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