Reduction of the Finsler gravity vacuum equation and dynamics for the cosmological Landsberg spacetimes

Christian Pfeifer; Fidel F. Villase\~nor

arxiv: 2606.24698 · v1 · pith:RRNLMDWMnew · submitted 2026-06-23 · 🌀 gr-qc · math-ph· math.DG· math.MP

Reduction of the Finsler gravity vacuum equation and dynamics for the cosmological Landsberg spacetimes

Christian Pfeifer , Fidel F. Villase\~nor This is my paper

Pith reviewed 2026-06-25 23:17 UTC · model grok-4.3

classification 🌀 gr-qc math-phmath.DGmath.MP

keywords Finsler gravityvacuum equationsRicci curvatureLandsberg spacetimescosmological modelsPalatini formulationmetric formulationlight cones

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The pith

The scalar Finsler gravity vacuum equation reduces to vanishing Finslerian Ricci curvature when some power of the Finsler function is regular on light cones and the Landsberg term vanishes.

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

The paper identifies exact conditions that let the Finsler gravity vacuum equation simplify exactly as Einstein's vacuum equations do, by reducing to the vanishing of the Finslerian Ricci curvature. This occurs in both the metric and Palatini versions whenever a power F^n of the Finsler function meets a regularity requirement that makes its associated metric non-degenerate on the light cones and the Landsberg term is zero. The result is then used to obtain explicit solutions for the dynamics of homogeneous and isotropic cosmological spacetimes that are of Landsberg type. A reader would care because Finsler geometry supplies a phase-space description of gravity for kinetic gases, and the reduction removes a major obstacle to solving the equations in that setting.

Core claim

When there exists a sufficiently regular power F^n of the Finsler function whose associated Finsler metric is non-degenerate on the light cones and the Landsberg term vanishes, the scalar Finsler gravity vacuum equation reduces to the vanishing of the Finslerian Ricci curvature; the same reduction holds in the Palatini formulation. This condition is applied to homogeneous and isotropic Finsler spacetime functions of Landsberg type to solve the resulting dynamics.

What carries the argument

The joint requirement of sufficient regularity for some power F^n (with non-degenerate metric on light cones) together with vanishing of the Landsberg term, which together produce the reduction of the scalar vacuum equation to Finslerian Ricci-flatness.

If this is right

The vacuum equations become solvable for homogeneous and isotropic Landsberg-type cosmological spacetimes.
The reduction applies equally to the purely metric formulation and the Palatini formulation.
The result covers Finsler functions for which F squared itself is not regular, thereby extending the range of models that can be treated directly.
The dynamics of such cosmological models are governed by the vanishing of the Finslerian Ricci curvature once the stated conditions hold.

Where Pith is reading between the lines

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

The same regularity-plus-vanishing criterion might be checked directly on other families of Finsler functions arising in kinetic-gas models.
If the Landsberg term can be shown to vanish for a larger class of spacetimes, the reduction would apply more broadly without additional assumptions.
One could test the reduction numerically by constructing explicit Finsler metrics on light cones and verifying non-degeneracy of the associated metric for some power n.

Load-bearing premise

That there exists some power of the Finsler function meeting the stated regularity and non-degeneracy conditions on the light cones and that the Landsberg term vanishes.

What would settle it

An explicit calculation for a homogeneous isotropic Landsberg spacetime in which the scalar equation fails to equal the Ricci term even though a suitable regular power F^n exists and the Landsberg term is zero.

read the original abstract

When solving the Einstein vacuum equations, a very helpful feature is that they reduce simply to the vanishing of the Ricci tensor. In Finsler gravity, a promising extension of general relativity that can describe the gravitational field of kinetic gases from a phase space perspective in terms of Finsler geometry, such a reduction is not as straightforward. In this article, we identify precise conditions under which the scalar Finsler gravity vacuum equation (in either its purely metric or its Palatini formulation) reduces to the vanishing of the Finslerian Ricci curvature. Through analytic arguments, we find that this happens if there exists some power $F^n$ of the Finsler function $F$ that is sufficiently regular and whose associated Finsler metric is non-degenerate on the light cones. Moreover, the Landsberg term in the scalar equation must vanish. This result significantly generalizes the findings of [Villasenor2024], where a reduction theorem was established under the quite strong assumption that $F^2$ is regular, which is not satisfied by many examples currently under consideration. We demonstrate the impact of our findings by applying them to solve the Finsler gravity equations for homogeneous and isotropic Finsler spacetime functions of Landsberg type.

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 generalizes the prior reduction theorem by allowing F^n regularity instead of just F^2, then applies the result to Landsberg cosmologies, but the examples need explicit verification that the new conditions actually hold.

read the letter

The core advance is relaxing the regularity requirement in the reduction theorem. Where Villasenor2024 needed F squared to be regular, this version shows that some power F^n suffices provided the associated metric stays non-degenerate on the light cones and the Landsberg term vanishes. That change matters because many Finsler functions used in cosmology fail the stronger F^2 condition but may satisfy the weaker one.

The paper then uses the generalized theorem to reduce the scalar vacuum equation for homogeneous isotropic Landsberg spacetimes and solves the resulting system. The analytic steps appear straightforward and the application is direct.

The soft spot is verification on the concrete examples. The reduction only goes through if a suitable n exists and the metric is non-degenerate on the light cones for those specific Finsler functions. The abstract states the result applies, but does not indicate whether the authors checked the conditions explicitly rather than assuming they hold. If the full text contains those checks, the application is solid; if not, the solved equations rest on an unconfirmed premise.

This is niche work aimed at researchers already inside the Finsler gravity program who need workable vacuum equations for cosmological models. It is not aimed at the broader GR community.

The generalization is a genuine technical step and the paper engages the literature honestly, so it deserves a serious referee even if the example verification needs tightening.

Referee Report

1 major / 1 minor

Summary. The paper claims to identify precise conditions under which the scalar Finsler gravity vacuum equation (in metric or Palatini form) reduces to vanishing of the Finslerian Ricci curvature: existence of some power F^n of the Finsler function F that is sufficiently regular with non-degenerate associated metric on the light cones, plus vanishing of the Landsberg term. This generalizes prior results that required regularity of F^2 and is applied to solve the equations for homogeneous isotropic Landsberg-type Finsler spacetimes.

