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.
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!
