Repeatable light paths in the shearfree normal cosmological models

Abstract

Conditions for the existence of repeatable light paths (RLPs) in the shearfree normal cosmological models are investigated. It is found that in the conformally nonflat models the only RLPs are radial null geodesics (in the spherical case) and their analogues in the plane- and hyperbolically symmetric cases. In the conformally flat Stephani models, there exist special spherically-, plane- and hyperbolically symmetric subcases, in which all null geodesics are RLPs. They are slightly more general than the Friedmann -- Lema\^{\i}tre -- Robertson -- Walker (FLRW) models of the corresponding symmetries: their curvature index function $k(t)$ and the scale factor $R(t)$ are expressed through a single function of time. In addition to that, there exist special cases of the Stephani solution in which some of the null geodesics are RLPs. All these special cases are identified.