170 research outputs found
Finsleroid-Finsler Space of Involutive Case
The Finsleroid-Finsler space is constructed over an underlying Riemannian
space by the help of a scalar and an input 1-form 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 may vary in the direction assigned by ,
such that with a scalar .
We show by required calculation that the involutive case realizes through the
-special relation the picture that instead of the Landsberg condition we have the vanishing \dot{\al}_{ijk}=0 with the normalized tensor
\al_{ijk}=A_{ijk}/||A||. Under the involutive condition, the derivative
tensor and the curvature tensor have explicitly been found,
assuming the input 1-form 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
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