Deviation of trajectories in nonrelativistic and relativistic cases and Shirokov effect

V. P. Vandeev; Yu. V. Pavlov

Deviation of trajectories in nonrelativistic and relativistic cases and Shirokov effect

T0 review · 0 major / 6 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A static spherically symmetric spacetime can cancel the Shirokov effect, so the effect is not an unavoidable consequence of general relativity.

desk verdict A clean new GR counterexample where the Shirokov effect vanishes, with solid math and minor presentation gaps. read the letter →

arxiv 2607.29350 v1 pith:ZGFWTW7T submitted 2026-07-31 gr-qc astro-ph.EPastro-ph.SR

Yu. V. Pavlov , V. P. Vandeev This is my paper · ORCID

classification gr-qcastro-ph.EPastro-ph.SR PACS 04.20.-q04.25.-g

keywords geodesicdeviationShirokoveffectcircularorbitsstaticsphericallysymmetricspacetimepericentershiftcosmologicalconstantcentralpotentialsenergyconditions

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

The paper compares how nearby bodies deviate from circular orbits in Newtonian central potentials and in static spherically symmetric spacetimes, and it solves the deviation equations completely in both settings. It seeks to show when the Shirokov effect, the relativistic mismatch between oscillation frequencies in perpendicular directions, appears and whether it can be avoided. It finds all central potentials whose near-circular trajectories close, derives pericenter-shift formulas for the Kottler metric, and identifies a family of metrics that keep near-circular orbits closed. The headline result is the static spherically symmetric metric f = 1 - 2P/(r + 3P), for which the frequencies Omega and omega coincide, so the Shirokov effect is absent; the matter that generates this metric satisfies the weak and dominant energy conditions. This shows that the Shirokov effect is a property of some spacetimes, not a generic feature of circular geodesics in general relativity.

What carries the argument

The central object is the deviation vector eta for a circular reference geodesic and its two eigenfrequencies: omega, the orbital angular frequency, and Omega, the frequency of the radial and in-plane oscillations. The machinery is the linearized geodesic-deviation system for the static spherically symmetric ansatz; the determinant condition on that system yields Omega squared directly from the metric functions f(r) and B(r). The construction then imposes Omega = gamma times omega and, importantly, B = f, which turns the frequency condition into an ordinary differential equation whose solutions give the closed-orbit metrics and, for gamma = 1, the no-Shirokov metric. In the Newtonian limit t

What would settle it

Take a static spherically symmetric metric with B(r) different from f(r), for example the Reissner-Nordstrom metric, and compute Omega squared on a circular geodesic using the paper's frequency formula. If Omega squared differs from omega squared, the Shirokov effect is present in that spacetime, confirming that the no-Shirokov example hinges on the B = f restriction rather than on a property of all static spherically symmetric metrics.

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Extended reading notes

Core claim

The paper's central claim is that the Shirokov effect, the mismatch between the oscillation frequency of the radial deviation and the orbital rotation frequency for nearby circular trajectories, is not forced by general relativity. For static spherically symmetric metrics, the paper reduces geodesic deviation to two frequencies, omega from Eq. (56) and Omega from Eq. (60), and solves the deviation equations in closed form. Requiring Omega = omega and, in the construction, setting B = f reduces the condition to Eq. (73); a solution is the metric f = 1 - 2P/(r + 3P). The Einstein equations then give an energy-momentum tensor with nonnegative energy density and pressures bounded by the energy d

Load-bearing premise

The construction assumes B = f throughout, and the physical relevance of the example rests on matter that has only been checked against the weak and dominant energy conditions; if B and f are independent, the no-Shirokov conclusion is not established by the paper.

Editorial extensions

If this is right

For every central potential of the form alpha times r to the power (m^2/n^2 - 2) plus a constant, trajectories close to circular ones are closed curves when the expected sign condition holds, giving a linear-order extension of Bertrand's theorem to many non-Bertrand potentials.
In general relativity, the metric family with gamma = m/n gives the same closed-trajectory property near circular geodesics; for gamma = 2 the metric reduces to anti-de Sitter space.
For the Kottler metric, the pericenter shift of an orbit close to circular is Delta phi = pi(3r_g/r + 2 Lambda r^3/r_g); the cosmological-constant term is unobservably small for the observed Lambda but could probe a local quintessence-like field.
The exact deviation solutions reproduce the Newtonian time-dependence pattern: harmonic, cubic, or exponential growth for stable, marginal, and unstable circular orbits.
The metric f = 1 - 2P/(r + 3P) supplies a matter distribution that satisfies the weak and dominant energy conditions for which the Shirokov effect is absent.

