arxiv: 2603.18416 · v2 · submitted 2026-03-19 · 🧮 math-ph · gr-qc· math.DG· math.MP

Recognition: no theorem link

On the Finsler variational nature of autoparallels in metric-affine geometry

Lehel Csillag , Nicoleta Voicu , Salah Elgendi , Christian Pfeifer

Authors on Pith no claims yet

Pith reviewed 2026-05-15 09:07 UTC · model grok-4.3

classification 🧮 math-ph gr-qcmath.DGmath.MP

keywords metric-affine geometryFinsler metrizabilityautoparallelsvectorial nonmetricitytorsion-free connectionsWeyl geometryvariational principlesFinsler geodesics

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The pith

Torsion-free affine connections with vectorial nonmetricity admit explicit Finsler Lagrangians that turn their autoparallels into geodesics.

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

The paper establishes necessary and sufficient conditions under which torsion-free connections carrying vectorial nonmetricity become Finsler metrizable. When the conditions hold, an algebraic Finsler Lagrangian built directly from the metric and the nonmetricity vector field is constructed explicitly. The resulting geodesics coincide with the autoparallels of the connection. This supplies a variational principle for a broad family of such connections, including the Weyl and Schrödinger cases as subfamilies.

Core claim

For torsion-free affine connections with vectorial nonmetricity, the existence of a Finsler Lagrangian that metrizes the connection (depending algebraically on the metric and the nonmetricity vector) is equivalent to a set of explicit algebraic conditions on the nonmetricity vector field; whenever these conditions are satisfied the Lagrangian can be written down directly and its geodesics reproduce the autoparallels.

What carries the argument

The algebraically constructed Finsler Lagrangian built from the metric and the nonmetricity defining vector field, whose extremals are required to coincide with the autoparallels.

If this is right

Autoparallels of the admissible connections acquire a variational formulation.
The same construction applies to the Weyl and Schrödinger connections whenever their nonmetricity vectors satisfy the derived algebraic relations.
The Finsler Lagrangian can be used to compute conserved quantities and to study the dynamics variationally.
Metric-affine field equations can be recast as Euler-Lagrange equations of the combined metric-plus-Finsler action.

Where Pith is reading between the lines

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

The algebraic dependence assumption may be relaxed to include mild differential dependence while preserving metrizability.
The same metrizability criterion could be tested numerically on concrete solutions of metric-affine gravity.
Connections outside the torsion-free vectorial class might still be metrizable if a different functional dependence on nonmetricity is allowed.

Load-bearing premise

The Finsler Lagrangian is required to depend only algebraically on the metric and on the nonmetricity defining vector field.

What would settle it

An explicit torsion-free connection with vectorial nonmetricity for which no algebraic Finsler Lagrangian reproduces the autoparallels as its geodesics would falsify the claimed necessary and sufficient conditions.

read the original abstract

In metric-affine geometry, autoparallels are generically non-variational, i.e., they are not the extremals of any action integral. The existence of a parametrization-invariant action principle for autoparallels is a long-standing open problem, which is equivalent to the so-called Finsler metrizability of the connection -- that is, to the fact that these autoparallels can be interpreted as Finsler geodesics. In this article, we address this problem for the class of torsion-free affine connections with vectorial nonmetricity, which includes, as notable subcases, Weyl and Schr\"odinger connections. For this class, we determine the necessary and sufficient conditions for the existence of a Finsler Lagrangian that metrizes the connection (and depends only algebraically on the metric and on the nonmetricity defining vector field). In the cases where such a Finsler Lagrangian exists, we construct it explicitly. In particular, we show that a broad class of such connections is in fact Finsler metrizable, i.e., the autoparallels of these connections are Finsler geodesics.

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 paper gives necessary and sufficient conditions plus explicit constructions for Finsler metrizability of torsion-free connections with vectorial nonmetricity.

read the letter

The main advance here is a clean set of conditions under which autoparallels of torsion-free metric-affine connections with vectorial nonmetricity become Finsler geodesics, together with explicit Lagrangians built algebraically from the metric and the nonmetricity vector. That covers Weyl and Schrödinger cases as subcases and turns a long-standing non-variational problem into a variational one for this family. The derivation stays close to the geometric definitions without obvious fitting or circular steps, and the abstract indicates they checked both directions of the necessary-and-sufficient claim. That is concrete progress on an open question in generalized gravity. The soft spot is the convexity requirement. The algebraic dependence on g and the nonmetricity vector guarantees the Euler-Lagrange equations reproduce the autoparallel equation, but positive-definiteness of the Hessian on the slit bundle is not automatic for every such connection; it has to be verified on the constructed L, and the stress-test note is right that this may fail globally even when the local matching holds. If the paper only shows the matching and leaves the convexity check to the reader for the broad class, that is a minor but real gap rather than a fatal one. The work is aimed at people who already care about metric-affine geometry and want action principles for autoparallels. A reader who needs the explicit Lagrangians or the precise boundary between metrizable and non-metrizable cases in this class will get direct value. It is coherent on its own terms and deserves a serious referee, even if the convexity verification needs tightening in revision.

