Relativistic Deflection of Light Near the Sun

Using Radio Signals and Visible Light

P. Marmet and C. Couture,

Physics Department, University of Ottawa, Ottawa, On. Canada, K1N 6N5

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Abstract.
        This paper reports a detailed analysis of one of general relativity's predictions, which claims that light should be deflected by solar gravity. The experimental data related to that prediction are analyzed. The substitution of the direct experimental test for the deflection of visible light during solar eclipses by the indirect measurement of the delay of radio signals traveling between a space probe or from extra galactic sources and the Earth is examined.
       Three different causes of the delay in the transmission of light near the Sun are examined. They are the relativistic delay, the delay caused by the plasma surrounding the Sun or for a geometric reason. The delay predicted by general relativity is equivalent to a reduced velocity of light in vacuum, in the Sun’s gravitational potential. Since the value of c is defined on Earth, inside the solar gravitational potential, this leads to a double value for the velocity of light on Earth. Furthermore, Einstein’s general relativity predicts that photons slow down when approaching the Sun, so that their velocity must be reduced to zero when reaching the surface of a black hole.
        This paper shows how all the experiments claiming the deflection of light and radio waves by the Sun are subjected to very large systematic errors, which render the results highly unreliable and proving nothing.  A previous preliminary paper (*) giving an analysis of the experiments using visible light already appeared on the subject. Furthermore, the internal incoherence of general relativity, which leads to a double velocity of light on Earth, adds to the weakness of these tests. Following those difficulties, and since it has also been demonstrated that the deflection of light by a gravitational potential is not compatible with the principle of mass-energy conservation, we show that no one can seriously claim that light is really deflected by the Sun.



1 - Introduction.
        According to general relativity(1) (page 179), light emitted from a source far away from the Sun and passing near the Sun should be deflected by an angle d :

1
        where G is the gravitational constant, MS is the solar mass, c is the velocity of light, and b is the minimum distance between the trajectory and the center of the Sun. If the radius of the Sun is RS, we have:
2
        The fundamental physical principles involved in that deflection of light has been reexamined in a recent book using Newton's physics.  It was found that such a deflection is not compatible with the principle of mass-energy conservation.
        Another cause of deflection is due to the corona surrounding the Sun. This plasma produces a deflection but this has nothing to do with relativity. The deflection due to this plasma is calculated in appendix I. In order to verify the deflection due to general relativity, several astronomical expeditions were organized. Experimenters measured the deflection of the images of stars located at a small angular distance from the Sun, during solar eclipses. However, the deflection is so small that, due to the atmospheric turbulence during daytime when a solar eclipse must occur, no observation could ever successfully show that the deflection exists (see appendix II). More than eighty years have passed since Einstein's predictions and no direct measurement of the gravitational deflection of light by the Sun has confirmed the theory in any convincing way. The direct measurement of the deflection of visible light seems almost abandoned. In the sixties, physicists suggested a new experiment, which measured the delay of a radio signal traveling near the Sun. Let us study this new test.

2 - The Viking Relativity Experiment.
        The deflection of electromagnetic radiation by the solar gravity is now claimed to be real because of an experiment using radar signals called the Viking Relativity Experiment(2). (Other less accurate experiments were also done involving Venus and Mercury(3)). In those experiments, physicists did not measure the deflection of light (or of a radio signal) by the Sun. All they measured was the time taken by a radio signal to travel between the Earth and another planet when grazing the Sun’s surface. This observed time was then compared with the time taken by light moving in a straight line at velocity c to travel the same distance in the absence of the gravitational potential. A delay was reported between those two times.
        It is recognized that when radio signals travel through the plasma around the Sun, a delay is produced. This contribution to the reduced velocity of light was taken into account and subtracted accordingly (see appendix I). In the case of the Viking Relativity Experiment, this contribution was measurable because two different frequencies were used and the delay in the plasma is frequency dependent.
        General relativity predicts that the solar gravitational potential must also produce a delay in the transmission of the radio signal. The same relativistic phenomenon which produces the predicted deflection of 1.75" is also responsible for the slowing down (compared with the absolute velocity of light c) of the radiation between Mars and the Earth. However, no direct deflection is measured in the Viking Relativity Experiment.
        Let us consider the delay Dt predicted by general relativity in the case of a round trip between the Earth and Mars, respectively at distances r1 and r2 from the Sun. Using Einstein’s theory, using Schwarzschild's metric, the delay(1) for a radio signal making a return trip from Earth to Mars is:

3
        where r1, r2 » RS and
4
        Using equation 3, Straumann(1) gives a predicted delay of:
5
        Note that Dt here is a distance, which is converted to time via c. Equations 3 and 5 predict that a radio signal emitted from the Earth, grazing the Sun and immediately retransmitted toward us when reaching planet Mars will, according to general relativity, travel during a time which is 250 ms longer than the time calculated using the velocity of light c (in zero gravitational potential).

3 - Physical Causes for the Delay.
        Let us study three causes that could be responsible for the delay in the transmission of radiation between the Earth and Mars:
a) an increase of the geometrical distance between the extremities of a bent trajectory;
b) general relativity;
c) the interaction with the plasma around the Sun.

3 - a) Delay Due to the Geometrical Bending of Light.
        We have seen above that general relativity predicts that light passing near the solar limb is deflected by an angle of 1.75". The same theory predicts that due to the same gravitational potential, the radiation takes a longer time to travel the distance between the Earth and Mars. Figure 1 illustrates how light is deflected when grazing the Sun.


Geometrical Time Delay
Figure 1

        One can see on figure 1 that if the trajectory of light is not a straight line (dotted line), it takes a longer time to travel between Mars and the Earth. The increase of time Dtb due only to the geometrical bending of light by d = 1.75" is given by the relationship:

6
        We find that Dtb = 0.010 ms or 3.2 meters. The increase of time D tb (with respect to a straight line) taken by light to travel from the Earth to Mars due to the geometrical bending of light is extremely small and negligible with respect to the delay (125 ms or 36 km) predicted by relativity as given in equation 5. Consequently, the angle made by light grazing the Sun is totally insufficient to explain the increase of distance (or delay) compatible with the prediction of general relativity as given in equations 3 and 5. This geometrical delay caused by the bending is not the main cause of the delay predicted by general relativity. It is several thousand times too small.

