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REVIEW 1 major objections 5 minor 76 references

Intrinsic Torsion, Extrinsic Torsion, and the Hubble Parameter

T0 review · 1 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Extrinsic torsion inevitably mixes with the Hubble parameter in metric-compatible torsional theories, and ignoring it makes inferred expansion rates too small.

desk verdict The paper's real contribution is a clean, apparently correct geometric identity that mixes the Hubble parameter with extrinsic torsion in the generalized Friedmann constraint; the Hubble-anomaly application rests entirely on an unproven tracking conjecture and is not established. read the letter →

arxiv 2412.08990 v2 pith:W7H3PS5N submitted 2024-12-12 gr-qc astro-ph.CO

Brett McInnes This is my paper · ORCID
classification gr-qcastro-ph.CO
keywords torsionextrinsicHubbleparametergeneralizedFriedmannequationinflationEinstein-Cartantheorysecondfundamentalformgeodesicdeviation
topics The Hubble Tension
open problems The Hubble Tension
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 develops a submanifold version of torsion, splitting the torsion of a spatial slice into intrinsic torsion, living inside the slice, and extrinsic torsion, arising from how the slice twists into spacetime. Its central result is that the second fundamental form of a spatial slice is exactly a sum of a term governed by the Hubble parameter and a term governed by the extrinsic torsion, $h = \tfrac12 \mathcal{L}_\xi g^* + \tfrac12 \kappa$. Because the generalized Friedmann equation follows from the Gauss equation, the Hubble parameter and extrinsic torsion enter it inseparably, with opposite signs: $16\pi G\rho/c^4 = \mathrm{Scal} + 6H^2 - \tfrac12 \kappa^2$. A sympathetic reader would take away that any metric-compatible torsional theory inevitably mixes $H$ with extrinsic torsion, so an observer who assumes the torsion is absent will systematically underestimate the expansion rate. The paper also argues that inflation can "inflate away" intrinsic torsion while "inflating up" extrinsic torsion, so $\kappa$ could be of order $H$ at reheating.

What carries the argument

The central object is the torsional Gauss-Codazzi equation, $T^*(X,Y)=T(X,Y)+\alpha(X,Y)-\alpha(Y,X)$, which defines intrinsic torsion $T$ and extrinsic torsion $\alpha(X,Y)-\alpha(Y,X)$. Combined with the second fundamental form decomposition $h=\tfrac12 \mathcal{L}_\xi g^* + \tfrac12 \kappa$, where $\kappa$ is the extrinsic torsion two-form, it shows that the antisymmetric part of the second fundamental form is exactly the extrinsic torsion. This identity is what carries the argument: it puts $H$ and $\kappa$ on equal footing in the generalized Friedmann constraint.

What would settle it

Take one cosmic epoch and measure $\rho$, the intrinsic scalar curvature $\mathrm{Scal}$, and $H$ independently; in Einstein-Cartan theory the residual $16\pi G\rho/c^4 - \mathrm{Scal} - 6H^2$ must equal $-\kappa^2/2$, so a positive residual, or a zero residual where independent spin-density bounds force $\kappa=0$, would rule out the mechanism.

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Extended reading notes

Core claim

The paper's central claim is that torsion has an extrinsic face that cannot be ignored. Splitting the spacetime torsion into terms tangential and normal to a spacelike hypersurface gives $T^*(X,Y)=T(X,Y)+\alpha(X,Y)-\alpha(Y,X)$, so the antisymmetric part of the second fundamental form is literally the extrinsic torsion. Because the second fundamental form also carries the Hubble parameter, the geometry forces a mixing: $h=\tfrac12 \mathcal{L}_\xi g^* + \tfrac12 \kappa$. Feeding this into the Gauss equation and the Einstein-Cartan field equations yields $16\pi G\rho/c^4=\mathrm{Scal}+6H^2-\tfrac12 \kappa^2$, with $H^2$ and $\kappa^2$ entering with opposite signs. The paper argues that the same combination survives in any metric-compatible torsional theory, and that inflation can naturally make $\kappa$ of order $H$ at reheating while wiping out the intrinsic torsion. It stops short of claiming to solve the Hubble tension, but asserts that extrinsic torsion is the most natural geometric source of Hubble-parameter anomalies.

Load-bearing premise

The whole mechanism depends on torsion 'tracking' curvature during inflation: what makes spatial curvature shrink must also make the twist of space into spacetime grow; if torsion does not follow curvature in this way, the extrinsic torsion could remain zero or negligible even at reheating.

