On horizon structure of bimetric spacetimes

TL;DR

The paper analyzes horizon structure in spacetimes with two metrics, addressing how Killing horizons and their surface gravities relate between the metrics without relying on field equations. It derives three propositions establishing when horizons must coincide and when bifurcation surfaces enforce equal surface gravities, using regularity arguments and the Racz–Wald framework. These results imply that the Vainshtein mechanism cannot generically recover black holes in diagonal two-metric setups and illuminate the global structure of known bigravity solutions, including cases with mixed diagonal/non-diagonal metrics and noncoincident horizons. The findings constrain the consistency of bimetric and massive gravity theories and clarify how horizons and bifurcation surfaces distribute across dual geometries, with concrete illustrations from Schwarzschild/flat examples and bigravity black holes.

Abstract

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.