Snowflakes

It took me two tries to get out of my flat this morning. I really ought to get into the habit of looking out the window in the morning; too often do I open the front door, ready myself to step out, only to turn back to fetch my umbrella. The annoying thing about snow is that I can’t hear it, unlike the pitter-patter of rain.

Somehow or another I ended up looking at Wilson Bentley’s micro-photographs of snow crystals. And a question forms in my mind, “Why are they all so symmetrical?” If all snowflakes were to look alike, then perhaps the dynamics leading to the formation of snow crystal is stable, and the global convergence unto a completely symmetrical pattern would not be surprising. But not all snowflakes look alike. In fact, colloquially we speak of them as each completely different from every other. This implies that the dynamics of snow crystal growth should be at least somewhat sensitive to atmospheric conditions in a local scale (and perhaps to the nucleus that seeds the crystal) so that the seemingly random to-and-fro dance as the snowflake falls from the sky can effect different shapes and branches.

Now, much experimental evidence has gone to show that the formation of ice crystals tends to by catalyzed by impurities. Pure water can be supercooled, in normal pressure conditions, to temperatures below 273 Kelvin. But in these situations a single mite of impurity dropped into the water can cause the entire beaker to freeze over suddenly. Similarly, ice crystals in the upper atmosphere tend to form around impurities: bacterium floating in the air, dust or ash, or perhaps particles introduced artificially. So one may surmise that the fact that all 6 branches of a snowflake grows in the same way because, somehow, the eventual shape of the snowflake is already encoded in the original formation of the central nucleus. Let me try to explain why this hypothesis is not very convincing. I’ll make one a priori assumption, that the growth of a crystal structure is purely local, and not due to some long-range interaction.

To draw an analogy, consider a large group of acrobats. They are trying to bring themselves into a formation around a leader. Disallowing long-range interaction can be thought of requiring that the leader cannot shout out orders to individual troupe members. But we can allow passing of information by short-range interactions, i.e. whispering instructions to the people already in formation. So the leader stands alone at the start. Then he grabs on a few people nearby to form the nucleus. Then he tells each of the people he grabbed a set of instructions on how to grab more people, where to put them, and what instructions to pass on to those people (included in these instructions are instructions to be passed on to the even more remote layer of acrobats and so on). Then if the instructions were passed correctly, a completely ordered pattern will form. But as anyone who has played the game of telephone can testify, in these scenarios some errors will always work its way into the instructions. In the physical case of snowflakes, these are thermodynamical fluctuations. So some irregularities should happen. Now, if the instructions the leaders were trying to pass down were very short and easy to remember, the errors tend not to build up, and the formation will, for the most part, be correct. But keeping the message short has the drawback that the total number of formations one can form is fewer. In the snowflake case, one can imagine somehow each small group of molecules in the snow crystal can encode some fixed amount of information. If the encoding is very redundant (so the total number of shapes is small), then the thermodynamical fluctuations will not be likely to break the symmetries between the arms. But considering the large number of possible shapes of snowflakes, such encoding of information should be taxed to the limit, and small fluctuation (errors in the game of telephone) should be able to lead one arm to look drastically different from the others. One possible way to get around this difficulty would be to use some sort of self similarity principle. But this will suggest the snowflakes are true fractals, which they are not.

The Kodama vector field and the gravitational red-shift

I was reading a survey article by Jaramillo and Gourgoulhon on mass and angular momentum when I come across the notion of the Kodama vector field. (Actually, I first heard of it from Mihalis Dafermos last summer; I’ve just completely forgotten about it until now.) I took notice of it because of a discussion yesterday in a seminar on the red/blue-shift effect. Since this post will focus mostly on the use of the Kodama vector field in analyzing gravitational red/blue-shift, I will start by an overview of the effect as well as some philosophical caveats.

