Critical remarks on Finsler modifications of gravity and cosmology by Zhe Chang and Xin Li

I do not agree with the authors of papers arXiv:0806.2184 and arXiv:0901.1023v1 (published in [Zhe Chang, Xin Li, Phys. Lett. B 668 (2008) 453] and [Zhe Chang, Xin Li, Phys. Lett. B 676 (2009) 173], respectively). They consider that “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 are not metric compatible. For instance, the Chern's one (being with zero torsion and “weak” compatibility on the base manifold of tangent bundle) is not generally compatible with the metric structure on total space. This results in a number of additional difficulties and sophistication in definition of Finsler spinors and Dirac operators and in additional problems with further generalizations for quantum gravity and noncommutative/string/brane/gauge theories. I conclude that standard physics theories can be generalized naturally by gravitational and matter field equations for the Cartan and/or any other Finsler metric compatible connections. This allows us to construct more realistic models of Finsler spacetimes, anisotropic field interactions and cosmology.     

Averaging out the Einstein equations

A general scheme to average out an arbitrary 4-dimensional Riemannian space and to construct the geometry of the averaged space is proposed. It is shown that the averaged manifold has a metric and two equi-affine symmetric connections. The geometry of the space is characterized by the tensors of Riemannian and non-Riemannian curvatures, an affine deformation tensor being the result of non-metricity of one of the connections. To average out the differential Bianchi identities, correlation 2-form, 3-form and 4-form are introduced and the differential relations on these correlations tensors are derived, the relations being integrable on an arbitrary averaged manifold. Upon assuming a splitting rule for the average of the product including a covariantly constant tensor, an averaging out of the Einstein equations has been carried out which brings additional terms with the correlation tensors into them. As shown by averaging out the contracted Bianchi identities, the equations of motion for the averaged energy-momentum tensor do also include the geometric correction terms. Considering the gravitational induction tensor to be the Riemannian curvature tensor (then the non-Riemannian one is the macroscopic gravitational field), a theorem that relates the algebraic structure of the averaged microscopic metric with that of the induction tensor is proved. Due to the theorem the same field operator as in the Einstein equations is manifestly extracted from the averaged ones. Physical interpretation and application of the relations and equations obtained to treat macroscopic gravity are discussed.     

A YANG–MILLS ELECTRODYNAMICS THEORY ON THE HOLOMORPHIC TANGENT BUNDLE

Considering a complex Lagrange space ([24]), in this paper the complex electromagnetic tensor fields are defined as the sum between the differential of the complex Liouville 1-form and the symplectic 2-form of the space relative to the adapted frames of the Chern–Lagrange complex nonlinear connection. In particular, an electrodynamics theory on a complex Finsler space is obtained.

We show that our definition of the complex electrodynamics tensors has physical meaning and these tensors generate an adequate field theory which offers the opportunity of coupling with the gravitation. The generalized complex Maxwell equations are written.

A gauge field theory of electrodynamics on the holomorphic tangent bundle is put over T′M and the gauge invariance to phase transformations is studied. An extension of the Dirac Lagrangian on T′M coupled with the electrodynamics Lagrangian is studied and it offers the framework for a unified gauge theory of fields.     

Towards a gravitation theory in Berwald-Finsler space

Finsler geometry is a natural and fundamental generalization of Riemann geometry. The Finsler structure depends on both coordinates and velocities. It is defined as a function on tangent bundle of a manifold. We use the Bianchi identities satisfied by the Chern curvature to set up a gravitation theory in Berwald-Finsler space. The geometric part of the gravitational field equation is nonsymmetric in general. This indicates that the local Lorentz invariance is violated.     

Spacetime tangent bundle with torsion

It is demonstrated explicitly that the bundle connection of the Finslerspacetime tangent bundle can be made compatible with Cartan's theory of Finsler space by the inclusion of bundle torsion, and without the restriction that the gauge curvature field be vanishing. A component of the contorsion is made to cancel the contribution of the gauge curvature field to the relevant component of the bundle connection. Also, it is shown that the bundle manifold remains almost complex, and that the almost complex structure can be made to have a vanishing covariant derivative if additional conditions on the torsion are satisfied. However, the Finsler-spacetime tangent bundle remains complex only if the gauge curvature field vanishes.     

Towards a gravitation theory in Berwald-Finsler space

Finsler geometry is a natural and fundamental generalization of Riemann geometry. The Finsler structure depends on both coordinates and velocities. It is defined as a function on tangent bundle of a manifold. We use the Bianchi identities satisfied by the Chern curvature to set up a gravitation theory in Berwald-Finsler space. The geometric part of the gravitational field equation is nonsymmetric in general. This indicates that the local Lorentz invariance is violated.     

