Hyperbolic trigonometry in the Einstein relativistic velocity model of hyperbolic geometry

Hyperbolic geometry is a fundamental aspect of modern physics. We explore in this paper the use of Einsteinʹs velocity addition as a model of vector addition in hyperbolic geometry. Guided by analogies with ordinary vector addition, we develop hyperbolic vector spaces, called gyrovector spaces, which provide the setting for hyperbolic geometry in the same way that vector spaces provide the setting for Euclidean geometry. The resulting gyrovector spaces enable Euclidean trigonometry to be extended to hyperbolic trigonometry. In particular, we present the hyperbolic law of cosines and sines and the Hyperbolic Pythagorean Theorem emerges when the common vector addition is replaced by the Einstein velocity addition.