Exotic Smoothness on Spacetime
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 (YangMills 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 nonexistence results can be stated. For example, it is shown that the ``exoticness'' can be confined to a timelike world tube, providing a possible model for an exotic source. Other suggestions and conjectures for future research are made.
 Publication:

Gravitation and Cosmology
 Pub Date:
 1997
 DOI:
 10.1007/9789401158121_27
 arXiv:
 arXiv:grqc/9604048
 Bibcode:
 1997ASSL..211..160B
 Keywords:

 General Relativity and Quantum Cosmology
 EPrint:
 To appear in proceedings of Pacific Conference on Gravitation and Cosmology, Seoul , 1996. LaTeX, 16 pages