In the beginning of the last century, the Nobel physics subcommittee – made up of experimentalists detached from the dramatic developments of theoretical physics on the continent – was surprized both by the quantum mechanics and the theory of relativity, but the biggest problems unfolded with Einstein’s concept of space and time, concepts which was considered to lie within the regime of philosophy. The committee´s struggle with the concepts has been described by the philosopher pf science professor Aant Elzinga (University of Gothenburg, Sweden) who was given access to the archives of the Nobel committee (see A. Elzinga, Einstein´s Nobel prize, a glimpse behind closed doors).

One of the complaints in a special report was concerning the perihelion change of the planet Mercury. The report claims that the theory contained a constant – and that this constant was set to zero (and that therefore the predictions for the planet was not valid or at least flawed in some way).

My question is simply: could the constant referred to the report possibly be Einstein´s cosmological constant?


I have a new source: Gårding L., Mathematics and Mathematicians, American Mathematical Soc., 1998.

He says:

“Gullstrand examines Einstein's equations for the movements of bodies and his explanation of the movement of the perihelion of the planet Mercury. His criticism was that Einstein’s equations for this phenomenon permit several solutions. As remarked by Oseen in a subsequent paper, Gullstrand did not observe that the choice of coordinates must be adjusted to the observer and that this gives the correct result.”

It looks more like a simple error, missing the adjustment of a constant.

  • $\begingroup$ This source appears to be a book, which is behind a paywall for most users. Do you have a link to an accessible version of a source for what you're saying? $\endgroup$ Dec 31, 2016 at 15:26

2 Answers 2


Let's go back to Einstein's original work, courtesy of Wikisource, where he reports his calculation of the perihelion shift of Mercury, purely under General Relativity:


I don't see the constant they're referring to anywhere in here. It's possible that they're referring to some other work of his; if so, their comments do not seem to be relevant. Admittedly, I haven't read up much on the history of science at this point, so there may have been some politics going on (as you seem to allude to). My point is, though, that their claim of an unnecessary constant does not seem to hold up under scrutiny.

  • $\begingroup$ Thanks. I believe it may be a case where the discussion on the cosmological constant was expected to occur everywhere in Einstein´s work and could be used to discredit him. $\endgroup$ Dec 31, 2016 at 19:17
  • $\begingroup$ Ah, so it was probably devious flimflammery after all... $\endgroup$ Dec 31, 2016 at 19:38
  • $\begingroup$ @probably_someone There is a reprint of Einstein's paper on the specific subject, with a translation and commentary, at gsjournal.net/old/eeuro/vankov.pdf. It also contains a letter (in German, no translation) from Scwarzchild to Einstein. There would certainly appear to be some issues with Einstein's math, and the physical interpretation of "constants" in the approximate solutions to the equations, independent of the "cosmological constant"... $\endgroup$
    – alephzero
    Dec 31, 2016 at 21:04
  • $\begingroup$ ... Incidentally, I remember this example from a course on classical mechanics and special relativity, as a math student in about 1970 - so even if Einstein solved the problem using the apparatus of GR, that is not actually necessary to get the same result. $\endgroup$
    – alephzero
    Dec 31, 2016 at 21:06
  • $\begingroup$ That's really strange. It seems that if GR accounts fully for the anomaly, then any additional corrections must be wrong. Either that, or the theories from which they arise are just GR again in some way. $\endgroup$ Jan 1, 2017 at 0:45

The question is correct. Einstein obtained the correct result for the Mercury perihelion shift in 1915 - before the cosmological constant. The Constant was introduced to GR two years later in 1917. Nowadays the interest to cosmological constant $\Lambda$ increased because of the fact of accelerated expansion of Universe. The influence of the cosmological term to the perihelion precession of a planet can be estimated using the perturbing potential. Basically, the additional potential due to $\Lambda$ is $$\Delta V=-\frac{\Lambda}{6}c^2 r^2\tag{1}$$ leading to the additional repulsive radial force as $$\Delta F=m \frac{\Lambda}{3}c^2 r\tag{2}$$ where $m$ is mass of a planet. One can write it as an addition to the celestial equation of motion to obtain the result for the planet's perihelion precession. The resulting additional precession per revolution due to the cosmological constant can be estimated as $$\theta=\frac{\pi \Lambda c^2 L^3}{G M}\tag{3}$$ where $L$ is semilatus rectum of the planet's orbit. The expression is given by G. Adkins "Orbital precession due to central-force perturbations" [https://arxiv.org/abs/gr-qc/0702015] - but it can be also deduced via perturbing force (2) using a similar approach. Knowing the Solar mass $M$ and estimated value for $\Lambda$ one can see that the impact of this term is rather small comparing to GR correction without the cosmological constant. My rough estimation gives somewhere around $10^{-10}$ of the GR precession which is rather small to have any real influence on planetary motion in the solar system.


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