Exotic Smoothness on Spacetime

TL;DR

Recent advances in differential topology reveal exotic smoothness structures on topologically trivial manifolds like ${S^7}$ and ${f R^4}$, which could imply new physical models of spacetime with distinct geometric and physical properties.

Contribution

This paper reviews recent discoveries of exotic smoothness structures on trivial manifolds and explores their potential implications for spacetime models in physics.

Findings

Exotic ${f R^4}$ structures can be confined to a time-like world tube.

Different exotic structures on the same topological manifold are not diffeomorphic.

Existence and non-existence results for exotic smoothness structures are discussed.

Abstract

Recent discoveries in differential topology are reviewed in light of their possible implications for spacetime models and related subjects in theoretical physics. Although not often noted, a particular smoothness (differentiability) structure must be imposed on a topological manifold before geometric or other structures of physical interest can be discussed. The recent discoveries of interest here are of various surprising ``exotic'' smoothness structures on topologically trivial manifolds such as ${S^7}$ and ${\bf R^4}$. Since no two of these are diffeomorphic to each other, each such manifold represents a physically distinct model of topologically trivial spacetime. That is, these are not merely different coordinate representations of a given spacetime. The path to such structures intertwines many branches of mathematics and theoretical physics (Yang-Mills and other gauge theories). An overview of these topics is provided, followed by certain results concerning the geometry and physics of such manifolds. Although exotic ${\bf R^4}$'s cannot be effectively exhibited by finite constructions, certain existence and non-existence results can be stated. For example, it is shown that the ``exoticness'' can be confined to a time-like world tube, providing a possible model for an exotic source. Other suggestions and conjectures for future research are made.