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A minimization problem for the lapse and the initial-boundary value problem for Einstein's field equations

Nagy, Gabriel and Sarbach, Olivier (2006) A minimization problem for the lapse and the initial-boundary value problem for Einstein's field equations. Classical and Quantum Gravity, 23 (16). S477-S504. ISSN 0264-9381. https://resolver.caltech.edu/CaltechAUTHORS:NAGcqg06

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Abstract

We discuss the initial-boundary value problem of general relativity. Previous considerations for a toy model problem in electrodynamics motivate the introduction of a variational principle for the lapse with several attractive properties. In particular, we show that the resulting elliptic gauge condition for the lapse together with a suitable condition for the shift and constraint-preserving boundary conditions controlling the Weyl scalar Ψ0 yields a priori estimates on the metric, connection and curvature fields. These estimates are expected to be useful for obtaining a well-posed initial-boundary value problem for metric formulations of Einstein's field equations which are commonly used in numerical relativity. To present a simple and explicit example, we consider the 3 + 1 decomposition introduced by York of the field equations on a cubic domain with two periodic directions and prove in the weak field limit that our gauge condition for the lapse, an a priori specified shift vector and our boundary conditions lead to a well-posed problem. The method discussed here is quite general and should also yield well-posed problems for different ways of writing the evolution equations, including first-order symmetric hyperbolic or mixed first-order second-order formulations. Well-posed initial-boundary value formulations for the linearization about arbitrary stationary configurations will be presented elsewhere.


Item Type:Article
Additional Information:© 2006 IOP Publishing Limited Received 26 January 2006, in final form 3 April 2006. Published 27 July 2006. Print publication: Issue 16 (21 August 2006) Special issue: Invited papers on numerical relativity, related to the Banff International Research Station programme 16–21 April 2005 and the Newton Institute programme 8 August–23 December 2005 It is a pleasure to thank H Beyer, G Calabrese, M Holst, L Lehner, H Pfeiffer, O Reula and M Tiglio for helpful discussions and suggestions and D Reynolds for reading the manuscript. This research 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:16
Record Number:CaltechAUTHORS:NAGcqg06
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:NAGcqg06
Alternative URL:http://dx.doi.org/10.1088/0264-9381/23/16/S11
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:5563
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:25 Oct 2006
Last Modified:02 Oct 2019 23:24

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