The Synergism of Analytic and Numerical Techniques in General Relativity: Calculation of Radiative Spacetimes
Abrahams, Andrew M.
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https://hdl.handle.net/2142/70670
Description
Title
The Synergism of Analytic and Numerical Techniques in General Relativity: Calculation of Radiative Spacetimes
Author(s)
Abrahams, Andrew M.
Issue Date
1988
Doctoral Committee Chair(s)
Smarr, Larry L.
Department of Study
Astronomy
Discipline
Astronomy
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Physics, Astronomy and Astrophysics
Abstract
The theoretical difficulty of identifying the radiative component of a general relativistic spacetime is manifested in numerical relativity as the problem of extracting asymptotic gravitational radiation waveforms at finite radii during simulations on spacelike slices. These waveforms can be contaminated by linear effects due to the choice of gauge and the wave's near zone field as well as by non-linear interactions. What is usually done is to extract the waveforms at large radii in the wave zone where these effects are assumed to be negligible using special radiative variables which have radiative terms rather than the mass monopole moment as the leading term in their asymptotic expansion. This approach can be wasteful of computer resources and is not very rigorous. We take a synergistic approach where a numerical solution of the Einstein equations coupled with hydrodynamics for the source and strong-field region is matched in the near zone or local wave zone onto a linear analytic solution used for the exterior region. No asymptotic flatness assumption is required; it is sufficient that the source be isolated enough that a local wave zone exists.
First we develop general solutions to the linearized vacuum Einstein equations expanded in multipole moments. These solutions are employed in three matching schemes for extracting waveforms. In the first an infinitesimal gauge transformation is used to express the analytic solution in several gauges useful for numerical relativity and gauge-independent combinations of multipole amplitudes are formed. In the second, variables are constructed which are invariant under gauge-transformations. The third method uses surface integrals of Riemann tensor components to avoid gauge effects. In all three methods the near-zone field terms are separated off with an ordinary differential equation integration over a timelike cylinder. We give a prescription for calculating initial values for these ODEs from data on a spacelike hypersurface.
These procedures are tested using simulated sinusoidal and non-sinusoidal pulsating neutron-star spacetimes. We demonstrate that each technique correctly matches the exterior and interior solutions to eliminate gauge effects and that the ODE integration separates off the near-zone field to yield asymptotic waveforms. We also present an application of these methods to Brill wave spacetimes. Discussion of non-linear effects in these spacetimes is also included.
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