Einstein Bitopological Gyrogroups

Source abstract

We introduce and investigate Einstein bitopological gyrogroups, namely Einstein gyrogroups endowed with two compatible topologies. The construction is motivated by the Einstein velocity addition law from special relativity and by the gyrogroup theory developed by Ungar. The additional topological structure permits one to distinguish two possibly different notions of continuity, convergence, compactness, and local regularity on the same Einstein gyrogroup. We first recall the Einstein addition law on the open unit ball and the associated gyrogroup structure. We then introduce bitopological Einstein gyrogroups and define pairwise continuous gyrogroup operations, pairwise continuous gyrations, and bitopological Einstein gyrohomomorphisms. Several basic properties are established, including pairwise continuity of translations, inversion, and gyrations. We further study pairwise Hausdorffness, pairwise compactness, separation properties, and the relationship between the two topologies. Examples are given in which the two topologies coincide and in which they are genuinely distinct. Finally, we discuss Einstein bitopological gyrovector spaces and indicate connections with hyperbolic geometry, Lorentz transformations, and future developments in differential and geometric structures on bitopological gyrogroups.