Jacobson's thermodynamic approach to classical gravity applied to non-Riemannian geometries: remarks on the simplicity of Nature
Pith reviewed 2026-05-16 09:30 UTC · model grok-4.3
The pith
Jacobson's thermodynamic approach selects Einstein-Hilbert action plus quadratic torsion term for non-Riemannian geometries without non-metricity.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Jacobson's thermodynamic approach to classical gravity, when applied to non-Riemannian geometries, shows that the Einstein-Hilbert action does not belong to the pool of gravitational theories available for Nature's selection except in the Riemannian case. Considering the hypotheses from Lanczos-Lovelock theories of gravity, the two approaches together point to the theory derived from the Einstein-Hilbert action plus a term quadratic in the torsion vector as the one selected by Nature in the non-Riemannian case without non-metricity, when the energy-momentum tensor is identified as its metric version. The same strategy cannot be followed in the full non-Riemannian case or when using the canon
What carries the argument
Jacobson's thermodynamic relation applied to local Rindler horizons in non-Riemannian spacetimes, required to be consistent with Lanczos-Lovelock gravity hypotheses.
If this is right
- The pure Einstein-Hilbert action is excluded as a candidate theory once torsion is present.
- A quadratic term in the torsion vector must be added to restore consistency with both thermodynamic and Lanczos-Lovelock requirements.
- The selection works only when non-metricity vanishes and the energy-momentum tensor is taken in its metric form.
- The thermodynamic and Lanczos-Lovelock approaches become mutually inconsistent once non-metricity is allowed or the canonical energy-momentum tensor is used.
- Nature would therefore select a unique modified theory rather than the standard Einstein-Hilbert action in the presence of torsion.
Where Pith is reading between the lines
- If the selection rule is robust, torsion may be required to appear explicitly in any thermodynamically consistent gravitational action beyond Riemannian geometry.
- The identified inconsistency when non-metricity is present suggests that a different thermodynamic construction would be needed to accommodate metricity violation.
- The result could be tested by comparing predictions of the modified action against observations in cosmological models that allow torsion.
- Similar thermodynamic derivations might be applied to other horizon types to see whether the same quadratic torsion term continues to be selected.
Load-bearing premise
The structural hypotheses used to formulate Lanczos-Lovelock theories can be applied directly to non-Riemannian geometries while preserving thermodynamic consistency only for specific energy-momentum identifications and in the absence of non-metricity.
What would settle it
An explicit derivation of the field equations from the thermodynamic first law on a horizon in a spacetime with torsion that yields equations different from those obtained from the Einstein-Hilbert action plus quadratic torsion term.
read the original abstract
Three decades ago, Ted Jacobson surprised us with a very appealing approach to classical gravity. According to him, the gravitational field equations are the consequence of the first law of thermodynamics applied to a Rindler observer. Jacobson's approach being formulated for Riemannian geometries, we have wondered what its consequences would be for non-Riemannian geometries. The results of our quest have been particularly appealing: we have found that the theory that derives from the Einstein-Hilbert action, arguably ``the simplest one'', does not belong to the pool of gravitational theories available for Nature's selection (except in the Riemannian case). In the search of a unique alternative, we have considered the hypotheses employed in the formulation of the Lanczos-Lovelock theories of gravity. Together, the two approaches point towards the theory that derives from the Einstein-Hilbert action plus a term quadratic in the torsion vector as the one that would be selected by Nature in the non-Riemannian case without non metricity (when the energy-momentum tensor is identified as its metric version). The same strategy cannot be followed in the full non-Riemannian case (and in the previous case when the energy-momentum tensor is identified as its canonical version) as the two approaches are mutually inconsistent.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript extends Jacobson's thermodynamic derivation of gravitational field equations from Rindler horizons to non-Riemannian geometries with torsion but no non-metricity. It concludes that the Einstein-Hilbert action is excluded from the set of theories selectable by Nature in this setting, but that supplementing it with a quadratic term in the torsion vector yields the unique theory consistent with both the thermodynamic first law and the Lanczos-Lovelock hypotheses, provided the energy-momentum tensor is identified with its metric version. The two approaches become mutually inconsistent when non-metricity is present or when the canonical energy-momentum tensor is used instead.
