An algorithm for computing geometric relative velocities through Fermi and observational coordinates

TL;DR

This paper introduces a numerical method to compute various types of relative velocities between a distant test particle and an observer using Fermi and observational coordinates, extending previous concepts to complex spacetime regions.

Contribution

The paper develops a numerical approach for calculating Fermi and observational coordinates and extends relative velocity concepts to non-convex neighborhoods in Schwarzschild and Kerr spacetimes.

Findings

Successfully computes relative velocities in Schwarzschild and Kerr spacetimes.

Extends relative velocity concepts to non-convex neighborhoods.

Provides convergence tests and an alternative computational method.

Abstract

We present a numerical method for computing the \textit{Fermi} and \textit{observational coordinates} of a distant test particle with respect to an observer. We apply this method for computing some previously introduced concepts of relative velocity: \textit{kinematic}, \textit{Fermi}, \textit{spectroscopic} and \textit{astrometric} relative velocities. We also extend these concepts to non-convex normal neighborhoods and we make some convergence tests, studying some fundamental examples in Schwarzschild and Kerr spacetimes. Finally, we show an alternative method for computing the Fermi and astrometric relative velocities.