Jeffrey Winicour: Boundary Conditions for an Isolated System

06/06/2008 - 11:30am to 12:30pm
Jeffrey Winicour
Speaker affiliation: 
University of Pittsburgh

The prime sources of gravitational radiation can be modeled as
isolated systems in which the waves propagate to infinity. However,
most numerical simulations of such sources introduce an artificial
outer boundary at a finite distance. There has been recent progress in
formulating boundary conditions for the Einstein equations which lead
to a constraint-preserving well-posed initial-boundary value problem.
I will review the basic ideas governing well-posedness of the
initial-boundary value problem for hyperbolic systems of second-order
wave equations, such as the harmonic formulation of Einstein's
equations and Maxwell's equations in the Lorentz gauge. For the
harmonic Einstein's equations, there are several options akin to
Sommerfeld boundary conditions.  There remains the problem of
assigning  appropriate boundary data. Unless the numerical evolution
is matched to an exterior solution, the only possibility is to give
homogeneous (vanishing) boundary data which approximate the asymptotic
behavior at infinity. One proposal is to use vanishing curvature data
(Psi_0 = 0 in the Newman-Penrose formalism). However, this results in
a second differential order boundary condition that does not fit the
theorems for well-posedness of the harmonic problem. I discuss how
Psi_0 = 0 data can be reformulated as first order data in Sommerfeld
form, the resulting deficiencies and alternative approaches.

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