Relativistic hyperbolic motion and its higher order kinematic quantities

TL;DR

This paper derives explicit formulas for all higher-order kinematic quantities of relativistic hyperbolic motion in Minkowski spacetime, providing general solutions and exploring modified Rindler worldlines.

Contribution

It presents the first general solutions for higher derivatives of relativistic hyperbolic motion, extending previous work by considering more general velocity parameterizations.

Findings

Explicit formulas for kinematic quantities up to the fourth derivative.

General solutions for the differential equations governing hyperbolic motion.

Identification of a class of modified Rindler hyperbolic worldlines.

Abstract

We investigate the kinematics of the motion of an observer with constant proper acceleration (relativistic hyperbolic motion) in 1+1 and 1+3 dimensional Minkowski spacetimes. We provide explicit formulas for all the kinematic quantities up to the fourth proper time derivative (the Snap). In the 1+3 case, following a recent work of Pons and de Palol [Gen. Rel. Grav. 51 (2019) 80], a vectorial differential equation for the acceleration is obtained which by considering constant proper acceleration is turned into a nonlinear second order differential equation in terms of derivatives of the radius vector. If, furthermore, the velocity is parameterized in terms of hyperbolic functions, one obtains a differential equation to solve for the argument f(s) of the velocity. Differently from Pons and de Palol, who employed the particular solution, linear in the proper time s, we obtain the general solution and use it to work out more general expressions for the kinematical quantities. As a byproduct, we obtain a class of modified Rindler hyperbolic worldlines characterized by supplementary contributions to the components of the kinematical quantities.