A Geometrical Model for the Unified Theory of Physical Fields

2015

Class. Quantum Grav.

Abstract. The Raychaudhuri equation and its consequences for chronality are studied in the context of Finsler spacetimes. It is proved that the notable singularity theorems of Lorentzian geometry extend to the Finslerian domain. Indeed, so do the theorems by Hawking, Penrose, Hawking and Penrose, Geroch, Gannon, Tipler, or Kriele, but also the Topological Censorship theorem and so on. It is argued that the notable results in causality theory connected to achronal sets, future sets, domains of dependence, limit curve theorems, length functional, Lorentzian distance, geodesic connectedness, extend to the Finslerian domain. Results concerning the spacetime asymptotic structure, horizons differentiability and conformal transformations are also included.

“…Let v = v ± in Eq. (38) and observe that ±g νµ Ω ,µ = v ν . SinceĨ δ is positive homogenous of zero degree in v, the second term vanishes.…”

Section: Asymptotics Singularities Differentiabilitymentioning

confidence: 96%

“…indeed, it is implied by almost every choice of dynamical equations that has been proposed so far, starting from the first proposal by Horvath [38,39]. Not all authors obtained this equation from a tensorial generalization of Einstein's.…”

Section: Introductionmentioning

confidence: 99%

Raychaudhuri equation and singularity theorems in Finsler spacetimes

2015

Class. Quantum Grav.

“…With the notation of Theorem (4),ρ = αρ. Equation (36) clarifies that N is still an affine sphere with mean curvatureȞ wherě…”

Section: Proper Affine Spheresmentioning

confidence: 99%

Affine Sphere Relativity

2016

Commun. Math. Phys.

We investigate spacetimes whose light cones could be anisotropic. We prove the equivalence of the structures: (a) Lorentz-Finsler manifold for which the mean Cartan torsion vanishes, (b) Lorentz-Finsler manifold for which the indicatrix (observer space) at each point is a convex hyperbolic affine sphere centered on the zero section, and (c) pair given by a spacetime volume and a sharp convex cone distribution. The equivalence suggests to describe (affine sphere) spacetimes with this structure, so that no algebraic-metrical concept enters the definition. As a result, this work shows how the metric features of spacetime emerge from elementary concepts such as measure and order. Non-relativistic spacetimes are obtained replacing proper spheres with improper spheres, so the distinction does not call for group theoretical elements. In physical terms, in affine sphere spacetimes the light cone distribution and the spacetime measure determine the motion of massive and massless particles (hence the dispersion relation). Furthermore, it is shown that, more generally, for Lorentz-Finsler theories non-differentiable at the cone, the lightlike geodesics and the transport of the particle momentum over them are well defined though the curve parametrization could be undefined. Causality theory is also well behaved. Several results for affine sphere spacetimes are presented. Some results in Finsler geometry, for instance in the characterization of Randers spaces, are also included.

“…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

2017

Phys. Rev. D

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