Design Error in the Brillet and Hall’s
Experiment
Revised,
Jan, 27, 2005
( Last
checked: 2023/081/21 - The estate of Paul Marmet )
Abstract.
Similarly to
the Michelson and Morley experiment, the Brillet and Hall experiment
is also designed to test the symmetry of space. Brillet and
Hall use a large Fabry-Pérot “etalon of length” to stabilize the
frequency of a He-Ne laser, rotating with respect to a fixed
reference laser. It is measured that there is no apparent
change of frequency between the two lasers, during the rotation of
the Fabry-Pérot etalon. There is an error in the Brillet and Hall
calculation, which is due to the fact that when the cavity moves
sideways, light inside the Fabry-Pérot cavity needs to use an
extended optical path to fill the cavity. When we take into
account that ignored phenomenon, we find that the conclusions
presented in the Brillet and Hall experiment are due to that
disregarded change of light path in the transverse direction.
We show here that the resonant frequency of the Fabry-Pérot cavity
does not change in Galilean space after a rotation of 900. Taking into account that
overlooked phenomenon inside the cavity, we find that the null
result in the Brillet and Hall experiment corresponds to Galilean
space, instead of a relativistic space. We must recall that
Einstein’s relativity is based on the erroneous belief that in
Galilean mechanics, there should appear a shift of frequencies in
the Brillet and Hall experiment and a shift of fringes in the
Michelson-Morley experiment when the instrument is rotated.
This paper shows that this is an error. Without any space
distortion, Galilean space is compatible with zero shifts during a
rotation. We must conclude that the null result observed by
Brillet and Hall is in agreement with an absolute Galilean
space.
1-
Distance Traveled by Light in Parallel 
Velocity.
The Brillet and
Hall experiment has much in common with the Michelson-Morley
experiment. In both experiments, we compare the time taken
by light to travel a constant distance moving parallel to the
moving frame, with the distance when they travel in perpendicular
directions. We have seen that, according to the
Michelson-Morley calculation, in Galilean space, the total time
light takes to travel the constant length of the etalon in the
parallel 
direction has been calculated to be different
from the corresponding total distance in the transverse 
direction. The symbol means that light
travels in a direction, which is parallel to the velocity of the
moving frame. The symbol corresponds to the
direction of light after a rotation of 900
of the moving frame.
The constant length of
the etalon in the frame moving at velocity v is defined as
L. We can see that in the parallel direction,
in the Brillet experiment, the total time [t(total)]
taken by light is equal to the time in the forward direction
t(forward), plus the time in the backward
direction t(backward), before completing the two-way trip
inside the cavity. This gives:
 |
1 |
Equation (1) can also be
written:
 |
2 |
In equation (2), we use
the approximation v<<c. Equation (2) is valid when light
moves parallel to the velocity of the moving
frame. We have t0, is the
time taken by light when the frame velocity is zero. Since
light moves at velocity c in the stationary frame, equation (2)
gives the two-way distance [D(total)] traveled by that
light, (as measured by the stationary observer) inside the
cavity. We get:
 |
3 |
Equation (3) gives the
two-way distance [D(total)], given by the time
multiplied by the velocity of light, in the case of a parallel
trajectory, as required in the Michelson-Morley experiment, as well
as in the Brillet and Hall experiment. The last bracket in
equation (3) represents the increase of path length of light, due to
the velocity of the frame.
The corresponding
parameters are also calculated in the transverse direction
when the moving frame has rotated by 900.
Then, light moves in the transverse direction ,
with respect to the frame velocity, and the light path makes a
narrow isoscele triangle in the stationary frame. Using
geometrical considerations (1, 2),
it is found that, in the transverse direction ,
the distance D traveled by light at velocity c, in Galilean
space, is:
 |
4 |
Equation (4) gives the
distance Dtraveled by light to travel the distance 2L in the
moving Fabry-Pérot etalon, when light moves in a transverse direction.
