Reasons to love the "Dark Energy Task Force"

It's official: the words 'dark,' 'energy,' 'task,' and 'force' have all been used in the title of a single scientific paper. The "Report of the Dark Energy Task Force" is available here on arxiv.org. Obviously, it's hard not to love a paper like this. The reasons appear to break down roughly as follows.

Of course, as I've noted before, there remain many alternatives to dark energy cosmology. But without a title like this, I'm afraid the competition is doomed.

Related Posts:

• Map of the cosmic acceleration literature
• Penrose in Pittsburgh
• No, there are STILL no deadly mini-black holes!

Rescher Prize for Contributions to Systematic Philosophy

Photo Credit: John D. Norton

The University of Pittsburgh has announced the establishment of the Nicholas Rescher Prize for Contributions to Systematic Philosophy. The details:

• Rescher Prize gold medal for work in philosophy
• $25,000 award
• Awarded every two years

The prize is being offered in part to establish an award in philosophy hoped to become comparable to the Field Medal, Pulitzer Prize, and Nobel Prizes in other fields. It is also being offered to combat the fracturing specialization of the field:

"The philosopher's key job is to integrate philosophy and to provide a systematic picture of the whole field: Systematic thinking across frontiers is not fashionable but nevertheless crucial. Virtually all major contributors to philosophy have been systematic thinkers." (Nicholas Rescher, press release)

Pittsburgh is honoring Rescher for his own contributions, which include over 100 books, 1000 articles, and half a century of systematic contributions from the perspective of American pragmatism. In return, Rescher is donating his extensive library to the University of Pittsburgh, which includes 40,000 pages of correspondence and volumes of original manuscripts by 20th century philosophers.

Thanks to Jonah at Choice & Inference for bringing this to my attention.

Springer Publishes Crank 'Proofs'

crank, n.
A pejorative term for a person who holds a belief that a vast majority of their contemporaries consider false. A "cranky" belief is so wildly at variance with commonly accepted truth as to be ludicrous, and arguing with cranks is useless, because they will invariably dismiss all evidence or arguments which contradict their unconventional beliefs. (Wikipedia.org)

Two propositions, Fermat's Last Theorem and the Goldbach Conjecture, truly stand out among cranky topics, in drawing the vast majority of bogus "proofs" (although the Riemann Hypothesis may be another contender.) This is well-known among publishers like arxiv.org and philsci-archive, which get plenty of crank submissions. These topics are generally treated with distrust. So it's just stunning that Springer, one of the most trustworthy scientific publishers, may have just printed a book containing crank "proofs" of both these propositions!

The author, Nico F. Benschop of Crank.net fame, has seen basically the same argument refuted by number theorist Robin Chapman seven years ago. (Chapman can be found dismissing this book anew in a sci.math post last week.) You can take a look at the work here through Google Books; on page 133, you'll find the following alleged "proof" of the Goldback conjecture:

Yes, it's three paragraphs of jargon, and it's not clear if it's worth the effort to decipher it. Thanks to John Baez for pointing this out -- Springer really managed to miss a whopper!

No, there are STILL no deadly mini-black holes!

Kentucky, who runs a fantastic blog over at arxivblog.org, has pointed out a new calculation about black hole creation here on earth. The paper (arxiv) suggests that mini-black holes might be created at CERN, and actually be long-lasting, existing for as long as a minute. Kentucky is worried about all of our well-beings.

The press for CERN is of course nice. But as I've argued before, there will be no deadly black holes created at CERN.

The theories that predict mini black holes are highly -- I repeat, highly -- speculative. For example, in the paper Kentucky mentions, the authors adopt a Randall-Sundrum (RS) brane world model. In these theories, gravitational interactions take place in 5-dimensional (4+1) anti de Sitter spacetime, in which there is a 4-dimensional embedded hypersurface (a brane) in which the particle interactions of the Standard Model occur.

In classical general relativity, typical gravitational collapse into a black hole happens for bodies with greater than 3 solar masses of mass-energy. That kind of energy would kind of blow the power-grid here on Earth. However, in the brane world model, the extra spatial dimension for gravity has the effect of vastly reducing the amount of mass-energy in our (4-dimensional brane) world needed to create a black hole -- by screwing with that extra dimension, you can still get the necessary curvature conditions. The five-dimensional black hole is then expected to accrete mass until a 4-dimensional black hole is created in our world.