Editorial extensions

If this is right

  • Any metric-compatible torsional theory has $h = \tfrac12 \mathcal{L}_\xi g^* + \tfrac12 \kappa$, so the Hubble parameter and extrinsic torsion are inseparable in the second fundamental form.
  • In Einstein-Cartan theory the generalized Friedmann constraint is $16\pi G\rho/c^4 = \mathrm{Scal} + 6H^2 - \tfrac12 \kappa^2$; an observer setting $\kappa=0$ infers an $H$ that is too small.
  • If intrinsic torsion tracks intrinsic curvature during inflation, then extrinsic torsion can grow to order $H$ by reheating, producing percent-level corrections to the theoretical Hubble parameter.
  • Fully isotropic spatial sections have identically zero extrinsic torsion, so nonzero extrinsic torsion at late times requires anisotropic pre-inflationary initial data.
  • The mixing is generic across torsional theories with zero non-metricity, not an artifact of Einstein-Cartan theory.

Reading between the lines

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

  • A precision measurement of $16\pi G\rho/c^4 - \mathrm{Scal} - 6H^2$ at a single epoch would directly bound $\kappa^2$; a negative residual would be a clean torsion signature, since no standard-matter contribution produces that sign in this combination.
  • The paper's 'torsion escarpment' phenomenon suggests that gravitational-wave or tensor-harmonic surveys should look for large geodesic deviations sourced by small but rapidly changing torsion, a signature that curvature alone cannot produce.
  • If torsional inflation can start from anisotropic initial data with large intrinsic torsion, the usual causality-based obstruction to inflation onset may fail; redoing that no-go argument with torsion is a concrete theoretical next step.
  • The same geometric mixing should apply around static compact objects: in a torsional Schwarzschild-like solution the extrinsic torsion does not vanish even though the Hubble term does, so solar-system tests of spatial geometry could probe torsion independently of cosmology.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper develops a geometric framework for intrinsic and extrinsic torsion of spacelike hypersurfaces in metric-compatible torsional spacetimes. It derives the torsional Gauss-Codazzi equation (11), the decomposition of the second fundamental form into a Hubble-related symmetric part and an extrinsic-torsion antisymmetric part (26), and, in Einstein-Cartan theory, a generalized Friedmann constraint (37) in which the Hubble parameter H and the extrinsic torsion parameter κ mix with opposite signs. It then conjectures that during inflation the intrinsic torsion tracks the intrinsic Levi-Civita curvature and inflates away, while the extrinsic torsion tracks the extrinsic curvature and inflates up, leading to a possible underestimate of the Hubble parameter if extrinsic torsion is ignored.

Significance. The purely geometric identities, especially Eq (26) and the decomposition of the second fundamental form, are clean, clearly derived, and likely useful for future work on torsional cosmology. The paper is commendably explicit about the distinction between the derived identities and the speculative dynamical assumptions. However, the central quantitative conclusion (the 4% Hubble underestimate) is not established: it rests on an unproven tracking conjecture that is not supported by any field equations or explicit model, and the paper itself shows that in fully isotropic sections the extrinsic torsion remains exactly zero. The significance of the work is therefore that of a well-formulated possibility, not a derivation of an observable effect.

major comments (1)
  1. [Section 4.2 (tracking conjecture)] The quantitative Hubble-parameter claim is not derived. The paragraph beginning 'It seems natural, however, to assume...' asserts that if intrinsic torsion tracks intrinsic Levi-Civita curvature, then extrinsic torsion tracks extrinsic Levi-Civita curvature, and the following 'concrete example' stipulates κ ≈ H at reheating. No field equations, evolution equations, or explicit torsional inflationary model are provided to support this tracking behaviour. This assumption is load-bearing because if κ decays during inflation or remains negligible, Eq (37) reduces to the standard Friedmann equation and the claimed anomaly disappears. The paper itself shows (Section 4.2, after Eq (40)) that in the fully isotropic case the extrinsic torsion is zero at all times; the proposed resolution, namely anisotropic pre-inflationary sections, is not realized by any concrete example. The 4% underestimate should therefore be presented as an illustrative possibility, not as a prediction, unless a supporting model is supplied.
minor comments (5)
  1. [Section 4.2, p. 23] The statement that 'H and κ will appear in the same combination even in such theories' is stronger than the derivation supports: Eq (37) is obtained from the Einstein-Cartan field equations (28)-(29), whereas the universal geometric result is Eq (26). Please restrict the generality claim to Eq (26) or provide a proof for a broader class of theories.
  2. [Section 4.2, Eq (35)] The symbol κ is used both for the two-form in Eq (25) and for the real parameter introduced after Eq (35); this reuse is potentially confusing, and a distinct notation (e.g., κ₀) would improve readability.
  3. [Abstract] The phrase 'extrinsic torsion is by far the most natural way to produce such anomalies' is a subjective assertion not accompanied by a comparison with other proposed mechanisms for Hubble-parameter anomalies; consider softening it.
  4. [Title page] The title currently displays stray spaces ('Intrinsic T orsion, Extrinsic T orsion'); these should be corrected in the final typeset version.
  5. [Section 3.2, after Eq (11)] The statement that torsion 'cannot be cancelled' by manipulating α is too absolute: Eq (11) shows the normal part of T* can be cancelled by an antisymmetric α, and cancellation is prevented only by the tangential part of T*. Suggest replacing 'cannot' with 'cannot in general'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Eq. (26) and Eq. (37) are derived from definitions and the Einstein-Cartan field equations, and the Hubble-underestimate estimate is explicitly conditional on a labeled tracking postulate rather than a fitted or self-referential prediction.

