A note on boundary conditions in Euclidean gravity
Abstract
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
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); https://doi.org/10.1142/9789811210679_0025