Geometric classification of the torsion tensor of space‐time

Torsion appears in literature in quite different forms. Generally, spin is considered to be the source of torsion, but there are several other possibilities in which torsion emerges in different contexts. In some cases a phenomenological counterpart is absent, in some other cases torsion arises from sources without spin as a gradient of a scalar field. Accordingly, we propose two classification schemes. The firstone is based on the possibility to construct torsion tensors from the product of a covariant bivector and a vector and their respective space‐time properties. The secondone is obtained by starting from the decomposition of torsion into three irreducible pieces. Their space‐time properties again lead to a complete classification. The classifications found are given in a U ₄, a four dimensional space‐time where the torsion tensors have some peculiar properties. The irreducible decomposition is useful since most of the phenomenological work done for torsion concerns four dimensional cosmological models. In the second part of the paper two applications of these classification schemes are given. The modifications of energy‐momentum tensors are considered that arise due to different sources of torsion. Furthermore, we analyze the contributions of torsion to shear, vorticity, expansion and acceleration. Finally the generalized Raychaudhuri equation is discussed.     

Solution to torsion relations in finsler-spacetime tangent bundle

In the Finsler-spacetime tangent bundle, a simple solution is determined to the torsion relations that were obtained previously to maintain (1) compatibility with Cartan's theory of Finsler space, (2) the almost complex structure, and (3) the vanishing of the covariant derivative of the almost complex structure.     

Modified Newton's gravity in Finsler space as a possible alternative to dark matter hypothesis

A modified Newton's gravity is obtained as the weak field approximation of the Einstein's equation in Finsler space. It is found that a specified Finsler structure makes the modified Newton's gravity equivalent to the modified Newtonian dynamics (MOND). In the framework of Finsler geometry, the flat rotation curves of spiral galaxies can be deduced naturally without invoking dark matter.     

A Class of Anisotropic (Finsler-)Space-time Geometries

A particular Finsler-metric proposed in [1, 2]and describing a geometry with a preferred nulldirection is characterized as belonging to a subclasscontained in a larger class of Finsler-metrics with one or more preferred directions (null, space- or timelike). The metrics are classified according to theirgroup of isometries. These turn out to be isomorphic tosubgroups of the Poincare (Lorentz-) group complemented by the generator of a dilatation.The arising Finsler geometries may be used for theconstruction of relativistic theories testing theisotropy of space. It is shown that the Finsler space with the only preferred null direction is the anisotropic space closest to isotropic Minkowski-spaceof the full class discussed.     

Weak gravitational field in Finsler–Randers space and Raychaudhuri equation

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 the weak field in a generalized Finsler spacetime for gravitational waves are given.     

Deformable gyroscope in a non-euclidean space classical,non-relativistic theory

The classical mechanics of structured test particles in a manifold with affine connection is studied. Gyroscopic rotations and homogeneous deformations are taken into account as internal degrees of freedom. Hence, in addition to the orbital motion of the centre of mass, the body undergoes affine rotations about the centre (“affinely-rigid body”). Configurations of particles are described mathematically by linear frames in an underlying manifold (physical space).Symmetries of the theory are discussed and in some special cases the equations of motion are derived. The orbital motion is found to be influenced by internal degrees of freedom which are dynamically coupled to the geometry of the manifold. For example, in a Riemann-Cartan space ([4], [8], [9], [25]), internal degrees of freedom interact with curvature and torsion tensors. Imposing then some holonomic constraints (orthonormal frames only), one gets the theory of test rigid bodies in a curved space with torsion. In a Riemannian case (no torsion) such a theory seems to coincide with non-relativistic (although spatially non-Euclidean) limit of theories studied by Dixon, Künzle, Tulczyjew and Papapetrou [5], [13], [26], [15].As for all standard techniques and notations of differential geometry, we follow mainly Kobayashi-Nomizu [12] and Sternberg [23].     

Towards a theory of macroscopic gravity

By averaging out Cartan's structure equations for a four-dimensional Riemannian space over space regions, the structure equations for the averaged space have been derived with the procedure being valid on an arbitrary Riemannian space. The averaged space is characterized by a metric, Riemannian and non-Rimannian curvature 2-forms, and correlation 2-, 3- and 4-forms, an affine deformation 1-form being due to the non-metricity of one of two connection 1-forms. Using the procedure for the space-time averaging of the Einstein equations produces the averaged ones with the terms of geometric correction by the correlation tensors. The equations of motion for averaged energy momentum, obtained by averaging out the contracted Bianchi identities, also include such terms. Considering the gravitational induction tensor to be the Riemannian curvature tensor (the non-Riemannian one is then the field tensor), a theorem is proved which relates the algebraic structure of the averaged microscopic metric to that of the induction tensor. It is shown that the averaged Einstein equations can be put in the form of the Einstein equations with the conserved macroscopic energy-momentum tensor of a definite structure including the correlation functions. By using the high-frequency approximation of Isaacson with second-order correction to the microscopic metric, the self-consistency and compatibility of the equations and relations obtained are shown. Macrovacuum turns out to be Ricci non-flat, the macrovacuum source being defined in terms of the correlation functions. In the high-frequency limit the equations are shown to become Isaacson's ones with the macrovauum source becoming Isaacson's stress tensor for gravitational waves.     