Significance. If the regularity, non-degeneracy, and Landsberg-vanishing conditions hold for the concrete cosmological examples, the reduction theorem would enable direct solution of the vacuum equations via Ricci vanishing, substantially broadening the class of Finsler functions to which the framework applies beyond the restrictive F^2-regular case.

major comments (1)

[Application to cosmological Landsberg spacetimes] Application section (demonstration for homogeneous isotropic Landsberg-type functions): the reduction theorem is invoked to solve the equations, yet the abstract (and by extension the demonstration) provides no explicit verification that a suitable n exists making F^n sufficiently regular with non-degenerate g_ij on the light cones, nor that the Landsberg term vanishes for the specific Finsler functions employed. Without such checks the solved equations rest on an unconfirmed premise and the claimed impact does not follow.

minor comments (1)

The abstract refers to 'analytic arguments' establishing the reduction but supplies no outline of the key steps or error-term estimates; adding a short roadmap in the introduction would improve accessibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and valuable feedback on our manuscript. We address the major comment point by point below.

read point-by-point responses

Referee: [Application to cosmological Landsberg spacetimes] Application section (demonstration for homogeneous isotropic Landsberg-type functions): the reduction theorem is invoked to solve the equations, yet the abstract (and by extension the demonstration) provides no explicit verification that a suitable n exists making F^n sufficiently regular with non-degenerate g_ij on the light cones, nor that the Landsberg term vanishes for the specific Finsler functions employed. Without such checks the solved equations rest on an unconfirmed premise and the claimed impact does not follow.

Authors: We agree that the application section would benefit from explicit verification of the conditions for the specific Finsler functions used in the homogeneous isotropic Landsberg spacetimes. In the revised version of the manuscript, we will add the necessary checks to confirm that there exists an appropriate n such that F^n is sufficiently regular with non-degenerate associated metric on the light cones, and that the Landsberg term vanishes for these functions. This will substantiate the application of the reduction theorem and clarify the impact of our results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; reduction theorem derived via analytic arguments independent of self-citation

full rationale

The paper presents a reduction theorem obtained through analytic arguments on the Finsler gravity equations, requiring existence of suitable F^n with non-degenerate metric on light cones plus vanishing Landsberg term. This generalizes but does not rely on the prior Villasenor2024 result for its validity. No fitted inputs renamed as predictions, no self-definitional loops, and no load-bearing self-citation chain; the central claim has independent mathematical content. The application to Landsberg spacetimes invokes the theorem under its stated hypotheses without reducing to input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only; the paper relies on standard assumptions of Finsler geometry (smoothness away from zero section, positive homogeneity) and the existence of a power F^n with the stated regularity properties. No free parameters or invented entities are mentioned.

axioms (2)

standard math Finsler function satisfies standard regularity and homogeneity properties of Finsler geometry
Invoked implicitly as background for the entire Finsler gravity setup.
domain assumption Existence of power F^n that is sufficiently regular with non-degenerate metric on light cones
This is the key condition stated in the abstract for the reduction to hold.

pith-pipeline@v0.9.1-grok · 5768 in / 1466 out tokens · 19255 ms · 2026-06-25T23:17:27.512437+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

73 extracted references · 1 canonical work pages

[1]

It is1-homogeneous,F(x, λ˙x)=λF(x,˙x)∀(x,˙x)∈T, λ>0
[2]

In other words, every time that a vectorξ=ξ µ ˙∂µ is tangent toΣ x, meaning thatF(x,˙x)=1 andξ µF⋅µ(x,˙x)=0, one has F⋅µ⋅ν(x,˙x)ξ µξν <0.(3)

At eachx∈M, let theindicatrixofFbeΣ x∶={˙x∈Tx∶F(x,˙x)=1}; on each indicatrix, the (vertical) Hessian F⋅µ⋅ν= ∂ ∂˙xµ ∂ ∂˙xν F(2) is negative definite. In other words, every time that a vectorξ=ξ µ ˙∂µ is tangent toΣ x, meaning thatF(x,˙x)=1 andξ µF⋅µ(x,˙x)=0, one has F⋅µ⋅ν(x,˙x)ξ µξν <0.(3)
[3]

4.Fextends continuously as0to the boundary∂TofT

EachT x is connected andF>0in all ofT. 4.Fextends continuously as0to the boundary∂TofT. This means that at eachx∈M, we require F(x,˙x)=0∀˙x∈(∂T) x ⊂TxM∖{0}.(4) TakingL=F 2 and focusing on 1 T, our definition here agrees with the one presented in [28]. However, it differs from other definitions in the literature [12, 17, 19, 20, 34, 35]. These different de...
[4]

ensures the reparameterization invariance of the length integral for curvesγ, S[γ]=∫dτ F(γ,˙γ),(6) which has the well-known interpretation of proper time along worldlines inM
[5]

They are interpreted as the set oftimelike vectors, and the sets Σx are the sets ofunit timelike vectorsatx∈M

guarantees that the conical setsT x are convex. They are interpreted as the set oftimelike vectors, and the sets Σx are the sets ofunit timelike vectorsatx∈M. They can be understood as the Finsler generalizations of the usual hyperboloids that are known from special and general relativity
[6]

It also says that no hypothetical lightlike or spacelike vector has been included in this domain; consequently, the pair(T, F)contains exactly the same information as Σ∶=⋃ x∈M Σx

states that among the components that might constitute the timelike cones atx, one is selected and identified as future pointing; this constitutes a smooth time orientation. It also says that no hypothetical lightlike or spacelike vector has been included in this domain; consequently, the pair(T, F)contains exactly the same information as Σ∶=⋃ x∈M Σx. Thu...
[7]

distance

ensures the existence of non-trivial null (or lightlike) directions that are the boundary of the timelike directions, the so-called light cone. In general, these might not be all null directions that exist. For example, in the case of spacetimes with multiple light cones, which describe the phenomenon of birefringence [36], further null directions exist. ...
[8]

This means that the curvature scalar is derived from thecanonicalnon-linear connection

The purely metric approach: As an action for the Finsler functionF(or any of its powersF n) alone,S=S[F]. This means that the curvature scalar is derived from thecanonicalnon-linear connection
[9]

This means that, while the volume form is defined in terms ofF, the curvature scalar is constructed from theindependentnon-linear connection coefficients ¯N µν

The Palatini approach: As an action for the Finsler functionFand an independent non-linear connection ¯N, namelyS=S[F, ¯N]. This means that, while the volume form is defined in terms ofF, the curvature scalar is constructed from theindependentnon-linear connection coefficients ¯N µν. We shall write this scalar as ¯R(its construction being formally identic...
[10]

The purely metric field equation: g(2)µν˙∂µ ˙∂νR− 6 F 2 R+2g (2)µν(2 ˙∂µ∇Pν−∇(˙∂µPν)) =0.(17)
[11]

6 Both types of field equations possess a part which contains only the curvature tensors

The Palatini field equations: 0=g (2)µν˙∂µ ˙∂ν ¯R− 6 F 2 ¯R,(18) 0=(−2Pµ+(6 ˙xν F 2 −2Cν)J ν µ−(˙∂νJ ν µ+ ˙∂µJ ν ν))(δ µ λ ˙xτ −δτ λ ˙xµ)−(˙∂ρJ τ λ−˙∂λJ τ ρ)˙xρ ,(19) whereJ µν = ¯N µν −Nµν is the difference between the connection coefficients of the independent non-linear connection and the canonical Cartan ones. 6 Both types of field equations possess a...
[12]

The interesting observation here is that the first term in (42) depends onF n, while in the second,uis defined fromF 2

The metric Finsler gravity equation Rewriting (39) in terms ofu, we find that for anyn, except 0 and 1, the metric gravity equation can be cast as follows, using for example (26), 0= n 2 F n−2g(n)µν˙∂µ ˙∂νR+ 2(1−2n) n−1 u(42) = n 2 F ng(n)µν˙∂µ ˙∂νu+2u ,(43) which we interpret as a partial differential equation determining the ˙x-dependence ofu. The inter...

