Differential equations for relativistic radiating stars.
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Date
2013
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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.
Description
Ph. D. University of KwaZulu-Natal, Durban 2013.
Keywords
Differential equations--Numerical solutions., Astrophysics., Lie groups., Symmetry (Physics), Stars., Theses--Applied mathematics.