Journal Article

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

Boudewijn F. Roukema

in Monthly Notices of the Royal Astronomical Society

Published on behalf of The Royal Astronomical Society

Volume 404, issue 1, pages 318-324
Published in print May 2010 | ISSN: 0035-8711
Published online April 2010 | e-ISSN: 1365-2966 | DOI:
There was movement that was stationary, for the four-velocity had passed around*

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Is the Doppler interpretation of galaxy redshifts in a Friedmann–Lemaître–Robertson–Walker (FLRW) model valid in the context of the approach to comoving spatial sections pioneered by de Sitter, Friedmann, Lemaître and Robertson, i.e. according to which the three-manifold of comoving space is characterized 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.

Keywords: relativity; reference systems; time; cosmology: theory

Journal Article.  5855 words.  Illustrated.

Subjects: Astronomy and Astrophysics

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