The Descartes Code (Spin Orbital Rotation of Photons)–II. The Gravitational Redshift



Abstract Views
234

Downloads
106

Citations

Share

##plugins.themes.bootstrap3.article.sidebar##

Submitted Dec 16, 2023
Published Jan 26, 2024
10.24018/ejphysics.2024.6.1.296

Abstract viewed = 234 times

Table of Contents

##plugins.themes.bootstrap3.article.main##

A new alternative model to the Schwarzschild’s solution of the Einstein's gravitational theory is presented. Descartes’s code is the old color theory (published in 1637) based on the rotation of “light globules”, however, it was rejected by Newton in 1672. In this model, the photon spin-orbital speed is modified by the gravitational field. The formulae for the wavelength, frequency, local time, momentum, energy, and temperature of photons in the gravitational field are identical in the first order both for the Schwarzschild solution and Descartes’ code. To experimentally determine the predictions of Descartes’ photon model we have to search for the experiments in the second order or for the situations where the standard model has to postulate some hidden elements (e.g., the dark matter). Some examples are presented here. The new interpretations of the bending of light by the Sun and the Shapiro time delay are given. The Anderson acceleration constant known as the Pioneer anomaly was derived as the effect of the Earth’s gravitational field on photons. Similar acceleration could be found in invaluable data from the Laser Lunar Ranging experiment. Stellar photons are modified by the Solar gravitational field at the distance of 1 AU and the acceleration constant identical to the Milgrom acceleration (MOND gravitational model) was derived. Therefore, we should study in more detail the influence of the Solar, Earth’s, and Moon’s gravitational fields on photons to avoid traps with false gravitational models.

Keywords: Descartes’ gravitational redshift Einstein’s gravitational redshift first order effects second order effects

Introduction

Einstein’s theory of gravity is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics [1]. Einstein used approximation methods in working out the initial predictions of his theory. In 1916 Karl Schwarzschild found the first non-trivial solution to the Einstein field equations-the Schwarzschild metric [2]. This Schwarzschild solution leads to quantitative predictions that were very intensively experimentally tested with impressive accuracy [3]. Since that time there were published many attempts as alternative scenarios to the original Einstein’s theory of gravity [4]. But none of these proposals can compete with predictions of the original Einstein’s theory.

Therefore, it seems almost impossible to appear with an alternative model describing the influence of the gravitational field on photon properties in the Euclidean space. This contribution brings to the focus an old, forgotten, and overlooked theory of colors that might newly quantitatively describe the photon colors–the gravitational redshift/blueshift-in the gravitational field.

The Significant Bifurcation Point in the History of Physics–Prague June 1911

Albert Einstein started his three semesters professorship at the German Karl-Ferdinand University in Prague on April 01, 1911, e.g. [5]–[10]. Several Czech physicists were aware of the historical turning point in the development of physics represented by Einstein.

Julius Suchý (1879–1920; study visit at H.A. Lorentz in Leyden during 1907–1909) on April 07, 1911 had written [11]: “A great struggle has just begun in physics between two scientific theories: the old, classical mechanics of Newton and the new mechanics based on the principle of relativity. The problems at stake are of fundamental importance; the implications of the principle of relativity directly affect the fundamental concepts of matter, time, and space! Therefore, the significance of this research is of importance for the world of natural science in general.” Later Julius Suchý on April 14, 1911 had written [11]: “Whichever way the final decision turns out: this much can be said today, that the theory built by Lorentz, Einstein and Minkowski is one of the most interesting and original attempts of the human spirit to penetrate to the knowledge of truth, if this is available to human at all.”

August Žáček (1886–1961; study visit at Gőttingen University at T. Th. Simon during 1910–1911) in May 1911 had written [12]: “Other applications of Einstein’s important principle cannot be dwelt upon here; the significance which it has for physics today is clear from the words of Planck: ´It is needless to say that this new definition of the concept of time makes the highest demands on the abstracting faculty of the physicist and his imagination. It surpasses in boldness everything that has been done in speculative natural science, and even in the philosophical theory of knowledge; the non-Euclidean geometry is a mere child’s play in comparison. And yet the principle of relativity makes a full claim to physical application, whereas we meet with non-Euclidean geometry in pure mathematics. The upheaval caused by him in all physical views can only be compared in depth and extent to that caused by the views of Copernicus.”

Albert Einstein in 1925 had written [6]: “In the quiet rooms of the Institute of Theoretical Physics of Prague’s German University in Viničná Street, I discovered that the principle of equivalence implies the deflection of light rays near the Sun by an observable amount, without at that time knowing that a similar result had been derived from Newton’s mechanics and his corpuscular theory of light. In Prague, I also discovered the shift of spectral lines towards the red, which is not completely confirmed".

