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 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

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This paper reviews the known boundary conditions in General Relativity on a spacetime of Euclidean signature, discussing the issues with the obvious Dirichlet boundary condition and proposing a more effective boundary condition for solving perturbation theory problems.