# Understanding Galilean structure

1. Aug 14, 2014

### 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).

2. Aug 14, 2014

### Staff: Mentor

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.

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.

3. Aug 14, 2014

### 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.

4. Aug 14, 2014

### hunc

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

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

5. Aug 14, 2014

### hunc

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

6. Aug 16, 2014

### Staff: Mentor

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.

7. Aug 17, 2014

### hunc

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?

8. Aug 17, 2014

### Staff: Mentor

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.

9. Aug 18, 2014

### hunc

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.

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10. Aug 18, 2014

### Staff: Mentor

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

11. Aug 18, 2014

### vanhees71

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

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

12. Aug 18, 2014

### hunc

Thanks still.

13. Aug 18, 2014

### hunc

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

14. Aug 18, 2014

### 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.