Spherically symmetric, asymptotically flat Berwald vacuum solutions in Finsler gravity

Christian Pfeifer; Diana - Maria Birla; Nicoleta Voicu

arxiv: 2606.05427 · v1 · pith:YYMJBWG5new · submitted 2026-06-03 · 🌀 gr-qc · math-ph· math.DG· math.MP

Spherically symmetric, asymptotically flat Berwald vacuum solutions in Finsler gravity

Nicoleta Voicu , Diana - Maria Birla , Christian Pfeifer This is my paper

Pith reviewed 2026-06-28 04:50 UTC · model grok-4.3

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

keywords Finsler gravityBerwald spacetimesspherical symmetryasymptotic flatnessvacuum solutionsnon-Ricci flatalpha-beta metrics

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The pith

Only one class of spherically symmetric Berwald spacetimes yields non-Ricci-flat asymptotically flat vacuum solutions in Finsler gravity.

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

The paper restricts attention to Berwald-Finsler spacetimes in spherical symmetry to solve the vacuum equation of Finsler gravity under asymptotic flatness. It shows that asymptotic flatness together with a well-defined causal structure selects only one class of such spacetimes. Within this class the vacuum equation is solved completely, yielding three families of solutions that are not Ricci flat. These (α,β)-Finsler geometries are built from a pseudo-Riemannian metric plus a one-form and represent the first explicit non-trivial exact spherically symmetric vacuum solutions. Such solutions could describe gravitational fields outside compact objects without reducing to the Riemannian case.

Core claim

Among all spherically symmetric Berwald spacetimes, only one class is compatible with asymptotic flatness and a well defined causal structure. For this class the Finsler gravity vacuum equation is solved completely, producing three families of non-Ricci flat solutions. These solutions are (α,β)-Finsler spacetimes constructed from a pseudo-Riemannian metric and a 1-form, and they represent the first non-trivial exact spherically symmetric vacuum solutions in Finsler gravity.

What carries the argument

The Berwald condition, under which the canonical nonlinear connection defines an affine connection on spacetime, allowing the vacuum equation to be reduced in spherical symmetry with asymptotic flatness.

If this is right

Existence of SO(3)-symmetric asymptotically flat vacuum solutions that are not Ricci flat is established.
The solutions are (α,β)-metrics and can be used to model gravitational fields around compact objects.
Three distinct families of such solutions are obtained explicitly.

Where Pith is reading between the lines

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

The reduction technique might extend to axisymmetric cases or non-vacuum sources.
Non-Ricci-flat character could produce measurable differences in light deflection or orbital motion.
Asymptotic flatness may act as a general filter for admissible Finsler structures.

Load-bearing premise

The spacetimes must be Berwald so that the nonlinear connection defines an affine connection allowing the vacuum equation to be reduced accordingly.

What would settle it

An explicit construction of a spherically symmetric asymptotically flat Berwald spacetime with well-defined causal structure outside the identified class, or a failure to obtain the three families by solving the vacuum equation inside that class.

read the original abstract

So-called Berwald-Finsler spacetimes are Finsler spacetimes that are closest to pseudo-Riemannian geometry, as their canonical nonlinear connection defines an affine connection on spacetime. In spherical symmetry, these geometries can be used to describe the gravitational field outside of compact objects. We solve the Finsler gravity vacuum equation for $SO(3)$-symmetric Berwald spacetimes that are asyptotically flat, but not Ricci flat. We find that among all spherically symmetric Berwald spacetimes, only one class is compatible with asymptotic flatness and a well defined causal structure. For this class, we completely solve the Finsler gravity vacuum equation and find three families of non-Ricci flat solutions -- which represent the first non-trivial, exact spherically symmetric vacuum solutions. They are so-called $(\alpha,\beta)$-Finsler spacetimes that are constructed from a pseudo-Riemannnian metric and a 1-form. In particular, we show, by providing a concrete example, that in Finsler geometry there exist $SO(3)$-symmetric, asymptotically flat vacuum solutions that are not Ricci flat; these solutions are promising candidates to model the gravitational field around compact objects, beyond their Riemannian description.

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

The paper supplies the first explicit non-Ricci-flat spherically symmetric Berwald vacuum solutions in Finsler gravity by solving the vacuum equation under asymptotic flatness.

read the letter

The core result is that only one class of SO(3)-symmetric Berwald spacetimes meets the asymptotic flatness and causal structure requirements, and on that class the authors solve the Finsler vacuum equation to obtain three families of non-Ricci-flat solutions. These are built as (α,β)-Finsler structures from a pseudo-Riemannian metric plus a 1-form.

The work is straightforward in its approach: restrict to Berwald spacetimes so the nonlinear connection reduces to an affine connection, impose spherical symmetry and asymptotic flatness on the connection coefficients, then solve the resulting PDE system. The abstract states they perform the direct substitution check and exhibit a concrete example. That is the actual advance over prior literature, which apparently lacked such explicit non-Ricci-flat vacuum solutions.

The main limitation is the Berwald restriction itself. It is stated up front and is the regime where the equations simplify, but it narrows the claim to a subclass of Finsler geometries. No hidden circularity appears in the logic, and the stress-test note aligns with the abstract's description of the steps.

This is for people working on exact solutions in modified gravity or Finsler extensions of GR who need concrete backgrounds for compact-object modeling. It is grounded enough and supplies falsifiable metric forms, so it deserves a serious referee even if revisions are needed on the explicit components or stability checks.

