On the stability of the differentiability of Cauchy horizons
\textit{J. K. Beem} [Gen. Relativ. Gravitation 27, 93-108 (1995; Zbl 0815.53073)] showed that, in general, the existence of Cauchy horizons in spacetimes is not stable under metric perturbations and gave some sufficient conditions for such stability. In this paper, the authors show that in a class of spacetimes containing compact Cauchy horizons and satisfying certain conditions, both the existence and differentiability of the horizon are stable under smooth perturbations of the metric. In the class of spacetimes under consideration, each contains a compact Cauchy horizon whose null generators admit a global Poincaré section. To prove stability, it is required both that a certain global quantity be sufficiently large and that the spacetime differentiability order be sufficiently large. The key idea of the proof is to use the implicit function theorem of Nash and Moser. The proof of the main result in this paper also suggests that if the sufficient conditions are not imposed, then the Cauchy horizon and its differentiability are not necessarily stable. The authors give an example to support this final claim. The results in this paper are of considerable interest since many theorems in general relativity concerning horizons require some degree of differentiability of the horizons.
scientific article; zbMATH DE number 1127777