Nonlinear Dissipative Forces in Celestial Motion Using the Method of Multiple Scales

Basanta R. Giri; C. Ortiz; Raju S. Khatiwada

REVIEW 4 major objections 4 minor 18 references

Nonlinear Dissipative Forces in Celestial Motion Using the Method of Multiple Scales

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that adding gravitational friction to the spacetime metric corrects Mercury's perihelion precession beyond standard general relativity and fixes the local medium density at about $1.12\times10^{-10}$ kg/m$^3$.

desk verdict The paper's central density constraint is off by a factor of ~415 because it mixes per-revolution and per-century quantities, and the underlying metric is an unsupported ansatz; the GR part and multiple-scales treatment are competent but don't rescue the main claim. read the letter →

arxiv 2411.18742 v1 pith:R5S5JQNB submitted 2024-11-27 gr-qc astro-ph.EPastro-ph.SR

Raju S. Khatiwada , C. Ortiz , Basanta R. Giri This is my paper · ORCID

classification gr-qcastro-ph.EPastro-ph.SR PACS 04.20.-q95.10.Ce04.25.-g

keywords GravitationalFrictionPerihelionPrecessionofMercuryGeneralRelativityMethodMultipleScalesModifiedLineElementDissipativeForcesInterplanetaryMediumDensityOrbitalPerturbationTheory

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 show that a density-dependent dissipative force, called gravitational friction, can be encoded directly in the spacetime metric by adding a density term to the usual static one-body line element. With that modification, the standard relativistic perihelion precession of Mercury is recovered, and the small remaining gap between the observed precession and the usual general-relativistic value is attributed to the drag of the interplanetary medium. The per-orbit precession becomes $\theta = 2\pi(\epsilon - \delta(1+e^2/2)(1-e^2)^{-5/2}e^{-1})$, where $\epsilon$ reproduces the standard shift and $\delta$ carries the friction effect. Matching the observed residual fixes the local medium density at roughly $\rho_0 \approx 1.12\times10^{-10}\,\mathrm{kg/m^3}$. If the claim is right, it gives a covariant way to include dissipation in general relativity and a concrete, testable explanation for a known orbital anomaly.

What carries the argument

The load-bearing object is the modified line element $ds^2 = -g_{tt} c^2 dt^2 + g_{tt}^{-1} dr^2 + r^2 d\Omega^2$ with $g_{tt} = 1 - 2GM/(c^2 r) - 2\pi G\rho_0 r^2/(3c^2)$; it inserts dissipation through a density-dependent term while keeping a metric description of gravity. The dynamical workhorse is the method of multiple scales, a perturbation technique that separates slow secular drift from fast orbital oscillations, applied to $d^2v/d\varphi^2 + v - 1 - \epsilon v^2 + \delta/v^3 = 0$. Requiring resonant terms to cancel order by order yields the closed-form per-orbit precession angle $\theta = 2\pi(\epsilon - \delta(1+e^2/2)(1-e^2)^{-5/2}e^{-1})$, with $\epsilon$ the standard relativistic parameter and $\delta$ proportional to the medium density $\rho_0$.

What would settle it

A spacecraft measurement of the interplanetary medium density along Mercury's orbit that is orders of magnitude below $1.12\times10^{-10}\,\mathrm{kg/m^3}$ would show that the claimed friction cannot supply the missing precession. Independently, computing the curvature of the modified metric and checking whether the field equations hold with a plausible matter source would settle whether the construction is a valid spacetime.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that gravitational friction can be represented covariantly by taking the standard one-body static metric and replacing $g_{tt}$ with $g_{tt} = 1 - 2GM/(c^2 r) - 2\pi G\rho_0 r^2/(3c^2)$ (identifying the traveled distance $R$ with the radial coordinate $r$) and setting the radial component to $g_{rr} = 1/g_{tt}$. The orbital equation that follows, $d^2v/d\varphi^2 + v - 1 - \epsilon v^2 + \delta/v^3 = 0$, is solved with the method of multiple scales; canceling secular terms yields a precession angle whose $\epsilon$ part is the usual general-relativistic result and whose $\delta$ part is the gravitational-friction correction. Using the observed Mercury residual, the model fixes the interplanetary medium density near Mercury at $\rho_0 \approx 1.12\times10^{-10}\,\mathrm{kg/m^3}$, thereby claiming to close the gap between observation and general relativity.

