Discover magazine is conducting a chat with Prof. Lisa Randall of Harvard University today at 2:00 PM EST:
I have reviewed Lisa’s excellent book, “Warped Passages”, here.
Poincare’s conjecture (1904) is related to the nature of three-dimensional manifolds, or spaces. It is in many ways deceptively simple. It asserts that if any loop in a certain kind of three-dimensional space can be shrunk to a point without ripping or tearing either the loop or the space, the space is equivalent to a sphere.
(The conjecture is fundamental to topology, and incidentally one of the $1 million Millenium Prize Problems awarded by The Clay Mathematics Institute of Cambridge, Massachusetts).
In three dimensions, it is hard to discern the overall shape of something; so for example, when we envision the surface of a sphere, we are really seeing a two-dimensional object embedded in three dimensions. To understand the problem it is therefore easier to start with two dimensions.
The topology of two-dimensional manifolds or surfaces has been well understood since the 19th century. In fact, there is a simple list of all possible smooth compact orientable surfaces. Any such surface has a well-defined genus g ≥ 0, which can be described intuitively as the number of holes; and two such surfaces can be put into a smooth one-to-one correspondence with each other if and only if they have
the same genus.
This means that all such two-dimensional surfaces are classified by a single integer, g. A surface of genus zero is topologically a sphere (an apple), a surface of genus one is a torus (a doughnut) and a surface of genus two is, well a dougnut with two handles (or whatever you want to call it), and so on.
In the case of two dimensions, it is easy to see how the conjecture should be understood: imagine a rubber band stretched around an apple or a doughnut; on the apple, the rubber band can be shrunk without limit, but on the doughnut it is stopped by the hole.
The corresponding question in higher dimensions is much more difficult. The most basic example of such a manifold is the three-dimensional unit sphere, that is, the set of all points (x,y,z,w) in four-dimensional Euclidean space which have distance exactly 1 from the origin: x^2 + y^2 + z^2 + w^2 = 1.
A distinguishing feature of the two-dimensional sphere is that every simple closed curve in the sphere can be deformed continuously to a point without leaving the sphere, or that “you can’t lasso a two-sphere”.
In mathematical terms, the Poincare Conjecture can be phrased as follows: If a compact three-dimensional manifold M has the property that every simple closed curve within the manifold can be deformed continuously to a point, does it follow that M is homeomorphic to the sphere S^3 ?
From the first, the apparently extremely simple nature of this statement has led mathematicians to overdo it. For about 100 years, several mathematicians have tried to prove the conjecture. But they all failed. Well, except from one, it seems.
Grigori Perelman’s proof is based on a study of the geometry of the infinite dimensional space consisting of all Riemannian metrics on a given smooth three-dimensional manifold.
The metric g_ij on a Riemannian manifold determines the length between points according to ds^2 = gij dx^i dx^j. (From the first and second derivatives of this metric tensor, one can compute the Ricci curvature tensor R_ij , and the scalar curvature R; for example, Euclidean flat space has R_ij = R = 0, while a usual three-dimensional sphere of radius r, has Ricci curvature R_ij = 2g_ij /r^2 and scalar curvature R = 6/r^2).
Perelman’s proof is based on the study of the Ricci flow, i.e. the solutions to the differential equation:
dg_ij/dt = −2R_ij ,
where t is like an external time-parameter. In other words, the metric is required to change with time so that distances decrease in directions of positive curvature.
Perelman’s succeeded in showing that certain generally round objects, like a head, would evolve into spheres under this process; however, the fates of more complicated objects were previously thought to be problematic. As the Ricci flow progressed, kinks and neck pinches – or singularities – could appear, pinch off and even shrink away. Perelman has showed that the singularities were all friendly. They turned into spheres or tubes. Moreover, they did it in a finite time once the Ricci flow started (his papers at arXiv.org).
