Hyperboloid model

Red circular arc is geodesic in Poincaré disk model; it projects to the brown geodesic on the green hyperboloid.

In geometry, the hyperboloid model, also known as the Minkowski model or the Lorentz model (after Hermann Minkowski and Hendrik Lorentz), is a model of n-dimensional hyperbolic geometry in which points are represented by the points on the forward sheet S⁺ of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space and m-planes are represented by the intersections of the (m+1)-planes in Minkowski space with S⁺. The hyperbolic distance function admits a simple expression in this model. The hyperboloid model of the n-dimensional hyperbolic space is closely related to the Beltrami–Klein model and to the Poincaré disk model as they are projective models in the sense that the isometry group is a subgroup of the projective group.

Minkowski quadratic form

If (x₀, x₁, …, x_n) is a vector in the (n + 1)-dimensional coordinate space Rⁿ⁺¹, the Minkowski quadratic form is defined to be

Q(x_{0},x_{1},\ldots ,x_{n})=x_{0}^{2}-x_{1}^{2}-\ldots -x_{n}^{2}.

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. The points of the n-dimensional hyperboloid model are the points on the forward sheet S⁺.

The Minkowski bilinear form B is the polarization of the Minkowski quadratic form Q,

B(\mathbf {u} ,\mathbf {v} )=(Q(\mathbf {u} +\mathbf {v} )-Q(\mathbf {u} )-Q(\mathbf {v} ))/2.

Explicitly,

B((x_{0},x_{1},\ldots ,x_{n}),(y_{0},y_{1},\ldots ,y_{n}))=x_{0}y_{0}-x_{1}y_{1}-\ldots -x_{n}y_{n}.

The hyperbolic distance between two points u and v of S⁺ is given by the formula

d(\mathbf {u} ,\mathbf {v} )=\operatorname {arcosh} (B(\mathbf {u} ,\mathbf {v} )),

where $arcosh$ 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 (non-empty) intersection of the hyperboloid with a two-dimensional linear subspace (including the origin) of the n+1-dimensional Minkowski space. If we take u and v to be basis vectors of that linear subspace with

B(\mathbf {u} ,\mathbf {u} )=1

B(\mathbf {v} ,\mathbf {v} )=-1

B(\mathbf {u} ,\mathbf {v} )=B(\mathbf {v} ,\mathbf {u} )=0

and use w as a real parameter for points on the geodesic, then

\mathbf {u} \cosh w+\mathbf {v} \sinh w

will be a point on the geodesic.^[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.

Isometries

The indefinite orthogonal group O(1,n), also called the (n+1)-dimensional Lorentz group, is the Lie group of real (n+1)×(n+1) 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 (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

{\begin{pmatrix}1&0&\ldots &0\\0&&&\\\vdots &&A&\\0&&&\\\end{pmatrix}}

Where $A$ belongs to the compact special orthogonal group SO(n) (generalizing the rotation group SO(3) for n = 3). It follows that the n-dimensional hyperbolic space can be exhibited as the homogeneous space and a Riemannian symmetric space of rank 1,

\mathbb {H} ^{n}=\mathrm {SO} ^{+}(1,n)/\mathrm {SO} (n).

The group SO⁺(1,n) is the full group of orientation-preserving isometries of the n-dimensional hyperbolic space.

History

In several papers between 1878-1885, Wilhelm Killing ^[2]^[3]^[4] used the representation he attributed to Karl Weierstrass for Lobachevskian geometry. In particular, he discussed quadratic forms such as $k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}$ or in arbitrary dimensions $k^{2}x_{0}^{2}+x_{1}^{2}+\dots +x_{n}^{2}=k^{2}$ , where $k$ is the reciprocal measure of curvature, $k^{2}=\infty$ denotes Euclidean geometry, $k^{2}>0$ elliptic geometry, and $k^{2}<0$ hyperbolic geometry. For details, see History of Lorentz transformations#Killing.

