Spacetime geometry in (2+1)-gravity via measurements with returning lightrays

TL;DR

This paper explores how an observer in a (2+1)-dimensional gravity spacetime can determine the full geometry by analyzing returning lightrays, providing explicit formulas and insights into the observables involved.

Contribution

It introduces explicit expressions for measurable quantities related to returning lightrays and demonstrates how these allow full reconstruction of spacetime geometry in (2+1)-gravity.

Findings

Derived formulas for return time, emission directions, and frequency shift.

Showed these quantities enable spacetime reconstruction within finite eigentime.

Connected these measurements to Dirac observables and Wilson loops.

Abstract

We consider an observer in a (2+1)-spacetime without matter and cosmological constant who measures spacetime geometry by emitting lightrays which return to him at a later time. We investigate several quantities associated with such lightrays: the return time, the directions into which light needs to be emitted to return and the frequency shift between the lightray at its emission and its return. We derive explicit expressions for these quantities as functions on the reduced phase space and show how they allow the observer to reconstruct the full geometry of the spacetime in finite eigentime. We comment on conceptual issues. In particular, we clarify the relation between these quantities and Dirac observables and show that Wilson loops arise naturally in these quantities.