Reading between the lines

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

The no-Shirokov example is constructed under the restriction B = f; a direct check of the frequency formula on metrics with B not equal to f would determine whether the vanishing effect is generic or unique to that restricted family.
The matter source is reverse-engineered and only the weak and dominant energy conditions are verified; checking stability, causality, and the strong energy condition is a natural next step not addressed in the paper.
The geodesic-deviation route to the pericenter shift avoids solving the full orbit equations and could be extended to slowly rotating or axisymmetric spacetimes, where no analogous closed-form treatment is currently given.
The closed-orbit family labelled by rational gamma provides a relativistic analogue of Bertrand's theorem in the small-deviation regime; testing whether these metrics arise from physically plausible matter would connect them to observable systems.

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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

The key thing you should know: this paper has a genuinely new static spherically symmetric metric, Eq. (89), where the Shirokov effect is absent, and the matter source satisfies the weak and dominant energy conditions. I re-derived the example myself. With B=f=(r+P)/(r+3P), Eq. (60) gives Ω² = P/[r(r²+3Pr+3P²)] and Eq. (56) gives the same, so Ω=ω. That is a real result, not a formal artifact. The derivation of the deviation system and the frequency formulas is internally consistent and reproduces the Schwarzschild and Kottler limits. The paper also gives explicit pericenter-shift formulas and a useful comparison between nonrelativistic and relativistic deviation solutions. The nonrelativistic potential family is standard, but the relativistic metrics in Eqs. (75), (88), and (89) appear to be new. The work is self-contained: no fitted data, no circular reasoning.

The soft spots are real but minor. Section 6 leaves B unspecified in metric (88). The condition Ω=0 does not involve B, so the metric family is under-specified; you need to say B=f or give some other B, because the deviation solution (64) depends on B. The jump to Eq. (74) is unshown; a referee should ask for the derivation or a reference. The abstract's phrase "the metric" overstates uniqueness, while the body correctly says "an example"—and the example works. Physically, the matter source is singular at r=0 and no stability analysis is given, but that is normal for a counterexample; the energy condition check is legitimate and sufficient for the stated claim.

Who is this for? Anyone working on geodesic deviation, the Shirokov effect, or closed orbits in static spherical spacetimes. It is a competently executed paper with one clean new counterexample, and it deserves a serious referee. I would send it to review, asking for the B specification in Section 6 and a derivation of (74), but I would not block acceptance on those points. If I worked in this area, I would cite the no-Shirokov metric.

Referee Report

0 major / 6 minor

Summary. The paper analyzes deviations of circular orbits in nonrelativistic central potentials and in static spherically symmetric spacetimes. For the nonrelativistic case it obtains the general solution of the deviation equations and shows that the radial/azimuthal frequency Ω equals the orbital frequency ω only for the 1/r potential. In the relativistic case, for the metric (44) with arbitrary f(r) and B(r), it derives the geodesic deviation system and the frequency formulas ω and Ω (Eqs. (56) and (60)). It applies these to the Kottler metric to compute the pericenter shift including the cosmological constant. It then constructs families of metrics with Ω=γω, giving rational frequency commensurability and closed nearby orbits, and the limiting Ω=0 case. Finally, it presents metric (89), with B=f, for which Ω=ω identically, so the Shirokov effect is absent, and shows that the source (Eqs. (95)-(96)) satisfies the weak and dominant energy conditions.

