Smashed by the cosmological constant

This post is an English rendering of an old post (May, 2019)  from my now abandoned Polish blog.

I have been interested in gravity since childhood. Then, as a student, I puzzled over Einstein's field equations. I shared my thoughts with my colleagues - one of them even immortalized it in a photo.

Einstein's equations of gravity

Then my interests became a bit more serious. I received a German Humboldt scholarship, during the scholarship I stayed in a hotel for scholarship holders, it was Europa Kolleg,

Europa-Kolleg Hamburg

near the DESY (Deutsches Elektronen-Synchrotron) research center in Hamburg, where I worked.

I was working on advanced gravity. In the hotel, in the same corridor, a couple of doors away, lived another grantee - a Russian. Since I knew Russian, every now and then he would invite me to his room for a discussion. On the table he would put a bottle of vodka, a loaf of bread, and we would start the discussion. The end of the discussion I usually did not remember, as well as the return to my room. I only remember that I was returning, as it were, against the laws of gravity and certainly not along the shortest (geodesic) line - as Einstein commanded.

Then I dealt with various things. For example, the basics of quantum mechanics. I wrote one book on theories of gravity and one on quantum mechanics and quantum fractals. Both books are apparently ahead of the era - no one reads them. Maybe in a hundred years.

Recently, somehow, I went from Clifford's algebras to gravity. I started thinking about antigravity and that's how I came across "bimetric theories." I began to correspond with authors of papers writing about these theories and negative masses. I wrote a short note criticizing the errors I noticed in their works. The result is that one of these authors stopped talking to me because I did not defend him in a dispute over priority with the other. However, I still have good contact with the other author. This one drew my attention to yet another theory of gravity - which seems to be becoming more and more fashionable lately . It is called unimodular gravity. This phenomenon escaped my attention, so I began to study what it was about. And I found out that it goes to the problem of the cosmological constant.

Nobel Prize-winning physicist Steven Weinberg wrote a nice review of "The cosmological constant problem." Weinberg's paper is from 1988, but is still cited as a classic. You can download it from here.

There is a small problem with this cosmological constant. We have two great theories, we (we physicists, and we humanity) are very proud of them. These are the General Theory of Relativity and Quantum Mechanics. We like to highlight their great successes. Time and again we read that quantum mechanics may be strange, may not really explain anything, but if we take the pragmatic attitude of "shut your mouth and get down to the calculus" - then quantum mechanics is never but never wrong. Similarly, the General Theory of Relativity. It predicted, as it should, the strange motion of Mercury, it predicted, as it should, the deflection of light rays by the Sun, it predicted various lensings and such, it predicted black holes, and recently a photograph of such one was even in the paper. So what is this little problem? This is what Weinberg wrote about.

When we combine the powers of these two theories together, we can calculate from here the value of the cosmological constant responsible for the observed course of expansion of our Universe. And we get the number as needed. This number obtained by using our best theories, of which we are so proud, we compared with observations. It turns out that it doesn't quite agree.

Perhaps it is for want of other crises to worry about that interest is increasingly centered on one veritable crisis: the theoretical expectations for the cosmological constant exceed observational limits by some 120 orders of magnitude.

The theory comes out with a number 10 to the power 120 times too large. How many times too big? By 10 times? 120 times? No, 1 with 120 zeros times too big. Weinberg writes that we have something to worry about. We have a CRISIS.

One of the ideas to get out of this crisis is precisely unimodular gravity. A good review paper came out not too long ago (2011), authors are Ellis and co, title: "On the trace-free Einstein equations as a viable alternative to general relativity".

That's what the newspapers say, that it's supposedly like in the picture, but what is it really like?

From this paper we learn, among other things, that from combining the forces of ordinary quantum theory and Einstein's theory of gravity, it follows that our Universe should have a diameter of 31 km:

"... one can easily compute that R = 31 km [17]. So this value drastically affects the solar system, since there would be no solar system.

It seems prudent to look for a way out."

 I'll just mention that the authors seem to show a bit of black humor here.

To lift your spirits, let me briefly say what the idea of unimodular gravity is all about. Normally, gravity is described by a 4x4 symmetric matrix of gravitational potentials - the space-time metric. These are the dynamical variables of the ordinary theory of gravity. Unimodular gravity arises when we impose constraints, i.e., when we find that not all these ten functions are independent functions. Constraints are the imposition of a condition on the determinant of a matrix, that it should have a predetermined value and that's it. For this "predetermined value" can be taken (in the chosen system of units) the value of 1 (variations then have trace zero). One , odin, uno - hence also the name "unimodular". When this is done, the cosmological constant appears only as an integration constant, not predicted by the theory, and then we can take its value as we are comfortable. In boxing, this is "evading the blow". It has its own value. Even if we lose the fight, it's only a matter of time and not right now.

It is not easy to be a physicist. In order to survive one needs black humor and faddishness. One such fad in our house is daily gymnastics, a combination of Chinese, Tibetan and Western exercises. We have our own repertoire. The picture below shows how we exercise. I myself am lying down and hiding my face in the grass on the top left. It's because of this cosmological constant.

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That was in May 2019. Today I am coming back to the idea of unimodular gravity. And unstable gravity waves. Another  nice review of the subject can be found here: Unimodular conformal and projective relativity, Kaća Bradonjić, John Stachel

And here is  one of my own old attempts at proposing a new theory of gravity based on a conformal structure rather than on Riemannian metric: A Note on Conformal Field Equations. Int. J. Theor. Phys. 18, (1979) p. 107-112.

I am going to return to this subject again and again.

P.S.1. Thursday Feb. 2. 2023, 12:40: As I wrote above, it is not easy to be a physicist. This very morning I was subjected to the direct attack - the smash by an unstable gravity wave (materialized as a variable g heavy hammer). Here is the result:

This event cause a number of secondary effects. The end result being: we won the war. The office chair has been fixed. And, at the end of the day, results is all that counts.

P.S.2. My present task: t will understand every line in the paper W. Kopczyński and L.S. Woronowicz, A geometrical approach to the twistor formalism, Rep. Math. Phys. Vol 2, pp. 35-51 (1971). It is a paper of an indescribable beauty!!! 

And I will understand it at least as deep as the authors of this true pearl understood it!