Finitary-Algebraic `Resolution' of the Inner Schwarzschild Singularity

Details Finitary-Algebraic `Resolution' of the Inner Schwarzschild Singularity Finitary-Algebraic `Resolution' of the Inner Schwarzschild Singularity

2004-08-16

arxiv.org/abs/gr-qc/0408045v6

Int.J.Theor.Phys. 45 (2006) 79-128

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

41 pages, to appear in the International Journal of Theoretical Physics

Scientific paper

A `resolution' of the interior singularity of the spherically symmetric Schwarzschild solution of the Einstein equations for the gravitational field of a point-particle is carried out entirely and solely by finitistic and algebraic means. To this end, the background differential spacetime manifold and Calculus-free purely algebraic (:sheaf-theoretic) conceptual and technical machinery of Abstract Differential Geometry (ADG) is employed via Sorkin's finitary (:locally finite) poset substitutes of continuous manifolds in their Gel'fand-dual picture in terms of discrete differential incidence algebras and the finitary spacetime sheaves thereof. It is shown that the Einstein equations hold not only at the finitary poset level of `discrete events', but also at a suitable `classical continuum limit' of the said finitary sheaves and the associated differential triads that they define ADG-theoretically. We infer that the law of gravity does not break down in any (differential geometric) sense in the vicinity of the locus of the point-mass as the usual manifold based analysis of spacetime singularities in General Relativity has hitherto maintained. Various possible implications that such a total evasion of smooth gravitational singularities, as well as some anticipations of the wider significance that the general ADG-framework, may have for certain current `hot' issues in both classical and quantum gravity research are briefly discussed at the end.

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