Unziker in 2015 [13] identified the significant bifurcation point in the history of physics with (1) derived briefly by Einstein in Bern in 1907 [14] and in detail in Prague in June 1911 [15]: (1)c(R)=c0(1−GMRc02)where c0 = 299792458 ms−1, G is the Newtonian gravitational constant, M is the mass of the source of the gravitational field, R is the distance from the center of that object, and c (R) depends on the position in the gravitational field. Unzicker stated that without the cracking of the true meaning of this formula, there is no way to return to classical physics.

Gunnard Nordstrőm in a paper submitted on October 20, 1912, wrote [16], [17]: “Einstein’s [1911] hypothesis that the speed of light c (R) depends upon the gravitational potential leads to considerable difficulties for the principle of relativity, as the discussion between Einstein and Abraham shows us [18]. Hence, one is led to ask if it would not be possible to replace Einstein’s hypothesis with a different one, which leaves c (R) constant and still adapts the theory of gravitation to the principle of relativity in such a way, that gravitational and inertial mass are equal.”

After hundreds of ad hoc hypotheses proposed during the last century, it seems that there is no way to discover any reasonable idea of how to express the light velocity in the gravitational field for the Euclidean space.

The Descartes Code: Spin-Orbital Rotation of Photons in the Gravitational Field

The very old (1637) Descartes’ color theory [19] based on the rotation of “light globules” can be the “lost key” how to crack the mysterious Einstein’s equation. This Descartes’ color theory was rejected by Newton in 1672 and was forgotten and overlooked in the coming centuries [20]–[30]. During the last centuries there were several important contributions to the description of color in physics that were newly re-interpreted based on the Descartes color theory, [31]–[33]. Fig. 1 shows some important turning points in our understanding of colors.

Fig. 1. Evolution of the color theories: Descartes (1637), Newton (1672), Malus (1800), Doppler (1842), Einstein (1911), Descartes (2023).

In the Descartes’s color theory the rotation speed of photons in the gravitational field can be described by (2): (2)crotational=clongitudinal+GMRclongitudinal=c+GMRcc=299792458 ms−1where the rotational speed of photons crotational increases with the intensity of the gravitational field and thus changes the photon frequency, wavelength, temperature, momentum, and energy–Table I.

Descartes’code–photon properties in the gravitational field
crotational=clongitudinal+GMRclongitudinal=c+GM/Rcc=299792458 ms−1
clongitudinal=c=λν=λ0ν0=299792458 ms−1
ν=ν01+GMRc21−(GMRc2)2frequencyofphoton
λ=λ01−GMRc21−(GMRc2)2wavelengthofphoton
t=t01−GMRc21−(GMRc2)2localtimeofphoton
p=hλ01+GMRc21−(GMRc2)2momentumofphoton
E=hν=hν01+GMRc21−(GMRc2)2energyofphoton
T=T01+GMRc21−(GMRc2)2temperatureofphoton
Table I. The Descartes Code–Photon Properties in the Gravitational Field

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. [34]–[41]. It could be very interesting to study the rotation of photons in the gravitational field as well.

The Descartes Code and the Schwarzschild Solution of the Einstein’s General Relativity

The Descartes’ code describes newly the properties of photons in the gravitational field. In the first step, we can compare the mathematical language of the Descartes code with the mathematical language of the Schwarzschild solution of Einstein’s theory–Table II.

Schwarzschild solution of GR for the frequency of photons
ν=ν011−2GMRc2≈ν0(1+GMRc2+32(GMRc2)2+…)
Descartes’ code for the frequency of photons
ν=ν01+GMRc21−(GMRc2)2≈ν0(1+GMRc2+12(GMRc2)2+…)
Difference of two models for the photon frequency
Δ(ν)=νD−νE≈−ν0(GMRc2)2
Schwarzschild solution of GR for the wavelength of photons
λ=λ01−2GMRc2≈λ0(1−GMRc2−12(GMRc2)2−…)
Descartes’ code for the wavelength of photons
λ=λ01−GMRc21−(GMRc2)2≈λ0(1−GMRc2+12(GMRc2)2−…)
Difference of two models for the photon wavelength
Δ(λ)=λD−λE≈λ0(GMRc2)2
Schwarzschild solution of GR for the local time of photons
t=t01−2GMRc2≈t0(1−GMRc2−12(GMRc2)2−…)
Descartes’ code for the local time of photons
t=t01−GMRc21−(GMRc2)2≈t0(1−GMRc2+12(GMRc2)2−…)
Difference of two models for the photon local time
Δ(t)=tD−tE≈t0(GMRc2)2
Table II. The Descartes Code and the Schwarzschild Solution of the Einstein’s Theory

Table II compares the mathematical predictions of the photon properties in the gravitational field for these two models. Both Descartes’ code and the Schwarzschild solution of Einstein’ theory are identical in the first order of formulae. To distinguish the predictions of these two models we have to test the photon properties in the second order of these formulae.