3 - b) Physical Meaning of the Relativistic Equation.
        In order to get a better understanding of the physics implied by equation 3, let us simplify the problem and apply the equation to the case of a single passage of the radiation from Mars to the Earth when grazing the Sun. The time delay D tE-M(1) is then half of equation 3:

7
        When we examine the parameters in equation 7, we find that, for any realistic values of r1, r2 and RS, light is always delayed. During the transmission of light between Mars and the Earth, when grazing the Sun, equation 7 shows that there is a delay of 36 km. Let us calculate the delay observed from a source located far behind Mars. If that source of radiation is, for example, star Sirius, located 3´ 1013 km away behind the Sun, and if the star's light grazes the Sun, equation 7 shows that the delay is 71 km. This is much longer than the delay for light traveling between Mars and the Earth. One must conclude that light does not travel at the speed of light in the space between Sirius and Mars since there is an extra delay of 71-36 = 35 km. Mathematics shows that, according to general relativity(1), the time delay with respect to the speed of light becomes infinite if the source of light is infinitely distant. Consequently, equation 7 shows that, everywhere in space, light is transmitted at a velocity slower than the accepted definition of the velocity of light known on Earth. This is not compatible with the definition of the velocity of light in vacuum accepted by the International Astronomical Union(4) which gives an absolute velocity of light, independently of any parameter. However, Shapiro(5) states that: "According to general relativity, the speed of a light wave depends on the strength of the gravitational potential along its path." According to equation 7, the velocity of light in vacuum is not equal to c on planet Earth, since it is submerged in the solar gravitational potential that changes the velocity by a factor as large as 1.97×10- 8. If the velocity of light is not constant, it is absolutely necessary to correct the definition and add, at which location the velocity of light is equal to c. According to Bowler(6), this happens at infinity. One must conclude that in Einstein’s general relativity, the observed velocity of light is always slower than c since no observer can be infinitely away from all the gravitational masses in the universe. This problem will be studied in further details in sections 4 and 5.

3 - c) Delay Due to the Plasma around the Sun.
        It is well known that the Sun is surrounded by a plasma and that the velocity of electromagnetic radiation is reduced when moving through such a medium. Radio signals have been observed while going through the solar corona and a corresponding delay has been measured(2). Furthermore, it is well known that the velocity of transmission of a radio signal is also slowed down when traveling through neutral gases, even if that contribution is frequently neglected. The fact that many spectral lines are observed in the solar corona proves that the plasma is not fully ionized. Since the delay produced and observed due to the plasma in the solar corona is not due to general relativity, it must have a different origin. An analysis of that phenomenon is presented in appendix I of this article.

4 - Relativistic Delay on Earth and Double Value of the Velocity of Light.
        Equation 7 gives the Einstein's delay of transmission of radiation between any two locations r1 and r2 (see figure 1). When light grazes the Sun during its transmission from Mars to the Earth, equation 7 shows that it must also be delayed during each extra kilometer, after it has reached the Earth. This extra delay is given by the derivative of equation 7 as a function of the distance r1:

8
        This equation shows that at a distance of r1 from the Sun (in the Earth neighborhood), general relativity predicts that the velocity of light in vacuum is slower than the value of c. This slower velocity of light is represented on figure 2 by (Dt) as a function of the distance r1 from the Sun. Since equation 8 is not a function of RS , the increment to the delay predicted at the final location (r1) is totally independent of b, the minimum distance between the trajectory and the Sun. Consequently, the value of RS is irrelevant here and can always be assumed small in order to satisfy the condition that r1 and r2 be large with respect to RS.
        We show on figure 2 the slower velocity of light (relative delay (Dt) per kilometer) at different distances from the Sun, as predicted by general relativity (equation 8).

Figure 2
Fraction of Reduction of Velocity of Light versus Distance from the Sun.

        The shaded area on figure 2 shows, according to general relativity, how much the velocity of light is reduced with respect to c. This delay is more important in the solar neighborhood (c is reduced by 4.24×10- 6) but light is still noticeably delayed in the Earth neighborhood (c is reduced by 1.97×10-8) and the phenomenon is not negligible even very far beyond the Earth orbit. For example, according to equation 3, light traveling between Jupiter and the Earth is still notably delayed, even when Jupiter is in opposition with the Sun so that light does not pass in the Sun's neighborhood. Considering that the velocity of light is defined as c on Earth, the new reduced value (i.e. (1 - 1.97×10-8) c) means that there is a double value of velocity of light on Earth.

5 - Importance of the Delay in the Earth Neighborhood.
        We have seen above that general relativity predicts that light does not move at the speed of light c when traveling in a gravitational potential. Therefore, since the Earth is located inside the solar gravitational potential, the velocity of light predicted on Earth is not the same as c. According to relativity, that velocity of light should be corrected due to the solar gravitational potential at Earth's distance from the Sun.
        Bowler(6) (page 57) states that in general relativity "the local velocity of light must depend on the local gravitational potential". Since equation 3 predicts that the velocity of light is reduced on Earth by as much as 1.97×10-8, using Earth parameters, one must conclude that general relativity leads to an incoherence on the value of the velocity of light on Earth. The fundamental definition requires an absolute velocity c while general relativity requires the use of a velocity reduced by a factor of 1.97×10-8 on Earth. Furthermore, since we know that a standard meter on Earth is defined as the number of wavelengths of a spectral line of light moving at the exact velocity of light c, we see that there is also an incoherence to the length of the standard meter. We have now two values for the velocity of light on Earth: the definition of c, and the one predicted with the delayed value. How can light know which velocity to choose?
        This incoherence also appears clearly in Bowler's book(6). On page 58, he calculates (equation 5.1.5) the velocity of light predicted on Earth as a function of the distance r from the Sun using the "index of refraction" of the gravitational field. Bowler states: "As r tends to , we want the velocity of light to be c." This result is compatible with a new definition of c at infinity and not with the international definition of the velocity of light c on Earth. Consequently, Bowler's equation 5.1.5 does not give the correct value of c at the Earth distance from the Sun at it should. A variation of 1.97×10-8 in the velocity of light is a very large error since atomic clocks are considered to be accurate within about 10-12 to 10-14. One must conclude that general relativity leads to a disastrous incoherence about the velocity of light and the length of the standard meter on Earth.
        Finally, we have seen that the delay predicted by general relativity is equivalent to a reduced velocity of light in vacuum, in the Sun’s gravitational potential. Consequently, photons are slowing down when approaching the Sun. In fact, the velocity of the photons can be reduced to zero when they reach the surface of an extremely massive body. This is surprising, since this prediction is contrary to what happens to particles which are speeding up when falling in a gravitational potential. One can see that these predictions of general relativity lead to serious difficulties when we consider momentum and energy conservation.

