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.

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,

$B((t,\; x\_1,\; ...\; x\_n),\; (s,\; y\_1,\; ...\; y\_n))\; =\; ts\; -\; x\_1\; y\_1\; -\; ...\; -\; x\_n\; y\_n$.

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

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

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

$int\_a^b\; sqrt\{-Q(C\text{'}(r))\}\; dr.$

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

That is, $mbox\{O\}(n,1)$ 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 $mbox\{O\}^\{+\}(n,1)$.

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

The orientation preserving isometry group $mbox\{SO\}^\{+\}(n,1)$ acts transitively and faithfully on $U$, by Witt's theorem. This is to say that $U$ is a homogeneous space for the action of $mbox\{SO\}^\{+\}(n,1)$. 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 $mbox\{SO\}(n)$; that is, $A$ is an $n\; times\; n$ orthogonal matrix with determinant +1. Hyperbolic space $U$ can therefore be identified with the quotient space $mbox\{SO\}^\{+\}(n,1)/mbox\{SO\}(n)$.

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

See also

• Klein model
• Poincaré disk model
• Hyperbolic space
• Hyperbolic geometry
• Minkowski space
• Hyperbolic quaternions

References

• , page 13
• , Chapter 3