Understanding Galilean structure

[hunc]

I just start to read Mathematical Methods of Classical Mechanics by Arnold. And I am sort of very puzzled by all the notion.
Firstly, if the universe is seen as a 4D affine space, why is time a mapping from $R^4→R$? I mean this kind of 4D contains time, right?
Secondly, I thought the kernel of such a mapping t should be the set of events simultaneous with a given event (affine apace), yet it saids kernel is a 3D linear subspace of a vector space $R^4$.
Thirdly, I never formally took a class on group theory, and google did not exactly answer me. But what is the dimension of the galilean group (or any other group).

Thanks in advance!

 

[mfb]

Firstly, if the universe is seen as a 4D affine space, why is time a mapping from $R^4→R$? I mean this kind of 4D contains time, right?

Every point in spacetime (R^4) gets exactly one real value "time". The mapping is trivial here, it will get more interesting in special relativity.

Secondly, I thought the kernel of such a mapping t should be the set of events simultaneous with a given event (affine apace), yet it saids kernel is a 3D linear subspace of a vector space $R^4$.

While "t=0" is nothing special for physics, it's still the kernel by definition, and it is a 3D linear subspace of R^4.

 

[dextercioby]

The Galilei group is a 10 dimensional Lie group. It's the basic symmetry group of non-relativistic physics, being more important to Quantum Mechanics than to Classical Mechanics.

 

[hunc]

Every point in spacetime (R^4) gets exactly one real value "time". The mapping is trivial here, it will get more interesting in special relativity.

I see. I guess I'll have to wait for the special relativity to come in.

While "t=0" is nothing special for physics, it's still the kernel by definition, and it is a 3D linear subspace of R^4.

I still don't get it. How does "t=0" provide a fixed origin for the space (if that is indeed what it takes to change a affine space to a linear space)?

Regards,
hunc

 

[hunc]

The Galilei group is a 10 dimensional Lie group. It's the basic symmetry group of non-relativistic physics, being more important to Quantum Mechanics than to Classical Mechanics.

I dig into a Lie group textbook and find the definition. Thanks.

 

[mfb]

I still don't get it. How does "t=0" provide a fixed origin for the space (if that is indeed what it takes to change a affine space to a linear space)?

What do you mean with "provide"? The definition of this point is arbitrary. In every (arbitrary) fixed system, t=0 is a linear 3D subspace.

 

[hunc]

What do you mean with "provide"? The definition of this point is arbitrary. In every (arbitrary) fixed system, t=0 is a linear 3D subspace.

Before t=0, we have an affine space. By t=0, we fixed a point in "time", not "space". So how should we end up with a linear space?

 

[mfb]

Before t=0, we have an affine space.

What does that mean?
We "have" a 4D space. This 4D space has some arbitrary point defined as (0,0,0,0) with arbitrary direction definitions and so on. Now you can consider all points (0,x,y,z). They form a linear subspace.

 

[hunc]

What does that mean? We "have" a 4D space.

Maybe I didn't put it clear enough. A 4D space is what I had in mind before reading Arnold. But in the book Mathematical Methods of Classical Mechanics, it introduce galilean space-time structure as a affine space A⁴. And it kind of offers the difference between the two case in the image. What puzzles me is how are such difference generated.

 

[mfb]

Sorry, I don't understand what the problem is.

 

[vanhees71]

I thought, mathematically Galileo spacetime is a fibrebundle, 3d affine euclidean spaces along the time axis. It's nicely explained in

R. Penrose, Road to reality.

Minkowski spacetime is a 4d pseudo-euclidean affine space with a fundamental form of signature (1,3).

 

[hunc]

Sorry, I don't understand what the problem is.

Thanks still.

 

[hunc]

I thought, mathematically Galileo spacetime is a fibrebundle, 3d affine euclidean spaces along the time axis. It's nicely explained in

R. Penrose, Road to reality.

Minkowski spacetime is a 4d pseudo-euclidean affine space with a fundamental form of signature (1,3).

I'll check it out. Thanks for the input.

 

[fisicist]

One important note: By R^4, Arnold does not denote RxRxRxR, but any four-dimensional real vector space (he states that very early). That's not really an issue, as they are isomorphic to each other, but once you forget about the construction via cartesian products, the time mapping is not "trivial" in the sense of canonical.