Space-time foam dense singularities and de Rham cohomology

In an earlier paper of the authors, it was shown that the sheaf theoretically based recently developed abstract differential geometry of the first author can, in an easy and natural manner, incorporate singularities on arbitrary closed nowhere dense sets in Euclidean spaces, singularities which therefore can have arbitrary large positive Lebesgue measure. As also shown, one can construct in such a singular context a de Rham cohomology, as well as a short exponential sequence, both of which are fundamental in differential geometry. In this paper, these results are significantly strengthened, motivated by the so-called space-time foam structures in general relativity, where singularities can be dense. In fact, this time one can deal with singularities on arbitrary sets, provided that their complementaries are dense, as well. In particular, the cardinal of the set of singularities can be larger than that of the nonsingular points

Special Relativity over the Field of Rational Numbers

We investigate the question: what structures of num- bers (as physical quantities) are suitable to be used in special relativity? The answer to this question depends strongly on the auxiliary assumptions we add to the basic assumptions of special relativity. We show that there is a natural axiom system of spe- cial relativity which can be modeled even over the field of rational numbers