Exotic Smoothness and Physics

TL;DR

Recent mathematical discoveries reveal that space-time models, especially ${\bf R^4}$, can have multiple, non-diffeomorphic differentiable structures, potentially introducing new global features into physical theories.

Contribution

The paper reviews the role of exotic differentiable structures in physics and discusses new results on their spatial localization and implications for physical theories.

Findings

Existence of multiple exotic smooth structures on ${\bf R^4}$

Potential for exotic structures to influence global space-time features

Suggestions of spatial localization of exotic structures

Abstract

The essential role played by differentiable structures in physics is reviewed in light of recent mathematical discoveries that topologically trivial space-time models, especially the simplest one, ${\bf R^4}$, possess a rich multiplicity of such structures, no two of which are diffeomorphic to each other and thus to the standard one. This means that physics has available to it a new panoply of structures available for space-time models. These can be thought of as source of new global, but not properly topological, features. This paper reviews some background differential topology together with a discussion of the role which a differentiable structure necessarily plays in the statement of any physical theory, recalling that diffeomorphisms are at the heart of the principle of general relativity. Some of the history of the discovery of exotic, i.e., non-standard, differentiable structures is reviewed. Some new results suggesting the spatial localization of such exotic structures are described and speculations are made on the possible opportunities that such structures present for the further development of physical theories.