Finsleroid-Finsler Space of Involutive Case

The Finsleroid-Finsler space is constructed over an underlying Riemannian space by the help of a scalar $g(x)$ and an input 1-form $b$ of unit length. Explicit form of the entailed tensors, as well as the respective spray coefficients, is evaluated. The involutive case means the framework in which the characteristic scalar $g(x)$ may vary in the direction assigned by $b$, such that $dg=\mu b$ with a scalar $\mu(x)$. We show by required calculation that the involutive case realizes through the $A$-special relation the picture that instead of the Landsberg condition $\dot A_{ijk}=0$ we have the vanishing \dot{\al}_{ijk}=0 with the normalized tensor \al_{ijk}=A_{ijk}/||A||. Under the involutive condition, the derivative tensor $A_{i|j}$ and the curvature tensor $R^i{}_k$ have explicitly been found, assuming the input 1-form $b$ be parallel. Key words: Finsler metrics, spray coefficients, curvature tensors

Finsleroid--Finsler Space and Spray Coefficients

In the previous work, the notion of the Finsleroid--Finsler space have been formulated and the necessary and sufficient conditions for the space to be of the Landsberg type have been found. In the present paper, starting with particular spray coefficients, we demonstrate how the Landsberg condition can explicitly appear in case of the Finsleroid--type metric function. Calculations are supplementing by a convenient special Maple--program. The general form of the associated geodesic spray coefficients is presented for such metric function under the condition of constancy of the Finsleroid charge. Key words: Finsler geometry, metric spaces, spray

Pseudo-Finsleroid metric function of spatially anisotropic relativistic type

The paper contributes to the important and urgent problem to extend the physical theory of space-time in a Finsler-type way under the assumption that the isotropy of space is violated by a single geometrically distinguished spatial direction which destroys the pseudo-Euclidean geometric nature of the relativistic metric and space. It proves possible to retain the fundamental geometrical property that the indicatrix should be of the constant curvature. Similar property appears to hold in the three-dimensional section space. The last property was the characteristic of three-dimensional positive-definite Finsleroid space proposed and developed in the previous work, so that the present paper lifts that space to the four-dimensional relativistic level. The respective pseudo-Finsleroid metric function is indicated. Numerous significant tensorial and geometrical consequences have been elucidated. \ses {\bf Keywords:} Finsler metrics, relativistic spaces

Finsleroid--Relativistic Space Endowed With Scalar Product

When a single time-like vector is distinguished geometrically to present the only preferred direction in extending the pseudoeuclidean geometry, the hyperboloid may not be regarded as an exact carrier of the unit-vector image. So under respective conditions one may expect that some time-assymetric figure should be substituted with the hyperboloid. To this end we shall use the pseudo-Finsleroid. The spatial-rotational invariance (the P-parity) is retained. The constant negative curvature is the fundamental property of the pseudo-Finsleroid surface. The present paper develops the approach in the direction of evidencing the concepts of angle, scalar product, and geodesics. In Appendices we shortly outline the basic aspects that stem from the choice of the Finsleroid-relativistic metric functions

Finslerian Extension of Lorentz Transformations and First-Order Censorship Theorem

Granted the post-Lorentzian relativistic kinematic transformations are described in the Finslerian framework, the uniformity between the actual light velocity anisotropy change and the anisotropic deformation of measuring rods can be the reason proper for the null results of the Michelson-Morley-type experiments at the first-order level.Comment: 6 pages, LaTeX. : Final version, accepted for publication in the April issue of Found. Phys. Let

Finslerian Post-Lorentzian Kinematic Transformations in Anisotropic-Space Case

The Finslerian post-Lorentzian kinematic transformations can explicitly be obtained under uni-directional breakdown of spatial isotropy, provided that the requirement that the relativistic unit hypersurface (indicatrix or mass shell) be a space of constant negative curvature is still fulfilled. The method consists in evaluating respective Finslerian tetrads and then treating them as the bases of inertial reference frames. The Transport Synchronization has rigorously been proven, which opens up the ways proper to favour the concept of one-way light velocity. Transition to the Hamiltonian treatment is straightforward, so that the Finslerian transformation laws for momenta and frequences, as well as due Finslerian corrections to Doppler effect, become clear. An important common feature of the ordinary pseudo-Euclidean theory of special relativity and of the Finslerian relativistic approach under study is that they both endeavour to establish a universal prescription for applying the theory to systems in differing states of motion

Finsleroid--Finsler Parallelism

The Finsleroid--induced scalar product, and hence the angle, proves to remain unchanged under the Finsleroid--type parallel transportation of involved vectors in the Landsberg case. The two--vector extension of the Finsleroid metric tensor is proposed

Finsleroid-Space Supplemented by Angle

Our previous exploration of the \cE_g^{PD}-geometry has shown that the field is promising. Namely, the \cE_g^{PD}-approach is amenable to development of novel trends in relativistic and metric differential geometry and can particularly be effective in context of the Finslerian or Minkowskian Geometries. The main point of the present paper is the tenet that the \cE_g^{PD}-space-associated one-vector Finslerian metric function admits in quite a natural way an attractive two-vector extension, thereby giving rise to angle and scalar product. The underlying idea is to derive the angular measure from the solutions to the geodesic equation, which prove to be obtainable in an explicit simple form. The respective investigation is presented in Part I. Part II serves as an extended Addendum enclosing the material which is primary for the \cE_g^{PD}-space. The Finsleroid, instead of the unit sphere, is taken now as carrier proper of the spherical image. The indicatrix is, of course, our primary tool

Two-axes pseudo-Finsleroid metrics: general overview and angle-regular solution

The class of the two-axes pseudo-Finslerian metrics which is specified by the condition of the angle-separation in the involved characteristic functions is proposed and studied. The complete Total Set of algebraic and differential equations is derived in all rigor which are necessary and sufficient in order that a pseudo-Finsleroid metric function belong to the class. It proves possible to solve the equations of the set. The angle-regular solution of the Finsleroid-in-pseudo-Finsleroid type is found and described in detail

Finslerian grounds for four--directional anisotropic kinematics

Upon straightforward four--directional extension of the special--relativistic two--dimensional transformations to the four--dimensional case we lead to convenient totally anisotropic kinematic transformations, which prove to reveal many remarkable group and invariance properties. Such a promise is shown to ground the basic manifold with the Finslerian fourth-root metric function to measure length of relativistic four--vectors. Conversion to the framework of relativistic four--momentum is also elucidated. The relativity principle is strictly retained. An interesting particular algebra for subtraction and composition of three-dimensional relative velocities is arisen. The correspondence principle is operative in the sense that at small relative velocities the transformations introduced tend approximately to ordinary Lorentzian precursors. The transport synchronization remains valid. Abbreviation RF will be used for (inertial) reference frames. {\bf Keywords:} special relativity, invariance, Finsler geometry