Basically, one tells me the difference is fundamental, the other tells me its not. Where am I going wrong?
No where really, there's just a detail I haven't been explicit on. I only did things in one frame.
When we Lorentz transform we go from one guys frame, let's call it F, to another guys frame, let's call it F'.
The coordinates in F are (t,x,y,z) and in F' they are (t',x',y',z').
Let's say the Lorentz transformation is just a boost in speed along the x-direction.
Then
t' = a
t + b
x and
x' = c
t + d
x. Where a,b,c and d are just some numbers, it doesn't matter about the details.
So you can see how the
t'(time in the new frame) is a mix of
t(time in the old frame) and
x(space in the old frame).
So space and time don't really matter, because they're frame dependant. Now what about the distance rule, ds^2=dx^2+dy^2+dz^2-dt^2? (I prefer to leave out c^2)
Well as you might imagine the rule in the F' frame is:
ds'^2 =
dx'^2+
dy'^2+
dz'^2-
dt'^2
Which we can compare with the distance rule in F:
ds^2 =
dx^2+
dy^2+
dz^2-
dt^2
And, believe it or not,
ds'^2 =
ds^2.
So the answer to your question is that although time is important in the distance rule, it doesn't matter whose frame we are in, everybody gets the same answer from the distance rule. Basically no particular person's time matters, it's just that their time is important in their distance rule and everybody’s distance rule gives the same answer.
The technical term for the fact that the distance rule gives the same answer in every frame is that the distance rule is Lorentz invariant.
Hopefully that might clear something up, any questions ask away.
Edited by Son Goku, : Slight modification.
™ Version 4.2
