Exotic Differentiable Structures and General Relativity

TL;DR

This paper reviews recent advances in differential topology, focusing on exotic differentiable structures on simple manifolds and exploring their potential significance and implications for general relativity and physical theories.

Contribution

It highlights the existence of exotic differentiable structures on familiar manifolds and discusses their possible physical implications, a novel perspective in the context of general relativity.

Findings

Existence of non-standard differentiable structures on simple manifolds.

Potential physical significance of exotic manifolds in general relativity.

Discussion of mathematical formalisms and conjectures on physical implications.

Abstract

We review recent developments in differential topology with special concern for their possible significance to physical theories, especially general relativity. In particular we are concerned here with the discovery of the existence of non-standard (``fake'' or ``exotic'') differentiable structures on topologically simple manifolds such as $S^7$, \R and $S^3\times {\bf R^1}.$ Because of the technical difficulties involved in the smooth case, we begin with an easily understood toy example looking at the role which the choice of complex structures plays in the formulation of two-dimensional vacuum electrostatics. We then briefly review the mathematical formalisms involved with differentiable structures on topological manifolds, diffeomorphisms and their significance for physics. We summarize the important work of Milnor, Freedman, Donaldson, and others in developing exotic differentiable structures on well known topological manifolds. Finally, we discuss some of the geometric implications of these results and propose some conjectures on possible physical implications of these new manifolds which have never before been considered as physical models.