A Deeper Understanding of the Mathematics of the Michelson - Morley Experiment

suggested source of diagram: Richard Feynman's " Lectures on Physics - Vol. I "

I. By the addition of velocities of light thru the aether, longitudinal time is:

II. Determining transverse time:

Now for some other arbitrary time the Michelson - Morley interferometer moves a longitudinal distance

and hence

.

Solving for transverse time:

IIa. Another derivation of solving for transverse time:

Each one of these right triangles is similar [ or proportional ]  to each other and hence the following ratios are true: 

III. Also the following relationships are rather interesting and will be used shortly: 

A.

B.

IV. Michelson - Morley Interferometer is rotated 90° and final interference fringe calculations are taken:

That is, the interference displacement fringe ratio must be at least 0.37 when the Michelson - Morley Interferometer is rotated 90° and given the fact that the instrument itself was itself sensitive to within 1/100th of a fringe ratio. 

Rather what the Michelson - Morley Interferometer produced for all degree turns of the instrument was

§ An ad - hoc assumption provides a partial solution:

The continuing conundrum of adhering to a luminiferous aether hypothesis ( which by definition should be producing fringe interferences! ) while repeated interference experiments were still producing 

,

was tentatively solved in 1889 by George Francis FitzGerald ( 1851 - 1901, born Dublin, Ireland ) and later elaborated upon in 1892 by Hendrik Antoon Lorentz ( 1853 - 1928, born Arnheim, Holland ) when together they conjectured that there existed a sort of "aether drag" causing all bodies ( including an observer's eyeballs! ) to be "squashed" or contracted in the rectilinear direction of motion thru the luminiferous aether by a so - called Lorentz factor !

FitzGerald's ad - hoc solution relied upon his understanding of Maxwell's Electromagnetic Wave Equations and how rectilinear motion effected these waves, in particular the Doppler Effect, whereas Lorentz more fully developed a "contraction theory" based upon his theory of electrons. 

§ The FitzGerald - Lorentz Contraction Hypothesis:

§ Ok, so how does the FitzGerald - Lorentz Contraction Equation solve the Michelson - Morley null result? 

Remember, at this stage of physics understanding ( 1895 ) the idea of the hypothetical luminiferous aether is still being maintained and that the FitzGerald - Lorentz Contraction Equation is being utilized in an ad-hoc fashion in order to rescue the concept of the aether! That is, the interpretation given by Lorentz for this "contraction effect" was wrong, nevertheless the equation was correct and it would shortly enter into a pantheon of other equations developed by Poincaré and others leading ultimately to the tensor mathematics of General Relativity. 

In truth the FitzGerald - Lorentz Contraction Equation only temporarily postponed the eventual abandonment of the concept of an invisible luminiferous aether. It will be Albert Einstein's publication in 1905 of the tensor mathematics of General Relativity and Einstein's concept of spacetime with the Lorentz Transformation Equations ( as the FitzGerald - Lorentz Contraction Equation came to be known ) at its core foundation to have finally solved  the Michelson - Morley null result.  

§ In the meantime, go to "Albert Michelson & the Aether - Part III - The FitzGerald - Lorentz Solution " to view the full solution to the null result of the Michelson - Morley Experiment.

§ For an updated version of the 1887 Michelson - Morley Experiment, go to "Limits on Violations of Lorentz Symmetry and the Einstein Equivalence Principle using Radio - Frequency Spectroscopy of Atomic Dysprosium ", by M. A. Hohensee, N. Leefer, D. Budker, C. Harabati, V. A. Dzuba, and V. V. Flambaum, Phys. Rev. Lett. 111, 050401 ( July 29, 2013 ).

Abstract:

"We report a joint test of local Lorentz invariance and the Einstein equivalence principle for electrons, using long - term measurements of the transition frequency between two nearly degenerate states of atomic dysprosium. We present many - body calculations which demonstrate that the energy splitting of these states is particularly sensitive to violations of both special and general relativity. We limit Lorentz violation for electrons at the level of 10^-17, matching or improving the best laboratory and astrophysical limits by up to a factor of 10, and improve bounds on gravitational redshift anomalies for electrons by 2 orders of magnitude, to 10^-8. With some enhancements, our experiment may be sensitive to Lorentz violation at the level of 9 x 10^-20."

Relativity Science Calculator Summary Conclusion:

The maximum speed of an electron ( theoretically the speed of light ) is invariant in all directions of earth's diurnal rotation ( using earth as a table - top platform, so to speak ) as well as demonstrating the constancy in the "clock" time - frequency of the two isotopes of the rare element dysprosium over a period of two years during which earth traversed its solar orbit, alternating between closer and further distances from the sun.

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