Explicit solutions for relativistic acceleration and rotation

TL;DR

This paper introduces a new variable called symmetric velocity to simplify relativistic acceleration and rotation equations, providing explicit solutions for charged particles in electromagnetic fields and describing transformations between accelerated and rotating frames.

Contribution

It presents a novel approach using symmetric velocity to derive explicit solutions for relativistic motion and transformations, enhancing understanding of accelerated and rotating systems.

Findings

Explicit solutions for charge motion in electric and magnetic fields.

Representation of relativistic transformations via conformal maps.

Description of space-time transformations between accelerated and rotating systems.

Abstract

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 dynamic equation. If we introduce a new dynamic variable, called symmetric velocity, the above representation becomes a representation by conformal, instead of projective maps. In this variable, the relativistic dynamic equation for systems with an invariant plane, becomes a non-linear analytic equation in one complex variable. We obtain explicit solutions for the motion of a charge in uniform, mutually perpendicular electric and magnetic fields. By assuming the Clock Hypothesis and using these solutions, we are able to describe the space-time transformations between two uniformly accelerated and rotating systems.