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arxiv: 2606.24365 · v1 · pith:JIPVB57Vnew · submitted 2026-06-23 · 🌀 gr-qc · hep-th

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

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

Pith reviewed 2026-06-25 23:00 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords fermionic tidal Love numbershigher-dimensional black holesReissner-Nordström metricDirac equationtidal responsehigher dimensionsextremal black holes
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0 comments X

The pith

Fermionic tidal Love numbers of higher-dimensional charged black holes remain non-zero for every angular momentum except in the extremal limit.

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

The paper solves the massless Dirac equation for perturbations of D-dimensional Reissner-Nordström black holes in ingoing Eddington coordinates. It isolates the regular solution branch at the horizon and reads the tidal Love number from the leading coefficient in the far-field expansion. The resulting numbers stay finite and non-vanishing for all D ≥ 4 and all half-integer l, unlike the bosonic Love numbers that vanish for selected l when D > 4. The dependence on l becomes weaker with rising D and vanishes completely when D tends to infinity. These properties hold for any non-extremal charge.

Core claim

We generalize fermionic tidal Love numbers to higher-dimensional Reissner-Nordström black holes by solving the massless Dirac equation with ingoing Eddington coordinates and regular tetrads. After selecting the regular solution branch, the Love numbers are extracted from the asymptotic behavior at infinity. The resulting TLNs remain non-zero for all l and all D ≥ 4 except for extremal black holes, and the l-dependence weakens with increasing D until it disappears in the infinite-dimensional limit.

What carries the argument

The regular branch of solutions to the massless Dirac equation in D-dimensional RN spacetime, whose leading far-field coefficient directly supplies the fermionic tidal Love number.

If this is right

  • Fermionic perturbations always produce a non-vanishing tidal response on non-extremal RN black holes in any dimension and for any multipole.
  • Only the extremal RN family has vanishing fermionic TLNs, independent of dimension.
  • In the infinite-D limit the fermionic Love number becomes independent of the angular momentum quantum number l.
  • The contrast with bosonic TLNs, which vanish for certain l when D > 4, persists in every finite dimension.

Where Pith is reading between the lines

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

  • The result suggests that fermionic and bosonic tidal responses remain qualitatively distinct even when extra dimensions are added.
  • One could check whether the same non-vanishing behavior appears for massive Dirac fields or for rotating higher-dimensional black holes.
  • The disappearance of l-dependence at large D may connect to the known simplification of black-hole dynamics in the infinite-dimensional limit.

Load-bearing premise

The regular solution at the horizon can be uniquely identified and its far-field leading coefficient isolates the pure tidal response without contamination from other modes or coordinate effects.

What would settle it

An explicit asymptotic expansion of the Dirac solution for a non-extremal five-dimensional RN black hole with l = 1/2 must produce a non-zero fermionic Love number, while the same calculation for the extremal case must give exactly zero.

read the original abstract

In this paper, we generalize our previous work on the fermionic tidal Love numbers (TLNs) to higher-dimensional Reissner-Nordstr\"om black holes. The massless Dirac equation is solved in $D$-dimensional spacetime using ingoing Eddington coordinates and regular tetrads. After identifying the regular solution branch, we extract the fermionic TLNs from its asymptotic behavior at infinity. The resulting TLNs exhibit a rich dimension-dependent structure that generalizes the four-dimensional case. Unlike bosonic TLNs, which vanish for certain values of the total angular momentum $l$ in dimensions $D>4$, fermionic TLNs remain non-zero for all $l$ and $D \geq 4$, except for extremal black holes. Moreover, the $l$-dependence weakens as $D$ increases, disappearing entirely in the infinite-dimensional limit. These results provide new insights into black hole responses to fermionic perturbations in higher-dimensional spacetimes.

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

0 major / 3 minor

Summary. The manuscript generalizes the computation of fermionic tidal Love numbers (TLNs) to higher-dimensional Reissner-Nordström black holes. It solves the massless Dirac equation in D-dimensional spacetime using ingoing Eddington coordinates and regular tetrads, identifies the regular solution branch at the horizon, and extracts the TLNs from the leading far-field asymptotic coefficient at infinity. The central results are that fermionic TLNs remain non-zero for all angular momentum l and all D ≥ 4 (except extremal cases), in contrast to bosonic TLNs which vanish for certain l when D > 4, and that the l-dependence of the fermionic TLNs weakens with increasing D and disappears entirely in the infinite-D limit.

Significance. If the calculations hold, the work provides a direct higher-dimensional extension of the authors' prior 4D fermionic TLN results, highlighting qualitative differences from the bosonic case and a simplifying behavior in the large-D limit. These findings could inform analyses of fermionic perturbations in higher-dimensional gravity and related holographic models. The reliance on the standard analytic procedure for the Dirac equation (with explicit branch identification) is a methodological strength.

minor comments (3)
  1. The abstract and introduction describe the D=4 reduction but supply no explicit comparison (numerical values or analytic match) to the authors' prior 4D fermionic TLN results; adding such a verification table or equation would confirm the higher-D implementation.
  2. The extraction of the TLN from the leading asymptotic coefficient is stated without accompanying error estimates or convergence checks for the analytic expressions in D > 4; including these would support the dimension-dependent claims.
  3. Notation for the total angular momentum l and the dimension D should be introduced with a brief reminder of the distinction from the bosonic case to improve readability for readers familiar with the bosonic TLN literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of our results on fermionic tidal Love numbers in higher-dimensional Reissner-Nordström spacetimes and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's derivation consists of directly solving the massless Dirac equation in D-dimensional RN spacetime using ingoing Eddington coordinates and regular tetrads, identifying the regular ingoing branch at the horizon, and extracting the TLN from the leading far-field asymptotic coefficient. This procedure is self-contained and does not reduce any reported TLN value to a fitted parameter, a self-citation, or a prior result by algebraic construction; the dimension dependence (non-vanishing for all l, weakening with D) follows from the explicit solution once the branch is fixed. The mention of generalizing prior 4D work is a contextual statement only and does not serve as a load-bearing premise that forces the higher-D outcomes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The calculation relies on standard differential-equation techniques and background assumptions of general relativity in D dimensions; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • standard math The Dirac equation in curved spacetime is well-defined and can be written using a regular tetrad in ingoing Eddington coordinates.
    Invoked to set up the perturbation equation whose solution yields the Love numbers.
  • domain assumption A unique regular solution branch exists at the horizon whose asymptotic behavior at infinity directly encodes the tidal response.
    Central to extracting the Love number from the far-field expansion.

pith-pipeline@v0.9.1-grok · 5700 in / 1495 out tokens · 32053 ms · 2026-06-25T23:00:32.976562+00:00 · methodology

discussion (0)

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Reference graph

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