The singularity at distance r → 0 at the center of a spherically symmetric non-rotating, uncharged mass of radius R, is considered here. Under inverse square law force, the Schwarzschild metric, needs to be modified, to include Newton’s Shell Theorem (NST). By including NST for r＜R, both Schwarzschild singularity at r = 2GM/c2 and at r → 0 singularities are removed from the metric. Near R → 0, the question of maximal density is considered based on Schwarzschild’s modified metric, and compared to the quantum limit of maximal mass density put by Planck’s quantum-based universal units. It is asserted, that General relativity, when combined with Planck’s universal units, inevitably leads to quantization of gravity.
References
[1]
Lemos, J.P.S. and Silva, D.L.F.G. (2020) Maximal Extension of the Schwarzschild Metric: From Painlevé-Gullstrand to Kruskal-Szekeres. Annals of Physics, 430, Article ID: 168497. https://doi.org/10.1016/j.aop.2021.168497
[2]
Painlevé, P. (1921) La mécanique classique et la théorie de la relativité. L’Astronomie, 36, 6-9.
[3]
Gullstrand, A. (1922) Allgemeine Lösung des statischen Einkörperproblems in der Einsteinschen Gravitationstheorie. Almqvist & Wiksell, Stockholm.
[4]
Eddington, A.S. (1924) A Comparison of Whitehead’s and Einstein’s Formula. Nature, 113, 192. https://doi.org/10.1038/113192a0
[5]
Finkelstein, D. (1958) Past-Future Asymmetry of the Gravitational Field of a Point Particle. Physical Review Journals Archive, 110, 965-967. https://doi.org/10.1103/PhysRev.110.965
[6]
Lemaître, G. (1933) L’Univers en expansion. Annales de la Société Scientifique de Bruxelles, 53A, 51-83.
[7]
Synge, J. (1950) The Gravitational Field of a Particle. Proceedings of the Royal Irish Academy, Section A: Mathematical and Physical Sciences, 53, 83-114.
[8]
Brown, K. (2010) Reflections on Relativity. https://scirp.org/reference/referencespapers?referenceid=3332982 https://philpapers.org/rec/BROROR-4
[9]
Landau, L.D. and Lifshitz, E.M. (1950) The Classical Theory of Fields. Pergamon Press, Oxford.
[10]
Einstein, A. (1939) On a Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses. Annals of Mathematics, 40, 922-936. https://doi.org/10.2307/1968902
[11]
Vaz, C. (2014) Black Holes as Gravitational Atoms. International Journal of Modern Physics D, 23, Article ID: 1441002. https://doi.org/10.1142/S0218271814410028
[12]
Corda, C. (2023) Schrödinger and Klein-Gordon Theories of Black Holes from the Quantization of the Oppenheimer and Snyder Gravitational Collapse. Communications in Theoretical Physics, 75, Article ID: 095405. https://doi.org/10.1088/1572-9494/ace4b2
[13]
Corda, C. (2023) Black Hole Spectra from Vaz’s Quantum Gravitational Collapse. Fortschritte der Physik, 71, Article ID: 2300028. https://doi.org/10.1002/prop.202300028
[14]
Buchdahl, H.A. (1985) Isotropic Coordinates and Schwarzschild Metric. International Journal of Theoretical Physics, 24, 731-739. https://doi.org/10.1007/BF00670880
[15]
Kruskal, M. (1959) Maximal Extension of Schwarzschild Metric. Physical Review Journals Archive, 119, 1743-1745. https://doi.org/10.1103/PhysRev.119.1743
[16]
Szekeres, G. (1959) On the Singularities of a Riemannian Manifold. Publicationes Mathematicae Debrecen, 7, 285-301. https://doi.org/10.5486/PMD.1960.7.1-4.26
[17]
Gkigkitzis, I., Haranas, I. and Ragos, O. (2014) Kretschmann Invariant and Relations between Spacetime Singularities, Entropy and Information. Physics International, 5, 103-111. https://doi.org/10.3844/pisp.2014.103.111
[18]
Henry, R.C. (2000) Kretschmann Scalar for a Kerr-Newman Black Hole. The Astrophysical Journal, 535, 350-353. https://doi.org/10.1086/308819
[19]
Newton, I. (1687) Philosophiae Naturalis Principia Mathematica. https://doi.org/10.5479/sil.52126.39088015628399
[20]
Arens, R. (1990) Newton’s Observations about the Field of a Uniform Thin Spherical Shell. Note di Matematica, X, 39-45.
[21]
Davie, R. Schwarzschild Metric 1-3. https://www.youtube.com/watch?v=D1mmLjR-szY https://www.youtube.com/watch?v=3NFqXgH-4tg
[22]
Gorelik, G.E. and Ya Frenkel, V. (1994) cGħ Physics in Bronstein’s Life. In: Gorelik, G.E. and Ya Frenkel, V., Eds., Matvei Petrovich Bronstein and Soviet Theoretical Physics in the Thirties, Birkhäuser, Basel, 83-121. https://doi.org/10.1007/978-3-0348-8488-4_5
[23]
Callender, C. and Huggett, N. (2004) Physics Meets Philosophy at the Planck Scale, Contemporary Theories in Quantum Gravity. Cambridge University Press, Cambridge.
[24]
Garay, L.J. (1995) Quantum Gravity and Minimum Length. International Journal of Modern Physics A, 10, 145-165. https://doi.org/10.1142/S0217751X95000085
[25]
Heger, A., Fryer, C.L., Woosley, S.E., Langer, N. and Hartmann, D.H. (2003) How Massive Single Stars End Their Life. Astrophysical Journal, 591, 288-300. https://doi.org/10.1086/375341