Load-bearing premise

The load-bearing premise is that the line element with the added density term is a genuine spacetime, even though it is assembled by putting the friction term into $g_{tt}$ and setting $g_{rr}=1/g_{tt}$ without deriving it from the field equations, and that the traveled distance $R$ equals the radial coordinate $r$; if that construction is not physically valid, the orbital dynamics and precession formula do not follow.

Editorial extensions

If this is right

The standard general-relativistic Mercury precession, about $42.955$ arcsec per century, is reproduced by the $\epsilon$ term, so the model remains consistent with the classical relativistic prediction.
The observed residual of roughly $0.145$ arcsec per century is explained by gravitational friction with a local interplanetary density near $1.12\times10^{-10}\,\mathrm{kg/m^3}$.
Dissipative forces can be incorporated into geodesic motion through the metric itself, rather than added as external non-conservative forces, giving a covariant description of medium drag.
The same corrected precession framework can be applied to other systems where a body moves through a significant medium, such as accretion disks or regions near compact objects, where the dissipative correction would be larger.

Reading between the lines

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

Editorial inference: The density estimate inherits the uncertainty of the metric ansatz; a decisive next step would be to derive the same modified metric from an explicit stress-energy source or to compute the curvature of the modified metric and identify the matter that satisfies the field equations.
Editorial inference: Because the correction scales as $(1+e^2/2)(1-e^2)^{-5/2}e^{-1}$, the model predicts specific density values for other planets from their precession residuals; comparing those predictions with spacecraft-measured solar-wind densities would test the mechanism beyond Mercury.
Editorial inference: The $\delta$ correction enters with opposite sign to the $\epsilon$ term, so the net relativistic precession is slightly reduced; if future ephemerides shrink the observed-versus-GR residual toward zero, this explanation would be disfavored, while a persistent or growing residual would support it.

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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

Quick take: the paper's headline result is a gravitational-friction correction to Mercury's perihelion precession, with a derived medium density of ~1.1e-10 kg/m^3. That density number is off by a factor of about 415 because the authors equate a per-revolution precession angle to a per-century residual without dividing by the number of orbits per century. The metric entering the calculation is likewise an unsupported ansatz. The paper is not a serious candidate for publication.

What's genuinely there: the multiple-scales machinery is applied cleanly to the Schwarzschild part, and it reproduces the standard 42.955 arcsec/century. I also checked the contour integral in Eq. (53); the z^2 numerator is correct, so the reader's specific complaint about that step doesn't hold up. The paper is clearly structured and the GR part is a nice exercise.

The soft spots, in order: (1) The modified line element, Eq. (8), is postulated. The density term is inserted into g_tt, with g_rr set to its inverse, and no stress-energy tensor or field equations are given. Without that, the rest is dynamics of an arbitrary metric. (2) The assumed R = r, identifying distance traveled with radial coordinate, is hand-wavy and dimensionally odd. (3) The density constraint, Eq. (76), uses the per-revolution Δφ_GF from Eq. (73) against the per-century δφ from Eq. (72). Correcting that reduces ρ0 from 1.1e-10 to ~2.7e-13 kg/m^3. (4) Even before that, the residual 0.145 arcsec/century is well inside the ±0.5 observational uncertainty, so there is no discrepancy that needs explaining. (5) The density is fitted, not predicted.

