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Spin precession frequencies of a test gyroscope around a naked singularity and quasi-periodic oscillations

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Pith's one-line read For a stationary observer in the equatorial plane, the spin precession frequency of a test gyroscope around a rotating naked singularity tends to a finite value as $r\to 0$, while for a Kerr naked singularity it diverges, allowing the two…

desk verdict A solid, workmanlike calculation of spin precession in a specific rotating naked-singularity spacetime; the discriminating claim is interesting but overstated, and the QPO fits rely heavily on Kerr-based priors. read the letter →

arxiv 2505.06443 v1 pith:KS5HVRCU submitted 2025-05-09 gr-qc

Tehreem Zahra (1) , Mubasher Jamil (1) , Mustapha Azreg-Aïnou (2) ((1) School of Natural Sciences , National University of Sciences , Technology , Islamabad , Pakistan , (2) Başkent University
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Engineering Faculty Bağlica Campus Ankara Turkey.)
This is my paper · ORCID
classification gr-qc
keywords nakedsingularityspinprecessiontestgyroscopeLense-Thirringgeodeticquasi-periodicoscillationsrelativisticmodelMCMCparameterestimation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that spin precession of a test gyroscope can tell a rotating naked singularity (RNS) apart from a Kerr naked singularity. For an observer on the equatorial plane, the general spin precession frequency remains finite as $r\to 0$ in the RNS spacetime, whereas in the Kerr naked singularity it blows up. The paper also shows that in the weak-field limit the RNS produces the same Lense-Thirring precession as a Kerr black hole, and that its non-rotating limit gives the same leading-order geodetic precession as Schwarzschild, so these naked singularities mimic black holes asymptotically. Using the relativistic precession model and Monte Carlo Markov Chain fits to five X-ray binaries, the authors find that RNS spin estimates for GRO J1655-40, XTE J1859+226, and GRS 1915+105 remain inconsistent with independent measurements, and that the inferred QPO emission radii are much larger than the ISCO radius, which they read as evidence against the RNS model for these sources.

What carries the argument

The central object is the general spin precession frequency one-form $\tilde{\Omega}_p = \frac{1}{2K^2}\star(\tilde{K}\wedge d\tilde{K})$ for a test gyroscope carried by a stationary observer moving along the Killing vector $K=\partial_t+\Omega\,\partial_\phi$, reduced to the vector form in Eq. (5). Applied to the rotating naked singularity metric of Eq. (1), it yields the explicit expression in Eq. (6) and, after parametrizing $\Omega$ by $k$ through $\Omega=k\Omega_++(1-k)\Omega_-$, the magnitude in Eq. (11) whose near-singularity limit is the key discriminator. The same machinery applied to the Kerr metric produces the divergent equatorial-plane behavior that the paper contrasts with the finite RNS limit; in the weak-field regime it reduces to the Lense-Thirring formula $\vec{\Omega}_{\mathrm{LT(weak)}} = (J/r^3)(2\cos\theta\,\hat{r}+\sin\theta\,\hat{\theta})$, which is the same as for Kerr.

What would settle it

Compute the limit of $\Omega_p$ from Eq. (11) for the RNS metric along $\theta=\pi/2$ as $r\to 0$ for several values of $k$ in the range $0<k\le 1$; if any such value gives a diverging limit, the paper's distinguishing claim fails. Independently, verify whether the metric of Eq. (1) is an exact solution of Einstein's equations sourced by matter that satisfies the energy conditions, and if it is not, the physical predictions of the paper lack a foundation.

