RELATIVISTIC DEFLECTION OF LIGHT NEAR THE SUN USING
RADIO SIGNALS AND VISIBLE LIGHT
(Paul Marmet, Christine Couture)
Abstract. This paper reports a detailed
analysis of one of general relativityís 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 Sunís 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, Einsteinís 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 by the Sun are subjected to
very large systematic errors, which render the results highly unreliable.
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 :
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:
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 Einsteinís
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 Sunís 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 D t 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 Einsteinís theory, using Schwarzschildís metric, the delay(1) for a radio signal making a return trip from Earth to Mars is:
where r1, r2 " RS and
Using equation 3, Straumann(1) gives a predicted delay of:
D t » 250 m s or 72 km. (5)
Note that D t 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 m s longer
than the time calculated using the velocity of light c (in zero gravitational
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
Geometrical Time Delay
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 D tb due only to the geometrical bending of light by d = 1.75" is given by the relationship:
We find that D tb
= 0.010 m s 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 m s 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
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:
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 starís 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-
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 Einsteinís 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
4 - Relativistic Delay on Earth and Double Value of
the Velocity of Light.
Equation 7 gives the Einsteinís 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:
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 ¶ (D t) 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
¶ (D t)
per kilometer) at different distances from the Sun, as predicted by general
relativity (equation 8).
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
but light is still noticeably delayed in the Earth neighborhood (c is reduced
by 1.97´ 10-
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 Sunís neighborhood. Considering
that the velocity of light is defined as c on Earth, the new reduced value
(i.e. (1- 1.97´
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 Earthís 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 Bowlerís 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 ® ¥ 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, Bowlerís 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
Sunís 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 Einsteinís general theory of relativity. Theoretically, using Fermatís 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 m s. 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 millions 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 Shapiroís teamís 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 Shapiroís teamís paper(2).
Calculations show that the expected relativistic correction that are needed to be applied to Newtonís elements of the orbit is much larger (» 2´ 10- 8) than the relative error claimed in the distances of the planets (0.36 ¸ 760 millions » 5´ 10- 10). Consequently, the delay claimed by Shapiroís 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 a 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 Shapiroís teamís 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.
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 D l 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 Lebachís 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 Einsteinís general relativity, the parameter
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.
Lebachís 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 seriously 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 Lebachís 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".
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 sunís corona.
9 - From Amateur Size Telescopes to Multi-Billion Dollar
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 Sunís 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 Einsteinís 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 Newtonís 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
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.
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:
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:
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):
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 D tm 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 m s or
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.
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:
Using the Earth and Mars distances and for rays passing at a distance equal to the Sunís radius, if the electron density is that of a quiet Sun, equation A-4 gives a deflection (in seconds of arc) equal to:
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.
The Gravitational Deflection of Light by the Sun during Solar Eclipses.
A - Introduction.
According to Einsteinís 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 Sunís gravitational field by an amount that is inversely proportional to the starís radial distance from the Sun (1.745íí at the Sunís limb). This amount (dubbed the full deflection) is twice the one predicted by Einstein in 1908(16) and in 1911(17) using Newtonís 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 Sunís gravitational field and if there is, which of the two theories mentioned above it follows. The expedition was claimed to be successful in proving Einsteinís full deflection(18,19). This test was crucial to the general approval that Einsteinís 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 Eddingtonís
influence worked for Einsteinís full displacement and against any other
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 Einsteinís 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 1íí 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 Sunís limb were drowned in the corona and could not be observed(18). Of the stars that were not drowned in the corona, Einsteinís theory predicts that k 2 Tauri should have the largest displacement, with 0.88íí. In Sobral, the displacement for that star was reported to be 1.00íí(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 1.25íí. This central spot is called the Airy disk. Since some of the results were presented with a claimed accuracy of the order of 0.01íí(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 1.25íí and/or because, as expected, the atmospheric turbulence was larger than 1.25íí, 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 teamís 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 0.01íí(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 Sunís 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 Sunís 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 Einsteinís 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 Sunís 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 Klüberís(20) article, which reviews all observations done before 1960) could easily be fitted by a straight line instead of Einsteinís deflection equation. Therefore, these data cannot prove any of Einsteinís 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 Einsteinís theory states, but sometimes deflected from the radial direction by as much as 35°! Nothing was said about that in Dysonís article(18). According to Silberstein: "If we had not the prejudice of Einsteinís 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 Einsteinís theory.
C - About Eddingtonís 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 0.01íí. In another meeting(23) (page 40), Oliver Lodge asked if it were possible to measure a deflection of 1/60íí (approximately 0.02íí) 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 Dysonís 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 Einsteinís 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 Einsteinís 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 Einsteinís 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 Einsteinís 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  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 worldís greatest scientist was based on the Mathematical Magic of Trimming
and Cooking of the eclipse data to present the illusion that Einsteinís
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 Einsteinís 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
Einsteinís 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. Einsteinís theory might
not have enjoyed the popularity it now does and a new more realistic theory
might have been found years ago.
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27. J. Munshi ² Weird but True² on the internet:
Captions for Figures
Geometrical Time Delay
Fraction of Reduction of Velocity of Light versus Distance from the Sun.
Differential Delay after Deflection of Light by the Sun.
Delay Due to the Solar Plasma
- - - - -
Paul Marmet, Ph. D. in Physics in 1960, in the field of Electron Spectroscopy, has published more than 100 papers in this subject. Officer of the Order of Canada (the highest honor given by the Canadian Government), Fellow of the Royal Society of Canada, National President of the Canadian Association of Physicists (1981-1982). He has been teaching and doing research in Physics at Laval University (1961-1984), and later at the University of Ottawa. He has spent seven years doing research at the Herzberg Institute for Astrophysics of the National Research Council of Canada (1984-1991). He is the author of: A New Non-Doppler Redshift, Physics Department, Laval University, Québec, 1980; Absurdities in Modern Physics: A Solution, Les Editions du Nordir, Ottawa, 1993; Einstein's Theory of Relativity versus Classical Mechanics, Newton Physics Books, Gloucester, 1997.
Web page: http://www.newtonphysics.on.ca/
Christine Couture, a bright Physics student,
is also an excellent pianist, and gives piano concerts.