Finitary-Algebraic `Resolution' of the Inner Schwarzschild Singularity

TL;DR

This paper presents a purely algebraic and finitistic approach to resolving the Schwarzschild singularity in General Relativity using Abstract Differential Geometry and finitary poset models, avoiding traditional manifold breakdowns.

Contribution

It introduces a novel algebraic framework employing ADG and finitary sheaves to resolve spacetime singularities without relying on smooth manifolds, extending Einstein equations to discrete structures.

Findings

Einstein equations hold at finitary poset level and in continuum limit.

Singularities can be resolved algebraically without manifold breakdown.

Implications for classical and quantum gravity research.

Abstract

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.