Space-Time Geometry: Consequences of Minkowski's Relativity

In summary: S. In summary, it can be argued that all of special relativity is a consequence of the Minkowski geometry of space-time. This is because the hyperbolic nature of Minkowski space-time provides us with all of the classic special relativity effects, and the geometry itself is considered to be physically meaningful. However, without more context, it is difficult to determine if a specific result can be considered a consequence of space-time geometry. It is also worth noting that there may be alternate sets of postulates that can lead to the same results in special relativity, highlighting the importance of the underlying geometry.
  • #1
bernhard.rothenstein
991
1
under which circumstances do we say that something in special relativity is the conseqeunce of space-time geometry (Minkowski)?
 
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  • #2
bernhard.rothenstein said:
under which circumstances do we say that something in special relativity is the conseqeunce of space-time geometry (Minkowski)?
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
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
bernhard.rothenstein said:
under which circumstances do we say that something in special relativity is the conseqeunce of space-time geometry (Minkowski)?

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
pervect said:
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.
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
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
"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
 

1. What is space-time geometry?

Space-time geometry is a mathematical framework used to describe the relationship between space and time. It combines the concepts of space and time into a single entity, where they are no longer separate and independent, but instead are intimately interconnected.

2. What are the consequences of Minkowski's relativity in space-time geometry?

Minkowski's relativity, also known as special relativity, has several consequences in space-time geometry. One of the most well-known is the concept of time dilation, where time appears to move slower for an observer in motion compared to an observer at rest. Another consequence is length contraction, where objects in motion appear shorter in the direction of their motion. Additionally, Minkowski's relativity predicts the equivalence of mass and energy, as described by the famous equation E=mc^2.

3. How does space-time geometry relate to Einstein's theory of general relativity?

Einstein's theory of general relativity builds upon Minkowski's relativity and expands it to include the effects of gravity. In this theory, spacetime is described as a curved four-dimensional structure, where the curvature is caused by the presence of mass and energy. This framework allows for a more accurate understanding of the relationship between space, time, and gravity.

4. Can space-time geometry be visualized?

While space-time geometry cannot be directly visualized, it can be represented through mathematical models and diagrams. These visual representations help scientists and researchers understand and analyze the complex relationships and concepts within space-time geometry.

5. How does space-time geometry impact our understanding of the universe?

Space-time geometry has greatly impacted our understanding of the universe by providing a mathematical framework to explain the relationship between space, time, and gravity. It has allowed scientists to make accurate predictions and measurements, and has led to the development of many important theories, such as the Big Bang theory and the concept of black holes. Additionally, space-time geometry has played a crucial role in the development of technologies such as GPS and satellite communications.

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