Rigidity and Parallelism in the spacetime

Abstract

The effect of the linear-fractional transformations on the parallel lines in the spacetime has been studied. Fock-Lorentz transformations maps a line to a line, from which one can obtain the combinations rule for the velocities in the Fock-Lorentz transformations. Rigidity is defined as a consequences of holding parallelism under the transformations. The Fock-Lorentz transformations do not preserve rigidity, which leads to some novel results such as growing distances alongside with advancing time. Also, it is shown that the time coordinates of events will come closer to each other in the transformed coordinates by going back in time