Markus Keel, a senpai of mine (who left over 10 years ago), is back in Princeton for the week teaching Nonlinear Wave Equations and stuff for the program "Analysis and Geometry", an NSF Research Training Group funded summer school for advanced undergraduates and beginning graduate students. I have to good fortune to be his TA for the class.
Today I finally met him, at about half-hour prior to his first lecture (shows how coördinated we are). He is a great guy, and tall (geez, seems like all Caucasian mathematicians in my field are much taller than I am [I know I'm short by western standards, but not that short]; Alex Ionescu has about 8 inches on me, so does my advisor, and Keel is about a foot over my head. The only exception seems to be Igor Rodnianski. I heard Sigmund Selberg is visiting next semestre, wonder how he will add to the statistics). And after discussing a bit about what the course would consists of, we parted ways and I went on to gather things for the class. (Thank god we use 100% recycled paper for the copy machine in Fine hall, else I'd feel so guilty: Just this morning for one of the hand-outs I had to make 27 copies of a 50-page document; with double spacing that comes out to 675 sheets! And last week there were two hand-outs that comes to about 500 pages each; with the homework sets and solutions and what nots I'd hate to think how many trees I am personally responsible for.)
After the stack of paper was distributed to the students, Keel went on to give the most awesome first lecture I've ever heard to a math class. He titled it:
From Black Cats to Black Holes: A Biased Look at Nonlinear Wave Equations
The presentation opened with a quote from Charles Darwin:
"A mathematician is a blind man in a dark room looking for a black cat which isn't there."To which he promised a retort by the end. There were a lot of interesting tid-bits scattered throughout the slides. He started by examining what a nonlinear wave equation is ("a p.d.e. [which he defined] ... whose solutions look wavy") and what do we want to find out about them (initial value problems, long time behaviour, asymptotics, etc.) To illustrate why he is taking a biased look (because the field is soooooo big) he showed a comic illustration that Sir James Lighthill used when he attended a conference at Courant Institute about wave and fluid mechanics. It took the outline of Warren Weaver Hall and divided it up into boxes, each one for a different sub-discipline the falls under the heading of the conference and then lists the attendees in that discipline.
From James Lighthill we learned some important life lessons: he was an avid swimmer, and he goes on adventure swims on his holidays. Once was around an island where an active volcano was spewing lava. At the age of 49, Lighthill became the first man to swim the entire nine miles around the Channel Island of Sark. His death eventually came in his 70s when he tried to repeat the feat for the seventh time. Kind of interesting, considering that he is an applied mathematician studying fluids.
From there a discussion of fluid equations was launched, and a particular fun result was mentioned: Stokes, in his 1880 treatise ["On the theory of oscillatory waves. Appendix B." Math. and Phys. Pap. 1, 225-228. Cambridge University Press] found a local solution to the equations for surface waves (such as those on water) that became well known as Stoke's Conjecture. It states that if the water wave has a pointy crest (like seen in the bathtub of early Sesame Street segments with Ernie and Bert), the angle of the crest must be 120 degrees.
From there, Burgers' equation was brought up. (I hadn't known that it was a fluid equation; learned something new.) Burgers' equation is a classical equation that exhibits shock propagation. It can be used as a model for the velocity field of a gas moving in one dimension. Imagine people standing in a line, with you standing somewhere in it. Suppose the people in front of you are moving forward by taking one step per second and the people behind you are moving forward by taking three steps per second, soon enough, the guy behind you will run into you. Now, if you behave like gas particles, you would tumble and push into the guy in front of you. For a situation like this in fluids (where we assume two guys can occupy roughly the same space), Burgers' equation predicts that a discontinuous solution exists such that the "accident area" where people bunches up would move forward at the average of the speed that the people behind you and in front of you are moving, in this case, 2 steps per second.
The question is, whether this solution is "physical"? It turns out that Burgers' equation also models very well the height of water in a long, narrow, and shallow channel of water. And lo-and-behold, Tidal Bores are exactly a physical manifestation of shock waves. Tidal bores are famous in the Qian Tang river mouth around Hangzhou, China, and at the mid-autumn festival (where the full moon pulls a strong tidal wave) people gather to watch the meters-high shock wave travel up the river, washing away any unfortunate visitors to the mid-river sandbars (there's an essay by Chinese writer Liang Shi Qiu testifying to his almost drowning for failing to cross the sand bar fast enough to catch the ferry before the wave went in). Slightly less, but almost very famous, is the bore on River Severn in England. Now what do you do if you are a bored (pun very much intended) English Colonel, living in the countryside of Gloucestershire, and while lounging on the banks one peaceful afternoon, said to yourself, "Ah... placid river... ah... serene river... ah... shock wave... ah... placid river..."? You decide that it might be very fun to surf it. As a testimony to the stability of the shockwave solution to the Burgers' equation, one Dave Lawson managed to surf 5.7 miles upstream, spending more than 35 minutes on the board.
