Logical Invalidity of the Postulate of the Constancy
of the Speed of Light
by
Ardeshir
Mehta
Ottawa, Canada
Wednesday, March 6, 2002
The Theory of Relativity is based on two postulates: the postulate that
the speed of light is a constant for all inertial observers, and the postulate
that all uniform rectilinear motion is relative.
The first postulate states that all inertial observers will obtain the
same result when measuring the speed of light, namely c, which is
calculated these days to be 299,792,458 metres per second.
However, the following simple thought-experiment will show that this
cannot possibly be the case.
Let there be two long cylindrical spaceships S and S',
each parallel to the other, in a region of space far away from any detectable
gravitational field. Let there be two observers O and O'
in the spaceships S and S' respectively, each carrying timepieces
accurate to one part in a million: that is, they show less than one millionth
of a second of difference after the passage of one second. Let the two
ends of the spaceship S be labelled A and B, and the
two ends of spaceship S' be labelled A' and B'. Let
S
and S' be identical in dimensions to an accuracy of one in a million
— i.e., let every metre of their dimensions be accurate to one-thousandth
of a millimetre.
Let each spaceship have a "proper length" of one light-second (i.e.,
let the distance AB as well as the distance A'B' be 299,792,458
metres in length, as measured by observers on the spaceships S and
S',
respectively). Let them each be one metre in diameter, and let them be
separated from one another by a distance of one metre. Let them be moving
along their long axes relative to one another at a velocity of exactly
one-thousandth of the speed of light, i.e., 299,792.458 m/s.
Note that at this speed, the Lorentz <gamma> factor, namely
1/(1-v2/c2)0.5, which when calculated
is 1/(1-0.000001)0.5, equals 1/0.9999995 or 1.0000005, accurate
to seven decimal places. Thus the difference in length between the two
spaceships due to any Relativistic length contraction, even if any exists,
would be smaller than the limits of accuracy of the measurements of the
spaceships, which as we said is one part in a million. So the spaceships
would not be measurably different in length, even if there were
such a thing as Relativistic length contraction.
And likewise, the timepieces carried by the observers O and O'
would not show any difference between one second elapsed on the timepiece
carried by O, for instance, as compared to one second elapsed on
the timepiece carried by O' — for any such difference, even if any
truly existed, would be too small to measure.
Now as the two spaceships pass one another there must come a point at
which A and A' are at their minimum distance apart, namely
one metre, and at that instant, so too must B and B' be at
their minimum distance apart. (See Fig. 1 — the grey arrow
represents the direction of movement of spaceship S' relative to
spaceship S.) When this happens, let there be a small and delicate
antenna jutting out from A at right angles to spaceship
S,
and another similar antenna jutting out from A' at right angles
to spaceship S', oriented in such a manner that as A and
A'
pass one another, the two antennae make brief contact. Let an electrical
current pass though the antennae when they make contact, creating a brief
spark as they make contact and then break away.
Fig. 1
Now the question is, how long should it take the light from
the spark to reach B and B', respectively, as calculated
by observers
O and O'?
If the Relativistic postulate of the constancy of the speed of light
for all inertial observers is correct, it should take exactly one second
for the light from the spark to reach both B and B', as calculated
by either observer. That’s because in the frame
of observer O, the distance from A to B is exactly
one light-second, and in the frame of the observer O', the distance
from A' to B' is also exactly one light-second.
(Remember that the time dilation <gamma> factor is too small
to show any difference between one second as measured by either
of the two timepieces, as compared with one second as measured by the other;
and so is the Relativistic length contraction factor to show any difference
in the length of the spaceships as measured by either of the two observers.)
However, since the spaceships are moving at a relative velocity equal
to one-thousandth c, as calculated by observer O, say, during
the one second the light from the spark takes to travel from A to
B,
the end B' of spaceship S' will have moved away from
B
and towards A by an amount equal to one-thousandth of a light-second,
or about 299 metres. (See Fig. 2.) Thus as calculated by
the observer O, the light from the spark would have reached B'
about a thousandth of a second before it reached B: which
means that as per the calculations of the observer
O, light would
take 0.999 seconds to reach B', and not exactly 1.000 seconds as
calculated by the observer O'.
Fig. 2
And as calculated by the observer O', the light from the spark
would have reached B' exactly 1.000 second after it left A',
but during that one second, B would have moved away from
both A' and B', again by an amount equal to one-thousandth
of a light-second. Thus according to the observer O', the light
from the spark would have reached B about one-thousandth of a second
after it reached B': which means that as per the calculations
of the observer
O', light should reach B 1.001 seconds after
the spark ignited.
But this is a contradiction, because as we calculated, the timepieces
of both O and O' must tick at the same rate (within the margin
of error of one millionth of a second for every second). If the calculations
performed by O show that light from the spark must reach B'
in 0.999 seconds, then the calculations performed by O' should show
the same thing; and if the calculations performed by O' show that
light from the spark should reach B in 1.001 seconds, so should
the calculations performed by O.
The following table (Table 1) illustrates the contradiction
very clearly. It is seen that the figures obtained by assuming that the
postulate of the constancy of the speed of light for all inertial observers
contradict the figures obtained by both observers as a result of the calculations
performed by them.
Table 1
