V0ODO0CH1LD said:
When studying the motion of particles in space, what are the mathematical considerations that have to made of spacetime? Could I say there exists a bijection between spacetime and ##\mathbb{R}^4##? Is the topology under consideration the usual product topology of ##\mathbb{R}^4##? Are there any other consideration if all I am concerned with is the kinematics of particles in space?
Also, what differs between the mathematical considerations of spacetime for non-relativistic classical mechanics and relativistic mechanics? Is the set not bijective to ##\mathbb{R}^4## anymore? Or is it only the topology defined on ##\mathbb{R}^4## that is different?
Space-time is a manifold. That means that LOCALLY there exists a bijection between event in spacetime and ##\mathbb{R}^4##. This bijection is one of the charts in the manifold. A manifold is a collection of such charts with certain conditions on how they are combined, which I'll describe non-rigorously as "sewn together smoothly".
http://en.wikipedia.org/wiki/Manifold
Although a manifold resembles Euclidean space near each point, globally it may not. For example, the surface of the
sphere is not a Euclidean space, but in a region it can be charted by means of
map projections of the region into the
Euclidean plane (in the context of manifolds they are called
charts).
The local group structure of space-time in general relativity, however, is the group structure of space-time for special relativity, the Poincare group.
http://en.wikipedia.org/wiki/Poincaré_group By "group structure of space-time" we mean the Lie group,
http://en.wikipedia.org/wiki/Lie_group,
a
group that is also a
differentiable manifold, with the property that the group operations are compatible with the
smooth structure
So we are back to manifolds again, you can't escape them for a serious discussion of space-time. Now, while the global topology of space-time depends on the manifold structure, we can compare the structure of classical and relativistic space-times by considering the local group structure, which is equivalent to the global structure if we consider only flat manifolds. So for the rest of the discusssion, I'll talk about this case (local structure, which is equivalent to the global structure for flat manifold).The group structure for classical space-time is the tensor product of Euclidean group E(3) that represents space, with another group, R(1) and/or E(1) (I think - corrections welcome) that represent time.
If you look at the generators of the groups, E(3) is generated by rotations, translations, and maybe reflections (depending on your exact definition). See
http://en.wikipedia.org/wiki/Euclidean_group.
Recall that generators of lie groups are the infinitesimal transformations (such as translations) that preserve the metric. If "preserving the metric" seems a bit technical and unintuitive, it's a lot like preserving distances. To be precise, however, actual distances are only preserved in classical mechanics, in relativity we preserve something that's closely related that we call the Lorentz interval. The Lorentz interval is a general name for what we call "proper distances" and "proper times". Hopefully this is all familiar, if it's not - hmmm, do some more reading and ask some questions, I guess...
For classical space-times, we say that two points are , say, 1 unit apart. Then we rotate our frame of reference. The two points are still 1 unit apart after the rotation, just like they were before. Thus rotations "preserve distances".
The generators of the group for time are simple, just translations and possibly reflections. The tensor product of the group representing space and the group representing time give the classical group structure of space-time.
For relativistic space-time, we have for the generators of the group rotations, translations, possibly reflections, and the boost symmetries. See the detailed description of the Poincare group,
http://en.wikipedia.org/wiki/Poincaré_group The boost symmetries, boost stands for "Lorentz boost", is the underlying symmetry that says that space-time looks the same no matter what velocity you are traveling at. These boost symmetries are not present in classical space-time, they are a feature of relativistic space-times.
Note that we don't have any separate "time group" or any tensor product in making up the Poincare group, the Poincare group is a 4 dimensional group.