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arxiv: 2605.29759 · v1 · pith:QXTKRJEOnew · submitted 2026-05-28 · 🌀 gr-qc

Gyroscopic Precession in Axisymmetric Kerr Spacetime: Horizon Regularity and Coordinate Effects

Paulami Majumder This is my paper

Pith reviewed 2026-06-29 06:41 UTC · model grok-4.3

classification 🌀 gr-qc
keywords gyroscopic precessionKerr spacetimeFrenet-Serret formalismevent horizoncoordinate effectsBoyer-Lindquist coordinatesKerr-Schild coordinatestimelike trajectories
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The pith

Divergence of gyroscopic precession near the Kerr horizon disappears in horizon-penetrating coordinates.

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

The paper calculates gyroscopic precession frequency for both Killing and non-Killing timelike trajectories in Kerr spacetime using the Frenet-Serret formalism. In Boyer-Lindquist coordinates the frequency diverges as trajectories approach the horizon, but the same calculation in Kerr-Schild coordinates yields a finite result for generic spiral paths. The authors conclude that finiteness is controlled by the timelike character of the trajectory, not by the presence of the event horizon. This shows the divergence cannot serve as a coordinate-independent signature of horizon structure.

Core claim

In axisymmetric Kerr spacetime the Frenet-Serret precession frequency along generic spiral trajectories diverges near the event horizon in Boyer-Lindquist coordinates yet remains finite in Kerr-Schild coordinates. The difference arises solely from the regularity properties of the coordinate chart at the horizon; once the trajectory remains timelike, the frequency stays well-defined regardless of whether an event horizon is crossed.

What carries the argument

Covariant Frenet-Serret formalism applied to timelike four-velocities in Boyer-Lindquist versus Kerr-Schild coordinates, isolating the effect of coordinate regularity on the precession frequency.

If this is right

  • Precession remains finite for any timelike observer that reaches or crosses the horizon in a regular coordinate system.
  • Divergence in the strong-field regime cannot be treated as an invariant diagnostic of horizon structure.
  • The same coordinate dependence appears for both prograde and retrograde motions.
  • Results hold for both geodesic (Killing) and accelerated (non-Killing) trajectories.

Where Pith is reading between the lines

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

  • Other proposed strong-field signatures that rely on frame or curvature quantities may need re-examination in multiple coordinate systems.
  • Observers using horizon-penetrating coordinates could measure finite precession rates even very close to the horizon.
  • The finding suggests that coordinate artefacts can mimic physical effects in any calculation that mixes frame-dependent quantities with singular charts.

Load-bearing premise

The Frenet-Serret equations remain valid and free of extra singularities for both Killing and non-Killing timelike paths after the coordinate change.

What would settle it

Explicit numerical evaluation of the precession frequency for one chosen spiral trajectory that crosses the horizon radius; the value must remain finite throughout the crossing when computed in Kerr-Schild coordinates.

Figures

Figures reproduced from arXiv: 2605.29759 by Paulami Majumder.

Figure 1
Figure 1. Figure 1: Variation of angular velocity ω± given in Eq. (7) as a function of r for different values of black hole spin parameter a. The difference between the two distinct roots given in Eq. (7) can be evaluated as, ω+ − ω− = − 2 q g 2 tϕ − gttgϕϕ gϕϕ . (8) The discriminator q g 2 tϕ − gttgϕϕ of Eq.(8) is positive for real roots. From our usual convention of metric signature, gϕϕ < 0. As a consequence, the differenc… view at source ↗
Figure 2
Figure 2. Figure 2: The norm of 4-velocity and FS parameters of co and counter-rotating observer [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The Logarithimic Spiral r = route λϕ 4.1 Standard Coordinate System In this section, we will study the precession frequency of a gyroscope moving along a timelike spiral trajectory in Kerr spacetime. For simplicity, we have used the logarith￾mic spiral trajectory for our study. Non-Killing logarithmic spiral trajectories provide useful toy models for matter motion in realistic accretion environments. The r… view at source ↗
Figure 4
Figure 4. Figure 4: Variation of allowed values of ω vs. r M along the timelike spiral orbit for different values of black hole spin parameter a and spiral path parameter λ. the Killing case. The allowed values of ω for the spiral trajectory are given by, ω∓ = −gtϕ ± q (gtϕ) 2 − (gϕϕ + grrr 2λ 2 ) gtt (gϕϕ + grrr 2λ 2 ) (15) Now we can discuss the possibilities of the existence of both co-rotating and counter￾rotating observe… view at source ↗
Figure 5
Figure 5. Figure 5: Norm of 4-velocity and FS parameters of co and counter-rotating gyroscope [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The norm of the 4-velocity and FS parameters of co and counter-rotating [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
read the original abstract

We investigate gyroscopic precession along Killing and non-Killing timelike trajectories in Kerr spacetime using the covariant Frenet--Serret formalism. The precession frequency is analyzed for both prograde and retrograde motion in Boyer--Lindquist and horizon-penetrating Kerr--Schild coordinate systems. For generic spiral trajectories, we show explicitly that the divergence of the precession frequency appearing near the horizon in Boyer--Lindquist coordinates disappears in Kerr--Schild coordinates. Our analysis demonstrates that the finiteness of the Frenet--Serret precession frequency is determined by the timelike character of the trajectory rather than by the existence of an event horizon itself. These results indicate that the divergence of gyroscopic precession in the strong-field regime is a coordinate artefact and therefore cannot serve as an invariant signature of horizon structure.

