  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.
