16,150 research outputs found
Exotic Differentiable Structures and General Relativity
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 , \R and
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.Comment: 11 pages, LaTe
Exotic Smoothness and Physics
The essential role played by differentiable structures in physics is reviewed
in light of recent mathematical discoveries that topologically trivial
space-time models, especially the simplest one, , possess a rich
multiplicity of such structures, no two of which are diffeomorphic to each
other and thus to the standard one. This means that physics has available to it
a new panoply of structures available for space-time models. These can be
thought of as source of new global, but not properly topological, features.
This paper reviews some background differential topology together with a
discussion of the role which a differentiable structure necessarily plays in
the statement of any physical theory, recalling that diffeomorphisms are at the
heart of the principle of general relativity. Some of the history of the
discovery of exotic, i.e., non-standard, differentiable structures is reviewed.
Some new results suggesting the spatial localization of such exotic structures
are described and speculations are made on the possible opportunities that such
structures present for the further development of physical theories.Comment: 13 pages, LaTe
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