Referee Report

0 major / 2 minor

Summary. The paper solves the Finsler gravity vacuum equation restricted to SO(3)-symmetric Berwald spacetimes that are asymptotically flat. It identifies a single compatible class under the additional requirements of asymptotic flatness and a well-defined causal structure, then derives three explicit families of non-Ricci-flat solutions. These solutions are (α,β)-Finsler spacetimes built from a pseudo-Riemannian metric plus a 1-form and are presented as the first non-trivial exact spherically symmetric vacuum solutions in Finsler gravity.

Significance. If the derivations hold, the work supplies the first concrete, non-Ricci-flat vacuum solutions in Finsler gravity for spherical symmetry. The explicit construction via Berwald reduction and the provision of a concrete example that satisfies the vacuum equation constitute a clear advance over prior abstract or perturbative treatments, offering candidate metrics for modeling gravitational fields outside compact objects beyond the Riemannian case.

minor comments (2)

Abstract, line 3: 'asyptotically' is a typographical error and should read 'asymptotically'.
The manuscript states that the three families are obtained by direct substitution into the vacuum equation, but the explicit component forms of the Finsler metric or the connection coefficients are not reproduced in the provided abstract; these should appear in the main text with verification steps.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript on spherically symmetric, asymptotically flat Berwald vacuum solutions in Finsler gravity. The recommendation for minor revision is noted. No major comments were provided in the report, so we address the absence of specific points below and will incorporate any minor suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation solves PDE on explicitly restricted domain

full rationale

The paper states the Berwald restriction explicitly as the domain where the nonlinear connection reduces to an affine connection and the vacuum equation simplifies. It then imposes SO(3) symmetry and asymptotic flatness, derives the compatible class from those conditions, and solves the resulting PDE system to obtain three explicit families, verified by substitution. No step reduces by construction to a fitted input, self-citation chain, or ansatz smuggled via prior work; the central claim rests on direct solution of the differential equation rather than re-labeling of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the definition of Berwald spacetimes and the form of the Finsler gravity vacuum equation; no new free parameters or invented entities are introduced in the abstract, but the restriction to Berwald geometry is an upstream modeling choice.

axioms (2)

domain assumption The Finsler spacetime is Berwald, so its canonical nonlinear connection defines an affine connection.
Stated in the first sentence of the abstract as the class under study.
domain assumption The spacetime is SO(3)-symmetric and asymptotically flat with a well-defined causal structure.
Used to select the compatible class before solving the vacuum equation.

pith-pipeline@v0.9.1-grok · 5768 in / 1424 out tokens · 27826 ms · 2026-06-28T04:50:17.332624+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

69 extracted references · 19 linked inside Pith

[1]

Computation ofRand ofa abR⋅ab 15
[2]

Branch 2:a abR⋅ab=0 19 ∗nico.voicu@unitbv.ro † diana.birla@unitbv.ro ‡ christian.pfeifer@zarm.uni-bremen.de arXiv:2606.05427v1 [gr-qc] 3 Jun 2026 2 VI

Integration of the field equation 17 B. Branch 2:a abR⋅ab=0 19 ∗nico.voicu@unitbv.ro † diana.birla@unitbv.ro ‡ christian.pfeifer@zarm.uni-bremen.de arXiv:2606.05427v1 [gr-qc] 3 Jun 2026 2 VI. Concrete example 19 VII. Conclusion and outlook 21 Acknowledgments 22 A. Connection coefficients and curvature components 22 B. Pseudo-Riemannian metric components 2...

Pith/arXiv arXiv 2026
[3]

Positive 2-homogeneity:L(x, λ˙x)=λ 2L(x,˙x),∀λ>0
[4]

We note that any pseudo-Finsler functionLcan be continuously prolonged as 0 at ˙x=0

Nondegeneracy: at any(x,˙x)∈Aand in one (then, in any) local chart around(x,˙x), the Hessian: gab(x,˙x)∶=1 2 ˙∂a ˙∂bL(2) is nonsingular. We note that any pseudo-Finsler functionLcan be continuously prolonged as 0 at ˙x=0. 1 A conic subbundle ofT Mis defined as an open subsetA⊂T M∖{0}with non-empty fibersA x =A∩TxMat all x∈M,and which is stable under posit...
[5]

The elements∈SO(3)leave the coordinatet(usually interpreted as time) invariant. In naturally induced coordinates(x,˙x)∶=(t, r, θ, ϕ,˙t,˙r, ˙θ, ˙ϕ) onT M,any spatially spherically sym- metric Finsler metric is known to be, [53], of the form L(x,˙x)=L(t, r, ˙t,˙r, w), w 2 = ˙θ2+ ˙ϕ2 sin2 θ . Nontrivially Finslerian (i.e., non-quadratic) Berwald metrics with...
[6]

Choosing Θ=1 in (41), one findsL=B 2, where B=e G 2 u=e G 2 ( ˙t−a˙r) .(44) AsBis linear in ˙x, it can be understood asB=b c(x)˙xc, induced by the 1-form b=e G 2 dt+−ae G 2 dr(45) onM. Moreover, as (42)-(43) guarantee that the Berwald conditionδ aL=δ aB2 =0 holds, this 1-form is absolutely parallel with respect to the affine connection onM, i.e., it satis...
[7]

Alternatively, choosing in (41), Θ∶=id., we find thatLis a quadratic function in ˙x: L=A∶=eG (ve−(G−2K)+Mu 2) =ve 2K+(eGM) u2 .(46) The functionAdefines a pseudo-Riemannian metric a=a bc(x)dx b⊗dxc (47) with componentsa bc = ˙∂b ˙∂cA; this is defined on and nondegenerate at least on a subset D⊂M(which can be specified by appropriately choosing the range o...
[8]