Of
course,
using
Galilean
coordinates,
that
distance
is
measured in the stationary frame so that the velocity of light is
equal to c for the stationary observer. Using a series
expansion of equation (4), the distance traveled by light, in the
transverse direction is:
 |
5 |
The equations above are
the fundamental relationships giving the times and the distances
traveled by light in the parallel and transverse directions,
to complete a two-way travel across a moving constant length 2L, as
applied previously in the calculation of the Michelson-Morley and
Brillet-Hall experiments. These equations will be used below
in order to determine the resonant frequency of the Fabry-Pérot
cavity used in the Brillet-Hall experiment. Let us explain
first the Brillet-Hall experiment.
2
- The Brillet and Hall’s Instrument.
The aim of the Brillet
and Hall experiment is to verify Einstein’s hypothesis, which
assumes that there is an asymmetric distortion of space (or
matter) when the frame is moving. In the case of the Michelson-Morley experiment (1),
such an asymmetric distortion has been claimed following the zero
shifts of the observed fringes. However, we
have seen (2) that this
zero shift must be reinterpreted.
The principle of the Brillet and Hall experiment (3),
consists first in having a constant reference length using a
Fabry-Pérot etalon. A He-Ne laser is servo stabilized
with respect to the Fabry-Pérot etalon as illustrated on figure
(1). Therefore, it is usually claimed that the stability of
the frequency of the He-Ne laser, which is servo controlled,
should be as good as the length of the etalon. The
Fabry-Pérot etalon with the servo stabilized He-Ne laser can
rotate, as illustrated in the lowest part of figure (1). The
frequency of the light signal transmitted on the axis of that
rotating frame is compared with a non-rotating reference laser
shown on the upper part of figure (1). In order to make sure
that the length of the Fabry-Pérot etalon is highly stable, it is
made of low expansion glass-ceramics and temperature stabilized
inside a vacuum tube. The very high stability of the length of
that etalon is hopefully replicated to the frequency of the
servostabilized He-Ne laser. Therefore, any change of length of
the etalon (or a change of velocity of light) should be detected
as a corresponding change of frequency of the rotating
servostabilized He-Ne laser, with respect to an independent
non-rotating exterior stable laser.
The mechanical length
of the Fabry-Pérot etalon in the Brillet and Hall experiment is
“L” as illustrated on figure (1). The conventional
demonstration of the Michelson-Morley experiment (1)has been interpreted as an
asymmetric space contraction in the transverse direction with
respect to the parallel direction. Therefore that assumed
space distortion, when measuring a moving length must also be
reproduced similarly in the Brillet and Hall experiment, using the
Fabry-Pérot etalon, after a rotation of the moving frame. Brillet and Hall report (3),
that their experiment gives a null result during a rotation.
Assuming that
everything above is right, this leads to believe that there is an
asymmetric space distortion between the parallel and transverse direction.
In this paper, we show here that that claim is erroneous because,
in their calculation, Brillet and Hall ignore the fact that in
Galilean space, they did not consider the needed change of path,
due to an angle that makes the light path [1/Cos a] times longer. In this paper, we
reconsider the calculation of the Brillet and Hall experiment,
taking into account the increase of length of the trajectory of
light inside the Fabry-Pérot etalon when it is moving sideways, as
explained below.
3 –
The Fabry-Pérot Etalon.
In the Brillet and Hall
experiment, the Fabry-Pérot etalon is moving at velocity v with
respect to the stationary frame. L is the distance between mirrors
A and B of the Fabry-Pérot etalon, as illustrated on figure
2. The Fabry-Pérot etalon is alternatively oriented so that
light, inside the etalon, in a Galilean space, moves either
parallel or perpendicular to the
velocity of the frame.
On figure 2A and 2B, a
large extended parallel beam of light, projects monochromatic
light (from the left hand side) through a pair of highly
reflecting parallel mirrors (A and B). On figure 2A, we see
that the incident light, which is parallel to the axis of the
mirrors, is reflected many times between mirrors A and B.
Some light emerges each time, through mirror A after each two-way
collision. Then, that light is focused by a lens on a light
detector, forming the central spot shown on figure 2A.
