Physics with and without the equivalence principle

A differential manifold (d-manifold, for short) can be defined as a pair (M, C), where M is any set and C is a family of real functions on M which is (i) closed with respect to localization and (ii) closed with respect to superposition with smooth Euclidean functions; one also assumes that (iii) M is locally diffeomorphic to Rⁿ. These axioms have a straightforward physical interpretation. Axioms (i) and (ii) formalize certain compatibility conditions which usually are supposed to be assumed tacitly by physicists. Axiom (iii) may be though of as a (nonmetric) version of Einstein's equivalence principle. By dropping axiom (iii), one obtains a more general structure called a differential space (d-space). Every subset of Rⁿ turns out to be a d-space. Nevertheless it is mathematically a workable structure. It might be expected that somewhere in the neighborhood of the Big Bang there is a domain in which space-time is not a d-manifold but still continues to be a d-space. In such a domain we would have a physics without the (usual form of the) equivalence principle. Simple examples of d-spaces which are not d-manifolds elucidate the principal characteristics the resulting physics would manifest.on leave of absence from the Institute of Nuclear Physics, Department of Theoretical Physics, ul. Radzikowskiego 152, 31–342 Cracow, Poland.