Now, 4-dimensional mini black holes quickly evaporate because of Hawking radiation. So if you believe the RS brane world model, then you're wondering: will the growth of the black hole in 5-dimensions outrun the evaporation of the black hole in 4-dimensions? According to the paper Kentucky discusses, the answer is: no! If black holes appear, they will still evaporate more quickly than they grow.

But here's the real reason why you shouldn't be worried: The RS brane world theory is still a fairy tale, not a confirmed theory!

There are plenty of competing theories of quantum gravity out there. One of them might even someday be shown to be the correct one, especially once the Large Hadron Collider gets up and running. But for now, there is simply no evidence confirming the possible existence of this 5th spatial dimension. So there is no empirical evidence to support the possibility of black hole creation at CERN.

Riemann Hypothesis Gets Proved Again

Yet another proof of the Riemann Hypothesis (RH) has been proposed by BYU mathematician Xian-Jin Li. Dr. Li posted his proof on the arXiv a few days ago. (Update, 11:05am.)

Why I'm Not Holding My Breath. Although it's tempting to get excited over a proof of RH, here are a few reasons why I'm not holding my breath:

1. Pitkanen's proof of RH (2001). Withdrawn by author due to errors.


2. Castro & Hahecha's proofs of RH (2001, 2002, 2006). Approach has been rejected.


3. Shi's proof of RH (2003). Proof contains errors.


4. de Brange's proof of RH (2004). Remember all the media attention this got? A counterexample was later produced.


5. Chun-Xuan's proof of RH (2005). Proof contains errors.


6. Aizenberg's counterexample to RH (2007). Withdrawn by author due to errors.


7. Madrecki's proofs of RH (2007a, 2007b, and 2007c). Proofs contain errors.


8. Han's proof of RH (2008). Proof contains errors.

These once-hopeful provers of RH are only the tip of the iceberg. But that's not what's really fearful about proposing a proof of RH. Check out the way that Stanford mathematician Brian Conrad layed into this nut job, for being arrogant about a purported proof!

I suppose you can't escape when duty calls. (A note on Conrad's last sentence: he's evidently not a historian of mathematics.) At any rate, the expert verdict on Dr. Li's proof will be out within a year or so. I'm willing to be patient on this one.

What is the Riemann Hypothesis? The Riemann Hypothesis is that the non-trivial roots of the Zeta function on C, given by


all have real component Re(s) = 1/2. In his famous 1859 paper, Riemann wrote that,

Without doubt it would be desirable to have a rigorous proof of this proposition; however I have left this research aside for the time being after some quick unsuccessful attempts, because it appears to be unnecessary for the immediate goal of my study...

You can read more about RH at Mathworld, about failed proofs at Matthew Watkins's homepage, and about the $1 million prize out for its solution at ClayMath.org.

Edit: Update, 11:05am. Field's medalist Terence Tao has pointed out an error in the proof. It is possible that there are more.

Group Theory Of Your Spine

In more examples of the unreasonable effectiveness of mathematics, there now exists a robust application of group theory to the human spinal cord.

Modeling the Human Spine. A few years ago, Australian DOD scientist Dr. Vladimir Ivancevic unveiled a new model of the human spine. The model, called the Human Biodynamics Engine, was designed to study how various stresses (like those on a soldier carrying a load) can result in spinal injury. However, while older models were built up from the physics of compression, bending and shear, Ivancevic built his model using group theory. He published his result last week on the arXiv.

The 'Spinal Cord' Group. The central idea is to describe the degrees of freedom of each vertebrae using a group. The group that does the trick turns out to be the Special Euclidean Group SE(3). SE(3) is equivalent to the direct product group SO(3) ⊗ R³, which is just the group of rotations around a fixed point:


together with the group of spatial translations:


(both in three dimensions).

Makes sense, right? Those are the degrees of freedom of a piece of your spine. From there, it's easy to model the spinal cord as one big direct product group, made up of as many copies of SE(3) as there are vertebrae.



Of course, to make this idea useful, Ivancevic must argue that most spinal injuries are due to what he calls an 'SE(3)-jolt.' This is, effectively, a certain kind of local perturbation of the forces acting on the 'spinal cord' group. However, if the data continues to support Ivancevic's hypothesis, then he may have discovered a very useful new way to think about these injuries.

