Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
1996-04-24
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
To appear in proceedings of Pacific Conference on Gravitation and Cosmology, Seoul , 1996. LaTeX, 16 pages
Scientific paper
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.
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