Finding Schwarzschild metric component grr and FLRW's k without solving the Einstein's equation, rather by a synergistic matching between geometric results enfranchised by Newtonian gravity

Abstract

As is well known, some aspects of General Relativity and Cosmology can be reproduced without even using Einstein's equation. As an illustration, the 0 - 0 component of the Schwarzschild space can be obtained by the requirement that the geodesic of slowly mov- ing particles match the Newtonian equation. Given this result, we shall show here that the remaining component (grr) can be obtained by requiring that the inside of a Newtonian ball of dust matched at a free falling radius with the external space of unspecified type. This matching determines the external space to be of Schwarzschild type. By this, it is also possi- ble to determine that the constant of integration that appears in the Newtonian Cosmology, coincides with the spatial curvature of the FLRW metric. All we assumed was some classical boundary conditions and basic assumptions.