Gravitational red/blue-shift
It is already well-known in folklore that a strong gravitational well causes light emitted to lengthen in wave-length. The standard, classical (not involving general relativity too heavily) explanation goes something like this: we know that the total energy is conserved in the motion of classical mechanics. We now take a hybrid quantum-mechanical/Newtonian-corpuscular view that light is carried by particles called photons, and that the intrinsic kinetic energy of the photon is , where is Planck’s constant, and is the angular frequency of the radiation. Suppose a photon is “produced” inside a gravitational well (say, near a very heavy star, or near a black hole). Its total energy will be . Now the photon escapes the gravitational well, to a point where the gravitational potential energy is higher. Then since total energy is conserved, the kinetic energy must correspondingly decrease. Looking back at the definition of the kinetic energy of a photon, this means a reduction in the frequency, which also means a increase in wavelength, since the product of frequency and wavelength is a fixed constant, the wave-speed, or the speed of light. This is why light escaping from the vicinity of a black hole will be said to be red-shifted. Analogously, a photon falling toward a black hole will gain energy and become blue-shifted.

Now we move toward general relativity. A problem immediately arises: there is no such thing as a preferred inertial frame. Whereas it makes sense in special relativity and in Galilean mechanics to compare two distant observers as “at rest with respect to each other”, the Principle of General Covariance makes the same comparison nonsensical in general relativity. Geometrically, this is the problem of having curvature: parallel transports of a vector is now path dependent, so there is no one unique way to define a “parallel vector” at a distant point B of some known vector here at point A. As is well-known, the frequency of light can also be shifted by the Doppler effect (in the acoustic sector, this is the effect where the siren of an approaching ambulance sounds higher pitched than that of an ambulance driving away). We will describe the geometric interpretation of this effect later. If there is no canonical way to pick an observer, there is also no way to say definitively what the frequency of light observed at a point is.

Mathematics and Jargon

Each profession has its own set of special language: some, like the medical profession, rely on long and precisely defined words, often with Latin, Greek, or German origins, to describe objects and events that we do not encounter on the everyday (I doubt anyone actually found the need to use pneumonoultramicroscopicsilicovolcanoconiosis in everyday conversation [it is a lung disease caused by inhalation of metallic dust]); some, like the legal profession, attach a preferred meaning to common everyday words (in addition to special terminology), and put certain heft on syntactical and grammatical specification to say exactly what they mean.

In the natural sciences, the tendency has been toward the coinage of words to refer to new objects or ideas. My guess is that this has to do with the earliest natural philosophers taking a cue from Adam and giving a name to everything they have not seen before. In mathematics (which, by the way, while may be arguably natural by certain definitions, is not a science), on the other hand, the tendency has been toward usurping everyday words for specialist purposes.

Managing papers with Subversion

Jumping through a link from Terry’s blog, I came across this article on the Secret Blogging Seminar describing how to use Subversion to keep track of one’s papers. (Better yet, this can also be implemented for collaboration.) This is one of those ideas that upon hearing, I slapped my forehead and went “why didn’t I think of it?”

In the past several months (after moving to my current position), I’ve just used rsync to download the files from the department server to my laptop before I start work, and to upload the changes afterward. This carries with it a slight headache: if I made some changes on the work computer, and if I bring my laptop to a coffeeshop with no wireless signals, I am in somewhat of a bind. I will have to edit my file, and be very careful that once I get internet access to merge the new version on my laptop against the version at work so all the changes are kept. With svn, I can look forward now to intelligent merging of the files automatically.

Subversion is what is known as a revision control system. It was developed in the land of computer science for manage computer code, especially across collaboration. The program makes it easy to keep track of revisions of a file: it knows which version is current and what changed between it and the previous versions. So instead of having to number each version of a file by hand and keeping track of the numbers, the computer does it for you. This reduces clutter in the workspace and minimizes the chance of mistakes. Furthermore, as we all know computer code tends to contain bugs (to err is human). With a system like this in place it is much easier to correct boneheaded mistakes: one just needs to revert to a previous version. The full documentation by the system of the changes made to the files also makes it easy to track down where and when a mistake happened.