Watch

Extended reading notes

Core claim

The paper's central claim is that the magnitude of the general spin precession frequency $\Omega_p$ of a test gyroscope attached to a stationary observer on the equatorial plane ($\theta=\pi/2$) of a rotating naked singularity (RNS) has a finite limit as $r\to 0$ for $0<k\le 1$, whereas for a Kerr naked singularity ($a>M$) the same quantity diverges on that plane. This makes spin precession a discriminator between the two horizonless spacetimes: approaching the central object along the equatorial plane, a gyroscope stays regular in RNS but blows up in Kerr naked singularity. The paper also derives that the weak-field Lense-Thirring precession of RNS matches that of a Kerr black hole, and that the asymptotic geodetic precession of the non-rotating limit (NNS) matches the Schwarzschild value, so both naked-singularity spacetimes mimic their black-hole counterparts in the regimes where they are hardest to tell apart. Finally, using the relativistic precession model and Monte Carlo Markov Chain fits to five X-ray binaries, the paper reports that the RNS spin parameter estimates for GRO J1655-40, XTE J1859+226, and GRS 1915+105 remain inconsistent with independently measured values, and that the best-fit QPO emission radii sit much farther from the center than the ISCO radius, which the paper treats as tension against the RNS model for these sources.

Load-bearing premise

The rotating naked singularity line element of Eq. (1), taken from Ref. [31], is assumed to be a physically realizable Einstein-matter spacetime satisfying the energy conditions; the paper does not derive this metric or verify its matter content, so all subsequent precession and QPO calculations would fail if this spacetime is not a legitimate solution.

Editorial extensions

If this is right

  • If the central claim is correct, a test gyroscope carried toward the center of a rotating naked singularity along the equatorial plane would measure a finite precession frequency all the way to $r=0$, while the same experiment near a Kerr naked singularity would show a divergence.
  • In the weak-field regime, the RNS and Kerr spacetimes produce identical Lense-Thirring precession, and the non-rotating NNS produces the same leading-order geodetic precession as Schwarzschild, meaning these naked singularities are black-hole mimickers at the level of gyroscope measurements.
  • Within the relativistic precession model, the RNS fit does not resolve the known tension in spin estimates for GRO J1655-40, XTE J1859+226, or GRS 1915+105, so the discrepancy between precession-based and spectral or continuum-fitting methods persists even with this alternative spacetime.
  • For all five X-ray binaries studied, the best-fit QPO emission radius is much larger than the corresponding ISCO radius, which challenges the assumption that these systems are well described by the RNS model.
  • Because the RNS spacetime has no event horizon and no ergoregion, the spin precession formula remains valid everywhere outside the singularity, unlike the Kerr black hole case where precession diverges at the horizon.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A consequence the paper leaves implicit is that a future measurement of gyroscope or pulsar spin precession that stays finite while approaching a compact object would be direct evidence for a horizonless spacetime, whereas a divergence at the would-be horizon would support a black hole interpretation.
  • The paper's finding that only counter-rotating orbits admit a photon sphere in RNS suggests a complementary test: combining spin precession with photon-orbit or shadow observations could distinguish RNS from Kerr even for observers outside the equatorial plane.
  • The persistence of the spin-parameter tension across both Kerr and RNS fits hints that the issue may lie in the relativistic precession model or in the identification of observed QPO frequencies, rather than in the spacetime geometry alone.
  • An extension of the present framework would be to compute how a nearby pulsar's spin axis precesses in the RNS spacetime and how that changes the observed pulse profile, something the paper only outlines as future work.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper considers the rotating naked singularity (RNS) spacetime of Ref. [31], obtained from the null naked singularity (NNS) via the Newman-Janis algorithm, and studies the spin precession frequency of a test gyroscope attached to a stationary observer. It derives general spin precession, Lense-Thirring (LT) precession, and geodetic precession, showing that the weak-field LT limit matches Kerr and the asymptotic geodetic limit matches Schwarzschild. The central claim is that on the equatorial plane the general spin precession of an observer with fixed angular-velocity parameter k tends to a finite value as r -> 0 in RNS, while for a Kerr naked singularity it diverges, providing a potential observational discriminator. The paper then derives the orbital, periastron, and nodal precession frequencies and the ISCO condition for RNS, and performs an MCMC fit of the relativistic precession model to QPO data from five X-ray binaries, concluding that RNS spin estimates for GRO J1655-40, XTE J1859+226, and GRS 1915+105 are inconsistent with continuum-fitting or spectral measurements, and that the RNS model is disfavored because the QPO emission radii are much larger than the tiny RNS ISCO radii.

