This work belongs to a series of studies on the tidal response of black holes. Neighboring papers show that bosonic Love numbers vanish for Kerr black holes, while dissipative numbers remain nonzero due to frame dragging. Our paper fills a gap: until now, the fermionic response had not been investigated, and we demonstrate for the first time its difference from the bosonic one.
We used the analytical solution of the Teukolsky equation in the static limit for arbitrary spin s. The response is extracted from the asymptotic behavior of the radial function at large distances. To remove coordinate ambiguity, we performed an analytic continuation of the angular momentum ℓ into the complex plane. The fermionic case corresponds to half-integer s, ℓ, m.
For fermionic perturbations (s = ±1/2, ±3/2), the response function turns out to be real and nonzero. In the Schwarzschild limit for spin 1/2, we obtain a simple expression: F = ±$4^{-2ℓ-1}$, independent of the azimuthal number m. For an extremal black hole, the response is finite and grows exponentially with ℓ when ℓ = m. Dissipative numbers for static fermionic perturbations are identically zero, contrasting with the bosonic case.
This work shows for the first time that black holes in general relativity can have a nonzero conservative response to fermionic fields. This distinguishes it from neighboring works, which considered only bosonic perturbations and always found vanishing Love numbers. The result indicates a breaking of the ladder symmetries that explain the zero bosonic Love numbers, and suggests the possible existence of fermionic "hair" on black holes.
It is natural to extend the analysis to charged black holes, where the coupling of fermionic and electromagnetic perturbations could give rise to mixed Love numbers. Also promising is embedding into an effective field theory worldline to compute the influence of fermionic Love numbers on the dynamics of binary systems and gravitational-wave signals.
The results may affect the phenomenology of strong gravitational fields, tests of general relativity, and the interpretation of gravitational-wave observations, especially for rapidly rotating black holes.
Check whether the nonzero fermionic Love numbers persist at the nonlinear level, and investigate the possibility of constructing a complete set of fermionic "hair" for all multipoles.
This work links the vanishing of bosonic Love numbers to the no-hair theorem and shows that fermions may bypass this restriction. The question of the existence of nontrivial fermionic configurations beyond the linear approximation remains open.
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
This paper computes the tidal response of a black hole to fermionic perturbations. The analysis uses the analytic solution of the Teukolsky equation in the static limit for arbitrary spin, and the response is extracted from the asymptotic behavior at infinity. It is shown that the conservative Love numbers for fermions are nonzero, while the dissipative ones vanish.
Where its strength is
For the first time, it is demonstrated that black holes in general relativity can have a nonzero conservative response to fermionic fields, indicating a breaking of the ladder symmetries responsible for the vanishing bosonic Love numbers. This opens a new direction in the study of black hole tidal properties.
Where it could go next
Worth trying
Why this may matter
The discovery of a nonzero fermionic response could change the picture of black hole 'no-hair' and point to new observational manifestations in astrophysics, for example in gravitational wave spectra or in neutrino fluxes from accretion disks.
Nearby works in our archive
The archive is sparse around this work: the closest review is only 0.54, against a usual 0.64. It looks like few have drilled nearby.
For a Schwarzschild black hole, the fermionic Love number for spin 1/2 equals ±1/16 in the lowest multipole, sharply contrasting with the zero bosonic response.