Cosmological Schwarzschild radii and Newtonian gravitational theory

TLDR: In this paper, the Friedmann-Robertson-Walker zero-pressure dust-filled universe is described using a Schwarzschild-like curvature spatial coordinate R along with the usual cosmological time coordinate t.

Abstract: We describe Friedmann–Robertson–Walker zero‐pressure dust‐filled universes using a Schwarzschild‐like curvature spatial coordinate R along with the usual cosmological time coordinate t. In terms of coordinates (R,t), the geodesic equations of general relativity for the motion of the galaxies comprising the universe satisfy exactly a Newtonian inverse‐square relationship. This allows us to formulate relativistic cosmology by starting first with Newtonian cosmology. After the mathematics of Newtonian cosmology is worked out, the general‐relativistic metric for the universe is constructed from a well‐defined prescription, and the behavior of light signals is then determined from the metric. It is found that certain radial light signals born at the Big Bang eventually reach a maximum distance in their journey through the universe, where they turn around and return to some arbitrarily chosen origin. The turning around takes place at an apparent horizon located at a Schwarzschild radius in the universe.