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) alone is sufficient to forbid the existence 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.
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