A ug 2 00 4 Finitary - Algebraic ‘ Resolution ’ of the Inner Schwarzschild Singularity

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, in extenso, 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’—as it were, when only finitely many ‘degrees of freedom’ of the gravitational field are involved, so that no infinity or uncontrollable divergence of the latter arises at all in our inherently finitistic-algebraic scenario, let alone that the law of gravity—still modelled in ADG by a differential equation proper—breaks down in any (differential geometric) sense in the vicinity of the locus of the point-mass as it is currently maintained in the usual manifold based analysis of spacetime singularities in General Relativity (GR), but also that it holds at a suitable ‘classical spacetime continuum limit’ of the said finitary sheaves and the associated differential triads that they define ADG-theoretically. We infer what has already been pointed out numerous times in the past during various applications of ADG to classical (GR) and quantum gravity (QG); namely, that the purely algebraico-categorical and background geometrical manifold independent differential geometric mechanism of ADG is in no way impeded by singularities, which are anyway built into our conventional C-smooth manifold model for spacetime, but on the contrary, that it applies in full effect in their very presence. Various possible implications that such a total evasion of 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 QG research are briefly discussed at the end. PACS numbers: 04.60.-m, 04.20.Gz, 04.20.-q