Schwarzschild and Friedmann Lema\^itre Robertson Walker metrics from Newtonian Gravitational collapse

TL;DR

This paper demonstrates how the full Schwarzschild and FLRW metrics can be derived from Newtonian gravitational principles, linking Newtonian collapse and cosmology to relativistic solutions.

Contribution

It shows that the entire Schwarzschild metric and the spatial curvature of FLRW models can be obtained from Newtonian gravitational collapse conditions.

Findings

Schwarzschild metric components derived from Newtonian matching conditions.

The Newtonian cosmological constant matches the FLRW spatial curvature.

External Schwarzschild space is uniquely determined by Newtonian dust collapse.

Abstract

As it is well known, the $0-0$ component of the Schwarzschild space can be obtained by the requirement that the geodesic of slowly moving particles match the Newtonian equation. Given this result, we show here that the remaining components can be obtained by requiring that the inside of a Newtonian ball of dust matched at a free falling radius with the external space determines that space to be Schwarzschild, if no pathologies exist. Also we are able to determine that the constant of integration that appears in the Newtonian Cosmology coincides with the spacial curvature of the FLRW metric.