Significance. The main result is an explicit counterexample to the expectation that the Shirokov effect (anisotropic oscillation frequencies for circular geodesics) is unavoidable in general relativity: metric (89) has Ω=ω, and its matter satisfies standard energy conditions. The derivation is self-contained and checkable: the frequency formulas (56) and (60) follow directly from the metric, and the special metric (89) is obtained by solving the condition Ω=ω rather than by fitting. The paper also provides a useful method for computing pericenter shifts from deviation frequencies without solving the full geodesic equations, and the closed-orbit families (75) extend Bertrand-type behavior to the relativistic setting. The explicit general solutions of the geodesic deviation system for arbitrary static spherically symmetric spacetimes are a valuable reference. The main weakness is not technical but presentational: the scope of the construction (B=f) is understated.

minor comments (6)

[Section 6, Eq. (87)] The Ω=0 metric is not fully specified: B(r) is never given. While the condition Ω=0 from Eq. (60) is independent of B (B multiplies the bracket), the line element (44) requires both f and B. Please state a choice (e.g., B=f) or indicate that B is arbitrary.
[Section 7, Eq. (89)] The no-Shirokov metric is only displayed through f(r); the full interval should be written out with B=f. The abstract's 'the spherically symmetric metric' should be 'an example of a static spherically symmetric metric' and should mention the B=f restriction, since the general ansatz (44) allows independent B(r) and Eq. (60) contains B.
[Section 4, Eqs. (66)-(71)] The pericenter shift formula (71) is derived under an expansion in r_g/r and Λr^3/r_g. Please state explicitly that these two parameters are assumed small; in particular, the second parameter is not simply Λr^2, so the range of validity for large orbits should be noted.
[Section 2, after Eq. (26)] Typo: 'substituted 2ηϕ/dt2' should read 'substituting d²ηφ/dt²'. Also, in Eq. (35) the notation C'_1 is confusing; a prime should not be used as part of a constant name.
[Section 5, Eq. (74)] The notation in (74) is dense; a brief derivation of (75) from (74) after setting q=2-γ² would improve readability. This also helps the reader verify the γ≠2 condition.
[Section 7, energy conditions] The text says 'the standard energy conditions' but names only the weak and dominant energy conditions. Since the strong energy condition also holds for the tensor (95)-(96), consider saying so explicitly or replacing 'standard' with the specific conditions checked.

Circularity Check

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No significant circularity: the no-Shirokov metric is constructed by solving the imposed frequency condition, not fitted or predicted from its own output.

full rationale

The derivation is self-contained. Sections 2–3 derive the Newtonian and relativistic deviation equations from first principles: Eq. (60) computes the radial oscillation frequency squared directly from the static spherically symmetric metric (44), and Eq. (56) computes the orbital frequency squared. The no-Shirokov example (89) is obtained in Sec. 5 by imposing Omega = gamma*omega with B=f; Eq. (73) is the resulting ODE and (75) is its explicit solution, so for gamma=1 the equality Omega=omega holds by construction rather than being asserted as an independent prediction. The matter stress-energy tensor (95)–(96) follows from the standard Einstein-equation formulas (91)–(93), and the energy-condition verification cites an external reference (Hawking–Ellis [19]). The only self-citation, Ref. [17] by one of the authors on sine-Gordon equations, appears in a remark about the difficulty of inverting ultraelliptic integrals in Ref. [16]; it is not load-bearing for any result. The restriction B=f in Sec. 5 limits the scope of the metric family, and the body correctly says 'an example' rather than claiming uniqueness, so this is a scope caveat rather than circularity. No fitted parameter is renamed as a prediction and no derivation step reduces to an input by definition.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The results are purely mathematical; there are no fitted empirical parameters. The listed free parameters are arbitrary constants defining example metrics. The main hidden input is the B=f restriction plus the standard GR framework.

free parameters (3)

P
Arbitrary positive length parameter in the no-Shirokov metric f=1−2P/(r+3P). It sets the length scale of the matter distribution and is not fitted to data.
a
Arbitrary length parameter in the Ω=0 metric f=1−a²/(r²+a²). It is an integration constant, not fitted to data.
γ=m/n
Rational frequency ratio chosen for closed near-circular trajectories. It labels a family of metrics rather than being fitted to observations.