Referee Report

1 major / 0 minor

Summary. The paper addresses the long-standing problem of whether autoparallels of torsion-free affine connections with vectorial nonmetricity (including Weyl and Schrödinger cases) admit a parametrization-invariant variational principle. It derives necessary and sufficient conditions for the existence of a Finsler Lagrangian L that depends only algebraically on the metric g and the nonmetricity vector field Q, constructs such Lagrangians explicitly when the conditions hold, and concludes that a broad class of these connections is Finsler metrizable, so that their autoparallels coincide with Finsler geodesics.

Significance. If the central claims hold, the work supplies the first explicit variational formulations for autoparallels in this geometrically natural class of metric-affine geometries. The algebraic dependence of L on g and Q yields concrete, checkable conditions rather than abstract existence results, which could be directly useful in modified-gravity models and in the study of non-metricity-driven dynamics.

major comments (1)

[Abstract / main existence theorem] The necessary-and-sufficient conditions derived for the Euler-Lagrange equations to reproduce the autoparallel equation do not automatically guarantee that the Hessian of L with respect to the velocity variable is positive definite on the slit tangent bundle. This convexity requirement is essential for L to define a genuine Finsler structure and must be verified separately on the explicit form of L; the manuscript does not provide this verification for the constructed family.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comment. The observation regarding the convexity requirement is valid and we will revise the paper to include an explicit verification that the constructed Lagrangians satisfy the positive-definiteness condition on the Hessian. We address the major comment below.

read point-by-point responses

Referee: The necessary-and-sufficient conditions derived for the Euler-Lagrange equations to reproduce the autoparallel equation do not automatically guarantee that the Hessian of L with respect to the velocity variable is positive definite on the slit tangent bundle. This convexity requirement is essential for L to define a genuine Finsler structure and must be verified separately on the explicit form of L; the manuscript does not provide this verification for the constructed family.

Authors: We agree that positive-definiteness of the Hessian is indispensable for L to induce a Finsler structure. Our necessary-and-sufficient conditions guarantee that the Euler-Lagrange equations of the constructed L reproduce the autoparallel equation, but convexity must indeed be checked separately. For the explicit algebraic family of Lagrangians we derive (which depend only on g, the nonmetricity vector Q, and the velocity), the Hessian can be computed directly. Under the stated conditions the resulting quadratic form is positive definite on the slit tangent bundle; this follows from the algebraic structure of L and the sign restrictions imposed by the metrizability conditions. We will add a dedicated subsection immediately after the construction of L that performs this explicit verification, including the expression for the Hessian and the proof of its positive-definiteness. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation from geometric definitions of torsion-free connections with vectorial nonmetricity

full rationale

The paper starts from the standard definitions of metric-affine geometry, torsion-free affine connections, and vectorial nonmetricity. It derives necessary and sufficient conditions for the existence of an algebraically dependent Finsler Lagrangian L(g, Q, v) whose geodesics reproduce the autoparallels. The construction is explicit and proceeds by solving the Euler-Lagrange equations to match the autoparallel equation; no fitted parameters are renamed as predictions, no self-citation chain is invoked as a uniqueness theorem, and no ansatz is smuggled in. The convexity requirement is noted as a separate check but is not part of the metrizability derivation itself. The central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard geometric restrictions to torsion-free connections whose nonmetricity is generated by a single vector field; these are domain assumptions rather than new postulates.

axioms (2)

domain assumption The affine connection is torsion-free
Restricts the analysis to the class containing Weyl and Schrödinger connections as stated in the abstract.
domain assumption Nonmetricity is vectorial
Nonmetricity is completely determined by a single vector field, enabling algebraic dependence of the Lagrangian.

pith-pipeline@v0.9.0 · 5526 in / 1356 out tokens · 56500 ms · 2026-05-15T09:07:06.296228+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

77 extracted references · 77 canonical work pages

[1]

What are the necessary and sufficient conditions to be satisfied by a given torsion-free connection with vectorial nonmetricity, such that it isFinsler metrizable, i.e., its autoparallels arise as arc-length parametrized geodesics of a Finsler Lagrangian?

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What are the corresponding Finsler Lagrangians, which depend algebraically on the components of the metric and the one-form defining the affine connection? Finsler Lagrangians whose geodesics arise as autoparallels of an affine connection on spacetime are known under the name ofBerwald-type[11, 30–33] ones. On the other hand, Finsler Lagrangians that depe...

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They are singled out in the class of generalized(α, β)-Finsler Lagragians as they do not depend on∣b∣

We first consider the so called(α, β)-Finsler Lagragians- which are the simplest and most commonly used Finsler functions in applications. They are singled out in the class of generalized(α, β)-Finsler Lagragians as they do not depend on∣b∣. For this class, we completely clarify under which conditions(α, β)-Finsler functions are of Berwald type and their ...

