Definitions

# Hyperboloid model

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

$Q\left(\left[t, x_1, cdots, x_n\right]\right) = t^2 - x_1^2 - cdots - x_n^2.$
The points of the n-dimensional hyperboloid model are then the vectors $v$ such that $Q\left(v\right) = 1$, 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 $\left(m+1\right)$-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 $\left(n+1\right)$-dimensional vector space (the Grassmannian) with flats of n-dimensional projective space; this leads to the related Klein model of hyperbolic geometry.

## Distance function

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

$B\left(u, v\right) = \left(Q\left(u+v\right)-Q\left(u\right)-Q\left(v\right)\right)/2$
Thus,
$B\left(\left(t, x_1, ... x_n\right), \left(s, y_1, ... y_n\right)\right) = ts - x_1 y_1 - ... - x_n y_n$.
In terms of this bilinear form the distance between any two points $u$ and $v$ on the hyperboloid model is given by
$d\left(u, v\right) = operatorname\left\{arccosh\right\}\left(B\left(u, v\right)\right).$

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

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

## 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.

## Isometries and symmetry

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

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

The action of $mbox\left\{O\right\}^\left\{+\right\}\left(n,1\right)$ on $mathbb\left\{R\right\}^\left\{n,1\right\}$ restricts to an action on $U$. This group clearly preserves the hyperbolic metric on $U$. In fact, $mbox\left\{O\right\}^\left\{+\right\}\left(n,1\right)$ is the full isometry group of $U$. This isometry group has dimension $n\left(n+1\right)/2$, 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 $mbox\left\{SO\right\}^\left\{+\right\}\left(n,1\right)$, which is the identity component of the full Lorentz group.

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

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

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

The bilinear form $B$ is the Cartan-Killing form, the unique $mbox\left\{SO\right\}^\left\{+\right\}\left(n,1\right)$-invariant quadratic form on $mbox\left\{SO\right\}^\left\{+\right\}\left(n,1\right)$.