# The cosmological constant and the precession of Mercury

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?

Edit:

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.

• 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? – probably_someone Dec 31 '16 at 15:26

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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