TY - JOUR

T1 - A note on boundary conditions in Euclidean gravity

AU - Witten, Edward

N1 - Funding Information:
Research supported in part by NSF Grant PHY-1606531.
Publisher Copyright:
© 2021 World Scientific Publishing Company.

PY - 2021/11/1

Y1 - 2021/11/1

N2 - 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.

AB - 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.

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U2 - 10.1142/S0129055X21400043

DO - 10.1142/S0129055X21400043

M3 - Article

AN - SCOPUS:85110713071

SN - 0129-055X

VL - 33

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

IS - 10

M1 - 2140004

ER -