Mathematical Considerations of Spacetime in Classical Mechanics

In summary, space-time is a manifold, and the topology under consideration is the usual product topology. 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). For relativistic space-time, we have for the generators of the group rotations, translations, possibly reflections, and the boost symmetries.
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V0ODO0CH1LD
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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?
 
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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.
 

What is spacetime in classical mechanics?

Spacetime in classical mechanics is a mathematical representation of the combination of space and time as a single entity. It is used to describe the motion of objects and the interactions between them.

What are the mathematical principles that govern spacetime in classical mechanics?

The mathematical principles that govern spacetime in classical mechanics are based on Newton's laws of motion, which describe how objects move in response to forces, and the concept of inertia, which states that objects will remain at rest or in motion unless acted upon by an external force.

How does classical mechanics differ from other theories of spacetime?

Classical mechanics differs from other theories of spacetime, such as relativity, in that it does not take into account the effects of gravity and the curvature of spacetime. Instead, it focuses on the motion of objects under the influence of external forces.

What is the role of mathematics in understanding spacetime in classical mechanics?

Mathematics plays a crucial role in understanding spacetime in classical mechanics. It provides the tools and equations necessary to describe and predict the motion of objects, and it allows for the precise analysis of the relationships between space, time, and forces.

Why is the concept of spacetime important in classical mechanics?

The concept of spacetime is important in classical mechanics because it allows for the accurate description and prediction of the motion of objects in our physical world. It also provides a framework for understanding the fundamental principles that govern the behavior of matter and energy.

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