For his work, Dr. Perelman has just been awarded the Fields Medal (an analog of the Nobel Prize) and as said before, also the Millenium Prize; but Perelman seems to have refused to accept the award (news.google.com). What a coolheaded guy 😉
In the context of string theory, the Ricci flow represent the world-sheet renormalization group (RG) flow; Perelman’s proof might have important implications for understanding space-time singularities.
To be continued…
Note: Technical details and discussion are written with small print and can be skipped in a first reading 😉
What is a bubble of “nothing”? The answer clearly depends on which area of reality you are thinking about. I’ll illustrate this with a few examples.
In physics, a “bubble of nothing” refers to an effect discovered by Witten twenty years ago. Witten showed that the standard so-called Kaluza-Klein vacuum, Minkowski spacetime cross a circle M x S^1, is unstable to nucleating a “bubble of nothing”. It is of course important to have some criteria for determining whether a ground state of the form M x S^1 is reasonable as a unification of gauge fields with general relativity.
First of all, one should impose that this vacuum should be stable at the classical and the semiclassical level. The Kaluza-Klein vacuum is classically stable but unstable against semiclassical decay. Now, even if a state is stable against small oscillations, it may be unstable at the semiclassical level. This can occur if the state is separated by only a finite barrier from a more stable state. It will then be unstable against decay by semiclassical barrier penetration. To look for a semiclassical instability of a vacuum state, one looks for a ”bounce” solution of the classical euclidean field equations.
How is this applied to the Kaluza-Klein vacuum? First you analytically continue the Kaluza-Klein vacuum to euclidean space, i.e. to
ds^2 = dx^2 + dy^2 + dz^2 + dt^2 + dphi^2,
where phi is a periodic variable running from 0 to 2pi R, so that dphi is the line element of the circle S^1. Equivalently,
ds^2 = dr^2 + r^2 dTheta^2 + dphi^2.
A solution to the Einstein equations with the same asymptotic behavior is
ds^2 = dr^2/(1-k/r^2) + r^2 dTheta^2 + (1-k/r^2) dphi^2.
This is actually the five-dimensional Schwarzschild solution (but should not be interpreted as a black hole). Regularity at the point where r^2 = k, requires that we set k = R^2, where R is the radius of the circle. The metric of the resulting space – continued to Minkowski space – is
ds^2 = dr^2/(1-R^2/r^2) + (1-R^2/r^2) dphi^2 – r^2dpsi^2 + cosh psi^2dOmega^2.
This space is nonsingular and geodesically complete (which roughly means that there are no possible light-rays that suddenly “end”, like there are in spacetimes with black holes); it is the space that the Kaluza-Klein vacuum decays into.
What happens in the decay of the Kaluza-Klein vacuum is that a hole spontaneously forms in space. As a function of time, t, the boundary of the hole is at x^2 = R^2 + t^2. After a very brief time, this hole – or bubble of “nothing” – is expanding to infinity at the speed of light. So, why is it called a “bubble of nothing”? This is because the Kaluza-Klein vacuum decays into nothing, or more precisely into a space which is bounded by a bubble of nothing – space does not exist “inside” this bubble – which is expanding to infinity and pushing to infinity anything it may meet!
In pseudoscience the examples of “bubbles of nothing” are abundant. I’ll just mention two examples. One is the crackpot book The Final Theory by Mark McCutcheon; another one seem to be the research carried out at The Quality of Life Research Center in Copenhagen, directed by holistic physician Søren Ventegodt.
In politics, there are plenty of examples of “bubbles of nothing”. Recently, there are the examples of various reactions to Israel’s extreme aggression against the palestinians and the people of Lebanon (the massacre in Qana being just one very recent example)- such as those by president Bush, Rice and many others (however strange, my friends over at Cosmic Variance have had nothing to say about this conflict). Another one is Bush’ veto against research in stem cells. (In an upcoming post I’ll explain why research in stem cells is so important).