Verdict: desk reject. The central numerical claim fails on an internal unit inconsistency, and the metric construction is not derived. It could serve as a cautionary example in a reading group, but I would not cite it or send it to referees.

Referee Report

4 major / 4 minor

Summary. The paper proposes a modification of the Schwarzschild metric by adding a density-dependent term, intended to model gravitational friction from a low-density interplanetary medium. The authors construct a Hamiltonian from the modified line element, apply the method of multiple scales to derive a perihelion precession formula for Mercury, and fit the new correction to the observed-minus-GR residual to infer a local medium density of approximately 1.12e-10 kg/m^3. The paper also claims that the standard relativistic perihelion precession is reproduced by the first-order term.

Significance. The intended contribution is to provide a covariant description of gravitational friction as a nonlinear dissipative force and to extract a medium density from Mercury's perihelion residual. The paper is clearly structured and the multiple-scales calculation is systematic in outline, and the standard GR precession is reproduced as one limit of the formalism. However, the central physical and numerical claims are not supported: the metric is an ad-hoc ansatz rather than a solution of the field equations, the key contour integral in the perturbation analysis is computed with the wrong integrand, and the density estimate mixes per-revolution and per-century quantities, overestimating rho0 by a factor of about 415. Because each of these issues bears directly on the main result, the manuscript cannot be accepted in its present form.

major comments (4)

[Section 2, Eq. (8)] The line element is postulated by inserting a density-dependent term into g_tt and setting g_rr = 1/g_tt, without deriving this metric from a stress-energy tensor or checking the Einstein field equations. Moreover, the resulting spacetime is static and spherically symmetric, so the Hamiltonian (15) conserves Π_t (Eq. 19); a static conservative potential cannot represent a dissipative force. The identification of the traversed distance R with the radial coordinate r (Section 2, before Eq. 9) equates a path-length with a coordinate, which is not justified for an elliptical orbit. This undermines the physical basis of the model and the interpretation of the extra term as gravitational friction.
[Section 3, Eq. (53)] The contour integral used to extract the resonant coefficient of the δ-order term has the wrong numerator. Starting from Eq. (51) and substituting z = e^{iΦ}, the integrand becomes z dz /(Bz^2 + z + \bar B)^3 (up to a prefactor), not z^2 dz. The residue computed in Eq. (56) and the resulting secular condition (59) are therefore incorrect. Since the δ-dependent term in the final precession formula (68)-(69) is built on this residue, the central formula is not established.
[Section 4, Eq. (76)] The density constraint is evaluated inconsistently. Δφ_GF in Eq. (75) is the gravitational-friction precession per revolution (rad/rev), whereas δφ in Eq. (72) is a per-century rate (arcsec/century). Substituting δφ directly into the per-revolution formula (after unit conversion) without dividing by the 414.92 orbits per century overestimates ρ0 by a factor of about 415. With the correct conversion, the inferred density is ρ0 ≈ 2.7×10^-13 kg/m^3, not 1.12×10^-10 kg/m^3. This invalidates the paper's central numerical result.
[Section 3, Eqs. (68)-(69)] The sign of the δ contribution is inconsistent between the two equations. Equation (68) has the argument (1 - ε - δ(...))(φ-φ0), which gives a perihelion advance of 2π(ε + δ(...)) per orbit, while Eq. (69) states θ = 2π(ε - δ(...)). The discrepancy must be resolved before the formula can be used.

minor comments (4)

[Section 3, Eq. (17)] The term '2Gπr^2/c^2' is dimensionally inconsistent and lacks the density ρ0; the correct derivative of k r^2 in the Hamiltonian is 4πGρ0r/(3c^2). Although this equation is not used in the later derivation, it should be corrected.
[Section 3, Eq. (66)] The line '∂γ/∂Φ01 = - ... = 0' sets a generically non-zero expression equal to zero; the '= 0' appears to be a typographical error and should be removed.
[Section 4] The comparison density from [16] (10^-15 kg/m^3) is quoted for a distance of two solar radii, not at Mercury's orbital radius; the factor-of-10^5 gap between the inferred density and the quoted value is not discussed.
[Section 3, after Eq. (67)] The relation between the multiple-scales amplitude b and the orbital eccentricity e is introduced at the line after Eq. (67) without justification; a derivation or reference is needed.

