Updated: Oct 31, 2022
The so-called red-shift has stood as a classical pillar of general relativity. Einstein deduced this in terms of a red-shift in frequency due to a change in the gravitational potential.
According to Sexl & Sexl, White Dwarfs and Black Holes, p 11:
"In 1911 Einstein for the first time theoretically predicted this shift in frequencies. Since then, it has been experimentally tested many time by measuring the red shift of spectral lines of the light emitted from the sun and from varying dense stars."
On closer inspection this does not appear to be at all true! In his "Report on the General Theory of Relativity" (1920) Eddington references the paper by Charles E St John, entitled, "The principle of the generalized relativity of the displacement of Fraunhofer lines toward the red", that appeared in the Astrophysical Journal, 49 (1917) 249-265.
In that paper St John tells a completely different story. Not only is the above conclusion contradicted, the origin of the effect, as a Doppler effect, is also shown to be a sham. St John gets the putative origin of the shift in lines to the red correct when he says
"the effect depends upon the difference in the gravitational potential between the gravitational field in which the emitting center is located and the terrestrial field where the radiation is received and measured."
He then has to correct Einstein's deduction of the phenomenon as a Doppler effect into on in which "the gravitational effect is proportional to wave-length." It is thus apparent that if such an effect exists "the displacement due to it should be more evident for lines of great wave-length."
The result that Einstein gets in terms of a relative frequency shift,
(n_0-n)/n_0 = 2 X 10^(-8),
must be transformed into one of wavelength. This St John does by "postulating the constant space-velocity of light" to get the shift in wavelength
lambda -lambda_0 = 2 X 10^(-6) lambda.
Unlike Einstein's other prediction which were too small to be measured at the time, the "calculated displacement of two parts in a million is well within the possibilities of observation, being 0.010 A for a wavelength of 5000."
Again if the result is to be transformed into a Doppler effect, it would "at the surface of the sun" correspond to a "displacement equivalent to the Doppler displacement produced by a radial velocity of 0.634 km per sec." This last statement is referenced to Eddington's paper in Monthly Notices 77 (1917) 380.
Since the red shift corresponds to a shift in wavelength due to the difference in gravitational potentials between the sun and earth, it should have been apparent that no radial velocities would have been measured. Even Schwarzschild's measurements, to which St John only has the abstract, confirm a downward velocity of 0.2 km per second "and that, if this downward movement is taken into account, there remain only slight displacements, which cannot be considered favorable to the Einstein theory." The downward velocity would, if it existed, be due to the displacement of the line between the center and limb of the sun that arises from convection currents. This corresponds to a Doppler effect with a velocity of 0.12 km/sec, which is a far cry from that predicted by Einstein of 0.634 km/sec.
St John concludes that
"within the limits of error there is no evidence in these observation that a displacement to longer wave-length, either at the center or at the limb of the sun, of the order of 0.008 A, as required by the principle of relativity."
He also claims that there is a limb-effect not due to the motion, but which involves pressure level. Now faced with such a negative result, what does Eddington do? He admits that the Einstein effect (which is really not his but Soldner's who "predicted" it a century earlier) "has been disproved by observations of the solar spectrum" (reference to St John's paper). But instead of dumping the theory he goes on to "measure" the shift is twice what Einstein predicted in 1911in the solar eclipse expedition of 1919. In the words of Sexl & Sexl,
"Einstein himself committed a well-known error by attempting to calculate the deflection without using his full theory. The answer he gave in 1911 turned out to be too small by a factor of 2. He had made used of Newton's theory, which, however is only applicable to speeds that are small compared to the speed of light."
Nothing could be farther from the truth! As we have seen the reddening of spectral lines is not a Doppler effect and velocities don't enter at all. The fact that Einstein doubles his previous result and claims a radial velocity of 1.268 km/sec should have raised eye-brows in view of St John's negative result. And who do we have to thank for this? No one else but Eddington!
Just listen to what he says:
"The difficulties of the test are so great that we may perhaps suspend judgment; but it would be idle to deny the seriousness of this apparent break-down of Einstein's theory. We shall therefore consider the phenomenon from a more elementary point of view."
This, supposedly, applies to St John's spectroscopic measurements at Mount Wilson Solar Observatory, but not to his solar eclipse measurements which have gone down in history as
the day (May 29, 1919) that "astronomer Sir Arthur Eddington verified Einstein's general theory of relativity, by observing how stars near the sun were displaced from their normal positions."
The rationale--or lack of it--gives insight into Eddington's choice of championing Einstein's theory even in the face of its of correspondence to reality. He claims that the red shift would occur in a uniform gravitational field. He compares two similar atoms separated by a distance "a" measured parallel to the field. The observer, together with his atoms are falling in a uniform field with acceleration g. The observer is at atom 1 which is separated from atom 2 by a distance "a". The impulse of radiation emitted by 2 will reach the observer at time t=a/c, where c is the speed of light, "by which time [the observer] O will have acquired a velocity ga/c relative to the unaccelerated axes. He will, therefore, correct his observation of the period of [atom 2] for the Doppler effect of this relative velocity and deduce a true period T/(1-ga).....Since ga is the difference in potential between 1 and 2, this agrees with"
t'/t = 1+ m/R = 1.00000212
where t' refers to time in a terrestrial laboratory and t to "an atom vibrating in the photosphere of the sun". What kind of clock would measure such a time, if not a vibrational frequency?
Frequencies are not involved and uniform acceleration is not a uniform velocity no matter what time "a" we multiply it with.
Eddington's conclusion that "If the displacement of solar lines were confirmed, it would be the first experimental evidence that relativity holds for quantum phenomena" is totally unjustified, and amounts to wishful thing. But, sadly, this is not how history interpreted it.
A recent book, Gravitation and Spacetime, 3rd ed (2013) by Ohanian and Ruffini describe the experimental measurements as
"The experimental results by these expeditions were judged to be in reasonable agreement with prediction, to within 10%, but the analysis of the photographs was somewhat arbitrary and questionable.
Since then, observations have been carried out during a dozen other eclipses. Unfortunately, all the eclipse observation are afflicted with large experimental errors, and despite the best efforts of astronomers over a span of half a century, the accuracy of these observations have barely improved over that of the first observations. All we can say is that the observed deflection is approximately 2 arcsecs, but the precise test of the predicted deflection is not possible by optical methods."
And this is criticisms is what Eddington lodged against the spectrophotometric measurements of St John!