The canonical affine connection of a Class 3 Berwald-Finsler functionLis the Levi-Civita connection of its pseudo-Riemannian constitutentA. 13 This Lemma means that geodesics of the Finsler metricLand those ofAare the same assets of points- i.e., judging by their shapes only, we could not tell what is the correct model for our universe - the Finslerian or...
[9]

There holds the identity Rab˜b b =0,(50) where tildes denote raising indices bya ab,that is: ˜b b =a bcbc
[10]

The quantity⟨b,b⟩∶=aabbabb is a constant
[11]

Finally, statement 3

The curvature componentsa 7 anda 11 obey: (ab+c)a 7 =ba 11.(51) Proof.The first result follows immediately from the Ricci identities ˜b c ;a;d−˜b c ;d;a =R c a bd ˜b a applied to the vector field ˜b= ˜b a ∂a and to the fact that this vector field is absolutely parallel with respect to Γ.The second one is also obtained as a direct consequence of the fact t...
[12]

Computation ofRand ofa abR⋅ab Throughout this section, we will prove Lemma 6In the casea abR⋅ab/=0,for all Class 3 metrics, we have⟨b,b⟩ /=0and R=2α(A− B2 ⟨b,b⟩), a abR⋅ab=12α,(61) whereα∈R/{0}is a constant. To justify the above statement, we proceed in several steps: •Step 1:Show that there existβ=β(t, r)andγ=γ(t, r)such that R=βA+γB 2∶(62) We note that,...
[13]

Let us analyze each factor

Integration of the field equation Using relations (61), the field equation (59) can be rewritten as: Φ= 3A⟨b,b⟩ A⟨b,b⟩−B2 ,which is simply: Φ= 3⟨b,b⟩ ⟨b,b⟩−s.(80) Substituting Φ from (60), a direct computation shows that this is equivalent (after discarding a common factor ofs), to: s[Ψ+(⟨b,b⟩−s)Ψ′]×(81) (3s2(Ψ′)2+3Ψ 2+3⟨b,b⟩ΨΨ′−6sΨΨ′−3⟨b,b⟩s(Ψ′)2−5s2ΨΨ′′...
[14]

Bridging high and low energies in search of quantum gravity (BridgeQG)

˙t2− 4e2α2 (4r2−K)r2 ˙r2−e2α2 r2 w2 (103) ⇔(aab)=diag(α 3α2 1,− 4e2α2 (4r2−K)r2 ,−e2α2 r2 ,−e2α2 r2 sin2 θ).(104) We note that(a ab)is in diagonal form, meaning that its inverse is simply (aab) =diag ⎛ ⎝ 1 α3α2 1 ,− (4r2−K) r2e−2α2 4 ,−r2e−2α2 ,−r2e−2α2 sin2 θ⎞ ⎠.(105) 21 Similarly, we get (ba)∶=(e G 2 ,−ae G 2 ,0,0)=(α 1,0,0,0)⇔B=α 1 ˙t(106) and accordin...
[15]

The hyperplane (H)∶u∶=˙t−a˙r=0.(C10)
[16]

Case (i):b/=0

The quadric (C)∶v+ρu 2 =0.(C11) Let us investigate these cones in each of the casesb/=0 andb=0. Case (i):b/=0. In this situation, we obtain, we have seen above thatρ= b2 c+2ab ,which leads to: v+ρu 2 = (b˙t+(ab+c)˙r ) 2 2ab+c −˙θ2−˙φ2 sin2 θ.(C12) Using the inequality 2ab+c<0,this implies thatv+ρu 2 ≤0 holds identically on the domain of definition ofL.In ...
[17]

Voje Johansen and F

N. Voje Johansen and F. Ravndal, On the discovery of Birkhoff’s theorem, Gen. Rel. Grav.38, 537 (2006), arXiv:physics/0508163

Pith/arXiv arXiv 2006
[18]

G. D. Birkhoff and R. E. Langer,Relativity and modern physics(1923)

1923
[19]

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
[20]

S´ anchez, On the foundations and applications of Lorentz-Finsler Geometry, Gen

M. S´ anchez, On the foundations and applications of Lorentz-Finsler Geometry, Gen. Rel. Grav.58, 47 (2026), arXiv:2511.04645 [gr-qc]

arXiv 2026
[21]

Bao, S.-S

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

2000
[22]

Miron and I

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

2007
[23]

Finsler, ¨Uber Kurven und Fl¨ achen in allgemeinen R¨ aumen, Ph.D

P. Finsler, ¨Uber Kurven und Fl¨ achen in allgemeinen R¨ aumen, Ph.D. thesis, Georg-August Universit¨ at zu G¨ ottingen (1918)

1918
[24]

J. K. Beem, Indefinite Finsler spaces and timelike spaces, Can. J. Math.22, 1035 (1970)

1970
[25]

M. A. Javaloyes and M. S´ anchez, On the definition and examples of cones and Finsler spacetimes, (2018), arXiv:1805.06978 [math.DG]

arXiv 2018
[26]

M. A. Javaloyes, E. Pend´ as-Recondo, and M. S´ anchez, An account on links between Finsler and Lorentz Geometries for Riemannian Geometers, (2022), arXiv:2203.13391 [math.DG]

arXiv 2022
[27]

Carvalho, C

P. Carvalho, C. Landri, R. Mistry, and A. Pinzul, Multimetric Finsler Geometry, (2022), arXiv:2208.03800 [math-ph]

arXiv 2022
[28]

