18 research outputs found
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
Absolute spacetime: the twentieth century ether
All gauge theories need ``something fixed'' even as ``something changes.''
Underlying the implementation of these ideas all major physical theories make
indispensable use of an elaborately designed spacetime model as the ``something
fixed,'' i.e., absolute. This model must provide at least the following
sequence of structures: point set, topological space, smooth manifold,
geometric manifold, base for various bundles. The ``fine structure'' of
spacetime inherent in this sequence is of course empirically unobservable
directly, certainly when quantum mechanics is taken into account. This issue is
at the basis of the difficulties in quantizing general relativity and has been
approached in many different ways. Here we review an approach taking into
account the non-Boolean properties of quantum logic when forming a spacetime
model. Finally, we recall how the fundamental gauge of diffeomorphisms (the
issue of general covariance vs coordinate conditions) raised deep conceptual
problems for Einstein in his early development of general relativity. This is
clearly illustrated in the notorious ``hole'' argument. This scenario, which
does not seem to be widely known to practicing relativists, is nevertheless
still interesting in terms of its impact for fundamental gauge issues.Comment: Contribution to Proceedings of Mexico Meeting on Gauge Theories of
Gravity in honor of Friedrich Heh
Localized Exotic Smoothness
Gompf's end-sum techniques are used to establish the existence of an infinity
of non-diffeomorphic manifolds, all having the same trivial
topology, but for which the exotic differentiable structure is confined to a
region which is spatially limited. Thus, the smoothness is standard outside of
a region which is topologically (but not smoothly) ,
where is the compact three ball. The exterior of this region is
diffeomorphic to standard . In a
space-time diagram, the confined exoticness sweeps out a world tube which, it
is conjectured, might act as a source for certain non-standard solutions to the
Einstein equations. It is shown that smooth Lorentz signature metrics can be
globally continued from ones given on appropriately defined regions, including
the exterior (standard) region. Similar constructs are provided for the
topology, of the Kruskal form of the Schwarzschild
solution. This leads to conjectures on the existence of Einstein metrics which
are externally identical to standard black hole ones, but none of which can be
globally diffeomorphic to such standard objects. Certain aspects of the Cauchy
problem are also discussed in terms of \models which are
``half-standard'', say for all but for which cannot be globally
smooth.Comment: 8 pages plus 6 figures, available on request, IASSNS-HEP-94/2
Nonminimal Couplings in the Early Universe: Multifield Models of Inflation and the Latest Observations
Models of cosmic inflation suggest that our universe underwent an early phase
of accelerated expansion, driven by the dynamics of one or more scalar fields.
Inflationary models make specific, quantitative predictions for several
observable quantities, including particular patterns of temperature anistropies
in the cosmic microwave background radiation. Realistic models of high-energy
physics include many scalar fields at high energies. Moreover, we may expect
these fields to have nonminimal couplings to the spacetime curvature. Such
couplings are quite generic, arising as renormalization counterterms when
quantizing scalar fields in curved spacetime. In this chapter I review recent
research on a general class of multifield inflationary models with nonminimal
couplings. Models in this class exhibit a strong attractor behavior: across a
wide range of couplings and initial conditions, the fields evolve along a
single-field trajectory for most of inflation. Across large regions of phase
space and parameter space, therefore, models in this general class yield robust
predictions for observable quantities that fall squarely within the "sweet
spot" of recent observations.Comment: 17pp, 2 figs. References added to match the published version.
Published in {\it At the Frontier of Spacetime: Scalar-Tensor Theory, Bell's
Inequality, Mach's Principle, Exotic Smoothness}, ed. T. Asselmeyer-Maluga
(Springer, 2016), pp. 41-57, in honor of Carl Brans's 80th birthda
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
The unexpected resurgence of Weyl geometry in late 20-th century physics
Weyl's original scale geometry of 1918 ("purely infinitesimal geometry") was
withdrawn by its author from physical theorizing in the early 1920s. It had a
comeback in the last third of the 20th century in different contexts: scalar
tensor theories of gravity, foundations of gravity, foundations of quantum
mechanics, elementary particle physics, and cosmology. It seems that Weyl
geometry continues to offer an open research potential for the foundations of
physics even after the turn to the new millennium.Comment: Completely rewritten conference paper 'Beyond Einstein', Mainz Sep
2008. Preprint ELHC (Epistemology of the LHC) 2017-02, 92 pages, 1 figur