Global Lorentzian geometry from lightlike geodesics: What does an observer in (2+1)-gravity see?

TL;DR

This paper demonstrates how an observer in (2+1)-gravity can measure light signals to reconstruct the entire geometry of flat Lorentzian 3D spacetimes using realistic light-based measurements.

Contribution

It introduces a method for an observer to determine the full geometry of (2+1)-gravity spacetimes through practical light-based measurements.

Findings

Holonomy variables can be measured via returning light signals.

Full spacetime geometry can be reconstructed in finite eigentime.

Measurements include elapsed time, emission directions, and frequency shifts.

Abstract

We show how an observer could measure the non-local holonomy variables that parametrise the flat Lorentzian 3d manifolds arising as spacetimes in (2+1)-gravity. We consider an observer who emits lightrays that return to him at a later time and performs several realistic measurements associated with such returning lightrays: the eigentime elapsed between the emission of the lightrays and their return, the directions into which the light is emitted and from which it returns and the frequency shift between the emitted and returning lightray. We show how the holonomy variables and hence the full geometry of these manifolds can be reconstructed from these measurements in finite eigentime.