ii Cauchy - Characteristic Matching In General Relativity

The problem of self-gravitating, isolated systems is undoubtedly an important and intriguing area of research in General Relativity. However, due to the involved nature of Einstein's equations physicists found themselves unable to fully explore such systems. From Einstein's theory we know that besides the electromagnetic spectrum , objects like quasars, active galactic nuclei, pulsars and black holes also generate a physical signal of purely gravitational nature. Now scientists involved in the Laser Interferometric Gravitational Observatory (LIGO) project are feverishly trying to build an instrument that will detect it. While the theory of gravitational radiation has been developed using sophisticated mathematical techniques, the actual form of the signal from a given source is impossible to determine analytically. The need to investigate this and related problems has led to the creation of the field of numerical relativity. Immediately two major approaches emerged. The first one formulates the gravitational radiation problem as a standard Cauchy initial value problem. The advantage of this approach is that it is able to handle regions of space-time where strong fields are present and caustics in the wavefronts are likely to form. But it suffers from some inherent disadvantages when it comes to the prediction of gravitational radiation waveforms. Another approach is the Characteristic Initial value problem. This method is unable to treat regions of space-time where caustics form but it is uniquely suited to study radiation problems because it describes space-time in terms of radiation wavefronts. The fact that the advantages and disadvantages of these two systems are complementary suggests that one may want to use the two of them together. In a full nonlinear problem it would be advantageous to evolve the inner (strong field) region using Cauchy evolution and the outer (radiation) region with the Characteristic approach. Cauchy Characteristic Matching enables one to evolve the whole space-time matching the boundaries of Cauchy and Characteristic evolution. The methodology of Cauchy Characteristic Matching has been successful in numerical evolution of the spherically symmetric Klein-Gordon-Einstein field equations as well as for 3-D non-linear wave equations. In this thesis the same methodology is studied in the context of the Einstein equations. iii ACKNOWLEDGMENTS First of all, I want to thank my advisor, Jeffrey Winicour. His wisdom and patience made this work possible. Without his continuous input I would not have been able to accomplish the work being done the last four years. In him I found not only an …