Updated on November
15th 2005 and August 30th 2008
Abstract
Although
different arguments argue in favour of the anisotropy of the speed of light
in the Earth frame, the value of this speed, measured with clocks
synchronized by means of the Einstein-Poincaré procedure, is always found
invariant and equal to C.
This
result can be explained if we realize that the measurement is affected by
systematic errors due to length contraction, clock retardation and the
synchronization itself.
Many
authors give credit to the slow clock transport procedure because they
ignore the absolute motion of the Earth. Taking account of this absolute
motion, we can demonstrate that this method entails the
same synchronism discrepancy effect as the Einstein-Poincaré procedure.
I - Measurement of the
speed of light with one or two clocks by means of the Einstein-Poincaré
procedure.
In
order to measure the speed of light, we can use one or two clocks. When we
use one clock, the signal is sent from the clock towards a mirror, and,
after reflection, comes back to its initial position. In this case, what we
measure is the average round trip velocity of the light signal.
As we
have seen in ref 1, even if we give credit to the Lorentz
postulates, which assume the anisotropy of the one-way speed of light in
the Earth frame, the theory demonstrates that the average velocity is
(erroneously) found equal to C in any direction of space. It also appears
independent of the relative speed of the frame in which it is measured,
with respect to the aether frame. (These results follow from the systematic
measurement distortions entailed by Lorentz contraction, clock retardation
and the method of synchronization itself).
Therefore,
a priori, it seems justified to use two clocks in order to accurately
measure the one way speed of light. To this end, we need, beforehand, to
synchronize two distant clocks A and B.
According
to the Einstein-Poincaré procedure (E.P) this requires two steps. First we
send a light signal from clock A to clock B at instant t0. After reflection
the signal comes back to A at instant t1. Then we send another
signal at instant . The clocks will be considered synchronous
if when the signal reaches clock B, the display of clock B is:
 is the 'apparent' average transit time of
the signal measured with the retarded clocks attached to the Earth frame.
But in
the E.P procedure it is identified with the one way transit time of light.
As we
have seen in ref 1, in any direction of space
(where  is the length that AB would assume if it were at rest in the aether
frame) and since, because of the contraction of the meter stick used to
measure it, the distance AB is always found equal to  , the
speed of light is found equal to C in the same way as when we use one
clock.
So,
even if the speed of light is given by the formulas  and  , (see ref 1)
the E.P procedure finds C.
Therefore
it appears justified to test another method, i.e the slow clock transport
procedure.
II - Measurement of the speed of light by means of the slow clock transport
procedure.
Many
physicists believe that one can obtain exact estimate of the speed of light
by means of the slow clock transport method. The procedure consists of
synchronizing two clocks A and B at a point O’ in the Earth frame,
and then transporting clock B to a distance from A at low speed
(v<<C).
Several
authors have envisaged the problem in different ways 2-10.
A
priori, it seems that, since the transport is very slow, and the motion
would have no perceptible influence on the time displayed by clock B, and
that the two clocks would remain almost synchronized all the time.
Is this
really the case?
- Point of view
of special relativity
If we
consider the assumptions of special relativity as indisputable, then
absolute speeds do not mean anything: only relative speeds exist. The
display of clock B will be*:
where t is the display of clock A. (Note that for convenience we have
supposed that the display of the two clocks at the initial instant was  ).
Once
clock B has stopped (at point P), its delay with respect to clock A will
remain constant. The synchronism discrepancy between clocks A and B is then
to first order.
where T designates the display of clock A when clock B reaches point P.
So the
speed of light will appear to be:
(1)
Since  expression
(1) reduces to  .
The
experimental value of the speed of light obtained by this method is C.
Since the
measurements of  and  are supposed to be exact,
special relativity concludes that the real value of the speed of light in
the Earth frame is C.
Therefore,
if we admit the assumptions of special relativity, the measurement of the
speed of light by means of the method of slow clock transport is in
agreement with the assumed hypotheses. But it does not allow these
hypotheses to be verified.
