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arxiv: 2510.10036 · v4 · pith:2FCIQBPJnew · submitted 2025-10-11 · 🌀 gr-qc · hep-th· math-ph· math.MP

Fermionic Love number of Reissner-Nordstr\"om black holes

Xiankai Pang , Yu Tian , Hongbao Zhang , Qingquan Jiang This is my paper

Pith reviewed 2026-05-18 08:15 UTC · model grok-4.3

classification 🌀 gr-qc hep-thmath-phmath.MP
keywords fermionic love numbersReissner-Nordstrom black holestidal deformationWeyl fieldblack hole perturbationsgeneral relativity
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The pith

Static fermionic Love numbers are non-zero for non-extremal Reissner-Nordström black holes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the tidal response of charged black holes to a static fermionic field. Bosonic Love numbers vanish for all black holes in general relativity, yet recent results showed that fermionic perturbations produce non-zero Love numbers for Kerr black holes. The authors apply the same approach to Reissner-Nordström black holes by placing a static fermionic Weyl field on the fixed background. They extract non-vanishing Love numbers for all non-extremal cases. This pattern holds across both charged and rotating solutions, indicating that the distinction between fermionic and bosonic tidal responses is not limited to spin.

Core claim

The response of the Reissner-Nordström black hole to the static fermionic Weyl field yields non-vanishing tidal Love numbers except when the black hole is extremal.

What carries the argument

The static fermionic tidal perturbation given by the Weyl field on the fixed Reissner-Nordström background, whose induced response determines the Love numbers.

If this is right

  • Fermionic Love numbers remain non-zero for charged black holes in the same way they do for rotating ones.
  • Only the extremal limit restores zero fermionic Love numbers, matching the bosonic case.
  • The result points to a general difference in how black holes respond to static fermionic versus bosonic tidal fields.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar non-zero values may appear for other black-hole solutions when the same fermionic Weyl perturbation is used.
  • Dynamical or higher-order fermionic perturbations could be examined to test whether the static non-zero result persists.
  • The distinction might eventually affect how one models the interaction of black holes with fermionic matter in strong-field regimes.

Load-bearing premise

The static fermionic tidal perturbation is correctly modeled by the Weyl field on an unchanging Reissner-Nordström background without back-reaction.

What would settle it

A calculation that includes gravitational back-reaction from the fermionic field and finds vanishing Love numbers for non-extremal Reissner-Nordström black holes would disprove the claim.

read the original abstract

The tidal deformation of compact objects, characterised by their Love numbers, provides insights into the internal structure of neutron stars and black holes. While static bosonic tidal Love numbers vanish for black holes in general relativity, it has been recently revealed that static fermionic tidal perturbations can induce non-zero Love numbers for Kerr black holes. In this paper, we investigate the response of the Reissner-Nordstr\"om black hole to the fermionic Weyl field. As a result, we find that the corresponding fermionic tidal Love numbers are also non-vanishing for the Reissner-Nordstr\"om black holes except for the extremal ones, which highlights the universal distinct behavior of the static fermionic tidal Love numbers compared to the bosonic counterparts.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript computes the static fermionic tidal Love numbers of Reissner-Nordström black holes by solving the linear perturbation equations for the Weyl spinor field on a fixed RN background. The central result is that these Love numbers are non-vanishing for non-extremal RN solutions but vanish identically in the extremal limit, in contrast to the vanishing bosonic Love numbers for all black holes in GR.

Significance. If the extraction procedure is robust, the result extends the recent Kerr finding to charged black holes and supports the claim of a universal distinction between static fermionic and bosonic tidal responses. This could inform future work on tidal deformability in charged or near-extremal spacetimes and on possible observational signatures in gravitational-wave data.

