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REVIEW 3 major objections 5 minor 103 references

Nethotrons: exploring the possibility of measuring relativistic spin precessions, from Earth's satellites to the Galactic Centre

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A pulsar orbiting the Milky Way's central black hole could show measurable relativistic spin precessions, while Earth satellites and the double pulsar cannot.

desk verdict A clear, honest feasibility map for 1pN spin-precession tests; the negative results are solid, but the pulsar–Sgr A* numbers sit on an unevaluated electromagnetic noise floor. read the letter →

arxiv 2506.10162 v1 pith:XOP5I2SY submitted 2025-06-11 gr-qc astro-ph.EPphysics.space-ph

Lorenzo Iorio This is my paper · ORCID
classification gr-qcastro-ph.EPphysics.space-ph PACS 04.80.Cc
keywords nethotronsdeSitterprecessionPugh-SchiffgravitomagnetismLAGEOSsatellitesSagittariusA*pulsardoublesatellitelaserranging
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 investigates whether the two post-Newtonian spin precessions of general relativity, the gravitoelectric de Sitter effect and the gravitomagnetic Pugh-Schiff effect, can be measured with objects other than the dedicated Gravity Probe B mission. For the laser-ranged satellites LAGEOS, LAGEOS 2 and LARES, it finds expected right-ascension shifts of tens of thousands and hundreds of milliarcseconds over decades, below the current roughly 0.1-degree measurement accuracy and swamped by Newtonian and non-gravitational torques. For a hypothetical millisecond pulsar in a half-year orbit around the supermassive black hole Sgr A*, it finds spin precessions that may reach tens or hundreds of degrees over ten years, a signal large enough to matter provided competing electromagnetic torques can be modelled. For the double pulsar, the spin-spin (Pugh-Schiff-type) precession is predicted to be about four orders of magnitude below current measurement accuracy. The paper thus maps which spin-precession tests are feasible with present or near-future technology.

What carries the argument

The machinery is the set of first-order post-Newtonian (1pN) spin precession equations in vector form, together with the orbital precession equations for the node and inclination. The de Sitter (gravitoelectric) rate is $\boldsymbol{\Omega}_{\rm dS} = (3 n_K \mu / 2 c^2 p) \hat h$, and the Pugh-Schiff (gravitomagnetic) rate is $\boldsymbol{\Omega}_{\rm PS} = (GJ / 2 c^2 a^3 (1-e^2)^{3/2})[3(J_l \hat l + J_m \hat m) - 2\hat J]$, with the spin axis tracked through its right ascension $\alpha$ and declination $\delta$; the orbital plane itself is allowed to precess under the primary's quadrupole moment $J_2$ and gravitomagnetic field, so the signatures are harmonic rather than linear trends. A Newtonian own-oblateness torque proportional to the nethotron's $J_2^s$ is included as the main classical competitor. The same equations are applied to laser-ranged satellites, a hypothetical pulsar around Sgr A*, and the double pulsar.

What would settle it

Discover a millisecond pulsar in a 0.5-year orbit around Sgr A*, track its spin-axis orientation over ten years, and subtract a detailed model of magnetic-dipole and ambient-field torques; a residual spin shift consistent with zero would falsify the predicted tens-to-hundreds-of-degrees relativistic precession.

Watch

Extended reading notes

Core claim

The paper's central quantitative claim is a feasibility ranking of three arenas for detecting relativistic spin precessions. Using the standard first-order post-Newtonian rates for the de Sitter and Pugh-Schiff precessions, parameterized by the spin axis' right ascension and declination, it numerically integrates the spin and orbital motion of LAGEOS, LAGEOS 2, and LARES over their lifetimes and finds accumulated right-ascension shifts of a few tens of thousands (de Sitter) and a few hundred (Pugh-Schiff) milliarcseconds, against a current spin-axis measurement accuracy of about 0.1 degree and Newtonian own-oblateness shifts of hundreds of millions of milliarcseconds. For a millisecond pulsar in a 0.5-year, highly eccentric orbit around Sgr A*, the same calculation yields de Sitter and Pugh-Schiff spin precessions of tens to hundreds of degrees over ten years, with the pulsar's own quadrupole torque negligible. For the double pulsar PSR J0737-3039A/B, the gravitomagnetic spin-spin precession of component B due to component A is computed as 0.00008 degrees per year, about four orders of magnitude below the current 0.6-0.3 degrees per year measurement accuracy. The paper concludes that only the Galactic-Centre pulsar scenario offers signals large enough relative to present measurement capabilities, and that electromagnetic torques there remain an unmodelled competing effect.

Load-bearing premise

Electromagnetic torques on the pulsar's spin axis, from magnetic-dipole emission, dipole inertia, and the strong magnetic field near Sgr A*, do not mask the post-Newtonian precession; the paper explicitly leaves their evaluation to future work.

