Hyperboloid model

The red arc of a circle is a geodesic in the Poincare disk model . It is projected onto a brown geodesic on a green hyperboloid.

The hyperboloid model , also known as the Minkowski model or the Lorentz model ( German Minkowski , Hendrik Lorentz ), is a model of n- dimensional Lobachevsky geometry in which each point is represented by a point on the upper surface $S^{+}$ ${\ displaystyle S ^ {+}}$ ${\ displaystyle S ^ {+}}$ the two-sheeted hyperboloid in the ( n +1) -dimensional Minkowski space and the m -planes are represented by the intersection of the ( m +1) -planes in Minkowski space with S ⁺ . The hyperbolic distance function in this model satisfies a simple expression. The hyperboloid model of an n- dimensional hyperbolic space is closely related to the Beltrami – Klein model and the Poincare disk model , since they are projective models in the sense that the is a subgroup of the projective group .

Minkowski quadratic form

If a $(x_{0},x_{1},\dots ,x_{n})$ ${\ displaystyle (x_ {0}, x_ {1}, \ dots, x_ {n})}$ $(x_{0},x_{1},\dots ,x_{n})$ are vectors in an ( n + 1) -dimensional coordinate space $\mathbb {R} ^{n+1}$ ${\ displaystyle \ mathbb {R} ^ {n + 1}}$ $\R^{n+1}$ , the quadratic Minkowski form is defined as

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

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

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

Vectors $v\in \mathbb {R} ^{n+1}$ ${\ displaystyle v \ in \ mathbb {R} ^ {n + 1}}$ $v\in \mathbb {R} ^{n+1}$ such that $Q(v)=1$ ${\ displaystyle Q (v) = 1}$ $Q(v)=1$ , form an n- dimensional hyperboloid S , consisting of two connected components , or sheets - the upper, or future, sheet $S^{+}$ ${\ displaystyle S ^ {+}}$ $S^{+}$ where $x_{0}>0$ ${\ displaystyle x_ {0}> 0}$ $x_{0}>0$ and bottom or past sheet $S^{-}$ ${\ displaystyle S ^ {-}}$ $S^{-}$ where $x_{0}<0$ ${\ displaystyle x_ {0} <0}$ $x_{0}<0$ . Points of an n- dimensional hyperboloid model are points on a sheet of the future $S^{+}$ ${\ displaystyle S ^ {+}}$ $S^{+}$ .

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

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

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

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

Or 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}.

{\ displaystyle 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}.}

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}.

Hyperbolic distance between two points u and v of space $S^{+}$ ${\ displaystyle S ^ {+}}$ $S^{+}$ is given by the formula $d(\mathbf {u} ,\mathbf {v} )=\operatorname {\mathrm {arch} } (B(\mathbf {u} ,\mathbf {v} ))$ ${\ displaystyle d (\ mathbf {u}, \ mathbf {v}) = \ operatorname {\ mathrm {arch}} (B (\ mathbf {u}, \ mathbf {v}))}$ $d(\mathbf {u} ,\mathbf {v} )=\operatorname {\mathrm {arch} } (B(\mathbf {u} ,\mathbf {v} ))$ ,

where arch is the inverse function of the hyperbolic cosine .

Direct

A straight line in a hyperbolic n- space is modeled by a geodesic on a hyperboloid. A geodesic on a hyperboloid is a (nonempty) intersection with a two-dimensional linear subspace (including the origin) of an n + 1-dimensional Minkowski space. If we take as u and v the basis vectors of a linear subspace with

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

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

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

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

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

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

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

\mathbf {u} \,\mathrm {sh} \,w+\mathbf {v} \,\mathrm {sh} \,w

{\ displaystyle \ mathbf {u} \, \ mathrm {sh} \, w + \ mathbf {v} \, \ mathrm {sh} \, w}

will be a point on the geodesic ^[1] .

More generally, a k -dimensional “plane” in a hyperbolic n- space will be modeled by the (nonempty) intersection of the hyperboloid with a k + 1-dimensional linear subspace (including the origin) of the Minkowski space.

Movement

The indefinite orthogonal group O (1, n ), also called the ( n +1) -dimensional Lorentz group , is a Lie group of real ( n +1) × ( n +1) matrices that preserves the Minkowski bilinear form. In other words, this is a group of linear motions of Minkowski space . In particular, this group preserves the hyperboloid S. Recall that indefinite orthogonal groups have four connected components corresponding to the inversion or preservation of orientation on each subspace (here, 1-dimensional and n- dimensional), and form a quadruple Klein group . The subgroup O (1, n ), which retains the sign of the first coordinate, is the Lorentz orthochronous group , denoted by O ⁺ (1, n ), and has two components corresponding to the conservation or reversal of the subspace orientation. 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 hyperbolic distance. This action is transitive and is the stabilizer of the vector (1,0, ..., 0), consisting of matrices of the form

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

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

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

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

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

The group SO ⁺ (1, n ) is a complete group of orientation-preserving motions of an n- dimensional hyperbolic space.

