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PrefaceThis book is about the physics of spacetime at a deep and fundamental level, encoded in the mathematical assumptions of differentiability. We understand the phrase, "physics of spacetime" in the sense that general relativity has taught us, namely that the geometry, the topology, and now the smoothness of our spacetime mathematical models have physical significance.Our aim is to introduce some of the exciting developments in the mathematics of differential topology over the last fifteen or twenty years to a wider audience than experts. In particular, we are concerned with the discoveries of "exotic" (sometimes called "fake" or "non-standard" ) smoothness (differentiable) structures on R4 and other topologically simple spaces. We hope to help physicists gain at least a superficial understanding of these results and their potential impact on physical theories involving spacetime models, i.e., all fundamental theories. Diffeomorphisms, the basic morphisms of differential topology, are the mathematical representations of the physical notion of transformations between reference frames. As we have learned from Einstein the investigation of these transformations can lead to deep insights into our physical world, as embodied for example in his General Relativistic theory of spacetime and gravity. What the mathematicians have discovered is that the global properties of diffeomorphisms are not at all trivial, even on topologically trivial spaces, such as R4. Yet in general relativity and other field theories physicists continue to assume that the global covering of such spacetime models with smooth reference frames is trivial. This is strongly reminiscent of the assumption of geometric triviality (flatness) of spacetime physics before Einstein. We hope that by presenting an overview of the mathematical discoveries, we may induce physicists to consider the possible physical significance of this newly discovered wealth Within each section we attempt to accompany the mathematical presentation with parallel narratives of related physical topics as well as more informal "physical" descriptions if feasible. The abundant cross-fertilization of physics and mathematics in recent differential topology makes this endeavor quite natural.Chapter 1 is an introduction, discussing the traditional interaction of physics and mathematics and speculations that this interaction extends to these differential topology results. We also review some "exotic" and unexpected or at least counter-intuitive facts in elementary topology and analysis to provide somewhat easier analogs to the more technically challenging "exotic" mathematics coming later. We then survey possible physical consequences of these unexpected structures. Chapter 2 begins with a review of some of the mathematical tools and techniques of algebraic topology. Chapter 3 concentrates on the notion of "smoothness" as defined by the introduction of differential structures on topological spaces. The field of differential topology is built on these constructions. In Chapter 4 we pr...

A procedure is described for obtaining a complete, invariant classification of the local, analytic geometries and matter fields in general relativity by a finite number of algebraic steps. The approach is based on an extension of the classification scheme to include differential invariants of all orders and to provide maximally determined standard frames of vectors at each point. It is further shown that the resultant invariant functions can be replaced, in a finite number of algebraic steps, by special invariant functions which, while still uniquely representative of the geometry, can be assigned arbitrarily to produce all possible local, analytic solutions to the Einstein equations, in this representation. It is suggested that this type and special function scheme, obtainable from ideal geometric measurements in a finite number of steps, could be useful in general relativity. Unfortunately, due to the extensive algebra involved, this scheme has not yet been explicitly calculated, even for empty spaces.