We prove that any compact Cauchy horizon with constant non-zero surface gravity in a smooth vacuum spacetime is a smooth Killing horizon. The novelty here is that the Killing vector field is shown to exist on both sides of the horizon. This generalises classical results by Moncrief and Isenberg, by dropping the assumption that the metric is analytic. In previous work by Rácz and the author, the Killing vector field was constructed on the globally hyperbolic side of the horizon. In this paper, we prove a new unique continuation theorem for wave equations through smooth compact lightlike (characteristic) hypersurfaces which allows us to extend the Killing vector field beyond the horizon. The main ingredient in the proof of this theorem is a novel Carleman type estimate. Using a well-known construction, our result applies in particular to smooth stationary asymptotically flat vacuum black hole spacetimes with event horizons with constant non-zero surface gravity. As a special case, we therefore recover Hawking's local rigidity theorem for such black holes, which was recently proven by Alexakis-Ionescu-Klainerman using a different Carleman type estimate.

我们证明了在光滑真空时空中具有恒定非零表面重力的任何致密柯西视界都是光滑的克林视界。这里的新颖之处在于，Killing 向量场显示存在于两个地平线的两侧。通过放弃度量是分析性的假设，这概括了 Moncrief 和 Isenberg 的经典结果。在 Rácz 和作者之前的工作中，Killing 向量场是在地平线的全局双曲线侧构建的。在本文中，我们通过光滑紧凑的类光（特征）超曲面证明了波动方程的一个新的独特的连续定理，这使我们能够将 Killing 向量场扩展到地平线之外。该定理证明的主要成分是新的卡尔曼类型估计。使用众所周知的构造，我们的结果特别适用于平滑平稳渐近平坦的真空黑洞时空，其事件视界具有恒定的非零表面重力。作为一个特例，我们因此恢复了霍金对此类黑洞的局部刚性定理，