OK, that was trivial...I knew it must be easy…if one arranges things the right way. I too was using the pythagorean theorem in order to get rid of some terms. But at the same time I was computing the difference u^2_x+u^2_y-c^2 to get 0…and yeah…somehow fell asleep.
Many thanks, George.
If you mean using the angles in u_x=u \cos \theta,\, u_y=u \sin \theta, \, u'_x=u' \cos \theta',\, u'_y=u' \sin \theta' …does it really help? I tried to plug them in, too…but same thing…the computation gets lengthier and seems to be getting nowhere. There is actually no more on that page, except...
Consider the law of addition of velocities for a particle moving in the x-y plane:
In the book by Szekeres on mathematical physics on p.238 it is said that if u'=c, then it follows from the above formulae that...
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...
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.
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...