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arxiv: 2601.14664 · v2 · submitted 2026-01-21 · 🌀 gr-qc · astro-ph.CO· math-ph· math.MP

Covariant interpretation of proper infall times in Kerr spacetime

Erick Pasten , Claudia Alvarez , Norman Cruz This is my paper

Pith reviewed 2026-05-16 12:43 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.COmath-phmath.MP
keywords proper infall timesKerr spacetimetimelike geodesicsRaychaudhuri equationblack hole spincovariant formalismequatorial planegeodesic congruence
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The pith

Kerr black hole spin can lengthen or shorten proper infall times compared to Schwarzschild depending on orbit direction and energy.

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

The paper compares the proper time experienced by test particles falling into Kerr and Schwarzschild black holes along equatorial timelike geodesics. It adopts an equal circumferential-radius prescription to ensure the comparison between rotating and non-rotating cases occurs between geometrically equivalent surfaces. Calculations for varying specific energy E, angular momentum L, and spin parameter a show that Kerr rotation produces either longer or shorter integrated proper times than the Schwarzschild case. The differences are traced to the Raychaudhuri evolution equation in the 1+3 covariant formalism, where rotation alters the competition between the radial change in expansion and the nonlinear focusing terms involving expansion and shear.

Core claim

Within the equal circumferential-radius prescription for equatorial timelike geodesics, the Kerr angular momentum a produces longer or shorter integrated proper infall times relative to the Schwarzschild case, depending on the orbital configuration and energy regime. These differences are encoded in the corresponding Raychaudhuri time integrand, which reflects a competition between the radial evolution of the expansion and the nonlinear focusing contribution driven by expansion and shear; black-hole rotation modifies both effects systematically, leading to distinct behaviors for prograde and retrograde infall.

What carries the argument

Equal circumferential-radius prescription applied to equatorial timelike geodesics, with differences interpreted via the 1+3 covariant formalism through the Raychaudhuri equation governing expansion and shear of the geodesic congruence.

If this is right

  • Kerr angular momentum systematically modifies the radial evolution of expansion in the Raychaudhuri integrand.
  • The nonlinear focusing term driven by expansion and shear receives distinct corrections for prograde versus retrograde configurations.
  • Integrated proper times therefore vary with spin parameter a in a manner that depends on both orbital direction and energy regime.
  • The covariant 1+3 analysis isolates the competition between expansion evolution and shear-enhanced focusing as the source of the Kerr-Schwarzschild difference.

Where Pith is reading between the lines

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

  • Frame-dragging induced by rotation changes the rate at which nearby geodesics converge or diverge in a direction-dependent way.
  • The same mechanism may influence the duration of signals emitted by matter spiraling inward from the equatorial plane.
  • Extensions beyond the equatorial plane could reveal whether polar components of spin alter the focusing balance further.

Load-bearing premise

The equal circumferential-radius prescription for comparing rotating and non-rotating black holes in the equatorial plane provides a geometrically consistent basis for the difference in proper times.

What would settle it

A numerical integration of proper time along an equatorial geodesic with fixed E and L from a chosen circumferential radius down to the horizon in Kerr spacetime that yields exactly the same value as the corresponding Schwarzschild integration at identical starting radius would falsify the claimed spin dependence.

Figures

Figures reproduced from arXiv: 2601.14664 by Claudia Alvarez, Erick Pasten, Norman Cruz.

Figure 1
Figure 1. Figure 1: Ratio between the Newtonian infall time and the Schwarzschild proper infall time for trajectories from [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Maps of the Kerr spin-induced deviation of the effective potential from Schwarzschild, [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Profiles of τ as a function of the spin parameter a for prograde (UP) and retrograde (DOWN) infall down to a fixed radius rf = 3M. Curves are shown for several values of the specific energy E and for two representative angular momenta, L = 0.3 and L = 1. For low and moderate energies, τ increase markedly with a for prograde orbits, while the trend is the opposite for retrograde trajectories. At higher ener… view at source ↗
Figure 4
Figure 4. Figure 4: Ratio between Kerr and Schwarzschild values of [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Parametric maps of Kerr–minus–Schwarzschild variation of the effective potential in the ( [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Differences between Kerr and Schwarzschild congruence kinematics for moderately relativistic infall with [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
read the original abstract

