New characterization of Robertson-Walker geometries involving a single timelike curve

TL;DR

This paper introduces a new geometric characterization of Robertson-Walker spacetimes using conditions on a single timelike curve, relaxing previous assumptions and providing a canonical form of the metric for arbitrary observers.

Contribution

It generalizes the known characterization of RW spacetimes by relaxing the expansion condition and introduces a canonical form of the metric adapted to any timelike curve.

Findings

Characterization of RW spacetimes via conditions on a single timelike curve.

Derivation of a canonical form of the RW metric for arbitrary observers.

Establishment of local conditions equivalent to zero energy flux and divergence on a single curve.

Abstract

Our aim in this paper is two-fold. We establish a novel geometric characterization of the Roberson-Walker (RW) spacetime and, along the process, we find a canonical form of the RW metric associated to an arbitrary timelike curve and an arbitrary space frame. A known characterization establishes that a spacetime foliated by constant curvature leaves whose orthogonal flow (the cosmological flow) is geodesic, shear-free, and with constant expansion on each leaf, is RW. We generalize this characterization by relaxing the condition on the expansion. We show it suffices to demand that the spatial gradient and Laplacian of the expansion on a single arbitrary timelike curve vanish. In General Relativity these local conditions are equivalent to demanding that the energy flux measured by the cosmological flow, as well as its divergence, are zero on a single arbitrary timelike curve. The proof allows us to construct canonically adapted coordinates to the arbitrary curve, thus well-fitted to an observer with an arbitrary motion with respect to the cosmological flow.