1927
[13]

In the Palatini approach, see Section II C, there exist two independent variables, namely the independent non-linear connection coefficients ¯N µν and the Finsler functionF

The Palatini Finsler gravity equation In order to conclude that the metric Finsler gravity equation (39) reduces toR=0, we required certain regularity conditions onF n. In the Palatini approach, see Section II C, there exist two independent variables, namely the independent non-linear connection coefficients ¯N µν and the Finsler functionF. Hence, to conc...
[14]

Lemma 2.Assume thatFsolves the metric Finslerian vacuum equation inTwithg (n)µνPµν =0, which means that (39)holds

The metric Finsler gravity equation We first establish the following lemma; it leads to the regularity condition(b)in Section IV A. Lemma 2.Assume thatFsolves the metric Finslerian vacuum equation inTwithg (n)µνPµν =0, which means that (39)holds. Suppose that for somen∈R∖{0,1}andB⊆∂T, the functionF n extends smoothly toT∪Bwithg (n) µν∣ B non-degenerate. I...
[15]

The Palatini Finsler gravity equation Here, we state the Palatini version of Theorem 3, in a similar fashion as how Theorem 2 is the Palatini version of Theorem 1. Again, the proof consists in repeating the steps from the previous subsubsection, basically word by word, replacing theF-based curvature scalarRby the ¯Rconstructed from an independent non-line...
[16]

Reduction of the equation In order to apply Theorem 3, we need to study the non-degeneracy ofg (n)and the smoothness ofF n uni itself. 17 As the parameterfmust be negative, it is convenient to introduceQ 2∶=−f, and to consider then-th power ofFuni withn=(Q 2+1)N: F (Q2+1)N uni = (−˙tQ2−a(t)w) Q2N (a(t)w−˙t) N ,(87) whereN>0 is another arbitrary parameter....
[17]

Consider the first light cone, ˙t=a(t)w, that can be interpreted as future pointing lightlike directions, as long as a(t)w>0 and thus ˙t>0. The determinant detg ((Q2+1)N) µν in (88) is non-vanishing and non-diverging there for N=1, for which F (Q2+1) uni = (−˙tQ2−a(t)w) Q2 (a(t)w−˙t) ,(90) which clearly is smooth (C ∞) on the causal cone under considerati...
[18]

The same argument holds when we consider the second light cone, ˙t=−a(t)w/Q 2, that can be interpreted as past pointing lightlike directions as ˙t<0 as long asa(t)w>0, for another choice ofN. The determinant (88) is non-vanishing and non-diverging there forN=1/Q 2, for which F (Q2+1)/Q2 uni = (−˙tQ2−a(t)w)( a(t)w−˙t) 1/Q2 .(91) Analogously as the previous...
[19]

Solving the Finsler gravity equation Spatially homogeneous and isotropic Finsler spacetimes have been studied in quite some detail [33, 43, 53]. Using the results from [33, (B.28)], we can write the curvature scalar for homogeneous and isotropic Finsler functions of the type F 2 = ˙t2h(t, s)2, s∶= w ˙t ,(93) as R w2 = Rµµν ˙xν w2 =−2κ−3h′′′2˙h′2 s24h′′4−9...
[20]

F. F. Villase˜ nor,Lorentz-Finsler geometry and Einstein equations, Phd thesis, University of Granada, Spain (2024), avail- able athttps://digibug.ugr.es/handle/10481/94842

2024
[21]

Leauthaud and A

A. Leauthaud and A. Riess, Looking beyond lambda, Nature Astron.9, 1123 (2025), arXiv:2509.00359 [astro-ph.CO]

arXiv 2025
[22]

Capozziello, H

S. Capozziello, H. Chaudhary, T. Harko, and G. Mustafa, Is dark energy dynamical in the DESI era? A critical review, Phys. Dark Univ.51, 102196 (2026), arXiv:2512.10585 [astro-ph.CO]

arXiv 2026
[23]

J. A. Devlin, Dark matter: what is it, and can quantum sensors help find it?, Contemp. Phys.65, 239 (2024)

2024
[24]

Bertone and D

G. Bertone and D. Hooper, History of dark matter, Rev. Mod. Phys.90, 045002 (2018), arXiv:1605.04909 [astro-ph.CO]

Pith/arXiv arXiv 2018
[25]

E. Abdallaet al., Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies, JHEAp34, 49 (2022), arXiv:2203.06142 [astro-ph.CO]

Pith/arXiv arXiv 2022
[26]

Di Valentinoet al.(CosmoVerse Network), The CosmoVerse White Paper: Addressing observational tensions in cos- mology with systematics and fundamental physics, Phys

E. Di Valentinoet al.(CosmoVerse Network), The CosmoVerse White Paper: Addressing observational tensions in cos- mology with systematics and fundamental physics, Phys. Dark Univ.49, 101965 (2025), arXiv:2504.01669 [astro-ph.CO]

Pith/arXiv arXiv 2025
[27]

Roger Penrose, Gravitational Collapse and Space-Time Singularities, Physical Review Letters14, 57 (1965)

1965
[28]

Hawking and Roger Penrose, The Singularities of Gravitational Collapse and Cosmology, Proceedings of the Royal Society of London

Stephen W. Hawking and Roger Penrose, The Singularities of Gravitational Collapse and Cosmology, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences314, 529 (1970)

1970
[29]

Addaziet al., Quantum gravity phenomenology at the dawn of the multi-messenger era—A review, Prog

A. Addaziet al., Quantum gravity phenomenology at the dawn of the multi-messenger era—A review, Prog. Part. Nucl. Phys.125, 103948 (2022), arXiv:2111.05659 [hep-ph]

arXiv 2022
[30]

Buoninfanteet al., Visions in quantum gravity, SciPost Phys

L. Buoninfanteet al., Visions in quantum gravity, SciPost Phys. Comm. Rep. , 11 (2024), arXiv:2412.08696 [hep-th]

arXiv 2024
[31]

J. K. Beem, Indefinite Finsler Spaces and Timelike Spaces, Canadian Journal of Mathematics22, 1035–1039 (1970)

1970
[32]

