arxiv: 2605.06410 · v1 · submitted 2026-05-07 · 🌀 gr-qc · hep-th

Recognition: unknown

Spin and Quadrupole Sectors in Nonrelativistic Gravity

Utku Zorba

Authors on Pith no claims yet

Pith reviewed 2026-05-08 07:09 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords large-c expansionGalilean limitADM formalismnonrelativistic gravitystationary solutionsspin sectorquadrupole sectorKerr metric

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The pith

Large-c expansion of GR in ADM variables produces Galilean solutions with spin and quadrupole up to NNLO including mixed corrections.

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

This paper develops the large-c expansion of general relativity using the ADM formulation of spacetime. It derives the field equations for the Galilean branch up to next-to-next-to-leading order and constructs stationary vacuum solutions that incorporate both rotational spin and quadrupole deformation. In the weak branch these solutions require extra corrections at the highest order due to nonlinear mixing between spin and quadrupole terms, while the strong branch allows simpler solutions for Kerr-like data. The resulting approximate metrics can describe the spacetime geometry around rotating compact objects. A reader would care because this approach offers a controlled expansion for studying strong gravity effects in a nonrelativistic setting without solving the full Einstein equations.

Core claim

Using a unified even ω-expansion in ADM variables, the Galilean limit yields NLO solutions in the weak branch that include Kerr-type, Hartle-Thorne-type and mixed data, extendable to higher multipoles; at NNLO the weak Kerr and Hartle-Thorne sectors solve separately but their combination requires additional lapse and spatial corrections to account for J²Q sources, producing a consistent mixed solution, while strong branch Kerr data solve the equations through NNLO.

What carries the argument

The even ω-expansion of the ADM action and constraints, with separation into weak and strong branches for the Galilean limit, allowing order-by-order construction of solutions with spin (J) and quadrupole (Q) moments.

If this is right

Weak-branch NLO equations permit extension to higher mass multipoles beyond quadrupole.
Naive combination of Kerr-type and Hartle-Thorne-type data fails at NNLO due to generated mixed source terms.
Strong-branch Kerr-type data remains a solution through NNLO without additional corrections.
ADM data can be reconstructed into approximate spacetime metrics that include spin, quadrupole, and mixed effects.
These metrics are applicable to modeling the spacetime around rotating black holes and neutron stars.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method could be extended to include time dependence or dynamical systems like orbiting binaries in the Galilean regime.
Similar expansions might apply to other limits such as Carrollian gravity for different physical regimes.
Numerical validation against full relativistic simulations could test the accuracy of these approximate metrics for compact object spacetimes.
The branch separation may suggest analogous structures in effective field theories of gravity.

Load-bearing premise

The assumption that the large-c expansion in ADM variables stays consistent through NNLO, with the branch separation and metric reconstruction not introducing extra inconsistencies from nonlinear terms.

What would settle it

Computing the Einstein tensor for the reconstructed NNLO mixed weak-branch metric and finding it does not vanish at the corresponding order, or observing that the strong-branch solution deviates from the Einstein equations at NNLO.

read the original abstract

We study the large-$c$ expansion of general relativity in ADM variables. Using a unified even $\omega$-expansion, the ADM formulation gives a common starting point for Galilean and Carrollian limits. We focus on the Galilean branch and derive the ADM action and field equations up to NNLO. We then construct stationary vacuum solutions in weak and strong branches. In the weak branch, we find NLO Kerr-type, Hartle-Thorne-type and mixed-type solutions. The NLO weak equations also allow a simple extension to higher mass multipoles. At NNLO, the weak Kerr-type and extended Hartle-Thorne-type sectors solve the equations separately, but their naive sum is not a solution. The nonlinear NNLO equations generate mixed $J^2Q$ source terms, which require additional corrections to the NNLO lapse and NNLO spatial tensor field. This gives a mixed weak-branch Galilean solution in the ADM gauge. In the strong branch, Kerr-type data solve the equations through NNLO while the strong Hartle-Thorne-type data solve the NLO equations. We also explain how the ADM data can be reconstructed into approximate spacetime metrics. Since these metrics include spin, quadrupole and mixed spin-quadrupole effects, they may be useful for studying the spacetime around rotating compact objects such as black holes and neutron stars.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper works out NNLO mixed spin-quadrupole solutions in the weak Galilean branch of nonrelativistic gravity via ADM variables and a unified even omega expansion.

