Spacetime symmetries and topology in bimetric relativity

TL;DR

This paper investigates the spacetime symmetries and topologies in Hassan-Rosen bimetric theory, revealing conditions under which the two metric sectors share or differ in symmetries, especially in the presence of matter.

Contribution

It analyzes the possible symmetry configurations in bimetric theory, including cases with different symmetries in each sector and the impact of topology constraints.

Findings

Vacuum solutions can have shared or separate symmetries in the two sectors.

Matter presence introduces a third symmetry configuration with different symmetries within the same sector.

Spacetime topology constrains the allowed metric combinations.

Abstract

We explore spacetime symmetries and topologies of the two metric sectors in Hassan-Rosen bimetric theory. We show that, in vacuum, the two sectors can either share or have separate spacetime symmetries. If stress-energy tensors are present, a third case can arise, with different spacetime symmetries within the same sector. This raises the question of the best definition of spacetime symmetry in Hassan-Rosen bimetric theory. We emphasize the possibility of imposing ans\"atze and looking for solutions having different Killing vector fields or different isometries in the two sectors, which has gained little attention so far. We also point out that the topology of spacetime imposes a constraint on possible metric combinations.