The Schwarzschild metric and the Friedmann equations from Newtonian Gravitational collapse

TL;DR

This paper demonstrates how key components of the Schwarzschild metric and Friedmann equations can be derived from Newtonian gravity principles, offering a pedagogical approach and insights into the generality of these results beyond Einstein's theory.

Contribution

It shows that the Schwarzschild metric and Friedmann equations can be obtained from Newtonian assumptions, bypassing Einstein's equations, highlighting their broader applicability.

Findings

Schwarzschild metric components derived from Newtonian matching conditions.

The Newtonian constant of integration matches the spatial curvature in FLRW models.

Results provide a pedagogical alternative to teaching aspects of General Relativity.

Abstract

As 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 spatial curvature of the FLRW metric. These results are of interest at least in two respects, one from the point of view of its pedagogical value of teaching General Relativity without in fact using Einstein's equation and second, the fact that some results attributed to General Relativity can be obtained without using General Relativity indicates that these results are more general than the particular dynamics specified by General Relativity.