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

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.

16 pages. arXiv admin note: substantial text overlap with arXiv:1105.0492

References in corpus (2)

• Accelerating relativistic reference frames in Minkowski space-time
• Dynamics of hydrogen-like atom bounded by maximal acceleration

Cited by in corpus (6)

• Uniform Acceleration in General Relativity
• A covariant formulation of relativistic mechanics
• Observers with constant proper acceleration, constant proper jerk, and beyond
• Synchronization in Uniformly Accelerated Frames
• Fermion and gluon spectral functions far from equilibrium
• On a scalar field of time and electromagnetism