On finsler spaces with kropina metric

The curvature tensors and other important tensors of a Finsler space with Kropina metric are studied. We find conditions for such a space to be affinely connected in Berwald's sense.     

Born’s Reciprocal Gravity in Curved Phase-Spaces and the Cosmological Constant

The main features of how to build a Born’s Reciprocal Gravitational theory in curved phase-spaces are developed. By recurring to the nonlinear connection formalism of Finsler geometry a generalized gravitational action in the 8D cotangent space (curved phase space) can be constructed involving sums of 5 distinct types of torsion squared terms and 2 distinct curvature scalars which are associated with the curvature in the horizontal and vertical spaces, respectively. A Kaluza-Klein-like approach to the construction of the curvature of the 8D cotangent space and based on the (torsionless) Levi-Civita connection is provided that yields the observed value of the cosmological constant and the Brans-Dicke-Jordan Gravity action in 4D as two special cases. It is found that the geometry of the momentum space can be linked to the observed value of the cosmological constant when the curvature in space is very large, namely the small size of P is of the order of . Finally we develop a Born’s reciprocal complex gravitational theory as a local gauge theory in 8D of the Quaplectic group that is given by the semi-direct product of U(1,3) with the (noncommutative) Weyl-Heisenberg group involving four coordinates and momenta. The metric is complex with symmetric real components and antisymmetric imaginary ones. An action in 8D involving 2 curvature scalars and torsion squared terms is presented.     

New geometrical approach to Rainich-Misner-Wheeler theory

We introduce a new principal fiber bundle, the bundle of biframes, associated with the geometry of bivectors on spacetime. It is shown that the biframe bundle is a natural geometric arena for modeling the already unified theory of Rainich, Misner, and Wheeler (RMW). The structure equations for the bitorsion inherent in the biframe bundle lead to a generalization of Rainich's algebraic conditions for electromagnetic-type stress tensors which includes sources in a natural way. Besides the usual complexion vector of the RMW theory, an additional new complexion-type vector is found. The generalized algebraic conditions reduce to the usual RMW conditions in the special case of no sources.     

Electromagnetic Field in Finsler and Associated Spaces

The electromagnetic field and its interaction with the leptons is introduced in Finsler space. This space is also considered as the microlocal space-time of the extended hadrons. The field equations for the Finsler space have been obtained from the classical field equations by quantum generalization of this space-time below a fundamental length-scale. On the other hand, the classical field equations are derived from a property of the fields on the autoparallel curve of the Finsler space. The field equations for the associated spaces of the Finsler space, which are macroscopic spaces, such as the large-scale space-time of the universe and the usual Minkowski space-time, can also be obtained for the case of Finslerian bispinor fields separable as the direct products of fields depending on the position coordinates with those depending on the directional arguments. The equations for the coordinate-dependent fields are the usual field equations with the cosmic time-dependent masses of the leptons. The other equations of the directional variable-dependent fields are solved here. Also, the lepton current and the continuity equation are considered. The form-invariance of the field equations under the general coordinate transformations of the Finsler spaces has been discussed.     

路径约束条件下车辆行为的时空演化模型

复杂地理环境下车辆运行线路具有空间三维特征, 它对车辆行为的约束是显然的, 非一维空间和二维空间所能描述. 将复杂地理环境下车辆运行线路抽象为空间曲线, 引入微分几何理论, 利用弧长、曲率和挠率等几何不变量参数建立沿空间曲线运动的Serret-Frenet活动标架; 然后, 对空间曲线上任意一点处Serret-Frenet标架具有时变属性的动态行为进行数学描述, 进而建立路径约束条件下车辆行为的时空演化模型, 并在数学上严格证明了所建时空演化模型适用于车辆(Serret-Frenet标架)直线运行和做匀速圆周运动的特殊情形. 为后续复杂地理环境中交通线路上的车辆跟驰、变道等微观行为和交通流宏观行为研究, 奠定了理论基础.     

A generalized second-order frame bundle for Fréchet manifolds

Working within the framework of Fréchet modelled infinite-dimensional manifolds, we propose a generalized notion of second-order frame bundle. We revise in this way the classical notion of bundles of linear frames of order 2, the direct definition and study of which is problematic due to intrinsic difficulties of the space models. However, this new structure keeps all the fundamental characteristics of a frame bundle. It is a principal Fréchet bundle associated (differentially and geometrically) with the corresponding second-order tangent bundle.