# Relativistic space-time geometry

What subject (suggest book titles, etc.) should I study to get a clear grasping of hypersurfaces, 2-surfaces, and integration on them, mostly in special relativity (I'm not messing with general relativity yet).

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This is a topic that usually uses the full machinery of differential geometry, and isn't very different for curved or non-curved manifolds, or will simply just be based on the Jacobian-style logic of a third semester Calculus course. What are you trying to do with this math? –  Jerry Schirmer Apr 26 '11 at 21:50
I want to get a solid understanding of hypersurfaces, normal vectors, 2-surfaces, their normal 2nd rank tensors, etc, and the standard integral theorems. I just want to understand this 4-dimensional stuff, just as deep as a 3rd semester calculus teaches you about them on 3 dimensions. –  becko Apr 26 '11 at 22:15
@becko: Perhaps you're really looking for just (hyper)planes rather than the full concept of general curved (hyper)surfaces? –  Qmechanic Apr 27 '11 at 14:51
@Qmechanic: I need hypersurface, but I'm not looking for a fully abstract mathematical treatment in n-dimensions, etc... I just need whatever is enough to get a feeling for Minkowski 4-dimensional space. –  becko Apr 27 '11 at 15:41
@becko: If you just want $2$-dimensional hypersurfaces embedded in $\mathbb{R}^3$, you may take a look at "Differential Geometry of Curves and Surfaces" by Manfredo Do Carmo. –  Qmechanic Apr 27 '11 at 16:04
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## 1 Answer

As others mentioned, special relativity (by definition really) doesn't have anything to do with curved surfaces! Special relativity has a particular metric (minkowski metric) which has no curvature. If your interested in manifolds (particularly integration on them, since integration in minkowski space is pretty trivial) and things like that, you really getting into general relativity.

Although I am sure many people will have pedagogical arguments against the following approach, I will say it anyway... You don't really need to do an in depth study of mathematical SR to do GR. Really once you understand the basic properties of minkowski space, know what a lorenz transformation is, understand things like length contraction, etc, you really should just jump right into GR. One sorta friendly way to go about it is apply SR to E&M (covariant formalization) as a way to play around with tensors a little.

As far as recommendations of what books to get, take a look at Sean Carroll's notes (http://preposterousuniverse.com/grnotes/) and if you like them get his book too! Perhaps also looking at the first two sections will give you a better idea of what you are looking for.

EDIT: After commenting, I thought I would also just post a link to O'Neills Amazon Page. His books are probably more mathematical than what you are looking for, but they could be very useful to someone else.

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