History

In several articles between 1878 and 1885, Wilhelm Killing ^[2] ^[3] ^[4] used the representation of Lobachevsky geometry , which he ascribes to Karl Weierstrass . In particular, he discusses quadratic forms such as $k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}$ ${\ displaystyle k ^ {2} t ^ {2} + u ^ {2} + v ^ {2} + w ^ {2} = k ^ {2}}$ or for arbitrary dimensions $k^{2}x_{0}^{2}+x_{1}^{2}+\dots +x_{n}^{2}=k^{2}$ ${\ displaystyle k ^ {2} x_ {0} ^ {2} + x_ {1} ^ {2} + \ dots + x_ {n} ^ {2} = k ^ {2}}$ where $k$ ${\ displaystyle k}$ is a dual measure of curvature, $k^{2}=\infty$ ${\ displaystyle k ^ {2} = \ infty}$ means Euclidean geometry , $k^{2}>0$ ${\ displaystyle k ^ {2}> 0}$ elliptical geometry , and $k^{2}<0$ ${\ displaystyle k ^ {2} <0}$ means hyperbolic geometry. For details, see .

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 discusses the invariance of a quadratic form $\xi ^{2}+\eta ^{2}-\zeta ^{2}=-1$ ${\ displaystyle \ xi ^ {2} + \ eta ^ {2} - \ zeta ^ {2} = - 1}$ ^[6] . Gray shows where the hyperboloid model is explicitly mentioned in the later works of Poincare ^[7] . For more details, see .

Also, Homersham Cox in 1882 ^[8] ^[9] used Weierstrass coordinates (without using this name) satisfying the relation $z^{2}-x^{2}-y^{2}=1$ ${\ displaystyle z ^ {2} -x ^ {2} -y ^ {2} = 1}$ as well as the relation $w^{2}-x^{2}-y^{2}-z^{2}=1$ ${\ displaystyle w ^ {2} -x ^ {2} -y ^ {2} -z ^ {2} = 1}$ . See the for more details.

Further, the model was used by Alfred Clebsch and Ferdinand von Lindeman in 1891 when discussing the relations $x_{1}^{2}+x_{2}^{2}-4k^{2}x_{3}^{2}=-4k^{2}$ ${\ displaystyle x_ {1} ^ {2} + x_ {2} ^ {2} -4k ^ {2} x_ {3} ^ {2} = - 4k ^ {2}}$ and $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}$ ${\ displaystyle x_ {1} ^ {2} + x_ {2} ^ {2} + x_ {3} ^ {2} -4k ^ {2} x_ {4} ^ {2} = - 4k ^ {2} }$ ^[10] . See for more details.

Weierstrass coordinates were also used by .

Later (1885) Killing argued that the phrase Weierstrass coordinate corresponds with the elements of the hyperboloid model as follows: if a scalar product is given $\langle \cdot ,\cdot \rangle$ ${\ displaystyle \ langle \ cdot, \ cdot \ rangle}$ on $\mathbb {R} ^{n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ , coordinates of Weierstrass point $x\in \mathbb {R} ^{n}$ ${\ displaystyle x \ in \ mathbb {R} ^ {n}}$ are equal

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

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

what can be compared with the expression

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

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

for the hemisphere model ^[11] .

As a metric space, the hyperboloid was considered by in the book Papers in Space Analysis (1894). He noted that points on a hyperboloid can be written as

\mathrm {sh} \,A+\alpha \,\mathrm {sh} \,A,

{\ displaystyle \ mathrm {sh} \, A + \ alpha \, \ mathrm {sh} \, A,}

where α is a base vector orthogonal to the axis of the hyperboloid. For example, he obtained the by using the ^[1] .

H. Jensen focused on the hyperboloid model in a 1909 article, "Representation of Hyperbolic Geometry on a Two-Cavity Hyperboloid" ^[12] . In 1993, W.F. Reynolds presented the early history of the model in an article published in the journal American Mathematical Monthly ^[13] .

Being a recognized model in the twentieth century, it was identified with Geschwindigkeitsvectoren (German, speed vectors) by German Minkowski in Minkowski space . Scott Walter in his 1999 article “The Non-Euclidean Style of the Special Theory of Relativity” ^[14] mentions Minkowski’s awareness, but leads the model to Helmholtz and not to Weierstrass or Killing.

In the early years, the relativistic hyperboloid model was used by to explain the physics of speed. In his report to the German Mathematical Society in 1912, he referred to the Weierstrass coordinates ^[15] .

Notes

↑ ¹ ² Macfarlane, 1894 .
↑ Killing, 1878 , p. 72-83.
↑ Killing, 1880 , p. 265-287.
↑ Killing, 1885 .
↑ Gray, 1986 , p. 271-2.
↑ Poincaré, 1881 , p. 132 -138.
↑ Poincaré, 1887 , p. 71-91.
↑ Cox, 1881 , p. 178-192.
↑ Cox, 1882 , p. 193-215.
↑ Lindemann, 1891 , p. 524.
↑ Deza E., Deza M., 2006 .
↑ Jansen, 1909 , p. 409-440.
↑ Reynolds, 1993 , p. 442-55.
↑ Scott, 1999 , p. 91–127.
↑ Varićak, 1912 , p. 103–127.

