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REVIEW 4 major objections 3 minor 1 cited by

The paper claims that Galilean, Carroll, string Carroll, and Einstein gravity all emerge as leading-order limits of a single unified expansion of the Einstein–Hilbert action, controlled by two integers (s,n) and a contraction parameter ε.

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2026-08-01 20:54 UTC pith:YENFCBJJ

load-bearing objection A genuinely unifying (s,n) expansion of Einstein gravity, where the advertised near-horizon string-Carroll checks are still asserted rather than demonstrated. the 4 major comments →

arxiv 2607.16459 v1 pith:YENFCBJJ submitted 2026-07-17 hep-th gr-qc

A unified expansion of Einstein's gravity

Arkachur Bhattacharya , Pushkar Soni This is my paper
classification hep-th gr-qc
keywords non-Lorentzian gravityGalilean gravityCarroll gravitystring Carroll gravityEinstein-Hilbert actionunified expansionnear-horizon geometryblack holes
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper claims that Galilean, Carroll, string Carroll, and Einstein gravity all arise as the leading-order limit of a single, unified expansion of the Einstein–Hilbert action. The expansion is parameterized by two integers (s,n) that fix the size of the scaled sector of the local metric and the placement of its timelike signature, plus a contraction parameter ε that is sent to zero. A single covariant computation then yields the action decomposition Eq. (5.10), and fixing (s,n) selects which gravitational theory governs the leading order. If correct, this replaces a patchwork of separate expansions with one calculation, and it supplies equations of motion for the string Carroll geometry that describes the near-horizon region of generic non-extremal black holes.

Core claim

The central result is the unified decomposition of the Einstein–Hilbert action, Eq. (5.10): S ≈ (1/16πG) ∫ dᵈx E [ ε^{n−4} L_(n−4) + ε^{n−2} L_(n−2) + εⁿ L_n + ε^{n+2} L_(n+2) − 2 ε^{Δ+n} Λ ], with the four Lagrangians defined in Eq. (5.11). Keeping s and n unfixed until the end, the same identity reduces at leading order to Galilean gravity for (s,n)=(0,d−1), Carroll gravity for (1,1), string Carroll gravity for (1,2), and Einstein gravity for (0,0) or (1,d). The paper further derives the equations of motion for string Carroll gravity (Eq. 7.8) and verifies that the near-horizon geometries of non-extremal Plebanski–Demianski black holes (with zero electric and magnetic charge) and of a 4D b

What carries the argument

The unified flat geometry: a block-diagonal Minkowski metric split into a scaling sector of size n carrying ε² and a non-scaling sector, with the placement s of the timelike signature. This data defines a unified algebra of infinitesimal isometries upon sending ε→0. The load-bearing computation is the unified decomposition of the Levi–Civita connection and Ricci scalar (Eqs. 5.1–5.7), leading to the action decomposition (5.10). The unified compatible connection (4.6) is the affine object that lets the leading-order variables v and h move freely through covariant derivatives.

Load-bearing premise

The expansion proceeds under the assumption that a suitable frame exists in which the metric and vielbeins are analytic in even powers of ε; if a physical near-horizon limit requires odd powers of ε, the leading-order action derived here is not the correct limit.

What would settle it

Take a non-extremal black hole and perform the near-horizon expansion in a coordinate system that produces odd powers of ε (an example the paper itself cites); if the extracted leading-order geometry does not satisfy the string Carroll equations (7.8), the even-power unified expansion does not describe that limit.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Galilean, Carroll, string Carroll, and Einstein gravity are all recovered from one action decomposition by fixing (s,n), removing the need for separate expansions.
  • The near-horizon limit of generic non-extremal black holes is governed by the string Carroll gravity equations (7.8), so the dynamics near any such horizon can be studied with those equations.
  • Sub-leading orders in ε can be extracted systematically from the same decomposition, providing a handle on corrections away from the strict limit.
  • The framework extends to higher-derivative gravity by combining α′ with ε, potentially reaching different sectors depending on the relative scaling.
  • In d=2, the Galilean and Carroll choices are dual by a time–space flip, hinting at dualities between these theories.