Editorial extensions

If this is right

  • Existing LAGEOS-type satellites cannot currently test the de Sitter or Pugh-Schiff precessions: the relativistic signals are at the milliarcsecond level while spin-axis orientation is known to about 0.1 degree, and Newtonian self-oblateness torques produce nominal shifts hundreds of millions of times larger.
  • A purpose-built passive satellite with Gravity Probe B-like spin and orbit geometry would accumulate a 40 mas/yr Pugh-Schiff right-ascension trend and a 7000 mas/yr de Sitter declination trend, requiring new measurement techniques and careful manufacturing to reduce classical torques.
  • A millisecond pulsar in a 0.5-year orbit around Sgr A* could show de Sitter and Pugh-Schiff precessions of tens to hundreds of degrees over ten years, making its spin axis a potential probe of the black hole's spacetime, provided electromagnetic torques can be modelled.
  • The double pulsar's spin-spin precession is far too small (0.00008 deg/yr) to be measured with current or foreseeable accuracy.

Reading between the lines

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

  • If a short-period pulsar around Sgr A* is eventually found, the spin-precession signal would add a second observable, independent of orbital timing, for constraining the black hole's dimensionless spin parameter and possibly testing the Kerr no-hair relation.
  • The paper's own admission that electromagnetic torques are unmodelled means the tens-to-hundreds-of-degrees figures are best read as an upper bound on the relativistic signal; estimating the pulsar's magnetic-dipole and external-field torques is the natural next calculation.
  • The same 1pN equations apply to any future pulsar around an intermediate-mass or supermassive black hole, so the feasibility map drawn here can be reused as new systems are discovered.
  • Satellite laser ranging would need an improvement of roughly two orders of magnitude in spin-axis determination to reach the milliarcsecond signals; solar-glint photometry, mentioned in the paper, is the only existing technique said to offer such gains.
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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

3 major / 5 minor

Summary. The paper explores whether post-Newtonian spin precessions—the gravitoelectric de Sitter effect and the gravitomagnetic Pugh–Schiff effect—could be measured using spinning objects other than Gravity Probe B. It derives orbit-averaged spin-precession equations from the Barker–O'Connell formalism, then numerically integrates them for the SLR satellites LAGEOS, LAGEOS 2, and LARES, for a hypothetical laser-ranged satellite (NethoSAT), for a hypothetical pulsar orbiting the supermassive black hole Sgr A*, and for the double pulsar PSR J0737-3039A/B. The paper reports that the LAGEOS-family relativistic spin shifts are tens of thousands (de Sitter) and hundreds (Pugh–Schiff) of milliarcseconds over decades—below the current ~0.1-degree measurement accuracy—and that the Newtonian oblateness torque is orders of magnitude larger. For the Sgr A* pulsar scenario, it reports de Sitter shifts of tens to hundreds of degrees over ten years for a 0.5-year orbit, while the double pulsar spin–spin precession is estimated at 0.00008 degrees per year, about four orders of magnitude below current measurement accuracy.

Significance. If the quantitative estimates are reliable, the paper provides a useful scoping map for future tests of relativistic spin precession and, in particular, suggests that the LAGEOS-family route is not competitive with the current measurement capabilities. The calculations are forward applications of published formulas with parameters taken from the literature; I found no fitted-parameter circularity in the central results, and the arithmetic is internally consistent. The negative conclusions for the LAGEOS satellites and the double pulsar are robust and are the most directly useful part of the paper. The Sgr A* pulsar scenario is the only quantitatively exciting possibility, but, as stated in the paper itself, its interpretation is currently blocked by an unquantified electromagnetic torque noise floor.

