I am writing something for the journal of the university on the Lorentz space, and I want to prove that the following definition of the hyperbolic distance on the upper sheet of the hyperboloid $v \cdot v=-1$ is a metric:
The hyperbolic distance between $u$ and $v$ is the only positive number $\eta (u,v)$ such that
$$\cosh(\eta(u,v))=-u \cdot v.
$$
The first properties are easy, but all the proofs of the triangle inequality that I have seen seem complicated. In Ratcliffe's Foundations of Hyperbolic Manifolds the lorentzian cross product is used along with many of its properties, and I would like not having to introduce the cross product just for this proof.
I have thought about showing that the hyperbolic angle is the same as the lorentzian arc-length distance, and then showing that the metric given by the arc length satisfies the triangle inequality, but this seems even more tedious. Is there a shorter and nicer proof of this?
Best Answer
We need to check $\eta(u,v)+\eta(v,w)\ge\eta(u,w)$. Introduce coordinates $x,y,z$ so that the form is $x^2+y^2-z^2$.
First, verify that there is a Lorentz map sending $v$ to $(0,0,1)$. Since it is an isometry, we may now assume that $v=(0,0,1)$. This is the main idea. For added convenience, you may also rotate the $xy$-plane so that the $y$-coordinate of $u$ equals 0.
Next, observe that the formula yields equality in the case when the projections of $u$ and $w$ to the $xy$-plane are endpoints of a segment containing (0,0). This is straigtforward if you write $u=(\sinh a,0,\cosh a)$ and $w=(-\sinh b,0,\cosh b)$ where $a,b\ge 0$.
Finally, rotate $w$ around the $z$-axis until it comes to a position as above. The product $u\cdot w$ grows down (it equals contant plus the scalar product of the $xy$-parts, since $z$-coordinates are fixed). Hence $\eta(u,w)$ grows up while the two other distances stay, q.e.d.
Of course, for writing purposes the last step is just an application of Cauchy-Schwarz for the scalar product in $\mathbb R^2$.
This was about two-dimensional hyperbolic plane, in higher dimensions just insert more coordinates (they will not actually show up in formulae).