Making the Relativistic Dynamics Equation Covariant: Explicit Solutions for Motion under a Constant Force

TL;DR

This paper derives a fully covariant 4D relativistic dynamics equation, providing explicit solutions for uniformly accelerated motion categorized into four Lorentz-invariant types, addressing a longstanding problem in relativistic physics.

Contribution

It introduces a fully Lorentz covariant 4D relativistic dynamics equation using an antisymmetric tensor, and computes explicit solutions for different types of uniformly accelerated motion.

Findings

Derived a fully covariant 4D relativistic dynamics equation.

Classified solutions into null, linear, rotational, and general types.

Provided explicit solutions for each type of uniformly accelerated motion.

Abstract

We derive a 4D covariant Relativistic Dynamics Equation. This equation canonically extends the 3D relativistic dynamics equation $\mathbf{F}=\frac{d\mathbf{p}}{dt}$, where $\mathbf{F}$ is the 3D force and $\mathbf{p}=m_0\gamma\mathbf{v}$ is the 3D relativistic momentum. The standard 4D equation $F=\frac{dp}{d\tau}$ 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. This solves a problem of Einstein and Planck. We compute explicit solutions for uniformly accelerated motion. The solutions are divided into four Lorentz-invariant 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.