Intrinsic definitions of "relative velocity" in general relativity

TL;DR

This paper introduces four intrinsic definitions of relative velocity in general relativity, applicable in various spacetime frameworks, and explores their properties, relations, and examples in specific spacetimes.

Contribution

It provides four coordinate-independent definitions of relative velocity in general relativity, connecting spacelike and lightlike simultaneity frameworks.

Findings

Two definitions based on spacelike simultaneity

Two definitions based on lightlike simultaneity

Relations between definitions in specific spacetimes

Abstract

Given two observers, we define the "relative velocity" of one observer with respect to the other in four different ways. All four definitions are given intrinsically, i.e. independently of any coordinate system. Two of them are given in the framework of spacelike simultaneity and, analogously, the other two are given in the framework of observed (lightlike) simultaneity. Properties and physical interpretations are discussed. Finally, we study relations between them in special relativity, and we give some examples in Schwarzschild and Robertson-Walker spacetimes.