Minkowski space-time and hyperbolic geometry

Abstract

It has become generally recognized that hyperbolic (i.e. Lobachevskian) space can be represented upon one sheet of a two-sheeted cylindrical hyperboloid in Minkowski space-time. This paper aims to clarify the derivation of this result and to describe some further related ideas. Firstly a simple justification is given of the stated property, which seems somewhat lacking in the literature. This is straightforward once it is shown that differential displacements on the hyperboloid surface are space-like elements in Minkowski space-time. This needs certain preliminary remarks on Minkowski space-time.Two other derivations are given which are valid in any pseudo-Euclidean space of the same type.An alternative view comes from regarding Minkowski space-time projectively as a velocity space. This is possible with Minkowski's original representation but is best seen when Minkowski space-time is regarded differentially as a special case of the metric of General Relativity. Here the space may also be considered as differential space-time in the sense of Minkowski'. It may be considered as a projective space and in this case, as a velocity space which is a Lobachevsky space with hyperboloid representation. Projection of the hyperboloid to a disc or spherical ball gives an associated Beltrami-Klein representation of velocity space. This geometrical representation has important application in physics being related to the hyperbolic theory of Special Relativity which was first proposed by Varićak in 1910 following Einstein's original 1905 paper. The Cayley metric for the velocity space representation leads to relativistic addition of two velocities. The paper emphasizes the importance of Weierstrass coordinates as they are highly appropriate to the relativity application. They also show the close relation between the hyperboloid representation and the equivalent spherical one, the hyperbolic space being then regarded as a sphere of imaginary radius which has historically been a guiding idea and one closely related to Special Relativity.<br/