Covariant Uniform Acceleration

We show that standard Relativistic Dynamics Equation F=dp/d\tau is only partially covariant. To achieve full Lorentz covariance, we replace the four-force F by a rank 2 antisymmetric tensor acting on the four-velocity. By taking this tensor to be constant, we obtain a covariant definition of uniformly accelerated motion. We compute explicit solutions for uniformly accelerated motion which are divided into four types: null, linear, rotational, and general. For null acceleration, the worldline is cubic in the time. Linear acceleration covariantly extends 1D hyperbolic motion, while rotational acceleration covariantly extends pure rotational motion. We use Generalized Fermi-Walker transport to construct a uniformly accelerated family of inertial frames which are instantaneously comoving to a uniformly accelerated observer. We explain the connection between our approach and that of Mashhoon. We show that our solutions of uniformly accelerated motion have constant acceleration in the comoving frame. Assuming the Weak Hypothesis of Locality, we obtain local spacetime transformations from a uniformly accelerated frame K' to an inertial frame K. The spacetime transformations between two uniformly accelerated frames with the same acceleration are Lorentz. We compute the metric at an arbitrary point of a uniformly accelerated frame. We obtain velocity and acceleration transformations from a uniformly accelerated system K' to an inertial frame K. We derive the general formula for the time dilation between accelerated clocks. We obtain a formula for the angular velocity of a uniformly accelerated object. Every rest point of K' is uniformly accelerated, and its acceleration is a function of the observer's acceleration and its position. We obtain an interpretation of the Lorentz-Abraham-Dirac equation as an acceleration transformation from K' to K.Comment: 36 page

Topological and Borel compactifications of Polish G-spaces

AbstractWe investigate the relationships between topological and Borel G-spaces, where G is a Polish group. We show that every Polish G-space can be topologically and equivariantly embedded into a compact Polish G-space iff G is locally compact. This answers a question of Kechris. It also provides a striking contrast to the recent result of Becker and Kechris which states that every Borel G-space can be Borel-embedded into a compact Polish G-space

Physical applications of homogeneous balls

One of the mathematical challenges of modern physics lies in the development of new tools to efficiently describe different branches of physics within one mathematical framework. This text introduces precisely such a broad mathematical model, one that gives a clear geometric expression of the symmetry of physical laws and is entirely determined by that symmetry. The first three chapters discuss the occurrence of bounded symmetric domains (BSDs) or homogeneous balls and their algebraic structure in physics. The book further provides a discussion of how to obtain a triple algebraic structure as

Physical Applications of Homogeneous Balls

One of the mathematical challenges of modern physics lies in the development of new tools to efficiently describe different branches of physics within one mathematical framework. This text introduces precisely such a broad mathematical model, one that gives a clear geometric expression of the symmetry of physical laws and is entirely determined by that symmetry. The first three chapters discuss the occurrence of bounded symmetric domains (BSDs) or homogeneous balls and their algebraic structure in physics. The book further provides a discussion of how to obtain a triple algebraic structure as

Constructing Tychonoff G-spaces which are not G-Tychonoff

Jan de Vries' compactification problem is whether every Tychonoff G- space can be equivariantly embedded in a compact G-space. In such a case, we say that G is a V -group. De Vries showed that every locally compact group G is a V -group. The first example of a non-V -group was constructed in 1988 by the first author. Until now, this was the only known counterexample. In this paper, we give a systematic method of constructing non-compactificable G-spaces. We show that the class of non-V -groups is large and contains all @0-bounded non-locally precompact groups. This establishes the existence of monothetic (even cyclic) non-V -groups, answering a question of the first author. As a related result, we obtain a characterization of locally compact groups in terms of "G-normality." 1

The Pre-Potential of a Field Propagating with the Speed of Light and Its Dual Symmetry

Relativity theory assumes that force fields propagate with the speed of light. We show that such force fields generated by a single source can be described by a pre-potential, which is a complex-valued function on spacetime outside the worldline of the source. The pre-potential is invariant under a spin-half representation of the Lorentz group acting on complexified spacetime. The complex four-potential of such a field is defined and calculated explicitly from the pre-potential without assuming any particular force law for the field. The real part of the obtained four-potential coincides with the known Liénard–Wiechert potential. The symmetry of the four-potential is described herein. The pre-potential satisfies the wave equation. The single source electromagnetic field derived from this four-potential is self-dual or anti-self-dual. The pre-potential and the four-potential are extended to a field with several sources