read the letter

The main point is that they construct stationary vacuum solutions up to NNLO in the Galilean limit, including the mixed J²Q terms that appear when you combine Kerr-type and Hartle-Thorne-type sectors. They start from the ADM formulation, apply the large-c expansion with even powers, derive the field equations to NNLO, and then build explicit NLO solutions plus the required corrections at NNLO to cancel the nonlinear mixing sources in the weak branch. The strong branch gets Kerr-type data through NNLO but only NLO for the Hartle-Thorne sector. They also outline how to reconstruct approximate spacetime metrics from the ADM data that carry spin, quadrupole, and mixed effects. This is useful for modeling rotating compact objects like black holes or neutron stars in a controlled expansion. The approach is systematic, stays within standard ADM and large-c methods, and produces concrete new solutions without free parameters or circular definitions. The NLO extension to higher mass multipoles is a nice practical addition. The soft spot is whether the added corrections to the lapse and spatial tensor really satisfy the complete nonlinear NNLO system in the weak branch or only fix the sourced components while leaving other cross terms. The abstract states that the result is a solution, but if the verification skips some equations the claim would not hold. The assumption that the expansion remains valid through NNLO without extra inconsistencies from nonlinear mixing also needs explicit checks. This is for specialists in nonrelativistic gravity expansions or post-Newtonian modeling of astrophysical sources. A reader who needs approximate metrics with controlled spin-quadrupole mixing would get direct value from the constructions. It deserves a serious referee because the results are new, the method is reproducible, and the mixed terms represent a clear step forward even if revisions are needed on the verification details. I would send it for peer review.

Referee Report

2 major / 3 minor

Summary. The manuscript derives the large-c expansion of general relativity in ADM variables using a unified even ω-expansion, obtaining the ADM action and field equations up to NNLO in the Galilean branch. It constructs stationary vacuum solutions in weak and strong branches: NLO Kerr-type, Hartle-Thorne-type, and mixed-type solutions in the weak branch (with extension to higher mass multipoles), while at NNLO the separate Kerr-type and extended Hartle-Thorne-type sectors solve the equations but their sum requires additional corrections to the lapse and spatial tensor to cancel J²Q mixing terms, yielding a mixed weak-branch solution. In the strong branch, Kerr-type data solve through NNLO and Hartle-Thorne-type at NLO. The paper also outlines reconstruction of the ADM data into approximate spacetime metrics incorporating spin, quadrupole, and mixed effects.

Significance. If the derivations hold, the work supplies a systematic nonrelativistic framework for GR with spin and quadrupole moments, yielding approximate metrics potentially useful for modeling spacetimes around rotating black holes and neutron stars. Strengths include the explicit construction of mixed solutions at NNLO via targeted corrections, the parameter-free character of the expansion (building directly on standard ADM and prior large-c literature), and the reconstruction procedure that makes the results applicable beyond pure ADM variables.

major comments (2)

[NNLO weak-branch mixed solution] In the NNLO weak-branch mixed solution (the paragraph following the statement that 'the nonlinear NNLO equations generate mixed J²Q source terms'), the corrections to the lapse and spatial tensor are presented as producing a full solution. Explicit verification is needed that these corrections satisfy the complete set of ADM constraints and evolution equations at NNLO, including any unsourced cross terms or components not directly generated by the J²Q mixing; if only a subset is addressed, the claim of a consistent mixed Galilean solution would not hold.
[Reconstruction of spacetime metrics] § on reconstruction of spacetime metrics: the procedure for obtaining approximate metrics from ADM data should include an explicit check that nonlinear mixing terms at NNLO do not introduce inconsistencies when the data are lifted back to the spacetime metric, particularly for the mixed J²Q contributions.

minor comments (3)

[Introduction] The introduction could clarify the rationale for restricting to even powers in the ω-expansion and how this choice unifies Galilean and Carrollian limits, as the current presentation assumes familiarity with the prior literature.
Notation for the ADM variables and the large-c expansion orders (NLO/NNLO) is generally clear but would benefit from a consolidated table of symbols and orders early in the manuscript to aid readers.
[Derivation of ADM action and field equations] A few intermediate steps in the derivation of the NNLO field equations from the ADM action appear abbreviated; adding one or two lines of algebra would improve traceability without lengthening the paper substantially.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major comment below and will revise the manuscript to incorporate the requested explicit verifications.

read point-by-point responses

Referee: [NNLO weak-branch mixed solution] In the NNLO weak-branch mixed solution (the paragraph following the statement that 'the nonlinear NNLO equations generate mixed J²Q source terms'), the corrections to the lapse and spatial tensor are presented as producing a full solution. Explicit verification is needed that these corrections satisfy the complete set of ADM constraints and evolution equations at NNLO, including any unsourced cross terms or components not directly generated by the J²Q mixing; if only a subset is addressed, the claim of a consistent mixed Galilean solution would not hold.

