I am going to analyse the situation in which, in the transformation
t' = px + qt,
the coefficient p is different from zero. This assumption leads to such
a chaos in the time variation between the two inertial systems that the
argument leading to this chaos amounts to redictio ad absurdum.
Therefore p = 0 is the only reasonable solution to the problem.
We have the traditional situation - two inertial frames, the primed
one moving in the positive X direction with a speed v with respect to
the unprimed one etc. The linearity condition
x' = ax + by + cz + dt
(and a similar one for t') is reduced to
x' = ax + dt /1/
t' = px + qt /2/
In the derivation of Lorentz transformations, Einstein introduces
the condition
x' = 0 <-> x = vt /3/
which can be read: if and only if x' = 0 then x = vt. Einstein does not
explain /3/ well but still it is reasonable. It says that the front of a
beam or any other process starting at the origin (x=x'=t=t'=0) and
moving along the y'-axis in the primed frame is characterized by x' = 0
and x = vt. So we adopt the condition /3/. Combining it with /1/ yields
d = -av /4/
Next we apply the principle of special relativity to the condition
/3/. If a beam moving along the y'-axis in the primed frame is
characterized by x = vt in the unprimed frame, then a symmetrical beam
moving along the y-axis in the unprimed frame (x = 0) will be
characterized by x' = -vt' in the primed frame:
x = 0 <-> x' = -vt' /5/
Combining /1/, /2/, /4/ and /5/ leads eventually to
q = a /6/
Then substituting /6/ into /2/ gives
t' = px + at /7/
Next we consider processes starting at the origin and developing
along a straight line IN THE UNPRIMED SYSTEM. For the front of such a
process,
x = kt /8/
where k could be positive, negative or zero. Substituting /8/ into /7/
gives
t' = pkt + at /9/
Now if t > t', /9/ becomes
t > pkt + at /10/
and so we obtain
1 - a > pk /11/
Eq. /11/ describes all cases of time dilation in the primed system.
If, for instance, a > 1 and p < 0 (as is the case in Lorentz
transformations), sufficiently large positive values of k will satisfy
/11/ and for those processes there will be time dilation in the primed
system. However if k is a small positive value, zero or a negative
value, /11/ reverses and for such processes there is time constriction
in the primed system (t < t'). Of course, there is some borderline k
such that t = t'.
If the above conclusions sound absurd, the absurdity could be
avoided by postulating p = 0. This forces us to advance THE ONLY
REASONABLE TRANSFORMATIONS compatible with /1/, /2/, /3/ and /5/. These
are
x' = a(x - vt) /12/
t' = at /13/
Obviously the transformations /12/ and /13/ are incompatible with the
postulate of the constancy of the speed of light (x=ct <-> x'=ct'). As
for the coefficient a, it could be determined as follows. If time
dilation is experimentally proved and processes in the moving system are
slower than in the system at rest, we can substitute, in /13/, a =
1/gamma
and then
x' = (1/gamma)(x - vt) /14/
t' = (1/gamma)t /15/
If time dilation is an unreliable experimental finding, we substitute a
= 1 and return to Galilean transformations.