The Deflection of Light by the Sun

The deflection of light by the Sun was experimentally very carefully studied and the Einstein prediction was confirmed [3], [42]–[52]. The same first-order formulae for the light bending can be derived using the Descartes code. Fig. 2 shows the evolution of the blueshift and redshift of the photons in the vicinity of the Sun.

Fig. 2. The bending of light by the Sun-the force acting on the photon.

The force acting between the photon and the Sun can be expressed in (3) as: (3)F=hνλ=hν0λ01+GMRc21−GMRc2≈hν0λ0(1+2GMRc2+…)

The change of this force during the flight of photons around the Sun equals in (4): (4)ΔFF0≈2GMRc2

Equation (4) can be integrated similarly as did Einstein in 1911 [16] but with the factor 2 * 2 = 4, e.g., [53]–[55]. This integration is given by (5): (5)α=∫ε=−π/2ε=+π/22GM⊙R⊙c2cosεdε=4GM⊙R⊙c2

In order to distinguish between these two models, which one describes better the true Nature, we have to realize tests in the second order. However, it will be difficult in some cases even with our excellent technology.

Gerber’s Retarded Potential and the Time Delay Hidden in the Descartes’ Code

Gerber in 1898 derived the precise formula for the description of the perihelion shift of Mercury based on his ad hoc formula for the retarded potential given by (6): (6)V=−GMr(1−1cdrdt)2where the gravitational potential V experienced by an object depends not only on the radial distance from the gravitating source but also on the derivative (with respect to time) of that distance. There was an intensive discussion about the physical reality of this formula, however, at the end of this discussion, this formula was rejected, e.g., [56]–[63].

We will try to derive the retarded time of photons on their path from the source to an object based on the Descartes’ code. Fig. 3 depicts the path of photons from the Sun to the planet Mercury.

Fig. 3. The retardation of time during the flight of photons from the Sun to the planet Mercury.

We can derive the retarded macro time of photons flying from the Sun to the planet Mercury as shown in (7) and (8): (7)(ctmacro)2=(GMRctmacro)2+(R1−GMRc2)2 (8)tmacro=Rc1(1−GMRc2)1−(GMRc2)2=t0(1−GMRc2)1−(GMRc2)2

In order to get the total retarted time we have to combine the macro time corrected with the local time of photons in the gravitational field–(9): (9)tretarted=t0(1−GMRc2)1−(GMRc2)21−(GMRc2)2(1−GMRc2)=t0(1−GMRc2)2

We should observe the effect of the delayed photon flight time in some events, e.g., in the Shapiro time delay, Pioneer anomaly, Laser Lunar ranging, etc., expressed by (10): (10)tretarted=t0(1−GMRc2)2≈t0(1+2GMRc2+3(GMRc2)2+…)

Shapiro Time Delay

In the Shapiro time delay experiment radar signals passing near the Sun travel slightly longer to a target and longer to return back to the source of that signal. In this scenario the photons search an optimum path through the gravitational field of the Sun in order to satisfy both the Fermat’s principle of the least time and the Maupertuis’ principle of the least time–Fig. 4.

Fig. 4. The Shapiro time delay. Photons follow Fermat’s principle of the least time and Maupertuis’ principle of the least path in the gravitational field of the Sun.

The photon time delay on the round-trip Earth–Venus–Earth could be evaluated from the retarted time in (11): (11)tretarted=2t0(1−GMRc2)2≈t0(1+2GMRc2+…)∗2where the expression 2 GM/c2 is mathematically identical to the Schwarzschild radius but this formula has a different meaning in Descartes’ code. The detailed derivation of the Shapiro time delay was published [3], [64], [65]. Quattrini [66] derived the Shapiro time delay in Euclidean space based on the model of the variable speed of light in (12): (12)tdelay=ttravel−t=2∗∫1c(1+2GMRc2)dx−2Lc=4GMc3∫dxR

This time delay for the photon path: the Earth–Venus–Earth gives around 200 μsec, i.e., the path increment is around 60 km. This is the optimal path increment in the length of the trajectory to fulfill Fermat’s principle of the least time and Maupertuis’s principle of the least path.

