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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!

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The symmetry group of the Minkowski (vector) space is the same as for hyperbolic space. The hyperbolic space carries all the information of the isometry, well, provided the isometry preserves the sense of time. It's the same idea as how one can study euclidean (vector space) symmetries by studying the action on the sphere. –  Ryan Budney Mar 15 '13 at 3:55
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I think of hyperbolicity as having multi-pronged connections to the real world. It comes up naturally in many situations, especially via the geometrization of 3-manifolds. In practice it seems to be the right object for succinctly describing certain types of "teeming" complexity. –  Ryan Budney Mar 15 '13 at 3:57
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Heads up: the way you're using the phrase "hyperbolicity of space time" is completely different than what relativists mean by the same words. To a relativist, a (globally) hyperbolic spacetime is essentially one in which solutions to Cauchy problems (e.g., for a wave equation within that spacetime) exist and are unique. See Hawking and Ellis, p. 206, and Geroch, J Math Phys 11 (1970) 437. –  Ben Crowell Mar 15 '13 at 22:23
    
Ben Crowell, I am not quite sure the two ways of using that world are actually unrelated. The solvability features of a wave equation (as opposed to, say, a heat equation) actually depend on the fact that the "hyperbolic" wave equation is a PDE whose symbol is a function with certain properties - the same properties that one would need to define a riemannian metric of negative curvature. –  Delio Mugnolo Mar 16 '13 at 4:49
    
I didn't say they were unrelated. However, they are completely different. –  Ben Crowell Mar 16 '13 at 14:22
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2 Answers

up vote 14 down vote accepted

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.

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Very nice to read this! Thanks for pointing out this connection! –  Suvrit Mar 15 '13 at 15:55
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For a creative (and more digestible) twist on this try http://gregegan.customer.netspace.net.au/ORTHOGONAL/00/PM.html

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