Isometries of spacetimes without observer horizons
Leonardo García-Heveling, Abdelghani Zeghib
TL;DR
This work identifies strong rigidity for the symmetry groups of causal spacetimes under the no future observer horizons condition. By proving that the time-orientation-preserving isometry group acts properly on the spacetime, the authors obtain an invariant Cauchy temporal function and a canonical split of the isometry group as $\operatorname{Isom}^[0m\u007fm(M,g) = L \ltimes N$ with $N$ compact and $L \in \{\text{trivial}, \mathbb{Z}, \mathbb{R}\}$; the connected component further splits when $L=\mathbb{R}$. This ties NFOH to global hyperbolicity with compact Cauchy surfaces and yields a concrete framework for understanding Lorentzian isometries via semi-direct product decompositions, with sharpness shown through product spacetimes and counterexamples like de Sitter. The paper also provides auxiliary results on uniform Lipschitz bounds for families of isometries and shows compactness results in spacetimes with a regular cosmological time function, broadening the applicability of these rigidity phenomena in Lorentzian geometry.
Abstract
We study the isometry groups of (non-compact) Lorentzian manifolds with well-behaved causal structure, aka causal spacetimes satisfying the ``no observer horizons'' condition. Our main result is that the group of time orientation-preserving isometries acts properly on the spacetime. As corollaries, we obtain the existence of an invariant Cauchy temporal function, and a splitting of the isometry group into a compact subgroup and a subgroup roughly corresponding to time translations. The latter can only be the trivial group, $\mathbb{Z}$, or $\mathbb{R}$.