Bulg. J. Phys. vol.45 no.2 (2018), pp. 180-202

Finding Schwarzschild Metric Component grr and FLRW's k without Solving the Einstein Equation, Rather by a Synergistic Matching between Geometric Results Enfranchised by Newtonian Gravity

E.I. Guendelman1,2,3, A. Rabinowitz1, A.P. Banik4
1Ben Gurion University of the Negev, Department of Physics, Beer-Sheva, Israel
2Bahamas Advanced Study Institute and Conferences, 4A Ocean Heights, Hill View Circle, Stella Maris, Long Island, The Bahamas
3Frankfurt Institute for Advanced Studies, Giersch Science Center, Campus Riedberg, Frankfurt am Main, Germany
4Department of Physics and Astronomy, University of Missouri -- Columbia, Columbia, MO, USA
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 moving 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 possible 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.

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