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 R 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) B ×R, where B is the compact three ball. The exterior of this region is diffeomorphic to standard R × S × R. 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, S × R 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 R Θ models which are “halfstandard”, say for all t < 0, but for which t cannot be globally smooth. PACS: 04.20.Cv, 02.40.+m This paper is concerned with smooth manifold models for space-time which have relatively trivial topology, e.g., R, or R × S, but non-standard, or “exotic” smoothness structures. By definition, such manifolds are not diffeomorphic to their standard smooth form, and hence, from the basic principles of general relativity, cannot be physically equivalent to any previously studied manifold with the corresponding simple topology. In the non-compact cases an important feature of many of these examples is that they require that the exotic part extend “to infinity”, as illustrated, for example, in figure 1. This fact has served as a deterrent to the consideration of such spaces as space-time models since classical observations of space-time are “large scale” in some sense, with resulting expectation of asymptotic regularity, including smoothness. In fact, all that the mathematics requires is that the exotic region not be contained in a compact set. However, to my knowledge, no example of an exotic manifold which is standard at spatial infinity has ever been published before now. The main result of this paper can be summarized informally: Result 1 There exists exotic smooth manifolds with R topology which are standard at spatial infinity, so that the exoticness can be regarded as spatially confined. A more precise statement of this result is provided in Theorem 1 below. The resulting manifold structures are illustrated in examples such as those shown in figures 3 and 4 where everything looks normal at spacelike infinity but the standard structure cannot be continued all the way in to spatial origin. This work is based on the remarkable mathematical breakthroughs of Milnor, Freedman, Donaldson, Gompf [1],[2],[3],[4], and others, establishing the surprising existence of such exotic structures on topologically trivial spaces, including R, together with the end-sum techniques of Gompf[7]. This result could have great significance in all fields of physics, not just relativity. Some model of space-time underlies every field of physics. It has now been proven that we cannot infer that space is necessarily smoothly standard from investigating what happens at spacelike infinity, even for topologically trivial R. It seems very clear that this is potentially very important to all of physics since it implies that there is another possible obstruction, in addition to material sources and topological ones, to continuing external vacuum solutions for any field equations from infinity to the origin. Of course, in the absence of any explicit coordinate patch presentation, no example can be displayed. However, this leads naturally to a conjecture, informally stated: Conjecture 1 This localized exoticness can act as a source for some externally regular field, just as matter or a wormhole can. Of course, the exploration of this conjecture will require more detailed knowledge of the global metric structure than is available at present. The notions of domains of dependence, Cauchy surfaces, etc., necessary for such studies cannot be fully explored with present differential geometric information on exotic manifolds. However, a beginning can be made with certain general existence results as established and discussed below.