Exotic Smoothness on Spacetime

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.