In the context of Finsler-Randers theory we consider, for a first time, the cosmological scenario of the varying vacuum. In particular, we assume the existence of a cosmological fluid source described by an ideal fluid and the varying vacuum terms. We determine the cosmological history of this model by performing a detailed study on the dynamics of the field equations. We determine the limit of General Relativity, while we find new eras in the cosmological history provided by the geometrodynamical terms provided by the Finsler-Randers theory.
Gravitational field equations in Randers-Finsler space of approximate Berwald type are investigated. A modified Friedmann equation and a new luminosity distance-redshift relation is proposed. A best-fit to the Type Ia supernovae (SNe) observations yields that the $Ω_Λ$ in the $Λ$-CDM model is suppressed to almost zero. This fact indicates that the astronomical observations on the Type Ia SNe can be described well without invoking any form of dark energy. The best-fit age of the universe is given. It is in agreement with the age of our galaxy.
Gravitational field equations in Randers-Finsler space of approximate Berwald type are investigated. A modified Friedmann model is proposed. It is showed that the accelerated expanding universe is guaranteed by a constrained Randers-Finsler structure without invoking dark energy. The geodesic in Randers-Finsler space is studied. The additional term in the geodesic equation acts as repulsive force against the gravity.
Kinematics in Finsler space is used to study the propagation of ultra high energy cosmic rays particles through the cosmic microwave background radiation. We find that the GZK threshold is lifted dramatically in Randers-Finsler space. A tiny deformation of spacetime from Minkowskian to Finslerian allows more ultra-high energy cosmic rays particles arrive at the earth. It is suggested that the lower bound of particle mass is related with the negative second invariant speed in Randers-Finsler space.
We perform a detailed comparison between the Finsler-Randers cosmological model and the Dvali, Gabadadze and Porrati braneworld model. If we assume that the spatial curvature is strictly equal to zero then we prove the following interesting proposition: {\it despite the fact that the current cosmological models have a completely different geometrical origin they share exactly the same Hubble expansion}. This implies that the Finsler-Randers model is cosmologically equivalent with that of the DGP model as far as the cosmic expansion is concerned. At the perturbative level we find that the Finsl…
We study the symmetries of the Lorentz violating Randers-Finsler spacetime. The privileged frame defined by the background vector diagonalises the deformed mass-shell and provides an anisotropic observer transformations. The particle Randers transformations are achieved through the Finslerian Killing equation which also shows the equivalence between observer and particle Randers transformation. For a constant background vector, the Randers transformations can be regarded as a deformed directional-dependent Poincaré transformation. The Randers algebra is a deformed Poincaré algebra whose struct…
In this article we present the cosmological equivalence between the relativistic Finsler-Randers cosmology, with dark energy and modified gravity constructions, at the background level. Starting from a small deviation from the quadraticity of the Riemannian geometry, through which the local structure of General Relativity is modified and the curvature theory is extended, we extract the modified Friedmann equation. The corresponding extended Finsler-Randers cosmology is very interesting, and it can mimic dark-energy and modified gravity, describing a large class of scale-factor evolutions, from…
The linearized form of the metric of a Finsler - Randers space is studied in relation to the equations of motion, the deviation of geodesics and the generalized Raychaudhuri equation are given for a weak gravitational field. This equation is also derived in the framework of a tangent bundle. By using Cartan or Berwald-like connections we get some types "gravito - electromagnetic" curvature. In addition we investigate the conditions under which a definite Lagrangian in a Randers space leads to Einstein field equations under the presence of electromagnetic field. Finally, some applications of th…
In this work, we obtain the Raychaudhuri equations for various types of Finsler spaces as the Finsler-Randers (FR) space-time and in a more general geometrical structure of the space-time manifold which contains two fibres that are two scalars which represent inflaton fields $ϕ^{(1)}, ϕ^{(2)}.$In addition, the energy-conditions are studied in a FR cosmology and are correlated with FRW model. Finally an application of Raychaudhuri equation for the model $M \times \{ ϕ^{(1)} \} \times \{ ϕ^{(2)} \}$ with M a FRW space is given.
Kinematics in Finsler space is investigated. It is showed that the result based on the kinematics with a special Finsler structure is in good agreement with the reported value of secular trend in the astronomical unit, $d{\rm AU}/dt=15\pm4[{\rm m/century}]$. The space deformation parameter $λ$ in this special structure is very small with scale of $10^{-6}$ and should be a constant. This fact is consistent with the reported value of an anomalous secular eccentricity variation of the Moon's orbit.
