New ideas about multiplication of tensorial distributions

TL;DR

This paper introduces a new geometric framework for multiplying tensor distributions that inherently includes covariant derivatives, emphasizing a physical perspective and applicability to piecewise smooth manifolds.

Contribution

It develops a canonical, covariant tensor distribution multiplication theory based solely on Colombeau equivalence, avoiding explicit geometric constructions.

Findings

Provides a geometric, covariant tensor multiplication framework

Enables formulation of physics on piecewise smooth manifolds

Offers higher symmetry and physical insight in tensor analysis

Abstract

There is a need in general relativity for a consistent and useful mathematical theory defining the multiplication of tensor distributions in a geometric (diffeomorphism invariant) way. Significant progress has been made through the concept of Colombeau algebras, and the construction of full Colombeau algebras on differential manifolds for arbitrary tensors. Despite the fact that this goal was achieved, it does not incorporate clearly enough the concept of covariant derivative and hence is of a limited use. We take a different approach: we consider any type of preference for smooth distributions (on a smooth manifold) as nonintuitive, which means all our approach must be based fully on the Colombeau equivalence relation as the fundamental feature of the theory. After taking this approach we very naturally obtain a canonical and geometric theory defining tensorial operations with tensorial distributions, including covariant derivative. This also happens because we no longer need any explicit canonical geometric construction of Colombeau algebras. The big advantage of our approach lies also in the fact that it brings a physical insight into the mathematical concepts used and naturally leads to formulation of physics on (what we call) piecewise smooth manifolds, rather than on smooth manifold. This brings to the language of physics much higher symmetry (in the same way as turning from Poincare invariance to diffeomorphism invariance), and is compatible with our intuition, that "pointwise" properties in some metaphorical sense "do not matter".