Remarks on "singularities"

TL;DR

This paper explores how abstract differential geometry (ADG) allows the inclusion of singularities in physical theories without altering fundamental equations, shifting focus from the underlying space to the objects themselves.

Contribution

It demonstrates that ADG enables the extension of classical differential-geometric relations over singularities without changing their form or core arguments.

Findings

Classical relations can be extended over singularities using ADG.

The base space becomes secondary, with focus on the objects rather than the manifold.

Standard equations like Einstein and Yang-Mills remain valid in this generalized context.

Abstract

We present herewith certain thoughts on the important subject of nowadays physics, pertaining to the so-called ``singularities'', that emanated from looking at the theme in terms of ADG (: abstract differential geometry). Thus, according to the latter perspective, we can involve ``singularities'' in our arguments, while still employing fundamental differential-geometric notions such as connections, curvature, metric and the like, retaining also the form of standard important relations of the classical theory (e.g. Einstein and/or Yang-Mills equations, in vacuum), even within that generalized context of ADG. To wind up, we can extend (in point of fact, {calculate) over singularities classical differential-geometric relations/equations, without altering their forms and/or changing the standard arguments; the change concerns thus only the way, we employ the usual differential geometry of smooth manifolds, so that the base ``space'' acquires now quite a secondary role, not contributing at all (!) to the differential-geometric technique/mechanism that we apply. Thus, the latter by definition refers directly to the objects being involved--the objects that ``live on that space'', which by themselves are not, of course, ipso facto ``singular''!