History: Applications of Groups. Mathematical groups began their history in the early 19th century, as a play-thing for pure mathematicians. Abel and Galois famously used them to study the roots of polynomials (especially the quintic).

Since then, applications of group theory have turned up in unexpected places. Group theory became inseparable from the study of both quantum theory and relativity in the mid-20th century, leading Wigner to famously call mathematics in general "unreasonably effective." (I recommend Octavio Bueno's very informative analysis of this history.)

In the last two decades, applications of group theory have surfaced in biology and in medicine as well. For example, groups have been applied to population genetics, epidemiology, neurobiology, and even anesthesiology. However, few of these applications have been very robust. Most deal only with combinatorial facts about the symmetric group S_n, the group of permutations of n objects.

Update on the LHC: Still No Mini-Black Holes

The new CERN press release has finally come out, with the latest report on the (nearly-go-for-launch) Large Hadron Collider.

No surprises, of course. The major new result is strong evidence that the kind of interaction that (poor arguments suggest) might be dangerous is in fact happening all the time in the Earth's atmosphere. This result agrees with my assessment in April, that nobody in their philosophically-right mind need worry about mini-black holes destroying the universe.

Was anyone holding out on breathing that sigh of relief? And... *sigh*.

Read the new safety report yourself: here.

Update, 8:21PM. A good friend of mine from Mexico is working on the Alice experiment at CERN. He is periodically reporting on it via his blog, which you can access here (Spanish only).

Fundamental Rubik's Cube Problem Is Nearly Solved

Question: What is the maximum number of moves required to solve a Rubik's cube?

Nobody knows the answer yet. But it's either 21, 22, or 23 moves, and we're about to find out which one: we've got a determined programmer with a huge super-computer on the case.

Rapid Recent Progress. In 2006, Tom Rokicki showed that there are cube configurations requiring at least 20 moves to solve; this gave us a lower bound. Then last year, Kunkle and Cooperman proved (pdf) that any cube configuration can be solved in 26 moves; this gave us an upper bound.

But in recent months, rapid progress has been made. In March, Rokicki developed and posted a highly effecient solution-finding algorithm, with which he was able to reduce the upper bound to 25 moves. He now says he's already reduced that number to 23, and is currently working on 22.

Clearly this will all be over soon. In the worst case, Rokicki will have to keep going until he shows that every cube position can be solved in 21 moves -- then this would be the maximum, and the problem would be solved.

Solving a Rubik's Cube. There are roughly 40 quintillion possible positions on a Rubik's cube -- that's 40 with 19 zeros after it. So even a super-computer that can check a trillion cube positions a second (which doesn't exist) would require a length of time equal to the current age of the Universe in order to check every position.

The plan of attack is thus to vastly reduce the number of cube positions to be checked, and then apply a very efficient algorithm. Rokicki's algorithm, together with 8 GB of memory, allowed him to clip through 16 million positions per second, and gave him the 25 move upper bound. This algorithm is based on a similar algorithm developed by Herbert Kociemba, and used in his fascinating Cube Explorer software (which you can download for free).

However, to produce his most recent results, Rokicki had to head over to Sony Pictures Imageworks. Sony lent Rokicki their super-computers during the idle-time between productions, which allowed him to grind through cube positions much more quickly. Yes, the very same computers that brought you Spiderman 3 now bring you "God's Algorithm" for the Rubik's Cube. Unfortunately, you won't be able to impress your friends with arbitrary 23-move solutions using your laptop -- the computing power you would need is enormous.

History of Solutions. The Rubik's cube was developed in the 70's in Hungary, and released internationally in 1980. The first popularly distributed solution was published in David Singmaster's (1981) Notes on Rubik's Magic Cube.

Operating a Rubik's cube forms a mathematical group, the Cube Group. Knot theorist Morwen Thisthlethwaite, who was Singmaster's office-mate, noticed that subgroups of the Cube Group could be systematically described as nested one within the other. He was able to exploit this structural feature in order to produce an algorithm that solves the cube in a maximum of 52 moves.

