Abstract
One of the three consequences of Einstein’s theory of general relativity was the curvature of light passing near a massive body. In 1911, he published a first value of the angle of deflection of light, then a second value in 1915, equal twice the first. In the early 1920s, when he received the Nobel Prize in Physics, a violent controversy broke out over this result. It was then disclosed that the first value he had obtained in 1911 had been calculated more than a century before by a German astronomer named Johann von Soldner. The aim of this article is therefore to compare the methods used by Soldner and then by Einstein leading to this first value and to explain the importance of the doubling of this value in the framework of Einstein’s theory of general relativity. Such a consequence of this theory lies at the intersection of several scientific fields such as Mathematics, Physics, Astronomy and Philosophy.
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Notes
The value \(\kappa = 8 \pi K / c^2\) where \(K = 6.67 \times 10^{-11} m^3 kg^{-1} s^{-2}\) is the Cavendish’s constant.
It might also be interesting to note that the original Annalen paper by Einstein of 1916 (Einstein 1916, pp. 819–822) has a factor of 2 error in its eq. (74), going back to a mistake in its eq. (70a). This misprint was corrected in the reprint that was included in the collection of papers published as Das Relativitätsprinzip, see Collected Papers of Albert Einstein (CPAE), Vol. 6 (Einstein et al. 1996, pp. 334–337) in German and (Einstein et al. 1997, pp. 196–199) in English. See also Collected Papers of Albert Einstein (CPAE), Vol. 10 Einstein to Carl Runge, 8.11.20, Doc. 195 (Einstein et al. 2006, pp. 306–307).
E. Gehrcke (1878–1960) was a German experimental physicist, Director of the Imperial Institute of Berlin. He was a Privatdozent at the Friedrich-Wilhelms-Universität from 1904 to 1921 and an außerordentlicher Professor (extraordinarius professor) from 1921 to 1946.
Oscar Edward Westin (1848-1930) was a Swedish engineer, professor of mechanical engineering at the Royal Institute of Technology in Stockholm.
See The New York Times, April 13, 1923. This event is also mentioned in Philipp Frank (Frank 1947, p. 202 and next).
See Sect. 3.
We will see in Sect. 3 that this statement is partially inaccurate.
“Thus, despite the aid of the principle of equivalence, one gets the old false classic value again! How is that possible?”
“Although we observe a slowing down of the earth from the earth, we will measure the same frequencies and the same speed of light with the same clocks on the sun as on the earth, because the clocks are slowed down to the same extent as the natural processes! Einstein achieved this logically correct view of things only through the deeper understanding that came with the general theory of relativity (after 1915).”
See also Eisenstaedt (1991).
Stefanini quotes Hans-Jüurgen Treder and Jackisch (1981).
“If one were to investigate by means of the given formula how much the moon would deviate a light ray when it goes by the moon and comes to the earth, then one must, after substituting the corresponding magnitudes and taking the radius of the moon for unity, double the value found through the formula, because a light ray, which goes by the moon and comes to the earth describes two arms of a hyperbola (Jaki 1978).”
“For this purpose we shall investigate the parallax of the moon, from the experiments of the length of a pendulum vibrating in a second, and shall compare it with astronomical observations. On the parallel on which the square of the sine of the latitude is 1/3, the space through which gravity causes a heavy body to descend in a second of time, is, according to the observations of the length of the pendulum, equal to \(3^{m}, 65648\), as we shall see in the third book; we have chosen this parallel, because the attraction of the earth on the corresponding points of its surface, when compared with that at the distance of the moon, is very nearly as the mass divided by the square of the distance from the centre of gravity of the earth. On this parallel, the force of gravity is less than that depending on the attraction of the earth, by two thirds of the centrifugal force, corresponding to the rotatory motion at the equator ; this force is \(\frac{1}{288}\) of gravity ; we must therefore increase the preceding space by its \(\frac{1}{432}\) part, to obtain the whole space arising from the attraction of the earth, which on this parallel is equal to the mass divided by the square of the radius of the earth : we shall therefore have \(3^{m}, 66394\) for this space (Bowditch 1829, p. 251).”
“Soldner further calculates the deflection for the earth, moon, and sun using the formula obtained and finds it very small. For the sun \(\omega = 0''.84\). (p. 170) (while in fact, according to his formula \(\omega /2 = 0''.84\), which seems to be consistent with experience, as far as the same goes today, such as it was explained in the preliminary remark).”
See also Brown (2017).
In his 1911 and 1915 articles Einstein also made use of first-order approximations as well as Soldner (1921) in 1921.
See Brown (2017).
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Author would like to thank the reviewer who considerably improved this work with his questions and remarks and also Pr. Franck Jovanovic for his helpful advices.
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Ginoux, JM. Albert Einstein and the Doubling of the Deflection of Light. Found Sci 27, 829–850 (2022). https://doi.org/10.1007/s10699-021-09783-4
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DOI: https://doi.org/10.1007/s10699-021-09783-4