Where Pith is reading between the lines

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

  • If the expansion is universal, the differences between Galilean, Carroll, and string Carroll gravity are not fundamental but reflect a choice of which spacetime directions are scaled; one could attempt a quantization program that treats all these limits as sectors of a single theory.
  • The even-power ansatz is a genuine limitation that the paper flags: for near-horizon limits whose coordinate expansion requires odd powers of ε, the leading-order action here may miss the correct theory. Testing a known odd-power example against Eq. (7.8) would delimit the framework.
  • The same decomposition can serve as a classification device: scanning the (s,n) grid in any dimension produces the full family of p-Galilei and p-Carroll gravity theories, including intermediate cases the paper does not study in detail.
  • Because the physical meaning of ε changes with (s,n) — inverse speed of light for Galilean, speed of light for Carroll, horizon nearness for string Carroll — one could look for a double-scaling limit that interpolates between these regimes within a single background, potentially exposing new dualities.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 3 minor

Summary. The paper constructs a two-parameter (s,n) family of non-Lorentzian limits of the Einstein–Hilbert action. Starting from a block-diagonal local metric with a scaling factor ε², it defines a unified flat geometry and algebra, expands the vielbeins in even powers of ε, introduces a compatible connection, and decomposes the Einstein–Hilbert action into Eq. (5.10). Choosing (s,n) is then claimed to reproduce, at leading order, general relativity, Galilean gravity, Carroll gravity, and string Carroll gravity. The final sections propose string Carroll equations (7.8) and assert that the near-horizon limits of the Plebanski–Demianski family and of a 4D black brane solve them.

Significance. If Eq. (5.10) is correct, the paper provides a single computational framework for several previously separate non-Lorentzian gravity limits and a candidate dynamical description of near-horizon regions of non-extremal black holes. The explicit reproduction of the known pre-non-relativistic and pre-ultra-local expansions of Refs. [46] and [8] is an independent check on the algebraic structure. The paper is also transparent about its main assumptions, especially the even-power frame existence and the lack of a rigorous near-horizon/string-Carroll identification. However, as detailed below, two load-bearing aspects are not established in the text: the uniqueness of the compatible connection used in the action decomposition, and the verification of the black-hole examples claimed to satisfy the string Carroll equations.

