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

In beyond-GR theories with a scalar field, a black hole's scalar charge is a crucial quantity for testing the theory. The scalar charge---given by an integral at spatial infinity---controls the dynamics and gravitational radiation of black hole--compact object binaries. Extracting this scalar charge would seem to require solving the field equations for a black hole spacetime. In this talk I will show that in the special case of Einstein-dilaton-Gauss-Bonnet theory, the scalar charge can instead be extracted from a horizon integral. This relationship between an integral at infinity and one on the horizon is reminiscent of mass and spin integrals in GR.