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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关于欧几里得引力边界条件的注释

我们回顾了广义相对论中关于欧几里得签名时空边界条件的已知知识。明显的 Dirichlet 边界条件（其中指定了边界几何）实际上不是椭圆的，并且通常不会导致明确定义的微扰理论。如果边界的外在曲率受到适当的约束，例如如果它是正定或负定，则表现更好。一个不同的边界条件，其中一个指定边界的共形几何和外在曲率的轨迹，是椭圆的，并且总是在形式上导致令人满意的微扰理论。这些事实可能对量子引力的半经典方法产生有趣的影响。2018 年 4 月