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[4]

torsion-free affine connection

After having presented a clear discussion of the simpler case of(α, β)-Finsler Lagragians, we pass to the most general case. More precisely, for a given connection with vectorial non-metricity, we find the necessary and sufficient conditions such that it is metrizable by ageneralized(α, β)-Finsler Lagragian. With this, we identify for the first time that ...

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[5]

Positive 2-homogeneity:L(x, λ˙x)=λ 2L(x,˙x),∀λ>0

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[6]

The pair(M, L)is called a(pseudo)-Finsler space

Nondegeneracy: the Hessiang µν(x,˙x)= 1 2 ∂2L ∂˙xµ∂˙xν(x,˙x)is non-singular at all(x,˙x)∈A. The pair(M, L)is called a(pseudo)-Finsler space. The above notion includes as subclasses classical Finsler spaces (whereA=T M∖{0}and(g µν)is everywhere positive definite), andFinsler spacetimes(where(g µν)has Lorentzian signature on an appropriate subset ofA). In w...

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[7]

Its arc-length parametrized geodesics coincide with the autoparallels of a symmetric affine connection∇onM, with coefficients Γµνρ(x): d2xµ ds2 +Γ µ νρ(x) dxν ds dxρ ds =¨xµ+Γ µ νρ(x)˙xν ˙xρ =0.(29) Berwald-type pseudo-Finsler spaces therefore provide a natural setting for adressing our main question: Given a symmetric affine connection with vectorial non...

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[8]

Saying that∇is pseudo-Finsler metrizable is equivalent to saying that there exists a nondegenerate, positively 2- homogeneous LagrangianL=L(x,˙x)(i.e., a reparametrization-invariant arc length functional) whose arc-length parametrized geodesics are the autoparallels of∇

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Riemann metrizability obviously implies Finsler metrizability. Yet, the converse statement is more nuanced: whereas metrizability by a classical (T M∖{0}-smooth, positive definite) Finsler metric does imply Riemann metrizability (a result known as Szabo’s Theorem, [50]), there exist multiple examples of connections that are metrizable by Finsler spacetime...

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If the nonmetricityQof∇is nonzero, then there exists at least one indexµ∈{0,1,2,3}such that both the left and right-hand sides of theµ-th equation (48) are not identically zero. Indeed, assuming the contrary and taking into account that none of Φ,Φ ′, AandBcan vanish identically (the vanishing of Φ′would entail the degeneracy ofL), this implies that, for ...

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Using these remarks, we obtain a second lemma

The nondegeneracy of the Riemannian metricaimplies that the polynomial in ˙xexpressionsAandBcannot have any common factors. Using these remarks, we obtain a second lemma. Lemma III.3.If the(α, β)-metric functionL=AΦ( B2 A )metrizes the symmetric affine connectionΓ= ○ ∇+Don M,then there exists a nonvanishing one-formρ=ρ µ(x)dx µ onMand constantsm, n, q∈Rwi...

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Contracting the first equation (59) with ˙x µ leads to ( 1 2 c1+c 2)AB+ c3 2 B3 =(ρ µ ˙xµ)(mA+qB 2) .(60) Taking, again, into account thatAandB(regarded as polynomial expressions in ˙x) have no common factors, we find (ρµ ˙xµ)q= c3 2 B,(ρ µ ˙xµ)m=( 1 2 c1+c 2)B.(61) Differentiation with respect to ˙xµ then implies ρµq= c3 2 bµ, ρ µm=( 1 2 c1+c 2)b µ ,(62)...

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Ifc 3 =0, then, the second equation of (58) impliesq=0. Further, fromm 2+q 2 ≠0 andn2+q 2 ≠0, we find that neitherm, norncan vanish, that is, we can write: ρµ = c1 2m bµ .(71) Substituting the above value into the last equation of (58), we get the differential constraint (42) with λ= m n ≠0.(72)

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Denoting: Λ∶=m q = c1 c3 , τ∶= p q ,(73) the differential constraint in (58) becomes precisely (44)

Ifc 3 ≠0, then, the third equation of (58) ensures thatq≠0 (elsewhere, we would get thatbidentically vanishes, which is excluded by hypothesis). Denoting: Λ∶=m q = c1 c3 , τ∶= p q ,(73) the differential constraint in (58) becomes precisely (44). Moreover, we have shown that the pseudo-Finsler structureL=AΦ, if any, must necessarily be obtained by integrat...

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This is done in four lemmas

In the first subsection, we rewrite the Berwald metrizability PDE system (31) using the ansatz (81) and use the contractions of the newly obtained PDEs withuand ˙xto deduce the constraints upon the derivatives of⟨b, b⟩ andu µ. This is done in four lemmas

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Then, using the obtained expressions, we proceed to the integration of the metrizability conditions. Before starting the proof, it is worth noting that extending our search to generalized(α, β)-metrics, we have completely eliminated the constraintc 2 =0; in particular, it turns out that, under appropriate conditions upon the defining one-form, Schr¨ oding...

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