The Structure of Galilean Space

[littleHilbert]

A Galilean transformation is defined as a transformation that preserves the structure of Galilean space, namely:

1. time intervals;
2. spatial distances between any two simultaneous events;
3. rectilinear motions.

Can anyone give a short argument for the fact that only measuring the distance between simultaneous events is relevant? I've just read one in the Course on Mathematical Physics by Szekeres, but I am not particularly enthusiastic about it. I'm willing to say a few words about that, but for the moment I just wonder what other people would say on this issue.

 

[Bill_K]

Can anyone give a short argument for the fact that only measuring the distance between simultaneous events is relevant?

For two events which are not simultaneous, by a choice of Galiean frame you can make the spatial distance between them to be anything you like.

 

[littleHilbert]

OK. Now I think that I somehow didn't get the point.
I thought the point is that two non-simultaneous events can be brought by a suitable choice of Galiliean frame to simultaneity, i.e. simply by time shift (adding a constant), so that their distance becomes purely spatial distance.

You assert that we can transform spatial distance independently of time, whereas I felt that it's all about transforming time independently of space coordinates. These two formulations seem to be one the same thing. I could have said that by measuring time intervals only events happening at the same place are of interest (referring to point 1. of the list), right?

The example in the above mentioned book (p. 54) concerns two observers, one of which is at point A and the other is on a train, traveling (with uniform velocity) to point B. It is said that the distance between the events "train departs from A" and "train arrives in B" is the spatial distance between A and B -- nonzero value -- for the observer in A. But for the observer on the train the spatial difference is zero (and time difference is non-zero).

That example seemed a bit weird as an explanation of the main point, which, again, for me was how to make measuring spatial distances between non-simultaneous events and measuring distances between simultaneous events in some sense equivalent.

 

[George Jones]

I've just read one in the Course on Mathematical Physics by Szekeres, but I am not particularly enthusiastic about it.

For two events which are not simultaneous, by a choice of Galiean frame you can make the spatial distance between them to be anything you like.

Szekeres just illustrates with a "train" example the point that Bill makes.

 

[littleHilbert]

Yes, as a matter of fact I recognized the example in his formulation. But the problem is that I saw a slightly different thing in it.

 

[George Jones]

Yes, as a matter of fact I recognized the example in his formulation. But the problem is that I saw a slightly different thing in it.

I am not sure what you mean.

 

[littleHilbert]

At this stage I actually meant nothing, except that I agreed that what Bill said was in fact exactly what Szekeres said by example, and I pointed to my previous post.

 

[littleHilbert]

I think I got now what the message is.

The message is simply that the spatial distance is not a well-defined function on non-simultaneous events, i.e. not independent under the Galilean transforms, for that is what we wish it to be -- the (classical) frame reference change must preserve lengths.

Is it what it's all about?

 

[robphy]

Yes.
Try this:
Compute the eigenvectors and eigenvalues of the Galilean Transformation. Interpret physically.