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arxiv: 2604.23677 · v1 · submitted 2026-04-26 · 🌀 gr-qc · hep-th

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Stationary solutions in the small-c expansion of GR

Emre Onur Kahya, Enes Bal, Ertu\u{g}rul Ekiz, Utku Zorba

Authors on Pith no claims yet

Pith reviewed 2026-05-08 05:38 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords small-c expansionADM formalismstationary vacuum solutionsgeneral relativitystrong-gravity branchweak-gravity branchLense-Thirring solutionHartle-Thorne solution
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The pith

The ADM formulation of the small-c expansion of general relativity admits exact stationary vacuum solutions at NNLO in both strong- and weak-gravity branches.

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

The paper investigates the small-c expansion of general relativity using ADM variables up to next-to-next-to-leading order, focusing on the stationary sector. This formulation simplifies the field equations, enabling explicit construction of vacuum solutions. In the strong-gravity branch, exact NLO solutions such as Lense-Thirring and rotating C-metric geometries are found, with corresponding NNLO extensions that match expansions of the Kerr and rotating C-metric metrics. In the weak-gravity branch, exact Hartle-Thorne-type solutions appear with independent quadrupole moments, spin-squared corrections, and extensions to higher multipoles up to ell=4. These findings establish that the full NLO/NNLO theory supports a richer set of stationary vacuum solutions than the magnetic Carroll truncation alone.

Core claim

In the stationary sector, the small-c expansion of general relativity in ADM variables yields exact vacuum solutions at both NLO and NNLO. The strong-gravity branch produces Lense-Thirring-type and C-metric-type geometries at NNLO, matching the small-c expansion of the Kerr and rotating C-metric metrics up to O(J) at NLO and O(J^3) at NNLO. The weak-gravity branch produces exact Hartle-Thorne-type solutions with an independent quadrupole moment, exact spin-squared corrections, and a mixed quadrupolar-rotating solution, with the ell=0,2 sector extended to include higher multipoles up to ell=4.

What carries the argument

The ADM formulation of the small-c expansion of GR up to NNLO, which makes the stationary field equations tractable for explicit solution building in strong- and weak-gravity branches.

If this is right

  • Exact Lense-Thirring-type and rotating C-metric-type vacuum geometries exist at NNLO in the strong-gravity branch.
  • Exact Hartle-Thorne-type solutions with independent quadrupole moment and spin-squared corrections exist in the weak-gravity branch.
  • The stationary vacuum sector at NLO/NNLO is richer than the magnetic Carroll truncation.
  • Higher multipoles up to ell=4 can be consistently included in the weak-gravity sector.
  • The ADM approach organizes rotational and higher-multipole deformations relevant to slowly 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 richer solution set could provide a systematic way to deform known GR metrics order by order in c without solving the full nonlinear equations at once.
  • Matching the weak-gravity solutions to astrophysical models might allow direct comparison of predicted multipole moments against observational constraints from pulsar timing or gravitational waves.
  • The tractability in the stationary sector suggests that similar expansions could be applied to axisymmetric or slowly time-dependent cases to model inspiraling binaries in the small-c limit.

Load-bearing premise

The small-c expansion of GR in ADM variables remains well-defined and tractable for constructing explicit stationary vacuum solutions at NNLO.

What would settle it

A direct check showing that the small-c expansion of the Kerr metric fails to satisfy the NNLO field equations up to O(J^3) would falsify the matching of the constructed Lense-Thirring-type solutions to known metrics.

read the original abstract

We study the small-$c$ expansion of general relativity in ADM variables up to next-to-next-to-leading order (NNLO). We show that, in the stationary sector, this formulation renders the field equations more tractable for explicit solution building. The stationary sector exhibits both strong- and weak-gravity branches, whose structure becomes richer at NNLO. In the strong-gravity branch, we first obtain exact vacuum solutions of NLO Carroll gravity, including the Lense--Thirring and rotating C-metric backgrounds. At NNLO, we then construct the corresponding Lense--Thirring-type and C-metric-type exact vacuum geometries. These solutions also arise from the small-$c$ expansion of the Kerr and rotating C-metric geometries around the strong-gravity background, up to $\mathcal{O}(J)$ at NLO and up to $\mathcal{O}(J^3)$ at NNLO. In the weak-gravity branch, we find exact Hartle--Thorne-type solutions with an independent quadrupole moment, together with exact spin-squared corrections and a mixed quadrupolar-rotating solution. We further extend the $\ell=0,2$ sector by including higher multipoles up to $\ell=4$, where $\ell$ denotes the multipole index. These results show that the full NLO/NNLO theory admits a richer stationary vacuum sector than the magnetic Carroll truncation. More broadly, the ADM formulation provides a practical framework for constructing and analyzing stationary backgrounds in the small-$c$ expansion of general relativity, and may also offer a useful framework for organizing rotational and higher-multipole deformations relevant to compact astrophysical objects such as slowly rotating 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.

