Modeling Extended Fluid Objects in General Relativity
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The purpose of this dissertation is to introduce and explore the notion of modeling extended fluid objects in numerical general relativity. These extended fluid objects, called Fat Particles, are proxies for compact hydrodynamic objects. Unlike full hydrodynamic models, we make the approximation that the details of the matter distribution are not as important as the gross motion of the Fat Particle's center of mass and its contribution to the gravitational field. Thus we provide a semi-analytic model of matter for numerical simulations of Einstein's equations, which may help in modeling gravitational radiation from candidate sources.
Our approach to carrying out these investigations is to begin with a continuum variational principle, which yields the desired hydrodynamic and gravitational equations for ideal fluids. Following our analysis of the related numerical technique, Smoothed Particle Hydrodynamics (SPH), we apply a set of discretization and smoothing rules to obtain a discrete action. Subsequent variations yield the Fat Particle equations.
Our analysis of a classical ideal fluid demonstrated that a Newtonian Fat Particle is capable of remaining at rest while generating its own gravitational field. We then developed analogous principles for describing relativistic ideal fluids in both covariant and ADM 3+1 forms. Using these principles, we developed analytic and numerical results from relativistic Fat Particle theory. We began with the Subscribe Only model, in which a Fat Particle of negligible mass moves in a fixed background metric. Corrections to its motion due to the extended nature of the Fat Particle, are obtained by summing metric contributions over its volume. We find a universal scaling law that describes the phase shift, relative to a test particle, that is independent of its size, shape, and distribution. We then show that finite-size effects eventually dominate radiation damping effects in describing the motion of a white dwarf around a more massive black hole. Finally, we derive the Publish and Subscribe model, which comprises a full back-reacting system. Comparison of the Fat Particle equations for a static, symmetric spacetime with their continuum analogs shows that the system supports a consistent density definition and holds promise for future development.