Einstein equation in lifted Finsler spaces

Ishikawa, Hirohisa

doi:10.1007/bf02729263

Cited by 13 publications

(12 citation statements)

References 10 publications

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“…2 Such surveys of second 2 For reviews on relativistic developments and MGTs, we cite [8,153,161,169,173,175,185,194] and references therein. Here we note that for local Euclidean signatures, the main ideas on Lagrange-Hamilton geometries formulated as generalized Finsler spaces on (co)tangent bundles were proposed and studied in a series of works due to J. Kern, M. Matsumoto, S. Ikeda, see [54][55][56][57][58]64,65,75,91,92]. There were elaborated theories on supersymmetric and higher order generalizations, almost Kähler models and applications in particle physics and gravity.…”

Section: Introductionmentioning

confidence: 99%

Axiomatic formulations of modified gravity theories with nonlinear dispersion relations and Finsler–Lagrange–Hamilton geometry

2018

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We develop an axiomatic geometric approach and provide an unconventional review of modified/nonlinear gravity theories, MGTs, with modified dispersion relations, MDRs, encoding Lorentz invariance violations, LIVs, classical and quantum random effects, anisotropies etc. There are studied Lorentz-Finsler like theories elaborated as extensions of general relativity, GR, and quantum gravity, QG, models and constructed on (co)tangent Lorentz bundles, i.e. (curved) phase spaces or locally anisotropic spacetimes. An indicator of MDRs is considered as a functional on various type functions depending on phase space coordinates and physical constants. It determines respective generating functions and fundamental physical objects (generalized metrics, connections and nonholonomic frame structures) for relativistic models of Finsler, Lagrange and/or Hamilton spaces. We show that there are canonical almost symplectic differential forms and adapted (non)linear connections which allow us to formulate equivalent almost Kähler-Lagrange/-Hamilton geometries. This way, it is possible to unify geometrically various classes of (non)commutative MGTs with locally anisotropic gravitational, scalar, non-Abelian gauge field, and Higgs interactions. We elaborate on theories with Lagrangian densities containing massive graviton terms and bi-connection and bi-metric modifications which can be modelled as Finsler-Lagrange-Hamilton geometries. An example of short-range locally anisotropic gravity on (co)tangent Lorentz bundles is analysed. We conclude that a large class of such MGTs admits a self-consistent causal axiomatic formulation which is similar to GR but

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Section: Introductionmentioning

confidence: 99%

Axiomatic formulations of modified gravity theories with nonlinear dispersion relations and Finsler–Lagrange–Hamilton geometry

2018

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“…Many authors proposed Finslerian generalizations of Einstein's general relativity (see [33,15,2,30,26,9,36,28,29] and their bibliographies). Unfortunately, there is no consensus on the set of equations as different authors gave different generalizations.…”

Section: Discussionmentioning

confidence: 99%

“…Finally, let u ∈ J α and let W be defined through Eq. (15). Suppose that W intersects J α , namely suppose that W is not α-spacelike, and let w ∈ W ∩ J α then 0 = g u (u, w) ≤ − −g u (u, u) −g w (w, w),…”

Section: Theorem 3 (Finslerian Reverse Cauchy-schwarz Inequalitymentioning

confidence: 99%

Light cones in Finsler spacetime

Minguzzi

2014

Preprint

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Some foundational results on the geometry of Lorentz-Minkowski spaces and Finsler spacetimes are obtained. We prove that the local light cone structure of a reversible Finsler spacetime with more than two dimensions is topologically the same as that of Lorentzian spacetimes: at each point we have just two strictly convex causal cones which intersect only at the origin. Moreover, we prove a reverse Cauchy-Schwarz inequality for these spaces and a corresponding reverse triangle inequality. The Legendre map is proved to be a diffeomorphism in the general pseudo-Finsler case provided the dimension is larger than two.

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“…The traditional approach in Finsler gravity theory consists in trying to build, if not a Lagrangian, some field equations directly from the various curvatures associated to the Berwald, Cartan or Chern-Rund Finsler connections. This approach has been followed by Horvath [23], Takano [58], Ishikawa [25,26], Ikeda [24], Asanov [2], Miron [45], Rutz [54], Li and Chang [33], Vacaru [59], Pfeifer and Wohlfarth [49], to mention a few. The author has also explored this route [38].…”

Section: B Dynamicsmentioning

confidence: 99%

Affine sphere spacetimes which satisfy the relativity principle

Minguzzi

2017

Phys. Rev. D

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In the context of Lorentz-Finsler spacetime theories the relativity principle holds at a spacetime point if the indicatrix (observer space) is homogeneous. We point out that in four spacetime dimensions there are just three kinematical models which respect an exact form of the relativity principle and for which all observers agree on the spacetime volume. They have necessarily affine sphere indicatrices. For them every observer which looks at a flash of light emitted by a point would observe, respectively, an expanding (a) sphere, (b) tetrahedron, or (c) cone, with barycenter at the point. The first model corresponds to Lorentzian relativity, the second one has been studied by several authors though the relationship with affine spheres passed unnoticed, and the last one has not been previously recognized and it is studied here in some detail. The symmetry groups are O + (3, 1), R 3 , O + (2, 1) × R, respectively. In the second part, devoted to the general relativistic theory, we show that the field equations can be obtained by gauging the Finsler Lagrangian symmetry while avoiding direct use of Finslerian curvatures. We construct some notable affine sphere spacetimes which in the appropriate velocity limit return the Schwarzschild, Kerr-Schild, Kerr-de Sitter, Kerr-Newman, Taub, and FLRW spacetimes, respectively. CONTENTS

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Einstein equation in lifted Finsler spaces

Cited by 13 publications

References 10 publications

Axiomatic formulations of modified gravity theories with nonlinear dispersion relations and Finsler–Lagrange–Hamilton geometry

Axiomatic formulations of modified gravity theories with nonlinear dispersion relations and Finsler–Lagrange–Hamilton geometry

Light cones in Finsler spacetime

Affine sphere spacetimes which satisfy the relativity principle

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