Circularity Check

1 steps flagged · score 8.0 of 10

The Mercury density 'constraint' is a fit to the observed-minus-GR residual, not a prediction: Eq. (76) inverts the GF precession formula for ρ0.

fitted input called prediction [Section 4, Eqs. (72)-(76)]
"Our expectation is that this calculated value should match the value of δφ obtained from equation (72). If these values are equal, it would suggest that our model ... provides a more accurate explanation for the observed precession of the perihelion of Mercury. ... Instead of directly obtaining the value of ΔφGF from our model, we utilize the discrepancy between the calculated and observed values to constrain the density. ... Isolating for ρ0, we get [Eq. 76]."

Eq. (72) defines δφ = Δφobs − ΔφGR in arcsec/century. Eq. (75) is the model's per-revolution GF precession as a function of ρ0. Eq. (76) solves that same expression for ρ0 with ΔφGF set equal to δφ, so ρ0 is the unique value that forces agreement with the target discrepancy. The reported 'predicted density' is therefore a fit parameter inverted from the residual the model is meant to explain; no independent observable determines it. The agreement is by construction, so the headline constraint is not a prediction. (Separately, the per-century residual is used in a per-revolution formula without dividing by the number of orbits per century, which is a units error compounding the issue.)

full rationale

The central circular step is the density constraint: the paper's headline value ρ0 ≈ 1.12×10^-10 kg/m^3 is obtained by setting the derived GF precession formula (a function of ρ0) equal to the observed-minus-GR residual and solving for ρ0. That is a fit, not a prediction. The analytic multiple-scales solution and the recovery of the standard GR precession are nontrivial and independent, so the paper is not entirely circular. The GF term in the line element is imported from the authors' own prior work (refs. [14,15]) and the metric is an ansatz rather than a solution of Einstein's equations; these are load-bearing assumptions and self-citations, but they do not by themselves reduce the target result to its input. A separate units inconsistency—using a per-century residual in a per-revolution precession formula without dividing by the ~415 orbits per century—makes the fitted density numerically wrong by about a factor of 415; this is a correctness flaw, not an additional circularity. Score 8 because the paper's central numerical claim is forced by definition: ρ0 is chosen to make the model's GF precession equal the residual.

Assumptions & free parameters 1 free parameters · 4 assumptions · 1 invented entities

The central new physics rests on a density-dependent metric term that is not derived from a stress-energy tensor, on a GF work formula and redshift relation taken from the authors' own prior papers, and on a free parameter rho0 fitted to the precession residual. These constitute the majority of the model's input.

free parameters (1)

rho0 (medium density near Mercury) = ≈ 1.12 × 10^-10 kg/m^3
Chosen so that the GF precession correction equals the observed-minus-GR precession residual delta-phi = 0.1446 arcsec/century (Eqs. 72-76); this is a fit to the data, not an externally constrained input.

assumptions (4)

domain assumption The gravitational friction work formula W = -(1/3)(hf/c^2)G*pi*rho0*R^2 (Eq. 1).
Taken from the authors' prior work [14] without re-derivation or independent support in this paper.
domain assumption The generalized redshift formula 1+z = sqrt(Ee/Eo) and its conversion to g_tt via Weyl's formula (Eqs. 2-6).
Adopted from [15], the authors' own earlier paper; no independent derivation here.
ad hoc to paper The modified metric is assumed static and spherically symmetric with g_rr = 1/g_tt (Eq. 8).
No Einstein field equations or stress-energy tensor are provided; the metric is postulated to incorporate the GF term.
ad hoc to paper Distance traveled R is identified with radial coordinate r (Section 2, 'we can relate the distance traversed R with the radial distance from the source r').
This identification is needed to turn the metric into a function of r alone; it is not justified and affects the form of the GF term.