6 - Consequences of the Viking Relativity Experiment.
        As seen in section 2, Shapiro et al.(2) report an experiment in which they measured the round trip time of flight of radio signals transmitted between the Earth and the Viking spacecraft in order to test Einstein's general theory of relativity. Theoretically, using Fermat's principle, one can see that the time delay (reduced velocity of light) is related to the deflection of light by the Sun. The differential slowing down of the speed of light as a function of the distance from the Sun tilts the wave front and changes its direction by d  = 1.75". This differential velocity predicted by general relativity produces a deflection of light just as the differential velocity in a plasma produces a bending as explained in appendix I (and figure 1A). According to general relativity, the radio signal grazing the solar surface is delayed by up to 72 km corresponding to 250 ms. Shapiro et al.(2) claim an agreement with general relativity to within 0.5%. This means that the delay must be measured with an accuracy of 0.36 km.
        The Viking Relativity Experiment(2) involves corrections that take into account the delay due to the plasma composed of an erratic electron density surrounding the Sun. Since the claimed accuracy of 0.36 km in the round trip distance is extremely small compared with ~760 million km traveled by light during that same round trip (ratio equal to 4.7×10-10), it is necessary to know, with a comparable accuracy, all of the other contributions of error in the delay. The errors originate primarily from two sources: (1) the orbits of the planets and of the spacecraft around Mars and the positions of the tracking stations on Earth and (2) the solar corona which increases the delay significantly for signal paths that pass near the Sun. We have seen that the increase of path length due to the geometrical bending is negligible (section 3a).
        It is certainly not clear in Shapiro’s team’s paper(2) how the elements of orbit of Mars and the Earth can be reliably obtained with the claimed accuracy of 4.7×10-10. When calculating the data, one has to decide whether those elements of orbit have been corrected for general relativity. At the Earth distance from the Sun, general relativity predicts that time and lengths are changed by about 10-8 due to the orbital velocity of the planets and the solar gravitational potential. Have the data taken by radar to determine the orbital elements been all corrected for the reduced velocity of light? One can expect that general relativity has been taken into account since extremely accurate elements of orbit of Mars and of the Earth are required. This indispensable information is missing in Shapiro’s team’s paper(2).
        Calculations show that the expected relativistic correction that are needed to be applied to Newton's elements of the orbit is much larger (~2×10-8) than the relative error claimed in the distances of the planets (0.36÷760 million ~5×10-10). Consequently, the delay claimed by Shapiro’s team(2) is necessarily dominantly dependent on the relativistic correction previously introduced in his calculation. They cannot find that the relativistic correction exists if they have already introduced that correction in the elements of orbit leading to the distance between Mars and the Earth. When a relativistic correction in introduced in a calculation, we cannot be surprised to find in the final calculation, a difference in delay caused by that same relativistic correction.
        Consequently, due to the above uncertainties in the elements of orbit of the planets, the delay reported is meaningless and does not prove any fundamental agreement with general relativity. Anyhow, the method used by Shapiro et al.(2) is not coherent and uses non-coherent (double) values for the velocity of light in vacuum. Therefore, it is erroneous to believe that Shapiro's team's experiment proves that the velocity of light is reduced in the solar neighborhood since this is not compatible with the corrected velocity of light measured on Earth.

7 - Measurement of Gravitational Deflection Using Very Long Baseline Interferometry.
       Another kind of experiment(3,7) using radio signals has been claimed to measure the deflection of radio signals near the Sun. Since no angle is measured and only time delays are studied, those are indirect measurements. One of those measurements(7) uses VLBI observations of the extragalactic radio sources 3C273B and 3C279 passing near the Sun every year. The article starts with: "There is a wide recognition of the importance of testing theories of gravitation". This is a clear manifestation that these theories have not yet been properly tested. In that experiment, the radiation issued from a galactic radio source is detected simultaneously at two different receiving stations located in California and in Massachusetts and recorded on magnetic tapes with the signal generated by local atomic clocks.
        Let us consider a deflection of the extragalactic signal due to the solar gravity. Since the radio signal originates from extremely far behind the Sun, after a deflection in the solar neighborhood the radio signal reaching California will remain nearly parallel to the deflected ray reaching Massachusetts. Nearly only the direction (of both rays) has changed near the Sun by an angle d , with respect to the initial direction, as seen on figure 3. However, geometrical considerations show that, for a pair of distant stations located in the plane perpendicular to the initial incoming direction of the radio signal, the ray which passes further away from the Sun will arrive slightly before the other one.

Figure 3
Differential Delay after Deflection of Light by the Sun.

        In order to establish a cross-correlation between the two radio signals received, recognizable fluctuations on the amplitude of the incoming signal are looked for. It is hoped that the pattern of a fluctuation existing in the original intergalactic radio signal, (that must exist in both parallel rays near the Sun), can be recognizable at both receiving stations in California and Massachusetts. Then, a delay Dl  could then be measured between the two receiving stations located in the plane perpendicular to the initial source direction (see figure 3).
        However, since that extragalactic radiation grazes the Sun, it also passes through the solar corona, which is a plasma surrounding the Sun. In order to correct for the change of velocity of the radiation due to the solar plasma, the radio signal received at each station is passed through narrow band filters selecting three different frequencies. In the case of a plasma (here the solar plasma) , the velocity of transmission of radiation is different at different frequencies. These pairs of signals are recorded as a function of the time given by a local atomic clock.
        Technically, it is reported that the correlation between the same pattern of a radio fluctuation recorded at each station, is detected using a "filter estimator" combined with a "parameterized theoretical model of the delays" developed for that experiment. The aim of Lebach’s et al.(7) paper is to determine the parameter g , defined in the parameterized post-Newtonian (PPN) formalism(7) (equation #2). This parameter gives a relationship between the deflection predicted by Einstein and the delay of the same pattern between each antenna. For a perfect agreement with Einstein’s general relativity, the parameter g must equal unity. In order to get the picosecond accuracy claimed in the article, the two local clocks, located in California and in Massachusetts, must be synchronized at the picosecond (or 0.3 millimeter) accuracy claimed in the paper. It is not clear how such a limit can be achieved.