G. S. Asanov,Finsler Geometry, Relativity and Gauge Theories, Fundamental Theories of Physics, Vol. 12 (D. Reidel Publishing Company, Dordrecht, Netherlands, 1985)

1985
[33]

S. F. Rutz, A Finsler generalisation of Einstein’s vacuum field equations, General Relativity and Gravitation25, 1139 (1993)

1993
[34]

A. P. Kouretsis, M. Stathakopoulos, and P. C. Stavrinos, The General Very Special Relativity in Finsler Cosmology, Phys. Rev. D79, 104011 (2009), arXiv:0810.3267 [gr-qc]. 21

Pith/arXiv arXiv 2009
[35]

Chang and X

Z. Chang and X. Li, Modified Friedmann model in Randers-Finsler space of approximate Berwald type as a possible alternative to dark energy hypothesis, Physics Letters B676, 173 (2009), arXiv:0901.1023

Pith/arXiv arXiv 2009
[36]

and S´ anchez, Miguel, Finsler metrics and relativistic spacetimes, International Journal of Geometric Methods in Modern Physics11, 1460032 (2014)

Javaloyes, Miguel A. and S´ anchez, Miguel, Finsler metrics and relativistic spacetimes, International Journal of Geometric Methods in Modern Physics11, 1460032 (2014)

2014
[37]

Stavrinos and O

P. Stavrinos and O. Vacaru and S. I. Vacaru, Modified Einstein and Finsler like theories on tangent Lorentz bundles, International Journal of Modern Physics D23, 1450094 (2014), arXiv:1401.2879

Pith/arXiv arXiv 2014
[38]

Caponio and G

E. Caponio and G. Stancarone, Standard static Finsler spacetimes, Int. J. Geom. Meth. Mod. Phys.13, 1650040 (2016), arXiv:1506.07451 [math.DG]

Pith/arXiv arXiv 2016
[39]

Minguzzi, An Equivalence of Finslerian Relativistic Theories, Reports on Mathematical Physics77, 45 (2016)

E. Minguzzi, An Equivalence of Finslerian Relativistic Theories, Reports on Mathematical Physics77, 45 (2016)

2016
[41]

L¨ ammerzahl and V

C. L¨ ammerzahl and V. Perlick, Finsler geometry as a model for relativistic gravity, Int. J. Geom. Meth. Mod. Phys.15, 1850166 (2018), arXiv:1802.10043 [gr-qc]

Pith/arXiv arXiv 2018
[42]

M. A. Javaloyes, M. S´ anchez, and F. F. Villase˜ nor, The Einstein-Hilbert-Palatini formalism in Pseudo-Finsler Geometry, Adv. Theor. Math. Phys.26, No. 10, pp., 3563 (2022), arXiv:2108.03197 [math.DG]

arXiv 2022
[43]

Heefer,Finsler Geometry, Spacetime & Gravity – From Metrizability of Berwald Spaces to Exact Vacuum Solutions in Finsler Gravity, Other thesis, (2024), arXiv:2404.09858 [gr-qc]

S. Heefer,Finsler Geometry, Spacetime & Gravity – From Metrizability of Berwald Spaces to Exact Vacuum Solutions in Finsler Gravity, Other thesis, (2024), arXiv:2404.09858 [gr-qc]

arXiv 2024
[44]

S´ anchez, On the foundations and applications of Lorentz-Finsler Geometry, (2025), arXiv:2511.04645 [gr-qc]

M. S´ anchez, On the foundations and applications of Lorentz-Finsler Geometry, (2025), arXiv:2511.04645 [gr-qc]

arXiv 2025
[45]

Bao, S.-S

D. Bao, S.-S. Chern, and Z. Shen,An Introduction to Finsler-Riemann Geometry(Springer, New York, 2000)

2000
[46]

and Bucataru, I.,Finsler Lagrange geometry(Editura Academiei Romane, 2007)

Miron, R. and Bucataru, I.,Finsler Lagrange geometry(Editura Academiei Romane, 2007)

2007
[47]

Hohmann, C

M. Hohmann, C. Pfeifer, and N. Voicu, Mathematical foundations for field theories on Finsler spacetimes, J. Math. Phys. 63, 032503 (2022), arXiv:2106.14965 [math-ph]

arXiv 2022
[48]

Nagasawa

A. Garc´ ıa-Parrado and E. Minguzzi, An anisotropic gravity theory, General Relativity and Gravitation54, 10.1007/s10714- 022-03039-7 (2022)

work page doi:10.1007/s10714- 2022
[49]

Hohmann, C

M. Hohmann, C. Pfeifer, and N. Voicu, Finsler gravity action from variational completion, Phys. Rev. D100, 064035 (2019), arXiv:1812.11161 [gr-qc]

arXiv 2019
[50]

Hohmann, C

M. Hohmann, C. Pfeifer, and N. Voicu, The kinetic gas universe, Eur. Phys. J. C80, 809 (2020), arXiv:2005.13561 [gr-qc]

arXiv 2020
[51]

Hohmann, C

M. Hohmann, C. Pfeifer, and N. Voicu, Relativistic kinetic gases as direct sources of gravity, Phys. Rev. D101, 024062 (2020), arXiv:1910.14044 [gr-qc]

arXiv 2020
[52]

Pfeifer, N

C. Pfeifer, N. Voicu, A. Friedl-Sz´ asz, and E. Popovici-Popescu, From kinetic gases to an exponentially expanding universe — the Finsler-Friedmann equation, Journal of Cosmology and Astroparticle Physics2025(10), 050
[53]

M. A. Javaloyes and M. S´ anchez, On the definition and examples of cones and Finsler spacetimes, RACSAM114, article 30 (2020), arXiv:1805.06978 [math.DG]

arXiv 2020
[54]

L¨ ammerzahl, V

C. L¨ ammerzahl, V. Perlick, and W. Hasse, Observable effects in a class of spherically symmetric static Finsler spacetimes, Phys. Rev. D86, 104042 (2012)

2012
[55]

Pfeifer, Finsler spacetime geometry in Physics, Int

C. Pfeifer, Finsler spacetime geometry in Physics, Int. J. Geom. Meth. Mod. Phys.16, 1941004 (2019), arXiv:1903.10185 [gr-qc]

arXiv 2019
[56]

Asanov, Finsleroid-Finsler spaces of positive-definite and relativistic types, Rep

G. Asanov, Finsleroid-Finsler spaces of positive-definite and relativistic types, Rep. Math. Phys.58, 275 (2006)

2006
[57]

Bao, David, On two curvature-driven problems in Riemann-Finsler geometry, inFinsler Geometry, Sapporo 2005: In Memory of Makoto Matsumoto, Adv. Stud. Pure Math., Vol. 48 (Mathematical Society of Japan, 2007) pp. 19–71

