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, ${\bf R^4}$, 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 ${\bf R^4}$ 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) ${\bf B^3}\times {\bf R^1}$, where ${\bf B^3}$ is the compact three ball. The exterior of this region is diffeomorphic to standard ${\bf R^1}\times {\bf S^2}\times{\bf R^1}$. 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, ${\bf S^2}\times {\bf R^2}$ 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 ${\bf R^4_\Theta}$\models which are ``half-standard'', say for all $t<0,$ but for which $t$ 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 $S^7$, \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.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