On the concept of relative velocity

Four different concepts of "relative velocity" are given intrinsically in general relativity. The kinematic and kinematic2 relative velocities are defined in the framework of spacelike simultaneity, obtaining some general properties and interpretations. The kinematic relative velocity generalizes the usual concept of relative velocity when the two observers beta, beta' are at the same event On the other hand, the kinemafic2 relative velocity does not generalize this concept, but it is physically interpreted as the variation of the relative position of beta' with respect to beta along the world line of beta. Analogously, the spectroscopic and astrometric relative velocities are defined and studied in the framework of observed (lightlike) simultaneity. Next, we give some relations between these concepts in special relativity. Finally, we show some fundamental examples in Schwarzschild and Robertson-Walker space-times.