In sports, a recent example of a “bubble of nothing” was apparently Floyd Landis’ “victory” in the 2006 Tour de France. But to be fair this doping “scandal” might in itself be a “bubble of nothing”. This is, however, not very likely (as recent findings showed that some of the testosterone in his body had come from an external source) and Floyd might end up being considered by historians as the most naive and stupid “winner” of the Tour de France.
The list of “bubbles of nothing” is endless. Other examples, anyone?
Update: Floyd Landis’ B sample was positive; welcome in the historybooks and adieu to the title as the winner of the 2006 Tour de France…;-)
(A completely lighthearted post about a – hopefully – serious paper.)
Some people claim that, what should correctly be called modern theoretical physics, is Not Even Wrong, some that it is Not Even Wrestling. Well, my answer here is that such people are Not Even Wetting. To prove that I’m correct, just take a look at this interesting hep-th paper:
Peter Woit’s book, “Not Even Wrong: The Failure of String Theory & the Continuing Challenge to Unify the Laws of Physics”, is out. The book description at Amazon.com says:
How does the world work and what is mathematics’ role in its description? An authoritative and well-reasoned account of string theory’s fashionable status among today’s theoretical physicists, and promising new directions, including the role of beauty in mathematics and physics.
As of yet, the book has not been reviewed in any major physics journals (and might never be?). There is a rather positive (but also rather uninformed) piece on the book in the Times; there is a positive review in the Sunday Times; there is also a comment on the book in the Financial Times (registration required); and finally, Lubos Motl has a very critical review at his blog.
There is also a recent review by John Horgan in the August issue of the British magazine Prospect.
One of Woit’s main objections to string theory seem to be, that it offers no possibility of producing experimental evidence. And that even the proposed Superconducting Super Collider would have failed to provide any clue as to whether the theory had merit.
The problem with this conclusion is that Woit seem to be implying that you need to reach something like the order of the Planck energy to “confirm” string theory.
In spite of all this, the book should be interesting to read; but I have to add that most likely, will we never see universes wrestling, not even wrestling (which Peter Woit might be disappointed about…).
PS1: I coined the term “Not Even Wrestling”, and then Lubos commentet that: “we might never observe Universes wrestling: not even wrestling”;
PS2: This post DOES NOT refer to the book: “Not Even Wrong: Adventures in Autism”, but for some, I guess, this is more or less the same.
Review of “The Final Theory” rejected with the reason that: “… the comments you submitted did not review the title itself. Instead, your comments focused on another reviewer and were more suited to a chat room discussion.”
Review of “The Final Theory” submitted to Amazon.com yesterday, the 5th of June around 1 PM.
As described earlier, I’ve been trying to post a review of the crackpot book “The Final Theory” by Mark McCutcheon. My review appeared on the website of Amazon.com on May 7th, only to disappear a few hours later. I’m sure this was because my review was a one-star review. As this review was deleted, I’ll try posting a new one and include it here, so that the author of this book, at least, – or whoever else – cannot make it disappear into oblivion 😉
“The Final Theory” (by Mark McCutcheon)
This book is “not even wrong” (to use a famous quote of physicist Pauli). I’ll explain what that is supposed to mean.
A meaningful statement can be said to be either correct, or wrong. “The Final Theory” is full of so many meaningless and wrongful statements, that I consider it to be not even wrong. And I cannot consider it as being anything else.
First of all, it does not contain any valid physical arguments. Plainly speaking, it misrepresents current theory again and again. For example, it argues that gravity violates the law of conservation of energy, because it causes kinetic energy. As an example, the author asks: “How does it [gravitation] cause falling objects and orbiting planets without drawing on any known power source?”. This is simply explained by the fact that it is the kinetic plus the potential energy which is conserved – a falling object decreases its potential energy as it increases its kinetic energy.
Secondly, the author confuses the basic concepts of work and energy (which you normally learn about in elementary school). For example, it is postulated that if you want to move an object, you must spend energy, and that this is the only way how energy may be invested. This is obviously wrong. When you try to push a wall, no work is done – since the displacement is zero – but surely it costs energy!
