Re: SR fundamental contradiction

mluttgens@xxxxxxxxxx wrote:
Brian Kennelly wrote:
mluttgens@xxxxxxxxxx wrote:
If you consider a length g (c-v)T at rest in the S'-frame, you
get a length (c-v)T in the S-frame, by applying length contraction
(this is what the tracking guru did).

If you consider a stick of length (c-v)T at rest in the S-frame,
you get a length g(c-v)T in the S'-frame according to the LT.
It is dilated in the S'-frame. Indeed, if the ends of the stick
are x1 = vT and x2 = cT in the S-frame,
Okay, then from the LT, we have:
vT=g(x1'+vt1')
cT=g(x2'+vt2')
This allows us to find x2' and x1' at the same t'

x1' = 0 and x2' = g(c-v)T,
There is no value of t' that makes both of these equations true.
When x1'=0, then (from the first equation):
vT=g(vt1')
So t1' = T/g

When x2'=g(c-v)T, then
cT=g(g(c-v)T+vt2')
So t2'=cT/vg - g(c-v)T/v
\=t1'

To find the length in S', we must measure the distance between the endpoints at the same time in S' (the same t')

Setting t1'=t2' in the LT equations above and subtracting we get
cT-vT=g(x2'-x1')
x2'-x1'=(c-v)T/g

The length is contracted.

hence x2' - x1' = g(c-v)T. Notice that x1' is always zero, its
value is independent of time.
The only way that x1' can be always zero is if it is at rest in S'. Because you stated that the stick is at rest in S, it, and therefore its endpoints, will be moving in S':
vT=g(x1'+vt')
x1'=vT/g-vt'

In particular, when x2'=g(c-v)T, then t'=cT/vg - g(c-v)T/v and x1' is:
x1' = vT/g-v(cT/vg - g(c-v)T/v)
= vT/g- cT/g +g(c-v)T
=g(c-v)T - (c-v)T/g

Subtracting this from x2' again gives the result derived above:
x2'-x1'=(c-v)T/g


This is obvious: if L = L'/g (as in the guru exemple), L' is
necessarily gL. The one who got L = L'/g AND L' = L/g made a logical
mistake somewhere.
No, L=L'/g and L'=L/g apply to two different scenarios. In the first, the stick is at rest in S', in the second, it is at rest in S.
.

Relevant Pages

  • Re: SR fundamental contradiction
    ... possible only if you know the endpoints of the stick in S': ... at rest in the S' frame. ... If you apply length contraction, ... Setting t1'=t2' in the LT equations above and subtracting we get ...
    (sci.physics.relativity)
  • Re: SR fundamental contradiction
    ... calculate the endpoints in S' from the fixed values in S, ... at rest in the S' frame. ... If you apply length contraction, ... Setting t1'=t2' in the LT equations above and subtracting we get ...
    (sci.physics.relativity)
  • Re: SR fundamental contradiction
    ... the endpoints at the same time in S' ... Subtracting this from x2' again gives the result derived above: ... at rest in the S' frame. ... If you apply length contraction, ...
    (sci.physics.relativity)
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