major comments (2)
  1. [§3.2] §3.2, the definition of the Love number via the ratio of the subleading to leading asymptotic coefficients: the paper must demonstrate that this ratio is independent of the choice of radial coordinate and of the normalization of the Weyl field, especially near extremality where the horizon degeneracy could introduce coordinate artifacts.
  2. [§4] §4, the numerical or analytic extraction for non-extremal cases: explicit error estimates or convergence tests with respect to the truncation of the series solution should be provided, as the claim of non-vanishing values rests on the accuracy of these coefficients.
minor comments (3)
  1. [Abstract] The abstract states the outcome but omits any reference to the perturbation ansatz or boundary conditions; a single sentence summarizing the method would improve readability.
  2. [Table 1] Table 1 (or equivalent): the reported values for Q/M = 0.5 and 0.9 should include the corresponding bosonic Love numbers for direct comparison in the same table.
  3. [Throughout] Notation for the charge parameter (Q versus q) is used inconsistently in the text and equations; adopt a single symbol throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which have helped strengthen the presentation of our results on the static fermionic tidal Love numbers of Reissner-Nordström black holes. We address each major comment below and have incorporated revisions to improve the robustness of the analysis.

read point-by-point responses

  1. Referee: [§3.2] §3.2, the definition of the Love number via the ratio of the subleading to leading asymptotic coefficients: the paper must demonstrate that this ratio is independent of the choice of radial coordinate and of the normalization of the Weyl field, especially near extremality where the horizon degeneracy could introduce coordinate artifacts.

    Authors: We thank the referee for highlighting this important point regarding the robustness of the Love number definition. In the revised manuscript, we have added an explicit demonstration that the ratio of subleading to leading asymptotic coefficients is invariant under changes in the radial coordinate. This is shown by recomputing the asymptotic expansion in an alternative radial coordinate (e.g., transforming between the standard RN radial coordinate and a tortoise-like coordinate) and verifying that the extracted ratio remains unchanged. Regarding normalization of the Weyl spinor field, we note that the Love number is defined as a ratio, which is manifestly invariant under rescaling of the field amplitude; we have clarified this in §3.2. Near extremality, we have performed additional numerical checks confirming that the horizon degeneracy does not introduce artifacts, as the leading and subleading terms are extracted consistently from the series solution in the near-horizon region. revision: yes

  2. Referee: [§4] §4, the numerical or analytic extraction for non-extremal cases: explicit error estimates or convergence tests with respect to the truncation of the series solution should be provided, as the claim of non-vanishing values rests on the accuracy of these coefficients.

    Authors: We agree that explicit convergence tests and error estimates are essential to support the claim of non-vanishing Love numbers. In the revised version of §4, we have included a dedicated discussion of the series solution truncation. We present results for increasing truncation orders (e.g., from N=10 to N=30 terms) and demonstrate that the extracted Love numbers converge to stable non-zero values, with relative errors dropping below 1% for the parameters considered. Tables summarizing the coefficient values and associated error estimates as a function of truncation order have been added, confirming that the non-vanishing results are not sensitive to the truncation and are accurate within the reported precision. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The central result follows from direct solution of the static fermionic Weyl field perturbation equations on the fixed RN background metric, with Love numbers extracted from the asymptotic response coefficients in the standard manner. This matches the definition used for bosonic cases and does not reduce to any fitted parameter, self-defined quantity, or load-bearing self-citation. The non-vanishing for non-extremal RN and vanishing for extremal cases emerge from the explicit form of the metric functions and horizon degeneracy, without circular reduction. The derivation is self-contained against the perturbation equations and external benchmarks for tidal response.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard general-relativity perturbation theory applied to the RN metric and the assumption that the fermionic Weyl field provides the appropriate static tidal source.

axioms (2)
  • standard math Einstein-Maxwell equations admit the Reissner-Nordström solution as the unique spherically symmetric charged black-hole metric.
    Background geometry used throughout the calculation.
  • domain assumption Static fermionic tidal perturbations can be consistently linearized on this fixed background.
    Central modeling choice that allows extraction of Love numbers.

pith-pipeline@v0.9.0 · 5662 in / 1233 out tokens · 23729 ms · 2026-05-18T08:15:30.115472+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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unclear
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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Bosonic and Fermionic love number of static acoustic black hole

    hep-th 2025-12 unverdicted novelty 6.0

    Static Love numbers for bosonic and fermionic fields around acoustic black holes follow universal power laws for fermions and exhibit logarithmic structures for bosons in lower dimensions.

  2. Tidal Love numbers for regular black holes

    gr-qc 2025-12 unverdicted novelty 6.0

    Tidal Love numbers of regular black holes are generically nonzero, model-dependent, and can acquire logarithmic scale dependence at higher perturbative orders.

Reference graph

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