Thirdly, the book relies on what one could call “common sense appeals”. The author seem to think that science shouldn’t be mysterious or hard to understand. From common sense we have learned many “important” things: that women are less intelligent than men, that homosexuality is “unnatural”, that the earth is flat, that the earth is the center of the universe, that airplanes cannot fly etc. etc. Serious scientist never use common sense as a guiding principle.
Amazingly, the book argues that modern physics – including the pillars of the special and general theories of relativity, and quantum mechanics – is incorrect. The actual situation is that the validity of the special theory of relativity and quantum mechanics has been experimentally established beyond any reasonable doubt; and there are numerous positive tests of the general theory of relativity.
The most complicated thing you’ll find in this book is the “Geometric Orbit Equation”, or
v^2 x R = K,
where v is the velocity, R is the distance separating two bodies and K is a universal constant. I find it very hard to believe that the fundamental workings of the universe can be understood from such a simple equation.
There is basically only one correct, and in the slightest degree, important statement in this book: It is, that we – including the author – do not know everything, or understand everything yet. But we physicists definitely know enough to say that this book is not even wrong.
Finally, let me mention something quite suspisious about the other reviews of this book. As of today (the 31st of May, 2006), there is a total of 95 reviews. 71 of these are 5-star reviews. This is – of course – quite stunning. Out of the 71 reviews, 63, or 90%, have written only one review in total; furthermore, one person wants to give a 1-star review, but is being counted as a 5-star review, twice; another person is counted with a 1-star and a 5-star review, and yet another 5-star review is counted twice. One top-10 reviewer grants the book another 5 stars, but as far as I can tell, all of this persons reviews (which there are more than 2500 of) are 5-star reviews.
If you really want to learn about modern physics, I recommend books by Weinberg, Randall, Greene or Hawking.
In conclusion, I cannot give this book anything more than one star. And sadly enough, nothing less.
Note: Updated on May 7th, 2006
The story about the crackpot book “The Final Theory” (by Mark McCutcheon) has taken a new – and rather surprising – turn.
As described earlier in Lubos’ blog, Anthony Kirmis’ tried to post a 1-star review of “The Final Theory” back in August 2005, but it did not survive for much more than one week. Many other people experienced that their 1-star or 2-star reviews were deleted. However, as of today, there are 289 reviews of the book, with a staggering 2-star average, mainly because of around 200 1-star reviews which suddenly have appeared on amazon.com.
I commented on this book earlier here. The book claims to describe the “final” theory of everything, ignoring – for example – the existence of gravity and using only extremely simple elementary-school arithmetic. Instead, the sensation that we feel as gravity is really the Earth expanding up and accelerating against our feet!
As I’ve said before, I think it is most likely, that McCutcheon originally arranged for the 1-star reviews to be deleted (but of course, he had to ask amazon.com to do it). Still, there are many phony things about the reviewing system of Amazon.com. As others have observed, many of the 1-star reviews seem to have been posted by fictive persons. Also, I noticed a 5-star review by John Matlock “Gunny”. This person is a top-ten reviewer, and has posted around 2575 reviews, but all of those which I have skimmed through are 5-star reviews! About “The Final Theory” he makes the comment, that:
I find the book highly amusing. It’s worth reading, but don’t take it too seriously.
But still he rewards it with five stars…!?
I, myself, thought about posting a review. But it should be very short. Something like: “This book is not even wrong, but at any rate I’ll give it a 1-star review for money-making!”
Update: As of today (May 6th) there are now only 7 one-star reviews and 6 two-star reviews. The average rating is now 4.5 stars! I think I’ll have to withdraw my statement, that Amazon.com appears to be “more science than junk-science”. The rating system is completely flawed and obviously highly dubious.
I’ve posted a review of the book here. (Well, until somebody made sure that it was deleted….).