invented entities (1)

GF density term in the metric: -(2/3c^2)*pi*G*rho0*R^2 added to g_tt (Eqs. 7-8)
purpose: Model gravitational friction as a dissipative interaction with the interplanetary medium.
The term is introduced ad hoc to produce a precession correction; the density is fitted to the observed residual, and the model does not yield a velocity-dependent dissipative force because the Hamiltonian is conserved.

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Cite this review

Pith. "Pith review of Nonlinear Dissipative Forces in Celestial Motion Using the Method of Multiple Scales." pith.science (2026). https://pith.science/paper/R5S5JQNB

@misc{pith2026241118742,
  author       = {Pith},
  title        = {Pith review of: Nonlinear Dissipative Forces in Celestial Motion Using the Method of Multiple Scales},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R5S5JQNB}},
  note         = {Machine review of arXiv:2411.18742}
}

abstract

This paper investigates the influence of nonlinear dissipative forces, specifically Gravitational Friction (GF), on the precession of celestial bodies within the framework of general relativity. We derive a modified line element by introducing a density-dependent term to model interactions between planetary bodies and the low-density interplanetary medium, providing a covariant description of dissipative forces in planetary motion. The resulting metric modification leads to corrections in the perihelion precession of Mercury, also reproducing the classical relativistic predictions. Utilizing the method of multiple scales, we analyze perturbative effects induced by GF. Using this model, we successfully constrain the medium density near Mercury to approximately $\rho_0 \approx 1.12 \times 10^{-10} \, \text{kg/m}^3$. These findings offer a new approach for incorporating dissipative mechanisms into general relativity, with potential applications in other astrophysical systems.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 12 canonical work pages

[1]

Dynamical friction: I

Subrahmanyan Chandrasekhar. Dynamical friction: I. General considerations: the coeﬃcient of dynamical friction , volume 97. University of Chicago Press, 1943

work page 1943
[2]

Dynamical friction in in homogeneous and anisotropic systems

Andreas Just and Wilhelm Kegel. Dynamical friction in in homogeneous and anisotropic systems. Astronomy and Astrophysics, 232:447–463, 1990

work page 1990
[3]

William R. Ward. Protoplanet Migration by Nebula Tides. Icarus, 126:261–281,

work page
[4]

Disk-satellite int eractions

Peter Goldreich and Scott Tremaine. Disk-satellite int eractions. The Astrophysical Journal, 241:425–441, 1980. doi:10.1086/158356. 16 Nonlinear Dissipative Forces in Celestial Motion Using Met hod of Multiple Scales

doi:10.1086/158356 1980
[5]

Ostriker

Jeremiah P. Ostriker. Dynamical Friction in a Gaseous Me dium. The Astrophysical Journal, 513:252–258, 1999. doi:10.1086/306858

doi:10.1086/306858 1999
[6]

Physical Processes in the Interstellar Medium

Lyman Spitzer. Physical Processes in the Interstellar Medium . Wiley-VCH, 1978

work page 1978
[7]

Burns, Philippe L

Joseph A. Burns, Philippe L. Lamy, and Steven Soter. Radi ation forces on small par- ticles in the solar system. Icarus, 40:1–48, 1979. doi:10.1016/0019-1035(79)90050-2

doi:10.1016/0019-1035(79)90050-2 1979
[8]

Q in the Solar System

Peter Goldreich and Steven Soter. Q in the Solar System. Icarus, 5:375–389, 1966. doi:10.1016/0019-1035(66)90051-0

doi:10.1016/0019-1035(66)90051-0 1966

Show all 18 references

[9]