Literature

Killing W. Ueber zwei Raumformen mit constanter positiver Krümmung // Journal für die reine und angewandte Mathematik. - 1878. - T. 86 . - S. 72-83 .
Killing W. Die Rechnung in den Nicht-Euklidischen Raumformen // Journal für die reine und angewandte Mathematik. - 1880. - T. 89 . - S. 265-287 .
Killing W. Die nicht-euklidischen Raumformen . - 1885.

Jeremy Gray. Linear differential equations and group theory from Riemann to Poincaré. - 1986.- S. 271-2.
Poincaré H. Sur les applications de la géométrie non-euclidienne à la théorie des formes quadratiques // Association française pour l'avancement des sciences. - 1881. - T. 10 . - S. 132 -138 .
Poincaré H. On the fundamental hypotheses of geometry // Collected works. - 1887. - T. 11. - S. 71-91.
Cox H. Homogeneous coordinates in imaginary geometry and their application to systems of forces // The quarterly journal of pure and applied mathematics. - 1881. - T. 18 , no. 70 . - S. 178-192 .
Cox H. Homogeneous coordinates in imaginary geometry and their application to systems of forces (continued) // The quarterly journal of pure and applied mathematics. - 1882. - T. 18 , no. 71 . - S. 193-215 .
Lindemann F. Vorlesungen über Geometrie von Clebsch II . - Leipzig, 1891 .-- S. 524.
Elena Deza, Michel Deza . Dictionary of Distances. - 2006.
Jansen H. Abbildung hyperbolische Geometrie auf ein zweischaliges Hyperboloid // Mitt. Math. Gesellsch Hamburg. - 1909. - Vol. 4 . - S. 409-440 .
Alexander Macfarlane . Papers on Space Analysis . - New York: B. Westerman, 1894.
Alekseevskij DV, Vinberg EB, Solodovnikov AS Geometry of Spaces of Constant Curvature. - Berlin, New York: Springer-Verlag , 1993. - (Encyclopaedia of Mathematical Sciences). - ISBN 3-540-52000-7 .
James Anderson. Hyperbolic Geometry. - 2nd. - Berlin, New York: Springer-Verlag , 2005. - (Springer Undergraduate Mathematics Series). - ISBN 978-1-85233-934-0 .
John G. Ratcliffe. Chapter 3 // Foundations of hyperbolic manifolds. - Berlin, New York: Springer-Verlag , 1994 .-- ISBN 978-0-387-94348-0 .
Miles Reid, Balázs Szendröi. Geometry and Topology. - Cambridge University Press , 2005. - C. Figure 3.10, p 45. - ISBN 0-521-61325-6 .
Patrick J. Ryan. Euclidean and non-Euclidean geometry: An analytical approach. - Cambridge, London, New York, New Rochelle, Melbourne, Sydney: Cambridge University Press , 1986. - ISBN 0-521-25654-2 .
William F. Reynolds. Hyperbolic geometry on a hyperboloid // American Mathematical Monthly . - 1993. - Vol. 100 .
Scott Walter. The non-Euclidean style of Minkowskian relativity // The Symbolic Universe: Geometry and Physics. - Oxford University Press, 1999. - S. 91–127.
Varićak V. On the Non-Euclidean Interpretation of the Theory of Relativity // Jahresbericht der Deutschen Mathematiker-Vereinigung. - 1912.- T. 21 .

[_8b6c4e962c17996f-1] ¹ ² Macfarlane, 1894 .

[_e708c24dfeffbed4-2] Killing, 1878 , p. 72-83.

[_5b9e773630f8f93b-3] Killing, 1880 , p. 265-287.

[_94f76cdc79eb5fc1-4] Killing, 1885 .

[_02d156df6ad92079-5] Gray, 1986 , p. 271-2.

[_4e15cb176242ad0c-6] Poincaré, 1881 , p. 132 -138.

[_da5c13400a2a58d6-7] Poincaré, 1887 , p. 71-91.

[_ccdc90bdb1056976-8] Cox, 1881 , p. 178-192.

[_a5ac7083c2d7122e-9] Cox, 1882 , p. 193-215.

[_832f914ad4db52d3-10] Lindemann, 1891 , p. 524.

[_e6b695a28eb05a83-11] Deza E., Deza M., 2006 .

[_31f6babd26d70d0f-12] Jansen, 1909 , p. 409-440.

[_c7aac6a1271d105e-13] Reynolds, 1993 , p. 442-55.

[_d213de2b97e3f86d-14] Scott, 1999 , p. 91–127.

[_ec3f46fcfe939aa8-15] Varićak, 1912 , p. 103–127.