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 is a vector in real -space, we may define the Minkowski quadratic form to be
The points of the n
-dimensional hyperboloid model are then the vectors
; that is, the upper or future sheet. Calling this
, the lines of the model are the intersections of planes through the origin with
, and in general the m
-flats of the model are the intersection of the
-dimensional subspace through the origin with
. The flats of
are therefore a subset of the flats of n
-dimensional projective space
, making the usual identification of subspaces of real
-dimensional vector space (the Grassmannian
) with flats of n
-dimensional projective space; this leads to the related Klein model
of hyperbolic geometry.
Corresponding to the Minkowski quadratic form there is a Minkowski bilinear form , defined by
In terms of this bilinear form the distance between any two points
on the hyperboloid model is given by
If is any parametrized curve on , , then the length of is
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 , 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.
Isometries and symmetry
The generalized orthogonal group is the Lie group of real matrices that preserve the bilinear form .
That is, is the group of isometries of Minkowski space fixing the origin. This group is sometimes called the -dimensional Lorentz group. The subgroup which preserves the sign of the first coordinate value (if ) is called the orthochronous Lorentz group, denoted .
The action of on restricts to an action on . This group clearly preserves the hyperbolic metric on . In fact, is the full isometry group of . This isometry group has dimension , 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 is the group , which is the identity component of the full Lorentz group.
The orientation preserving isometry group acts transitively and faithfully on , by Witt's theorem. This is to say that is a homogeneous space for the action of . The isotropy group of the vector is a matrix of the form
1 & 0 & ldots & 0
0 & & &
vdots & & A &
0 & & &
where is a matrix in the rotation group ; that is, is an orthogonal matrix with determinant +1. Hyperbolic space can therefore be identified with the quotient space .
The bilinear form is the Cartan-Killing form, the unique -invariant quadratic form on .