Figure 2B is just
added here to complete the illustration of the Fabry-Pérot cavity,
when the incident light arrives at an angle with respect to the
axis of the interferometer. On figure 2B, we see then that that
light coming out of the interferometer can produce a circle around
the central spot of the Fabry-Pérot interferometer. Other
concentric circles are formed, corresponding to a larger integer
number of wavelengths. However, on figure 2B, we notice that the
distance traveled by light when completing the two-way travel is
one wavelength longer than in figure 2A. Therefore the
natural resonant frequency of the cavity, which is compatible with
that extra wavelength (N+1) on figure 2B is lower, because the
light path inside the cavity is longer due to the angle inside the
cavity. In reality, figures 2A and 2B are generally
superimposed. Figure 2B is not directly relevant here, but it
helps to understand the fundamental mechanism taking place inside
the Fabry-Pérot etalon.
4
–Resonant Frequency of the Cavity of a Rotating
Resonator.
Returning to figure 2A,
we need to calculate, the total time t taken by
light to travel the two-way distance inside the cavity, in both
parallel and perpendicular directions. We
consider here on figure 2A, the instant when the distance traveled
by light during a two-way reflection on mirrors A and B, is
exactly equal to an integer number N of wavelengths. Number
N must be an integer, in order to obtain a constructive
interference, so that the phase of the wave could be the same, at
the same spot, after multiple two-way reflections. Then, the
two-way distance D(total) traveled by light is given
by the number of wavelengths N, times the fundamental
resonant wavelength l. This gives:
 |
6 |
Therefore, after
each individual pair of internal reflections, the number of
wavelengths is an integer equal to N, and the central spot is formed
by the sum of all light passing through mirror A. In the
Brillet and Hall experiment, we read that the Fabry-Pérot cavity
controls the frequency of the He-Ne laser. This means that the
He-Ne laser is servo controlled, so that the constant frequency of
the etalon would be transmitted to the He-Ne laser. For
example, we can adjust the light of the He-Ne laser on the cavity,
so that the intensity of light becomes maximum at the central spot
of the cavity, which means that the number N of wavelengths
corresponds to an integer number N of wavelengths. The maximum
intensity of the peak observed at the central spot of the
interferogram is a good reference point for explaining how to
control the frequency of the He-Ne laser. If the frequency of
the He-Ne laser is changing, the intensity of light at the central
spot on the cavity will change, because there will no longer be an
integer number N of wavelengths in the Fabry-Pérot cavity.
This change of intensity of the central spot is detected and used as
a feedback to restore the “expected” correct
frequency of the He-Ne laser. This is one way the Fabry-Pérot
etalon can control the frequency of the servo controlled He-Ne
laser. Therefore the servo controlled He-Ne laser always
brings back the same resonant frequency as the Fabry-Pérot
cavity.
4
-A – Resonant Frequency in Mode.
In this problem, the
simplest way to calculate the resonant frequency of a cavity is
finding time light takes to travel a full two-way trip inside the
cavity corresponding to one cycle of oscillation. We know
that the resonant frequency “F” of a cavity is the inverse of the
time taken by light to complete its two-way reflection inside the
opposite mirrors of the cavity between mirrors B and A. In
the Brillet-Hall experiment, we wish to determine whether the
resonant frequency of the cavity is the same in the parallel as in the perpendicular directions.
We have F(cavity) is the resonant frequency of the
cavity in the parallel mode. Therefore the simple knowledge
of the periods P(cavity) and P(cavity) is
sufficient to determine the validity of the Brillet-Hall
experiment. In the parallel direction,
the period “P(cavity)” of the cavity is:
 |
7 |
Let us consider figure 2A, when the
frame moves toward the right hand side. Light transmitted
through mirror B (which is moving at velocity v) instantaneously
takes velocity c (with respect to Galilean space) and moves toward
mirror A (which is running away). After reflection on mirror
A, light returns to mirror B at velocity c (with respect to
stationary space), while mirror B approaches that incoming reflected
wave. We find that equation (1) corresponds exactly to that
phenomenon. Since we are in a Galilean space, we find that
when the moving observer sends a beam of light through mirror B, the
time taken to complete the two-way reflection is given by equation
(2). Therefore the period P(cavity) is:
 |
8 |
We recall that (v <<
c). Using equations (7) and (8), the corresponding
resonant frequency F(cavity) is then:
 |
9 |
Where F0 is the resonant frequency of the
stationary cavity. Of course, this same quantity can be
calculated using the parameters of the moving observer. In
Galilean space, we know that light traveling across the moving
distance 2L, has a slower average velocity for the moving
observer. Such a decrease of average velocity of light
explains the longer time interval taken by light inside the cavity
for a two-way reflection, as given in equation (8).
