# Isometries of Spacetimes Without Observer Horizons

## TL;DR

This paper investigates the symmetry properties of spacetimes without observer horizons and shows that their isometry groups behave in a structured way.

## Contribution

The paper proves that the isometry group acts properly and splits into specific subgroups under the no observer horizons condition.

## Key findings

- The time orientation-preserving isometry group acts properly on the spacetime.
- The isometry group splits into a compact subgroup and a subgroup resembling time translations.
- The time translation subgroup can only be trivial, Z, or R.

## 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, \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {Z}$$\end{document}Z, or \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}$$\end{document}R.