Pioneer Anomaly

The very well-known effect–Pioneer anomaly-is an extremely small acceleration towards the Sun or the Earth of the value (8.73 ± 1.33) * 10−10 m/s2, [67], [68]. This Pioneer anomaly was discovered by J.D. Anderson after many years of analysis of well-defined experimental data. This apparent anomaly was a matter of much interest for many years. Today’s official interpretation is anisotropic radiation pressure caused by the spacecraft’s heat loss [69], [70].

Now, we can propose a new interpretation based on Descartes’ code. Fig. 5 depicts the photon path from the Earth towards Pioneer 10/11 and back towards the surface of the Earth.

Fig. 5. The Pioneer anomaly. Photons are slightly blueshifted during their round-trip between the Earth and the Pioneers 10/11.

We propose to determine the Anderson constant (expressed in (13) and (14)) characterizing that observed change of blueshift of returned photons from the second order effect in (10): (13)Δνpioneerν0=2∗3(GM⊕R⊕c2)2≈2.901∗10−18 (14)apioneer=2∗3(GM⊕R⊕c2)21sec∗c≈8.698∗10−10[ms−2]where G is the Newtonian gravitational constant, M⊕ is the mass of the Earth, R⊕ is the radius of the Earth, and c is the speed of light. The famous Pioneer anomaly is explained in this Descartes model of photons as the influence of the Earth´s gravitational field during the round-trip of photons from the Earth towards both Pioneers 10/11 and back to the surface of the Earth per one second.

Laser Lunar Ranging

The experiment termed as the Lunar Laser Ranging is the measuring the distance between the surfaces of the Earth and the Moon. There were placed three retroreflectors by the United States Apollo program (11, 14, and 15), two by the Soviet Lunokhod 1 and 2 missions, and one by India’s Chandrayaan-3 mission, [71]. This experiment could be very valuable for the estimation of the reality of Descartes´ code. The project I CURE “(India, China, United States of America, Russia, and the European Union) could bring new data for the properties of photons in the gravitational field of the Earth and the Moon. Fig. 6 shows the principle of this experiment.

Fig. 6. Laser lunar ranging experiment. Photons in the gravitational field of the Earth and the Moon.

Laser Lunar ranging experiment delivered highly accurate measurements of the time taken by photons on the round-trip between the Earth and the Moon. This experiment was repeated many times during the last fifty years, e.g. [72]. It was found that this round-trip time was steadily delayed by 0.255 ± 0.005 nsec per year (=8.081 * 10−18 s/s). The standard interpretation of this time delay is the Moon spiraling away from the Earth at a rate of 3.82 cm/year, e.g., [73], [74].

In this new approach, we can model the time delay of photons in the gravitational fields of the Earth and the Moon as the modification of the photon orbital speed–the Descartes’ code-the local time of photons. We should obtain a similar effect to the Pioneer anomaly. The estimated acceleration of spin-orbital speed of photons aLLR [75] returning to the Earth can be evaluated by (15): (15)aLLR=c22DEM2.55∗10−1031556926≈9.447∗10−10[ms−2]where DEM is the average distance between Earth–Moon 3.84399 * 108 m, seconds in the year 31556926 sec/year. From these data, we get the estimation for the change of the frequency of those LLR photons similar to the Pioneer anomaly–(16): (16)aLLRc≈3.151∗10−18[s−1]=3.151[attohertz:aHz]

The observed time delay during the Laser Lunar Ranging is caused by the local time of photons in the gravitational fields of the Earth and the Moon and can be expressed in units of attohertz [aHz]. This type of experiment can be done by any well-equipped observatory because those six retroreflectors on the Moon are freely available.

Interaction of Stellar Photons with the Solar Gravitational Field at 1 AU

The experimental determination of blueshift or redshift of Stellar photons occurs at the Earth at a distance of 1 AU from the Sun. Is there a possibility how quantifying the Solar gravitational field effect on those Stellar photons? Fig. 7 shows that the Solar gravitational field has to modify the total value of the blueshift/redshift of those Stellar photons entering into the Solar System.

Fig. 7. Stellar photons entering into the solar system should be influenced by the solar gravitational field.