A side benefit of subversion is that the files can be reliably merged across two separate working copies. This allows me to keep a central copy of my papers on the department server, and synchronize any changes between there, my laptop at home, and any other computer I may choose to use.

There are very many different ways of using subversion. Here I’m only going to describe the way I use it. (Just took about 20 minutes for me to read the relevant documentation and implement it.) Here I assume the user has

• A personal computer running linux, with svn-client software installed.
• Access (via ssh) to a server running linux/unix, with svn software installed.

Quick set-up instructions:

1. First we need to create a repository. A repository is a directory where all the version information will be held. Log into the server via ssh (or if you have local access, just log in). Let us just give the repository a generic name, say, “WorkSVN”. Issue
   

   svnadmin create ~/WorkSVN

   This will create the repository under your home directory.

2. Next we need to create a project. A project is a directory in which you hold a collection of files you want the svn system to keep track of. Suppose you already have a bunch of TeX files sitting in a directory called “PaperDrafts” under your home directory. We’ll just import the entire directory as your project
   

   svn import ~/PaperDrafts file:///[PATH TO HOME DIRECTORY]/WorkSVN/PaperDrafts

   Notice that inside the URL line, one cannot use the tilde-expansion for home directories. To find out the path to home directory, you can “cd” to the home directory, and issue

   pwd

   This should return a path like

   /home/XXYY

   Then the URL in the command above should be

   file:///home/XXYY/WorkSVN/PaperDrafts

3. If you want a local working copy of the files, you can now first remove the PaperDrafts, and “download” it from the svn server.
   

   mv PaperDrafts PaperDrafts.bkup
   svn co file:///home/XXYY/WorkSVN/PaperDrafts ~/PaperDrafts

   The first command moves your old copy out of the way. The second command downloads, from the subversion system, all the files in the project you just created. Now double check all the files are there! Then you can clean up the back up copy with

   rm -rf PaperDrafts.bkup

4. Now, to get a copy of the files on your laptop, we have to download it from svn. Suppose your server has a server name “Unseen.uni.am”. (This should be the same name you used to ssh.) Then just go to your computer, make sure you don’t already have a folder named PaperDrafts, and issue
   

   svn co svn+ssh://XXYY@Unseen.uni.am/home/XXYY/WorkSVN/PaperDrafts ~/PaperDrafts

   where the “XXYY” before the @ symbol is taken to mean the username you used to log into your server. The “/home/XXYY” after the server name is the path to your home directory on the server. Now after entering your password, you should have a copy of the papers!

To download the files to any other computer, just repeat the last step above.

Now, suppose you made some edits, what do you do? The principal commands to know are “add, del, copy, move, update, commit”.

• If a file is modified in place, then you don’t need to do anything special: svn already know to keep track of the file.
• del, copy, move: to remove, copy, or move/rename files inside the working directory, one needs to use the svn tools instead of the native linux tools. This is so that the svn system knows that the files are to be changed. So to remove a file, issue
  

  svn del [path to file]

  to copy a file,

  svn copy [path from] [path to]

  and similarly for move.

• If you create a new file or directory, you need to let svn know that this file/directory needs to be kept track of. So you issue
  

  svn add [path to file/directory]

• When you’ve done all the edits you want. You need to synchronize the local copy with the server. To do so just issue
  

  svn commit ~/PaperDrafts

  and any changes (deletions, copies/moves, new files, modified files) will be propagated to your server.

• If you now go to another computer onto which you’ve already downloaded the TeX files before and want to update it to reflect any changes you’ve made since, you can issue
  

  svn update ~/PaperDrafts

Lastly, for commits and adds and deletes, it is often prudent to use the “-m” flag to svn to add a comment about what is being done and why you are doing it. Just in case you will look at it later.

The usage described here is very simple and primitive. For more complex operations, read the documentation for Subversion. It is available on its website.