Finsler geometry is a natural arena to investigate the physics of spacetimes with local Lorentz violating. The directional dependence of the Finsler metric provides a way to encode the Lorentz violating effects into the geometric structure of spacetime. Here, a classical field theory is proposed in a special Finsler geometry, the so-called Randers-Finsler spacetime, where the Lorentz violation is produced by a background vector field. By promoting the Randers-Finsler metric to a differential operator, a Finsler-invariant action for the scalar, gauge and fermions are proposed. The theory contai…
In this paper we return to the subject of Jacobi metrics for timelike and null geodsics in stationary spactimes, correcting some previous misconceptions. We show that not only null geodesics, but also timelike geodesics are governed by a Jacobi-Maupertuis type variational principle and a Randers-Finsler metric for which we give explicit formulae. The cases of the Taub-NUT and Kerr spacetimes are discussed in detail. Finally we show how our Jacobi-Maupertuis Randers-Finsler metric may be expressed in terms of the effective medium describing the behaviour of Maxwell's equations in the curved spa…
We obtain some results in both Lorentz and Finsler geometries, by using a correspondence between the conformal structure (Causality) of standard stationary spacetimes on $M=\R\times S$ and Randers metrics on $S$. In particular, for stationary spacetimes, we give a simple characterization of when they are causally continuous or globally hyperbolic (including in the latter case, when $S$ is a Cauchy hypersurface), in terms of an associated Randers metric. Consequences for the computability of Cauchy developments are also derived. Causality suggests that the role of completeness in many results o…
In this work, we extend for the first time the spherically symmetric Schwarzschild and Schwarzschild-De Sitter solutions with a Finsler-Randers-type perturbation which is generated by a covector $A_γ$. This gives a locally anisotropic character to the metric and induces a deviation from the Riemannian models of gravity. A natural framework for this study is the Lorentz tangent bundle of a spacetime manifold. We apply the generalized field equations to the perturbed metric and derive the dynamics for the covector $A_γ$. Finally, we find the timelike, spacelike and null paths on the Schwarzschil…
In this article, we study further applications of the Schwarzschild-Finsler-Randers (SFR) model which was introduced in a previous work. In this model, we investigate curvatures and the generalized Kretschmann invariant which plays a crucial role for singularities. In addition, the derived path equations are used for the gravitational redshift of the SFR-model and these are compared with the GR model. Finally, we get some results for different values of parameters of the generalized photonsphere of the SFR-model and we find small deviations from the classical results of general relativity (GR)…
The weak field approximation of geodesics in Randers-Finsler space is investigated. We show that a Finsler structure of Randers space corresponds to the constant and sunward anomalous acceleration demonstrated by the Pioneer 10 and 11 data. The additional term in the geodesic equation acts as "electric force", which provides the anomalous acceleration.
We consider a triality between the Zermelo navigation problem, the geodesic flow on a Finslerian geometry of Randers type, and spacetimes in one dimension higher admitting a timelike conformal Killing vector field. From the latter viewpoint, the data of the Zermelo problem are encoded in a (conformally) Painleve-Gullstrand form of the spacetime metric, whereas the data of the Randers problem are encoded in a stationary generalisation of the usual optical metric. We discuss how the spacetime viewpoint gives a simple and physical perspective on various issues, including how Finsler geometries wi…
We present Finslerian perturbation for the $Λ$CDM model, which breaks the isotropic symmetry of the universe. The analysis on the Killing vectors shows that the Randers-Finsler spacetime breaks the isotropic symmetry even if the scalar perturbations of the FRW metric vanish. In Randers-Finsler spacetime, the modified geodesic equation deduces a modified Boltzmann equation. We propose a perturbational version of the gravitational field equation in Randers-Finsler spacetime, where we have omitted the curvature tensor that does not belong to the base space of the tangent bundle. The gravitational…