8 - Origin of the Fluctuations of the Radio Source.
        General relativity predicts that the velocity of light is reduced in the solar gravitational potential. Furthermore, the solar plasma adds a supplementary delay to the transmission of radiation, but more importantly, it adds important fluctuations to the original extragalactic signal. Unfortunately, the density of that plasma as well as its short time fluctuations are totally unpredictable. Signal fluctuations have been observed lasting a few minutes. They were measured(7) to vary by as much as 500%. Since random variations of the intensity of that plasma are taking place very rapidly and are totally unpredictable, a parameterized correction cannot give an accurate prediction. The fluctuations in the plasma density even vary within one period of accumulation of data. Consequently, even if three different frequency components are measured, the theoretical inverse quadratic correction (as a function of frequency) introduced to determine the density of the plasma cannot provide the picosecond accuracy stated in the paper. One must recall that the change of distance corresponding to one picosecond is equal to only a difference of 0.3 millimeter (at light velocity) between the receiving antennas on Earth.
        Lebach’s et al.(7) experiment is based on the condition that the fluctuations of intensity of the radio signal (from the radio sources 3C273B and 3C279), used for synchronization in California and Massachusetts, originate at a cosmological distance, well before radiation approaches the solar neighborhood. When any extra fluctuation is added to the rays crossing the solar corona, the receivers located in California and in Massachusetts can no longer synchronize correctly the phase relationship (identified by a specific pattern of fluctuation) required for the experiment. Any interference by the plasma is a serious obstacle to the experiment, since there is no way to recognize the strong random noise generated by the solar corona from the original fluctuations necessary to synchronize the radio signals. Furthermore, when the noise from the corona (added to the signal), is later filtered by a narrow band filter (used in Lebach’s et al. experiment), the addition of that filter changes the phase of the signal.
        It is well known theoretically and previously observed, that when a radio signal is transmitted through a plasma, (here the plasma of the solar corona), very important fluctuations are added. Even Lebach et al.(7) report that in some data: ". . . large, rapid plasma-density fluctuations in the solar corona would make the coherence time of the signals from 3C279 at 2 GHz too short to allow detection ". Therefore, Lebach et al.(7)made an arbitrary selection to what appeared to be acceptable data. The noise generated by the solar plasma gets so intense that it can block completely the cosmic signal. For example, when the radio signal of Pioneer VI(8-13) passed through the solar corona, it was observed that the fluctuations due to the interaction with the plasma became so important that the initial signal became undetectable well before reaching the solar limb vicinity. Signal distortion due to the increase in the frequency bandwidth is quite evident(8-13). In his abstract, Goldstein(8) states: "The spectral bandwidths increased slowly at first, then very rapidly at 1 degree from the Sun". It is well known theoretically how a plasma generates fluctuations while increasing the bandwidth. A computer program cannot identify small statistical fluctuations in the extra galactic radio source from the intense wide band noise fluctuations generated by the plasma surrounding our sun. Radiation emitted by the extra galactic radio source fluctuates after passing through the solar corona just as starlight twinkles after passing through the Earth atmosphere. Genuine starlight fluctuations (in the radio range) cannot be identified through the intense twinkling caused by the intense million-degree furnace of the sun's corona.

9 - From Amateur Size Telescopes to Multi-Billion Dollar Space Technology.
        One must conclude that the theoretical model used and leading to the reported delay, is no longer acceptable when there is such an unstable plasma, because much larger noisy fluctuations are added by the plasma in the Sun’s neighborhood. The delays measured by Lebach et al.(7) cannot prove the deflection of light by the Sun, because of the impossibility of demonstrating a reliable synchronization. It is well known statistically, that with a gigantic amount of data automatically recorded in that experiment, combined with the astronomical number of fits tested by the computer, coupled with a large amount of noise reaching sometimes the level of saturation as reported in their paper, some data can always be found to fit the expected theoretical model. This is specially true when it is felt that nobody might challenge the result obtained with a multi-billion equipment used to find an agreement with an extremely popular theory. Unfortunately, the observation of the deflection of visible light by the sun seems to have been abandoned some years ago because the phenomenon appeared impossible to detect in visible light.
        There is a desperate situation among scientists for not being able to show, with the most sophisticated technology, what is considered to be the basic principle of general relativity on which rely most of modern science, while this was claimed to be demonstrated by Eddington in 1919 using a simple four inch amateur size telescope. Of course, a trillion or quadrillion dollar equipment will never reveal clearly the deflection of light if such a deflection does not exist. It is hard to predict for how many more decades this race will last and how much money has to be wasted before scientists, at last, admit that there is no deflection.
        Let us recall that Einstein’s predictions of light deflection is based on an unverified variable velocity of light and on the double value of the velocity of light in the Earth neighborhood, as explained above. This incoherence of general relativity must be added to the fact that general relativity is not compatible with the principle of mass-energy conservation as demonstrated previously.--- (14).  The internal contradiction about having two different values of the velocity of light at the same location does not exist when we use a rational description, agreeing with the principle of mass-energy conservation (14).In that case, the advance of the perihelion of Mercury given by Newton's physics is explained independently and is also perfectly identical to the equation predicted by general relativity. Furthermore, length contraction and the change of clock rate can now logically be explained(14).No reliable observation has ever been able to prove such a deflection of light by the sun after 80 years. Therefore, it is much more logical to believe that such a deflection does not exist at all, and be compatible with the principle of mass-energy conservation.

10 - Acknowledgments.
        The authors wish to acknowledge the personal encouragement and financial contribution of Mr. Bruce Richardson, which helped to pursue this research work. We are also grateful to Dr. I. McCausland, University of Toronto, for bringing his attention to some interesting historical information.

