Uniformly accelerated traveler in an FLRW universe

TL;DR

This paper derives a general analytical solution for accelerated and geodesic motion in FLRW universes using conformal time transformations, and explores conditions for return journeys in de Sitter spacetime.

Contribution

It introduces a method to convert second order equations to first order in conformal spacetime, providing closed-form solutions for accelerated motion in curved FLRW universes.

Findings

Derived closed-form solutions for accelerated motion in FLRW spacetime.

Identified conditions for return journeys in de Sitter universe.

Applied the method to spatially flat FLRW cases.

Abstract

This paper provides an analytical treatment of accelerated and geodesic motion within the framework of the Friedmann -Lemaitre-Robertson-Walker (FLRW) spacetime. By employing conformal time transformations we manage to convert second order differential equations of motion in FLRW spacetime to first order equations in the conformally transformed spacetime. This allows us to derive a general analytical solution in closed-form for accelerated motion in spatially curved FLRW spacetime. We provide few examples of this general solution for the spatially flat cases. The last part of our work focuses on the return journey for a traveler exploring a FLRW universe. We derive certain condition for de Sitter universe that must be satisfied in order to have an actual return journey.