Design Error in the Brillet and Hall’s Experiment

by Paul Marmet (1932-2005)

        Revised,  Jan, 27, 2005
( Last modified: 2012/03/17 - The estate of Paul Marmet )
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          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, Ogilvie Rd. Ottawa, Canada, K1J7N4,   200 pages, (1997).  Also on the Web at:
http://www.newtonphysics.on.ca/EINSTEIN/Chapter7.html

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To be published in Physics Essays
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Version,  January 8,  2005
Corrected, Jan. 27,  2005

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Paul Marmet

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