Constraining quadrupole deformations with relativistic effects

Ainur Urazalina; Anuar Idrissov; Daniya Utepova; Gulzada Baimbetova; Hernando Quevedo; Kuantay Boshkayev; Serzhan Momynov

REVIEW 3 major objections 4 minor 79 references

Constraining quadrupole deformations with relativistic effects

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper shows that in the q-metric a quadrupole deformation shifts the maximum Shapiro time delay linearly at first post-Newtonian order, giving a direct observable for compact-object shapes.

desk verdict The Shapiro-delay quadrupole term looks like a coordinate artifact, and the Shirokov derivation has internal inconsistencies; the paper needs major revision before its main claims can be trusted. read the letter →

arxiv 2506.10437 v1 pith:VO4O7W23 submitted 2025-06-12 gr-qc

Daniya Utepova , Kuantay Boshkayev , Serzhan Momynov , Anuar Idrissov , Gulzada Baimbetova , Hernando Quevedo , Ainur Urazalina This is my paper

classification gr-qc PACS 04.20.-q04.25.Nx

keywords q-metricZipoy-VoorheesspacetimeShapirotimedelayShirokoveffectgeodesicdeviationquadrupoledeformationpost-Newtonianexpansioncompactobjects

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 works in the q-metric, a static, axially symmetric spacetime that adds a quadrupole deformation parameter $q$ to Schwarzschild. It derives how $q$ affects two relativistic observables: the Shirokov effect of small oscillations inside a satellite, and the Shapiro time delay of radar signals passing a massive body. The central result is that the maximum Shapiro delay contains a term linear in $q$ at first post-Newtonian order, in contrast to a recent analysis that found the delay insensitive to the deformation. If this is right, precision timing of radar or pulsar signals could constrain the quadrupole deformation of neutron stars and other compact objects. For the Shirokov effect, the quadrupole appears only at second post-Newtonian order.

What carries the argument

The engine of the paper is the q-metric (also called the Zipoy-Voorhees or gamma-metric), a static, axially symmetric vacuum solution that reduces to Schwarzschild when $q=0$ and whose angular part differs from Schwarzschild by a factor $g^{q(2+q)}$; the total ADM mass is $M = m(1+q)$, and $q>0$ corresponds to oblate, $q<0$ to prolate deformations. For the Shirokov effect, the load-bearing mechanism is the system of geodesic deviation equations, whose compatibility condition yields the radial and transverse oscillation frequencies, expanded in $1/r$ and $q$. For the Shapiro delay, the mechanism is the null-geodesic time-of-flight integral, expanded to first order in $m$ and $q$; the linear-in-$q$ piece of that expansion is what produces Eq. (51).

What would settle it

Recompute the maximum time delay after rewriting the q-metric in terms of the circumferential radius $R = r f^{-q/2}$ and re-expressing $r_A$, $r_B$, and $R_\odot$ in Eq. (51) in that variable; if the delay remains linear in $q$ the claim stands, while if the $q$-dependence cancels, the claimed contradiction with [36] is a coordinate artifact.

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Extended reading notes

Core claim

Starting from the q-metric line element, the authors solve the null geodesic equations and integrate the flight time between two points at coordinate radii $r_A$ and $r_B$ with closest approach $r_C$. Keeping only first-order terms in the mass and in $q$, they obtain the maximum round-trip delay when $r_C = R_\odot$: $\delta t_{\max} = \frac{2R_s}{c}\left[1+q+\ln\!\left(\frac{4r_A r_B}{R_\odot^2}\right)\right]$, with $R_s = 2GM/c^2$ and total mass $M = m(1+q)$. The quadrupole parameter therefore enters the delay linearly at the same order as the Schwarzschild logarithmic term: oblate deformations ($q>0$) increase the delay and prolate ones ($q<0$) decrease it. The paper states that the earlier result in [36], which reported no such dependence, is counterintuitive. For the Shirokov effect, $q$ enters only at second post-Newtonian order, for instance in the period difference $\Delta T = T_0\left[-\frac{3M}{r}-\frac{M^2(18+7q)}{2r^2}\right]$, making the Shapiro delay the more direct probe of deformation.

