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 inand outgoing linearized gravitational radiation. We then consider the Bianchi equations on a subset of flat spacetime of the form [0, T ] × BR , where BR is a ball of radius R, and analyse different kinds of boundary conditions on ∂BR . Our main results are as follows. (i) We give an explicit analytic example showing that boundary conditions obtained from freezing the incoming characteristic fields to their initial values are not compatible with the constraints. (ii) With the help of the exact solutions constructed, we determine the amount of artificial reflection of gravitational radiation from constraintpreserving boundary conditions which freeze the Weyl scalar 0 to its initial value. For monochromatic radiation with wave number k and arbitrary angular momentum number 2, the amount of reflection decays as (kR)−4 for large kR. (iii) For each L 2, we construct new local constraint-preserving boundary conditions which perfectly absorb linearized radiation with L. (iv) We generalize our analysis to a weakly curved background of mass M and compute first-order corrections in M/R to the reflection coefficients for quadrupolar odd-parity radiation. For our new boundary condition with L = 2, the reflection coefficient is smaller than that for the freezing 0 boundary condition by a factor of M/R for kR > 1.04. Implications of these results for numerical simulations of binary black holes on finite domains are discussed. PACS numbers: 04.20.−q, 04.25.−g, 04.25.Dm (Some figures in this article are in colour only in the electronic version) 0264-9381/06/236709+36$30.00 © 2006 IOP Publishing Ltd Printed in the UK 6709 6710 L T Buchman and O C A Sarbach