Related papers: Visualizing Imaginary Rotations and Applications i…
In present work the generalization of Einstein's special theory of relativity on 5-dimentional space is considered, in which as fifth coordinates we consider the interval s of a particle. 5-dimentional vectors in this space are isotropic…
We discuss the most elementary properties of the hyperbolic trigonometry and show how they can be exploited to get a simple, albeit interesting, geometrical interpretation of the special relativity. It yields indeed a straightforword…
Robotic motion optimization often focuses on task-specific solutions, overlooking fundamental motion principles. Building on Riemannian geometry and the calculus of variations (often appearing as indirect methods of optimal control), we…
The Lorentz transformations are represented by Einstein velocity addition on the ball of relativistically admissible velocities. This representation is by projective maps. The Lie algebra of this representation defines the relativistic…
In geometric algebra, the rotation of a vector is described using rotors. Rotors are phasors where the imaginary number has been replaced by a oriented plane element of unit area called a unit bivector. The algebra in three dimensional…
We provide a one-to-one map between the spin correlations and certain three-dimensional shapes, analogous to the map between single spins and Bloch vectors, and demonstrate its utility. Much as one can reason geometrically about dynamics…
This article develops a variational formulation of relativistic nature applicable to the quantum mechanics context. The main results are obtained through basic concepts on Riemannian geometry. Standards definitions such as vector fields and…
We attempt to see how closely we can formally obtain the planetary and light path equations of General Relativity by employing certain operations on the familiar Newtonian equation. This article is intended neither as an alternative to nor…
We compare the nature of electromagnetic fields and of gravitational fields in linearized general relativity. We carry out this comparison both mathematically and visually. In particular the "lines of force" visualizations of…
We combine bare perturbation theory with the imaginary time evolution technique to study one-loop radiative corrections to various components of angular momentum of the electron. Our investigations are based on the canonical decomposition…
The principles of the special theory of relativity are extremely simple. A knowledge of the Pythagorean theorem and an ability to perform the simplest algebraic operations are sufficient to be conversant with the kinematics of the special…
The use of Hilbert curves to visualize massive vector of data is revisited following previous authors. The Hilbert curve mapping preserves locality and makes meaningful representation of the data. We call such visualization as Hilbert…
Lorentz's group represented by the hypercomplex system of numbers, which is based on dirac matrices, is investigated. This representation is similar to the space rotation representation by quaternions. This representation has several…
High-resolution numerical simulations are utilized to examine isotropic turbulence in a compressible fluid when long wavelength velocity fluctuations approach light speed. Spectral analysis reveals an inertial sub-range of relativistic…
General relativistic quantum dynamics of twisted (vortex) Dirac particles is constructed. The Hamiltonian and equations of motion in the Foldy-Wouthuysen representation are derived for a twisted relativistic electron in arbitrary electric…
The article reviews some of the (fairly scattered) information available in the mathematical literature on the subject of angles in complex vector spaces. The following angles and their relations are considered: Euclidean, complex, and…
Applications of the three-dimensional transformation for rotating coordinate systems to quantum mechanics, general theory relativity and optics are considered.
We consider the role of the velocity in Lorentz-violating fermionic quantum theory, especially emphasizing the nonrelativistic regime. Information about the velocity will be important for the kinematical analysis of scattering and other…
We find relations between quantities defining geometry and quantities defining the length of a curve in geometries underlying Electromagnetism and unified model of Electromagnetism and Gravitation. We show that the length of a vector…
It is demonstrated how the right hand sides of the Lorentz Transformation equations may be written, in a Lorentz invariant manner, as 4--vector scalar products. This implies the existence of invariant length intervals analogous to invariant…