Ladder symmetries of black holes. Implications for love numbers and nohair theorems
Abstract
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 nontrivial tidal response and scalar hair must vanish in order for the symmetries to be preserved.
 Publication:

Journal of Cosmology and Astroparticle Physics
 Pub Date:
 January 2022
 DOI:
 10.1088/14757516/2022/01/032
 arXiv:
 arXiv:2105.01069
 Bibcode:
 2022JCAP...01..032H
 Keywords:

 GR black holes;
 gravity;
 High Energy Physics  Theory;
 Astrophysics  Cosmology and Nongalactic Astrophysics;
 Astrophysics  High Energy Astrophysical Phenomena;
 General Relativity and Quantum Cosmology
 EPrint:
 Some sentences rephrased for clarity. Equations and conclusions unchanged