Ricci solitons

22/01/2009

These days I am looking at all this area of mathematical research born with Richard Hamilton and put at maturity with the works of Grisha Perelman. As all of you surely know the conclusion was that the Thurston conjecture, implying Poincare’ conjecture, is a theorem. These results present the shocking aspect of a deep truth waiting for an understanding by physicists and, I think that this comes out unexpectedly, statisticians (do you know Fischer information matrix and Cramer-Rao bound?).

One of the most shocking concept mathematicians introduced working with Ricci flow is a Ricci soliton. I will use some mathematics to explain this. A Ricci flow is given by

$\frac{\partial g_{ik}}{\partial t}= -2R_{ik}$

a Ricci soliton is a metric solving the equation

$R_{ik}-L_X g_{ik} = \Lambda g_{ik}$

where I have used an awkward notation for the Lie derivative along a field X but if this field is a scalar than one has a gradient soliton. I think that all of you will recognize these equations that for a Lorentzian metric are just Einstein equations in vacuum with a cosmological constant! Now, I have found a beautiful paper about all this question on arxiv (see here). This paper gives the first meaningful application to physics of this striking concept. Ricci solitons are resembling a kind of behavior of the metric under the flow that can be expanding, collapsing or static depending on the cosmological constant.

As time goes by we learn something deeper about Einstein equations. Their very nature seems rooted in quite recent concepts coming from differential geometry and it is my personal view that whatever quantum gravity theory we will formulate, these are the questions we have to cope with.

Ted Jacobson and quantum gravity

15/01/2009

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

$\frac{\partial g_{ik}}{\partial t}=-2R_{ik}$

being $R_{ik}$ 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 $R_{ik}=0$ 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

$\frac{\partial g}{\partial t}=\triangle g$

where $\triangle$ 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 $\frac{\partial g}{\partial t}=0$ 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?