4
-B – Resonant Frequency in Mode.
Let us now calculate
how the “effective” length of the Fabry-Pérot
cavity changes, when moving in the transverse direction in
Galilean space, after a rotation from the parallel direction, to
the transverse direction. Only the rotation
of the moving frame at a constant velocity “v” needs to be
considered here. We need to calculate only light reaching
the “central spot of the interference pattern”.
The
corresponding problem of light moving parallel to the moving
frame is already calculated above in equations 7, 8 and 9 and
illustrated in figure 2A.
Let us now examine the
light path inside the etalon when light travels in the
transverse direction, as seen by the stationary
observer for which the velocity of light is always equal to c in
the Galilean frame. Figure 3 shows an illustration of the
light path inside the moving Fabry-Pérot etalon. The mirrors
A and B are highly reflective and slightly transparent. On
figure 3, the small concentric circles on the upper left of the
image represent the light source. The Fabry-Pérot etalon
appears as six vertical cylinders on figure 3, moving toward the
right hand side, as illustrated at six different instants.
Let us notice that the multiple reproductions of the same moving
Fabry-Pérot cylinder having a length L, are ended by mirrors A and
B (see figure 3).
At time t=0, light
entering the etalon, passes through the partially transparent
mirror B. The time interval between drawings corresponds to half
the time taken by light to travel the two-way distance D, between
mirrors B and A. After light reaches mirror A, at time t=1,
light is reflected toward mirror B and reaches it at time
t=2. After another reflection at time t=2 by mirror B,
toward mirror A at time t=3, these two last steps have completed
the two-way reflection between mirrors A and B.
We see that the
distance traveled by light while making the two-way reflection in
Galilean space, between each ends of the cavity “is not
equal” to twice the length of the cavity as assumed by
Brillet and Hall. We must recall that the real problem now
is the calculation of the resonant frequency of the cavity.
The Brillet-Hall experiment does not use any external light path
as in the Michelson-Morley experiment, but instead, uses the
effective length of the cavity in their attempt to compare the
velocity of light in perpendicular directions. Since this
experiment relies on a resonant cavity, we recall that the natural
frequency of a cavity does not depend only on the length
of the cavity, but also on the velocity of the wave
with respect to the moving cavity. In fact, the most
practical way to calculate the resonant frequency of a cavity is
to calculate the time light takes, to complete the two-way travel
inside the cavity as explained above in section 4-A.
That “Period”, is the inverse of the natural
frequency of the cavity. Let us consider the time P(cavity)
taken by light to complete the two-way travel inside the cavity
moving in the transverse direction.
Distance
Traveled inside the Moving Cavity.
Let us consider light
passing from mirror B at time t=0, to mirror A at time t=1, as
shown on figure 3.
a – First, we must notice that, in
Galilean space, light takes more time to travel the distance
between mirror “B at time t=0” and mirror “A at time t=1”, because
light travels a longer distance in the stationary frame, which is
the side of the isosceles triangle. Light always travel at
velocity c in the Galilean frame. Therefore, along the
side of that triangle, the distance D, becomes (1/Cos a) times longer. This gives:
 |
10 |
Velocity of the
Wave in the Moving Cavity.
b – Let us consider on figure 3,
light traveling between mirror B at time t=0, and mirror A at time
t=1. We know that the observer using the Fabry-Pérot cavity is
moving sideways at velocity v with respect to the Galilean
frame. Since the cavity is moving at velocity v with the
observer, it is necessary to get a total accumulation of the wave
inside the moving cavity in order to get the correct resonant
frequency F(cavity) of the moving cavity, as observed by the
moving observer.
