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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:
- 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.
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.
–Resonant Frequency of the Cavity of a Rotating
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:
-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:
-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.
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:
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:
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:
To be published in Physics Essays
Version, January 8, 2005
Corrected, Jan. 27, 2005