Load-bearing premise

The calculation assumes the coordinate $r$ that appears in Eq. (51), including the Sun's radius $R_\odot$ and the planet distances $r_A, r_B$, equals the physically measured distance, even though in the q-metric the circumference at coordinate $r$ is $2\pi r f^{-q/2}$ and a coordinate redefinition could absorb the linear $q$ term.

Editorial extensions

If this is right

A measurement of the Shapiro delay around a deformed compact object would constrain the sign and magnitude of its quadrupole parameter at first post-Newtonian order.
Pulsar timing and spacecraft ranging, which already use the Shapiro delay, become tools for probing the spacetime geometry around neutron stars and black hole mimickers.
For the Shirokov effect, detecting the quadrupole requires second-post-Newtonian sensitivity, but combining it with Shapiro delay could help separate mass from deformation.
In the limit $q\to 0$, the derived expressions reduce to the Schwarzschild results, so the formulas form a continuous deformed family anchored to the known general-relativistic baseline.

Reading between the lines

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

A natural follow-up is to re-express Eq. (51) using the circumferential radius $R = r f^{-q/2}$, which would isolate the physically meaningful part of the linear $q$ dependence.
If the linear term survives invariant re-expression, the same first-order quadrupole correction should appear in gravitational lensing deflection and in lensed quasar time delays, so existing lensing data could provide an independent check.
Applying the same expansion to rotating, spinning compact objects could separate a static quadrupole from spin-induced frame-dragging effects, since the two effects studied here carry $q$ at different post-Newtonian orders.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Utepova et al. calculate the Shirokov effect and the Shapiro delay in the q-metric, extending earlier work on geodesic deviation to a quadrupolar spacetime and adding a vertical-displacement estimate that is a nice touch. The Shirokov part, if the final expansions are correct, shows q enters the period difference only at second post-Newtonian order, which is physically plausible. But the headline Shapiro result does not hold up. Eq (51) gives a first-order q term in the maximum delay, but the derivation never fixes the physical meaning of the coordinate r. On the equatorial plane, the redefinition ρ = r f^{-q/2} makes g_φφ = ρ² exactly, and to first order in 1/ρ the metric is Schwarzschild with ADM mass M = m(1+q). The first-order Shapiro delay in those coordinates is the standard one, so the O(q) term in Eq (51) is an artifact of treating r as a physical distance. The paper does not reconcile this with the opposite conclusion of Chakrabarty and Tang for the same metric family. That is the load-bearing problem.

The Shirokov section has separate inconsistencies. Eq (27) should reduce to the Schwarzschild radial epicyclic frequency when q=0, but it gives a 9m/r correction instead of the standard 3m/r; the later expansions (28)–(32) are consistent with the standard result, so Eq (27) appears wrong. Eq (4) also looks incomplete: the full covariant derivative of the deviation vector includes products of connection coefficients, which are missing here. These are not cosmetic issues; they undercut the derivation, even if the final Shirokov formulas are salvageable from the authors' earlier work.

The paper is clearly motivated and well-organized, and the q-metric is a legitimate arena for these questions. But as submitted, the central new claim is not established, and the Shirokov derivation needs repair. I would not cite Eq (51). A reading group could use this as a case study in coordinate artifacts in timing measurements. If I were the editor, I would still send it to a referee—the claim is non-trivial and deserves expert judgment—but I expect a careful referee to reject the main result in its current form.

Referee Report

3 major / 4 minor

Summary. The manuscript studies two general-relativistic effects in the static, axially symmetric Zipoy–Voorhees (q-metric) spacetime. In Section III the authors derive geodesic-deviation equations for a circular equatorial orbit, compute oscillation frequencies and periods for the Shirokov effect, and give a quadrupole correction to the period difference. In Section IV they derive the Shapiro time delay for light propagating in the equatorial plane, obtaining Eq. (51), according to which the maximum extra delay is δtmax = (2Rs/c)[1+q+ln(4rA rB/Rsun^2)] to first order. They contrast this with the result of Chakrabarty and Tang [36] that the delay is insensitive to the deformation parameter, and conclude that [36] is counterintuitive. The paper's central claim is the O(q) modification of the first post-Newtonian Shapiro delay.