major comments (4)
  1. [Appendix A, Eq. (A.11)–(A.14); Eq. (4.3)] In App. A the system for the spin connections is explicitly underdetermined: Ω^A_{μC}E^μ_b and Ω^A_{μb}E^μ_A are constrained by a single equation (A.9b), and the authors state that "infinitely many solutions exist" before choosing the particular solution that yields (4.3). This compatible connection is then used in the action decomposition (5.10) through \hat R_{μν} in L_{(n-2)} and L_{(n)}. Without a proof that the decomposition is invariant under this ambiguity, or a uniqueness criterion (e.g., minimal torsion) selecting (4.3), the leading-order equations (7.8) and the Galilean reduction in App. E are contingent on an arbitrary choice. This needs to be fixed or explicitly justified as an intrinsic part of the framework.
  2. [Section 3, after Eq. (3.1); Sections 7.1–7.2] The whole expansion assumes an ε²-even analytic frame. The paper flags this assumption and cites Ref. [60] for examples with odd powers, but it does not verify that the geometries used in the central application satisfy the assumption. In §7.1, r=r_h+ερ² with ε=ϵ² is explicitly even, but in §7.2 the relation between ε and ϵ is not stated, and the data (7.18)–(7.19) are read off without checking the vielbein analyticity. Since these examples are the only concrete evidence for the near-horizon/string-Carroll statement, the assumption must be checked for them, or the unification claim must be restricted to the stated even-power class.
  3. [Sections 7.1 and 7.2 (after Eqs. (7.15) and (7.19))] The statement that substituting (7.15) and (7.19) into (7.8) verifies the equations is asserted, not demonstrated. The system (7.8) is a set of nontrivial second-order PDEs with multiple curvature-like terms; the verification is not visually obvious. Since the paper itself admits (Sec. 7) that no rigorous proof of the near-horizon/string-Carroll identification is available, these examples carry the evidential weight. Please include the explicit computation, or a supplementary notebook, showing that all of (7.8a)–(7.8c) hold for the stated data, including the choice of inverse data and any identities used.
  4. [Section 4, Eq. (4.10)] The trace formula (4.10) is used in (4.11)–(4.12) to simplify the divergence terms in the action decomposition. The derivation is deferred to "Appendix E of Ref. [61]", a preprint by the same group. Because this identity enters the derivation of (5.10), the paper should either give a self-contained derivation (at least for the n=2 case) or clearly state the degree of reliance on [61]. A referee cannot verify a load-bearing formula by citation to an unpublished companion paper.
minor comments (3)
  1. [Section 7.2, Eq. (7.18b)] The limit expression "lim_{ϵ→ εg^{-1}" is missing its arrow target and should read lim_{ϵ→0} ε g^{-1}=... Also, the relation between the near-horizon parameter ε and the expansion parameter ϵ is not stated in §7.2, in contrast to Eq. (7.11) in §7.1. Please clarify.
  2. [Section 2, display quote after Eq. (2.11)] The quoted display "The upshot of this exercise is that any computation performed with unified structure..." and the ⊛ markers used elsewhere are informal formatting; they should be converted to normal prose for a journal submission.
  3. [Sections 6.2–6.3] The extracted actions are determined only up to an overall scale; the text notes that "we need to put an overall scale" but does not state how this scale is fixed within the expansion. This should be stated explicitly as a property of the limiting procedure, or the scale should be derived.

Circularity Check

0 steps flagged

No significant circularity: the unified decomposition is derived algebraically from Einstein–Hilbert, with self-citations only providing context and a stated derivation gap rather than recycled inputs.

full rationale

The central chain is self-contained: the local metric ansatz (2.1) and vielbein expansion (3.1) define the data; the compatible connection (4.3) and the Ricci-scalar decomposition (5.4)–(5.6) are algebraic consequences; eq. (5.10) is obtained by multiplying the decomposed Ricci scalar by sqrt(-g)=epsilon^n E. The L's in (5.11) are defined by those contractions, not by the target Galilean/Carroll/string-Carroll actions. Fixing (s,n) only selects which epsilon-power is leading; the resulting actions (6.28), (6.38) and (7.4) then follow without fitting any parameter to the known results. The string-Carroll equations (7.8) are Euler--Lagrange equations of (7.4), so they are not derived from the claim that near-horizon geometries are string-Carroll. The near-horizon examples are genuine independent inputs: the data (7.15) and (7.18)--(7.19) come from coordinate limits of the PD and black-brane metrics. The paper does not display the substitution into (7.8) ('On substituting eq. (7.15) in eq. (7.8), one can see...'), which is an omitted verification, not a circular reduction; the same applies to the choice of inverse data in footnote (30), which is justified by local-symmetry invariance rather than by assuming the result. The self-citations ([61] for the n=2 trace formula and for the non-extremal near-horizon=string-Carroll phenomenology, and [65] for BTZ) are contextual: the identification is also attributed to the external papers [56,57], and the paper explicitly states 'we do not yet have a rigorous proof of this identification'. The even-power expansion in Section 3 is an explicitly stated assumption, with odd-power references [60] acknowledged; an assumption is a correctness risk, not circularity. Overall, the main derivation does not reduce to its inputs, and no fitted quantity is relabelled as a prediction.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The central claim rests on the block-split metric ansatz, an assumed analytic frame, a non-unique connection choice, and an externally sourced string-Carroll identification. No numerical constants are fitted to data, but the cosmological constant expansion power Δ is a hand-chosen knob.