Referee Report

1 major / 2 minor

Summary. The manuscript examines the small-c expansion of general relativity in ADM variables up to NNLO, restricted to the stationary sector. It constructs exact vacuum solutions in a strong-gravity branch that match the small-c expansions of the Kerr and rotating C-metric geometries (to O(J) at NLO and O(J^3) at NNLO), and in a weak-gravity branch it obtains exact Hartle-Thorne-type solutions with independent quadrupole moments, spin-squared corrections, and extensions to higher multipoles up to l=4. The ADM formulation is argued to render the stationary equations tractable order-by-order, yielding a richer set of solutions than the magnetic Carroll truncation alone.

Significance. If the explicit constructions hold, the work demonstrates that the NLO/NNLO theory supports a broader stationary vacuum sector than previously considered truncations, with direct matching to known exact metrics providing external validation. The tractability of the ADM approach for building solutions order-by-order, including independent multipole deformations, is a concrete strength that could organize rotational corrections relevant to slowly rotating compact objects.

major comments (1)
  1. The abstract and introduction state that NNLO solutions in the strong-gravity branch arise from the small-c expansion of Kerr and rotating C-metric geometries up to O(J^3). However, without explicit intermediate steps showing how the ADM equations are solved order-by-order and how the matching is verified at each order (particularly the J^3 terms), the support for the exactness claim at NNLO remains difficult to assess fully.
minor comments (2)
  1. Clarify the precise definition of the strong- and weak-gravity branches in the introduction, including how the branches are distinguished in the ADM variables.
  2. The multipole index l is used up to l=4 in the weak-gravity sector; add a brief statement on why higher l are not included or whether the pattern extends.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, positive evaluation of its significance, and recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: The abstract and introduction state that NNLO solutions in the strong-gravity branch arise from the small-c expansion of Kerr and rotating C-metric geometries up to O(J^3). However, without explicit intermediate steps showing how the ADM equations are solved order-by-order and how the matching is verified at each order (particularly the J^3 terms), the support for the exactness claim at NNLO remains difficult to assess fully.

    Authors: We appreciate the referee's observation that greater transparency in the derivation would strengthen the presentation. The manuscript constructs the NLO and NNLO solutions by positing stationary axisymmetric ansätze in ADM variables, substituting into the expanded field equations, and solving the resulting system order by order; the matching to the small-c expansions of Kerr and the rotating C-metric is performed by direct coefficient comparison after expanding those metrics around the strong-gravity background. Nevertheless, we agree that the intermediate algebraic steps, especially for the O(J^3) contributions at NNLO, are not displayed in full detail. In the revised manuscript we will add an appendix that (i) lists the ADM equations at each order, (ii) shows the order-by-order solution procedure for the Lense-Thirring-type and C-metric-type ansätze, and (iii) tabulates the explicit verification that the constructed solutions coincide with the expanded Kerr and rotating C-metric geometries through O(J^3). This addition will make the exactness claim fully verifiable without altering the results themselves. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives stationary solutions by solving the small-c expanded Einstein equations in ADM variables order-by-order at NLO and NNLO. Strong-gravity branch solutions are constructed directly from the expanded field equations and then cross-checked by explicit expansion of the known Kerr and rotating C-metric metrics, which supplies independent external grounding rather than internal fitting. Weak-gravity branch solutions (Hartle-Thorne-type with independent multipoles) are likewise obtained by direct integration of the truncated equations. No step reduces by definition to a fitted parameter, self-citation chain, or ansatz smuggled from prior work; the ADM formulation simply renders the stationary sector algebraically tractable. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the standard validity of the ADM 3+1 decomposition of GR and the perturbative small-c expansion being applicable and solvable in the stationary sector. No free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math General relativity admits a well-defined ADM 3+1 decomposition
    The entire expansion is performed in ADM variables.
  • domain assumption The small-c expansion remains consistent and tractable for stationary spacetimes up to NNLO
    The paper focuses on constructing solutions in this limit.

pith-pipeline@v0.9.0 · 5622 in / 1440 out tokens · 62820 ms · 2026-05-08T05:38:38.161828+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Spin and Quadrupole Sectors in Nonrelativistic Gravity

    gr-qc 2026-05 unverdicted novelty 7.0

    Derives NLO Kerr-type and Hartle-Thorne-type solutions plus NNLO mixed spin-quadrupole solutions in the Galilean branch of nonrelativistic gravity.

Reference graph

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