistic astrophysics.