Kinematic relative velocity with respect to stationary observers in Schwarzschild spacetime

TL;DR

This paper investigates the kinematic relative velocity of test particles in Schwarzschild spacetime, revealing its independence from the observer and its dependence solely on the orbit radius for circular geodesics.

Contribution

It introduces a new analysis of kinematic relative velocity in Schwarzschild spacetime, showing its invariance across observers and its relation to orbit radius.

Findings

Kinematic relative velocity modulus is observer-independent.

For circular geodesic orbits, velocity modulus depends only on orbit radius.

Results generalize previous findings on stationary and radial free-falling particles.

Abstract

We study the kinematic relative velocity of general test particles with respect to stationary observers (using spherical coordinates) in Schwarzschild spacetime, obtaining that its modulus does not depend on the observer, unlike Fermi, spectroscopic and astrometric relative velocities. We study some fundamental particular cases, generalizing some results given in other work about stationary and radial free-falling test particles. Moreover, we give a new result about test particles with circular geodesic orbits: the modulus of their kinematic relative velocity with respect to any stationary observer depends only on the radius of the circular orbit, and so, it remains constant.