Workshop From
geometry to numerics
IHP, Paris, 20-24
November 2006
Luisa Buchman (Univ. Texas, Austin, USA)
Towards
absorbing outer boundaries in General Relativity
We construct exact
solutions to the Bianchi equations on a flat spacetime background. When
the constraints are satisfied, these solutions represent in- and
outgoing linearized gravitational radiation. We then consider the
Bianchi equations on a subset of flat spacetime of the form [0,T]
x BR, where BR
is a ball of radius R, and analyze different kinds of boundary
conditions on \partial BR.
With the help of the exact solutions constructed, we determine the
amount of artificial reflection of gravitational radiation from
constraint-preserving boundary conditions which freeze the Weyl
scalar Psi0 to its initial value. For monochromatic
radiation with wave number k and arbitrary angular momentum
number l>=
2, the amount of reflection decays as (k R)-4
for large k R. For each L >= 2, we construct new local
constraint-preserving boundary conditions which perfectly absorb
linearized radiation with l<=
L. We generalize our analysis to a weakly curved background of
mass M (to first order in M/R and for quadrupolar
radiation).
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