- Point of view of the
Fundamental aether theory
Now, it
is interesting to see whether the previous results can be obtained with
basic hypotheses different from special relativity.
Today,
there are some strong arguments in favour of the
Lorentz assumptions. According to Lorentz, the speed of light is C
exclusively in the aether frame.
Notice
that, if absolute speeds are taken into consideration, then there is no
real slow clock transport since the absolute motion of the Earth is added
to that of the transported clock.
If the
method is reliable, it should give a value of the one way speed of light in
accordance with the assumed hypotheses.
Let us
verify this point. Two cases will be considered successively.
1
– The light ray travels along the direction of motion of the Earth
frame with respect to the aether frame.
Consider
two "inertial" co-ordinate systems S0 and S1, S0
is at rest in the Cosmic Substratum, and S1 is firmly linked to
the Earth frame. Initially the two frames are coincident. At this initial
instant, a vehicle, equipped with a clock, starts from the common origin
and moves slowly and uniformly along the x axis of frame S1,
towards a point P in this frame. We suppose that the x axis is aligned
along the direction of motion of the Earth with respect to the Comic
Substratum (see figure 1).
Figure 1
 is the speed of the Earth with respect to the fundamental frame S0,
 is the speed of the vehicle with respect to S0 and  the speed of
the vehicle with respect to S1.
(Note
that, during a short time, the motion of the Earth with respect to the
Cosmic Substratum can be considered as rectilinear and uniform. If this
were not the case, the bodies standing on the Earth platform would be
submitted to perceptible accelerations).
The
duration of the transport should be short enough so that the orbital and
rotational motions of the Earth would not significantly affect the
measurement.
When
the vehicle reaches point P, it stops. The real time needed to reach point
P is given by
where  is the length of O’P (which is
contracted because of the motion of the Earth with respect to the Cosmic
Substratum);  is the length that O’P would assume
if it were at rest in the aether frame; tr is the real transit
time of the vehicle from O’ to P. (It is the time that a clock in the
aether frame, opposite the vehicle at the instant when this one reaches
point P, would display).
(Let us
bear in mind that in the fundamental aether theory, real speeds obey the Galilean
law of composition of velocities).
But the
clock in the vehicle (B) is slow relative to the clock in frame S0,
and will display the apparent time
Now the
clock placed at the origin O’ of the Earth system (A) slows down with
respect to a clock in frame S0 opposite it. When the vehicle
reaches point P, it will display the apparent time:
(This implies
that, for an instantaneous event occurring at point P, all the clocks which
would be attached to the Cosmic Substratum would display the same time.
There is no measurement distortion in the aether frame). So, between clock
B and clock A, a synchronism discrepancy exists which is equal to:
(2)
we can see that, once the vehicle has stopped, the discrepancy will remain
constant.
-
Speed of light
If we
assume the Lorentz postulates, the real time of light transit along the
distance is theoretically
(Note
that we suppose here, a priori, that the speed of light with respect to
frame S1 is  . This is intentional since we
want here to check whether the method is reliable and if the results are in agreement with the premises).
Now, as
a result of clock retardation, (and without making allowance for lack of
synchronism) the display of a clock in frame S1 placed at point
P when the signal reaches this point should be:
If, in
addition, we take account of the synchronism discrepancy given by formula
(2), the apparent (measured) time of light transit will be:
 (3)
Ignoring
the terms of high order, expression (3) reduces to
Now,
since the measured length of O'P is always found equal to  , the
apparent speed of light will be
Since  is taken
as small as possible, the apparent speed of light is found equal to C.
Therefore, even if the real speed of light is  , the slow
clock transport method will (erroneously) find C in the same way as
Einstein-Poincaré method.
Therefore
the two methods can be considered equivalent.
2–
General case
Let us
now measure the speed of light along a rod O’B making an angle  with
respect to the x axis of a system of coordinates S1, firmly
linked to the Earth frame (see figure 2).
Figure 2
Note that
the x axis of S1 is aligned along the direction of motion of the
Earth with respect to the Cosmic Substratum.
(For a
short period of time this motion can be considered as rectilinear and
uniform).
