The Descartes Code (Spin Orbital Rotation of Photons)–I. The FourthOrder Effects in the Michelson Interferometer
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In 1887 Michelson and Morley published their very important paper with the null result in the Michelson interferometer. Since that time there were published several hundred papers with various ad hoc hypotheses to interpret this unexpected result but none of these attempts was accepted to offer an alternative to Einstein’s 1905 theory. In order to avoid this trap with ad hoc models we have combined knowledge of Old Masters and derived a new description of the longitudinal and transverse arms of the Michelson interferometer. In this model the fourthorder effect of (v/c)^{4} was derived where v is the interferometer velocity in the stationary luminiferous ether and c is the light speed. This formula cannot be experimentally tested in the original Michelson interferometer with the short light path of 11 meters. There is one very good opportunity to test this model with the LIGO technology with the length of both arms 4 km and the Fabry Perot cavities where the laser beam in each arm bounces between two mirrors about 300 times before being merged with the beam from the other arm. The predicted fringe shift for the LIGO interferometer is about n ≈ 5.64 × 10−^{5}.
Introduction
The MichelsonMorley experiment with the null result in the secondorder effects played a very significant role during the transition of classical physics to the modern physics, e.g., [1]–[10]. Albert Einstein formulated the theory of special relativity in 1905 based on the relativity postulate and the constancy of the speed of light. Until now Einstein’s theory is the best theory for the explanation of the MichelsonMorley null result. Since that time there were published hundreds alternative ad hoc models to find some other ways to this topic, e.g., [11]–[24]. The feature of these models is usually an ad hoc background and more over they do not forecast any experimental verifications in higherorders of (v/c)^{n}.
In our approach we tried to describe the time scales in the longitudinal and transverse arms of the Michelson interferometer using the knowledge of Old Masters in order to penetrate through the secondorder effects barrier. This new model enables to test the predicted fourthorder effects in the LIGO interferometer (Laser Interferometer Gravitational Wave Observatory).
Michelson Morley Experiment Based on the SecondOrder Effects
The basic knowledge about the interpretation of secondorder effects in the longitudinal and transverse arms of the Michelson interferometer is known to all scholars in physics. Fig. 1 schematically depicts the Michelson Morley experimental arrangement. Table I summarizes the second order effects in the Michelson Morley experiment.
Interpretation of the Michelson Morley experiment for the secondorder effects of (v/c)^{2} and the prediction for the fourthorder effects (v/c)^{4} _{,} t_{L} – time in the longitudinal arm, t_{T} – time in the transverse arm, L – length of the arm, λ – wavelength of the used photons  

Time difference  Δt=tL−tT=2L/c(11−v2c2−11−v2c2) 
Path difference  Δλ1=2L(11−v2c2−11−v2c2)≈2Lv22c2=Lv2c2 
Fringe shift after the rotation 90°  n=Δλ1−Δλ2λ≈2Lv2λc2≈2×11500×10−9(30/300000)2≈0.44 
MM experiment in 1887  n≤0.01 
Experiments in 1920–1930  n≤0.0002 
Prediction for the fourthorder effect  n≈0.44×10−8 
Old Masters Knowledge Used in the Michelson Interferometer
In order to avoid the trap with an ad hoc hypothesis, we were inspired by the knowledge of Old Masters to describe time events in the Michelson interferometer. Table II surveys the key ideas of Old Masters that were used to describe time events in the Michelson interferometer. The first key idea is the Descartes theory of colors based on the spin orbital rotation of photons. Unfortunately, this model was rejected by Newton and only recently was applied to describe color events that cannot be described by the Newtonian model [25], [26]. The recent strong growing interest in the Spin Angular Momentum (SAM) and the Orbital Angular Momentum (OAM) of photons brings many new experimental data, e.g., [27]–[34].
Old master  Knowledge 

Descartes  Spin orbital rotation of photons 
Fermat  The principle of the least time 
Maupertuis  The principle of the shortest path 
Herschel  Heat effect of photons 
Doppler  Doppler effect 
Lorentz  Local time of photons 
Einstein  Relativistic Doppler effect 
Michelson and Morley  Null result with the secondorder of (v/c)^{2} 
The second key idea was the Fermat principle of the least time and the Maupertuis principle of the shortest path–it is necessary to find times and paths in both the longitudinal and transverse arms to obtain the minimum both in the total time and paths of photons in those arms.
The Descartes’ colors are depicted in Fig. 2–during the photon path through the Michelson interferometer the photon momentum changes its value after the contact with individual mirrors and the beamsplitter. Tables III and IV show photon properties in the longitudinal arm: forward direction → and backward direction ←, and the transverse arm: forward direction ↑ and backward direction ↓. The known relativistic Doppler formulae are so selected for all directions so that the flight times of photons are the minimum in this situation. (We could not find a more efficient combination in times and paths).
Energy of descartes’ cool and warm colors  Descartes’ photon temperature  Descartes’ photon momentum 

