arxiv: 2604.22538 · v2 · submitted 2026-04-24 · 🧮 math.DG · gr-qc· math-ph· math.AP· math.MG· math.MP

Recognition: 3 theorem links

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Timelike Ricci curvature lower bounds via optimal transport for Orlicz-type Lorentzian costs

Argam Ohanyan , Marta S\'alamo Candal

Authors on Pith no claims yet

Pith reviewed 2026-05-14 21:31 UTC · model grok-4.3

classification 🧮 math.DG gr-qcmath-phmath.APmath.MGmath.MP

keywords optimal transportLorentzian geometrytimelike Ricci curvaturerelative entropyOrlicz costsu-separationdisplacement convexity

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The pith

Timelike Ricci curvature lower bounds are characterized by convexity of relative entropy along geodesics from Orlicz-type optimal transport.

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

The paper shows that in globally hyperbolic spacetimes, a lower bound on timelike Ricci curvature holds if and only if the relative entropy is convex along geodesics induced by the optimal transport problem with cost function u composed with the time-separation function ℓ, where u is a suitable monotonically increasing concave function. This extends earlier characterizations that were limited to power-type costs u_p(x) = x^p / p for p in (0,1) and later p less than 0. The proof relies on a new u-separation condition on pairs of measures that guarantees strong duality for the Orlicz-type transport problem and reduces the curvature bound to an entropy-convexity statement along the resulting geodesics.

Core claim

In globally hyperbolic spacetimes, timelike Ricci curvature is bounded from below if and only if the relative entropy functional is convex along the geodesics of the optimal transport problem whose cost is u ∘ ℓ, whenever the source and target measures satisfy the u-separation property.

What carries the argument

The Orlicz-type Lorentzian cost u ∘ ℓ together with the u-separation property on pairs of measures, which ensures strong duality and reduces the curvature bound to displacement convexity of relative entropy.

If this is right

Timelike Ricci lower bounds imply displacement convexity of relative entropy along all geodesics arising from u ∘ ℓ optimal transport.
The characterization applies uniformly to any admissible concave increasing u, recovering the power cases as special instances.
Strong duality holds for the Orlicz-type transport problem once the u-separation condition is satisfied.
The result supplies a synthetic criterion for curvature bounds that can be checked via optimal transport without direct computation of the curvature tensor.

Where Pith is reading between the lines

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

The same convexity statement might serve as a definition of lower Ricci bounds in more singular Lorentzian settings where the curvature tensor is not classically defined.
Numerical schemes for optimal transport on Lorentzian manifolds could be used to test or approximate timelike curvature bounds in concrete spacetimes.
The u-separation condition may admit further relaxation or geometric interpretations that connect to causal structure alone.

Load-bearing premise

The u-separation property must hold for the pairs of measures being transported.

What would settle it

A globally hyperbolic spacetime in which relative entropy fails to be convex along some u ∘ ℓ geodesic for u-separating measures, yet timelike Ricci curvature remains bounded from below (or the converse).

read the original abstract

We study the optimal transport problem on globally hyperbolic spacetimes associated with Orlicz-type Lorentzian cost functions of the form $u \circ \ell$, where $u$ is a suitable monotonically increasing and concave function, and $\ell$ is the time separation. Our work encompasses and generalises the case $u(x) = u_p(x) = p^{-1}x^p$ for $p \in (0,1)$, as well as the more recent $p < 0$, which have been the only examples considered so far in the literature. A fundamental notion for our purposes is the property of $u$-separation for a pair of measures, which generalises McCann's $p$-separation and for which we are able to obtain strong duality to the full Orlicz-type optimization problem. In our main results, we characterise timelike Ricci curvature lower bounds via the convexity of the relative entropy along geodesics arising from the Orlicz-type optimal transport with cost $u \circ \ell$, which is a far-reaching generalisation of McCann's seminal work in the case $u = u_p$, $p \in (0,1)$.

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 extends McCann's entropy-convexity characterization of timelike Ricci bounds from power costs to general Orlicz-type u, but the u-separation condition may limit how broadly the equivalence applies.

read the letter

The main point is a generalization of McCann's work on timelike Ricci lower bounds. The authors replace the specific p-power costs with a broader class of Orlicz-type functions u composed with the time separation ℓ, and they introduce u-separation for pairs of measures to secure strong duality. Under that condition they recover the equivalence between Ricci bounds and convexity of relative entropy along the induced geodesics. This is a clear step beyond the p in (0,1) and p<0 cases already in the literature, and the setup follows standard optimal transport techniques on globally hyperbolic spacetimes without obvious reinvention of the wheel. The proofs appear to handle the duality step cleanly once u-separation is assumed, which is a useful technical device. The soft spot is the scope of u-separation itself. If it only holds for a narrow class of measure pairs and is not automatically satisfied or dense when the spacetime obeys a Ricci bound, then the claimed characterization becomes conditional rather than unconditional. The abstract presents it as a full characterization, so the paper needs to show either that u-separation is inherited from the curvature hypothesis or that it covers enough cases to be useful in practice. Without that, the result risks being narrower than advertised. This is specialized work for people already using optimal transport to study curvature in Lorentzian geometry. Readers who know the McCann p-case will see the value in the extension, but only if the u-separation restriction does not shrink the statement too far. The paper shows clear thinking and honest engagement with the existing results, so it deserves a serious referee to check the details around duality and the range of applicability.

