  It is well known that asymptotically flat black holes in general relativity
have a vanishing static, conservative tidal response. We show that this is a
result of linearly realized symmetries governing static (spin 0,1,2)
perturbations around black holes. The symmetries have a geometric origin: in
the scalar case, they arise from the (E)AdS isometries of a dimensionally
reduced black hole spacetime. Underlying the symmetries is a ladder structure
which can be used to construct the full tower of solutions, and derive their
general properties: (1) solutions that decay with radius spontaneously break
the symmetries, and must diverge at the horizon; (2) solutions regular at the
horizon respect the symmetries, and take the form of a finite polynomial that
grows with radius. Taken together, these two properties imply that static
response coefficients -- and in particular Love numbers -- vanish. Moreover,
property (1) is consistent with the absence of black holes with linear
(perturbative) hair. We also discuss the manifestation of these symmetries in
the effective point particle description of a black hole, showing explicitly
that for scalar probes the worldline couplings associated with a non-trivial
tidal response and scalar hair must vanish in order for the symmetries to be
preserved.
