Geometric boundary data for the gravitational field

An outstanding issue in the treatment of boundaries in general relativity is the lack of a local geometric interpretation of the necessary boundary data. For the Cauchy problem, the initial data is supplied by the 3-metric and extrinsic curvature of the initial Cauchy hypersurface, subject to constr...

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Bibliographic Details
Published inClassical and quantum gravity Vol. 31; no. 6; pp. 65004 - 65022
Main Authors Kreiss, H-O, Winicour, J
Format Journal Article
LanguageEnglish
Published IOP Publishing 21.03.2014
Subjects
Boundaries
boundary conditions
Boundary value problems
Curvature
Degrees of freedom
Einstein's equations
Evolution
gravitational field
Initial value problems
Mathematical analysis
Quantum gravity
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Summary:An outstanding issue in the treatment of boundaries in general relativity is the lack of a local geometric interpretation of the necessary boundary data. For the Cauchy problem, the initial data is supplied by the 3-metric and extrinsic curvature of the initial Cauchy hypersurface, subject to constraints. This Cauchy data determine a solution to Einstein's equations which is unique up to a diffeomorphism. Here, we show how three pieces of unconstrained boundary data, which are associated locally with the geometry of the boundary, likewise determine a solution of the initial-boundary value problem which is unique, up to a diffeomorphism. Two pieces of this data constitute a conformal class of rank-2 metrics, which represent the two gravitational degrees of freedom. The third piece, constructed from the extrinsic curvature of the boundary, determines the dynamical evolution of the boundary.
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ISSN:0264-9381
1361-6382
1361-6382
DOI:10.1088/0264-9381/31/6/065004