Variational Principles in General Relativity

Abstract

This thesis details an effort to generate astrophysically interesting solutions to the two-body problem in General Relativity. 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. 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 simulations with relatively high grid densities. Both the variational principle and its application are used to generate accurate estimates of the orbital parameters.