# Space-time geometry

1. Feb 25, 2006

### bernhard.rothenstein

under which circumstances do we say that something in special relativity is the conseqeunce of space-time geometry (Minkowski)?

2. Feb 27, 2006

### pmb_phy

I can't think of anything and I seriously doubt that such an assertion that something "is the conseqeunce of space-time geometry" is meaningful.

Go back to Newtonian Mechanics. Under Newtoanian mechanics when can it be said that something "is the conseqeunce of geometry"?

I've seen people make that statement before but I've never seen anyone provide a meaningful answer without repeating themselves in a different way. It has said that GR is all about spacetime geometry. That make s me wince. Even Einstein didn't like that statement and neither does Weinberg in his GR text.

Pete

3. Feb 27, 2006

### dicerandom

I'm of the opinion that, in a sense, all of SR is a result of the Minkowski geometry. Moving observers disagree on length scales, time scales (really the same as length scales...), and simultenaity because of the way velocity vectors rotate and the way we define perpindicular in Minkowski geometry.

The hyperbolic nature of Minkowski space-time provides us with all of the classic SR effects, and indeed I would argue that it is the geometry itself which is physically meaningful.

I think I can give an example of something which, under Newtonian mechanics, is the consequence of geometry: F=ma. It's a beautiful statement really, it says that if you want an object to go in something other than a straight line through four dimensions then you must apply a force to it. Why does an object with no forces acting upon it go in a straight line? Because all directions are indistinguishable, how could it possibly decide one way to curve over another? Why does an object with a force on it curve? The force has established a preferred direction and thus pushed the object off of its straight path.

4. Feb 27, 2006

### pervect

Staff Emeritus
We'd need some context to fully understand the remark.

Without any more context, I would say that anything that can be computed or derived from the metric is a consequence of "space-time geometry". But it's hard to be sure if that's the author's intent without more information.

5. Feb 27, 2006

### bernhard.rothenstein

in order to be more specific I would ask if something can be derived using only the two relativistic postulates without any other relativistic ingredients can be considered as a consequence of space-time geometry?
(if i derive mass and momentum without using conservation laws the result is a consequence of space-time geometry?)

6. Feb 28, 2006

### robphy

Of course, there is, implicitly, more to special relativity than the two relativistic postulates. There is an underlying smooth manifold with lots of symmetries, namely, R4, and a riemannian-type metric. Such symmetries probably allow an alternate set of postulates to be used. That is to say, one can probably obtain SR without explicitly using Einstein's postulates by using an alternate set of postulates which are equivalent in light of all of the symmetries abound.

To me (in accord with dicerandom), the structure of SR is faithfully encoded in the geometry of Minkowski spacetime.

Maybe the question can be posed this way.

Is a given result exclusive to SR, or does it also work in galilean relativity? If it doesn't work in both, then the result is probably based on a feature of SR (i.e., a feature of Minkowski spacetime) that is not found in galilean relativity.

For example, along an inertial observer's worldline in SR, time displacements are additive: tA to C=tA to B+tB to C, where A,B,C are events on his inertial worldline. However, this result also holds true in galilean relativity. So, the result is not exclusive to SR or to the geometry of Minkowski spacetime.

7. Feb 28, 2006

### leandros_p

"Geometry" in general is the picture of physical structure.

A physical theory/model is the picture of physical dynamics.

In this context, geometry and physical models are related in the same way that snapshot image is related to cinematic flow.

Leandros