Q:

How does hyperbolicity of space time affect our lives?

My main research has been in hyperbolic geometry and geometric group theory. I always thought that the only real "application" of my work was that the universe is a 3-manifold.
But recently I found out that hyperbolic 3-space arises in a natural way from relativity: according to the work of Einstein and others, there is a (3,1) quadratic form on space-time that is invariant under transformations. In the Riemannian pseudometric obtained from this form, the "sphere" of radius -1 is a hyperboloid; the pseudo metric on this hyperboloid (or on one branch of it) becomes a real metric, making it a copy of hyperbolic 3-space.
I found this very exciting, because it meant that my research applied to real life. But now, I've had difficulty seeing exactly how it applies. The physical interpretation of the hyperboloid is that it is the set of all points in spacetime that an observer starting at the origin can reach in one unit of their own, "proper" time. It is difficult to imagine the physical meaning of the hyperbolic metric.
That leads into my question. What are the physical meanings of the basic tools of hyperbolic geometry? For nstance, what does the thin triangles condition say about spacetime? Do the existence of closed hyperbolic 3-manifolds imply the existence of spacetime universes with bounded space coordinates? Thanks for your help!

A:

I would suggest John Barrett's essay on The Hyperbolic Theory of Special Relativity as a comprehensive answer.

The principle of relativity corresponds to the hypothesis that the kinematic space is a space of constant negative curvature. The value of the radius of curvature is the speed of light. 
  The relativistic law of combination of velocities can be interpreted as the triangle of velocities in hyperbolic space.
  New formulations are given in
  optics relating hyperbolic velocity
  with logarithmic redshift and in
  dynamics including a reformulation of
  Newton's 2nd law in terms of
  hyperbolic acceleration.

