Fermionic Love Numbers in Kerr Geometry

• Fermionic Love Numbers are dimensionless quantities that quantify the nonzero, conservative tidal response of Kerr black holes to static fermionic (half-integer spin) perturbations.
• They are derived by analytically continuing the static Teukolsky equation into a hypergeometric form and enforcing ingoing boundary conditions at the horizon.
• The resulting nonzero response indicates a breaking of the ladder symmetry present in bosonic cases, offering new insights into black hole microstructure and effective field theory couplings.

Fermionic Love numbers quantify the conservative tidal response of a black hole to static external perturbations by fermionic (half-integer spin) fields. In contrast to the vanishing tidal Love numbers for static scalar, electromagnetic, and gravitational (bosonic) perturbations of black holes, the fermionic sector exhibits a nonzero, real Love number, reflecting a fundamental asymmetry in the response properties of black-hole spacetimes to bosonic versus fermionic fields. This distinction arises from the breaking of hidden symmetries—specifically, the ladder symmetry present in the static Teukolsky equation for integer spin—which enforces the vanishing of bosonic Love numbers but not their fermionic counterparts in Kerr backgrounds.

1. Definition and Conceptual Foundation

The classical notion of a tidal Love number, originating in Newtonian gravitational theory, relates the induced multipole moment of a gravitating body to the applied external tidal field. Love numbers are dimensionless quantities encoding information about the internal structure and deformability of compact objects. Extending this to General Relativity, the relativistic Love numbers characterize the static, conservative susceptibility of a spacetime to external perturbations, with precise gauge-invariant definitions enabling their computation in strongly curved backgrounds (0906.1366).

For black holes, all bosonic (spin-0, spin-1, spin-2) Love numbers vanish identically—both for Schwarzschild and Kerr metrics—reflecting profound constraints from symmetry and boundary conditions. However, the introduction of half-integer spin—i.e., fermionic—fields reveals nontrivial conservative susceptibilities. The fermionic Love numbers specifically parametrize the amplitude of the subleading, decaying term in the asymptotic expansion of the field at infinity, normalized by the incident amplitude, for static fermionic perturbations of a Kerr black hole (Chakraborty et al., 27 Aug 2025).

2. Mathematical Derivation in Kerr Geometry

The determination of fermionic Love numbers proceeds via the Teukolsky master equation, governing perturbations of type spin $s$ in the Kerr spacetime. After separation of variables in Boyer–Lindquist coordinates, the static ($\omega\to0$) radial equation is transformed into hypergeometric form via the substitution $z = (r - r_+)/(r_+ - r_-)$:

$$R_s(z) = c_1 z^{-iP} {}_2F_1(A, B; C; -z) + c_2 z^{s + iP} {}_2F_1(A', B'; C'; -z)$$

where $P = a m/(r_+ - r_-)$, with $a$ the black hole's spin parameter, $m$ the azimuthal quantum number, and $r_+$, $r_-$ the outer and inner horizons of Kerr.

Boundary conditions require regularity on the future horizon (selecting the ingoing mode) and control of the asymptotic behavior at spatial infinity. The asymptotic expansion at large radius reads: $$R_s \sim \mathcal{E}_{s\ell m} \, r^{\ell - s} [1 + \cdots] + \mathcal{F}_{s\ell m} \, r^{-\ell - s - 1} [1 + \cdots]$$ The fermionic Love number is then defined as the ratio $\mathcal{F}_{s\ell m}/\mathcal{E}_{s\ell m}$, capturing the amplitude of the response to the static perturbation.

A crucial step is the analytic continuation of the multipole index $\ell$ from integer to half-integer values. In the bosonic case (integer $\ell$), a hidden ladder symmetry in the static Teukolsky equation enforces perfect cancellation of the conservative response (all Love numbers vanish). In the fermionic case (half-integer $\ell$; e.g., for the Dirac equation $s=\pm1/2$, $\ell=1/2$), the ladder symmetry is broken, leading to a nonvanishing result. The closed-form expression found for the response coefficient (and thus the Love number) is:

$$\mathcal{F}_{s\ell m} = (-1)^{\ell-s} \frac{(\ell-s)! (\ell+s)!}{(2\ell)! (2\ell+1)! } \left( \frac{2am}{r_+} \right)^{2\ell+1} \prod_{k=1}^{\ell-1/2} \left[ k^2 (1 - r_-/r_+)^2 + \left( \frac{2am}{r_+} \right)^2 \right]$$

This expression depends nontrivially on the black hole spin, the quantum numbers, and the geometry of the Kerr horizon.