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

1 major / 2 minor

Summary. The manuscript uses the covariant Frenet-Serret formalism to compute gyroscopic precession frequencies along Killing and non-Killing timelike trajectories in Kerr spacetime. It compares prograde and retrograde motion in Boyer-Lindquist versus horizon-penetrating Kerr-Schild coordinates and claims to show explicitly that the divergence of the precession frequency near the horizon in Boyer-Lindquist coordinates is absent for generic spiral trajectories in Kerr-Schild coordinates. The central conclusion is that this divergence is a coordinate artifact and therefore cannot serve as an invariant signature of horizon structure.

Significance. If the compared trajectories are physically identical worldlines, the result would establish that Frenet-Serret precession divergence is not a coordinate-independent diagnostic of event horizons, with implications for strong-field GR phenomenology. The explicit coordinate comparison and use of an invariant formalism constitute a concrete test of coordinate dependence; however, the significance is conditional on verification that the trajectories match under the coordinate transformation.

major comments (1)
  1. [Trajectory definitions and comparison sections] The generic spiral trajectories are defined via coordinate-dependent expressions in each chart. Because the Frenet-Serret scalars (including the precession frequency) are coordinate invariants, any genuine discrepancy between charts requires that the worldlines be identical. The coordinate transformation between Boyer-Lindquist and Kerr-Schild coordinates changes the asymptotic behavior of the time and azimuthal coordinates near the horizon, so identical coordinate parametrizations do not guarantee identical physical curves. No invariant characterization of the trajectories (e.g., via conserved Killing quantities, proper-time parametrization, or geodesic deviation independent of chart) is provided to establish equivalence, which is load-bearing for the claim that the divergence is purely a coordinate artifact.
minor comments (2)
  1. [Abstract and §1] Clarify in the abstract and introduction whether the analysis applies only to non-geodesic trajectories or also includes geodesics, and specify the range of spin parameters and radii considered for the 'generic spiral' paths.
  2. [Figures] Ensure that all figures plotting precession frequency versus radius explicitly label the coordinate system used and indicate the location of the horizon in each chart.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and for identifying the key requirement that the compared trajectories must be physically identical worldlines. We address this point directly below.

read point-by-point responses

  1. Referee: The generic spiral trajectories are defined via coordinate-dependent expressions in each chart. Because the Frenet-Serret scalars (including the precession frequency) are coordinate invariants, any genuine discrepancy between charts requires that the worldlines be identical. The coordinate transformation between Boyer-Lindquist and Kerr-Schild coordinates changes the asymptotic behavior of the time and azimuthal coordinates near the horizon, so identical coordinate parametrizations do not guarantee identical physical curves. No invariant characterization of the trajectories (e.g., via conserved Killing quantities, proper-time parametrization, or geodesic deviation independent of chart) is provided to establish equivalence, which is load-bearing for the claim that the divergence is purely a coordinate artifact.

    Authors: We agree that the physical equivalence of the trajectories is essential to the central claim. The manuscript defines the spiral trajectories through their coordinate velocities in each chart separately, without an explicit demonstration that the resulting worldlines coincide under the coordinate transformation. In the revised version we will add an invariant characterization: we will express both families of trajectories using the same set of conserved Killing quantities (energy and angular momentum per unit mass) together with a common proper-time parametrization, and we will verify that the four-velocity components transform consistently between the two charts. This will establish that the worldlines are identical and that the observed difference in precession-frequency behavior arises solely from the coordinate representation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation compares independent coordinate calculations without reduction to inputs or self-citations.

full rationale

The paper computes Frenet-Serret precession frequencies explicitly in Boyer-Lindquist and Kerr-Schild coordinates for trajectories labeled as generic spirals in each chart. The central result—that divergence near the horizon vanishes in the horizon-penetrating system—follows directly from these separate calculations and the coordinate transformation properties, without any fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations. The Frenet-Serret scalars are treated as invariants within each coordinate system, and the conclusion that the effect is a coordinate artifact rests on the explicit disappearance rather than on any imported uniqueness theorem or ansatz from prior work. No step reduces the claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard assumptions of general relativity and the applicability of the Frenet-Serret formalism; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • standard math Kerr spacetime is described by the standard metric in general relativity
    The paper investigates gyroscopic precession in Kerr spacetime.

pith-pipeline@v0.9.1-grok · 5694 in / 1163 out tokens · 48050 ms · 2026-06-29T06:41:01.566709+00:00 · methodology

discussion (0)

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Reference graph

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