free parameters (2)
  • Delta (cosmological constant expansion power) = chosen per class: -4 (Galilean/SC), -2 (Carroll/GR(1,d)), 0 (GR(0,0))
    Eq. (5.9) introduces Delta as a free integer controlling whether the cosmological constant survives in the leading-order action; in each specialization Delta is selected by hand to produce the desired leading-order theory, rather than derived.
  • Overall scale of non-relativistic action = unspecified (to be fixed by dimension counting, e.g. in d=4)
    Sections 6.2-6.3 state an overall scale must be introduced for dimension counting but do not determine it; this leaves the normalization of the Galilean and Carroll actions incomplete.
axioms (5)
  • ad hoc to paper Even-power analytic frame exists
    Section 3: 'We therefore assume the existence of a suitably well-behaved frame or coordinate system in which the expansion remains well defined.' The whole extraction of the leading-order theory depends on this.
  • domain assumption Block-split local metric with scaling sector of size n and timelike signature controlled by s
    Eqs. (1.1)/(2.1): this is the defining assumption of the class of non-Lorentzian limits considered; it restricts the paper to contractions of Poincaré that preserve this block structure.
  • ad hoc to paper Particular solution for undetermined spin connections
    Appendix A: compatibility leaves infinitely many spin connections; the authors choose the one reproducing Ref [8] in the Carroll case. Physical independence from this choice is not shown.
  • domain assumption Near-horizon geometry of generic non-extremal BHs is string Carroll
    Section 7: 'Although we do not yet have a rigorous proof of this identification...' This is inherited from Refs [56,57,61] and needed for the application.
  • ad hoc to paper Cosmological constant expansion ansatz
    Eq. (5.9) introduces Δ by hand; the paper states it is 'a matter of choice' and that it is determined by the particular gravitational theory one wishes to obtain at leading order.

pith-pipeline@v1.3.0-alltime-deepseek · 32822 in / 17910 out tokens · 169990 ms · 2026-08-01T20:54:17.126236+00:00 · methodology

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read the original abstract

Non-Lorentzian theories of gravity, most common of which are Galilean and Carrollian gravity, arise from General relativity under suitable scalings. General relativity can be obtained by gauging the Poincar\'e algebra. A convenient formulation of non-Lorentzian gravity follows the contraction of the Poincar\'e algebra in the tangent space to its non-Lorentzian counterparts, e.g. Galilean and Carrollian algebras. In existing literature, different non-Lorentzian theories of gravity have been addressed separately. In this paper, we introduce a single unified framework of expansion to address all these different theories. We show that different scalings can be unified into a single covariant form parametrized by $(s,n)$, alongside the contraction parameter $\epsilon$. Keeping these parameters unfixed in the limit $\epsilon \to 0$ defines a $\textit{unified flat geometry}$ and its $\textit{unified algebra}$, which reduces to a specific non-Lorentzian geometry for a particular choice of $s$ and $n$. Using this setup in the tangent space, we systematically expand the Einstein-Hilbert action in even powers of $\epsilon$, which we call a $\textit{unified expansion}$ of Einstein's gravity, whose leading-order theory is fixed by $(s,n)$. This reproduces various classes of gravitational theories, including Einstein gravity (the trivial case), Galilean gravity, Carroll gravity, all of which can be extracted from this expansion. Using the expansion, we then formulate String Carroll (SC) gravity, where the local metric has two vanishing eigenvalues. The near-horizon region of generic non-extremal black holes has been recently shown to be a SC geometry. By considering explicit examples, we confirm that these near-horizon geometries constitute solutions of SC gravity, paving the way of understanding physics near the horizon of generic black holes in terms of SC gravity.

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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. Carrollian bosonic supergravity at order $\alpha'$ and the universal cancellation of higher-curvature divergences

    hep-th 2026-07 conditional novelty 5.0

    Four-derivative and higher pure-gravitational α' corrections to bosonic supergravity admit a finite Carrollian limit, with an explicit action and a universal finiteness criterion for Riem^N terms.

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