Derivation and Properties of a Modified Relativistic Klein-Gordon Equation for Late Time Evolution of Complex Scalar Fields in the Vicinity of Schwarzschild and Near-Extremal Reissner-Nordström Black Holes
Finite Element (Difference) Method
Partial Differential Equations
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PublisherThe University of Arizona.
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AbstractThis dissertation has provided a framework for black hole perturbation theory, aimed at the study of the stability of black holes. We consider linear perturbations of the Einstein Field Equations and have derived a modified relativistic Klein-Gordon model in observable spacetime coordinates in the presence of Schwarzschild and Reissner-Nordström geometries. The second order partial differential equation for complex fields is dispersive hyperbolic, with its characteristics identical to the null geodesics of the standard ingoing Eddington-Finkelstein coordinates. The equation preserves several important features of the original system such as decaying tail and quasi-normal ringing in the presence of a nonlinear potential introduced due to self-interaction. The dispersion relation for the frozen coefficient equation further exhibits amplification/damping and the origin of the tail. Additionally, it displays convergence to the standard wave equation as waves move further away from the event horizon. The reversal of the sign in the group velocity, near the event horizon, explains the long tail present in the linear case. We have applied both finite differences and finite element methods (of arbitrary spatial polynomial order) to study effects of the charge for fully complex fields as well as effects of the nonlinear self-interaction. A transparent boundary condition was utilized with an experimentally determined advection velocity.We have validated the codes using the method of manufactured solutions and provided a rigorous framework for convergence in the finite element contents. Finally, we have developed an open-source body of code via the DEAL.II software package, which is released on Github at https://github.com/nholtzerresearch/SCRN_Code.
Degree ProgramGraduate College