There was movement that was stationary, for the four-velocity had passed around

TL;DR

This paper questions the validity of the Doppler interpretation of galaxy redshifts within FLRW models when considering non-trivial cosmic topologies, revealing contradictions through holonomy transformations and closed spatial loops.

Contribution

It introduces holonomy transformations for FLRW models with different curvatures and links cosmic topology to redshift interpretation issues.

Findings

Doppler interpretation leads to contradictions in multiply-connected FLRW models.

Inertial observers appear to recede from themselves at non-zero speeds.

Contradictions occur for all redshifts in flat and spherical cases, and large redshifts in hyperbolic cases.

Abstract

Is the Doppler interpretation of galaxy redshifts in a Friedmann-Lemaitre-Robertson-Walker (FLRW) model valid in the context of the approach to comoving spatial sections pioneered by de Sitter, Friedmann, Lemaitre and Robertson, i.e. according to which the 3-manifold of comoving space is characterised by both its curvature and topology? Holonomy transformations for flat, spherical and hyperbolic FLRW spatial sections are proposed. By quotienting a simply-connected FLRW spatial section by an appropriate group of holonomy transformations, the Doppler interpretation in a non-expanding Minkowski space-time, obtained via four-velocity parallel transport along a photon path, is found to imply that an inertial observer is receding from herself at a speed greater than zero, implying contradictory world-lines. The contradiction in the multiply-connected case occurs for arbitrary redshifts in the flat and spherical cases, and for certain large redshifts in the hyperbolic case. The link between the Doppler interpretation of redshifts and cosmic topology can be understood physically as the link between parallel transport along a photon path and the fact that the comoving spatial geodesic corresponding to a photon's path can be a closed loop in an FLRW model of any curvature. Closed comoving spatial loops are fundamental to cosmic topology.