Redshift propagation equations in the $β' \neq 0$ Szekeres models

Abstract: The set of differential equations obeyed by the redshift in the general $\beta' \neq 0$ Szekeres spacetimes is derived. Transversal components of the ray's momentum have to be taken into account, which leads to a set of 3 coupled differential equations. It is shown that in a general Szekeres model, and in a general Lema^{\i}tre -- Tolman (L--T) model, generic light rays do not have repeatable paths (RLPs): two rays sent from the same source at different times to the same observer pass through different sequences of intermediate matter particles. The only spacetimes in the Szekeres class in which {\em all} rays are RLPs are the Friedmann models. Among the proper Szekeres models, RLPs exist only in the axially symmetric subcases, and in each one the RLPs are the null geodesics that intersect each $t =$ constant space on the symmetry axis. In the special models with a 3-dimensional symmetry group (L--T among them), the only RLPs are radial geodesics. This shows that RLPs are very special and in the real Universe should not exist. We present several numerical examples which suggest that the rate of change of positions of objects in the sky, for the studied configuration, is $10^{-6} - 10^{-7}$ arc sec per year. With the current accuracy of direction measurement, this drift would become observable after approx. 10 years of monitoring. More precise future observations will be able, in principle, to detect this effect, but there are basic problems with determining the reference direction that does not change.