Significance. The paper is a fully analytic study of two relativistic effects in the q-metric. It has the strength of recovering the Schwarzschild limit in several places and of using the ADM mass M=m(1+q) consistently in later expansions. If the central Shapiro result were correct, an O(q) correction to the first post-Newtonian delay would be a genuinely interesting observable for constraining quadrupole deformations of compact objects. However, the central result is not invariant: under the radial redefinition that restores the canonical angular part, the first-order metric is Schwarzschild with mass M, so the O(q) term in Eq. (51) is a coordinate artifact rather than a physical prediction. The Shirokov section additionally contains algebraic errors in Eqs. (27) and (36). Thus the paper addresses a relevant topic, but its main claims are not established.

major comments (3)

[§IV, Eq. (51)] The central claim is not invariant under the radial coordinate transformation that makes the angular part canonical. On the equatorial plane, the q-metric has g_φφ = r^2 f^{-q}, so r is not the circumferential radius. Defining ρ = r f^{-q/2} gives g_φφ = ρ^2. Expanding the metric in m/ρ with M = m(1+q), one obtains g_tt = -1 + 2M/ρ + O(m^2/ρ^2) and g_ρρ = 1 + 2M/ρ + O(m^2/ρ^2); this is exactly the first-order Schwarzschild metric in areal radius. The first post-Newtonian Shapiro delay is therefore the standard Schwarzschild formula in ρ coordinates, with no O(q) term. The O(q) contribution in Eq. (51) is an artifact of holding the coordinate r fixed while changing q, and the paper never specifies how r_A, r_B, and R_sun relate to physically measured distances. The contradiction with [36] is asserted, not demonstrated in invariant language; this is load-bearing for the paper's main conclusion.
[§III, Eq. (27)] The frequency formula in Eq. (27) cannot be correct. For q = 0 it reduces to ω^2 = m(r - 3m)(r - 6m)/r^3, which is dimensionless in units G = c = 1 (frequency squared should have dimension 1/length^2) and does not reduce to the standard Schwarzschild radial epicyclic frequency κ^2 = m(r - 6m)/r^4. Consequently the period expansions in Eqs. (28)–(32) and all Shirokov results derived from them are unreliable, including the claimed quadrupole corrections.
[§III, Eq. (36)] Equation (36) gives an incorrect orbital period. For q = 0 it yields T = T0 sqrt(r/(r-3m)) ≈ T0(1 + 3m/(2r) + ...), whereas the coordinate orbital period for a circular Schwarzschild geodesic is exactly T0 and the proper period is T0 sqrt(1 - 3m/r) ≈ T0(1 - 3m/(2r) + ...). The expression in Eq. (36) is the reciprocal of the correct proper-period factor. This error propagates into the vertical-shift estimate Eq. (37) and undermines the proposed Shirokov observable.

minor comments (4)

[§IV, Eq. (51)] The two forms in Eq. (51) are equal only after dropping O(q^2). Since the paper claims a first-order approximation, this truncation should be stated explicitly rather than written as an equality.
[§IV, Eq. (44)] The placement of the factors f^{2(1+q)} and g^{q(2+q)} in the integrand is ambiguous as printed; please insert parentheses to show whether the integrand is g^{q(2+q)}/[√(...) f^{2(1+q)}] or the inverse.
[References and text] Reference [16] contains a typo ('d'Invemo' should be 'd'Inverno'), and in the Introduction 'Lense-Thiring' should be 'Lense-Thirring'.
[§II–III] The paper should state explicitly which parts of the geodesic-deviation equations and frequency results are taken from [72] and [73] and which are new; the current text cites those papers for the same formulas without a clear attribution.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Shapiro-delay and Shirokov-effect derivations are analytic computations from the input q-metric, with no fitted parameters and no load-bearing self-citations.

