Ladder symmetries of black holes. Implications for love numbers and no-hair theorems

• Published in: Journal of cosmology and astroparticle physics 2022-01, Vol.2022 (1), p.32
• Main Authors: Hui, Lam, Joyce, Austin, Penco, Riccardo, Santoni, Luca, Solomon, Adam R.
• Format: Article
• Language: English
• Subjects: ASTRONOMY AND ASTROPHYSICS; Black holes; Couplings; GR black holes; gravity; Hair; Horizon; Love number; Perturbation; Polynomials; Relativity
• ISSN: 1475-7516; 1475-7516
• DOI: 10.1088/1475-7516/2022/01/032

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.