Thistlethaite's approach is the kind of solution that one's inner-mathematician would really like to see: a solution that actually teaches us something about the structure of the Rubik's cube. Rokicki's brute-force combinatorics approach offers little in this regard, despite its practical effectiveness.

Toward the mathematical end, Douglas Hofstadter suggested in 1981 (reprinted here, chapter 14) that there are sophisticated group theoretic arguments pointing to an upper bound of 22 or 23 moves. Frey and Singmaster also conjectured in 1982 that the maximum number of moves required to solve the cube may be in the low 20's. These arguments are in intriguing agreement with Rokicki's results. However, the group theorists have made little progress on the problem since then, and so a super-computer generated solution may be all we can expect in the near future.

Strange Fermilab Code Nearly Cracked

The Fermi National Accelerator Laboratory, or Fermilab, hosts the second largest particle accelerator in the world (next to the LHC), and employs hundreds of physicists from around the world.

Hundreds of physicsts, who had no idea what the hell it meant when they received this in the mail last year:So they decided to publish it a few days ago in Symmetry magazine, requesting that curious code-crackers take up the case.

The code has now been nearly cracked by an army of random internet cryptographers. The most progress that I've seen has been done by this graduate student, although the Fermilab website says it had several submissions with the message decrypted.

The message was composed of three paragraphs, each encoded with a different cypher. The first and third turned out to be very simple substitution cyphers, with hexadecimal and binary numbers representing letters. At this point, it's still not clear how the second paragraph was encrypted.

The upshot of the encrypted message so far is:

Frank Shoemaker would call this noise... employee number basse 16... AFC
Frank Shoemaker was an employee at Fermilab, and AFC may stand for Absorber Focus Coil. The employee number (base 16) appears to be S252, which belongs to Pierre Piroue, another former Fermilab employee who was friends with Shoemaker.

Both Shoemaker and Piroue say they know nothing the mysterious message.

Solving the Black Hole Information "Paradox"

In January, Abhay Ashtekar posted a short preprint in which he (along with two collaborators) proposed a new solution to the so-called black hole information paradox in 1+1 dimensions. Their essential trick is to propose a quantum-gravity inspired framework in which there is no black hole singularity, which leads to no loss in quantum information. This paper received a lot of superficial attention in the media last week; here's a little background on what's going on.

Background: Black Hole Entropy.The theory of black hole entropy is a well accepted combination of general relativity and quantum theory, which exploits the following close analogy between black hole physics and thermodynamics:

• black hole mass :: energy of a thermodynamic system;


• surface area of the event horizon :: entropy;


• surface gravity k :: temperature.

The last analogy is more than that: a black hole's surface gravity is literally its temperature. A black hole radiates energy exactly like a black body, with temperature k/2π, where k is the surface gravity of the black hole. This effect, called Hawking radiation (which I have discussed before), entails that eventually, all black holes will radiate away their mass, until they become boring everyday objects.

The Problem. The problem that many have with this process is that, according to a semi-classical analysis first argued by Stephen Hawking, it entails an unusual loss of quantum coherence. Here's a very informal sketch of how it happens.

Consider two particles that are initially correlated. One particle enters the event horizon of the black hole, while the other remains outside. Informally, it appears that half of the initially correlated state will disappear into the black hole singularity. So after the black hole has completely evaporated, we will have "lost" all information about the correlation. The result: the correlation is broken, and we are left with a mixed state.

In ordinary quantum mechanics, there is no mechanism for a state to evolve from pure to mixed in this way. As a result, many take the appearance of this phenomenon in black hole physics to be a fundamental problem for quantum gravity. The phenomenon is even sometimes referred to as a "paradox," although this is pretty inappropriate, since it doesn't seem to give rise to any contradiction. After all, the analysis doesn't actually use ordinary quantum mechanics, but rather quantum field theory.

In quantum field theory, the evolution of a state from pure to mixed is a pretty ordinary phenomenon. It certainly isn't unique to the highly curved spacetimes around black holes; in fact, curved spacetimes aren't even required to produce the effect! For example, Wald (1994, chapter 7.3) shows how this phenomenon can occur in (flat) Minkowski spacetime. Here's the trick.

