A tale of two velocities: Threading vs Slicing
Two principal definitions of a 3-velocity assigned to a test particle following timelike trajectories in stationary spacetimes are introduced and analyzed systematically. These definitions are based on the $1+3$ (threading) and $3+1$ (slicing) spacetime decomposition formalisms and defined relative to two different sets of observers. After showing that Synge's definition of spatial distance and 3-velocity are equivalent to those defined in the $1+3$ (threading) formalism, we exemplify differences between the two definitions, by calculating them for particles in circular orbits in axially symmetric stationary spacetimes. Illustrating its geometric nature, the relative linear velocity between the corresponding observers is obtained in terms of the spacetime metric components. Circular particle orbits in the Kerr spacetime, as the prototype and the most well known of stationary spacetimes, are examined with respect to these definitions to highlight their observer-dependent nature. We also examine the Kerr-NUT spacetime in which the NUT parameter contributing to the off-diagonal terms in the metric is mainly interpreted not as a rotation parameter but as a gravitomagnetic monopole charge. Finally, in a specific astrophysical setup which includes rotating black holes, it is shown how these local definitions are related to the velocity measurements made by distant observers using spectral line shifts.