Significance. If the RNS metric is physically admissible for the parameters used, the paper offers a new strong-field observable distinction between two classes of naked singularities and a concrete, falsifiable prediction. The analytic derivations are self-contained, the weak-field Kerr and Schwarzschild limits are verified, and the MCMC pipeline is standard, with explicit tables of data, priors, and posteriors. The paper is also honest in stating that its QPO analysis disfavors the RNS model. The main reservations are the precise class of observers for which the finite precession limit holds, the unverified energy-condition range for the metric parameters used (especially a/M = 1.2 in Fig. 6), and the potentially prior-biased MCMC setup.

major comments (4)
  1. [§III, Eq. (11), Eq. (17), Fig. 3] The paper states that the general spin precession frequency is finite for all observers and, after Fig. 3, that it diverges for the ZAMO in the equatorial plane. These statements are mutually inconsistent, and neither matches the scaling of Eq. (11). At θ = π/2 the numerator factor (a²+r²)²(M+r)² − a²(a²(M+r)²+r⁴) scales as r² and ρ⁷ scales as r⁷, so Ωp scales as |Y|/r⁵; with the printed ZAMO expression for Y in Eq. (14) this tends to zero, and even with the general Y of Eq. (6) it tends to a finite value for fixed k ∈ (0,1). The divergence in Eq. (17) is a property of the static-observer limit Ω = 0, which corresponds to k → 0, not to observers with fixed k. Because Section IV's discriminator relies on finiteness for equatorial observers, the paper must specify the exact class of observers for which the finite-limit claim holds and correct the conflicting statements.
  2. [§II and §IV (Fig. 6)] The line element (1) is imported from Ref. [31], and the only support for its physical admissibility is the sentence citing Eqs. (27)-(29) of that reference. Since the central distinguishing claim of Section IV is illustrated at a/M = 1.2 (Fig. 6), and the QPO fits use various a/M values, the manuscript should state explicitly the parameter range over which the energy conditions of Ref. [31] are satisfied and show that the values used here lie inside that range. Without this, the RNS-versus-Kerr prediction may concern a spacetime that is not a physically realizable Einstein-matter solution.
  3. [§VI, Table II] The MCMC priors in Table II are Gaussian distributions centered on Kerr-spacetime RPM estimates from Refs. [98,99], with very small standard deviations, e.g., a/M = 0.286 ± 0.003 for GRO J1655-40. Using posteriors obtained under the Kerr assumption as priors for the RNS model can bias the RNS parameter estimates toward Kerr values, weakening the subsequent comparison with continuum-fitting and spectral spin measurements. The authors should either repeat the analysis with wide, uninformative priors or demonstrate that the results are insensitive to the prior choice.
  4. [§V, Eq. (49) and text after Fig. 9] Equation (49) is presented as the ISCO condition, and for a = 0 it formally yields r = 0. The text then states, following Ref. [33], that no ISCO exists in the non-rotating NNS limit because the stationary point of the effective potential is a minimum. The paper should reconcile this a → 0 limit of its own ISCO equation with the no-ISCO claim for NNS, or explain why the r = 0 root is not physically acceptable.
minor comments (5)
  1. [Table III, XTE J1859+226 row] The quoted uncertainty in rISCO/M is 0.07, which exceeds the central value 0.0189; this is likely a typographical error for 0.0007 and should be corrected.
  2. [§III, Fig. 2 caption] The phrase "remains finite for all observers except at the singularity" is ambiguous about whether the limit as r → 0 is finite or the value at r = 0 is undefined; please state the limiting statement explicitly.
  3. [§III, text before Eq. (7)] The statement that Eq. (5) holds for a "limited range" of Ω is vague; the allowed range for timelike observers is an open interval, and this should be stated precisely.
  4. [§VII, first paragraph] The phrase "the null- and timelike- geodesics" contains stray hyphens and should read "the null and timelike geodesics".
  5. [§II, horizon discussion] The sentence beginning "To demonstrate this non-existence statement" refers to the absence of an event horizon, but the wording is awkward; please rephrase for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spin-precession and QPO derivations are self-contained given the adopted RNS metric, and the cited metric/energy-condition checks are external inputs rather than target results.

