Cumulative gravitational time dilations across the universe

Matter dilates time in its surroundings, typically quantified by the Schwarzschild metric. Extrapolating, mass-energy in the universe creates a cumulative time dilation at the Earth, a dilation different from the commonly invoked $(1 + z)$ ( 1 + z ) cosmological redshift. In Newtonian cosmology, it is shown that the diverging amount of distant mass-energy with distance causes a similarly diverging cumulative time dilation at the Earth. In several relativistic Friedmann-Lemaitre-Robertson-Walker cosmologies, the Newtonian gauge is used to estimate a conceptually similar cumulative time dilation by successively replacing mass-energy shells by voids and tallying the cumulative difference. As a function of redshift, it is found that this cumulative time dilation does not diverge in standard concordance cosmology, but as a function of comoving distance, it generally does diverge in several flat cosmologies.