A note on boundary conditions in Euclidean gravity

We review what is known about boundary conditions in General Relativity on a spacetime of Euclidean signature. The obvious Dirichlet boundary condition, in which one specifies the boundary geometry, is actually not elliptic and in general does not lead to a well-defined perturbation theory. It is be...

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Published inReviews in mathematical physics Vol. 33; no. 10
Main Author Witten, Edward
Format Journal Article
LanguageEnglish
Published Singapore World Scientific Publishing Company 01.11.2021
World Scientific Publishing Co. Pte., Ltd
Subjects
Boundary conditions
Curvature
Dirichlet problem
Perturbation theory
Quantum gravity
Relativity
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ISSN0129-055X
1793-6659
DOI10.1142/S0129055X21400043

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Summary:We review what is known about boundary conditions in General Relativity on a spacetime of Euclidean signature. The obvious Dirichlet boundary condition, in which one specifies the boundary geometry, is actually not elliptic and in general does not lead to a well-defined perturbation theory. It is better-behaved if the extrinsic curvature of the boundary is suitably constrained, for instance if it is positive- or negative-definite. A different boundary condition, in which one specifies the conformal geometry of the boundary and the trace of the extrinsic curvature, is elliptic and always leads formally to a satisfactory perturbation theory. These facts might have interesting implications for semiclassical approaches to quantum gravity. April, 2018
Bibliography:https://doi.org/10.1142/9789811210679_0025
This paper is reproduced from the book Roman Jackiw: 80th Birthday Festschrift, edited by Antti Niemi, Terry Tomboulis and Kok Khoo Phua (World Scientific, 2020)
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ISSN:0129-055X
1793-6659
DOI:10.1142/S0129055X21400043