Universe Expanding vs Non-Expanding Interpretation of Friedmann Lemaître Robertson Walker (FLRW) Metric

One of the keys to understanding the universe is the proper measurement of the relation between the luminosity distance dL and the angular diameter distance dA. In 1933, Etherington deduced from general relativity the reciprocity equation dL=dA(1+z)γ, with γ=1 for a local, i.e. non-expanding, universe. This equation was adapted to an expanding universe –through the comoving distance concept– with the value γ=2. On the other hand, the feasibility of an expanding universe rest on the Friedmann-Lemaître-Robertson-Walker (FLRW) metric that meets the Einstein’s Field Equations for an homogeneous and isotropic universe. The FLRW metric includes a time dependent factor a(t) – unlike the Einstein’s static universe– that accounts for the cosmological redshift. In this work, we show that the radial coordinate (r) of FLRW metric admits two different interpretations driving to an expanding and non-expanding universes. When r is considered has a radial comoving coordinate, the integration on r produces the comoving distance, and hence the metric drives to an expanding universe. Nevertheless, when r is considered simply has a radial coordinate, the integration on r produces the luminosity distance, and hence the metric drives to a non-expanding universe. Thus, weeding out the comoving concept, a non-expanding nature of FLRW model emerges, where a(t) accounts for the cosmological redshift in the form of a time downscaling or as a magnetic permeability growth with cosmic time, rather than as space scaling. Even more, the presence of the time varying factor a(t) drives to the same Friedmann equations, which guarantees the stability of the non-expanding universe, unlike the Einstein’s static universe.