The “Hoop Conjecture” of Kip Thorne and Spherically Symmetric Space-times

Abstract. (This being a rather long post, I feel the need to write one.) In the post I first gather some miscellaneous thoughts on what the hoop conjecture is and why it is difficult to prove in general. After this motivation, I show also how the statement becomes much easier to state and prove in spherical symmetry: the entire argument collapses to an exercise in ordinary differential equations. In particular, I demonstrate a theorem that is analogous, yet slightly different, from a recent result of Markus Khuri, using much simpler machinery.

The Hoop conjecture is a proposed criterion for when a black-hole will form under gravitational collapse. Kip Thorne, in 1972 [see Thorne, Nonspherical Gravitational Collapse: a Short Review in Magic without Magic] made the conjecture that (I paraphrase here)

Horizons form when and only when a mass gets compacted into a region whose circumference in EVERY direction is bounded by .

This conjecture, now widely under the name of “Hoop conjecture”, is deliberately vague. (This seemed to have been the trend in physics, especially in general relativity. Conjectures are often stated in such a way that half the effort spent in proving said conjectures are used to find the correct formulation of the statement itself.)

Joint Maths Meetings 2010

Last week I was in San Francisco, California, at the AMS/MAA/SIAM Joint Maths Meetings 2010. This year it was held at the Moscone Center. I was there in a position of quasi-responsibility: with Paul Allen, Michael Eichmair, Gustav Holzegel, and Jared Speck, I helped organize the AMS Special Session on Mathematical Challenges of Relativity, which is something that grew out of the Mathematical Research Communities program from last summer.

I saw many familiar faces there, though it was a shame that Michael could not make it in person. Unlike the MRC program from last year, where it was mostly focused on learning the tricks of the trade, the special session is more about presentation of new research work by young mathematicians. And I picked up quite a few things for ruminating later. Paul moderated the morning session and I did the afternoon (I hope I didn’t butcher the names of too many people!). Rather unfortunately there was still the usual hyperbolic versus elliptic split in the community.

The study of mathematical general relativity branched into several rather specialised disciplines in the past century. On the physics side there are of course studies of matter models and cosmology and whatnot, but even on the mathematical side there are deep divides. For starters there are numerical simulations: for a long while the field is largely ridiculed by more theory minded mathematicians, since a stable and robust method of simulation was lacking–when given the initial data for a static black hole, some of the numerical recipes will evolve it into all sorts of craziness. But recently (in the past 5 or 6 years) a lot of progress has been made, and we can speak of bona fide predictions based on numerical data now. But putting the theory versus numerics split aside, there is still the big divide between the hyperbolic/Lorentzian camp and the elliptic/Riemannian camp. One of the interesting features of general relativity is that, due to the strong coupling between gravitational effects and matter models, not all configurations of matter distribution and space-time gravitational field are admissible. There are some constraints to what an instantaneous snapshot of the space-time may look like. So unlike the study of many other equations from physics, where “now” can be freely prescribed and the only interest is in finding out about the “future”, general relativity has two aspects: the evolutionary problem of predicting what the “future” may be given information about “now”, and the constraint problem of what “now” can possibly be. The partial differential equations of the former are of hyperbolic type, while for the latter they are elliptic; the geometry for the former is Lorentzian, while for the latter is Riemannian.

As it so happens, in many ways the development of elliptic partial differential equations (and by extension Riemannian geometry) is eons ahead of that of hyperbolic equations. There exist more robust methods and more diverse tools. This unfortunately lead to the two branches becoming disjoint; and some of the current open questions asked by the Riemannian geometers, while many generations ago may have been inspired by a question in general relativity, have now only tenuous relations to the physics. (This is not to say that those problems are not interesting; just that they are not really motivated by physics.) This fact was somewhat reflected in the program for our special session. (As a side note, this was one of the major failures for the MRC program. An unstated goal I think was for the young mathematicians educated in the two different disciplines to intermingle and learn about each others’ fields. In actual implementation, the two groups more-or-less self-segregated.)