In an analogy with Gerber’s retarted potential [56], [57] we can write for the acceleration of the Solar gravitational field at 1 AU using (17): (17)a=GM⊙(AU)2(1−GM⊙(AU)c2)=GM⊙(AU)2(1+2GM⊙(AU)c2+…)

For the second order of the Solar acceleration, we get a value identical to the Milgrom acceleration known in the so-called MOND model [76] as aMILGROM = (1.2 ± 0.1) * 10−10 ms−2 in (18): (18)aMILGROM=2G2M⊙2(AU)3c2≈1.171∗10−10[ms−2]

Therefore, the Stellar photons at the distance 1 AU from the Sun are modified as given in (19): (19)aMILGROMc≈3.905∗10−19[s−1]=0.3905[aHz]

We should experimentally study in more details the influence of the Solar, Earth, and Moon gravitational fields of the blueshift/redshift of Stellar photons in order to avoid some wrong cosmological models.

Conclusion

This contribution is based on the old, forgotten, and rejected Descartes’ color theory based on the spin-orbital rotation of “light globules”. This Descartes’ model of photon spinning in the gravitational fields leads to simple but powerful formulae. Photons in the gravitational fields choose such times and paths on their trip so that they can penetrate through the first-order effect barrier but can be observed at the higher-order effects. This masking effect of Nature might confuse observers working with the first-order effects in the gravitational fields–“the mathematical camouflage”.

  1. Descartes’ model of rotating globules (based on the spin-orbit momentum of photons) was selected as the potential candidate to interpret new effects of the gravitational fields on photons.
  2. Descartes’ formulae and Schwarzschild’s solutions to Einstein’s gravitational theory are identical in the first-order experiments.
  3. Light bending by the Sun and the Shapiro time delay were newly interpreted.
  4. The Gerbers retarded potential and time delay was newly derived.
  5. The Anderson acceleration known as the Pioneer anomaly was derived as the effect of the Earth’s gravitational field on the photons.
  6. The Laser Lunar Ranging experiment and the measured time delay were newly interpreted.
  7. Milgrom acceleration constant known in the MOND model was derived as the effect of the Solar gravitational field at the distance of 1 AU on the Stellar photons.
  8. The momentum, energy, and temperature of photons in the gravitational fields should be experimentally tested in more detail.

