Geodesic models generated by Lie symmetries

TL;DR

This paper applies Lie symmetry analysis to a radiating star model, deriving new exact solutions for the boundary conditions, including self-similar and traveling wave solutions, with explicit gravitational potentials.

Contribution

It identifies Lie symmetries of the junction condition, generates an optimal system, and finds new invariant solutions, extending previous models including the Friedmann dust case.

Findings

Derived new exact solutions for the boundary condition

Connected solutions to the Friedmann dust model

Explicit gravitational potentials and line elements

Abstract

We study the junction condition relating the pressure to the heat flux at the boundary of a shearing and expanding spherically symmetric radiating star when the fluid particles are travelling in geodesic motion. The Lie symmetry generators that leave the junction condition invariant are identified and the optimal system is generated. We use each element of the optimal system to transform the partial differential equation to an ordinary differential equation. New exact solutions, which are group invariant under the action of Lie point infinitesimal symmetries, are found. We obtain families of traveling wave solutions and self-similar solutions, amongst others. 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, and we can connect our results to earlier investigations.