Special Theory of Relativity in Hyperbolic Functions

Karapetoff, Vladimir

doi:10.1103/revmodphys.16.33

Cited by 5 publications

(14 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, in Minkowski's theory, rapidity is an imaginary quantity iθ ′ and has a quasi-Euclidean dependence, which does not allow for an analysis of the dynamics of a relativistic particle using the real part of the particle velocity. The second method [12][13][14][15][16] uses Lobachevsky space, where the rapidity θ is a real and positive quantity. The second method will be applied in this article, since in Lobachevsky space, the negative curvature and radius of curvature of space are equal to the speed of light.…”

Section: Introductionmentioning

confidence: 99%

“…Varicak [14] continued to develop the application of hyperbolic functions in special relativity. Karapetoff also made a significant contribution to special relativity with hyperbolic functions, demonstrating their advantages for describing various physical processes [15,16].…”

Section: Introductionmentioning

confidence: 99%

“…The main novelty of References [15,16] was that the connection between proper spacetime coordinates in different inertial systems was presented in terms of the Lorentz boost with argument θ. However, the proper coordinates and proper time in the inertial system K were considered to be dynamic quantities independent of θ, and the connection q/t = tanhθ was only introduced for a special case.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Special Relativity in Terms of Hyperbolic Functions with Coupled Parameters in 3+1 Dimensions

Akintsov,

Nevecheria,

Kopytov

et al. 2024

Symmetry

View full text Add to dashboard Cite

This paper presents a method for parameterizing new Lorentz spacetime coordinates based on coupled parameters. The role of symmetry in rapidity in special relativity is explored, and invariance is obtained for new spacetime intervals with respect to the Lorentz transformation. Using the Euler–Hamilton equations, an additional angular rapidity and perpendicular rapidity are obtained, and the Hamiltonian and Lagrangian of a relativistic particle are expanded into rapidity spectra. A so-called passage to the limit is introduced that makes it possible to decompose physical quantities into spectra in terms of elementary functions when explicit decomposition is difficult. New rapidity-dependent Lorentz spacetime coordinates are obtained. The descriptions of particle motion using the old and new Lorentz spacetime coordinates as applied to plane laser pulses are compared in terms of the particle kinetic energy. Based on a classical model of particle motion in the field of a plane monochromatic electromagnetic wave and that of a plane laser pulse, rapidity-dependent spectral decompositions into elementary functions are presented, and the Euler–Hamilton equations are derived as rapidity functions in 3+1 dimensions. The new and old Lorentz spacetime coordinates are compared with the Fermi spacetime coordinates. The proper Lorentz groups SO(1,3) with coupled parameters using the old and new Lorentz spacetime coordinates are also compared. As a special case, the application of Lorentz spacetime coordinates to a relativistic hydrodynamic system with coupled parameters in 1+1 dimensions is demonstrated.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Special Relativity in Terms of Hyperbolic Functions with Coupled Parameters in 3+1 Dimensions

Akintsov,

Nevecheria,

Kopytov

et al. 2024

Symmetry

View full text Add to dashboard Cite

show abstract

“…*l Firstly, to satisfy Eq. (2), a relation (3) between the two constants W and p is required. In order that the relation _.,.…”

Section: §I Introductionmentioning

confidence: 99%

“…(6) and Dirac 4-4 matrices. Here making much of real quantities, we prefer the Hermitian matrices a, (3 to the so-called r"', as the expressions of the Dirac matrices. To begin with, for the density ¢*¢ and the expectation values of a:c, ay, az, we have (8) This quartet forms obviously a 4-vector and may be abbreviated as follows, i) the 4-vector :…”

Section: §I Introductionmentioning

confidence: 99%

Explicit Representation of the Transformation Property of Physical Quantity of Dirac Electron with the Hyperbolic Metric of Velocity

Yamasaki

1965

Prog. Theor. Phys.

View full text Add to dashboard Cite

This paper aims, in the theory of Dirac electron, at the deliberate use of the hyperbolic metric of velocity, the effectiveness of which Smorodinski1 has recently emphasized in relativity. The metric of velocity is fully used together w~th a proper spin direction vector, to represent explicitly spinor cf; and vector, tensor, etc., of cf;* (matrix) cf;, of free Dirac electron. The 3-vector polarization operator is confirmed to correspond to the proper spin vector. For the electron in a square well potential, cf; is a product of hyperbolic function of the metric with trigonometric or the Bessel function, etc., and the metric turns out to be imaginary or complex in the case of the bound state or free negative energy state. A classical model of Dirac electron is established concretely by obtaining simple time derivatives of the metric and spin vector, from the covariant equations of a spinning particle under general fields, suggested by Frenkel, Kramers and others. The so-called Thomas factor 1/2 is brought inevitably in the equation of the spin vector itself, regardless of the time average motion. These explicit representations may facilitate the visualization of Dirac electron. §I. IntroductionThe hyperbolic metric V of velocity, the so-called rapidity, may be expressed asfor the relativistic velocity v = dr/ dt. In the representation of the Lorentz transformation, the imaginary rotation angle or hyperbolic one which has been preferably used in the forms of cosh V, sinh V and cosh (V /2), sinh (V /2), etc., stands exactly for this metric. The significance of the usage of this metric has been clarified in the old text of Pauli 1 ) on relativity, and also in the recent one of F ock. 2 ) In 1944, Karapetoff 3 l used actually this metric in the regions of relativistic mechanics and electrodynamics. More recently, Srnorodinski1 4 l has explained that the relativistic velocity space is isomorphic to the Lobachevskian space, namely, hyperbolic space, and has applied this fact to the calculations of collisions of particles. The conscious use of this metric, however, is not yet prevalent in most of many works on relativity, although the convenience of this *> Some parts of this paper are the revised translation of the author's work published in Rep.

show abstract

References

1971

Additive and Polynomial Representations

View full text Add to dashboard Cite

Special Theory of Relativity in Hyperbolic Functions

Cited by 5 publications

References 11 publications

Special Relativity in Terms of Hyperbolic Functions with Coupled Parameters in 3+1 Dimensions

Special Relativity in Terms of Hyperbolic Functions with Coupled Parameters in 3+1 Dimensions

Explicit Representation of the Transformation Property of Physical Quantity of Dirac Electron with the Hyperbolic Metric of Velocity

References

Contact Info

Product

Resources

About