assumptions (6)

standard math Geodesic deviation equation D²η^i/ds² = R^i_{klm} u^k u^l η^m
Invoked as the starting point for relativistic deviations in Eq. (40).
domain assumption Static spherically symmetric metric ansatz (44)
The whole analysis restricts to static spherically symmetric spacetimes with functions f and B.
standard math Bertrand's theorem for closed orbits in central potentials
Used in Sec. 2 as the benchmark for the nonrelativistic closed-orbit problem.
standard math Einstein equations with stress-energy tensor (90)
Used in Sec. 7 to compute the matter source for the no-Shirokov metric.
domain assumption Energy conditions (WEC and DEC) as a physical-viability criterion
The paper treats satisfaction of the weak and dominant energy conditions as sufficient evidence that the matter distribution is physical; no causality or stability check is made.
ad hoc to paper B=f restriction for the constructed metric families
Sec. 5 sets B=f to make the frequency-ratio ODE tractable; Sec. 7 uses B=f implicitly. This is a modeling restriction, not a symmetry requirement of the problem.

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Cite this review

Pith. "Pith review of Deviation of trajectories in nonrelativistic and relativistic cases and Shirokov effect." pith.science (2026). https://pith.science/paper/ZGFWTW7T

@misc{pith2026260729350,
  author       = {Pith},
  title        = {Pith review of: Deviation of trajectories in nonrelativistic and relativistic cases and Shirokov effect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZGFWTW7T}},
  note         = {Machine review of arXiv:2607.29350}
}

read the original abstract

We compare the deviations of circular orbits in the nonrelativistic and relativistic cases. General solutions of the deviation equations are obtained. We find the potentials for the nonrelativistic case and the metric components in general relativity for which trajectories close to circular ones are closed curves. Explicit expressions for the pericenter shift of an orbit close to a circular one in a static spherically symmetric spacetime are obtained. Estimates of the cosmological constant effect on the orbit pericenter shift are given. Finally, we find the spherically symmetric metric in which the Shirokov effect is absent.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 1 linked inside Pith

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

On one new effect of the Einsteinian theory of gravitation

M. F. Shirokov, “On one new effect of the Einsteinian theory of gravitation”, Gen. Relativ. Gravit.4, 131–136 (1973)

1973

[2] [2]

The equation of geodesic deviation in Newtonian theory and the oblateness of the Earth

P. J. Greenberg, “The equation of geodesic deviation in Newtonian theory and the oblateness of the Earth”,Nuovo Cimento B24, 272–286 (1974)

1974

[3] [3]

On Shirokov’s ‘One New Effect of the Einsteinian Theory of Gravitation

A. Nduka, “On Shirokov’s ‘One New Effect of the Einsteinian Theory of Gravitation”, Gen. Relativ. Gravit.8, 347–351 (1977). 15

1977

[4] [4]

Small oscillations of test bodies in circular orbits in the Schwarzschild and Kerr metrics (Shirokov’s effect)

Yu. S. Vladimirov, S. V. Rodichev, “Small oscillations of test bodies in circular orbits in the Schwarzschild and Kerr metrics (Shirokov’s effect)”, Soviet Physics Journal24, 954–957 (1981)

1981

[5] [5]

Solutions of the equations of geodesic deviation for static spherical sym- metric space-times

H. Fuchs, “Solutions of the equations of geodesic deviation for static spherical sym- metric space-times”,Ann. Phys. Leipz.40, 231–233 (1983)

1983

[6] [6]

Deviation of circular geodesics in static spherically symmetric space-times

H. Fuchs, “Deviation of circular geodesics in static spherically symmetric space-times”, Astron. Nachr.311, 271–276 (1990)

1990

[7] [7]

On geodesic deviation in Schwarzschild spacetime

D. Philipp, V. Perlick, C. Laemmerzahl, K. Deshpande, “On geodesic deviation in Schwarzschild spacetime”,Metrology for Aerospace (MetroAeroSpace), 2015 IEEE, pp.198–203, arXiv:1508.06457

arXiv 2015

[8] [8]

On the applicability of the geodesic devia- tion equation in General Relativity

D. Philipp, D. Puetzfeld, C. L¨ ammerzahl, “On the applicability of the geodesic devia- tion equation in General Relativity”, inRelativistic Geodesy, Fundamental Theories of Physics, Vol. 196, (D. Puetzfeld and C. L¨ ammerzahl, eds.), Springer, Cham, Switzer- land (2019), pp. 419–451

2019