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

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 show that both these issues are evaded by requiring reality and general covariance of the equations. First 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 restricts the allowed metrics to geometries that admit compatible notions of space and time. It also selects 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.