Recovering the geometry of a flat spacetime from background radiation

TL;DR

This paper demonstrates that in flat spacetimes, the observed frequency of a uniform light signal encodes information about the spacetime's initial singularity and can be used to recover its geometry and topology, especially in 2+1 dimensions.

Contribution

It introduces a method to recover the geometry and topology of flat spacetimes from background radiation signals, with stability results in 2+1 dimensions.

Findings

Frequency function is well-defined and bounded.

Frequency function contains geometric and topological information.

Approximate frequency functions computed for simple models.

Abstract

We consider globally hyperbolic flat spacetimes in 2+1 and 3+1 dimensions, in which a uniform light signal is emitted on the $r$-level surface of the cosmological time for $r\to 0$. We show that the frequency of this signal, as perceived by a fixed observer, is a well-defined, bounded function which is generally not continuous. This defines a model with anisotropic background radiation that contains information about initial singularity of the spacetime. In dimension 2+1, we show that this observed frequency function is stable under suitable perturbations of the spacetime, and that, under certain conditions, it contains sufficient information to recover its geometry and topology. We compute an approximation of this frequency function for a few simple examples.