Making the Relativistic Dynamics Equation Covariant: Explicit Solutions   for Motion under a Constant Force
  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. 
Yaakov Friedman and Tzvi Scarr
physics.gen-ph