Isometries of spacetimes without observer horizons

TL;DR

This paper investigates the symmetry groups of certain Lorentzian manifolds without observer horizons, establishing proper action of isometries and structural decompositions of the isometry group.

Contribution

It proves that the isometry group acts properly on these spacetimes and describes its structure, including existence of invariant functions and group splitting.

Findings

Isometry group acts properly on the spacetime

Existence of an invariant Cauchy temporal function

Isometry group splits into specific subgroup types

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 $\mathbb{R}$.