2005
[58]

Shen, On a class of Landsberg metrics in Finsler geometry, Canadian Journal of Mathematics61, 1357–1374 (2009)

Z. Shen, On a class of Landsberg metrics in Finsler geometry, Canadian Journal of Mathematics61, 1357–1374 (2009)

2009
[59]

Elgendi, On the problem of non-Berwaldian Landsberg spaces, Bulletin of the Australian Mathematical Society102, 331 (2020)

S. Elgendi, On the problem of non-Berwaldian Landsberg spaces, Bulletin of the Australian Mathematical Society102, 331 (2020)

2020
[60]

and Kozma, L´ aszl´ o,(α, β)-Metrics Satisfying theT-Condition or theσT-Condition, The Journal of Geometric Analysis31, 7866 (2021)

Elgendi, Salah G. and Kozma, L´ aszl´ o,(α, β)-Metrics Satisfying theT-Condition or theσT-Condition, The Journal of Geometric Analysis31, 7866 (2021)

2021
[61]

Elgendi, Solutions for the Landsberg unicorn problem in Finsler geometry, Journal of Geometry and Physics159, 103918 (2021)

S. Elgendi, Solutions for the Landsberg unicorn problem in Finsler geometry, Journal of Geometry and Physics159, 103918 (2021)

2021
[62]

Heefer, Sjors and Pfeifer, Christian and Reggio, Antonio and Fuster, Andrea, A Cosmological unicorn solution to Finsler gravity, Phys. Rev. D108, 064051 (2023)

2023
[63]

Ikeda, On the theory of fields in Finsler spaces, Journal of Mathematical Physics22, 1215 (1981)

S. Ikeda, On the theory of fields in Finsler spaces, Journal of Mathematical Physics22, 1215 (1981)

1981
[64]

Voicu, New considerations on Hilbert action and Einstein equations in anisotropic spaces, inAIP Conference Proceedings, Vol

N. Voicu, New considerations on Hilbert action and Einstein equations in anisotropic spaces, inAIP Conference Proceedings, Vol. 1283 (2010) pp. 249–257, arXiv:0911.5034

Pith/arXiv arXiv 2010
[65]

Pfeifer and M

C. Pfeifer and M. N. R. Wohlfarth, Finsler geometric extension of Einstein gravity, Phys. Rev. D85, 064009 (2012), arXiv:1112.5641 [gr-qc]

Pith/arXiv arXiv 2012
[66]

Pfeifer and M

C. Pfeifer and M. N. R. Wohlfarth, Causal structure and electrodynamics on Finsler spacetimes, Phys. Rev. D84, 044039 (2011), arXiv:1104.1079 [gr-qc]

Pith/arXiv arXiv 2011
[67]

Eberhard Hopf, Elementare Bemerkungen ¨ uber die L¨ osungen partieller Differentialgleichungen zweiter Ordnung vom el- liptischen Typus, Sitzungsberichte der Preußischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse19, 147 (1927)

1927
[68]

Gilbarg and N

D. Gilbarg and N. S. Trudinger,Elliptic partial differential equations of second order, reprint of the 1998 ed. (Springer, 2001)

1998
[69]

Fuster, Andrea and Pabst, Cornelia and Pfeifer, Christian, Berwald spacetimes and very special relativity, Phys. Rev. D 98, 084062 (2018)

2018
[70]

Pfeifer, S

C. Pfeifer, S. Heefer, and A. Fuster, Identifying Berwald Finsler geometries, Differential Geometry and its Applications 22 79, 101817 (2021)

2021
[71]

Friedl-Sz´ asz, E

A. Friedl-Sz´ asz, E. Popovici-Popescu, N. Voicu, C. Pfeifer, and S. Heefer, Cosmological Landsberg-Finsler spacetimes, Phys. Rev. D111, 044058 (2025), arXiv:2410.18197 [math-ph]

arXiv 2025
[72]

Hohmann, C

M. Hohmann, C. Pfeifer, and N. Voicu, Cosmological Finsler Spacetimes, Universe6, 65 (2020), arXiv:2003.02299 [gr-qc]

arXiv 2020
[73]

J. M. Lee,Introduction to smooth manifolds, 2nd ed. (Springer, 2012)

2012
[74]

Krantz and Harold R

Steven G. Krantz and Harold R. Parks,A Primer of Real Analytic Functions, 2nd ed., Birkh¨ auser Advanced Texts: Basler Lehrb¨ ucher (Birkh¨ auser Boston, 2012)

2012

[1] [1]

It is1-homogeneous,F(x, λ˙x)=λF(x,˙x)∀(x,˙x)∈T, λ>0

[2] [2]

In other words, every time that a vectorξ=ξ µ ˙∂µ is tangent toΣ x, meaning thatF(x,˙x)=1 andξ µF⋅µ(x,˙x)=0, one has F⋅µ⋅ν(x,˙x)ξ µξν <0.(3)

At eachx∈M, let theindicatrixofFbeΣ x∶={˙x∈Tx∶F(x,˙x)=1}; on each indicatrix, the (vertical) Hessian F⋅µ⋅ν= ∂ ∂˙xµ ∂ ∂˙xν F(2) is negative definite. In other words, every time that a vectorξ=ξ µ ˙∂µ is tangent toΣ x, meaning thatF(x,˙x)=1 andξ µF⋅µ(x,˙x)=0, one has F⋅µ⋅ν(x,˙x)ξ µξν <0.(3)

[3] [3]

4.Fextends continuously as0to the boundary∂TofT

EachT x is connected andF>0in all ofT. 4.Fextends continuously as0to the boundary∂TofT. This means that at eachx∈M, we require F(x,˙x)=0∀˙x∈(∂T) x ⊂TxM∖{0}.(4) TakingL=F 2 and focusing on 1 T, our definition here agrees with the one presented in [28]. However, it differs from other definitions in the literature [12, 17, 19, 20, 34, 35]. These different de...

[4] [4]

ensures the reparameterization invariance of the length integral for curvesγ, S[γ]=∫dτ F(γ,˙γ),(6) which has the well-known interpretation of proper time along worldlines inM

[5] [5]

They are interpreted as the set oftimelike vectors, and the sets Σx are the sets ofunit timelike vectorsatx∈M

guarantees that the conical setsT x are convex. They are interpreted as the set oftimelike vectors, and the sets Σx are the sets ofunit timelike vectorsatx∈M. They can be understood as the Finsler generalizations of the usual hyperboloids that are known from special and general relativity

[6] [6]

It also says that no hypothetical lightlike or spacelike vector has been included in this domain; consequently, the pair(T, F)contains exactly the same information as Σ∶=⋃ x∈M Σx

states that among the components that might constitute the timelike cones atx, one is selected and identified as future pointing; this constitutes a smooth time orientation. It also says that no hypothetical lightlike or spacelike vector has been included in this domain; consequently, the pair(T, F)contains exactly the same information as Σ∶=⋃ x∈M Σx. Thu...