Referee Report

1 major / 1 minor

Summary. The paper studies the optimal transport problem on globally hyperbolic spacetimes with Orlicz-type Lorentzian costs of the form u ∘ ℓ, where u is monotonically increasing and concave. It introduces the u-separation property for pairs of measures (generalizing McCann's p-separation) to obtain strong duality, and claims to characterize timelike Ricci curvature lower bounds via convexity of the relative entropy along the resulting u∘ℓ-geodesics. This is presented as a far-reaching generalization of McCann's results for the power costs u_p with p ∈ (0,1) and p < 0.

Significance. If the central equivalence holds without restrictive caveats, the work would substantially extend the optimal-transport characterization of timelike Ricci bounds to a broad family of concave Orlicz costs, unifying and generalizing prior results while providing new tools for Lorentzian geometry. The introduction of u-separation and the duality proof under that hypothesis are technically substantive contributions.

major comments (1)

[Main results / characterization theorem] Main characterization theorem: the claimed equivalence between timelike Ricci lower bounds and relative-entropy convexity along u∘ℓ-geodesics is obtained only after imposing the u-separation condition on the pair of measures. While strong duality is proved under u-separation, the manuscript does not establish that u-separation holds for a dense class of compactly supported probabilities or for all pairs whose supports are chronologically separated. Consequently the convexity statement remains conditional on an auxiliary assumption that is not automatically inherited from the curvature hypothesis, weakening the claimed characterization.

minor comments (1)

[Abstract and Introduction] The abstract states that the work encompasses the case p < 0, but the precise range of admissible u functions and any new concrete examples beyond the power cases should be stated explicitly in the introduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We address the main comment on the characterization theorem below, and we will make revisions to strengthen the presentation of our results.

read point-by-point responses

Referee: Main characterization theorem: the claimed equivalence between timelike Ricci lower bounds and relative-entropy convexity along u∘ℓ-geodesics is obtained only after imposing the u-separation condition on the pair of measures. While strong duality is proved under u-separation, the manuscript does not establish that u-separation holds for a dense class of compactly supported probabilities or for all pairs whose supports are chronologically separated. Consequently the convexity statement remains conditional on an auxiliary assumption that is not automatically inherited from the curvature hypothesis, weakening the claimed characterization.

Authors: We acknowledge that our main characterization is formulated for u-separated pairs of measures, as this condition is necessary for the strong duality result that underpins the optimal transport geodesics. This is analogous to the p-separation condition in McCann's original work on the Riemannian case, which was also required for duality. To address the concern, we will add a new subsection demonstrating that the set of u-separated pairs is dense among pairs of compactly supported probability measures with chronologically separated supports, in the weak topology. This density, combined with continuity arguments for the relative entropy and the geodesics, allows extension of the convexity statement to all such pairs, thereby making the characterization hold unconditionally for the relevant class of measures. We will also clarify in the introduction and statement of the main theorem that u-separation is a technical but mild condition satisfied by a dense class. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to McCann; u-separation is newly introduced and enables independent generalization without definitional reduction

full rationale

The derivation introduces the u-separation property as a generalization of McCann's p-separation, proves strong duality under this assumption, and then obtains the entropy-convexity characterization of timelike Ricci bounds. This adds new content rather than reducing the central claim to a fitted input or self-referential definition. The only connection to prior work is a standard citation to McCann's results on p-separation, which is not load-bearing for the Orlicz extension. No equations or steps in the provided chain equate the Ricci characterization to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions from Lorentzian geometry and optimal transport; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The spacetime is globally hyperbolic
Explicitly stated as the setting for the optimal transport problem.
domain assumption u is monotonically increasing and concave
Required for the cost function u ∘ ℓ to satisfy the necessary properties for duality and convexity.

pith-pipeline@v0.9.0 · 5526 in / 1313 out tokens · 43636 ms · 2026-05-14T21:31:16.307174+00:00 · methodology

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discussion (0)

Lean theorems connected to this paper

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IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We characterise timelike Ricci curvature lower bounds via the convexity of the relative entropy along geodesics arising from the Orlicz-type optimal transport with cost u ∘ ℓ
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

u-separation for a pair of measures, which generalises McCann's p-separation
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

globally hyperbolic spacetimes

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

K., and McCann, R

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[15]Kell, M., and Suhr, S.On the existence of dual solutions for Lorentzian cost functions

[14]Kell, M.On interpolation and curvature via Wasserstein geodesics.Advances in Calculus of Variations 10, 2 (2017), 125–167. [15]Kell, M., and Suhr, S.On the existence of dual solutions for Lorentzian cost functions. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire 37, 2 (2020), 343–372. [16]Kunzinger, M., and S ¨amann, C.Lorentzian length spaces.Ann. Gl...

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[25]Mondino, A., and Suhr, S.An optimal transport formulation of the Einstein equations of general relativity.J

[24]Mondino, A., and S ¨amann, C.Lorentzian Gromov-Hausdorff convergence and pre- compactness.arXiv:2504.10380(2025). [25]Mondino, A., and Suhr, S.An optimal transport formulation of the Einstein equations of general relativity.J. Eur. Math. Soc. (JEMS) 25, 3 (2023), 933–994. [26]O’Neill, B.Semi-Riemannian geometry, vol. 103 ofPure and Applied Mathematics...

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I.Acta Math

[29]Sturm, K.-T.On the geometry of metric measure spaces. I.Acta Math. 196, 1 (2006), 65–131. [30]Sturm, K.-T.On the geometry of metric measure spaces. II.Acta Math. 196, 1 (2006), 133–177. [31]Sturm, K.-T.Generalized Orlicz spaces and Wasserstein distances for convex-concave scale functions.Bull. Sci. Math. 135, 6-7 (2011), 795–802. [32]Suhr, S.Theory of...

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