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Asymptotic Solutions of Radiating Stars

R. S. Bogadi, G. Leon, M. Govender, K. S. Govinder, S. Maharaj, A. Paliathanasis

Abstract

We investigate the evolution of the surface of radiating stars by studying the asymptotic behaviour of exact solutions initiated via the stationary boundary condition. This boundary condition leads to a master equation in the form of a second-order nonlinear differential equation that describes the evolution of the scale factor. We examine this master equation by introducing a set of dimensionless dynamical variables, motivated by similar approaches in cosmological settings. We derive the stationary points of the system in the presence of charge and a cosmological constant. Furthermore, we construct criteria for the initial conditions in order that the asymptotic limit approaches a static geometry.

Asymptotic Solutions of Radiating Stars

Abstract

We investigate the evolution of the surface of radiating stars by studying the asymptotic behaviour of exact solutions initiated via the stationary boundary condition. This boundary condition leads to a master equation in the form of a second-order nonlinear differential equation that describes the evolution of the scale factor. We examine this master equation by introducing a set of dimensionless dynamical variables, motivated by similar approaches in cosmological settings. We derive the stationary points of the system in the presence of charge and a cosmological constant. Furthermore, we construct criteria for the initial conditions in order that the asymptotic limit approaches a static geometry.

Paper Structure

This paper contains 10 sections, 27 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Radiating Stars
  3. Asymptotic analysis
  4. Case A: $~A^{\prime}\neq0,~\gamma=0,~\delta=0$
  5. Poincaré Variables
  6. Case B: $~A^{\prime}\neq0,~\gamma\neq0,~\delta=0$
  7. Poincaré Variables
  8. Case C: $~A^{\prime}\neq0,~\gamma\neq0,~\delta\neq0$
  9. Poincaré Variables
  10. Conclusion

Figures (2)

  • Figure 1: Phase-Space for the dynamical system (\ref{['ca.01']}), (\ref{['ca.02']}) in the compactified variables. The unstable asymptotic solutions are marked with red points, the attractor at the infinity is marked with blue, and the green line describes the family of points $P_{A}^{2}$.
  • Figure 2: Phase-Space for the dynamical system (\ref{['c.01']}), (\ref{['c.02']}), (\ref{['c.03']}) and (\ref{['c.04']}) in the compactified variables, on the surface where $A^{\prime }\left( r\right) =0$, that is, $\Omega_{\alpha}=0$ and $\Omega_{\beta}=0$. The unstable asymptotic solutions are marked with red points, the attractor $P_{C}^{4}$ is marked with blue.