Redshift propagation equations in the ${\beta }^{\prime }\ne 0$ Szekeres models

Abstract

The set of differential equations obeyed by the redshift in the general ${\beta }^{\prime }\ne 0$ 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\^ıtre -- 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.

23 pages, 3 figures. Replacement matches the published text. Changes made in response to referee comments: 2 new figures and a more realistic numerical example added, explanation of the numerical procedure added as Appendix H

References in corpus (8)

• Szekeres Swiss-Cheese model and supernova observations
• The Szekeres Swiss Cheese model and the CMB observations
• Structure formation in the quasispherical Szekeres model
• Dark Energy or Apparent Acceleration Due to a Relativistic Cosmological Model More Complex than FLRW?
• Cosmic spherical void via coarse-graining and averaging non-spherical structures
• Physical and Geometrical Interpretation of the epsilon <= 0 Szekeres Models
• Volume averaging in the quasispherical Szekeres model
• Is the shell-focusing singularity of Szekeres space-time visible?

Cited by in corpus (1)

• Cosmic spherical void via coarse-graining and averaging non-spherical structures