Authors: We agree that an explicit verification of the full set of ADM constraints and evolution equations is necessary to confirm the consistency of the mixed solution. In the original derivation, the corrections to the NNLO lapse and spatial tensor were constructed specifically to cancel the J²Q source terms appearing in the nonlinear equations. However, to rigorously address potential unsourced cross terms or other components, we will add an explicit component-by-component check in the revised manuscript, demonstrating that the corrected fields satisfy all NNLO equations without requiring further adjustments. revision: yes
Referee: [Reconstruction of spacetime metrics] § on reconstruction of spacetime metrics: the procedure for obtaining approximate metrics from ADM data should include an explicit check that nonlinear mixing terms at NNLO do not introduce inconsistencies when the data are lifted back to the spacetime metric, particularly for the mixed J²Q contributions.

Authors: We concur that verifying the consistency of the reconstruction procedure under nonlinear mixing is important for the applicability of the results. The manuscript outlines the general reconstruction from ADM data to approximate spacetime metrics, but does not perform an explicit check for NNLO J²Q mixing terms. In the revision, we will include a detailed verification step showing that the lifted metric remains free of inconsistencies or spurious contributions from these mixing terms, thereby strengthening the claim that the approximate metrics are reliable for modeling spacetimes around rotating compact objects. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows from standard ADM expansion and perturbative solution

full rationale

The paper derives the ADM action and field equations at NNLO from the large-c expansion of GR in ADM variables (standard starting point, not self-referential). It then solves the resulting perturbative equations order-by-order: separate Kerr-type and Hartle-Thorne-type sectors satisfy the equations independently at NLO/NNLO, their superposition generates explicit J²Q source terms in the nonlinear equations, and corrections to lapse and spatial tensor are introduced to cancel those sources. This is ordinary perturbative construction, not a reduction of the claimed mixed solution to its inputs by definition or by a fitted parameter. No load-bearing self-citation chain or ansatz smuggling is evident in the provided derivation steps; the result remains falsifiable against the derived equations themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard assumptions of general relativity in the ADM 3+1 decomposition and the validity of the even ω-expansion for large c. No new free parameters or invented entities are introduced; the work extends existing frameworks.

axioms (2)

domain assumption The large-c expansion of GR in ADM variables is valid for the Galilean branch up to NNLO.
Invoked as the common starting point for deriving the ADM action and field equations.
domain assumption Stationary vacuum solutions can be constructed separately in weak and strong branches with reconstruction to spacetime metrics.
Assumed to find and validate the Kerr-type and Hartle-Thorne-type solutions.

pith-pipeline@v0.9.0 · 5531 in / 1365 out tokens · 33571 ms · 2026-05-08T07:09:15.343907+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 20 canonical work pages · 1 internal anchor

[1]

Van den Bleeken,Torsional Newton–Cartan gravity from the large c expansion of general relativity,Class

D. Van den Bleeken,Torsional Newton–Cartan gravity from the large c expansion of general relativity,Class. Quant. Grav.34(2017) 185004, [1703.03459]

work page arXiv 2017
[2]

Hansen, J

D. Hansen, J. Hartong and N. A. Obers,Action Principle for Newtonian Gravity,Phys. Rev. Lett.122(2019) 061106, [1807.04765]

work page arXiv 2019
[3]

D. Van den Bleeken,Torsional Newton-Cartan gravity and strong gravitational fields, in15th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories, 3, 2019,1903.10682, DOI

work page arXiv 2019
[4]

Hansen, J

D. Hansen, J. Hartong and N. A. Obers,Non-Relativistic Gravity and its Coupling to Matter, JHEP06(2020) 145, [2001.10277]

work page arXiv 2020
[5]

Hartong, N

J. Hartong, N. A. Obers and G. Oling,Review on Non-Relativistic Gravity,Front. in Phys.11 (2023) 1116888, [2212.11309]

work page arXiv 2023
[6]

Elbistan, E

M. Elbistan, E. Hamamci, D. Van den Bleeken and U. Zorba,A 3+1 formulation of the 1/c expansion of General Relativity,JHEP02(2023) 108, [2210.15440]

work page arXiv 2023
[7]

Non-Relativistic Gravitation: From Newton to Einstein and Back

B. Kol and M. Smolkin,Non-Relativistic Gravitation: From Newton to Einstein and Back, Class. Quant. Grav.25(2008) 145011, [0712.4116]

work page Pith review arXiv 2008
[8]

B. Kol, M. Levi and M. Smolkin,Comparing space+time decompositions in the post-Newtonian limit,Class. Quant. Grav.28(2011) 145021, [1011.6024]

work page arXiv 2011
[9]

Kol and M

B. Kol and M. Smolkin,Einstein’s action and the harmonic gauge in terms of Newtonian fields, Phys. Rev. D85(2012) 044029, [1009.1876]

work page arXiv 2012
[10]