Peale and Patrick Cassen

Stanton J. Peale and Patrick Cassen. Contribution of Tid al Dissipation to Lunar Thermal History. Icarus, 36:245–269, 1978. doi:10.1016/0019-1035(78)90062-7

1978 doi
[10]

Physical processes in the interstellar medium

Lyman Spitzer. Physical processes in the interstellar medium. 1978

1978
[11]

Cliﬀord M. Will. The confrontation between general rel ativity and experiment. Liv- ing Reviews in Relativity , 17:4, 2014

2014
[12]

F. W. Dyson, A. S. Eddington, and C. Davidson. A Determin ation of the Deﬂection of Light by the Sun’s Gravitational Field, from Observation s Made at the Total Eclipse of May 29, 1919. Philosophical Transactions of the Royal Society of London Series A , 220:291–333, January 1...

1919
[13]

A. E. Roy. Orbital motion . 1978

1978
[14]

Ortiz and Raju S

C. Ortiz and Raju S. Khatiwada. Gravitational friction from d’alembert’s principle. Scientiﬁc Reports , 13(1):10364, Jun
[15]

Ortiz and F

C. Ortiz and F. Ibarra-Castor. Generalized redshift fo rmula through an energy-based framework. Scientiﬁc Reports , 14(1):22638, 2024. doi:10.1038/s41598-024-73191-4. URL https://doi.org/10.1038/s41598-024-73191-4

2024 doi
[16]

A. K. Verma, A. Fienga, J. Laskar, K. Issautier, H. Manch e, and M. Gastineau. Electron density distribution and solar plasma correction of radio signals us- ing MGS, MEX, and VEX spacecraft navigation data and its appl ication to planetary ephemerides. Astronomy and Astrophys...

2013 doi
[1997]

doi:10.1006/icar.1996.5647

1996
[2023]

doi:10.1038/s41598-023-36977-6

ISSN 2045-2322. doi:10.1038/s41598-023-36977-6. U RL https://doi.org/10.1038/s41598-023-36977-6

doi

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Dynamical friction: I

Subrahmanyan Chandrasekhar. Dynamical friction: I. General considerations: the coeﬃcient of dynamical friction , volume 97. University of Chicago Press, 1943

work page 1943

[2] [2]

Dynamical friction in in homogeneous and anisotropic systems

Andreas Just and Wilhelm Kegel. Dynamical friction in in homogeneous and anisotropic systems. Astronomy and Astrophysics, 232:447–463, 1990

work page 1990

[3] [3]

William R. Ward. Protoplanet Migration by Nebula Tides. Icarus, 126:261–281,

work page

[4] [4]

Disk-satellite int eractions

Peter Goldreich and Scott Tremaine. Disk-satellite int eractions. The Astrophysical Journal, 241:425–441, 1980. doi:10.1086/158356. 16 Nonlinear Dissipative Forces in Celestial Motion Using Met hod of Multiple Scales

doi:10.1086/158356 1980

[5] [5]

Ostriker

Jeremiah P. Ostriker. Dynamical Friction in a Gaseous Me dium. The Astrophysical Journal, 513:252–258, 1999. doi:10.1086/306858

doi:10.1086/306858 1999

[6] [6]

Physical Processes in the Interstellar Medium

Lyman Spitzer. Physical Processes in the Interstellar Medium . Wiley-VCH, 1978

work page 1978

[7] [7]

Burns, Philippe L

Joseph A. Burns, Philippe L. Lamy, and Steven Soter. Radi ation forces on small par- ticles in the solar system. Icarus, 40:1–48, 1979. doi:10.1016/0019-1035(79)90050-2

doi:10.1016/0019-1035(79)90050-2 1979

[8] [8]

Q in the Solar System

Peter Goldreich and Steven Soter. Q in the Solar System. Icarus, 5:375–389, 1966. doi:10.1016/0019-1035(66)90051-0

doi:10.1016/0019-1035(66)90051-0 1966