A note on the computation of geometrically defined relative velocities

TL;DR

This paper examines how different geometrically defined relative velocities in general relativity depend on observer and test particle properties, highlighting differences and proposing open problems for intrinsic expressions of Fermi and astrometric velocities.

Contribution

It clarifies the dependence of various relative velocities on observer and test particle data and introduces an open problem for deriving intrinsic expressions without evolving relative positions.

Findings

Kinematic and spectroscopic velocities depend only on 4-velocities.

Fermi and astrometric velocities depend on acceleration and relative position.

An open problem is proposed for intrinsic expressions of Fermi and astrometric velocities.

Abstract

We discuss some aspects about the computation of kinematic, spectroscopic, Fermi and astrometric relative velocities that are geometrically defined in general relativity. Mainly, we state that kinematic and spectroscopic relative velocities only depend on the 4-velocities of the observer and the test particle, unlike Fermi and astrometric relative velocities, that also depend on the acceleration of the observer and the corresponding relative position of the test particle, but only at the event of observation and not around it, as it would be deduced, in principle, from the definition of these velocities. Finally, we propose an open problem in general relativity that consists on finding intrinsic expressions for Fermi and astrometric relative velocities avoiding terms that involve the evolution of the relative position of the test particle. For this purpose, the proofs given in this paper can serve as inspiration.