# Velocities of distant objects in General Relativity revisited

## TL;DR

This paper revisits the definitions of velocities of distant objects in General Relativity, generalizing existing formulas and highlighting counter-intuitive properties to guide correct application.

## Contribution

It generalizes the velocity relation in FRW metrics to arbitrary spherically symmetric metrics and clarifies their counter-intuitive properties.

## Key findings

- Generalized velocity formula for spherically symmetric metrics.
- Identified counter-intuitive properties of velocity definitions.
- Provided guidelines for appropriate use of these velocity concepts.

## Abstract

We consider two most popular definitions of velocities of remote objects in General Relativity. Our work has two motivations. From a research point of view, we generalize the formula connecting these two velocities in FRW metrics found by Chodorowski to arbitrary synchronous spherically symmetric metrics. From a methodological point of view, our goal is to outline certain counter-intuitive properties of the definitions in question, which would allow to use them when it is reasonable and to avoid incorrect statements, based on inappropriate use of intuition.