Black Holes in a Gravitational Field: The Non-linear Static Love Number of Schwarzschild Black Holes Vanishes

We show that the static tidal Love number of Schwarzschild black holes in four dimensions and in the vacuum vanishes at any order in the external tidal force. We also identify the underlying non-linear symmetry which is responsible for this result and becomes manifest when the black hole metric is written in axsymmetric static Weyl coordinates.