Special Issue "Finsler Modification of Classical General Relativity"

A special issue of Universe (ISSN 2218-1997). This special issue belongs to the section "Gravitation".

Deadline for manuscript submissions: closed (31 December 2020).

Special Issue Editor

Dr. Volker Perlick
E-Mail Website
Guest Editor
University of Bremen, Bremen, Germany
Interests: classical general relativity; gravitational lensing; classical electrodynamics; nonlinear electrodynamics; Finsler modification of classical general relativity

Special Issue Information

Dear Colleagues,

General relativity is a very successful theory that is in agreement with all observations to date (if one accepts the existence of dark matter). Nonetheless, there are good reasons for investigating possible modifications of this theory. One modification is in replacing the pseudo-Riemannian metric on which general relativity is based by a Finslerian metric. There are at least two motivations for considering such a modification: The first motivation comes from the fact that a slight modification of the Ehlers–Pirani–Schild axiomatic approach to spacetime theory leads in a very natural way to a Finsler structure. The second motivation is based on the idea that some effects from a still-to-be-found quantum theory of gravity could be described in terms of a classical ("effective") Finslerian spacetime theory.

This Special Issue is devoted to all aspects of such Finsler-modified classical spacetime theories. Among other things, this includes (a) the question of which definition of a Finsler spacetime is most appropriate from the viewpoint of physics; (b) the Finsler-analogue of Einstein's  field equation; (c) the Maxwell, Klein-Gordon and Dirac equations on Finsler manifolds; and (d) experimental and observational tests of Finsler  spacetime theories in the laboratory, in the solar system, on galactic scales and in cosmology.    

Dr. Volker Perlick
Guest Editor

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Keywords

  • Finsler geometry
  • modified gravity
  • modified Einstein equation

Published Papers (5 papers)

