Tidally induced multipole moments of a nonrotating black hole vanish to all post-Newtonian orders

The tidal Love numbers of a black hole vanish, and this is often taken to imply that the hole's tidally induced multipole moments vanish also. An obstacle establishing link between these statements individual bodies are not defined in general relativity, when subjected mutual gravitational interaction. In previous publication [Phys. Rev. D 103, 064023 (2021)] I promoted view can be interaction sufficiently weak described by post-Newtonian expansion. view, compact body perceived far away as skeletonized object with structure, then related body's numbers. expand on demonstrate all static, induced, mass nonrotating orders. proof rests perturbative solution Einstein-Maxwell equations describes an electrically charged particle placed presence hole. attraction balanced electrostatic repulsion, system equilibrium state. provides environment hole, for environment. argue vanishing robust, applies slowly-varying environments. charge small desired (though identically zero); continuity, neutral will continue vanish.