Causal Structures in Linear Spaces

Abstract

Linear topological spaces with partial ordering (linear kinematics) are studied. They are defined by a set of 8 axioms implying that topology, linear structure and ordering are compatible with each other. Most of the results are valid for both the finite-dimensional and infinite-dimensional case. In applications to physics, partial ordering is interpreted as the causal structure. Both Newtonian and the special relativity causal structures are studied, and some other possible types of causality are discussed. Linear topological spaces with pseudometric which satisfies the time inequality instead of the triangle inequality are studied (3 axioms). Pseudometric (which is determined by a pseudonorm) is shown to define a topology on a linear space, it being a continuous mapping in this topology. Proved that for a space with pseudometric to be a linear kinematics it is necessary and sufficient that mapping of multiplication by -1 (i.e. time reversion) be continuous. Minkovskii space of the special relativity theory proved to be a pseudometric linear kinematics.