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Research

Article
Cosmological Finsler Spacetimes
Universe 2020, 6(5), 65; https://doi.org/10.3390/universe6050065 - 05 May 2020
Cited by 10 | Viewed by 832
Abstract
Applying the cosmological principle to Finsler spacetimes, we identify the Lie Algebra of symmetry generators of spatially homogeneous and isotropic Finsler geometries, thus generalising Friedmann-Lemaître-Robertson-Walker geometry. In particular, we find the most general spatially homogeneous and isotropic Berwald spacetimes, which are Finsler spacetimes [...] Read more.
Applying the cosmological principle to Finsler spacetimes, we identify the Lie Algebra of symmetry generators of spatially homogeneous and isotropic Finsler geometries, thus generalising Friedmann-Lemaître-Robertson-Walker geometry. In particular, we find the most general spatially homogeneous and isotropic Berwald spacetimes, which are Finsler spacetimes that can be regarded as closest to pseudo-Riemannian geometry. They are defined by a Finsler Lagrangian built from a zero-homogeneous function on the tangent bundle, which encodes the velocity dependence of the Finsler Lagrangian in a very specific way. The obtained cosmological Berwald geometries are candidates for the description of the geometry of the universe, when they are obtained as solutions from a Finsler gravity equation. Full article
(This article belongs to the Special Issue Finsler Modification of Classical General Relativity)
Article
On the Non Metrizability of Berwald Finsler Spacetimes
Universe 2020, 6(5), 64; https://doi.org/10.3390/universe6050064 - 01 May 2020
Cited by 8 | Viewed by 957
Abstract
We investigate whether Szabo’s metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its Chern–Rund connection defines an affine connection on the [...] Read more.
We investigate whether Szabo’s metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its Chern–Rund connection defines an affine connection on the underlying manifold), then it is affinely equivalent to a Riemann space, meaning that its affine connection is the Levi–Civita connection of some Riemannian metric. We show for the first time that this result does not extend to general Finsler spacetimes. More precisely, we find a large class of Berwald spacetimes for which the Ricci tensor of the affine connection is not symmetric. The fundamental difference from positive definite Finsler spaces that makes such an asymmetry possible is the fact that generally, Finsler spacetimes satisfy certain smoothness properties only on a proper conic subset of the slit tangent bundle. Indeed, we prove that when the Finsler Lagrangian is smooth on the entire slit tangent bundle, the Ricci tensor must necessarily be symmetric. For large classes of Finsler spacetimes, however, the Berwald property does not imply that the affine structure is equivalent to the affine structure of a pseudo-Riemannian metric. Instead, the affine structure is that of a metric-affine geometry with vanishing torsion. Full article
(This article belongs to the Special Issue Finsler Modification of Classical General Relativity)
Article
On the Analyticity of Static Solutions of a Field Equation in Finsler Gravity
Universe 2020, 6(4), 59; https://doi.org/10.3390/universe6040059 - 22 Apr 2020
Cited by 8 | Viewed by 825
Abstract
It is well-known that static vacuum solutions of Einstein equations are analytic in suitable coordinates. We ask here for an extension of this result in the context of Finsler gravity. We consider Finsler spacetimes that retain several properties of static Lorentzian spacetimes, are [...] Read more.
It is well-known that static vacuum solutions of Einstein equations are analytic in suitable coordinates. We ask here for an extension of this result in the context of Finsler gravity. We consider Finsler spacetimes that retain several properties of static Lorentzian spacetimes, are Berwald and have vanishing Ricci scalar. Full article
(This article belongs to the Special Issue Finsler Modification of Classical General Relativity)
Article
Numerical Modelling of Satellite Downlink Signals in a Finslerian-Perturbed Schwarzschild Spacetime
Universe 2020, 6(4), 57; https://doi.org/10.3390/universe6040057 - 20 Apr 2020
Cited by 1 | Viewed by 1120
Abstract
The work presented in this paper aims to contribute to the problem of testing Finsler gravity theories by means of experiments and observations in the solar system. Within a class of spherically symmetric static Finsler spacetimes we consider a satellite with an on-board [...] Read more.
The work presented in this paper aims to contribute to the problem of testing Finsler gravity theories by means of experiments and observations in the solar system. Within a class of spherically symmetric static Finsler spacetimes we consider a satellite with an on-board atomic clock, orbiting in the Finslerian-perturbed gravitational field of the earth, whose time signal is transmitted to a ground station, where its receive time and frequency are measured with respect to another atomic clock. This configuration is realized by the Galileo 5 and 6 satellites that have gone astray and are now on non-circular orbits. Our method consists in the numerical integration of the satellite’s orbit, followed by an iterative procedure which provides the numerically integrated signals, i.e., null geodesics, from the satellite to the ground station. One of our main findings is that for orbits that are considerably more eccentric than the Galileo 5 and 6 satellite orbits, Finslerian effects can be separated from effects of perturbations of the Schwarzschild spacetime within the Lorentzian geometry. We also discuss the separation from effects of non-gravitational perturbations. This leads us to the conclusion that observations of this kind combined with appropriate numerical modelling can provide suitable tests of Finslerian modifications of general relativity. Full article
(This article belongs to the Special Issue Finsler Modification of Classical General Relativity)
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Article
Foundations of Finsler Spacetimes from the Observers’ Viewpoint
Universe 2020, 6(4), 55; https://doi.org/10.3390/universe6040055 - 16 Apr 2020
Cited by 14 | Viewed by 1143
Abstract
Physical foundations for relativistic spacetimes are revisited in order to check at what extent Finsler spacetimes lie in their framework. Arguments based on inertial observers (as in the foundations of special relativity and classical mechanics) are shown to correspond with a double linear [...] Read more.
Physical foundations for relativistic spacetimes are revisited in order to check at what extent Finsler spacetimes lie in their framework. Arguments based on inertial observers (as in the foundations of special relativity and classical mechanics) are shown to correspond with a double linear approximation in the measurement of space and time. While general relativity appears by dropping the first linearization, Finsler spacetimes appear by dropping the second one. The classical Ehlers–Pirani–Schild approach is carefully discussed and shown to be compatible with the Lorentz–Finsler case. The precise mathematical definition of Finsler spacetime is discussed by using the space of observers. Special care is taken in some issues such as the fact that a Lorentz–Finsler metric would be physically measurable only on the causal directions for a cone structure, the implications for models of spacetimes of some apparently innocuous hypotheses on differentiability, or the possibilities of measurement of a varying speed of light. Full article
(This article belongs to the Special Issue Finsler Modification of Classical General Relativity)
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