E←=hν01−v/c1−v2c2  T←=T01−v/c1−v2c2  P←=hλ01−v/c1−v2c2 
E↑=hν01−v2c2  T↑=T01−v2c2  P↑=hλ01−v2c2 
E0=hν0  T0  ν0 
E↓=hν011−v2c2  T↓=T011−v2c2  P↓=hλ011−v2c2 
E→=hν01+v/c1−v2c2  T→=T01+v/c1−v2c2  P→=hλ01+v/c1−v2c2 
Magenta correction of the transverse path L_{T} in the perpendicular direction towards the motion of the michelson interferometer 
Descartes’ time of the photon orbit (local time)  Descartes’ photon wavelength  Descartes’ photon frequency  Constant light speed 

t←=t01−v2c21−v/c  λ←=λ01+v/c1−v2c2  ν←=ν01−v/c1−v2c2  c=λ←×ν← 
t↑=t011−v2c2  λ↑=λ011−v2c2  ν↑=ν01−v2c2  c=λ↑×ν↑ 
t0=1ν0  λ0  ν0  c=λ0×ν0 
t↓=t01−v2c2  λ↓=λ01−v2c2  ν↓=ν011−v2c2  c=λ↓×ν↓ 
t→=t01−v2c21+v/c  λ→=λ01−v/c1−v2c2  ν→=ν01+v/c1−v2c2  c=λ→×ν→ 
Magenta correction of the transverse path L_{T} in the perpendicular direction towards the motion of the michelson interferometer 
Table V summarizes the most effective combination of flight times of photons – the macro times formulae are corrected by the micro times of photons (the local times of photons in those paths). The combination of those times taken at 0° and 90° orientation of the Michelson interferometer towards to the stationary ether gives the prediction that could be tested in the famous LIGO instrument.
Interpretation of the Michelson Morley experiment for the fourthorder effects (v/c)^{4}  

Time in the longitudinal arm  tL=t→+t←=Lc(1−v/c)(1−v/c1−v2c2)+Lc(1+v/c)(1+v/c1−v2c2)=L/c21−v2c2 
Time in the longitudinal arm  tL=t→+t←=L/c21−v2c2≈L/c(2+v2c2+3/4v4c4+5/8v6c6…) 
Time in the transverse arm  tT=t↑+t↓=Lc1−v2c2(11−v2c2)+Lc1−v2c2(1−v2c2)=L/c(11−v2c2+1) 
Time in the transverse arm  tT=t↑+t↓=L/c(11−v2c2+1)≈L/c(2+v2c2+v4c4+v6c6…) 
Time difference  Δt=tT−tL≈1/4L/cv4c4 
Path difference  Δλ1≈L/4v4c4 
Fringe shift after the rotation 90°  n=Δλ1−Δλ2λ≈Lv42λc4 
Predicted fringe shift for LIGO  n=Δλ1−Δλ2λ≈Lv42λc4≈300×40002×1064×10−9(30/300000)4≈5.64×10−5 
The predicted fringe shift based on the fourthorder effects in the Michelson interferometer can be experimentally tested using the LIGO instrument [35]–Fig. 3. There is another possibility how to use the excellent LIGO, LIGO India, Virgo, KAGRA, GEO600 technology [36]–to evaluate the heating effect of “Herschel’s” photons in both arms. In this model not only optical properties of photons could be studied but the energy and momentum of photons in the individual paths can be evaluated. This possibility was already noticed by Einstein in his famous 1905 paper [4].
Albert A. Michelson’s quote [37] is still actual: “While it is never safe to affirm that the future of Physical Science has no marvels in store even more astonishing than those of the past, it seems probable that most of the grand underlying principles have been firmly established and that further advances are to be sought chiefly in the rigorous application of these principles to all the phenomena which come under our notice. It is here that the science of measurement shows its importance — where quantitative work is more to be desired than qualitative work. An eminent physicist remarked that the future truths of physical science are to be looked for in the sixth place of decimals”.
Conclusion
This contribution is based on the knowledge of Old Masters. The photon trip in the longitudinal forward and backward directions, and in the transverse forward and backward directions in the Michelson interferometer were described in order to get the least possible time for the combination of all photon paths. There is a hidden photon harmony in the Michelson interferometer. Photons choose such times and paths on their trip so that they are able to penetrate through the secondorder effect barrier but can be observed at the fourthorder effects. This masking effect of Nature might confuse observers working with the secondorder effects – “the mathematical camouflage”.
 The Descartes’ model of rotating globules (based on the spinorbit momentum of photons) was selected as the potential candidate to interpret the missing “hiddenvariable” in the Michelson interferometer.
 The Fermat’s principle of the least time and the Maupertuis principle of the shortest path served as guides to define the flying times of photons in both the longitudinal and transverse arms of the Michelson interferometer.
 The predicted fourthorder effect fringe shift can be experimentally observed in the LIGO instruments and/or their sister instruments.
 The very sensitive LIGO type instruments can evaluate also energy, momentum, and heat effect of photons in the longitudinal and transverse arms of the Michelson interferometer.
 This hidden photon harmony can document the old quote of Heraclitus: “Nature loves to hide”.
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