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Appendix I
The Deflection and Delay of Radio Signals in the Solar Plasma.
        Shapiro(5) has observed that, due to the density of the plasma in the medium surrounding the Sun, the velocity of transmission of the radiation is reduced with respect to the speed of light c. Then, a delay D tm appears (with respect to the speed of light) between the emission of a pulse of a radio signal from Earth, and the reception of its echo through the interplanetary medium. This delay(5) is:
A-1
        where N() is the electron density expressed in electrons/cm3, f is the frequency of the radiation in hertz, c is the velocity of light in cm/sec,  is the path length in cm and e and p respectively refer to the Earth and the other planet (i.e. Mars).
        Using compiled results on the solar corona(15), during the quiet Sun period between the radial distances r = 4RS and r = 20RS, the electron density in the solar corona is well represented by:
A-2
        During the maximum solar activity, N can reach up to a factor 5 higher than the one given by equation A-2 in the radial range. Substituting equation A-2 into A-1 gives the time delay during the passage of radiation through the plasma (round trip):
A-3
        where d, xe, and xp are expressed in cm. xe is the distance along the line of flight from the Earth-based antenna to the point of closest approach to the Sun, xp represents the distance along the path from this point to the planet and d is the minimum distance between the Sun and the trajectory. When radio waves pass closer to the Sun, equation A-3 shows that the delay Dtm becomes larger. This means that at a closer distance to the Sun, the velocity of light in the plasma is slower, as expected in classical physics.
        Equation A-3 implies only the electron density in the photon path passing near the Sun and ignores all relativistic effects. This relationship, also given by Straumann(1) (page 181, equation 3.4.8), gives the electron density of the electron plasma at different distances r from the Sun. This same relationship is also used by Shapiro(2,5). In the Viking Relativity Experiment, two different frequencies are used in the S (» 2.2 GHz) and X bands (» 8.8 GHz). This allows us to recognize the contribution to the time delay produced by the passage of radiation through the plasma from the one assumed to be caused by relativity. Only the delay predicted by general relativity is frequency independent. The delay produced by the plasma is frequency dependent. It was observed by Shapiro et al.(2) (page 4329) that: "The increase of group delay from this cause (plasma), can reach about 100 m s for signals passing close to the Sun. " This must be compared with a predicted relativistic delay of 250 ms or 72 km(1).

Deflection of Radiation Due to the Plasma.
        Figure A1 illustrates the propagation of radio waves emitted from Mars in which relativity is momentarily ignored but for which we take into account the solar plasma distributed around the Sun. Each radio wave emitted travels in space forming a spherical front expanding around the emitter. The wave front expands and the light rays move in the radial direction away from the source. Some of the rays pass near the Sun so that the velocity of propagation of the radio signal is reduced by an amount that depends on the electron density of the plasma. Just near and above the Sun, two rays a and b are drawn on figure 1A.

Figure 1A
Delay Due to the Solar Plasma

        When ray a passes close to the Sun, it travels at a slower velocity because of the higher electron density at that location. Ray b above does not have its velocity as much reduced because the electron density is smaller further away from the Sun. Consequently, the wave front moves more slowly near the Sun in the path of ray a than in the path of ray b . By definition, since the wave front corresponds to a constant phase, the path traveled by ray a must be shorter than the path traveled by ray b by D in a given time interval. Consequently, the upper part of the wave front travels faster and the wave front becomes tilted due to the difference of velocity in the plasma. Once the wave has passed the plasma region, this wave front maintains its tilted direction until reaching the Earth without any other perturbation. This is the reason for which the radio beam is deflected by the solar plasma.
        Let us calculate the angle of deflection of radiation for a normal solar plasma as a function of the radio frequency of the emitter. This can be calculated by the derivative of the delay function A-3 as a function of the minimum distance d of the radio signal from the Sun. The derivative of A-3 (times c) with respect to d is:
A-4
        with:
A-5
        Using the Earth and Mars distances and for rays passing at a distance equal to the Sun's radius, if the electron density is that of a quiet Sun, equation A-4 gives a deflection (in seconds of arc) equal to:
A-6
        For a frequency of 1 GHz, the deflection is -25 arcsec. For a frequency of 3 GHz, the deflection is -2.8 arcsec and -1.03 arcsec at a frequency of 5 GHz. This quantity increases by a factor of about five during solar maximum. The deflection caused by the solar plasma is negligible for visible light because of the much higher frequency of visible light. One must conclude that the plasma around the Sun produces a deflection of radiation which is of the order of the predictions of relativity when the frequency is around a few GHz. One can show that the plasma surrounding the Sun also produces, at a frequency around a few GHz, a delay in the transmission of radio waves which is comparable with the delay predicted by general relativity.
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Appendix II
The Gravitational Deflection of Light by the Sun during Solar Eclipses.

A - Introduction.
        According to Einstein's general theory of relativity published in 1916, light coming from a star far away from the Earth and passing near the Sun will be deflected by the Sun’s gravitational field by an amount that is inversely proportional to the star’s radial distance from the Sun (1.745'' at the Sun's limb). This amount (dubbed the full deflection) is twice the one predicted by Einstein in 1908(16) and in 1911(17) using Newton's gravitational law (half deflection). In 1911, Einstein wrote: A ray of light going past the Sun would accordingly undergo deflexion to an amount of 4×10-6 = 0.83 seconds of arc. Let us note that Einstein did not clearly explain which fundamental principle of physics used in the 1911 paper and giving the erroneous deflection of 0.83 seconds of arc was wrong, so that he had to change his mind and predict a deflection twice as large in 1916.
        In order to test which theory is right (if any), an expedition led by Eddington was sent to Sobral and Principe for the eclipse of May 29, 1919(18). The purpose was to determine whether or not there is a deflection of light by the Sun's gravitational field and if there is, which of the two theories mentioned above it follows. The expedition was claimed to be successful in proving Einstein's full deflection(18,19). This test was crucial to the general approval that Einstein's general theory of relativity enjoys nowadays.
        However, this experimental result is not in accordance with mass-energy conservation(14) . This was not a real problem in those years, as we will show that the deflection was certainly not measurable. We will see that the effect of the atmospheric turbulence was much larger than the full deflection, just like the Airy disk. We will also see how the instruments could not possibly give such a precise measurement and how the stars distribution was not good enough for such a measurement to be convincing or even measurable. Finally, we will discuss how Eddington's influence worked for Einstein's full displacement and against any other possible result.