After the small detour to surfing, Keel went back to one of the other objects in the title of his talk: black holes. In particular, Einstein's equations is one that shows off how many more things are still waiting to be discovered! Of course he mentioned the Klainerman-Christodoulou result, as it is really the only established nonlinear stability result in general relativity. (And in the process he started poking fun at physicists who study linear stability.) To show that even the stability result (which is manifestly a small data perturbation result for vacuum space-time) can have bearing on our rather non-empty universe, he mentioned the Christodoulou effect for gravitational waves.
Gravitational waves were postulated to exist since 1916, when Einstein published his GR theory. It was confirmed to exist in 1974 by Hulse and Taylor (of Princeton University) by observing PSR B1913+16, a binary pair that is spiraling toward each other. The evolution of the orbit suggests radiation loss of energy consistent with predictions of gravitational waves. For this discovery, Hulse and Taylor won the Nobel prize in physics of '93. (They won basically for that graph shown in the Wikipedia article above.) Since there has already been discussion of direct detection of gravitational waves in the 1970s by interferometry methods, people were seriously starting to consider building an observatory at that time.
By the early 1990s, LIGO, the Laser Interferometer Gravitational-Wave Observatory, was one of the many physics projects competing for funding and congressional support. As such, fights broke out among the physics community: opponents of LIGO (often also proponents of High Energy research) claim that the effect that LIGO aim to measure is too small to be detected by the then current instrumentation, and that it is a waste of money, while supporters of LIGO would make snide comments of the Superconducting Supercollider that was rapidly becoming a white elephant. LIGO was almost cancelled along with the SSC as a result: the physicists, in their in-fighting, had always thought that funding for physics research is self evident, and never thought to explain to congress that at least one of the projects should be funded. By the effort of Kip Thorne and colleagues, the project was kept alive and is now online.
One of the arguments put forth by the supporters of LIGO was the fact that the gravitational waves should be observable, and not too difficultly so, because of the memory effect. First we need to discuss how the interferometer works: LIGO is basically three huge concrete blocks hanging on threads. On each block are some mirrors. The concrete blocks, being huge, exert some minute gravitational attraction to each other. A laser bounces on the mirrors to measure the distance between the blocks. The idea is that if a gravitational wave passes through the instrument, it would affect the gravitational attraction between the blocks and cause changes in the distance between the blocks. If the gravitational waves just passes through the blocks linearly, once the wave passes, the blocks would return to normal, and it would be hard to tell the signal from the noise (say, trucks moving around the area). The memory effect is one that comes from the non-linearity of the wave: even after the wave passes through, some lingering effects of the wave still gives a permanent relative displacement on the concrete blocks, making detection more plausible.
Part of this memory effect is explained by Christodoulou using his work with Klainerman as a basis. When Kip Thorne heard about this, he wrote an article [Phys. Rev. D 45, 520-524 (1992)] examining what this "Christodoulou effect" (which even made New York Times as part of its coverage of the LIGO controversy) is. The best part (for mathematicians), though, is an admission by Thorne in the opening paragraphs of the second section of the article:
The author (who is an advocate of simple physical explanations for important physical effects) has long thought he understood fully the memories of gravitational-wave bursts. It has been a salubrious experience, therefore, for the author to be shown by a mathematician (Christodoulou, who uses elegant mathematics that is rather far from the physics) that he (the author) had missed a very important physical effect: There is a contribution to the memory that arises from the nonlinearities in the Einstein field equation, a contribution that at first sight seems not to be included in Eq. (1).Keel summarised the statement (jokingly) as: "Oops, I forgot that Einstein's equation is nonlinear... now let me see how I can cover my behind gracefully."
The main purpose of this paper is to show that Christodoulou's effect, in fact, is included in the general expression (1) for the memory, but the author had missed it there due to his obtuseness.
Anyway, after much merriment at the expense of physicists, Keel concluded his talk asserting that
1) Darwin is Dead.
2) Mathematicians are blind man in a dark room looking for a black cat which is rather nonlinear, only that physicists keep asserting that the black cat is not only linear by in fact a cricket, and half of the physicists end up wondering why they cannot hear the cricket chirp and the other half end up pondering by what devilry is the cricket made to sound like that.