Non-linear numerical Schemes in General Relativity

Abstract

First we present the first long-term stable second order convergent Cauchy characteristic matching code in cylindrical symmetry including both gravitational degrees of freedom. Compared with previous work we achieve a substantial simplification of the evolution equations as well as the relations at the interface by applying the method of Geroch decomposition to both the inner and outer region. We use analytic vacuum solutions with one and two gravitational degrees of freedom to demonstrate the accuracy and convergence properties of the code.

In the second part we numerically solve the equations for static and dynamic cosmic strings of infinite length coupled to gravity and provide the first fully non-linear evolutions of cosmic strings in curved spacetimes. The inclusion of null infinity as part of the numerical grid allows us to apply suitable boundary conditions on the metric and the matter fields to suppress unphysical divergent solutions. The resulting code is checked for internal consistency by a convergence analysis and also by verifying that static cosmic string initial data remain constant when evolved. The dynamic code is also shown to reproduce analytic vacuum solutions with high accuracy. We then study the interaction between a Weber-Wheeler pulse of gravitational radiation with an initially static string. The interaction causes the string to oscillate with frequencies proportional to the masses of its scalar and vector field. After the pulse has largely radiated away, the string continues to ring but the oscillations slowly decay and eventually the variables return to their equilibrium values.

In the final part of the thesis we probe a numerical approach for highly accurate evolutions of neutron star oscillations in the case of radial oscillations of spherically symmetric stars. For this purpose we decompose the problem into a static background governed by the Tolman-Oppenheimer-Volkoff equations and time dependent perturbations. In contrast to conventional treatments, the fully non-linear form of the resulting perturbative equations is used.

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