Design Error in the Brillet and Hall’s
Experiment
Revised,
Jan,
27,
2005
( Last
modified:
2009/11/7 )
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”. Web address:
http://www.newtonphysics.on.ca/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, 2401 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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