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
  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.