full rationale

The central precession result (Sections III-IV) starts from the line element (1), which is adopted from Ref. [31] with energy-condition checks cited to Eqs. (27)-(29) of that paper, and then derives Eqs. (5)-(11) by applying the standard Killing-vector spin-precession formula. The finite-versus-divergent distinction in Fig. 6 is an algebraic consequence of the two metrics, not an assumed conclusion. Similarly, Section V derives the three fundamental frequencies (42)-(44) and the ISCO condition (49) from the geodesic equation and effective potential for metric (1), with no use of the frequencies to construct the metric. The MCMC section fits M, a/M, and r/M to observed QPO frequencies with Gaussian priors from Refs. [98,99]; the posterior spin for XTE J1859+226 (0.050 +/- 0.002) differs strongly from its prior (0.149 +/- 0.005), showing that the fit is data-driven rather than a prior clone. The paper's own conclusion that the large emission radius compared with r_ISCO challenges the RNS model is an honest external consistency check, not a circular step. The coauthor self-citations (e.g., Refs. [32,76,78,97]) provide standard algorithms, formulas, or fitting methods; none is a uniqueness theorem or an unverified premise that reduces the target claim to itself. The only vulnerability, whether metric (1) is a legitimate Einstein-matter solution with plausible energy conditions, is an external-support and correctness issue, not circularity.

Assumptions & free parameters 16 free parameters · 5 assumptions · 0 invented entities

The central derivation requires the RNS metric to be a valid Einstein-matter solution and the RPM identification to hold; both are imported from prior work, not derived here. No new physical entities are introduced. The MCMC fits introduce 15 fitted parameters (mass, spin, radius for five sources) plus a kinematic parameter k.

free parameters (16)
  • M (GRO J1655-40) = 5.01±0.04 M_sun
    Fitted via MCMC to the three QPO frequencies of GRO J1655-40 under the RNS metric and RPM.
  • a/M (GRO J1655-40) = 0.271±0.002
    Fitted via MCMC; prior mean 0.286 taken from Kerr RPM results.
  • r/M (GRO J1655-40) = 5.48±0.03
    Emission radius at which the three QPOs are produced, fitted via MCMC.
  • M (XTE J1550-564) = 8.48+0.24-0.23 M_sun
    Fitted via MCMC to two QPO frequencies of XTE J1550-564 under the RNS metric and RPM.
  • a/M (XTE J1550-564) = 0.318±0.007
    Fitted via MCMC; prior mean 0.34 from Kerr RPM results.
  • r/M (XTE J1550-564) = 5.17±0.11
    Emission radius fitted via MCMC; the source has only two observed QPOs, so this is prior-dominated.
  • M (XTE J1859+226) = 14.03+0.35-0.35 M_sun
    Fitted via MCMC to three QPO frequencies of XTE J1859+226 under the RNS metric and RPM.
  • a/M (XTE J1859+226) = 0.050±0.002
    Fitted via MCMC; differs significantly from the Kerr RPM value 0.149.
  • r/M (XTE J1859+226) = 3.69±0.08
    Emission radius fitted via MCMC; much closer to the center than the Kerr-based estimate.
  • M (GRS 1915+105) = 12.73+0.51-0.50 M_sun
    Fitted via MCMC to two QPO frequencies of GRS 1915+105 under the RNS metric and RPM.
  • a/M (GRS 1915+105) = 0.255±0.015
    Fitted via MCMC; prior mean 0.29 from Kerr RPM results.
  • r/M (GRS 1915+105) = 5.55+0.19-0.18
    Emission radius fitted via MCMC; the source has only two observed QPOs, so this is prior-dominated.
  • M (H1743-322) = 9.86+0.31-0.30 M_sun
    Fitted via MCMC to three QPO frequencies of H1743-322 under the RNS metric and RPM.
  • a/M (H1743-322) = 0.245+0.010-0.009
    Fitted via MCMC; consistent with the range from reflection spectroscopy.
  • r/M (H1743-322) = 5.16±0.12
    Emission radius fitted via MCMC.
  • k
    Dimensionless parameter 0<k<1 that selects the observer's angular velocity Ω=kΩ+ +(1-k)Ω-; used in the precession plots, not fitted to data.
assumptions (5)
  • domain assumption The line element (1) is a valid solution of Einstein's field equations with matter satisfying the energy conditions, as established in Ref. [31].
    The paper does not re-derive the RNS metric or its matter content; all precession and geodesic results depend on this metric being physical.
  • domain assumption The relativistic precession model identification νU=νϕ, νC=νn, νL=νp (Eq. 52) relates observed QPO frequencies to orbital and epicyclic frequencies.
    This mapping is the basis for converting observed frequencies into constraints on mass, spin, and radius; if the RPM identification is wrong, the MCMC estimates are invalid.
  • domain assumption The MCMC priors (Table II) are Gaussian with means and widths adopted from previous Kerr-based RPM analyses [98, 99].
    The posterior estimates for sources with only two QPO frequencies are strongly influenced by these priors.
  • standard math Standard parallel transport equation (20) and the formula for spin precession (5) from Refs. [42, 78] are assumed.
    These are standard results in stationary spacetimes; the paper follows the established derivation.
  • standard math The existence of timelike stationary observers with angular velocity within the range (7) is assumed.
    Stationary observers are integral curves of K = ∂t + Ω∂ϕ; the domain of validity is the standard causality condition.