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 $\xi ^{2}+\eta ^{2}-\zeta ^{2}=-1$ .^[6] Gray shows where the hyperboloid model is implicit in later writing by Poincaré.^[7] For details, see History of Lorentz transformations#Poincare.

Also Homersham Cox in 1882^[8]^[9] used Weierstrass coordinates (without using this name) satisfying the relation $z^{2}-x^{2}-y^{2}=1$ as well as $w^{2}-x^{2}-y^{2}-z^{2}=1$ . 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 $x_{1}^{2}+x_{2}^{2}-4k^{2}x_{3}^{2}=-4k^{2}$ and $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}$ .^[10] For details, see History of Lorentz transformations#Lindemann.

Weierstrass coordinates were also used by Gérard (1892) and Hausdorff (1899) and Woods (1903) and Liebmann (1905).

Following Killing’s later (1885) attribution, the phrase Weierstrass coordinates has been associated with elements of the hyperboloid model as follows: given an inner product $\langle \cdot ,\cdot \rangle$ on $\mathbb {R} ^{n}$ , the Weierstrass coordinates of $x\in \mathbb {R} ^{n}$ are

(x,{\sqrt {1+\langle x,x\rangle }})\in \mathbb {R} ^{n+1},

which can be compared to

(x,{\sqrt {1-\langle x,x\rangle }})\in \mathbb {R} ^{n+1}

for the hemisphere model.^[11]

The hyperboloid was explored as a metric space by Alexander Macfarlane in his Papers in Space Analysis (1894). He noted that points on the hyperboloid could be written as

\cosh A+\alpha \sinh A,

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.^[1]

H. Jansen made the hyperboloid model the explicit focus of his 1909 paper "Representation of hyperbolic geometry on a two sheeted hyperboloid".^[12] In 1993 W.F. Reynolds recounted some of the early history of the model in his article in the American Mathematical Monthly.^[13]

Being a commonplace model by the twentieth century, it was identified with the Geschwindigkeitsvectoren (velocity vectors) by Hermann Minkowski in his Minkowski space of 1908. Scott Walter, in his 1999 paper "The Non-Euclidean Style of Special Relativity"^[14] 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.^[15]

Notes and references

1 2 Alexander Macfarlane (1894) Papers on Space Analysis, B. Westerman, New York, weblink from archive.org
↑ Killing, W. (1878) [1877]. "Ueber zwei Raumformen mit constanter positiver Krümmung". Journal für die reine und angewandte Mathematik. 86: 72–83.
↑ Killing, W. (1880) [1879]. "Die Rechnung in den Nicht-Euklidischen Raumformen". Journal für die reine und angewandte Mathematik. 89: 265–287.
↑ Killing, W. (1885). Die nicht-euklidischen Raumformen. Leipzig.
↑ Linear differential equations and group theory from Riemann to Poincaré (pages 271,2)
↑ Poincaré, H. (1881). "Sur les applications de la géométrie non-euclidienne à la théorie des formes quadratiques" (PDF). Association française pour l'avancement des sciences. 10: 132–138.
↑ See also Poincaré: On the fundamental hypotheses of geometry 1887 Collected works vol.11, 71-91 and referred to in the book of B.A. Rosenfeld A History of Non-Euclidean Geometry p.266 in English version (Springer 1988).
↑ Cox, H. (1881). "Homogeneous coordinates in imaginary geometry and their application to systems of forces". The quarterly journal of pure and applied mathematics. 18 (70): 178–192.
↑ Cox, H. (1882) [1881]. "Homogeneous coordinates in imaginary geometry and their application to systems of forces (continued)". The quarterly journal of pure and applied mathematics. 18 (71): 193–215.
↑ Lindemann, F. (1891) [1890]. Vorlesungen über Geometrie von Clebsch II. Leipzig. p. 524.
↑ Elena Deza and Michel Deza (2006) Dictionary of Distances
↑ Abbildung hyperbolische Geometrie auf ein zweischaliges Hyperboloid Mitt. Math. Gesellsch Hamburg 4:409–440.
↑ Reynolds, William F. (1993) "Hyperbolic geometry on a hyperboloid", American Mathematical Monthly 100:442–55, URL: https://www.jstor.org/stable/2324297
↑ Walter, Scott (1999), "The non-Euclidean style of Minkowskian relativity" (PDF), in J. Gray, The Symbolic Universe: Geometry and Physics, Oxford University Press, pp. 91–127 (see page 17 of e-link)
↑ Varićak, V. (1912), "On the Non-Euclidean Interpretation of the Theory of Relativity", Jahresbericht der Deutschen Mathematiker-Vereinigung, 21: 103–127