References

  1. Einstein A. Über Die Spezielle Und Die Allgemeine Relativitätstheorie. (On the Special and General Theory of Relativity). Braunschweig, German: Vieweg; 1917.
     Google Scholar
  2. Schwarzschild K. Uber das gravitationsfeld eines massenpunktes nach der einsteinschen theorie. (On the gravitational field of a point of mass according to Einstein´s theory). In Sitzungsberichte Der K˝oniglich Preussischen Akademie Der Wissenschaften (Berlin), German: Deutsche Akademie der Wissenschaften zu Berlin, 1916. pp. 189–96.
     Google Scholar
  3. Will CM. Theory and Experiment in Gravitational Physics. Cambridge: Cambridge University Press; 2018. ISBN 978-1-107-11744-0.
     Google Scholar
  4. Alternatives to general relativity. Wikipedia. December 12, 2023. Available from: https://en.wikipedia.org/wiki/Alternatives_to_general_relativity.
     Google Scholar
  5. Frank P. Einstein, His Life and Times. New York: A.A. Knopf; 1947.
     Google Scholar
  6. Bičák J. Einstein in prague: then and now. In General Relativity, Cosmology and Astrophysics Perspectives 100 Years After Einstein´s Stay in Prague, Bičák J, Ledvinka T, Eds. Berlin: Springer, 2014, pp. 33–63, ISBN: 978-3-319-06348-5.
     Google Scholar
  7. Těšínská E. Theoretical physics at the Prague University and its connections with Prague stay of A. Einstein 100 years ago. In Pokroky Matematiky, Fyziky A Astronomie, vol. 57, Czech: JCˇMF in Prague, 2012, pp. 146–68.
     Google Scholar
  8. Gordin MD. Einstein in Bohemia. Princeton University Press, Princeton; 2020. ISBN 978-0-691-17737-3.
     Google Scholar
  9. Podolský J, Cejnar P, Daniš S, Valenta J. Einstein in Prague Again, Physics in the Genius Series. Czech: MatfyzPress, Charles University; 2021. ISBN 978-80-7378-445-4.
     Google Scholar
  10. Těšínská E. Albert Einstein and Gustav Jaumann on the balance (Negotiations for the professorship of theoretical physics at the German University in Prague in 1910–1911. Proceedings of the International Conference Cosmology on Small Scales 2022, Dark Energy and the Local Hubble Expansion Problem. Eds. Křížek M and Dumin YV., pp. 145–68, Prague, September 21–24, 2022. Available from: https://css2022.math.cas.cz/proceedingsCSS2022.pdf.
     Google Scholar
  11. Suchý J. The principle of relativity. Týdenník Vˇenovaný Veˇrejným Otázkám. 1911;9:401–21, Czech.
     Google Scholar
  12. Zacek A. On the principle of relativity in physics. Ziva. 1911;21:135–7, Czech.
     Google Scholar
  13. Unzicker A. Einstein´s lost key. How we overlooked the best idea of the 20th century. Printed by Amazon in France. 2022. ISBN-13: 978-1519473431.
     Google Scholar
  14. Einstein A. On the relativity principle and the conclusions drawn from it. Jahrbuch Der Radioaktivität Und Elektronik. 1907;4:411–62.
     Google Scholar
  15. Einstein A. On the influence of gravitation on the propagation of light. Annalen Der Physik. 1911;35:898–908.
     Google Scholar
  16. Nordström G. The principle of relativity and the gravitation. In PhysikalischeZeitschrift, vol. 13. German: S.HirzelVerlag, 1912, pp. 1126–9.
     Google Scholar
  17. Weinstein G. The Einstein-Nordstr˝om theory. 2012, Arxiv:1205.5966. Accessed on December 10, 2023.
     Google Scholar
  18. Abraham M. Relativity and gravitation. Reply to a comment of Mr. Einstein´s reply. Annalen Der Physik. 1912;38:1056–8.
     Google Scholar
  19. de Descartes DRO. Les meteors. In Discours. Adam C, Tannery P, Eds. vol. 8, Paris: French, 1897–1910, pp. 331–5.
     Google Scholar
  20. Rosenfeld L. La theorie des couleurs de Newton et ses adversaires. (The color theory of Newton and his contemporaries). Isis. 1927;9(1):44–65, French.
     Google Scholar
  21. Westfall RS. The development of Newton´s theory of color. Isis. 1962;53(3):339–58.
     Google Scholar
  22. Westfall RS. Newton and his critics on the nature of colors. Arch Int d´Hist Des Sci. 1962;15:47–52.
     Google Scholar
  23. Sabra AI. Theories of light from descartes to Newton. London; 1967.
     Google Scholar
  24. Shapiro AE.Newton´s definition of a light ray and the diffusion theories of chromatic dispersion. Isis. 1975;66(2):194–220.
     Google Scholar
  25. Nakajima H. Two kinds of modification theory of light: some new observations on the Newton-Hooke controversy of 1672 concerning the nature of light. Ann Sci. 1984;41(3):261–78.
     Google Scholar
  26. Guerlac H. Can there be colors in the dark? Physical color theory before Newton. J History Ideas. 1986;47(1):3–20.
     Google Scholar
