On horizon structure of bimetric spacetimes

We discuss the structure of horizons in spacetimes with two metrics, with applications to the Vainshtein mechanism and other examples. We show, without using the field equations, that if the two metrics are static, spherically symmetric, nonsingular and diagonal in a common coordinate system, then a Killing horizon for one must also be a Killing horizon for the other. We then generalize this result to the axisymmetric case. We also show that the surface gravities must agree if the bifurcation surface in one spacetime lies smoothly in the interior of the spacetime of the other metric. These results imply for example that the Vainshtein mechanism of nonlinear massive gravity theories cannot work to recover black holes if the dynamical metric and the non-dynamical flat metric are both diagonal. They also explain the global structure of some known solutions of bigravity theories with one diagonal and one non-diagonal metric, in which the bifurcation surface of the Killing field lies in the interior of one spacetime and on the conformal boundary of the other.