Fuster and C

A. Fuster and C. Pabst, Finsler pp-waves, Phys. Rev.D94, 104072 (2016), arXiv:1510.03058 [gr-qc]

Pith/arXiv arXiv 2016
[29]

Heefer and A

S. Heefer and A. Fuster, Finsler gravitational waves of(α, β)-type and their observational signature, Classical and Quantum Gravity40, 184002 (2023)

2023
[30]

X. Li, S. Wang, and Z. Chang, Anisotropic inflation in the Finsler spacetime, Eur. Phys. J.C75, 260 (2015), arXiv:1502.02256 [gr-qc]

Pith/arXiv arXiv 2015
[31]

Minas, E

G. Minas, E. N. Saridakis, P. C. Stavrinos, and A. Triantafyllopoulos, Bounce cosmology in generalized modified gravities, Universe5, 74 (2019), arXiv:1902.06558 [gr-qc]

Pith/arXiv arXiv 2019
[32]

Papagiannopoulos, S

G. Papagiannopoulos, S. Basilakos, A. Paliathanasis, S. Savvidou, and P. C. Stavrinos, Finsler–Randers cosmology: dynamical analysis and growth of matter perturbations, Class. Quant. Grav.34, 225008 (2017), arXiv:1709.03748 [gr-qc]. 28

Pith/arXiv arXiv 2017
[33]

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
[34]

E. N. Saridakis, R. Lazkoz, V. Salzano, P. Vargas Moniz, S. Capozziello, J. Beltr´ an Jim´ enez, M. De Lau- rentis, and G. J. Olmo, eds.,Modified Gravity and Cosmology(Springer, 2021)

2021
[35]

Stavrinos, O

P. Stavrinos, O. Vacaru, and S. I. Vacaru, Modified einstein and finsler like theories on tangent lorentz bundles, International Journal of Modern Physics D23, 1450094 (2014)

2014
[36]

Tavakol, Geometry of spacetime and Finsler geometry, International Journal of Modern Physics A 24, 1678 (2009)

R. Tavakol, Geometry of spacetime and Finsler geometry, International Journal of Modern Physics A 24, 1678 (2009)

2009
[37]

Triantafyllopoulos and P

A. Triantafyllopoulos and P. C. Stavrinos, Weak field equations and generalized FRW cosmology on the tangent Lorentz bundle, Class. Quant. Grav.35, 085011 (2018), arXiv:1903.12521 [gr-qc]

Pith/arXiv arXiv 2018
[38]

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
[39]

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
[40]

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, JCAP10, 050, arXiv:2504.08062 [gr-qc]

arXiv
[41]

Fuster, S

A. Fuster, S. Heefer, C. Pfeifer, and N. Voicu, On the non metrizability of Berwald Finsler spacetimes, Universe6, 64 (2020), arXiv:2003.02300 [math.DG]

arXiv 2020
[42]

Csillag, N

L. Csillag, N. Voicu, S. Elgendi, and C. Pfeifer, On the Finsler variational nature of autoparallels in metric-affine geometry, (2026), arXiv:2603.18416 [math-ph]

Pith/arXiv arXiv 2026
[43]

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
[44]

Gibbons, J

G. Gibbons, J. Gomis, and C. Pope, General very special relativity is Finsler geometry, Phys.Rev. D76, 081701 (2007), arXiv:0707.2174 [hep-th]

Pith/arXiv arXiv 2007
[45]

Kostelecky, Riemann-Finsler geometry and Lorentz-violating kinematics, Phys

A. Kostelecky, Riemann-Finsler geometry and Lorentz-violating kinematics, Phys. Lett.B701, 137 (2011), arXiv:1104.5488 [hep-th]

Pith/arXiv arXiv 2011
[46]

I. P. Lobo and C. Pfeifer, Reaching the Planck scale with muon lifetime measurements, Phys. Rev. D 103, 106025 (2021), arXiv:2011.10069 [hep-ph]

arXiv 2021
[47]

N. E. Mavromatos, V. A. Mitsou, S. Sarkar, and A. Vergou, Implications of a Stochastic Microscopic Finsler Cosmology, Eur. Phys. J.C72, 1956 (2012), arXiv:1012.4094 [hep-ph]

Pith/arXiv arXiv 1956
[48]

Zhu and B.-Q

J. Zhu and B.-Q. Ma, Lorentz-violation-induced arrival time delay of astroparticles in Finsler spacetime, Phys. Rev. D105, 124069 (2022), arXiv:2206.07616 [gr-qc]

arXiv 2022
[49]

Zhu and B.-Q

J. Zhu and B.-Q. Ma, Lorentz Violation in Finsler Geometry, Symmetry15, 978 (2023), arXiv:2304.12767 [gr-qc]

arXiv 2023
[50]

M. C. Werner, Gravitational lensing in the Kerr-Randers optical geometry, Gen. Rel. Grav.44, 3047 (2012), arXiv:1205.3876 [gr-qc]

Pith/arXiv arXiv 2012
[51]

Caponio and M

E. Caponio and M. A. Javaloyes, On the formulations of the Fermat principle in general relativity and beyond, (2026), arXiv:2605.01532 [gr-qc]

Pith/arXiv arXiv 2026
[52]

Hohmann, C

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

arXiv 2022
[53]

M. ´A. Javaloyes, M. S´ anchez, and F. F. Villase˜ nor, The einstein–hilbert–palatini formalism in pseudo- finsler geometry, Advances in Theoretical and Mathematical Physics26, 3563 (2022)

2022
[54]