  27. Steinle F. Newton´s rejection of the modification theory of colour. In Hegel and the Sciences. Cohen RS, Wartofsky MW, Eds. vol. 64, Dordrecht, Boston, Hingham, MA: D. Reidel, 1994, pp. 547–56.
     Google Scholar
  28. Zemplén G. Newton’s rejection of the modificationist tradition. Form, Zahl. Ordnung: studien zur Wissenschafts-und Technikgeschichte: ivo Schneider zum 65. Geburtstag. 2004;48:481–503.
     Google Scholar
  29. Garben MD. Chymical wonders of light: j. Marcus Marci´s seventeenth-century Bohemian optics. Early Sci Med. 2005;10(4):478–509.
     Google Scholar
  30. Zemplén GA. The history of vision, colour, & light theories: introductions, texts, problems. In Bern Studies in the History and Philosophy of Science, Bern. 2005. ISBN 3-9522882-1.
     Google Scholar
  31. Stávek J. A new interpretation of the physical color theory based on the descartes’ rotation energy of visible, ultraviolet, and infrared photons. Eur J Appl Phys. 2023;5(5):29–38.
     Google Scholar
  32. Stávek J. The element of physical reality hidden in the letter of Malus to Lancret in 1800 can solve the EPR paradox (Malus thermochromatic loophole). Eur J Appl Phys. 2023;5(6):10–6.
     Google Scholar
  33. Stávek J. The descartes code (spin orbital rotation of photons). I. The fourth-order effects in the michelson interferometer. Eur J Appl Phys. 2023;5(6):25–30.
     Google Scholar
  34. Garetz BA. Angular doppler effect. J Opt Soc Am. 1980;71:609–11.
     Google Scholar
  35. Dholakia K. An experiment to demonstrate the angular doppler effect on laser light. Am J Phys. 1998;66:1007–10.
     Google Scholar
  36. Yao AM, Padgett MJ. Orbital angular momentum: origins, behavior and applications. Adv Opt Photonics. 2011;3(2): 161–204.
     Google Scholar
  37. Zhou H, Fu D, Dong J, Zhang P, Zhang X. Theoretical analysis and experimental verification on optical rotational Doppler effect. Opt Express. 2016;24(9):10053.
     Google Scholar
  38. Li G, Zentgraf T, Zhang S. Rotational Doppler effect in nonlinear optics. Nat Phys. 2016;12:736–41.
     Google Scholar
  39. Pan D, Xu H, Javier García de Abajo F. Rotational Doppler cooling and heating. 2019. Arxiv:1908.07973v1.
     Google Scholar
  40. Shen Y, Wang X, Xie Z, Min C, Fu X, Liu Q, et al. Optical vortices 30 years on: OAM manipulation from topological chargé to multiple singularities. Light: Sci Appl. 2019;8:90.
     Google Scholar
  41. Emile O, Emile J. Rotational Doppler effect: a review. Annalen Der Physik. 2023;535(11):2300250.
     Google Scholar
  42. Dyson FW, Eddington A, Davidson CR. A determination of the deflection of light by the sun´s gravitational field, from observations made at the total eclipse of May 29, 1919. Philos Trans R Soc A: math, Phys Eng Sci. 1920;220:291–333.
     Google Scholar
  43. Cambell WW, Trumpler RJ. Observations on the deflection of light in passing through the sun´s gravitational field. Lick Obs Bull. 1923;11:41–54.
     Google Scholar
  44. Chant CA, Young RK. Evidence of the bending of the rays of light on passing the sun, obtained by the Canadian expedition to observe the Australian eclipse. Publ Dom Astrophys Obs Vic. 1923;2:275–85.
     Google Scholar
  45. Dodwell GF, Davidson CR. Determination of the deflection of light by the sun´s gravitational field from observations made at Cardillo downs, South Australia, during the total eclipse of 1922 September 21. Mon Not R Astron Soc. 1924;84:150–62.
     Google Scholar
  46. Freundlich E, von Klüber H, von Brunn A. Ergebnisse der potsdamer expedition zur beobachtung der sonnenfinsternis von 1929, Mai 9. In Takengon (Nordsumatra). (Results of the Potsdam´s Expedition for the Observation of the Solar Eclipse on 1929, May 9, in Takengon (North Sumatra)). Zeitschrift Für Astrophysik, vol. 3, German: Springer Verlag, 1931, pp. 171–98.
     Google Scholar
  47. Mikhailov AA. The deflection of light by the gravitational field of the sun (George Darwin lecture).Mon Not R Astron Soc. 1959;119:593–608.
     Google Scholar
  48. Matukuma T. On the Einstein effect derived from the observations of the total solar eclipse of June 19th in 1936. Jpn J Astron Geophys. 1951;18:51–72.
     Google Scholar
  49. Van Biesbroeck G. The Einstein shift at the eclipse of May 20, 1947, in Brazil. Astron J. 1950;55:49–53.
     Google Scholar
  50. Van Biesbroeck G. The relativity shift at the 1952 February 25 eclipse of the sun. Astron J. 1953;58:87–8.
     Google Scholar
  51. Schmeidler F. Neuer Versuch einer Messung der relativistischer Lichtablenkung. (New attempt of a measurement of relativistic light deflection). Astron Nachr. 1963;287(1):7–16, German.
     Google Scholar
