Noncontinuous Minkowskian spacetime (Q1891486)

Let a smooth manifold be defined as a pair \((M,C)\) where \(C\) is a family of real functions on a set \(M\), satisfying the axioms: (i) \(C\) is closed with respect to localization; (ii) \(C\) is closed with respect to composition of the set \(\varepsilon\) of all \(C^\infty\)-function on \(\mathbb{R}^n\); (iii) \(M\) is locally diffeomorphic to \(\mathbb{R}^n\). The pair \((M,C)= \alpha\) is called \(d\)-space. The authors consider \(M = \left\{ {n\over m} (r^2 + s^2), {n\over m} (r^2 - s^2)\right\} \subset \mathbb{R}^2\) \((r, s, n, m \in \mathbb{Z} \setminus \{0\})\) and the Lorentz \(d\)-space \((M, C, G)\) where \(G\) is the scalar product. The Lorentz \(d\)-space \((M, C, G)\) possesses the important properties of special relativity, and it is interpreted in the context of general relativity.