Non-coordinate time/velocity pairs in special relativity

Abstract

Motions with respect to one inertial (or ``map'') frame are often described in terms of the coordinate time/velocity pair (or ``kinematic'') of the map frame itself. Since not all observers experience time in the same way, other time/velocity pairs describe map-frame trajectories as well. Such coexisting kinematics provide alternate variables to describe acceleration. We outline a general strategy to examine these. For example, Galileo's acceleration equations describe unidirectional relativistic motion {\it exactly} if one uses $V=\frac{dx}{dT}$, where $x$ is map-frame position and $T$ is clock time in a chase plane moving such that $γ^{\prime }=γ\sqrt{\frac 2{γ+1}}$. Velocity in the traveler's kinematic, on the other hand, has dynamical and transformational properties which were lost by coordinate-velocity in the transition to Minkowski space-time. Its repeated appearance with coordinate time, when expressing relationships in simplest form, suggests complementarity between traveler and coordinate kinematic views.