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Towards absorbing outer boundaries in general relativity

Buchman, Luisa T. and Sarbach, Olivier C. A. (2006) Towards absorbing outer boundaries in general relativity. Classical and Quantum Gravity, 23 (23). pp. 6709-6744. ISSN 0264-9381. doi:10.1088/0264-9381/23/23/007.

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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] × 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 constraint-preserving boundary conditions which freeze the Weyl scalar Ψ0 to its initial value. For monochromatic radiation with wave number k and arbitrary angular momentum number ell ≥ 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 ell ≤ 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 Ψ0 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.

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Additional Information:© 2006 IOP Publishing Limited 2006 Received 13 August 2006, in final form 16 September 2006. Published 18 October 2006. Print publication: Issue 23 (7 December 2006) It is a pleasure to thank J Bardeen, L Lehner, L Lindblom, J Novak, O Rinne, J Stewart, S Teukolsky, and M Tiglio for helpful discussions. LTB was supported by a NASA postdoctoral program fellowship at the Jet Propulsion Laboratory. OCAS was supported in part by NSF grant PHY-0099568, by a grant from the Sherman Fairchild Foundation to Caltech and by NSF DMS Award 0411723 to UCSD.
Issue or Number:23
Record Number:CaltechAUTHORS:BUCcqg06
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:5562
Deposited By: Archive Administrator
Deposited On:25 Oct 2006
Last Modified:08 Nov 2021 20:27

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