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Consequences of Spacetime Topology

Consequences of Spacetime Topology - CONSEQUENCES OF...

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CONSEQUENCES OF SPACETIME TOPOLOGY * Rafael D. Sorkin Department of Physics Syracuse University Syracuse, NY 13244-1130 In the last decade or so, topological questions have increasingly attracted the interest of physicists. Indeed the amount of work devoted to topological issues has grown so explosively that even a review talk restricting itself to the domain of Gen- eral Relativity can’t hope to give an adequate account of recent work. The review I am about to give will therefore be very superficial, although I will try to dwell a bit longer on one or two areas that I am most familiar with. As indicated by my title, I will also try, insofar as I can, to emphasize the physical consequences of spacetime topology - the possible effects that have excited much of the recent intense interest of which I spoke. Finally let me apologize in advance for the incompleteness of the references. Rather than attempt a comprehensive bibliography for this lecture, I am just going to give a relatively small number of unsystematically chosen cita- tions, hoping in particular to mention a few papers that you can follow up for more information 1) . Topology enters General Relativity through the fundamental assumption that spacetime exists and is organized as a manifold. This means in the first place that spacetime has a well-defined dimension, but it also carries with it the inherent possibility of modified “patterns of global connectivity”, such as distinguish a sphere from a plane, or a torus from a surface of higher genus. Such modifications can be present in the spatial topology without affecting the time direction, but they can also have a genuinely spacetime character in which case the spatial topology changes with time , if indeed it is well-defined at all. * Published in A. Coley, F. Cooperstock, B. Tupper (eds.), Proceedings of the Third Canadian Conference on General Relativity and Relativistic Astrophysics held May, 1989, Victoria, Canada, 137-163 (World Scientific, 1990). 1
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Strictly speaking the spacetime of General Relativity is a differentiable mani- fold rather than just a topological one, the extra structure being required in order that the manifold can play host to a Riemannian (or rather Lorentzian) metric. The contemplation of a modified topology thus leads naturally to the idea of a modified differentiable structure, and indeed the latter has recently shown signs of mathe- matical life, especially in four dimensions. However, nobody seems to understand yet the physical consequences of replacing, for instance, the standard R 4 by one of its more exotically differentiable variants. In any case there will be no time to say more about such possibilities in this talk. 2) Another set of related possibilities which I will have to neglect involves the topology, not solely of spacetime itself, but of the non-gravitational fields which live in it. Topological effects involving such fields have proved important in many settings, as have effects involving the topology of quantum configuration spaces and of non-trivial bundles over these configuration spaces. In themselves such possibil-
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