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Many books on special relativity eventually mention that the geometry of spacetime is special because the metric has a signature $(-,+,+,+)$ which is non-Euclidean. I have encountered many ways this makes it different from normal Euclidean geometry, for example, there is more than one null vector.

I want to study the mathematics of this new geometry in order to develop some intuition for it. I understand that the new geometry is called Hyperbolic geometry. Unfortunately, the information I find about that is all about negatively-curved saddles and Poincare disks, etc, which while interesting, seems quite different!

Can someone point to a good resource for learning just the geometry that underlies SR?

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The SR geometry isn't hyperbolic. SR spacetime is flat while hyperbolic spacetime is negatively curved. If you're asking about the maths underlying the geometry you're probably better of posting in the math SE. –  John Rennie Sep 10 '13 at 6:53
If it isn't hyperbolic, then is it euclidean or spherical? My understanding is that a geometry must be one of these three. –  xuanji Sep 10 '13 at 6:58
The term hyperbolic is a little misleading in this case. Minkowski space isn't hyperbolic in the same sense that the saddle and Poincare disk are. Actually Minkowski space is a flat Lorentzian manifold. Hyperbolic, euclidean and spherical are all Riemannian manifolds, i.e. they all have positive definite signature. They are all spaces, but Minkowski is a spacetime. It is the - sign in the Minkowski metric that sets it apart. People use the term "hyperbolic" in this case to refer to the fact that Minkowski "spheres" when plotted look like hyperbolas, not that the spacetime is curved. –  Michael Brown Sep 10 '13 at 7:14
SR geometry is an affine geometry, so is flat like Euclidean geometry, but has a different symmetry group. –  Henry Sep 10 '13 at 7:17
What is it that you really want to know about the geometry? In other words, how could you phrase your question to not ask for a resource? –  David Z Sep 10 '13 at 7:28
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1 Answer

up vote 2 down vote accepted

The geometry of special relativity is called Lorentzian geometry, or in full: the "pseudo-Riemannian geometry of Minkowsk spacetime". This is also the Cartan geometry of the Lorentz group inside the Poincaré group.

See on the nLab at Lorentzian geometry for further pointers. See the References there for introductions and surveys.

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@Qmechanic: it is common to use "Lorentzian geometry" to refer to just the SR case. To pick any random reference: for instance first paragraph of Graciela S. Birman and Katsumi Nomizu, "Trigonometry in Lorentzian Geometry" The American Mathematical Monthly Vol. 91, No. 9 (Nov., 1984), pp. 543-549 jstor.org/stable/2323737 –  Urs Schreiber Sep 10 '13 at 9:13
$\uparrow$ @Urs Schreiber: I agree. –  Qmechanic Sep 10 '13 at 10:41
Ooh the "Trigonometry in Lorentzian Geometry" article is basically what I'm looking for in this question, exploring the math without letting the physics interfere. Unfortunately I can only preview the first page. –  xuanji Sep 11 '13 at 12:43
@zodiac: contact me by email –  Urs Schreiber Sep 11 '13 at 13:51
I managed to obtain the paper, but thanks anyway :) I think you could include the paper in your answer –  xuanji Sep 11 '13 at 13:58
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