The Descartes Code (Spin Orbital Rotation of Photons)–I. The Fourth-Order Effects in the Michelson Interferometer

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 fourth-order 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 Michelson-Morley experiment with the null result in the second-order 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 Michelson-Morley 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 higher-orders 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 second-order effects barrier. This new model enables to test the predicted fourth-order effects in the LIGO interferometer (Laser Interferometer Gravitational Wave Observatory).

Michelson Morley Experiment Based on the Second-Order Effects

The basic knowledge about the interpretation of second-order 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.

Fig. 1. Michelson Morley experiment [1]: a–source of the light, b–the beamsplitter, c–the compensation glass, d–mirrors, e adjustment mirror for the length in the transverse arm, f–the observation of the fringe shift, Lorentz length contraction model.

Table I. The Mathematical Events in the Michelson Morley Experiment based on the Second-Order Effects

Interpretation of the Michelson Morley experiment for the second-order effects of (v/c)2 and the prediction for the fourth-order effects (v/c)4 , tL – time in the longitudinal arm, tT – time in the transverse arm, L – length of the arm, λ – wavelength of the used photons
Time difference	$\mathrm{\Delta }t=t_L-t_T=2L/c({\displaystyle \frac{1}{1-{\displaystyle \frac{v^2}{c^2}}}}-{\displaystyle \frac{1}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}})$
Path difference	$\mathrm{\Delta }{\lambda }_1=2L({\displaystyle \frac{1}{1-{\displaystyle \frac{v^2}{c^2}}}}-{\displaystyle \frac{1}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}})\approx 2L{\displaystyle \frac{v^2}{2c^2}}=L{\displaystyle \frac{v^2}{c^2}}$
Fringe shift after the rotation 90°	$n={\displaystyle \frac{\mathrm{\Delta }{\lambda }_1-\mathrm{\Delta }{\lambda }_2}{\lambda }}\approx {\displaystyle \frac{2L{v}^2}{\lambda c^2}}\approx {\displaystyle \frac{2\times 11}{500\times {10}^{-9}}}{\left(30/300000\right)}^2\approx 0.44$
MM experiment in 1887	$n\le 0.01$
Experiments in 1920–1930	$n\le 0.0002$
Prediction for the fourth-order effect	$n\approx 0.44\times {10}^{-8}$

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].

Table II. The Key Ideas used to Formulate Time Events in the Michelson Interferometer

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 second-order 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).

Fig. 2. Michelson Morley experiment [1]: a–source of the light, b–the beamsplitter, c–the compensation glass, d–mirrors, e adjustment mirror for the length in the transverse arm, f–the observation of the fringe shift, Descartes’ colors depict the change of the orbital angular momentum of photons during their contacts with beamsplitter and mirrors. Colors are significantly exaggerated.

Table III. Descartes’ Cool and Warm Colors of Photons in the Individual Paths in the Michelson Interferometer

Energy of descartes’ cool and warm colors	Descartes’ photon temperature	Descartes’ photon momentum
$E_{\leftarrow }=h{\nu }_0{\displaystyle \frac{1-v/c}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}$	$T_{\leftarrow }=T_0{\displaystyle \frac{1-v/c}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}$	$P_{\leftarrow }={\displaystyle \frac{h}{{\lambda }_0}}{\displaystyle \frac{1-v/c}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}$
$E_{\uparrow }=h{\nu }_0\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}$	$T_{\uparrow }=T_0\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}$	$P_{\uparrow }={\displaystyle \frac{h}{{\lambda }_0}}\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}$
$E_0=h{\nu }_0$	$T_0$	${\nu }_0$
$E_{\downarrow }=h{\nu }_0{\displaystyle \frac{1}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}$	$T_{\downarrow }=T_0{\displaystyle \frac{1}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}$	$P_{\downarrow }={\displaystyle \frac{h}{{\lambda }_0}}{\displaystyle \frac{1}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}$
$E_{\to }=h{\nu }_0{\displaystyle \frac{1+v/c}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}$	$T_{\to }=T_0{\displaystyle \frac{1+v/c}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}$	$P_{\to }={\displaystyle \frac{h}{{\lambda }_0}}{\displaystyle \frac{1+v/c}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}$
Magenta correction of the transverse path LT in the perpendicular direction towards the motion of the michelson interferometer

Table IV. Descartes’ Cool and Warm Colors of Photons in the Individual Paths in the Michelson Interferometer

