Relativistic motion under constant force: velocity and acceleration behavior

TL;DR

This paper explores the behavior of velocity and acceleration in relativistic motion under constant force, revealing new insights into their directional relationships and the non-monotonic nature of acceleration decay.

Contribution

It provides a detailed analysis of relativistic particle motion under constant force, correcting previous misconceptions about acceleration behavior and velocity limits.

Findings

Final velocity is always parallel to the force, regardless of initial velocities.

Negative acceleration components can exist in the direction of the largest force.

Acceleration does not decrease monotonically to zero as previously claimed.

Abstract

In relativistic dynamics, force and acceleration are no longer parallel. In this article, we revisit the relativistic motion of a particle under the action of a constant force, $\boldsymbol{f}$. \ For a two-dimensional motion, the final velocity in each axis is $(f_{i}/f)c,$ independently of the initial velocities, yielding an asymptotic velocity always parallel to the force. The particular case in which the force is applied in a single axis is analyzed in detail, with the behavior of velocity and acceleration being exhibited for several configurations. Some previous results of the literature concerning velocity and acceleration behavior are improved and better explored. Differently from which was previously claimed, it is shown that a negative acceleration component can exist in the direction of the biggest force component and that acceleration does not decrease monotonically to zero.