# What is the geometry behind special relativity?

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?

• 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

• $\uparrow$ @Urs Schreiber: I agree. – Qmechanic Sep 10 '13 at 10:41