Guerrieri and R

A. Guerrieri and R. F. Sobreiro,Non-relativistic limit of gravity theories in the first order formalism,JHEP03(2021) 104, [2010.14918]

work page arXiv 2021
[11]

E. Ekiz, O. Kasikci, M. Ozkan, C. B. Senisik and U. Zorba,Non-relativistic and ultra-relativistic scaling limits of multimetric gravity,JHEP10(2022) 151, [2207.07882]

work page arXiv 2022
[12]

E. Bal, E. Ekiz, E. O. Kahya and U. Zorba,Stationary solutions in the small-cexpansion of GR,2604.23677

work page internal anchor Pith review Pith/arXiv arXiv
[13]

Elbi ˙ stan,The 1/c expansion of general relativity in a 3+1 formulation, revisited,Turk

M. Elbistan,The 1/c expansion of general relativity in a 3+1 formulation, revisited,Turk. J. Phys.50(2026) 37–58, [2508.03548]

work page arXiv 2026
[14]

Hansen, N

D. Hansen, N. A. Obers, G. Oling and B. T. Sogaard,Carroll Expansion of General Relativity, SciPost Phys.13(2022) 055, [2112.12684]

work page arXiv 2022
[15]

F. J. Ernst,New formulation of the axially symmetric gravitational field problem,Phys. Rev. 167(1968) 1175–1179

1968
[16]

F. J. Ernst,New formulation of the axially symmetric gravitational field problem. ii,Phys. Rev. 168(1968) 1415–1417

1968
[17]

V. S. Manko and I. D. Novikov,Generalizations of the kerr and kerr-newman metrics possessing an arbitrary set of mass-multipole moments,Class. Quant. Grav.9(1992) 2477–2487

1992
[18]

Frutos-Alfaro and M

F. Frutos-Alfaro and M. Soffel,On the post-linear quadrupole-quadrupole metric,Rev. Mate. Teor. Aplic.24(2017) 239, [1507.04264]. – 53 –

work page arXiv 2017
[19]

R. L. Arnowitt, S. Deser and C. W. Misner,The Dynamics of general relativity,Gen. Rel. Grav.40(2008) 1997–2027, [gr-qc/0405109]

work page Pith review arXiv 2008
[20]

Ergen, E

M. Ergen, E. Hamamci and D. Van den Bleeken,Oddity in nonrelativistic, strong gravity,Eur. Phys. J. C80(2020) 563, [2002.02688]

work page arXiv 2020
[21]

Kol´ aˇ r, D

I. Kol´ aˇ r, D. Kubiznak and P. Tadros,Rotating Carroll black holes: A no go theorem,Phys. Rev. D112(2025) L121504, [2506.10451]

work page arXiv 2025
[22]

J. B. Hartle,Slowly rotating relativistic stars. 1. Equations of structure,Astrophys. J.150 (1967) 1005–1029

1967
[23]

J. B. Hartle and K. S. Thorne,Slowly Rotating Relativistic Stars. II. Models for Neutron Stars and Supermassive Stars,Astrophys. J.153(1968) 807

1968
[24]

Lense and H

J. Lense and H. Thirring, ¨Uber den einfluss der eigenrotation der zentralk¨ orper auf die bewegung der planeten und monde nach der einsteinschen gravitationstheorie,Physikalische Zeitschrift19(1918) 156–163

1918
[25]

F. D. Ryan,Gravitational waves from the inspiral of a compact object into a massive, axisymmetric body with arbitrary multipole moments,Phys. Rev. D52(1995) 5707–5718

1995
[26]

Baub¨ ock, E

M. Baub¨ ock, E. Berti, D. Psaltis and F. ¨Ozel,Relations Between Neutron-Star Parameters in the Hartle-Thorne Approximation,Astrophys. J.777(2013) 68, [1306.0569]

work page arXiv 2013
[27]

R. P. Geroch,Multipole moments. II. Curved space,J. Math. Phys.11(1970) 2580–2588

1970
[28]

R. O. Hansen,Multipole moments of stationary space-times,J. Math. Phys.15(1974) 46–52

1974
[29]

N. A. Collins and S. A. Hughes,Towards a formalism for mapping the space-times of massive compact objects: Bumpy black holes and their orbits,Phys. Rev. D69(2004) 124022, [gr-qc/0402063]

work page arXiv 2004
[30]

Glampedakis and S

K. Glampedakis and S. Babak,Mapping spacetimes with lisa: inspiral of a test-body in a quasi-kerr field,Class. Quant. Grav.23(2006) 4167–4188

2006
[31]

J. R. Gair, C. Li and I. Mandel,Observable properties of orbits in exact bumpy spacetimes, Phys. Rev. D77(2008) 024035

2008
[32]

R. P. Kerr,Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett.11(1963) 237–238. – 54 –

1963