Significance. If the central selection claim is substantiated, the work would supply a thermodynamic criterion for choosing among torsion-inclusive gravitational actions, reinforcing the idea that Nature selects the simplest consistent extension beyond Riemannian geometry. The manuscript contains no machine-checked proofs or reproducible code, but it attempts a parameter-free derivation based on consistency requirements.
major comments (2)
- [Lanczos-Lovelock-type construction] The uniqueness of the Einstein-Hilbert plus quadratic-torsion term rests on extending the Lanczos-Lovelock axioms (diffeomorphism invariance, second-order equations of motion) unchanged to a connection with torsion. The manuscript does not demonstrate that the resulting Lovelock-like densities remain the only candidates once the torsion vector enters the curvature scalars and the variational principle is performed with respect to both metric and connection (see the load-bearing step identified in the skeptic note).
- [Energy-momentum tensor identification] Consistency between the thermodynamic and Lanczos-Lovelock approaches is asserted only for the metric version of the energy-momentum tensor. The paper must supply the explicit derivation showing mutual inconsistency for the canonical identification, including the relevant first-law application and variation, to support the abstract claim.
minor comments (2)
- [Abstract] The phrasing 'the results of our quest have been particularly appealing' in the abstract is informal; replace with a neutral statement of the findings.
- Clarify the precise definition of the quadratic torsion term (e.g., its coefficient and contraction) and ensure all equations are numbered for reference.
Simulated Author's Rebuttal
We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the manuscript to provide the requested clarifications and explicit derivations.
read point-by-point responses
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Referee: The uniqueness of the Einstein-Hilbert plus quadratic-torsion term rests on extending the Lanczos-Lovelock axioms (diffeomorphism invariance, second-order equations of motion) unchanged to a connection with torsion. The manuscript does not demonstrate that the resulting Lovelock-like densities remain the only candidates once the torsion vector enters the curvature scalars and the variational principle is performed with respect to both metric and connection (see the load-bearing step identified in the skeptic note).
Authors: We agree that the uniqueness argument requires a more explicit demonstration. In the revised version we will add a dedicated subsection that starts from the general curvature scalar built from the torsionful connection, imposes diffeomorphism invariance and second-order equations of motion, and shows by direct variation with respect to both metric and connection that the only admissible term quadratic in the torsion vector is the one we identified. This will make the load-bearing step fully transparent and substantiate the selection claim under the stated hypotheses. revision: yes
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Referee: Consistency between the thermodynamic and Lanczos-Lovelock approaches is asserted only for the metric version of the energy-momentum tensor. The paper must supply the explicit derivation showing mutual inconsistency for the canonical identification, including the relevant first-law application and variation, to support the abstract claim.
Authors: We accept that the explicit calculation for the canonical energy-momentum tensor was only outlined. We will insert a new subsection that (i) applies the first law to the Rindler horizon using the canonical EMT, (ii) performs the metric and connection variations of the Einstein-Hilbert plus quadratic-torsion action, and (iii) demonstrates the resulting mismatch with the Lanczos-Lovelock conditions. The same explicit steps will be given both for vanishing non-metricity and for the general non-Riemannian case, thereby supporting the abstract statement. revision: yes
Circularity Check
No significant circularity in derivation chain
full rationale
The paper applies Jacobson's thermodynamic derivation to non-Riemannian geometries and then invokes the standard Lanczos-Lovelock hypotheses (diffeomorphism invariance, second-order equations) to identify a consistent extension containing a quadratic torsion term. This consistency is explicitly conditioned on the absence of non-metricity and a metric identification of the energy-momentum tensor; the resulting selection is presented as a conditional outcome rather than a tautology. No quoted step reduces by construction to the inputs, no parameter is fitted and relabeled as a prediction, and the LL axioms are treated as external rather than self-defined or smuggled via author self-citation. The central claim therefore retains independent content and remains open to external verification.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption The first law of thermodynamics applies to Rindler observers in non-Riemannian geometries.
- domain assumption Hypotheses employed in Lanczos-Lovelock theories remain valid for non-Riemannian geometries.
Lean theorems connected to this paper
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IndisputableMonolith/Foundation/AlexanderDuality.leanalexander_duality_circle_linking contradicts?
contradictsCONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.
we have assumed the spacetime is four dimensional
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IndisputableMonolith/Foundation/RealityFromDistinction.leanreality_from_one_distinction contradicts?
contradictsCONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.
the theory that derives from the Einstein-Hilbert action plus a term quadratic in the torsion vector
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
discussion (0)
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