For the moving cavity
(and the observer), light travels between mirror B at time t=1 and
mirror A at time t=1. Therefore, the time taken for light to
travel that distance is the same as the time between mirror B at
time t=0 and mirror A at time t=1. However, we see on figure
3, that the distance between mirror B at time t=1 and mirror A at
time t=1 is Cos(a) times smaller, due to
the proper velocity of the moving cavity with respect to the moving
wave in the stationary Galilean frame. Consequently, in the
moving frame, the relative velocity of light with respect to the
moving cavity is reduced. It takes more time for light to fill
up the cavity with the wave, and complete the full period P(cavity)
inside the cavity. Therefore light reaches mirror A (at time t=1) at
a slower velocity than c, because it is moving away. The relative
velocity (inside the cavity) between the incoming light and the
running away moving frame is VF..
 |
11 |
Therefore, there are two
(not one) physical mechanisms a
and b, responsible for the
fact that light takes more time to complete the two-way trip inside
the moving cavity in the direction. The first
one a, is related to the increase
of distance and the second b,
related to the slower velocity of the incoming wave due to the
running away frame.
Using equations (10) and
(11), we find inside the Fabry-Pérot cavity, the total time T(total)
light
needs
to
make
the
two-way
trip.
This
time interval is the time, to complete one full cycle of the
resonant frequency F of the cavity. This resonant
period P(cavity) of the cavity between mirror A and
B is equal to the distance traveled inside the cavity (equation 10)
divided by the velocity (equation (11) of the wave with respect to
the cavity. Since the cavity is moving away from the wave, the
period P(cavity) is longer. This gives:
 |
12 |
Using a series expansion,
we find that using equations (10), (11) in (12), the period P(cavity)
of
the
wave
inside
the
Fabry-Pérot
corresponding
to
the resonant frequency is:
 |
13 |
Similarly to equation (9), using
equation (13), the resonant frequency F(cavity) of the same cavity
in the transverse direction is:
 |
14 |
Let us now compare the natural
resonant frequency of the Fabry-Pérot cavity in the transverse direction
given
by
equation
(9),
with
the
resonant
frequency
of the cavity in the parallel direction given by
equation (14) after a rotation of 900
of the Brillet-Hall apparatus. The resonant frequencies of the
same cavity are the same along either parallel
and perpendicular directions. The distances
traveled by light in the moving frame oriented in the parallel
or in the perpendicular directions are exactly the
same in Galilean space. Therefore, when there is a rotation of
900 in the moving Fabry-Pérot
etalon, there is no drift of frequencies in the Brillet-Hall
experiment (3), in the case
of Galilean space as shown clearly in equations (9) and (14).
This result is always valid when there is an absolute velocity of
light equal to c in a Galilean space. Therefore, in agreement
with observations, there is no change of frequency in the Brillet
and Hall experiment. We must conclude that the null result observed
by Brillet-hall is in agreement with an absolute Galilean
space.
It is well
known that Einstein’s relativity has been based on the erroneous
belief that in Galilean mechanics, there should appear a shift of
frequencies in the Brillet and Hall experiment and a shift of
fringes in the Michelson-Morley experiment when the instrument is
rotated. This is an error. Without any space distortion,
Galilean physics is compatible with zero shifts during a
rotation. On the contrary, in order to be compatible with
Einstein’s space distortion, observations should record a well
determined positive change of frequency, which has never been
observed. Observations are perfectly compatible with the
existence of an absolute frame of reference in Galilean space.
5 - Acknowledgments.
The author
acknowledges the precious help from my son Nicolas Marmet and
Dennis O’Keefe for reading and commenting this paper, so that that
paper can be completed.
6 - References
1 - Albert A. Michelson, and Edward W. Morley, The
American Journal of Science, “On the Relative Motion of the Earth
and the Luminiferous Ether”. No: 203, Vol. 134, P. 333-345,
Nov. 1887.
2 – Paul Marmet, (to be published). Also “The Overlooked Phenomena in the
Michelson-Morley Experiment”. USB Drive address: michelson/index.html
3 – A. Brillet and J. L. Hall, “Improved
Laser Test of the Isotropy of Space”.
Phys. Rev. Letters, Vol. 42, No: 9, Page 549-552, 1979.
4 – Marmet Paul, “Einstein’s Theory of
Relativity versus Classical Mechanics” Newton Physics Books,
Ogilvie Rd. Ottawa, Canada, K1J7N4, 200 pages,
(1997). Also on the Web at:
http://www.newtonphysics.on.ca/einstein/chapter7.html
----------------------------------
To be published in Physics Essays
----------------------------------
Version, January 8, 2005
Corrected, Jan. 27, 2005
-------------------------------------------------------------
---------------------------
Paul Marmet
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