S´ anchez and F

M. S´ anchez and F. F. Villase˜ nor, The ladder of Finsler-type objects and their variational problems on spacetimes (2025) arXiv:2504.14710 [math.DG]

arXiv 2025
[55]

Berwald, Untersuchung der Kr¨ ummung allgemeiner metrischer R¨ aume auf Grund des in ihnen herrschenden Parallelismus, Mathematische Zeitschrift25, 40 (1926)

L. Berwald, Untersuchung der Kr¨ ummung allgemeiner metrischer R¨ aume auf Grund des in ihnen herrschenden Parallelismus, Mathematische Zeitschrift25, 40 (1926)

1926
[56]

Szilasi, R

J. Szilasi, R. L. Lovas, and D. C. Kertesz, Several ways to Berwald manifolds - and some steps beyond, Extracta Math.26, 89 (2011), arXiv:1106.2223 [math.DG]

Pith/arXiv arXiv 2011
[57]

Szab´ o, Positive definite berwald spaces, Tensor, New Series35, 25 (1981)

Z. Szab´ o, Positive definite berwald spaces, Tensor, New Series35, 25 (1981)

1981
[58]

Crampin, On the construction of riemannian metrics for berwald spaces by averaging, Houston Jour

M. Crampin, On the construction of riemannian metrics for berwald spaces by averaging, Houston Jour. Math.40, 737 (2014)

2014
[59]

Fuster, C

A. Fuster, C. Pabst, and C. Pfeifer, Berwald spacetimes and very special relativity, Phys. Rev.D98, 29 084062 (2018), arXiv:1804.09727 [gr-qc]

Pith/arXiv arXiv 2018
[60]

Pfeifer, S

C. Pfeifer, S. Heefer, and A. Fuster, Identifying Berwald Finsler geometries, Differ. Geom. Appl.79, 101817 (2021), arXiv:1909.05284 [math.DG]

arXiv 2021
[61]

Cheraghchi, C

S. Cheraghchi, C. Pfeifer, and N. Voicu, Four-dimensional SO(3)-spherically symmetric Berwald Finsler spaces, Int. J. Geom. Meth. Mod. Phys.20, 2350190 (2023), arXiv:2212.08603 [math.DG]

arXiv 2023
[63]

Voicu, C

N. Voicu, C. Pfeifer, and S. Cheraghchi, Birkhoff theorem for Berwald-Finsler spacetimes, Phys. Rev. D108, 104060 (2023), arXiv:2306.07866 [math.DG]

arXiv 2023
[64]

Voicu and S

N. Voicu and S. G. Elgendi, Metrizability of -invariant connections: Riemann versus Finsler, Class. Quant. Grav.41, 155012 (2024), arXiv:2404.02980 [math.DG]

arXiv 2024
[65]

H. R. F. Aurel Bejancu,Geometry of Pseudo-Finsler Submanifolds(Springer, Dodrecht, 2000)

2000
[66]

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
[67]

Lammerzahl and V

C. Lammerzahl and V. Perlick, Finsler geometry as a model for relativistic gravity (2018) arXiv:1802.10043 [gr-qc]

Pith/arXiv arXiv 2018
[68]

Minguzzi, The connections of pseudo-Finsler spaces, Int

E. Minguzzi, The connections of pseudo-Finsler spaces, Int. J. Geom. Meth. Mod. Phys.11, 1460025 (2014), [Erratum: Int. J. Geom. Meth. Mod. Phys.12,no.7,1592001(2015)], arXiv:1405.0645 [math-ph]

Pith/arXiv arXiv 2014
[69]

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
[70]

Voicu, A

N. Voicu, A. Friedl-Sz´ asz, E. Popovici-Popescu, and C. Pfeifer, The Finsler Spacetime Condition for (α,β)-Metrics and Their Isometries, Universe9, 198 (2023), arXiv:2302.09937 [math.DG]

arXiv 2023

[1] [1]

Computation ofRand ofa abR⋅ab 15

[2] [2]

Branch 2:a abR⋅ab=0 19 ∗nico.voicu@unitbv.ro † diana.birla@unitbv.ro ‡ christian.pfeifer@zarm.uni-bremen.de arXiv:2606.05427v1 [gr-qc] 3 Jun 2026 2 VI

Integration of the field equation 17 B. Branch 2:a abR⋅ab=0 19 ∗nico.voicu@unitbv.ro † diana.birla@unitbv.ro ‡ christian.pfeifer@zarm.uni-bremen.de arXiv:2606.05427v1 [gr-qc] 3 Jun 2026 2 VI. Concrete example 19 VII. Conclusion and outlook 21 Acknowledgments 22 A. Connection coefficients and curvature components 22 B. Pseudo-Riemannian metric components 2...

Pith/arXiv arXiv 2026

[3] [3]

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

[4] [4]

We note that any pseudo-Finsler functionLcan be continuously prolonged as 0 at ˙x=0

Nondegeneracy: at any(x,˙x)∈Aand in one (then, in any) local chart around(x,˙x), the Hessian: gab(x,˙x)∶=1 2 ˙∂a ˙∂bL(2) is nonsingular. We note that any pseudo-Finsler functionLcan be continuously prolonged as 0 at ˙x=0. 1 A conic subbundle ofT Mis defined as an open subsetA⊂T M∖{0}with non-empty fibersA x =A∩TxMat all x∈M,and which is stable under posit...

