The Schwarzschild metric and the Friedmann equations from Newtonian Gravitational collapse

University of Graz, Institute of Physics, 8010 Graz, Austria(Dated: April 2, 2015)As is well known, the 0 −0 component of the Schwarzschild space can be obtained by the re-quirement that the geodesic of slowly moving particles match the Newtonian equation. Given thisresult, we show here that the remaining components can be obtained by requiring that the inside ofa Newtonian ball of dust matched at a free falling radius with the external space determines thatspace to be Schwarzschild, if no pathologies exist. Also we are able to determine that the constantof integration that appears in the Newtonian Cosmology, coincides with the spatial curvature of theFLRW metric. These results are of interest at least in two respects, one from the point of viewof its pedagogical value of teaching General Relativity without in fact using Einstein’s equationand second, the fact that some results attributed to General Relativity can be obtained withoutusing General Relativity indicates that these results are more general than the particular dynamicsspecified by General Relativity.This essay has been written for the Gravity Research Foundation 2015 essay competitionI. INTRODUCTION