# On the local structure of spacetime in ghost-free bimetric theory and massive gravity

## TL;DR

This paper analyzes the local spacetime structure in ghost-free bimetric theory, establishing conditions for metric compatibility, ensuring well-defined initial value problems, and ruling out certain acausal solutions.

## Contribution

It proves that reality and covariance constraints lead to a unique, compatible metric structure in ghost-free bimetric theory, clarifying its local spacetime properties.

## Key findings

- Metrics must have intersecting null cones for reality of the square root.
- General covariance constrains metrics to admit compatible 3+1 decompositions.
- Certain acausal solutions are ruled out by the compatibility conditions.

## Abstract

The ghost-free bimetric theory describes interactions of gravity with another spin-2 field in terms of two Lorentzian metrics. However, if the two metrics do not admit compatible notions of space and time, the formulation of the initial value problem becomes problematic. Furthermore, the interaction potential is given in terms of the square root of a matrix which is in general nonunique and possibly nonreal. In this paper we prove that the reality of the square root matrix leads to a classification of the allowed metrics in terms of the intersections of their null cones. Then, the requirement of general covariance further constrains the allowed metrics to admit compatible notions of space and time. It also leads to a unique definition of the square root matrix. The restrictions are compatible with the equations of motion. These results ensure that the ghost-free bimetric theory can be defined unambiguously and that the two metrics always admit compatible 3+1 decompositions, at least locally. In particular, these considerations rule out certain solutions of massive gravity with locally Closed Causal Curves, which have been used to argue that the theory is acausal.