[5] [5]

The elements∈SO(3)leave the coordinatet(usually interpreted as time) invariant. In naturally induced coordinates(x,˙x)∶=(t, r, θ, ϕ,˙t,˙r, ˙θ, ˙ϕ) onT M,any spatially spherically sym- metric Finsler metric is known to be, [53], of the form L(x,˙x)=L(t, r, ˙t,˙r, w), w 2 = ˙θ2+ ˙ϕ2 sin2 θ . Nontrivially Finslerian (i.e., non-quadratic) Berwald metrics with...

[6] [6]

Choosing Θ=1 in (41), one findsL=B 2, where B=e G 2 u=e G 2 ( ˙t−a˙r) .(44) AsBis linear in ˙x, it can be understood asB=b c(x)˙xc, induced by the 1-form b=e G 2 dt+−ae G 2 dr(45) onM. Moreover, as (42)-(43) guarantee that the Berwald conditionδ aL=δ aB2 =0 holds, this 1-form is absolutely parallel with respect to the affine connection onM, i.e., it satis...

[7] [7]

Alternatively, choosing in (41), Θ∶=id., we find thatLis a quadratic function in ˙x: L=A∶=eG (ve−(G−2K)+Mu 2) =ve 2K+(eGM) u2 .(46) The functionAdefines a pseudo-Riemannian metric a=a bc(x)dx b⊗dxc (47) with componentsa bc = ˙∂b ˙∂cA; this is defined on and nondegenerate at least on a subset D⊂M(which can be specified by appropriately choosing the range o...

[8] [8]

The canonical affine connection of a Class 3 Berwald-Finsler functionLis the Levi-Civita connection of its pseudo-Riemannian constitutentA. 13 This Lemma means that geodesics of the Finsler metricLand those ofAare the same assets of points- i.e., judging by their shapes only, we could not tell what is the correct model for our universe - the Finslerian or...

[9] [9]

There holds the identity Rab˜b b =0,(50) where tildes denote raising indices bya ab,that is: ˜b b =a bcbc

[10] [10]

The quantity⟨b,b⟩∶=aabbabb is a constant

[11] [11]

Finally, statement 3

The curvature componentsa 7 anda 11 obey: (ab+c)a 7 =ba 11.(51) Proof.The first result follows immediately from the Ricci identities ˜b c ;a;d−˜b c ;d;a =R c a bd ˜b a applied to the vector field ˜b= ˜b a ∂a and to the fact that this vector field is absolutely parallel with respect to Γ.The second one is also obtained as a direct consequence of the fact t...

[12] [12]

Computation ofRand ofa abR⋅ab Throughout this section, we will prove Lemma 6In the casea abR⋅ab/=0,for all Class 3 metrics, we have⟨b,b⟩ /=0and R=2α(A− B2 ⟨b,b⟩), a abR⋅ab=12α,(61) whereα∈R/{0}is a constant. To justify the above statement, we proceed in several steps: •Step 1:Show that there existβ=β(t, r)andγ=γ(t, r)such that R=βA+γB 2∶(62) We note that,...

[13] [13]

Let us analyze each factor

Integration of the field equation Using relations (61), the field equation (59) can be rewritten as: Φ= 3A⟨b,b⟩ A⟨b,b⟩−B2 ,which is simply: Φ= 3⟨b,b⟩ ⟨b,b⟩−s.(80) Substituting Φ from (60), a direct computation shows that this is equivalent (after discarding a common factor ofs), to: s[Ψ+(⟨b,b⟩−s)Ψ′]×(81) (3s2(Ψ′)2+3Ψ 2+3⟨b,b⟩ΨΨ′−6sΨΨ′−3⟨b,b⟩s(Ψ′)2−5s2ΨΨ′′...

[14] [14]

Bridging high and low energies in search of quantum gravity (BridgeQG)

˙t2− 4e2α2 (4r2−K)r2 ˙r2−e2α2 r2 w2 (103) ⇔(aab)=diag(α 3α2 1,− 4e2α2 (4r2−K)r2 ,−e2α2 r2 ,−e2α2 r2 sin2 θ).(104) We note that(a ab)is in diagonal form, meaning that its inverse is simply (aab) =diag ⎛ ⎝ 1 α3α2 1 ,− (4r2−K) r2e−2α2 4 ,−r2e−2α2 ,−r2e−2α2 sin2 θ⎞ ⎠.(105) 21 Similarly, we get (ba)∶=(e G 2 ,−ae G 2 ,0,0)=(α 1,0,0,0)⇔B=α 1 ˙t(106) and accordin...

[15] [15]

The hyperplane (H)∶u∶=˙t−a˙r=0.(C10)

[16] [16]

Case (i):b/=0

The quadric (C)∶v+ρu 2 =0.(C11) Let us investigate these cones in each of the casesb/=0 andb=0. Case (i):b/=0. In this situation, we obtain, we have seen above thatρ= b2 c+2ab ,which leads to: v+ρu 2 = (b˙t+(ab+c)˙r ) 2 2ab+c −˙θ2−˙φ2 sin2 θ.(C12) Using the inequality 2ab+c<0,this implies thatv+ρu 2 ≤0 holds identically on the domain of definition ofL.In ...