[7] [7]

distance

ensures the existence of non-trivial null (or lightlike) directions that are the boundary of the timelike directions, the so-called light cone. In general, these might not be all null directions that exist. For example, in the case of spacetimes with multiple light cones, which describe the phenomenon of birefringence [36], further null directions exist. ...

[8] [8]

This means that the curvature scalar is derived from thecanonicalnon-linear connection

The purely metric approach: As an action for the Finsler functionF(or any of its powersF n) alone,S=S[F]. This means that the curvature scalar is derived from thecanonicalnon-linear connection

[9] [9]

This means that, while the volume form is defined in terms ofF, the curvature scalar is constructed from theindependentnon-linear connection coefficients ¯N µν

The Palatini approach: As an action for the Finsler functionFand an independent non-linear connection ¯N, namelyS=S[F, ¯N]. This means that, while the volume form is defined in terms ofF, the curvature scalar is constructed from theindependentnon-linear connection coefficients ¯N µν. We shall write this scalar as ¯R(its construction being formally identic...

[10] [10]

The purely metric field equation: g(2)µν˙∂µ ˙∂νR− 6 F 2 R+2g (2)µν(2 ˙∂µ∇Pν−∇(˙∂µPν)) =0.(17)

[11] [11]

6 Both types of field equations possess a part which contains only the curvature tensors

The Palatini field equations: 0=g (2)µν˙∂µ ˙∂ν ¯R− 6 F 2 ¯R,(18) 0=(−2Pµ+(6 ˙xν F 2 −2Cν)J ν µ−(˙∂νJ ν µ+ ˙∂µJ ν ν))(δ µ λ ˙xτ −δτ λ ˙xµ)−(˙∂ρJ τ λ−˙∂λJ τ ρ)˙xρ ,(19) whereJ µν = ¯N µν −Nµν is the difference between the connection coefficients of the independent non-linear connection and the canonical Cartan ones. 6 Both types of field equations possess a...

[12] [12]

The interesting observation here is that the first term in (42) depends onF n, while in the second,uis defined fromF 2

The metric Finsler gravity equation Rewriting (39) in terms ofu, we find that for anyn, except 0 and 1, the metric gravity equation can be cast as follows, using for example (26), 0= n 2 F n−2g(n)µν˙∂µ ˙∂νR+ 2(1−2n) n−1 u(42) = n 2 F ng(n)µν˙∂µ ˙∂νu+2u ,(43) which we interpret as a partial differential equation determining the ˙x-dependence ofu. The inter...

1927

[13] [13]

In the Palatini approach, see Section II C, there exist two independent variables, namely the independent non-linear connection coefficients ¯N µν and the Finsler functionF

The Palatini Finsler gravity equation In order to conclude that the metric Finsler gravity equation (39) reduces toR=0, we required certain regularity conditions onF n. In the Palatini approach, see Section II C, there exist two independent variables, namely the independent non-linear connection coefficients ¯N µν and the Finsler functionF. Hence, to conc...

[14] [14]

Lemma 2.Assume thatFsolves the metric Finslerian vacuum equation inTwithg (n)µνPµν =0, which means that (39)holds

The metric Finsler gravity equation We first establish the following lemma; it leads to the regularity condition(b)in Section IV A. Lemma 2.Assume thatFsolves the metric Finslerian vacuum equation inTwithg (n)µνPµν =0, which means that (39)holds. Suppose that for somen∈R∖{0,1}andB⊆∂T, the functionF n extends smoothly toT∪Bwithg (n) µν∣ B non-degenerate. I...

[15] [15]

The Palatini Finsler gravity equation Here, we state the Palatini version of Theorem 3, in a similar fashion as how Theorem 2 is the Palatini version of Theorem 1. Again, the proof consists in repeating the steps from the previous subsubsection, basically word by word, replacing theF-based curvature scalarRby the ¯Rconstructed from an independent non-line...

[16] [16]

Reduction of the equation In order to apply Theorem 3, we need to study the non-degeneracy ofg (n)and the smoothness ofF n uni itself. 17 As the parameterfmust be negative, it is convenient to introduceQ 2∶=−f, and to consider then-th power ofFuni withn=(Q 2+1)N: F (Q2+1)N uni = (−˙tQ2−a(t)w) Q2N (a(t)w−˙t) N ,(87) whereN>0 is another arbitrary parameter....

[17] [17]

Consider the first light cone, ˙t=a(t)w, that can be interpreted as future pointing lightlike directions, as long as a(t)w>0 and thus ˙t>0. The determinant detg ((Q2+1)N) µν in (88) is non-vanishing and non-diverging there for N=1, for which F (Q2+1) uni = (−˙tQ2−a(t)w) Q2 (a(t)w−˙t) ,(90) which clearly is smooth (C ∞) on the causal cone under considerati...

[18] [18]

The same argument holds when we consider the second light cone, ˙t=−a(t)w/Q 2, that can be interpreted as past pointing lightlike directions as ˙t<0 as long asa(t)w>0, for another choice ofN. The determinant (88) is non-vanishing and non-diverging there forN=1/Q 2, for which F (Q2+1)/Q2 uni = (−˙tQ2−a(t)w)( a(t)w−˙t) 1/Q2 .(91) Analogously as the previous...

[19] [19]

Solving the Finsler gravity equation Spatially homogeneous and isotropic Finsler spacetimes have been studied in quite some detail [33, 43, 53]. Using the results from [33, (B.28)], we can write the curvature scalar for homogeneous and isotropic Finsler functions of the type F 2 = ˙t2h(t, s)2, s∶= w ˙t ,(93) as R w2 = Rµµν ˙xν w2 =−2κ−3h′′′2˙h′2 s24h′′4−9...