We investigate proper infall times in the Schwarzschild and Kerr spacetimes from a covariant perspective, focusing on the role of black--hole rotation in the focusing properties of timelike geodesic congruences.To perform a geometrically consistent comparison between rotating and non--rotating black holes, we analyse infall trajectories between surfaces of equal circumferential radius in the equatorial plane. Using equatorial timelike geodesics in the test--particle limit, we compute and compare the corresponding proper infall times for different values of the specific energy $E$, specific angular momentum $L$, and black--hole spin parameter $a$. Within the equal circumferential-radius prescription adopted here, we show that Kerr angular momentum $a$ can produce longer or shorter integrated proper infall times relative to the Schwarzschild case, depending on the orbital configuration and energy regime considered. We then interpret these results within the covariant $1+3$ formalism of general relativity, in terms of the expansion, shear, and Raychaudhuri evolution of timelike congruences. Our analysis shows that the Kerr--Schwarzschild differences in proper infall times are encoded in the corresponding Raychaudhuri time integrand, which reflects a competition between the radial evolution of the expansion and the nonlinear focusing contribution driven by expansion and shear. Black--hole rotation modifies both effects in a systematic way, leading to distinct behaviours for prograde and retrograde infall configurations.

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 / 2 minor

Summary. The paper investigates proper infall times along equatorial timelike geodesics in Kerr and Schwarzschild spacetimes from a covariant viewpoint. It adopts surfaces of equal circumferential radius (defined via √g_φφ at θ=π/2) for comparison, computes integrated proper times for varying specific energy E, angular momentum L, and spin a, and finds that a can yield either longer or shorter infall times than the Schwarzschild case depending on orbital configuration and energy regime. These differences are then interpreted in the 1+3 formalism as arising from the Raychaudhuri integrand, reflecting competition between the radial evolution of the expansion scalar and nonlinear focusing terms involving expansion and shear, with rotation modifying both contributions systematically for prograde and retrograde cases.

Significance. If the central comparison holds, the work provides a covariant link between proper-time differences and the focusing properties of geodesic congruences, showing how black-hole spin alters the Raychaudhuri evolution in a quantifiable way. This could inform studies of particle dynamics near rotating black holes and the geometric origin of spin-dependent infall effects. The analysis employs standard GR tools (geodesic equations and 1+3 decomposition) and explores parameter space, offering concrete numerical comparisons that could be tested against more invariant radial measures.

major comments (2)
  1. [§2] §2 (comparison prescription): The equal-circumferential-radius identification (√g_φφ = constant at θ=π/2) maps to different Boyer-Lindquist coordinate intervals for Kerr versus Schwarzschild because the Kerr circumferential radius includes explicit a-dependent terms (r² + a² + 2Ma²/r). This makes the starting and ending surfaces non-equivalent under a coordinate-independent geometric quantity such as proper radial distance from the horizon or the area radius of the 2-sphere. Consequently the reported sign changes in Δτ may partly reflect this mapping rather than a pure effect of the Raychaudhuri integrand; an explicit check against an invariant radial coordinate is needed to confirm the claim.
  2. [§4] §4 (Raychaudhuri interpretation): The statement that Kerr-Schwarzschild differences are 'encoded in the corresponding Raychaudhuri time integrand' requires an explicit decomposition showing how the integrand (involving θ, σ, and their radial derivatives) differs term-by-term with a. Without the full derivation or tabulated integrand values at fixed circumferential radius, it is unclear whether the competition between expansion evolution and nonlinear focusing is demonstrated independently of the coordinate choice.
minor comments (2)
  1. [Abstract] Abstract and §1: The phrase 'geometrically consistent comparison' should be qualified by noting that the prescription is coordinate-dependent; a brief justification or reference to prior literature on circumferential-radius comparisons would clarify the choice.
  2. [Figures] Figure captions (e.g., those showing Δτ vs. a): Labels should explicitly state the fixed circumferential radius value used for each curve and whether E and L are held constant in the Kerr or Schwarzschild frame.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and constructive comments. We address each major point below, clarifying our choice of comparison surfaces and strengthening the explicit link to the Raychaudhuri integrand.