[17] [17]

Voje Johansen and F

N. Voje Johansen and F. Ravndal, On the discovery of Birkhoff’s theorem, Gen. Rel. Grav.38, 537 (2006), arXiv:physics/0508163

Pith/arXiv arXiv 2006

[18] [18]

G. D. Birkhoff and R. E. Langer,Relativity and modern physics(1923)

1923

[19] [19]

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

[20] [20]

S´ anchez, On the foundations and applications of Lorentz-Finsler Geometry, Gen

M. S´ anchez, On the foundations and applications of Lorentz-Finsler Geometry, Gen. Rel. Grav.58, 47 (2026), arXiv:2511.04645 [gr-qc]

arXiv 2026

[21] [21]

Bao, S.-S

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

2000

[22] [22]

Miron and I

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

2007

[23] [23]

Finsler, ¨Uber Kurven und Fl¨ achen in allgemeinen R¨ aumen, Ph.D

P. Finsler, ¨Uber Kurven und Fl¨ achen in allgemeinen R¨ aumen, Ph.D. thesis, Georg-August Universit¨ at zu G¨ ottingen (1918)

1918

[24] [24]

J. K. Beem, Indefinite Finsler spaces and timelike spaces, Can. J. Math.22, 1035 (1970)

1970

[25] [25]

M. A. Javaloyes and M. S´ anchez, On the definition and examples of cones and Finsler spacetimes, (2018), arXiv:1805.06978 [math.DG]

arXiv 2018

[26] [26]

M. A. Javaloyes, E. Pend´ as-Recondo, and M. S´ anchez, An account on links between Finsler and Lorentz Geometries for Riemannian Geometers, (2022), arXiv:2203.13391 [math.DG]

arXiv 2022

[27] [27]

Carvalho, C

P. Carvalho, C. Landri, R. Mistry, and A. Pinzul, Multimetric Finsler Geometry, (2022), arXiv:2208.03800 [math-ph]

arXiv 2022

[28] [28]

Fuster and C

A. Fuster and C. Pabst, Finsler pp-waves, Phys. Rev.D94, 104072 (2016), arXiv:1510.03058 [gr-qc]

Pith/arXiv arXiv 2016

[29] [29]

Heefer and A

S. Heefer and A. Fuster, Finsler gravitational waves of(α, β)-type and their observational signature, Classical and Quantum Gravity40, 184002 (2023)

2023

[30] [30]

X. Li, S. Wang, and Z. Chang, Anisotropic inflation in the Finsler spacetime, Eur. Phys. J.C75, 260 (2015), arXiv:1502.02256 [gr-qc]

Pith/arXiv arXiv 2015

[31] [31]

Minas, E

G. Minas, E. N. Saridakis, P. C. Stavrinos, and A. Triantafyllopoulos, Bounce cosmology in generalized modified gravities, Universe5, 74 (2019), arXiv:1902.06558 [gr-qc]

Pith/arXiv arXiv 2019

[32] [32]

Papagiannopoulos, S

G. Papagiannopoulos, S. Basilakos, A. Paliathanasis, S. Savvidou, and P. C. Stavrinos, Finsler–Randers cosmology: dynamical analysis and growth of matter perturbations, Class. Quant. Grav.34, 225008 (2017), arXiv:1709.03748 [gr-qc]. 28

Pith/arXiv arXiv 2017

[33] [33]

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

[34] [34]

E. N. Saridakis, R. Lazkoz, V. Salzano, P. Vargas Moniz, S. Capozziello, J. Beltr´ an Jim´ enez, M. De Lau- rentis, and G. J. Olmo, eds.,Modified Gravity and Cosmology(Springer, 2021)

2021

[35] [35]

Stavrinos, O

P. Stavrinos, O. Vacaru, and S. I. Vacaru, Modified einstein and finsler like theories on tangent lorentz bundles, International Journal of Modern Physics D23, 1450094 (2014)

2014

[36] [36]

Tavakol, Geometry of spacetime and Finsler geometry, International Journal of Modern Physics A 24, 1678 (2009)

R. Tavakol, Geometry of spacetime and Finsler geometry, International Journal of Modern Physics A 24, 1678 (2009)

2009

[37] [37]

Triantafyllopoulos and P

A. Triantafyllopoulos and P. C. Stavrinos, Weak field equations and generalized FRW cosmology on the tangent Lorentz bundle, Class. Quant. Grav.35, 085011 (2018), arXiv:1903.12521 [gr-qc]

Pith/arXiv arXiv 2018

[38] [38]

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

[39] [39]

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

[40] [40]

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, JCAP10, 050, arXiv:2504.08062 [gr-qc]

arXiv

[41] [41]

Fuster, S

A. Fuster, S. Heefer, C. Pfeifer, and N. Voicu, On the non metrizability of Berwald Finsler spacetimes, Universe6, 64 (2020), arXiv:2003.02300 [math.DG]

arXiv 2020

[42] [42]

Csillag, N

L. Csillag, N. Voicu, S. Elgendi, and C. Pfeifer, On the Finsler variational nature of autoparallels in metric-affine geometry, (2026), arXiv:2603.18416 [math-ph]

Pith/arXiv arXiv 2026

[43] [43]

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

[44] [44]

Gibbons, J

G. Gibbons, J. Gomis, and C. Pope, General very special relativity is Finsler geometry, Phys.Rev. D76, 081701 (2007), arXiv:0707.2174 [hep-th]

Pith/arXiv arXiv 2007

[45] [45]

Kostelecky, Riemann-Finsler geometry and Lorentz-violating kinematics, Phys

A. Kostelecky, Riemann-Finsler geometry and Lorentz-violating kinematics, Phys. Lett.B701, 137 (2011), arXiv:1104.5488 [hep-th]

Pith/arXiv arXiv 2011

[46] [46]

I. P. Lobo and C. Pfeifer, Reaching the Planck scale with muon lifetime measurements, Phys. Rev. D 103, 106025 (2021), arXiv:2011.10069 [hep-ph]

arXiv 2021

[47] [47]

