  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
nondiagonal 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.