  52. Schmeidler F. Messung der Lichtablenkung während der Sonnenfinsternis am 15. Februar 1961. (Measurement of relativistic light deflection on February 15, 1961). Astron Nachr. 1985;306(2):71–6, German.
     Google Scholar
  53. Jaki SL. Johann Georg von Soldner and the gravitational bending of light, with an English translation of his essay on it published in 1801. Found Phys. 1978;8:927–50.
     Google Scholar
  54. Ginoux JM. Albert Einstein and the doubling of the deflection of light. Found Sci. 2022;27(3):829–50. doi: 10.1007/s10699-021-09783-4.
     Google Scholar
  55. Sauer TS. Einstein, gravitational light deflection and factors of two. Annalen Der Physik. 2021;533:2100203. doi: 10.1002/andp.202100203.
     Google Scholar
  56. Gerber P. Die räumliche und zeitliche Ausbreitung der gravitation. (The spatial and temporal propagation of gravity). Zeitschrift F˝ur Mathematik Und Physik. 1898;43:93–104, German.
     Google Scholar
  57. Gerber P. Die fortpflanzungsgeschwindigkeit der gravitation. 1902. (The propagation speed of gravity). In Programmabhandlung Des Städtischen Realgymnansiums Zu Stargard. German: Gymnasium zu Stargard, 1902.
     Google Scholar
  58. Laue M. Die fortpflanzungsgeschwindigkeit der gravitation. Bemerkung zur gleichnamigen Abhandlung von P. Gerber. (The propagation speed of gravity.Acomment to the paper of P. Gerber). Annalen Der Physik. 1917;53(11):214–6, German.
     Google Scholar
  59. Oppenheim S. Zur frage nach der fortpflanzungsgeschwindigkeit der gravitation. (On the question of the propagation speed of the gravity). Annalen Der Physik. 1917;53(10):163–8, German.
     Google Scholar
  60. Seelinger H, Bemerkungen zu P. Gerbers aufsatz: die fortpflanzungsgeschwindigkeil der gravitation. (A comment to the approach of P. Gerber: the propagation speed of gravity). Annalen Der Physik. 1917;53(9):31–2.
     Google Scholar
  61. Silberstein L. The motion of the perihelion of mercury deduced from the classical theory of relativity. Mon Not R Astron Soc. 1917;77:503–10.
     Google Scholar
  62. Roseveare NT. Mercury’s Perihelion, from Leverrier to Einstein. Oxford: University Press; 1982. pp. 137–4.
     Google Scholar
  63. Anonymous. Gerber’s gravity. 2023, Accessed on December 13. 2023. Available from: https://www.mathpages.com/home/kmath527/kmath527.htm.
     Google Scholar
  64. Shapiro II. Fourth test of general relativity. Phys Rev Lett. 1964;13(26):789–91.
     Google Scholar
  65. Shapiro II, Pettengill GH, Ash ME, Stone ML, Smith WB, Inglls RP, et al. Fourth test of general relativity: preliminary results. Phys Rev Lett. 1968;20(22):1265.
     Google Scholar
  66. Quattrini S. Shapiro time delay, curved 4D space-time or variable speed of light. 2020. doi: 10.13140/RG.2.2.11819.39202/1.
     Google Scholar
  67. Anderson JD, Laing PA, Lau EL, Liu AS, Nieto MM, Turyshev SG. Indication, from Pioneer 10/11, Galileo, and Ulysses data of an apparent anomalous, weak, long/range acceleration. Phys Rev Lett. 1998;81(14):2858–61.
     Google Scholar
  68. Anderson JD, Laing PA, Lau EL, Liu AS, Nieto MM, Turyshev SG. Study of anomalous acceleration of Pioneer 10 and 11. Phys Rev D. 2002;65(8):82004.
     Google Scholar
  69. Turyshev SG, Toth VT, Kinsella G, Lee SC, Lok SM, Ellis J. Support for the thermal origin of the Pioneer anomaly. Phys Rev Lett. 2012;108(24):241101.
     Google Scholar
  70. Bertolami O, Francisco F, Gil PJS, Páramos J. The contribution of thermal effects to the acceleration of the deep-space Pioneeer spacecraft. Phys Rev Lett. 2012;107(8):81103.
     Google Scholar
  71. Laser Lunar Ranging Experiment.Wikipedia. 2023, Accessed on December 14, 2023. Available from: https://en.wikipedia.org/wiki/Lunar_Laser_Ranging_experiments.
     Google Scholar
  72. Dickey JO, Bender PL, Faller JE, Newhall XX, Ricklefs RL, Riesi JG, et al. Lunar laser ranging–A continuous legacy of the Apollo program. Sci. 1994;265(5171):482–90.
     Google Scholar
  73. Williams JG, Boggs DH. Secular tidal changes in lunar orbit and Earth rotation. Celest Mech Dyn Astron. 2016;126(1): 89–129.
     Google Scholar
  74. Colmenares NR, Battat JBR, Gonzales DP, Murphy TW Jr, Sabhlok S. Fifteen years of millimeter accuracy lunar laser ranging with APOLLO: data reduction and calibration. 2023, Arxiv: 2304.11174v1. Accessed on December 14, 2023.
     Google Scholar
  75. Sanejouand YH. Empirical evidences in favor of a varying-speed-of-light. 2009, Arxiv:0908.0249v4. Accessed on December 14, 2023.
     Google Scholar
  76. Milgrom M. A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. Astrophys J. 1983;270:365–70.
     Google Scholar


Most read articles by the same author(s)

1 2 3 > >>