N. E. Mavromatos, V. A. Mitsou, S. Sarkar, and A. Vergou, Implications of a Stochastic Microscopic Finsler Cosmology, Eur. Phys. J.C72, 1956 (2012), arXiv:1012.4094 [hep-ph]

Pith/arXiv arXiv 1956

[48] [48]

Zhu and B.-Q

J. Zhu and B.-Q. Ma, Lorentz-violation-induced arrival time delay of astroparticles in Finsler spacetime, Phys. Rev. D105, 124069 (2022), arXiv:2206.07616 [gr-qc]

arXiv 2022

[49] [49]

Zhu and B.-Q

J. Zhu and B.-Q. Ma, Lorentz Violation in Finsler Geometry, Symmetry15, 978 (2023), arXiv:2304.12767 [gr-qc]

arXiv 2023

[50] [50]

M. C. Werner, Gravitational lensing in the Kerr-Randers optical geometry, Gen. Rel. Grav.44, 3047 (2012), arXiv:1205.3876 [gr-qc]

Pith/arXiv arXiv 2012

[51] [51]

Caponio and M

E. Caponio and M. A. Javaloyes, On the formulations of the Fermat principle in general relativity and beyond, (2026), arXiv:2605.01532 [gr-qc]

Pith/arXiv arXiv 2026

[52] [52]

Hohmann, C

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

arXiv 2022

[53] [53]

M. ´A. Javaloyes, M. S´ anchez, and F. F. Villase˜ nor, The einstein–hilbert–palatini formalism in pseudo- finsler geometry, Advances in Theoretical and Mathematical Physics26, 3563 (2022)

2022

[54] [54]

S´ anchez and F

M. S´ anchez and F. F. Villase˜ nor, The ladder of Finsler-type objects and their variational problems on spacetimes (2025) arXiv:2504.14710 [math.DG]

arXiv 2025

[55] [55]

Berwald, Untersuchung der Kr¨ ummung allgemeiner metrischer R¨ aume auf Grund des in ihnen herrschenden Parallelismus, Mathematische Zeitschrift25, 40 (1926)

L. Berwald, Untersuchung der Kr¨ ummung allgemeiner metrischer R¨ aume auf Grund des in ihnen herrschenden Parallelismus, Mathematische Zeitschrift25, 40 (1926)

1926

[56] [56]

Szilasi, R

J. Szilasi, R. L. Lovas, and D. C. Kertesz, Several ways to Berwald manifolds - and some steps beyond, Extracta Math.26, 89 (2011), arXiv:1106.2223 [math.DG]

Pith/arXiv arXiv 2011

[57] [57]

Szab´ o, Positive definite berwald spaces, Tensor, New Series35, 25 (1981)

Z. Szab´ o, Positive definite berwald spaces, Tensor, New Series35, 25 (1981)

1981

[58] [58]

Crampin, On the construction of riemannian metrics for berwald spaces by averaging, Houston Jour

M. Crampin, On the construction of riemannian metrics for berwald spaces by averaging, Houston Jour. Math.40, 737 (2014)

2014

[59] [59]

Fuster, C

A. Fuster, C. Pabst, and C. Pfeifer, Berwald spacetimes and very special relativity, Phys. Rev.D98, 29 084062 (2018), arXiv:1804.09727 [gr-qc]

Pith/arXiv arXiv 2018

[60] [60]

Pfeifer, S

C. Pfeifer, S. Heefer, and A. Fuster, Identifying Berwald Finsler geometries, Differ. Geom. Appl.79, 101817 (2021), arXiv:1909.05284 [math.DG]

arXiv 2021

[61] [61]

Cheraghchi, C

S. Cheraghchi, C. Pfeifer, and N. Voicu, Four-dimensional SO(3)-spherically symmetric Berwald Finsler spaces, Int. J. Geom. Meth. Mod. Phys.20, 2350190 (2023), arXiv:2212.08603 [math.DG]

arXiv 2023

[62] [63]

Voicu, C

N. Voicu, C. Pfeifer, and S. Cheraghchi, Birkhoff theorem for Berwald-Finsler spacetimes, Phys. Rev. D108, 104060 (2023), arXiv:2306.07866 [math.DG]

arXiv 2023

[63] [64]

Voicu and S

N. Voicu and S. G. Elgendi, Metrizability of -invariant connections: Riemann versus Finsler, Class. Quant. Grav.41, 155012 (2024), arXiv:2404.02980 [math.DG]

arXiv 2024

[64] [65]

H. R. F. Aurel Bejancu,Geometry of Pseudo-Finsler Submanifolds(Springer, Dodrecht, 2000)

2000

[65] [66]

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

[66] [67]

Lammerzahl and V

C. Lammerzahl and V. Perlick, Finsler geometry as a model for relativistic gravity (2018) arXiv:1802.10043 [gr-qc]

Pith/arXiv arXiv 2018

[67] [68]

Minguzzi, The connections of pseudo-Finsler spaces, Int

E. Minguzzi, The connections of pseudo-Finsler spaces, Int. J. Geom. Meth. Mod. Phys.11, 1460025 (2014), [Erratum: Int. J. Geom. Meth. Mod. Phys.12,no.7,1592001(2015)], arXiv:1405.0645 [math-ph]

Pith/arXiv arXiv 2014

[68] [69]

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

[69] [70]

Voicu, A

N. Voicu, A. Friedl-Sz´ asz, E. Popovici-Popescu, and C. Pfeifer, The Finsler Spacetime Condition for (α,β)-Metrics and Their Isometries, Universe9, 198 (2023), arXiv:2302.09937 [math.DG]

arXiv 2023