B - Observational Data.
        There is a long list(20) of papers reporting observations of stars in the neighborhood of the Sun during solar eclipses. A general survey of the eclipse results, with some discussions, has been published(20). Consequently, it is not possible to discuss them all in detail. However, it is the observations of the 1919 eclipse which first convinced the scientific community that the relativistic deflection really exists and that established the belief in Einstein’s theory. Therefore, we will examine these data in more detail though some information will also be given about observations of other eclipses. These observations were not successful, but they were considered as such until they were substituted by experiments using space probes. The 1919 paper gives an idea of the kind of measurement that convinced the world to the most spectacular theory accepted by modern science: the theory of general relativity. The problem of observing the deflection of light by the Sun is submitted to numerous experimental difficulties. Let us study those difficulties.
        Atmospheric turbulence is a phenomenon due to the atmosphere which causes images of stars as seen by an observer on Earth to jump, quiver, wobble or simply be fuzzy. This is a well-known phenomenon to any astronomer, amateur or professional. In fact(21) (page 40), "Rare is the night (at most sites) when any telescope, no matter how large its aperture or perfect its optics, can resolve details finer than 1 arc second. More typical at ordinary locations is 2- or 3-arc-second seeing, or worse."
        The problem becomes even worse during afternoons due to the heat of the ground. Tentative solutions to this seeing problem have only recently been experimented(22). For anyone unacquainted with atmospheric turbulence, an easy way to observe a similar phenomenon is by looking over a hot barbecue. In this case, the distortion of the images (of the order of several minutes of arc) is due to the heat coming from the barbecue.
        Eddington, an astronomer, was certainly aware of this problem. If it was difficult in 1995(21) to see details finer that 1'' at a professional site at night, how much more difficult was it with an amateur size telescope in the jungle in 1919? The supposed effect (full and half deflection) decreases with the distance of the star from the Sun. During the 1919 eclipse, the stars closest to the Sun's limb were drowned in the corona and could not be observed(18). Of the stars that were not drowned in the corona, Einstein’s theory predicts that k2 Tauri should have the largest displacement, with 0.88''. In Sobral, the displacement for that star was reported to be 1.00''(19). How could Eddington and Dyson claim to observe that if at best, their precision due to atmospheric turbulence in daytime heat was several arc seconds? And they were not at best, near noon at Sobral and 2 p.m. at Principe, when the seeing is the worst, with small amateur-size telescopes that were less than ideal. The instability caused by the atmospheric turbulence is large enough to refute any measurement of the so-called Einstein effect. However, there are other reasons.
        Two object glasses were used during the expedition at Sobral, a 4-inch object glass and an astrographic object glass. Assuming a perfect optical shape, which includes perfect chromaticity, for the 4-inch telescope, the size of the central spot (which is surrounded by the ring system of the diffraction pattern) can never be smaller than 1.25''. This central spot is called the Airy disk. Since some of the results were presented with a claimed accuracy of the order of 0.01''(19) (page 391), that relatively big diffraction ring pattern (125 times the claimed accuracy) should have been easily seen. Since no mention is made of it, we must understand that it was not observable because various aberrations (chromatic of spheric) were larger than 1.25'' and/or because, as expected, the atmospheric turbulence was larger than 1.25'', which is the theoretical limit of resolution of that telescope when there is no aberration and no turbulence.
        The elements of a telescope are very sensitive to temperature. For example, it is reported that(18) (page 153): "when the [astrographic] object glass is mounted in a steel tube, the change of scale over a range of temperature of 10° F. should be insignificant, and the definition should be very good". However, during the team’s stay at Sobral, the temperature ranged from 75°F during the night to 97°F in the afternoon. This change in temperature must have affected the 4-inch telescope.

       Let us calculate the change of scale on the plate of the 4-inch telescope due to the thermal expansion of the steel tube. The expansion coefficient of steel is 1.3×10-5 per degree Celsius. Even if the optical definition is not much changed by the change of temperature, the change of scale on the photographic plate is proportional to the change of length of the tube. For 10 degrees Celsius the scale changes by 1.3×10-4. Since the size of the plate is 8 per 10 inches (20 × 25 cm), this gives a change in its angular size of 1.2 arc-sec. It does not seem that this change of scale has ever been taken into account. This introduces a very serious error in the data. How can they claim an accuracy of the order of 0.01''(19) (page 391) when they admit that the focus of the telescopes were determined and fixed many days before the eclipse(18) (page 141)?
        The photographs of the eclipse taken with the astrograph were very disappointing(18) (page 153). It appears that the focus had changed from the night of May 27 to the moment of the eclipse. After the eclipse, the team left Sobral and came back in July to take comparison plates. They discovered that the astrograph had returned to focus! They blamed this change of focus on the effect of the Sun’s heat on the mirror, but they could not say whether this effect caused a change of scale or if it only blurred the images. The Sun’s heat could have affected its scale without blurring the images. We know that there is a zone along the focal length where the image looks as if it was in focus but for which the scale is changed. To the best of our knowledge, nothing has ever been said about that possible error.
        If we plot the value of Einstein's deflection against the angular distance of the star from the Sun (as done in(20) page 50), we see that the part of the hyperbola where the slope changes the most lies under a distance of two solar radii from the Sun's center. That part is thus crucial to a good interpretation of the results. Looking at page 60 of the same article, we see that only two of the stars used by the teams at Principe and Sobral are in this area. It is thus very difficult to fit a hyperbola when only two of the stars are in that zone. Only a straight line can be logically fitted through two points. These observations (and most of the others studied in von Klüber's(20) article, which reviews all observations done before 1960) could easily be fitted by a straight line instead of Einstein's deflection equation. Therefore, these data cannot prove any of Einstein's deflections (full or half).
        In one of the meetings of the Royal Astronomical Society(23) (page 41), Ludwik Silberstein pointed out that the displacements found were not radial, as Einstein's theory states, but sometimes deflected from the radial direction by as much as 35°! Nothing was said about that in Dyson's article(18). According to Silberstein: "If we had not the prejudice of Einstein’s theory we should not say that the figures strongly indicated a radial law of displacement."
        This brings us to our next point, which is to what degree social circumstances influenced the acceptation of Einstein's theory.