Descartes’ time of the photon orbit (local time)	Descartes’ photon wavelength	Descartes’ photon frequency	Constant light speed
$t_{\leftarrow }=t_0{\displaystyle \frac{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}{1-v/c}}$	${\lambda }_{\leftarrow }={\lambda }_0{\displaystyle \frac{1+v/c}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}$	${\nu }_{\leftarrow }={\nu }_0{\displaystyle \frac{1-v/c}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}$	$c={\lambda }_{\leftarrow }\times {\nu }_{\leftarrow }$
$t_{\uparrow }=t_0{\displaystyle \frac{1}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}$	${\lambda }_{\uparrow }={\lambda }_0{\displaystyle \frac{1}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}$	${\nu }_{\uparrow }={\nu }_0\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}$	$c={\lambda }_{\uparrow }\times {\nu }_{\uparrow }$
$t_0={\displaystyle \frac{1}{{\nu }_0}}$	${\lambda }_0$	${\nu }_0$	$c={\lambda }_0\times {\nu }_0$
$t_{\downarrow }=t_0\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}$	${\lambda }_{\downarrow }={\lambda }_0\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}$	${\nu }_{\downarrow }={\nu }_0{\displaystyle \frac{1}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}$	$c={\lambda }_{\downarrow }\times {\nu }_{\downarrow }$
$t_{\to }=t_0{\displaystyle \frac{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}{1+v/c}}$	${\lambda }_{\to }={\lambda }_0{\displaystyle \frac{1-v/c}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}$	${\nu }_{\to }={\nu }_0{\displaystyle \frac{1+v/c}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}$	$c={\lambda }_{\to }\times {\nu }_{\to }$
Magenta correction of the transverse path LT 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.

Table V. The Mathematical Events in the Michelson Morley Experiment Based on the Fourth-Order Effects

Interpretation of the Michelson Morley experiment for the fourth-order effects (v/c)4
Time in the longitudinal arm	$t_L=t_{\to }+t_{\leftarrow }={\displaystyle \frac{L}{c(1-v/c)}}\left({\displaystyle \frac{1-v/c}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}\right)+{\displaystyle \frac{L}{c(1+v/c)}}\left({\displaystyle \frac{1+v/c}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}\right)=L/c{\displaystyle \frac{2}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}$
Time in the longitudinal arm	$t_L=t_{\to }+t_{\leftarrow }=L/c{\displaystyle \frac{2}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}\approx L/c(2+{\displaystyle \frac{v^2}{c^2}}+3/4{\displaystyle \frac{v^4}{c^4}}+5/8{\displaystyle \frac{v^6}{c^6}}\dots )$
Time in the transverse arm	$t_T=t_{\uparrow }+t_{\downarrow }={\displaystyle \frac{L}{c\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}\left({\displaystyle \frac{1}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}\right)+{\displaystyle \frac{L}{c\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}\left(\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}\right)=L/c({\displaystyle \frac{1}{1-{\displaystyle \frac{v^2}{c^2}}}}+1)$
Time in the transverse arm	$t_T=t_{\uparrow }+t_{\downarrow }=L/c({\displaystyle \frac{1}{1-{\displaystyle \frac{v^2}{c^2}}}}+1)\approx L/c(2+{\displaystyle \frac{v^2}{c^2}}+{\displaystyle \frac{v^4}{c^4}}+{\displaystyle \frac{v^6}{c^6}}\dots )$
Time difference	$\mathrm{\Delta }t=t_T-t_L\approx 1/4L/c{\displaystyle \frac{v^4}{c^4}}$
Path difference	$\mathrm{\Delta }{\lambda }_1\approx L/4{\displaystyle \frac{v^4}{c^4}}$
Fringe shift after the rotation 90°	$n={\displaystyle \frac{\mathrm{\Delta }{\lambda }_1-\mathrm{\Delta }{\lambda }_2}{\lambda }}\approx {\displaystyle \frac{L{v}^4}{2\lambda c^4}}$
Predicted fringe shift for LIGO	$n={\displaystyle \frac{\mathrm{\Delta }{\lambda }_1-\mathrm{\Delta }{\lambda }_2}{\lambda }}\approx {\displaystyle \frac{L{v}^4}{2\lambda c^4}}\approx {\displaystyle \frac{300\times 4000}{2\times 1064\times {10}^{-9}}}{\left(30/300000\right)}^4\approx 5.64\times {10}^{-5}$

The predicted fringe shift based on the fourth-order 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].

Fig. 3. LIGO experiment [35] with the Michelson interferometer and the Fabry Perot cavities with 300 bounces of observed photons.

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 second-order effect barrier but can be observed at the fourth-order effects. This masking effect of Nature might confuse observers working with the second-order effects – “the mathematical camouflage”.

1. The Descartes’ model of rotating globules (based on the spin-orbit momentum of photons) was selected as the potential candidate to interpret the missing “hidden-variable” in the Michelson interferometer.
2. 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.
3. The predicted fourth-order effect fringe shift can be experimentally observed in the LIGO instruments and/or their sister instruments.
4. 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.
5. This hidden photon harmony can document the old quote of Heraclitus: “Nature loves to hide”.