Natural Science Forum / Physics / Relativity / October 2006

SR fundamental contradiction
------------------------------------------

Luttgens:

Let x = ct.
Then x' = g(x - vt), where gamma = 1/sqrt(1-v^2/c^2), becomes
x' = g(c-v)t
What represents the length (c-v)t?
Is that length "dilated" by g?

Van de Moortel:

Consider the event E on the light signal with x = c t for some
chosen value of t.
Then c t - v t is the distance between the origin of S' (the 'moving
observer') and the light signal, as seen at time t in the S-frame
(the 'stationary frame'), and, by the way, so c - v is by definition
the closing velocity between the two.

For this event E, as seen in S', the light signal has covered the
distance
        x' = c t' = g (c - v) t
This is a distance of the event E in the S' frame.

Now imagine a stick with this particular length
        x' = g (c-v) t
at rest in the S' frame.
What is the length of such a stick in the S-frame?
If you apply length contraction, you find that this length
would be
       x' / g = (c - v) t
in the S frame.

Luttgens:

Any object (stick) measures shorter in terms of a frame relative to
which it is moving with velocity v that it does as measured in a frame
relative to which it is at rest, the ratio of shortening being
sqrt(1-v^2/c^2).
This is a relation between measurements referred to different frames.

If a stick of length x' = g(c-v)t is at rest in the S' frame,
it is moving at v relative to the frame S. So, measured in S, its
length is contracted by 1/g and becomes x = g(c-v)t * 1/g = (c-v)t.
This corresponds to Van de Moortel's reasoning, which is circular.
Indeed, x' = g(c-v)t has been obtained *by applying the LT* to
x = (c-v)t, when S' was considered as moving relative to S, and thus
relative to the stick. No wonder that one gets back  x = (c-v)t
when S is afterwards considered as moving wrt S'.

One has instead to consider a stick of length x = (c-v)t at rest
in the frame S. Relative to the frame S', such stick is moving
at v, hence its length, measured in S', is shortened by 1/g wrt its
length measured in S. Thus, x' = (c-v)t / g.

But *according to the LT*, x' = (c-v)t * g !

Such contradiction demonstrates the falseness of the Lorentz
transformation, falseness whose origin lies in the postulate
that when x = ct, x' = ct'.

Without such postulate, the LT become

x' = (c-v)t / g
t' = t / g

and the stick of length (c-v)t at rest in S is indeed shortened by
sqrt(1-v^2/c^2) in S'.

Derivation of the correct transformation:
----------------------------------------

Let's consider two frames of reference, S and S', each in uniform
translatory motion relative to the other, the velocity of S' relative
to S being v.

  Basis relations:

  x' = ax + bt
  t' = ex + kt

  At the origin of S', x' = 0 and x = vt.
  Hence, 0 = (av+b)t, whence b = -av

  The basis relations are now

  (1) x' = a(x - vt)
  (2) t' = ex + kt

  Now, let's suppose that a light signal, starting from the coincident

  origins of frames S and S' at t = t' = 0, travels toward positive x.
  After a time t, it will be at
  x = ct + vt, and also at
  x' = ct'

  Substituting these values of x and x' in relations
  (1) and (2), and eliminating t and t', one gets

  (3) a - ec - ev - k = 0

  If the signal travels toward negative x,
  x = -ct + vt and x' = -ct'

  Substituting these values of x and x' in relations
  (1) and (2), and eliminating t and t', one gets

  (4) a + ec - ev - k = 0

  From (3) and (4), one gets e = 0

  With e = 0, relations (3) or (4) reduce to a = k

  Hence, relations (1) and (2) become

  (1) x' = k (x - vt)
  (2) t' = k t

  Now, a light signal follow the y' axis. Relatively to S,
  it travels obliquely, for, while the signal goes
  a distance ct, the y'-axis advances a distance x = vt.
  Thus c^2t^2 = v^2t^2 + y^2, whence y = sqrt(c^2 - v^2) * t.
  But, also, y' = ct' = c * k * t.

  Equating y' to y, one gets k = sqrt(1 - v^2/c^2)

  The transforms obtained without the *bold* Einstein's postulate that
  the speed of light is the same in all frames are thus

   (5) x' = sqrt(1 - v^2/c^2) * (x - vt) = (x - vt) / g
   (6) t' = sqrt(1 - v^2/c^2) * t = t / g,

       (g corresponds to Einstein's gamma).

Transform (5) straightforwardly tells us that any body measures shorter
in terms of a frame relative to which it is moving with speed v than
it does as measured in a frame relative to which it is at rest.

Transform (6) implies that when two physical systems are in uniform
relative translation at speed v, the effects produced by system A
on system B are modified just as if all natural processes on A were
slowed down in the ratio sqrt(1 - v^2/c^2). (i.e., the so-called time
"dilation").

Those transforms, contrary to Einstein's LT, don't allow to claim that
simultaneity is relative (i.e., that events that are considered to
be simultaneous in one reference frame are not simultaneous in another
reference frame moving with respect to the first, cf. Wikipedia).

Marcel Luttgens