full rationale

The paper's central formulas (Eqs. 28-37, 47, and 51) are derived by direct substitution into the geodesic deviation and null geodesic equations for the q-metric (Eq. 1), with m and q serving as stated inputs. No parameter is fitted to data and then renamed as a prediction; the q-dependence of the time delay is a computed consequence of the metric, not an input assumed into the result. The self-citations [72] and [77] are not load-bearing: [72] is referenced only for a more detailed discussion of tidal forces, while the geodesic-deviation coefficients and frequencies are derived in the paper (Eqs. 5-14, 21, and 27), and [77] is cited for the standard conserved energy and angular momentum of static axisymmetric spacetimes, which are also written explicitly. The external comparison [36] is used only as a point of contrast. Even if the physical interpretation of Eq. (51) is contested on coordinate grounds, that is a correctness concern, not circularity: the derivation does not assume the delay it claims to derive.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

No data fitting is performed. The central results depend on the assumed q-metric, small-q truncation, and the geodesic-deviation equation as written in Eq. (4), which appears incomplete. The coordinate interpretation of r is an unstated physical assumption affecting the Shapiro delay claim.

free parameters (1)

q (quadrupole deformation parameter) = not fitted (model parameter)
q is the deformation parameter in the q-metric; the paper expands in small q and treats it as a free parameter of the spacetime, not fitted to data.

assumptions (3)

domain assumption The q-metric is the correct exterior spacetime for a static axisymmetric compact object with quadrupole deformation.
Used throughout the paper; the q-metric has a naked singularity for q != 0 and the associated source is not specified, so its physical applicability is an assumption.
domain assumption The deformation parameter q is small enough to truncate the expansions at linear order.
Eqs. (28)-(32) and (51) keep only terms linear in q; the paper does not quantify the validity range of this truncation.
ad hoc to paper Eq. (4) is a valid form of the geodesic deviation equation.
Eq. (4) omits connection-squared terms that appear in the standard Jacobi equation; the paper uses it to derive the Shirokov frequencies, so the correctness of this assumption is load-bearing.

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Pith. "Pith review of Constraining quadrupole deformations with relativistic effects." pith.science (2026). https://pith.science/paper/VO4O7W23

@misc{pith2026250610437,
  author       = {Pith},
  title        = {Pith review of: Constraining quadrupole deformations with relativistic effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VO4O7W23}},
  note         = {Machine review of arXiv:2506.10437}
}

abstract

We investigate two general relativistic effects - namely, the Shirokov and Shapiro effects - within the framework of the Zipoy-Voorhees spacetime ($q$-metric), which generalizes the Schwarzschild solution by incorporating a quadrupole moment. By analyzing the geodesic deviation equations, we explore the oscillatory motion of test particles and demonstrate how the source's quadrupole parameter influences the Shirokov effect. Furthermore, we derive an expression for the Shapiro time delay in this deformed spacetime and examine the quadrupole moment's impact on the gravitational time delay experienced by radio waves propagating near a massive object. The first-order approximation reveals a pronounced effect of the quadrupole parameter on the time delay, in contrast to similar recent analyses. These findings deepen our understanding of how deviations from spherical symmetry influence gravitational phenomena, with potential implications for the study of compact astrophysical objects such as neutron stars and naked singularities or ''black hole mimickers'' that exhibit significant multipolar structures.

Figures

Figures reproduced from arXiv: 2506.10437 by the authors.

**Figure 2.** FIG. 2: Dimensionless transverse oscillation frequency [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3: Dependence the periods [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4: The same as in Fig. 3 but here as a heatmap showing the dependence of the orbital periods [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 6.** Figure 6: FIG. 6: Schematic illustration of Shapiro time delay. The [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 8.** Figure 8: FIG. 8: The time delay Eq.(51) with respect to the [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

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AI is grateful for the kind hospitality of the Rudolf Peierls Centre for Theoretical Physics at the University of Oxford, where some work on this research was done, and extends appreciation to the Yessenov Foundation for its financial support

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