full rationale

The central derivation chain is self-contained. Equation (26) is obtained in Appendix 1 from the definition of the second fundamental form and metric compatibility; it is an identity, not a fit. Equation (37) is derived from the Einstein-Cartan field equation (28) together with the Gauss equation (14), so the appearance of 6H^2 - (1/2)kappa^2 is a consequence of the stated assumptions rather than an input. The quantitative Hubble-anomaly statement in Sec. 4.2 is explicitly conditional: the paper postulates that intrinsic torsion 'tracks' intrinsic Levi-Civita curvature and inflates away, and that extrinsic torsion then 'tracks' extrinsic Levi-Civita curvature and inflates up; this is labeled as 'we postulate' and 'It seems natural, however, to assume', and the paper itself notes that in the fully isotropic case (Eqs. (39)-(40)) the extrinsic torsion remains zero at all times, so the mechanism is not automatic. The 4% underestimate is therefore an illustrative stipulation rather than a derived prediction, and underdetermination is not circularity. The only self-citation ([58]) appears in an aside offering a possible topological explanation of small intrinsic curvature and is not load-bearing for the paper's main claims. No fitted input is renamed as a prediction, and no load-bearing assertion is justified solely by a self-citation.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new particles or forces. The only new object is the extrinsic torsion parameter kappa, which is a component of the standard second fundamental form, so it is not an invented entity. The main load-bearing inputs are the metric-compatibility restriction and the tracking conjecture.

free parameters (1)
  • tracking ratio r = kappa/H at reheating = r approximately 1 (illustrative example)
    Introduced by hand in Section 4.2 to produce the illustrative 4 percent underestimate; not derived from any specific torsional theory.
assumptions (3)
  • domain assumption The spacetime connection is metric-compatible (zero non-metricity).
    Stated in Section 1: the paper excludes non-metricity because submanifold theory becomes complicated; this restricts applicability of equations (24)-(37) to metric-compatible torsional theories.
  • ad hoc to paper The pre-inflationary spatial sections are sufficiently anisotropic to support a non-zero intrinsic torsion.
    Section 4.2 notes that in the fully isotropic case extrinsic torsion remains zero, so the Hubble-anomaly scenario requires anisotropic initial data.
  • standard math The Gauss-Codazzi equations (11), (13), (14) extend unchanged to metric-compatible connections with torsion.
    Used throughout Section 3.2 and Appendix 1; the paper claims all explicit torsion terms drop out of the curvature Gauss equation, but only sketches the derivation of (14).

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Cite this review

Pith. "Pith review of Intrinsic Torsion, Extrinsic Torsion, and the Hubble Parameter." pith.science (2026). https://pith.science/paper/W7H3PS5N

@misc{pith2026241208990,
  author       = {Pith},
  title        = {Pith review of: Intrinsic Torsion, Extrinsic Torsion, and the Hubble Parameter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W7H3PS5N}},
  note         = {Machine review of arXiv:2412.08990}
}
abstract

We study the intrinsic and extrinsic torsions (defined by analogy with the intrinsic and extrinsic curvatures) of the spatial sections of torsional spacetimes. We consider two possibilities. First, that the intrinsic torsion might prove to be directly observable. Second, that it is not observable, having been ``inflated away'' in the early Universe. We argue that, even in this second case, the extrinsic torsion may grow during the inflationary era and be non-negligible at reheating and thereafter. Even if the spatial intrinsic curvature and torsion are too small to be detected directly, then, the extrinsic torsion might not be. We point out that, if its presence is not recognised, the extrinsic torsion could lead to anomalies in the theoretical estimate of the Hubble parameter -- $\,$ a result with obvious potential applications. We stress that extrinsic torsion is by far the most natural way to produce such anomalies, simply because it mixes naturally with the Hubble parameter; that is, the second fundamental form of a spacelike section depends on a sum of two terms, one determined by the Hubble parameter, the other by the extrinsic torsion.

Discussion (0). Continue with ORCID to comment.

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