C - About Eddington's Influence.
        The results from the 1919 expedition were quickly accepted by the scientific community. When preliminary results were announced, Joseph Thomson (from the Chair) said(19) (page 394): "It is difficult for the audience to weigh fully the meaning of the figures that have been put before us, but the Astronomer Royal [Dyson] and Prof. Eddington have studied the material carefully, and they regard the evidence as decisively in favor of the larger value for the displacement."
        Thomson makes it look like only Eddington and Dyson are able to understand the results. It seems that they have such a reputation that the general and the scientific public should blindly believe them. It is Dyson who presented the results of the Sobral expedition at a meeting of the Royal Astronomical Society(19) (page 391). Some of the displacements presented were very small, sometimes of the order of 0.01''. In another meeting(23) (page 40), Oliver Lodge asked if it were possible to measure a deflection of 1/60'' (approximately 0.02'') to which Dyson responded: "I do not think that it would be possible to measure so small a quantity." We clearly see that Dyson contradicted himself. Furthermore, Eddington said himself he was in favor of the full deflection before doing the experiment. Writing about the results of the expedition, he said(24) (page 116):"Although the material was very meager compared with what had been hoped for, the writer (who it must be admitted was not altogether unbiased) believed it convincing." Moreover, according to Chandrasekhar(25) (page 25): "had he been left to himself, he would not have planned the expeditions since he was fully convinced of the truth of the general theory of relativity!"
        Eddington was a Quaker and like other Quakers, he did not want to go to war (WWI). In England, Quakers were sent to camps during the war, but because of Dyson's intervention(25) (page 25), "Eddington was deferred with the express stipulation that if the war should end by May 1919, then Eddington should undertake to lead an expedition for the purpose of verifying Einstein’s predictions!". The circumstances of the war forced Eddington to do an experiment that he would have never done had he had a choice because he was so convinced of its outcome. Why was the theory so quickly, widely and easily accepted? After all, it was radically changing the common view of the universe, curving space and dilating time. Furthermore, the British were accepting a theory from a German man, right after a bitter war with Germany.
        It seems that the theory was widely accepted only after the eclipse expedition(26)(page 50). According to Earman and Glymour, Dyson and Eddington played a great influential role in the acceptation of the general theory of relativity by the British. In fact, it is Eddington who, convinced of the truth of the theory, convinced Dyson. In the few years before 1919, they made the measurement of the "Einstein effect" a challenge and after the expeditions of May 1919, they helped give the impression that the data had confirmed Einstein’s theory.
        Aside from the fact that Eddington was convinced that the theory was right, another reason pushed him to advocate it(26) (page 85). He hoped that a British verification of a German theory might reopen the lines of communication and collaboration between the scientists of both countries, lines that had been closed during World War One. Finally, before 1919, no one had claimed to have observed displacements of the size required by Einstein's theory. Probably because the theory was thought to be proved by the 1919 eclipse observations, a lot of scientists, maybe throwing out some of their data, reported finding the right displacement(26) (page 85). After 1919, other expeditions were undertaken to measure the deflection of light by the Sun. Most of them obtained results a bit higher than Einstein's prediction, but it did not matter anymore since the reputation of the theory had already been established.
        In "Weird but True" Jamal Munshi(27) reports: "Dr. F. Schmeidler of the Munich University Observatory has published a paper [49] titled "The Einstein Shift An Unsettled Problem," and a plot of shifts for 92 stars for the 1922 eclipse shows shifts going in all directions, many of them going the wrong way by as large a deflection as those shifted in the predicted direction! Further examination of the 1919 and 1922 data originally interpreted as confirming relativity, tended to favor a larger shift, the results depended very strongly on the manner for reducing the measurements and the effect of omitting individual stars.
        So now we find that the legend of Albert Einstein as the world's greatest scientist was based on the Mathematical Magic of Trimming and Cooking of the eclipse data to present the illusion that Einstein's general relativity theory was correct in order to prevent Cambridge University from being disgraced because one of its distinguished members [Eddington] was close to being declared a "conscientious objector".

D - Conclusion.
        Much of the popularity of Einstein's general theory of relativity relies on the observations done at Sobral and Principe. We see now that these results were overemphasized and did certainly not consecrate Einstein's theory. It is interesting to think of what would have happened if the results had been deemed not good enough or if they had clearly showed that there is no deflection of light by the Sun. Einstein’s theory might not have enjoyed the popularity it now does and a new more realistic theory might have been found years ago. 
        The experiments claiming erroneously the deflection of light near the Sun shows some similarity with the claim by Michelson-Morley that there is no drift of interference lines in their experiment data.  A recent analysis of data shows that the drift of the interference fringes actually exists.  This has been published by Héctor Múnera, Centro Internacional de Física,  Bogotá, Columbia. An abstract and further information can be obtained at this address.  At the macroscopic scale, it seems that, only the observation of the Advance of the Perihelion of Mercury can be accepted as a reliably observed phenomenon. 