[20] [20]

F. F. Villase˜ nor,Lorentz-Finsler geometry and Einstein equations, Phd thesis, University of Granada, Spain (2024), avail- able athttps://digibug.ugr.es/handle/10481/94842

2024

[21] [21]

Leauthaud and A

A. Leauthaud and A. Riess, Looking beyond lambda, Nature Astron.9, 1123 (2025), arXiv:2509.00359 [astro-ph.CO]

arXiv 2025

[22] [22]

Capozziello, H

S. Capozziello, H. Chaudhary, T. Harko, and G. Mustafa, Is dark energy dynamical in the DESI era? A critical review, Phys. Dark Univ.51, 102196 (2026), arXiv:2512.10585 [astro-ph.CO]

arXiv 2026

[23] [23]

J. A. Devlin, Dark matter: what is it, and can quantum sensors help find it?, Contemp. Phys.65, 239 (2024)

2024

[24] [24]

Bertone and D

G. Bertone and D. Hooper, History of dark matter, Rev. Mod. Phys.90, 045002 (2018), arXiv:1605.04909 [astro-ph.CO]

Pith/arXiv arXiv 2018

[25] [25]

E. Abdallaet al., Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies, JHEAp34, 49 (2022), arXiv:2203.06142 [astro-ph.CO]

Pith/arXiv arXiv 2022

[26] [26]

Di Valentinoet al.(CosmoVerse Network), The CosmoVerse White Paper: Addressing observational tensions in cos- mology with systematics and fundamental physics, Phys

E. Di Valentinoet al.(CosmoVerse Network), The CosmoVerse White Paper: Addressing observational tensions in cos- mology with systematics and fundamental physics, Phys. Dark Univ.49, 101965 (2025), arXiv:2504.01669 [astro-ph.CO]

Pith/arXiv arXiv 2025

[27] [27]

Roger Penrose, Gravitational Collapse and Space-Time Singularities, Physical Review Letters14, 57 (1965)

1965

[28] [28]

Hawking and Roger Penrose, The Singularities of Gravitational Collapse and Cosmology, Proceedings of the Royal Society of London

Stephen W. Hawking and Roger Penrose, The Singularities of Gravitational Collapse and Cosmology, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences314, 529 (1970)

1970

[29] [29]

Addaziet al., Quantum gravity phenomenology at the dawn of the multi-messenger era—A review, Prog

A. Addaziet al., Quantum gravity phenomenology at the dawn of the multi-messenger era—A review, Prog. Part. Nucl. Phys.125, 103948 (2022), arXiv:2111.05659 [hep-ph]

arXiv 2022

[30] [30]

Buoninfanteet al., Visions in quantum gravity, SciPost Phys

L. Buoninfanteet al., Visions in quantum gravity, SciPost Phys. Comm. Rep. , 11 (2024), arXiv:2412.08696 [hep-th]

arXiv 2024

[31] [31]

J. K. Beem, Indefinite Finsler Spaces and Timelike Spaces, Canadian Journal of Mathematics22, 1035–1039 (1970)

1970

[32] [32]

G. S. Asanov,Finsler Geometry, Relativity and Gauge Theories, Fundamental Theories of Physics, Vol. 12 (D. Reidel Publishing Company, Dordrecht, Netherlands, 1985)

1985

[33] [33]

S. F. Rutz, A Finsler generalisation of Einstein’s vacuum field equations, General Relativity and Gravitation25, 1139 (1993)

1993

[34] [34]

A. P. Kouretsis, M. Stathakopoulos, and P. C. Stavrinos, The General Very Special Relativity in Finsler Cosmology, Phys. Rev. D79, 104011 (2009), arXiv:0810.3267 [gr-qc]. 21

Pith/arXiv arXiv 2009

[35] [35]

Chang and X

Z. Chang and X. Li, Modified Friedmann model in Randers-Finsler space of approximate Berwald type as a possible alternative to dark energy hypothesis, Physics Letters B676, 173 (2009), arXiv:0901.1023

Pith/arXiv arXiv 2009

[36] [36]

and S´ anchez, Miguel, Finsler metrics and relativistic spacetimes, International Journal of Geometric Methods in Modern Physics11, 1460032 (2014)

Javaloyes, Miguel A. and S´ anchez, Miguel, Finsler metrics and relativistic spacetimes, International Journal of Geometric Methods in Modern Physics11, 1460032 (2014)

2014

[37] [37]

Stavrinos and O

P. Stavrinos and O. Vacaru and S. I. Vacaru, Modified Einstein and Finsler like theories on tangent Lorentz bundles, International Journal of Modern Physics D23, 1450094 (2014), arXiv:1401.2879

Pith/arXiv arXiv 2014

[38] [38]

Caponio and G

E. Caponio and G. Stancarone, Standard static Finsler spacetimes, Int. J. Geom. Meth. Mod. Phys.13, 1650040 (2016), arXiv:1506.07451 [math.DG]

Pith/arXiv arXiv 2016

[39] [39]

Minguzzi, An Equivalence of Finslerian Relativistic Theories, Reports on Mathematical Physics77, 45 (2016)

E. Minguzzi, An Equivalence of Finslerian Relativistic Theories, Reports on Mathematical Physics77, 45 (2016)

2016

[40] [41]

L¨ ammerzahl and V

C. L¨ ammerzahl and V. Perlick, Finsler geometry as a model for relativistic gravity, Int. J. Geom. Meth. Mod. Phys.15, 1850166 (2018), arXiv:1802.10043 [gr-qc]

Pith/arXiv arXiv 2018

[41] [42]

M. A. Javaloyes, M. S´ anchez, and F. F. Villase˜ nor, The Einstein-Hilbert-Palatini formalism in Pseudo-Finsler Geometry, Adv. Theor. Math. Phys.26, No. 10, pp., 3563 (2022), arXiv:2108.03197 [math.DG]

arXiv 2022

[42] [43]

Heefer,Finsler Geometry, Spacetime & Gravity – From Metrizability of Berwald Spaces to Exact Vacuum Solutions in Finsler Gravity, Other thesis, (2024), arXiv:2404.09858 [gr-qc]

S. Heefer,Finsler Geometry, Spacetime & Gravity – From Metrizability of Berwald Spaces to Exact Vacuum Solutions in Finsler Gravity, Other thesis, (2024), arXiv:2404.09858 [gr-qc]

arXiv 2024

[43] [44]

S´ anchez, On the foundations and applications of Lorentz-Finsler Geometry, (2025), arXiv:2511.04645 [gr-qc]

M. S´ anchez, On the foundations and applications of Lorentz-Finsler Geometry, (2025), arXiv:2511.04645 [gr-qc]

arXiv 2025

[44] [45]

Bao, S.-S

D. Bao, S.-S. Chern, and Z. Shen,An Introduction to Finsler-Riemann Geometry(Springer, New York, 2000)

2000

[45] [46]

and Bucataru, I.,Finsler Lagrange geometry(Editura Academiei Romane, 2007)

Miron, R. and Bucataru, I.,Finsler Lagrange geometry(Editura Academiei Romane, 2007)

2007

[46] [47]

Hohmann, C

M. Hohmann, C. Pfeifer, and N. Voicu, Mathematical foundations for field theories on Finsler spacetimes, J. Math. Phys. 63, 032503 (2022), arXiv:2106.14965 [math-ph]

arXiv 2022

[47] [48]

Nagasawa

A. Garc´ ıa-Parrado and E. Minguzzi, An anisotropic gravity theory, General Relativity and Gravitation54, 10.1007/s10714- 022-03039-7 (2022)

work page doi:10.1007/s10714- 2022

[48] [49]