major comments (3)
  1. [Section 4, Figure 5] The paper's headline positive scenario is not yet interpretable as a measurement of relativistic precession. The text states that 'the magnetic torque should be a major competing effect also in this scenario' and that 'an evaluation of such potentially relevant competing effects is outside the scopes of the present work,' naming three spin-axis torques: magnetic-dipole radiation, the inertia of the magnetic dipole moment, and the strong external magnetic field around Sgr A*. These torques act on the same spin axis as the de Sitter and Pugh–Schiff precessions, so the tens-to-hundreds-of-degrees shifts shown in Figure 5 can be attributed to general relativity only if the electromagnetic contributions are shown to be subdominant on the 10-year timescale or are separately modelable. Please add at least an order-of-magnitude estimate for the fiducial 5 ms pulsar, using the quoted surface field strengths and the Sgr A* magnetic field, or explicitly reframe the Sgr A* result as a conditional upper bound rather than a predicted GR signal.
  2. [Section 4, Eqs. (47)–(50) and Figure 5] The parameter selection for the Sgr A* pulsar is almost entirely unconstrained, and Figure 5 uses one specific configuration from Eq. (50) together with an eccentricity fixed only through r_min = 12.4 r_Sch. Equations (49) and (50) identify configurations that maximize the instantaneous de Sitter and Pugh–Schiff rates, but the figure uses Eq. (50) also for the de Sitter panels, so the quoted 'tens or hundreds of degrees' is not the maximum cumulative de Sitter shift over the parameter space. A brief parameter scan, or at least a sensitivity statement, is needed to support the claim that the shifts 'may be as large as' the displayed values rather than being a single illustrative point estimate.
  3. [Section 3, Table 1 and Figures 1–3] The Newtonian oblateness shifts in Figures 1–3 reach hundreds of millions of milliarcseconds because the spin period is modeled as a linear trend with the Pdot_s values from Table 1, but no uncertainty or covariance is attached to that extrapolation. This does not change the paper's negative conclusion for the LAGEOS family, since the current spin-axis measurement accuracy is already about 0.1 degree, but the presentation would be more informative if the authors showed how the Newtonian signal changes under plausible variations of Pdot_s or displayed a bounding envelope for the nominal values.
minor comments (5)
  1. [Section 3, NethoSAT paragraph] The word 'graviteoectric' should be 'gravitoelectric' in the sentence describing the NethoSAT de Sitter precession.
  2. [Section 4, after Eq. (46)] The phrase 'the are' should be 'there are', and 'possess' should be 'possess' in the paragraph on pulsar magnetic moments.
  3. [Table 1] The units of the Pdot_s column are not specified; please indicate whether the values are in s/s, s/day, or another unit, since they enter directly into the Newtonian oblateness integrations.
  4. [Figure 5] The lower-left axis label '5.×10 -8' is missing a multiplication dot and has awkward spacing; using proper LaTeX notation (e.g., 5\times10^{-8}) would improve readability.
  5. [Section 2, Eqs. (18)–(19) and (26)–(27)] The paper calls these quantities dα/dt and dδ/dt while describing them as averaged precessions per orbit; the averaging convention and the resulting units should be stated explicitly at their first occurrence.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the central predictions are forward evaluations of standard published formulas with literature-based parameters; the only self-citation is a non-load-bearing numerical input.

full rationale

The paper's quantitative claims are forward numerical integrations of the de Sitter and Pugh-Schiff formulas (Eqs. 15-27) with orbital and spin parameters taken from the cited literature (Visco and Lucchesi 2018 for LAGEOS-family spin parameters; Peissker et al. 2022 for Sgr A* mass and spin; Laarakkers and Poisson 1999 for neutron-star quadrupole moments). No parameter is fitted in this paper, and no predicted signal is defined in terms of the measured quantity it is compared with. The only self-citation entering a numerical result is ref. [102], used in Section 5 to set the range of pulsar A's spin direction when computing the spin-spin precession of Eq. (52). This is a legitimate use of a previously published input parameter, not a derivation of the result from itself: the order-of-magnitude conclusion (about four orders below the current measurement accuracy) does not hinge on the precise orientation, and [102] is not invoked to forbid alternatives or to justify an ansatz. The paper's own stated limitation in Section 4, that electromagnetic torques on a pulsar near Sgr A* are not evaluated, concerns the physical isolation and interpretation of the signal, not circularity of the derivation. Therefore no step reduces by construction to its inputs, and the one self-citation does not make the central calculations circular.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The free parameters listed are scenario choices taken from the literature or adopted to maximize displayed signals; they are not fits to data. The axioms show the paper leans on standard GR results (Kerr no-hair, Barker-O'Connell formulas) and on one unverified assumption the paper itself flags: that electromagnetic torques on the pulsar do not mask the post-Newtonian precession. No new physical entities are introduced; 'nethotron' is a neologism for existing objects and 'NethoSAT' is a hypothetical satellite design.