read point-by-point responses

  1. Referee: [§2] §2 (comparison prescription): The equal-circumferential-radius identification (√g_φφ = constant at θ=π/2) maps to different Boyer-Lindquist coordinate intervals for Kerr versus Schwarzschild because the Kerr circumferential radius includes explicit a-dependent terms (r² + a² + 2Ma²/r). This makes the starting and ending surfaces non-equivalent under a coordinate-independent geometric quantity such as proper radial distance from the horizon or the area radius of the 2-sphere. Consequently the reported sign changes in Δτ may partly reflect this mapping rather than a pure effect of the Raychaudhuri integrand; an explicit check against an invariant radial coordinate is needed to confirm the claim.

    Authors: We agree that the equal-√g_φφ surfaces correspond to different Boyer-Lindquist r intervals because g_φφ carries explicit a dependence. Nevertheless, √g_φφ itself is the proper circumferential length of the equatorial circle and is therefore a coordinate-invariant geometric quantity. This prescription was chosen precisely to compare infall from surfaces of equal proper circumference, which provides a geometrically natural and covariant notion of “same radial location” for the purpose of contrasting rotating and non-rotating black holes. To address the referee’s concern directly, we will add in the revision an explicit comparison performed at fixed proper radial distance from the horizon (computed via the integral of the spatial metric along equatorial radial geodesics). This supplementary check will confirm that the qualitative sign changes in Δτ persist under the alternative invariant measure. revision: partial

  2. Referee: [§4] §4 (Raychaudhuri interpretation): The statement that Kerr-Schwarzschild differences are 'encoded in the corresponding Raychaudhuri time integrand' requires an explicit decomposition showing how the integrand (involving θ, σ, and their radial derivatives) differs term-by-term with a. Without the full derivation or tabulated integrand values at fixed circumferential radius, it is unclear whether the competition between expansion evolution and nonlinear focusing is demonstrated independently of the coordinate choice.

    Authors: The manuscript already derives the 1+3 Raychaudhuri equation for the timelike congruence and identifies the modifications to θ and σ induced by the Kerr metric. To make the term-by-term competition fully explicit, we will insert in the revised §4 a direct decomposition of the integrand evaluated at fixed circumferential radius. For representative values of a, E and L we will tabulate the separate contributions of dθ/dr, θ²/3 and σ_{ab}σ^{ab} and show how the spin parameter a systematically alters each piece for prograde versus retrograde orbits. This will demonstrate that the observed Δτ differences arise from the modified focusing dynamics independently of the particular radial coordinate label. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the claimed derivation

full rationale

The paper computes proper infall times directly from the standard geodesic equations for equatorial timelike geodesics in the Kerr and Schwarzschild metrics, integrated between surfaces of equal circumferential radius as an explicit modeling choice. These results are then interpreted using the standard 1+3 covariant formalism and the Raychaudhuri equation applied to the computed expansion and shear. No parameters are fitted to subsets of data and then relabeled as predictions. No self-citations are invoked to establish uniqueness theorems or to smuggle in ansatze. The derivation chain is self-contained against the Einstein equations and does not reduce any central claim to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes only standard general-relativistic geodesic equations and the 1+3 covariant formalism; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (2)
  • standard math Timelike geodesics in Kerr and Schwarzschild spacetimes obey the standard geodesic equation derived from the metric.
    Used to compute proper times along equatorial trajectories.
  • standard math The 1+3 covariant decomposition yields the Raychaudhuri equation for the expansion and shear of timelike congruences.
    Invoked to interpret the integrated proper-time differences.

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