Now a word about the organization of the conference: our approach was to focus on making a more compelling narrative, and hence we grouped talks on similar topics together. We started the morning with two talks on decay estimates for wave equations on Schwarzschild background, one talk about wave equation on spaces with cosmological constant, and then two talks about the breakdown criterion in general relativity … you get the idea. The main advantage I found was that this keeps the focus of the audience. Of course, a downside is that shifts in the numbers of audience members is more noticeable: some people just stayed for the “block” that interested them. Perhaps we could’ve also done the grouping while mixing the talks so that there were some elliptic talks and some hyperbolic talks in each session, rather than having what ended up more as a hyperbolic morning plus an elliptic afternoon. This is something I’ll have to think about if I organize a conference again in the future.

An unfortunate thing about the JMM (and many AMS conferences actually) is that the non-plenary talks are restricted to something like 20 – 30 minutes. While it is not a problem in certain fields, for disciplines in which analysis plays a heavy role (I’m sure it is also the case for other fields, it is just that I am more familiar with analysis), such as differential geometry, partial differential equations, etc. the time limit really makes the presentation a short announcement of result plus an advertisement of why we should care: something like the 5 minute talks in other sciences. (Incidentally, the general contributed papers sessions limit their talks to ten minutes. I learned at JMM that it is not possible to be fashionably late to one of those talks. A previous talk ended a couple minutes late, I had a bit of difficulty finding where the next talk I wanted to hear was happening, plus a lack of completely synchronized watches [nothing to do with relativity, mind you] resulted in my walking into the talk when the audience started applauding.)

In a separate post I will write about the interesting things I learned at the JMM while listening to talks not in my field. (In particular, the Friday morning session on mathematical origami was spectacular.)

A little Hilbert space problem

First let us consider the following question on a finite dimensional vector space. Let be a -dimensional Hermitian-product space. Let be an orthonormal basis for . Let be the linear operator defined by when , and . Does there exist any non-trivial vector such that and ?

The answer, in this case, is no. Present where are complex numbers. Let be the smallest index such that and . Similarly let be the largest non-vanishing index. If , then is a multiple of a standard basis element, and so is trivial. So assume . Now, by the requirement , we see that , which contradicts our assumption that are the minimum and maximum non-vanishing indices. In this proof, we used crucially that is finite dimensional, so that a largest element can exist.

Now, onto the real question

Question
Take the complex Hilbert space , i.e. the set of all complex sequences satisfying . Let latex T$ be the right shift operator: and . Then is an orthonormal basis of , and we have . Does there exist non-trivial elements of for which hold?

The answer is yes by the way.

Newton-Cartan part 3: gravitating particles

As a simple example of a physical theory on a Galilean manifold, let us consider the physics of a collection of massive particles that do not interact except for their gravitational interaction. In other words, let us consider a collisionless kinetic theory coupled to Newtonian gravity.

Vlasov system
The Vlasov system is a transport equation describing the free flow of collisionless particles. Let be a manifold with an affine connection that represents the spacetime. We postulate Newton’s first law:

Physical assumption 1
The motion of a free particle is geodesic.

Therefore the motion of a free particle is described by the following system of equations: let denote proper time as experienced by the particle, and the world-line of the particle (its spacetime trajectory) parametrized by , then we have the hyperbolic system of equations.

Equation 2

and

Parallel volume forms

In the previous post on Newton-Cartan theory and Galilean geometry, I showed that the Galilean manifolds admit a preferred volume form. After some discussion with my old officemate Pin Yu, and a bit of digging on the internet, I found that this notion of a parallel volume form is a rather well-developed one in classical differential and affine geometry.

Newton-Cartan, Part 2

After writing up the previous post on Newton-Cartan theory, I came to realize that it is actually a very nice exercise for myself to dig into the geometry more. So here goes a bit more on the implications of the Galilean geometry and the Newton-Cartan theory.