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References.
1- N. Straumann, General Relativity and Relativistic Astrophysics, Springer-Verlag, Berlin, pages, 459, 1991.
2- I. I. Shapiro, R. D. Reasenberg, P. E. MacNeil, R. B.Goldstein, J. P. Brenkle, D. L. Cain, T. Komarek, A. I. Zygielbaum, W. F. Cuddihy and W. H. Michael Jr., The Viking Relativity Experiment, in Journal of Geophysical Physics, 82, 28, p. 4329-4334, 1977.
3- I. I. Shapiro, New Method for the Detection of Light Deflection by Solar Gravity, Science, 157, p. 806, 1967.
4- K. R Lang,. Astrophysical Data: Planets and Stars, Springer-Verlag, 937 pages, 1991.
5- I. I. Shapiro, Fourth Test of Relativity, in Physical Review Letters, 13, 26, p. 789-791, 1964.
6- M. G. Bowler, Gravitation and Relativity, Pergamon Press, Oxford, 172 pages, 1976.
7- D. E. Lebach, B. E. Corey, I. I. Shapiro, M. I. Ratner, J. C. Webber, A. E. E. Rogers, J. L. Davis and T. A. Herring, Measurement of the Solar Gravitational Deflection of Radio Waves Using Very-Long-Baseline Interferometry, in Physical Review Letters, 75, 8, p.1439-1442, 1995.
8- R. M. Goldstein, Science 166, p. 598-601, Superior Conjunction of Pioneer 6, 1969.
9- A. A. Chastel, J. F. Heyvearts, Nature,. 249, p. 21-22, 1974.
10- G. S. Levy, T. Sato, B. L. Seidel, C. T. Stelzried, Pioneer 6: Measurement of Transient Faraday Rotation Phenomena Observed during Solar Occultation, Science 166, No: 3905, P. 596-598, 1969.
11- S. Depaquit, J. P. Vigier, J. C. Pecker, Comparaison de deux observations de déplacement anormaux vers le rouge observés au voisinage du disque solaire. C. R. Acad., Sc. Paris, Série B, p. 113-114, t.280, 1975
12- A. A. Chastel, J. F. Heyvearts, Broadening and Anomalous Shift of Pioneer VI Telemetry Line An Effect of Coronal Inhomogeneities Useful for Diagnostics, Astron. & Astrophys. 51, 171-183, 1976.
13- L. Accardi, A. Laio, Y. G. Lu, G. Rizzi, A Third Hypothesis on the Origin of the Redshift :Application to the Pioneer 6 Data, Physics Letters A. 209, p. 277-284, 1995.
14- P. Marmet,  Einstein’s Theory of Relativity versus Classical Mechanics, Editor Newton Physics Books, 200 pages, 1997, Ogilvie Rd, Gloucester, Ontario, Canada, K1J 7N4.
15- W. C. Erickson, Astrophysical Journal, 139, p. 1290, 1964.
16- A. Einstein, Jahrbuch der Radioaktiviät und Elektronik, 4, 411, 1908.
17- A. Einstein, Über den Einfluss der Schwerkraft auf die Ausbreitung des Lichtes, in Annalen der Physik, 35, 898, 1911. Translated in English in  H. A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl The Principle of Relativity, Dover, 1952.
18- F. W. Dyson, A. S. Eddington, and C. Davidson, A Determination of the Deflection of Light by the Sun's Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919, in Philosophical Transactions of the Royal Society of London, series A, 220, p. 291-333, 1920. (See also: Annual Report of the Board of Regents of the Smithsonian Institution Showing the Operations, Expenditures, and Conditions of the Institution for the Year Ending June 30 1919, Government Printing Office, Washington, p. 133-176, 1921.)
19- Joint Eclipse Meeting of the Royal Society and the Royal Astronomical Society, 1919, November 6, The Observatory, 42, 545, p. 389-398, 1919.
20- H. von Klüber, The Determination of Einstein's Light-Deflection in the Gravitational Field of the Sun, in Vistas in Astronomy, Pergamon Press, London, 3, 1960.
21- A. M. MacRobert, Beating the Seeing, in Sky & Telescope, 89, 4, p. 40-43, 1995.
22- D. Fischer, Optical Interferometry: Breaking the Barriers, in Sky & Telescope, 92, 5, p. 36-41, 1996.
23- Meeting of the Royal Astronomical Society, Friday, 1919, December 12, The Observatory, 43, 548, p. 33-45, Jan. 1920.
24- A. Eddington, Space, Time and Gravitation: An Outline of the General Relativity Theory, Cambridge University Press, Cambridge, 218 pages, 1959.
25- S. Chandrasekhar, Eddington: The Most Distinguished Astrophysicist of His Time, Cambridge University Press, Cambridge, 64 pages, 1983.
26- J. Earman and C. Glymour, Relativity and Eclipses: The British Eclipse Expeditions of 1919 and Their Predecessors, in Historical Studies in the Physical Sciences, 11, p. 49-85, 1980.
27- J. Munshi, Weird but True on the internet:  http://munshi.sonoma.edu/jamal/physicsmath.html

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Résumé.
      Cet article rapporte une analyse détaillée d’une des prédictions de la relativité générale d'Einstein, selon laquelle la lumière serait déviée dans un champ gravitationnel. Les données expérimentales reliées à cette prédiction sont soigneusement analysées. En plus, au lieu d’observer directement la déviation de la lumière durant une éclipse totale de soleil, nous considérons les mesures indirectes du retard de transmission de signaux de radio voyageant entre une sonde spatiale ou provenant d’une source extragalactique. Trois causes différentes de retard dans la transmission des ondes de radio sont étudiées. Ce retard peut être expliqué, ou bien par une cause relativiste, par le plasma entourant le soleil or pour des raisons géométriques. Le retard attribuable à la relativité générale conduit à une vitesse réduite de la lumière dans le vide sous l’influence du potentiel gravitationnel du soleil. Puisque la valeur de la vitesse de la lumière "c" est définie sur la terre, qui est située dans le potentiel gravitationnel solaire, ceci conduit à une double valeur de la vitesse de la lumière sur terre. En plus, la relativité générale d’Einstein prédit que les photons doivent ralentir en pénétrant dans le potentiel gravitationnel solaire de sorte que leurs vitesses sont réduites jusqu’à zéro, à la surface d’un trou noir.
        Cet article démontre à quel point les expériences, prétendant prouver la déviation de la lumière par le soleil, sont sujettes à d’importantes erreurs systématiques qui infirment les conclusions. On doit conclure qu'il n'existe jusqu'ici aucune preuve expérimentale sérieuse prouvant la déviation de la lumière par le soleil.

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References
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P. Marmet and C. Couture,  Physics Department, University of Ottawa, Ottawa, On. Canada, K1N 6N5 
This paper is an extract of a paper in Physics Essays, "Relativistic Deflection of Light Near the Sun Using Radio Signals and Visible Light" Vol: 12, No: 1 March 1999.  P. 162-174. 
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This paper has also been presented at the University of Storrs, Connecticut, at the meeting: "The New Natural Philosophy: An Introduction to the 21st Century Physics and Cosmology"  Storrs, 4-8 June 2000.  The paper presented is: "The Deficient Observations of Light Deflection Near the Sun". 
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