- #1
nonequilibrium
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Hello,
I was reading Einstein's RELATIVITY (his non-mathematical 1916 (or so) book).
Having shown the Lorentz transformation formulas, i.e.
t' = \gamma( t - vx/c^2)
x' = \gamma(x - vt),
he then talks about length contraction. He takes a meter-rod fixed in a certain coordinate system S' so that the left end is at x_1' = 0 and the right end is at x_2' = 1 and he then tells us to view these two points from system S at a certain time for which he takes t = 0. We then get that in S x_1 = 0 and x_2 = 1/ \gamma (we're ignoring units here). Thus we get that if we see a rod passing us at a speed v, the length at any given moment is \gamma times smaller than the so-called eigenlength of S' where the stick is fixed. (this is, of course, the famous Lorentz contraction formula)
Now in this case it makes sense to say "the length of the stick in S is smaller than the length of the stick in S' ". But notice that in our previous case, t_1' \neq t_2' (because t_1 = t_2 and x_1 \neq x_2). This was okay because the two points of the rod were fixed in S' anyway, so it didn't matter what "time it was" for a certain point of the rod (i.e. the position of the rod was independent of t'). But imagine the situation where we want to compare the length of a certain rod, moving relatively to us at c/2 and relatively to somebody else at c/3. It has no meaning to say the length I see is smaller/larger than what he sees; is this correct? By taking a certain time for both points in, say, S, then the two points in S' can't be at the same instant, but then you can't compare "two photographs" of each situation... (Then again, I just came to think of the fact that each of these systems can be compared with the eigenlength, thus offering a comparison after all by then cancelling the eigenlength out of those two equations... How come two of my reasonings lead to different statements? Where is there a mistake?)
Thank you.
I was reading Einstein's RELATIVITY (his non-mathematical 1916 (or so) book).
Having shown the Lorentz transformation formulas, i.e.
t' = \gamma( t - vx/c^2)
x' = \gamma(x - vt),
he then talks about length contraction. He takes a meter-rod fixed in a certain coordinate system S' so that the left end is at x_1' = 0 and the right end is at x_2' = 1 and he then tells us to view these two points from system S at a certain time for which he takes t = 0. We then get that in S x_1 = 0 and x_2 = 1/ \gamma (we're ignoring units here). Thus we get that if we see a rod passing us at a speed v, the length at any given moment is \gamma times smaller than the so-called eigenlength of S' where the stick is fixed. (this is, of course, the famous Lorentz contraction formula)
Now in this case it makes sense to say "the length of the stick in S is smaller than the length of the stick in S' ". But notice that in our previous case, t_1' \neq t_2' (because t_1 = t_2 and x_1 \neq x_2). This was okay because the two points of the rod were fixed in S' anyway, so it didn't matter what "time it was" for a certain point of the rod (i.e. the position of the rod was independent of t'). But imagine the situation where we want to compare the length of a certain rod, moving relatively to us at c/2 and relatively to somebody else at c/3. It has no meaning to say the length I see is smaller/larger than what he sees; is this correct? By taking a certain time for both points in, say, S, then the two points in S' can't be at the same instant, but then you can't compare "two photographs" of each situation... (Then again, I just came to think of the fact that each of these systems can be compared with the eigenlength, thus offering a comparison after all by then cancelling the eigenlength out of those two equations... How come two of my reasonings lead to different statements? Where is there a mistake?)
Thank you.
