Conformal coordinates associated with space-like motions

Abstract

Conformal transformations in two dimensions provide a simple extension of the Lorentz transformation. The velocity of light appears in such transformations as a singular velocity rather than as an upper limit for the velocity. A well-ordered branch of the theory exists for velocities in excess of the velocity of light. If the velocity of a point exceeds the singular velocity in an inertial system, then the conformal representation of the motion is no longer uniform, but contains a folded region. However, the branching of the transformation may be determined so that the elapsed time along the path of such a motion remains positive. Kinematic relations on the other side of the singular velocity seem to complement the usual results of relativity theory in an interesting way. Thus it is known that motion at the speed of light occurs along a null geodesic, and hence corresponds in a certain sense to motion at infinite velocity (that is, in the sense of proper time elapsed). The complementary relation is that a motion of infinite velocity corresponds in the same sense to motion at the speed of light.