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 -