An Alternative Framework of Geometry and Topology in Relativity

TL;DR

This paper explores a novel geometric-topological framework to better understand the fundamental forces, including gravity and dark energy, proposing a new classification scheme that enhances the conceptual unity of physics and mathematics.

Contribution

It introduces an innovative approach to classify fundamental forces using geometry and topology, addressing the dark energy problem and unifying physical and mathematical structures.

Findings

A new classification of the five fundamental forces.

Formalization of the geometry-topology complementarity as a basic theorem.

Enhanced understanding of the relationship between physical reality and mathematical structures.

Abstract

We study geometry and topology as complementary and dual aspects of the mathematical space. The same is used to get a better understanding of the Cosmological Constant. Having failed so far to include gravity in a proper unified framework with the other three fundamental gauge forces, we are now faced with an additional unwanted fifth force of repulsion, also envisaged as the Dark Energy problem. How does one understand this 3+1+1 fundamental force dilemma? We introduce here a novel idea of the Fundamental Forces. This will give us an additional and an all-encompassing way of classifying these five fundamental forces in a consistent manner and thereby strengthen the geometry-topology complementarity concept. This also helps us to understand as to what one actually means when substituting ${g_{\mu \nu}} = {\eta_{\mu \nu}} + {h_{\mu \nu}}$ with the last term being "small" in General Relativity. This provides an understanding of the generic relationship and the complementarity of the geometric and the topological structures which is then formalized as a basic theorem enabling us to understand the underlying connection between the physical reality and the pure mathematical structures.