Tidal deformations of slowly spinning isolated horizons

It is generally believed that tidal deformations of a black hole in an external field, as measured using its gravitational field multipoles, vanish. However, this does not mean that the black hole horizon is not deformed. Here we shall discuss the deformations of a black hole horizon in the presence of an external field using a characteristic initial value formulation. Unlike existing methods, the starting point here is the black hole horizon itself. The effect of, say, a binary companion responsible for the tidal deformation is encoded in the geometry of the spacetime in the vicinity of the horizon. The near horizon spacetime geometry, i.e. the metric, spin coefficients, and curvature components, are all obtained by integrating the Einstein field equations outwards starting from the horizon. This method yields a reformulation of black hole perturbation theory in a neighborhood of the horizon. By specializing the horizon geometry to be a perturbation of Kerr, this method can be used to calculate the metric for a tidally deformed Kerr black hole with arbitrary spin. As a first application, we apply this formulation here to a slowly spinning black hole and explicitly construct the spacetime metric in a neighborhood of the horizon. We propose natural definitions of the electric and magnetic surficial Love numbers based on the Weyl tensor component $\Psi_2$. From our solution, we calculate the tidal perturbations of the black hole, and we extract both the field Love numbers and the surficial Love numbers which quantify the deformations of the horizon.