From the Chalkboard 

Topic # 01,  One-Way Velocities  &  Sagnac Effect

{Posted May 2001}

 

We shall perform a Gedankenexperiment now.  Those who have the book please refer to  Appendix II, the figure on page 6A, also see figure below.  Assume we have 3 kids standing on a slowly rotating table as depicted in the figure.  Also, the kids have identical strength and are able to pitch a ball at the same velocity c relative to themselves.  They are in a triangular pattern a, b, c fixed to the rotating platform.  Kid a throws the ball to b which will have moved to position  b' during flight of our hypothetical photon, the ball.  Meanwhile, kid  b'  connects with  c  which will have moved to a new position c''   during the transit time from  a to b' to c'' .    Kid c'' connects with a''' , to b4 and to c5, etc.

 

It is to be noted that the sources of the photon balls have fixed distances relative to one another, since that are rigidly attached to the platform.  The distances between a, b and c are not dependent on the rotational rate of the table, but the paths of the photons are directly dependent.  We note that a triangular pattern would take place for a stationary (not rotating) table.  For increased angular rate, the path of the photons deviate from that of a triangular pattern.  It is also noted that the total transit time required for the balls to make a complete round from  a to b to c and then back to a has an increase time, greater than that for the stationary case, for the path in the clockwise direction.  If the kids throw the ball in the counter clockwise direction, a decreased transit time would be noted, that is less than that of the stationary case.  Also note that the kids throwing the balls will have intelligence with intuitive grasp to connect one another with the balls, anticipating that position where the receiver will be when the ball arrives so that the ball is given the proper trajectory.  If thrown towards the position where the receiver is positioned at that instant in time, the pass will NOT connect.  However, if the kids could throw out a tremendously large number of balls, i.e., a spherical wave of many balls, they would not have to worry about the trajectories at all.  It is also very important to note here, a point that is missed in too many other interpretations of the experiment, that only those balls that make the connection out of the large number of balls in a spherical wave, participate in the effect.  Those photons that fly by the receiver have nothing at all to do with the effect. 

It is important to note, that all electromagnetic photons move in a rectilinear fashion;  on a straight line path.  This has been too often ignored in the calculations and determinations of the outcomes of many optical experiments.  The Sagnac effect calculated on page 15 of the book uses 4 points to represent the re-emitters instead of 3 points as illustrated with our kids.  It is illustrated with the optical gyroscopes on page 9 thru 13 which are based on the principle of Sagnac, that the number N of  N point re-emitters does not effect in any way the outcome of the experiment, i.e., for N=3, N=4, N=6 or whether N is a very large number, the equation derived for the effect is exactly the same.

Let us assume now, in a continuation of our Gedankenexperiment, that our planet Earth with communication between New York and Los Angeles represent such a Sagnac mechanism.  Assume that the atoms and molecules of the air and the Earth's atmosphere represent the N point re-emitters. This time the N is a very large number.  The geographical latitudes, positions of New York and Los Angeles and the center of the Earth represent the geometrical configuration of our Sagnac device now.  The very large number N of the re-emitters in the Earth atmosphere now participate in the Sagnac effect, re-emitting the photons on the many short stretches between the N re-emitters participating in the effect.  

Taking the rotational rate of the Earth of (2x3.14159 / (24x3600sec)) = 7.27E-5 radians per sec, the radius of Earth, 6.37E+6 meters, the area A = 1.275E+14 sq meters enclosed by a signal that would circle the globe around the equator, the Sagnac effect (4A/c2) x angular rate of Earth yields 412 ns, which represents the difference in transit times of the signals in opposite directions.  Taking a projection of the area enclosed by New York, Los Angeles and the Earth center onto the plane of the equator, one gets approximately 28 ns for the difference between signals in opposite directions between New York and Los Angeles.  This time difference is routinely observed in the extremely high data rate communication systems where, the pulse rates are more than sufficiently high enough to resolve the time differences between the transit times of the opposing directions.

Note, this effect is clearly due to emissions and re-emissions of electromagnetic signals on a rectilinear path by N point emitters rigidly attached in a rotating frame.  The expanding spherical wave front made up of the constituent photons responsible for the effect does not rotate or move along any curved path or along any path deviating from that of a purely rectilinear path, between points of re-emission.