If is a vector in the -dimensional coordinate spaceRn+1, the Minkowski quadratic form is defined to be The vectors such that form an n-dimensional hyperboloid Sconsisting of two connected components, or sheets: the forward, or future, sheet S+, where x0>0 and the backward, or past, sheet S−, where x0<0. The points of the n-dimensional hyperboloid model are the points on the forward sheet S+. The Minkowski bilinear formB is the polarization of the Minkowski quadratic form Q, Explicitly, The hyperbolic distance between two points u and v of S+ is given by the formula where inverse hyperbolic function| is the inverse function of hyperbolic cosine.
Straight lines
A straight line in hyperbolic n-space is modeled by a geodesic on the hyperboloid. A geodesic on the hyperboloid is the intersection of the hyperboloid with a two-dimensional linear subspace of the n+1-dimensional Minkowski space. If we take u and v to be basis vectors of that linear subspace with and use w as a real parameter for points on the geodesic, then will be a point on the geodesic. More generally, a k-dimensional "flat" in the hyperbolic n-space will be modeled by the intersection of the hyperboloid with a k+1-dimensional linear subspace of the Minkowski space.
Isometries
The indefinite orthogonal group O, also called the -dimensional Lorentz group, is the Lie group of real × matrices which preserve the Minkowski bilinear form. In a different language, it is the group of linear isometries of the Minkowski space. In particular, this group preserves the hyperboloid S. Recall that indefinite orthogonal groups have four connected components, corresponding to reversing or preserving the orientation on each subspace, and form a Klein four-group. The subgroup of O that preserves the sign of the first coordinate is the orthochronous Lorentz group, denoted O+, and has two components, corresponding to preserving or reversing the orientation of the spatial subspace. Its subgroup SO+ consisting of matrices with determinant one is a connected Lie group of dimension n/2 which acts on S+ by linear automorphisms and preserves the hyperbolic distance. This action is transitive and the stabilizer of the vector consists of the matrices of the form Where belongs to the compact special orthogonal group SO. It follows that the n-dimensional hyperbolic space can be exhibited as the homogeneous space and a Riemannian symmetric space of rank 1, The group SO+ is the full group of orientation-preserving isometries of the n-dimensional hyperbolic space.
According to Jeremy Gray, Poincaré used the hyperboloid model in his personal notes in 1880. Poincaré published his results in 1881, in which he discussed the invariance of the quadratic form. Gray shows where the hyperboloid model is implicit in later writing by Poincaré. For details, see History of Lorentz transformations#Poincare.
Also Homersham Cox in 1882 used Weierstrass coordinates satisfying the relation as well as. For details, see History of Lorentz transformations#Cox.
Further exposure of the model was given by Alfred Clebsch and Ferdinand Lindemann in 1891 discussing the relation and. For details, see History of Lorentz transformations#Lindemann.
Weierstrass coordinates were also used by Gérard and Hausdorff and Woods and Liebmann.
The hyperboloid was explored as a metric space by Alexander Macfarlane in his Papers in Space Analysis. He noted that points on the hyperboloid could be written as where α is a basis vector orthogonal to the hyperboloid axis. For example, he obtained the hyperbolic law of cosines through use of his Algebra of Physics. H. Jansen made the hyperboloid model the explicit focus of his 1909 paper "Representation of hyperbolic geometry on a two sheeted hyperboloid". In 1993 W.F. Reynolds recounted some of the early history of the model in his article in the American Mathematical Monthly. Being a commonplace model by the twentieth century, it was identified with the Geschwindigkeitsvectoren by Hermann Minkowski in his 1907 Göttingen lecture 'The Relativity Principle'. Scott Walter, in his 1999 paper "The Non-Euclidean Style of Minkowskian Relativity" recalls Minkowski's awareness, but traces the lineage of the model to Hermann Helmholtz rather than Weierstrass and Killing. In the early years of relativity the hyperboloid model was used by Vladimir Varićak to explain the physics of velocity. In his speech to the German mathematical union in 1912 he referred to Weierstrass coordinates.
