| A fully relativistic description of the two-body problem in General Relativity is one of the outstanding unsolved problems in Einstein's theory. A detailed description of the orbital parameters of binary systems has become even more pressing with the advent of gravitational wave detection schemes. This thesis details our efforts to generate astrophysically interesting solutions to the two-body problem. The thesis consists of two main parts.; The first part presents an analytical variational principle for describing binary neutron stars undergoing irrotational fluid flow. The variational principle is a powerful tool for generating accurate estimates of orbital parameters for neutron stars in quasi-equilibrium orbits.; The second part of the thesis details the numerical application of the variational principle by solving the initial value problem for binary black holes in quasi-equilibrium circular orbits. The analysis draws from the novel “puncture” method of describing the black holes, and relies on nonlinear adaptive multigrid techniques for generating numerical results. These tools allow us to generate sequences of circular orbits, which in turn allow us to approximate the orbital parameters for the system up to and including the innermost stable circular orbit. We compare our numerical results to post-Newtonian expectations and we generate numerical data regarding the geometry and the gravitational radiation emitted by the system.; We arrive at two important conclusions. First, the analytical variational principle describing binary neutron stars in irrotational motion provides a road map for future numerical simulations, and also lends credence to previous simulations by other authors. Second, the numerical application and description of binary black holes in quasi-equilibrium circular orbits simplifies the analyses of previous authors, and allows for the imposition of realistic boundary data in the simulations with relatively high grid densities. Both the variational principle and its application may be used to help generate accurate estimates of the orbital parameters for waveform templates needed by gravitational wave observatories around the world. |
