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The Flaw
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Here we derive the time interval tB - tA
of the travel of the photon from end A to end B of the rod.
During the movement of the rod its end A moves according to the following
law of motion (classical
Newton's law of motion)
A(t) = A(tA) + v(t - tA),
the point A being at point A(tA) in the stationary
system at the moment tA.
Then,
the law of motion (classical
Newton's law of motion when the source of the photon is at rest with
the stationary system)
of the photon before reaching B is
F(t) = F(tA) + c(t - tA)
where the photon F is in point F(tB) at the moment
tB.
We know, from the condition of the problem, that the photon F at the
moment tB will be in the point
F(tB)
= F(tA) + c(tB
- tA)
(recall that for any moment
t the photon F will be in the point F(t) = F(tA)
+ c(t - tA) or
F(t)
= A(tA) + c(t - tA)
because
F(tA) = A(tA))
When we see where point A is at the moment tB
we obtain
A(tB) = A(tA)
+ v(tB - tA)
Let us now calculate the distance between A(tB)
and F(tB) at the moment tB.
We write:
F(tB) - A(tB)
= A(tA) + c(tB
- tA) - A(tA)
- v(tB - tA)
= (c - v)(tB - tA),
However, when at time tB the
photon reaches B the position of the photon and
B coincide, therefore then F(tB)
= B(tB). Also, B(tB)
- A(tB)
= rAB -- See
here why ...
Therefore,
tB - tA = (B(tB)
- A(tB))/(c - v)
= rAB/(c - v),
or
tB - tA = rAB/(c
- v)
The above formula invalidates the second postulate
at once
We do not need to go far to see the contradiction
between the absolute truth that Newton's mechanics is the valid mechanics
in the stationary system on the one hand and on the other the requirement
of STR that that
the light always propagates in free space with a constant velocity c
= cons which is independent of the state of motion of the emitting body
(second postulate of STR)
That contradiction
is seen at once when inspecting the above-derived
quantity tB - tA in the
formula tB - tA = rAB/(c
- v), a formula given by Einstein himself in §2
of the 1905
paper. In this formula the quantity tB
- tA is exactly the time interval for which nothing
else but light travels
from the one end A to the other end B of a rod having length rAB.
Therefore, the meaning of the quantity (c - v) can only be velocity
of that light. This is in direct contradiction
with the second postulate requiring the velocity of light to be c =
const under all circumstances. According to the second postulate it
should be of no significance whether or not the rod moves when the light
travels from its one end to its other end. The distance rAB,
which is the length of the rod, should always be traveled by light with
one and the same velocity c = const which, according to the second postulate,
should be the velocity of light when traveling along any distance. The
distance traveled by light should have no significance. If the velocity
of light is postulated to be always c = const then every distance, including
the distance rAB, must necessarily be traveled
with velocity c = const by the light and not with velocity (c - v) (or
with velocity (c + v)) as the formulae adopted by Einstein suggest.
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