Alekseevskij, D.V.; Vinberg, E.B.; Solodovnikov, A.S. (1993), Geometry of Spaces of Constant Curvature, Encyclopaedia of Mathematical Sciences, Berlin, New York: Springer-Verlag, ISBN 3-540-52000-7
Anderson, James (2005), Hyperbolic Geometry, Springer Undergraduate Mathematics Series (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-1-85233-934-0
Ratcliffe, John G. (1994), Foundations of hyperbolic manifolds, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94348-0 , Chapter 3
Miles Reid & Balázs Szendröi (2005) Geometry and Topology, Figure 3.10, p 45, Cambridge University Press, ISBN 0-521-61325-6, MR 2194744.
Ryan, Patrick J. (1986), Euclidean and non-Euclidean geometry: An analytical approach, Cambridge, London, New York, New Rochelle, Melbourne, Sydney: Cambridge University Press, ISBN 0-521-25654-2

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[Macfarlane-1] 1 2 Alexander Macfarlane (1894) Papers on Space Analysis, B. Westerman, New York, weblink from archive.org

[2] Killing, W. (1878) [1877]. "Ueber zwei Raumformen mit constanter positiver Krümmung". Journal für die reine und angewandte Mathematik. 86: 72–83.

[3] Killing, W. (1880) [1879]. "Die Rechnung in den Nicht-Euklidischen Raumformen". Journal für die reine und angewandte Mathematik. 89: 265–287.

[4] Killing, W. (1885). Die nicht-euklidischen Raumformen. Leipzig.

[5] Linear differential equations and group theory from Riemann to Poincaré (pages 271,2)

[6] Poincaré, H. (1881). "Sur les applications de la géométrie non-euclidienne à la théorie des formes quadratiques" (PDF). Association française pour l'avancement des sciences. 10: 132–138.

[7] See also Poincaré: On the fundamental hypotheses of geometry 1887 Collected works vol.11, 71-91 and referred to in the book of B.A. Rosenfeld A History of Non-Euclidean Geometry p.266 in English version (Springer 1988).

[8] Cox, H. (1881). "Homogeneous coordinates in imaginary geometry and their application to systems of forces". The quarterly journal of pure and applied mathematics. 18 (70): 178–192.

[9] Cox, H. (1882) [1881]. "Homogeneous coordinates in imaginary geometry and their application to systems of forces (continued)". The quarterly journal of pure and applied mathematics. 18 (71): 193–215.

[10] Lindemann, F. (1891) [1890]. Vorlesungen über Geometrie von Clebsch II. Leipzig. p. 524.

[11] Elena Deza and Michel Deza (2006) Dictionary of Distances

[12] Abbildung hyperbolische Geometrie auf ein zweischaliges Hyperboloid Mitt. Math. Gesellsch Hamburg 4:409–440.

[13] Reynolds, William F. (1993) "Hyperbolic geometry on a hyperboloid", American Mathematical Monthly 100:442–55, URL: https://www.jstor.org/stable/2324297

[14] Walter, Scott (1999), "The non-Euclidean style of Minkowskian relativity" (PDF), in J. Gray, The Symbolic Universe: Geometry and Physics, Oxford University Press, pp. 91–127 (see page 17 of e-link)

[15] Varićak, V. (1912), "On the Non-Euclidean Interpretation of the Theory of Relativity", Jahresbericht der Deutschen Mathematiker-Vereinigung, 21: 103–127