Hohmann, C

M. Hohmann, C. Pfeifer, and N. Voicu, Finsler gravity action from variational completion, Phys. Rev. D100, 064035 (2019), arXiv:1812.11161 [gr-qc]

arXiv 2019

[49] [50]

Hohmann, C

M. Hohmann, C. Pfeifer, and N. Voicu, The kinetic gas universe, Eur. Phys. J. C80, 809 (2020), arXiv:2005.13561 [gr-qc]

arXiv 2020

[50] [51]

Hohmann, C

M. Hohmann, C. Pfeifer, and N. Voicu, Relativistic kinetic gases as direct sources of gravity, Phys. Rev. D101, 024062 (2020), arXiv:1910.14044 [gr-qc]

arXiv 2020

[51] [52]

Pfeifer, N

C. Pfeifer, N. Voicu, A. Friedl-Sz´ asz, and E. Popovici-Popescu, From kinetic gases to an exponentially expanding universe — the Finsler-Friedmann equation, Journal of Cosmology and Astroparticle Physics2025(10), 050

[52] [53]

M. A. Javaloyes and M. S´ anchez, On the definition and examples of cones and Finsler spacetimes, RACSAM114, article 30 (2020), arXiv:1805.06978 [math.DG]

arXiv 2020

[53] [54]

L¨ ammerzahl, V

C. L¨ ammerzahl, V. Perlick, and W. Hasse, Observable effects in a class of spherically symmetric static Finsler spacetimes, Phys. Rev. D86, 104042 (2012)

2012

[54] [55]

Pfeifer, Finsler spacetime geometry in Physics, Int

C. Pfeifer, Finsler spacetime geometry in Physics, Int. J. Geom. Meth. Mod. Phys.16, 1941004 (2019), arXiv:1903.10185 [gr-qc]

arXiv 2019

[55] [56]

Asanov, Finsleroid-Finsler spaces of positive-definite and relativistic types, Rep

G. Asanov, Finsleroid-Finsler spaces of positive-definite and relativistic types, Rep. Math. Phys.58, 275 (2006)

2006

[56] [57]

Bao, David, On two curvature-driven problems in Riemann-Finsler geometry, inFinsler Geometry, Sapporo 2005: In Memory of Makoto Matsumoto, Adv. Stud. Pure Math., Vol. 48 (Mathematical Society of Japan, 2007) pp. 19–71

2005

[57] [58]

Shen, On a class of Landsberg metrics in Finsler geometry, Canadian Journal of Mathematics61, 1357–1374 (2009)

Z. Shen, On a class of Landsberg metrics in Finsler geometry, Canadian Journal of Mathematics61, 1357–1374 (2009)

2009

[58] [59]

Elgendi, On the problem of non-Berwaldian Landsberg spaces, Bulletin of the Australian Mathematical Society102, 331 (2020)

S. Elgendi, On the problem of non-Berwaldian Landsberg spaces, Bulletin of the Australian Mathematical Society102, 331 (2020)

2020

[59] [60]

and Kozma, L´ aszl´ o,(α, β)-Metrics Satisfying theT-Condition or theσT-Condition, The Journal of Geometric Analysis31, 7866 (2021)

Elgendi, Salah G. and Kozma, L´ aszl´ o,(α, β)-Metrics Satisfying theT-Condition or theσT-Condition, The Journal of Geometric Analysis31, 7866 (2021)

2021

[60] [61]

Elgendi, Solutions for the Landsberg unicorn problem in Finsler geometry, Journal of Geometry and Physics159, 103918 (2021)

S. Elgendi, Solutions for the Landsberg unicorn problem in Finsler geometry, Journal of Geometry and Physics159, 103918 (2021)

2021

[61] [62]

Heefer, Sjors and Pfeifer, Christian and Reggio, Antonio and Fuster, Andrea, A Cosmological unicorn solution to Finsler gravity, Phys. Rev. D108, 064051 (2023)

2023

[62] [63]

Ikeda, On the theory of fields in Finsler spaces, Journal of Mathematical Physics22, 1215 (1981)

S. Ikeda, On the theory of fields in Finsler spaces, Journal of Mathematical Physics22, 1215 (1981)

1981

[63] [64]

Voicu, New considerations on Hilbert action and Einstein equations in anisotropic spaces, inAIP Conference Proceedings, Vol

N. Voicu, New considerations on Hilbert action and Einstein equations in anisotropic spaces, inAIP Conference Proceedings, Vol. 1283 (2010) pp. 249–257, arXiv:0911.5034

Pith/arXiv arXiv 2010

[64] [65]

Pfeifer and M

C. Pfeifer and M. N. R. Wohlfarth, Finsler geometric extension of Einstein gravity, Phys. Rev. D85, 064009 (2012), arXiv:1112.5641 [gr-qc]

Pith/arXiv arXiv 2012

[65] [66]

Pfeifer and M

C. Pfeifer and M. N. R. Wohlfarth, Causal structure and electrodynamics on Finsler spacetimes, Phys. Rev. D84, 044039 (2011), arXiv:1104.1079 [gr-qc]

Pith/arXiv arXiv 2011

[66] [67]

Eberhard Hopf, Elementare Bemerkungen ¨ uber die L¨ osungen partieller Differentialgleichungen zweiter Ordnung vom el- liptischen Typus, Sitzungsberichte der Preußischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse19, 147 (1927)

1927

[67] [68]

Gilbarg and N

D. Gilbarg and N. S. Trudinger,Elliptic partial differential equations of second order, reprint of the 1998 ed. (Springer, 2001)

1998

[68] [69]

Fuster, Andrea and Pabst, Cornelia and Pfeifer, Christian, Berwald spacetimes and very special relativity, Phys. Rev. D 98, 084062 (2018)

2018

[69] [70]

Pfeifer, S

C. Pfeifer, S. Heefer, and A. Fuster, Identifying Berwald Finsler geometries, Differential Geometry and its Applications 22 79, 101817 (2021)

2021

[70] [71]

Friedl-Sz´ asz, E

A. Friedl-Sz´ asz, E. Popovici-Popescu, N. Voicu, C. Pfeifer, and S. Heefer, Cosmological Landsberg-Finsler spacetimes, Phys. Rev. D111, 044058 (2025), arXiv:2410.18197 [math-ph]

arXiv 2025

[71] [72]

Hohmann, C

M. Hohmann, C. Pfeifer, and N. Voicu, Cosmological Finsler Spacetimes, Universe6, 65 (2020), arXiv:2003.02299 [gr-qc]

arXiv 2020

[72] [73]

J. M. Lee,Introduction to smooth manifolds, 2nd ed. (Springer, 2012)

2012

[73] [74]

Krantz and Harold R

Steven G. Krantz and Harold R. Parks,A Primer of Real Analytic Functions, 2nd ed., Birkh¨ auser Advanced Texts: Basler Lehrb¨ ucher (Birkh¨ auser Boston, 2012)

2012