Fermionic Love number of higher-dimensional Reissner-Nordström black holes

Technical Summary: Fermionic Love Number of Higher-Dimensional Reissner-Nordström Black Holes

Problem Statement
Tidal Love numbers (TLNs) quantify the deformation of compact objects under external tidal fields and serve as critical probes for distinguishing black holes from neutron stars in gravitational-wave astronomy. While it is well-established that static bosonic TLNs (for scalar, electromagnetic, and gravitational perturbations) vanish identically for four-dimensional asymptotically flat black holes due to hidden ladder symmetries, non-trivial TLNs can arise in higher dimensions, modified gravity, or for charged scalar fields. Recently, it was discovered that static fermionic perturbations can induce non-zero TLNs even in four-dimensional Schwarzschild black holes. However, the behavior of these fermionic TLNs in higher-dimensional spacetimes, particularly for charged (Reissner-Nordström) black holes, remained unexplored. This paper addresses the gap by generalizing the study of static fermionic TLNs to $D$-dimensional Reissner-Nordström (RN) black holes.

Methodology
The authors employ a rigorous analytical approach to solve the massless Dirac equation in $D$-dimensional RN spacetime:

1. Metric and Coordinates: The study utilizes the RN metric in ingoing Eddington coordinates ($v, r$) to ensure regularity at the horizon. The electromagnetic potential is included for completeness, though the fermion is electrically neutral.
2. Conformal Transformation: To handle the spinor field's behavior near the horizon, the authors apply a conformal transformation ($\tilde{g}_{\mu\nu} = \Omega^2 g_{\mu\nu}$ with $\Omega = 1/r$). This allows the Dirac equation to be rewritten in a conformally transformed background where the spinor components remain finite at the horizon if they are non-singular in the transformed frame.
3. Variable Separation: Using a specific choice of regular tetrads and explicit representations of gamma matrices for both even and odd dimensions, the Dirac operator is decomposed. The angular part is separated using eigen-spinors of the Dirac operator on the $(D-2)$-dimensional unit sphere, characterized by a total angular momentum $l$ (a half-integer). This reduces the problem to an effective two-dimensional spacetime.
4. Radial Equation: For massless and static perturbations ($m=\omega=0$), the radial equation is derived. By introducing a dimensionless variable $z$ related to the horizon radii, the equation is transformed into a hypergeometric differential equation.
5. Extraction of TLNs: The authors identify the regular solution branch at the outer horizon by imposing boundary conditions that prevent divergence. The static fermionic TLNs are then extracted from the asymptotic behavior of this regular solution at spatial infinity ($z \to \infty$), specifically by isolating the coefficient of the decaying tidal response term.

Key Contributions and Results
The paper derives the analytical expression for the static fermionic TLNs ($F_{\pm 1/2, lm}$) for $D$-dimensional RN black holes. The results reveal a distinct structure compared to bosonic perturbations:

• Non-Vanishing Nature: Unlike bosonic TLNs, which vanish for specific integer values of the angular momentum in dimensions $D > 4$ (specifically when $l/(D-3)$ is an integer) and identically in $D=4$, the static fermionic TLNs are non-zero for all total angular momenta $l \geq 1/2$ and all dimensions $D \geq 4$, provided the black hole is non-extremal.
• Dimensional Dependence: The TLNs exhibit a rich dependence on the spacetime dimension $D$. The magnitude of the TLNs decreases monotonically as the dimensionless charge $q = Q/M$ increases, vanishing only in the extremal limit ($q \to 1$).
• Angular Momentum Sensitivity: As the spacetime dimension $D$ increases, the sensitivity of the TLNs to the total angular momentum $l$ weakens. In the infinite-dimensional limit ($D \to \infty$), the static fermionic TLNs become independent of $l$.
• Extremal Limit: The paper confirms that for extremal RN black holes ($Q=M$), the static fermionic TLNs vanish, consistent with the behavior of the regular solution at the horizon in the extremal case.

Significance and Claims
The authors claim that these results provide new insights into the tidal deformability of black holes in higher-dimensional spacetimes. The primary significance lies in demonstrating that fermionic tidal responses possess a robust, non-vanishing structure across all dimensions and angular momenta (except for the extremal case), contrasting sharply with the restrictive vanishing conditions found in bosonic cases. This suggests that fermionic perturbations offer a unique window into black hole physics in higher dimensions, potentially distinguishing them from bosonic responses. The paper concludes by noting that while the current analysis is limited to static, massless fermions in non-rotating RN black holes, it opens avenues for future research into rotating black holes, massive fermions, and the connection between these static responses and quantum gravity effects.