  The longitudinal Doppler shift is a measure of hyperbolic distance.
Transformations of uniform motion are determined by the Doppler shift, while
its square root transforms to a uniformly accelerated frame. A time-velocity
space metric is derived, by magnifying the Beltrami coordinates with the
geometric time, which is similar to the one obtained by Friedmann using
Einstein's equations in which the mass tensor describes a universe of dust at
zero pressure. No such assumption nor any approximation in which the
coordinates increase with time (i.e., constant velocities) need be made. The
hyperbolic velocities are related to the sides of a Lambert quadrilateral. In
the limit when the acute angle becomes an ideal point, the case of uniform
acceleration arises. The relations to Hubble's law, and to the exponential red
shift, are discussed.
