Hyperboloid model

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In geometry, the hyperboloid model, also known as the Minkowski model or the Lorentz model, is a model of hyperbolic geometry in which the points are points on one sheet of a hyperboloid of two sheets.

If [t, x_1, \cdots, x_n] is a vector in real (n + 1)-space, we may define the Minkowski quadratic form to be

Q([t, x_1, \cdots, x_n]) = t^2 - x_1^2 - \cdots - x_n^2.

The points of the n-dimensional hyperboloid model are then the vectors v such that Q(v) = 1, where t > 0; that is, the upper or future sheet. Calling this U, the lines of the model are the intersections of planes through the origin with U, and in general the m-flats of the model are the intersection of the (m + 1)-dimensional subspace through the origin with U. The flats of U are therefore a subset of the flats of n-dimensional projective space, making the usual identification of subspaces of real (n + 1)-dimensional vector space (the Grassmannian) with flats of n-dimensional projective space; this leads to the related Klein model of hyperbolic geometry.

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[edit] Distance function

Corresponding to the Minkowski quadratic form Q there is a Minkowski bilinear form B, defined by

B(u,v) = (Q(u + v) − Q(u) − Q(v)) / 2

Thus,

B((t,x1,...xn),(s,y1,...yn)) = ts − x1y1 − ... − xnyn.

In terms of this bilinear form the distance between any two points u and v on the hyperboloid model is given by

d(u, v) = \operatorname{arccosh}(B(u, v)).

If C(r) is any parametrized curve on U, a \le r \le b, then the length of C is

\int_a^b \sqrt{-Q(C'(r))} dr.

[edit] The hyperboloid model and relativity

In terms of special relativity, if we use units of years for time and light years for space, then the points of four-dimensional U, which are the points in the model of three-dimensional hyperbolic geometry, are all of the points reached after one year of travel, ship time, in a straight line at a constant velocity. Hence they can also be identified with velocites.

[edit] Isometries and symmetry

The generalized orthogonal group O(n,1) is the Lie group of (n+1)\times (n+1) real matrices that preserve the bilinear form B(u,v).

That is, O(n,1) is the group of isometries of Minkowski space \mathbb{R}^{n,1} fixing the origin. This group is sometimes called the (n + 1)-dimensional Lorentz group. The subgroup which preserves the sign of the first coordinate value t (if Q(u) > 0 ) is called the orthochronous Lorentz group, denoted O + (n,1).

The action of O + (n,1) on \mathbb{R}^{n,1} restricts to an action on U. This group clearly preserves the hyperbolic metric on U. In fact, O + (n,1) is the full isometry group of U. This isometry group has dimension n(n + 1) / 2, the maximal dimension of the isometry group of a Riemannian manifold. Therefore, hyperbolic space is said to be maximally symmetric. The group of orientation preserving isometries of U is the group SO + (n,1), which is the identity component of the full Lorentz group.

The orientation preserving isometry group SO + (n,1) acts transitively and faithfully on U, by Witt's theorem. This is to say that U is a homogeneous space for the action of SO + (n,1). The isotropy group of the vector (1,0,\ldots,0) is a matrix of the form

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

where A is a matrix in the rotation group SO(n); that is, A is an n \times n orthogonal matrix with determinant +1. Hyperbolic space U can therefore be identified with the quotient space SO + (n,1) / SO(n).

The bilinear form B is the Cartan-Killing form, the unique SO + (n,1)-invariant quadratic form on SO + (n,1).

[edit] See also

[edit] References

  • 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 , page 13
  • Anderson, James (2005), Hyperbolic Geometry (2nd ed.), Springer Undergraduate Mathematics Series, 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
  • 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