The geometry of deformed isolated horizons and quasi-local perturbation theory

It is generally believed that tidal deformations of a black hole in an external field, measured using the gravitational field multipoles, vanish. However, this does not mean that the black hole horizon is not deformed. In this talk, I shall discuss the deformations of a black hole horizon in the presence of an external field using a new method based on the characteristic initial value formulation. Unlike standard methods, the starting point here is the black hole horizon itself. The presence of the companion binary responsible for the tidal deformation is encoded on the geometry of the spacetime in the vicinity of the horizon, which is obtained by integrating the Einstein fields equations analytically outwards starting from the horizon. This method yields a powerful reformulation of black hole perturbation theory in a neighborhood of the horizon, which is quasi-local, geometrical, and completely general. For instance, by specializing the horizon geometry to be a perturbation of Kerr yields the metric for a tidally deformed Kerr black hole with arbitrary spin. In this talk, I will discuss the essential ideas in the application of the characteristic initial value formalism to black hole perturbation theory, and summarize the latest results.