Black hole scalar charge from a topological horizon integral in Einstein-dilaton-Gauss-Bonnet gravity

In theories of gravity that include a scalar field, a compact object's scalar charge is a crucial quantity since it controls dipole radiation, which can be strongly constrained by pulsar timing and gravitational wave observations. However in most such theories, computing the scalar charge requires simultaneously solving the coupled, nonlinear metric and scalar field equations of motion. In this article we prove that in linearly-coupled Einstein-dilaton-Gauss-Bonnet gravity, a black hole's scalar charge is completely determined by the horizon surface gravity times the Euler characteristic of the bifurcation surface, without solving any equations of motion. Within this theory, black holes announce their horizon topology and surface gravity to the rest of the universe through the dilaton field. In our proof, a 4-dimensional topological density descends to a 2-dimensional topological density on the bifurcation surface of a Killing horizon. We also comment on how our proof can be generalized to other topological densities on general G-bundles, and to theories where the dilaton is non-linearly coupled to the Euler density.