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Pith. "Pith review of Spin precession frequencies of a test gyroscope around a naked singularity and quasi-periodic oscillations." pith.science (2026). https://pith.science/paper/KS5HVRCU

@misc{pith2026250506443,
  author       = {Pith},
  title        = {Pith review of: Spin precession frequencies of a test gyroscope around a naked singularity and quasi-periodic oscillations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KS5HVRCU}},
  note         = {Machine review of arXiv:2505.06443}
}
read the original abstract

Various studies show that the gravitational collapse of inhomogeneous matter clouds leads to naked singularity formation. We investigate here the spin precession frequency of a test gyroscope attached to a stationary observer in a rotating naked singularity spacetime. In the weak field limit, Lense-Thirring precession for rotating naked singularity and geodetic precession in the asymptotic limit for null naked singularity are found to be equal to that of a Kerr black hole and a Schwarzschild black hole, respectively. In addition, we can distinguish a rotating naked singularity and a Kerr naked singularity for an observer in the equatorial plane using spin precession. To this end, we have found the constraints on the parameters of rotating naked singularity by employing the Monte Carlo Markov Chain simulation and using the observation from five quasi-periodic sources within the relativistic precession model. Our analysis shows that the measurement of spin parameter estimate for GRO J1655-40 is in disagreement with the value found from the continuum-fitting method, while for XTEJ1859+226 and GRS 1915+105, it is inconsistent with spectral analysis results.

Figures

Figures reproduced from arXiv: 2505.06443 by the authors.

Figure 1
Figure 1. FIG. 1: Roots of angular velocities Ω [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: General spin precession frequency for a RNS for [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Magnitude of the general spin precession plotted for a ZAMO by taking ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: FIG. 4: LT precession frequency vector fields for RNS, plotted in Cartesian coordinates corresponding to ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Magnitude of LT precession frequency at [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Magnitude of general spin precession frequency for Kerr naked singularity and RNS at [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Parameters [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Radius of photon orbit for counter-rotating orbit taking [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: Note that, in [33], the presence of radius of the ISCO was indicated for NNS and it was shown that [considering the representation r˙ 2 = E2 − Ueff(r)] the stationary point of effective potential i.e., L 2 M is the ISCO which extends up to the singularity at r = 0 as L…
Figure 10
Figure 10. Figure 10: FIG. 10: Variation of [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Variation of [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: One-dimensional posterior distribution for each parameter along diagonal and contour plots showing the correlation on other [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Posterior distributions of the MCMC simulation for the parameters of RNS with XTE J1550-564 (orange contours), XTE [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]

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