In this work, we extend the study of Schwarzschild-Finsler-Randers (SFR) spacetime previously investigated by a subset of the present authors (Triantafyllopoulos et al. in Eur Phys J C 80(12):1200, 2020; Kapsabelis et al. in Eur Phys J C 81(11):990, 2021). We will examine the dynamical analysis of geodesics which provides the derivation of the energy and the angular momentum of a particle moving along a geodesic of SFR spacetime. This study allows us to compare our model with the corresponding of general relativity (GR). In addition, the effective potential of SFR model is examined and it is c…
By a Randers' structure on a manifold $M$ we mean a Finsler structure $L^*=L+α$, where $L$ is a Riemannian structure and $α$ is a 1-form on $M$. This structure was first introduced by Randers ~\cite{[8]} from the standpoint of general relativity. In this paper, we replace $L$ by a Finsler structure, calling the resulting manifold a generalized Randers manifold. On one hand, we develop in some depth generalized Randers manifolds. On the other hand, we apply the results obtained in a foregoing paper ~\cite{[12]} to generalized Randers manifolds to obtain some new results in that domain. Among ma…
We study for the first time the dynamical properties and the growth index of linear matter perturbations of the Finsler-Randers (FR) cosmological model, for which we consider that the cosmic fluid contains matter, radiation and a scalar field. Initially, for various FR scenarios we implement a critical point analysis and we find solutions which provide cosmic acceleration and under certain circumstances we can have de-Sitter points as stable late-time attractors. Then we derive the growth index of matter fluctuations in various Finsler-Randers cosmologies. Considering cold dark matter and negl…
We investigate the properties of a real scalar field in the Finslerian Randers spacetime, where the local Lorentz violation is driven by a geometrical background vector. We propose a dynamics for the scalar field by a minimal coupling of the scalar field and the Finsler metric. The coupling is intrinsically defined on the Randers spacetime, and it leads to a non-canonical kinetic term for the scalar field. The nonlinear dynamics can be split into a linear and nonlinear regimes, which depend perturbatively on the even and odd powers of the Lorentz-violating parameter, respectively. We analyze t…
On a Finsler manifold $(M,L)$, we consider the change $L\longrightarrow\bar{L}(x,y)=e^{σ(x)}L(x,y)+β(x,y)$, which we call a $β$-conformal change. This change generalizes various types of changes in Finsler geometry: conformal, $C$-conformal, $h$-conformal, Randers and generalized Randers changes. Under this change, we obtain an explicit expression relating the Cartan connection associated to $(M,L)$ and the transformed Cartan connection associated to $(M,\bar{L})$. We also express some of the fundamental geometric objects (canonical spray, nonlinear connection, torsion tensors, ...etc.) of $(M…
A new geometric method to determine the deflection of light in the equatorial plane of the Kerr solution is presented, whose optical geometry is a surface with a Finsler metric of Randers type. Applying the Gauss-Bonnet theorem to a suitable osculating Riemannian manifold, adapted from a construction by Nazım, it is shown explicitly how the two leading terms of the asymptotic deflection angle of gravitational lensing can be found in this way.
We consider the cosmological evolution in an osculating point Barthel-Randers type geometry, in which to each point of the space-time manifold an arbitrary point vector field is associated. This Finsler type geometry is assumed to describe the physical properties of the gravitational field, as well as the cosmological dynamics. For the Barthel-Randers geometry the connection is given by the Levi-Civita connection of the associated Riemann metric. The generalized Friedmann equations in the Barthel-Randers geometry are obtained by considering that the background Riemannian metric in the Randers …
We investigate the bounce realization in the framework of generalized modified gravities arising from Finsler and Finsler-like geometries. In particular, the richer intrinsic geometrical structure is reflected in the appearance of extra degrees of freedom in the Friedmann equations that can drive the bounce. We examine various Finsler and Finsler-like constructions. In the cases of general very special relativity as well as of Finsler-like gravity on the tangent bundle we show that a bounce cannot be easily obtained. However, in the Finsler-Randers space the induced scalar anisotropy can fulfi…
We analyze the foundations of Finsler gravity theories with metric compatible connections constructed on nonholonomic tangent bundles, or (pseudo) Riemannian manifolds. There are considered "minimal" modifications of Einstein gravity (including theories with violation of local Lorentz invariance) and shown how the general relativity theory and generalizations can be equivalently re--formulated in Finsler like variables. We focus on Finsler branes solutions and perspectives in modern cosmology.