(Note
also that the rod is in the plane x, y, but obviously, provided thatremains
the same, the following reasoning would be identical in any plane passing
by the x0, x axis).
We can
choose a system of coordinates S0 in the Cosmic Substratum such
that, initially, S0 and S1 are coincident. At this
very instant, a vehicle leaves the common origin, and moves slowly and
uniformly along the rod towards point B.
As we
have seen in ref 1, due length contraction along the x0,
x axis, the length of the rod is given by
where  is the speed of the Earth with respect to
the fundamental frame S0.
Let us
designate as v the real speed of the vehicle with respect to S1,
and V its real speed with respect to S0 (see figure 2).
The
real time needed by the vehicle to reach point B is  , but the
apparent time in frame S1 after making allowance for clock
retardation is
The
apparent time as measured with a clock inside the vehicle is
Therefore,
the synchronism discrepancy between the apparent time displayed by a clock
attached to frame S1 placed at point O’ and the clock in
the vehicle is
we easily find
and  (4)
Taking
account of the inequality  , expression (4) reduces to
 (see
fig 2)
Therefore,
to first order,  becomes
-
Measurement of the speed of light along O’B by means of the slow
clock transport procedure.
Let us
now suppose that in O’ and B stand two clocks which have been (apparently)
synchronized by means of the slow clock transport method. In fact there is
an error of synchronism equal to  .
The
real speed of light along the rod from O’ to B is (see ref 1).
As a
result of clock retardation but without synchronism discrepancy effect, the
apparent time needed by the light ray to reach point B should be
But we
must take account of the synchronism discrepancy, so that the apparent
(measured) transit time of light will be:
(5)
Ignoring
the terms of high order, expression (5) reduces to
Since
the rod O’B is measured with a contracted meter stick, it appears
equal to  .
The
apparent speed of light is then
since 
which is different from its real value  as seen previously.
Therefore,
contrary to what is often believed 11, the slow clock transport
method does not allow an exact measurement of the speed of light.
It is
approximately equivalent to the method of Einstein-Poincaré, and like this
method, gives, erroneously, the value C for all measurements.
It is
interesting to note that, even if the speed of light is not constant, it is
found constant when standard methods of synchronization are used.
References
1. J.
Lévy "How the apparent speed of light invariance follows from Lorentz
contraction" Proceedings of the PIRT VII, 15-18 September 2000 (late
papers). Imperial College London. Updated in the web site www.levynewphysics.com
2. A.S. Eddington, “The
mathematical theory of relativity” 2nd ed, Cambridge University
Press, Cambridge
(1924).
3. H.
Reichenbach, “The philosophy of space and time”, Dover, New
York (1958).
4. A. Grünbaum, “Philosophical
problems of space and time”; A. Knopf,
New York (1963).
5. P.
W. Bridgman, “A Sophisticate’s primer of relativity”,
Wesleyan University Press, Middletown,
(1962).
6. B.
Ellis and P. Bowman, “Conventionality in distant simultaneity”,
Phil, Sci, 34, 116-136 (1967).
7. A. Grünbaum, “Simultaneity
by slow clock transport in the special theory of relativity”, Phil
Sci, 36, 5-43, (1969).
8. Yu.
B. Molchanov “On a permissible definition of simultaneity by slow
clock transport” (in Russian Einstein Studies, Nauka, Moskow (1972)).
9. J.
A. Winnie “Special relativity without one-way velocity assumptions”
Phil sci, 37, 81-89, 223-238 (1970).
10.
R.G. Zaripov, “Convention in defining simultaneity by slow clock
transport”. Galilean Electrodynamics, 10, 57, May June 1999.
11. R.
Anderson et al, Physics reports p 93 – 180 (1998). See in particular p
100, where the authors criticize some attempts to measure the one way speed
of light by means of the slow clock transport procedure. References to
Krisher et al, Nelson et al, Will, Haughan et al and Vessot.
* Note that the value of the speed of light is supposed to be known.
So the measurement consists in verifying whether the results obtained by
this method are in agreement with the premises, and if they enable to account for the experimental facts.
|