free parameters (4)
  • Sgr A* dimensionless spin chi = 0.5 (literature range up to ~0.9)
    Adopted from ref. [78] in Section 4; linearly scales the Pugh-Schiff rate APS (Eq. 25) and hence the pulsar PS signature, so the large uncertainty in chi propagates directly into the headline numbers.
  • Pulsar spin/orbital configuration (Eq. 50) = alpha=279.3 deg, delta=12.1 deg, zeta=184.2 deg, i=67.2 deg, I=67.2 deg, Omega=274.2 deg
    The system is undiscovered and unconstrained, so the maximum-PS configuration is adopted for Figure 5; the tens-of-degrees result is an upper envelope, not a typical expectation. The paper discloses this choice.
  • Neutron star quadrupole parameter xi_star = 3.507
    Upper end of the range from ref. [82]; enters only the negligible Newtonian oblateness torque (O(c^-4)), so the choice does not affect conclusions.
  • LAGEOS-family spin period slowdown rates Pdot_s = 1.4e-6 to 3e-6 s/s (Table 1)
    Inferred from ref. [59] and used to model the satellites' spin period lengthening; this choice drives the huge oscillatory Newtonian torque baselines in Figures 1-3 but does not change the negative feasibility conclusion.
assumptions (5)
  • domain assumption The Kerr metric and no-hair relations (Eqs. 38-40) describe the Sgr A* spacetime and its multipole moments.
    Section 4 uses M=4.1e6 Msun, chi=0.5 and relations (38)-(40) to characterize the central black hole; this is standard GTR but is an unproved conjecture applied to the actual Sgr A* system.
  • domain assumption The 1pN test-gyroscope precession formulas of Barker and O'Connell (Eqs. 15-37) apply to the extended bodies considered here.
    Equations (15)-(37) are taken from refs. [1,8] without re-derivation; for the pulsar case the paper argues the multipolar self-gravity contribution is negligible (O(c^-4)), supporting the approximation.
  • domain assumption The orbital plane evolution of the Earth satellites in the spin integrations is governed by Eqs. (10)-(11), i.e. the J2 and gravitomagnetic node precessions.
    Used for the numerical integrations of Section 3; non-gravitational orbital perturbations and higher zonal harmonics are not included in the spin integration.
  • ad hoc to paper Electromagnetic torques on the pulsar's spin axis are subdominant or separable (unverified).
    Section 4 identifies the magnetic torque as 'a major competing effect' but defers its evaluation; the positive pulsar scenario implicitly requires that these torques do not swamp the post-Newtonian signal.
  • domain assumption The moment of inertia of the LAGEOS-type satellites is treated as that of a homogeneous sphere (Eq. 33).
    Explicitly assumed in the Newtonian oblateness torque AJs2; this simplification is used throughout Section 3.

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Pith. "Pith review of Nethotrons: exploring the possibility of measuring relativistic spin precessions, from Earth's satellites to the Galactic Centre." pith.science (2026). https://pith.science/paper/XOP5I2SY

@misc{pith2026250610162,
  author       = {Pith},
  title        = {Pith review of: Nethotrons: exploring the possibility of measuring relativistic spin precessions, from Earth's satellites to the Galactic Centre},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XOP5I2SY}},
  note         = {Machine review of arXiv:2506.10162}
}
abstract

By ``nethotrons'', from the ancient Greek verb for to ``spin'', it is meant here a natural or artificial rotating object, like a pulsar or an artificial satellite, whose rotational axis is cumulatively displaced by the post-Newtonian static (gravitoelectric) and stationary (gravitomagnetic) components of the gravitational field of some massive body around which it freely moves. Until now, both relativistic effects have been measured only by the dedicated space-based mission Gravity Probe B in the terrestrial environment. It detected the gravitoelectric de Sitter and gravitomagnetic Pugh-Schiff spin precessions of four superconducting gyroscopes accumulated in a year after about 50 years from conception to completion of data analysis at a cost of 750 million dollars to $0.3$ and $19$ per cent accuracy, respectively. The perspectives to measure them also with long-lived Earth's laser-ranged geodetic satellites, like those of the LAGEOS family or possibly one or more of them to be built specifically from scratch, and pulsars orbiting the supermassive black hole in the Galactic Centre, yet to be discovered, are preliminarily investigated. The double pulsar PSR J0737-3039A/B is examined as well.

Figures

Figures reproduced from arXiv: 2506.10162 by the authors.

Figure 1
Figure 1. Numerically integrated time series, in mas, of the dS (upper row), PS (middle row) and Newtonian (lower row) RA and decl. shifts ∆α (t) and ∆δ (t) of the spin axis of LAGEOS over 49 yr. The initial values of the spin and orbital parameters were retrieved from [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Numerically integrated time series, in mas, of the dS (upper row), PS (middle row) and Newtonian (lower row) RA and decl. shifts ∆α (t) and ∆δ (t) of the spin axis of LAGEOS 2 over 33 yr. The initial values of the spin and orbital parameters were retrieved from [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Numerically integrated time series, in mas, of the dS (upper row) and PS (lower row) RA and decl. shifts ∆α (t) and ∆δ (t) of the spin axis of a hypothetical NethoSAT over 10 yr. The initial values of the spin and orbital parameters of GP-B, retrieved from [PITH_FULL_…
Figure 5
Figure 5. Figure 5: Numerically integrated time series, in ◦ , of the dS (upper row), PS (middle row) and Newtonian (lower row) RA and decl. shifts ∆α (t) and ∆δ (t) of the spin axis of a millisecond pulsar in a 0.5 yr orbit around the SMBH in Sgr A∗ over 10 yr. While the eccentricity was…

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