Hyperboloid model
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of n-dimensional hyperbolic geometry in which points are represented by points on the forward sheet S+ of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and m-planes are represented by the intersections of (m+1)-planes passing through the origin in Minkowski space with S+ or by wedge products of m vectors.
If (x0, x1, ..., xn) is a vector in the (n + 1)-dimensional coordinate space Rn+1, the Minkowski quadratic form is defined to be The vectors v ∈ Rn+1 such that Q(v) = -1 form an n-dimensional hyperboloid S consisting 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.
In n+1 dimensional Minkowski space, there are two choices for the metric with opposite signature, in the 3-dimensional case either (+, −, −) or (−, +, +).
If the signature (−, +, +) is chosen, then the scalar square of chords between distinct points on the same sheet of the hyperboloid will be positive, which more closely aligns with conventional definitions and expectations in mathematics.
Then n-dimensional hyperbolic space is a Riemannian space and distance or length can be defined as the square root of the scalar square.
If the signature (+, −, −) is chosen, scalar square between distinct points on the hyperboloid will be negative, so various definitions of basic terms must be adjusted, which can be inconvenient.
Nonetheless, the signature (+, −, −, −) is also common for describing spacetime in physics.
A straight line in hyperbolic n-space is modeled by a geodesic on the hyperboloid.
[1] More generally, a k-dimensional "flat" in the hyperbolic n-space will be modeled by the (non-empty) intersection of the hyperboloid with a k+1-dimensional linear subspace (including the origin) of the Minkowski space.
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 (here 1-dimensional and n-dimensional), and form a Klein four-group.
The subgroup of O(1,n) that preserves the sign of the first coordinate is the orthochronous Lorentz group, denoted O+(1,n), and has two components, corresponding to preserving or reversing the orientation of the spatial subspace.
Its subgroup SO+(1,n) consisting of matrices with determinant one is a connected Lie group of dimension n(n+1)/2 which acts on S+ by linear automorphisms and preserves the hyperbolic distance.
This action is transitive and the stabilizer of the vector (1,0,...,0) consists of the matrices of the form Where
In more concrete terms, SO+(1,n) can be split into n(n-1)/2 rotations (formed with a regular Euclidean rotation matrix in the lower-right block) and n hyperbolic translations, which take the form where
The general form of a translation in 3 dimensions along the vector
This extends naturally to more dimensions, and is also the simplified version of a Lorentz boost when you remove the relativity-specific terms.
In several papers between 1878-1885, Wilhelm Killing[2][3][4] used the representation he attributed to Karl Weierstrass for Lobachevskian geometry.
According to Jeremy Gray (1986),[5] 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
[6] Gray shows where the hyperboloid model is implicit in later writing by Poincaré.
[7] Also Homersham Cox in 1882[8][9] used Weierstrass coordinates (without using this name) satisfying the relation
Further exposure of the model was given by Alfred Clebsch and Ferdinand Lindemann in 1891 discussing the relation
[10] Weierstrass coordinates were also used by Gérard (1892),[11] Felix Hausdorff (1899),[12] Frederick S. Woods (1903)],[13] Heinrich Liebmann (1905).
For example, he obtained the hyperbolic law of cosines through use of his Algebra of Physics.
Reynolds recounted some of the early history of the model in his article in the American Mathematical Monthly.
[16] Being a commonplace model by the twentieth century, it was identified with the Geschwindigkeitsvectoren (velocity vectors) by Hermann Minkowski in his 1907 Göttingen lecture 'The Relativity Principle'.
Scott Walter, in his 1999 paper "The Non-Euclidean Style of Minkowskian Relativity"[17] 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.
