Naturalness of vanishing black-hole tides

Abstract: We provide a symmetry argument for the vanishing and non-renormalization of static Love numbers for spherically symmetric black holes at full nonlinear order in four-dimensional General Relativity. The symmetry is realized both in full GR and in the worldline EFT, allowing for a unified treatment and proving both vanishing and non-renormalization to all orders. This closes some loop-holes in previous arguments that neglected nonlinearities in the worldline EFT, and extends previous vanishing results to all nonlinear static tides. When extended to higher-dimensional gravity, these arguments also explain the pattern of vanishing and running static Love numbers of electric and tensor type, and predict new results at the nonlinear order. We also apply our findings to the tidal response of shift-symmetric scalar fields, predicting the vanishing of even order nonlinear static Love numbers and unifying these statements with the no-hair theorem (and its violations).