We study the geometric and physical foundations of Finsler gravity theories with metric compatible connections defined on tangent bundles, or (pseudo) Riemannian manifolds). There are analyzed alternatives to Einstein gravity (including theories with broken local Lorentz invariance) and shown how general relativity and modifications can be equivalently re-formulated in Finsler like variables. We focus on prospects in modern cosmology and Finsler acceleration of Universe. All known formalisms are outlined - anholonomic frames with associated nonlinear connection structure, the geometry of the L…
General Very Special Relativity (GVSR) is the curved space-time of Very Special Relativity (VSR) proposed by Cohen and Glashow. The geometry of GVSR possesses a line element of Finsler Geometry proposed by Bogoslovsky. We calculate the Einstein field equations and derive a modified FRW cosmology, for an osculating Riemannian space. The Friedman equation of motion leads to an explanation of the cosmological acceleration in terms of an alternative non-Lorentz invariant theory. A first order approach for a primordial spurionic vector field introduced into the metric, gives back an estimation of t…
We construct a cosmological toy model based on a Finslerian structure of space-time. In particular, we are interested in a specific Finslerian Lorentz violating theory based on a curved version of Cohen and Glashow's Very Special Relativity. The osculation of a Finslerian manifold to a Riemannian leads to the limit of Relativistic Cosmology, for a specified observer. A modified flat FRW cosmology is produced. The analogue of a zero energy particle unfolds some special properties of the dynamics. The kinematical equations of motion are affected by local anisotropies. Seeds of Lorentz Violations…
We discuss the most general Finsler spacetime geometry obeying the cosmological symmetry group $\mathrm{SO}(4)$. On this background geometry we derive the equations of motion for the most general kinetic fluid obeying the same cosmological symmetry. For this purpose we propose a set of coordinates on the tangent bundle of the spacetime manifold which greatly simplifies the cosmological symmetry generators.
This is our reply to "Critical remarks on Finslerian modifications of gravity and cosmology by Zhe Chang and Xin Li", Sergiu I. Vacaru, Phys. Lett. B 690 (2010) 224. It is pointed out that the Finslerian modifications of gravity and cosmology (Zhe Chang and Xin Li, Phys. Lett. B 676 (2009) 173; {\it ibid} 668 (2008) 453) is a suggestion on the generalization of Einstein's gravity and cosmology, but not a proof for theorems in geometry. False or true of the theory should be tested by experiments or observations. We show that the arguments of Sergiu I. Vacaru were based a wrong logic. A personal…
Finsler geometry is an important extension of Riemann geometry, in which to each point of the spacetime manifold an arbitrary internal variable is associated. Interesting Finsler geometries, with many physical applications, are the Randers and Kropina type geometries, respectively. A subclass of Finsler geometries is represented by the osculating Finsler spaces, in which the internal variable is a function of the base manifold coordinates only. In an osculating Finsler geometry one introduces the Barthel connection, which has the remarkable property that it is the Levi-Civita connection of a R…
Within the context of supersymmetric space-time (D-particle) foam in string/brane-theory, we discuss a Finsler-induced Cosmology and its implications for (thermal) Dark Matter abundances. This constitutes a truly microscopic model of dynamical space-time, where Finsler geometries arise naturally. The D-particle foam model involves point-like brane defects (D-particles), which provide the topologically non-trivial foamy structures of space-time. The D-particles can capture and emit stringy matter and this leads to a recoil of D-particles. It is indicated how one effect of such a recoil of D-par…
We generalize the kinetic theory of fluids, in which the description of fluids is based on the geodesic motion of particles, to spacetimes modeled by Finsler geometry. Our results show that Finsler spacetimes are a suitable background for fluid dynamics and that the equation of motion for a collisionless fluid is given by the Liouville equation, as it is also the case for a metric background geometry. We finally apply this model to collisionless dust and a general fluid with cosmological symmetry and derive the corresponding equations of motion. It turns out that the equation of motion for a d…
I do not agree with the authors of papers arXiv:0806.2184 and arXiv:0901.1023v1 (published in Phys. Lett., respectively, B668 (2008) 453 and B676 (2009) 173). They consider that \textit{"In Finsler manifold, there exists a unique linear connection - the Chern connection ... It is torsion freeness and metric compatibility ... "}. There are well known results (for example, presented in monographs by H. Rund and R. Miron and M. Anastasiei) that in Finsler geometry there exist an infinite number of linear connections defined by the same metric structure and that the Chern and Berwald connections \…
We consider models of accelerating Universe elaborated for Finsler like gravity theories constructed on tangent bundles to Lorentz manifolds. In the osculating approximation, certain locally anisotropic configurations are similar to those for f(R) gravity. This allows us to generalize a proposal (by Nojiri, Odintsov and Saez-Gomez, arXiv: 1108.0767) in order to reconstruct and compare two classes of Einstein-Finsler gravity, EFG, and f(R) gravity theories using modern cosmological data and realistic physical scenarios. We conclude that EFG provides inflation, acceleration and little rip evolut…
We show that the equations of motion governing the dynamics of strings in a compact internal space can be written as dispersion relations, with a local speed that depends on the velocity and curvature of the string in the large dimensions. From a $(3+1)$-dimensional perspective these can be viewed as dispersion relations for waves propagating in the string interior and are analogous to those for current-carrying topological defects. This allows us to construct a unified framework with which to study and interpret the internal structure of various field-theoretic and fundamental string species,…