3. Physical Implications and Distinctions

The discovery of nonzero fermionic Love numbers has significant conceptual and practical consequences:

• For black holes, the absence of tidal deformability to static bosonic fields is contrasted by their finite, conservative tidal response to static fermionic fields. This highlights a sectoral breaking of hidden symmetries and points to deeper structure in the field-theoretic description of black-hole physics.
• The conservative nature of the fermionic tidal response is underscored by the vanishing of the dissipative Love (or dissipation) numbers. Unlike bosonic fields, which can exhibit superradiant amplification and associated dissipative response in the presence of rotation, fermionic (Dirac or higher-half-integer) fields do not support superradiance. Consequently, the response function is entirely real (Chakraborty et al., 27 Aug 2025).
• For neutron stars and other compact objects with a polytropic equation of state, both electric-type ($k_{el}$) and magnetic-type ($k_{mag}$) relativistic Love numbers are nonzero and encode the internal structure (compactness, EOS stiffness) (0906.1366). In the black-hole limit, these reduce to zero in the bosonic sector but remain nonzero for fermionic perturbations in Kerr.

4. Computational Procedure and Key Results

The computation of fermionic Love numbers for a Kerr black hole involves:

1. Starting from the static Teukolsky equation in the desired spin sector;
2. Solving the radial ODE as a hypergeometric equation, with analytic continuation in $\ell$ to handle half-integer cases;
3. Imposing ingoing boundary conditions at the horizon and extracting the large-$r$ expansion;
4. Calculating the ratio $\mathcal{F}_{s\ell m}/\mathcal{E}_{s\ell m}$ as the Love number, with closed-form expressions in terms of elementary functions, factorials, and products over mode indices;
5. Verifying, via algebraic identities, the vanishing of the ratio for all integer $\ell$ (bosons) and its nonvanishing value for half-integer $\ell$ (fermions) (Chakraborty et al., 27 Aug 2025).

In the context of neutron stars, the relativistic Love numbers are obtained by integrating the perturbed field equations inside the star for a given polytropic equation of state, matched at the surface to an exterior analytic solution. High-precision numerical methods (e.g., Bulirsh–Stoer, embedded Runge–Kutta) achieve accuracy at the $10^{-9}$ level in the extracted Love numbers (0906.1366).

5. Interpretation in Symmetry and Effective Field Theories

The presence or absence of tidal response in the conservative sector reflects underlying symmetry structures of the field equations in curved spacetime:

• The vanishing of bosonic Love numbers in Kerr is linked to a hidden ladder symmetry (or conformal symmetry) in the near-horizon region. Integer-spin static perturbations possess a ladder structure that enforces the precise cancellation of the induced moment/response.
• The fermionic sector (half-integer spin) breaks this ladder symmetry, permitting a finite, nonzero response. This suggests a fundamental difference in the near-horizon conformal properties experienced by fermions as compared to bosons.
• From the perspective of effective field theory, the existence of nonvanishing point-particle response coefficients (Love numbers) for black holes perturbed by fermions opens the possibility of additional worldline couplings, distinct from those controlling the bosonic tidal sector.

A plausible implication is that future probes sensitive to fermionic fields—potentially realized in extensions of black-hole perturbation theory or fundamental-physics-motivated observations—could access new information about black-hole microstructure and hidden symmetry breaking.

6. Broader Impact and Future Directions

The identification and closed-form characterization of fermionic Love numbers for rotating black holes represent an advancement in understanding the sectoral properties of tidal responses in General Relativity. This provides analytical footholds for examining:

• The role of hidden symmetries in horizon physics and their breaking for discrete spin classes;
• The detailed microphysical response of black holes, potentially relevant in high-energy and quantum extensions of gravity;
• The foundations of black-hole effective field theory, as the nonzero fermionic Love numbers signal additional response operators absent for bosonic fields.

While no direct observational consequences are currently feasible—owing to the absence of known astrophysical fermionic fields coupling to black-hole environments at appreciable strength—this suggests a new avenue of theoretical investigation into the interface between spin, symmetry, and horizon physics. The contrast with bosonic responses also mandates a re-examination of the role of spin-statistics in gravitational and quantum-gravitational dynamics.

Field Type	Spin	Love Number	Superradiance	Symmetry Broken
Scalar (boson)	0	0	Yes (spinning BH)	No
Electromagnetic	1	0	Yes	No
Gravitational	2	0	Yes	No
Fermionic	1/2	Nonzero (real)	No	Yes

The distinction in response elucidated by the fermionic Love numbers is a window into the deeper structure of spacetime and field interactions in strong gravity environments.