There are some days when concepts are there running round and round in my head. I have taken a look at the Poincare’ conjecture and was really impressed by the idea of the Ricci’s flow. People with some background in mathematics should read this paper that contains a 493 pages long discussion of the Perelman proof and gives all technical details about that and the mathematics behind Ricci’s flow. If you have a manifold endowed with a metric g then Ricci’s flow satisfies the equation

being the Ricci tensor and t is taken to be time for convention. People knowing differential geometry should be accustomed with the fact that a flat manifold is not given by taking the Ricci tensor to be zero, rather is the Riemann tensor that should be null. But Einstein equations in vacuum are given by whose most known exact solution is Schwarschild solution. So, what has the Ricci’s flow so shocking to interest physicists?

Consider a two dimensional manifold that has only conformal metrics. In this case the Ricci’s flow takes a very simple form

where is the Laplace-Beltrami operator. This is a Fokker-Planck equation or, if you prefer, the heat equation. Fokker-Planck equations enter into statistical physics to describe a system approaching equilibrium and are widely discussed in the study of Brownian motion. So, Einstein equations seem to be strongly related to some kind of statistical equilibrium given by the solution of a Fokker-Planck like equation taking and, in some way, a deep relation seems to exist between thermodynamics and Einstein equations .

Indeed Einstein equations are an equation of state! This striking result has been obtained by Ted Jacobson. I point out to you a couple of papers by him where this result is given here and here. This result has the smell of a deep truth as also happens for the Bekenstein-Hawking entropy of a black hole. The next question should be what is the partition function producing such an equation of state? Here enters the question of quantum gravity in all its glory.

So, an equilibrium solution of an heat equation produces Einstein equations as seen from the Ricci’s flow. Does it exist in physics a fundamental model producing a Ricci’s flow? The answer is a resounding yes and this is the non-linear sigma model. This result was firstly obtained by Daniel Friedan in a classical paper that was the result of his PhD work. You can get a copy of the PhD thesis at his homepage. Ricci’s flow appears as a renormalization group equation in the quantum theory of the non-linear sigma model with energy in place of time and the link with thermodynamics and equations of state does not seem so evident. This result lies at the foundations of string theory.

Indeed, one can distinguish between a critical string and a non-critical string. The former corresponds to a non-linear sigma model in 26 dimensions granting a consistent quantum field theory. The latter is under study yet but *il va sans dire* that the greatest success went to the critical string. So, we can see that if we want to understand the heat operator describing Ricci’s flow in physics we have to buy string theory at present.

Is this an unescapable conclusion? We have not yet an answer to this question. Ricci’s flow seems to be really fundamental to understand quantum gravity as it represents a typical equation of a system moving toward equilibrium in quest for the identification of microstates. Fundamental results from Bekenstein, Hawking and Jacobson prove without doubt that things stay this way, that is, there is a more fundamental theory underlying general relativity that should have a similar link as mechanical statistics has with thermodynamics. So, what are quanta of space-time?

Dear Marco.

In your concern, space time quanta are the equivalent of atoms in Boltzmann statistics. Did I understand well?

Right.

Marco

Marco,

I’m sure your idea is very interesting. In a more general approach, it could be needed to have a more complex metric in order to take into account quantum phenomena. Maybe space – time is not enough. Is in this approach still valid the idea that in our universe the Weil tensor is null? If you put the cosmological term in Einstein equation, this means that the sistem is not at equilibrium, i believe.

Dear hermann,

Thank you for your comments. I agree with you that maybe something more general that our four dimensional manifold we are accustomed to must be considered. Some other proposals exist besides strings and loops and some approaches does seem rather successful as the one of Renate Loll (see http://www.phys.uu.nl/~loll/Web/title/title.html). It seems that the underlying substrate to space-time is 2-dimensional and so neither strings nor loops can be ruled out. This explains the success of the non-linear sigma model and why both strings and loops appears as viable options.

Jacobson obtains Einstein equations with the cosmological constant. Non-equilibrium implies higher order terms as you can see in one of his papers I cited in the post.

Marco

Thank you for your answer. There is a very simple, but instructive link to Ricci flow and heat equation in: http://www.claymath.org/programs/summer_school/2005/notes/chow/clayss02.pdf

Dear hermann,

Thank you for the link. I wonder if, in the heat equation for Ricci, Weyl plays some role.

Marco

Dear Marco.

Maybe it would be interesting for you to read the following little note of an italian mathematician.

cvgmt.sns.it/HomePages/cm/ricci/notes/Weyl.ps

Bye

Hermann

Dear hermann,

Thanks for the paper. Very nice. Let me put here the homepage of this Italian mathematician, Carlo Mantegazza, that is doing a lot of interesting work

http://cvgmt.sns.it/HomePages/cm/

It is worthwhile to notice that things are moving on very fast since the cornerstone works of Perelman. This convinces me once more that this mathematical theory should have a serious impact on studies in quantum gravity.

Ciao,

Marco

It is an amazing post (and references also) Marco!

Please, could you give me your oppinion about this:

http://lanl.arxiv.org/abs/gr-qc/0407048v1

it seems to bring some interesting insight about GR-EM unification… but I am not sure.

Thanks!

Rafael

Dear Rafael,

Thank you for appreciating my posts. The paper you cite starts with a mathematical choice that can undergo strong criticisms and builds all on this. As you can see, this paper did not go published. This in turn is not a disproof of the author’s arguments but just a confirmation of the fact that the community may find hard to digest a questionable starting point.

Marco

Hi Marco,

I am interested in reading that book about Ricci Flow and Perelman proof since I got this article:

http://arxiv.org/abs/0810.2356v1

A mechanics for the Ricci flow

Authors: S. Abraham, P. Fernandez de Cordoba, J.M. Isidro, J.L.G. Santander

(Submitted on 14 Oct 2008 (v1), last revised 17 Oct 2008 (this version, v2))

Abstract: We construct the classical mechanics associated with a conformally flat Riemannian metric on a compact, n-dimensional manifold without boundary. The corresponding gradient Ricci flow equation turns out to equal the time-dependent Hamilton-Jacobi equation of the mechanics so defined.

The most strinking stuff it is that GR naturaly arises from EM and QM, when certain conditions are obeyed in order to use Ricci flow.

See, for example, what he concludes (p.6).:

“Thus, as announced, on a compact, conformally flat Riemannian configuration space without boundary, Einstein–Hilbert gravity arises from Schroedinger quantum mechanics, from Perelman’s functional for the Ricci flow, and from the Coulomb functional.”

Cheers,

Daniel

[…] ago I published a post about Ted Jacobson and his deep understanding of general relativity (see here). Jacobson proved in 1995 that Einstein equations can be derived from thermodynamic arguments as […]