Consider a massless scalar field that is spread out across all of space (i.e., on a Cauchy spacelike hypersurface); call this the initial (pure) state. Suppose this field evolves into a hyperboloid, or any other non-Cauchy hypersurface; call this the final state (see figure below). Then instead of radiation falling into a black hole, there is radiation propagating out to infinity. As a result, the correlation of the field on the interior domain of dependence of the hyperboloid is only "correlated" with radiation at infinity, and so is really in a mixed state.

Wald's Construction.
This is the same "paradox" of quantum information, but in flat spacetime: a pure quantum state evolves into a mixed state. It's not a paradox that is somehow unique to strong gravitational systems. It is a very general feature of quantum field theory.

If one simply accepts this fact, then it is far from clear that the Black Hole Information "Paradox" is much of a problem at all.

Solving the Problem Anyway. If it's not clear that there's a problem, then why are people proposing so many solutions? Well, perhaps the rhetoric of solving a problem is misplaced here. Perhaps a better characterization would be that most physicists are trying to extend a theory. In both black hole evaporation and in the hyperboloid field evolution described above, the essence of the phenomenon is this:

the final state of some quantum field provides an "incomplete" description of the original field.

So any effort to solve the problem is really an effort to give a more complete description of the field. And much of the physics community now agrees that such a description should be available.

Now that we've got a better grip on what we're actually "solving" here, we can ask: what have Ashtekar and his collaborators brought to this discussion? The solution of Ashtekar et al. draws on ideas from Loop Quantum Gravity (LQG), one of the major competitors of string theory. LCG has recently received attention for its apparent ability to resolve certain kinds of singularities (for example, LCG can resolve the big bang singularity).

What Ashtekar et al. have shown is that, given a suitable quantization of spacetime, black hole singularities can also be resolved. In this framework, no part of the original field "disappears" into a singularity because there is no singularity. Therefore, no information is lost, and a pure state remains a pure state.

As discussed above, this doesn't mean that quantum information loss can't happen; it only suggests a framework in which it doesn't happen as a result of black hole evaporation. Whether or not any such framework can prevent quantum information loss altogether remains to be seen.

LHC Black Holes: Why I'm Not Holding My Breath

The attention that these two nut-jobs are receiving is a bit discouraging.

Some people do hope to see mini-black-holes at CERN, it's true. Some calculate that we will see thousands. But: (1) if you have any empiricist scruples, then you won't believe in these mini-black-holes at CERN; alternatively, (2) if you have no such empiricist scruples, then you'll agree that all mini-black-holes at CERN are short-lived and harmless. Here's why:

Let's begin with (2), and suppose that you're not too hardcore about your empiricism. Consider a Schwarzschild blackhole (spherically symmetric and non-rotating, simplified idealization of what is expected at CERN) with mass M. Then dM/dt = -K/M^2, where K is a (very large) constant (See Hobson et al, pp. 277). Let M get very, very small, since we are dealing with particle collisions and not collapsing super-structures. Then dM/dt will become an enormous negative number. In other words, any emergent mini-black-hole will quickly decay into a boring everyday particle. These black holes are harmless and short-lived!

This effect is called Hawking radiation. But should we really believe it will happen? It hasn't ever been observed. However, the result is far from speculative. It is derived from well-verified results of basic quantum theory and of general relativity. From GR, we need only the causal structure of black holes (which is now well accepted, and if we're wrong about it, then there are no black holes anyway). And from QT we need little more than quantum fluctuations (consisting of particle/anti-particle pairs), which we have good empirical reason to believe in. So this isn't one of those weird fringe cases where "quantum theory and gravity don't mix." As things currently stand, there are great betting odds in favor of Hawking radiation.


Image Credit: Universe Review (2008)

But now let's suppose that you're a hardcore empiricist and you still don't buy it. If that's the case, then you don't have to worry about mini-black-holes at CERN in the first place, as there is absolutely no empirical reason to believe they will appear.

Black holes appear when a sufficiently large mass-energy to be crammed into a sufficiently small radius, which in our example is called the Schwarzschild radius (R_s) of that mass. This is not expected to happen at CERN according to any well-confirmed quantum theory, for reasons that have to do with uncertainty, and our consequent inability to squish that much mass-energy into lengths of the order of a very small R_s.

However, some string theorists think that our four dimensions are just one surface of a many-dimensional world that we apparently can't access. One consequence they derive is a much larger value for R_s for a given mass in these situations (from what I understand, they think it gets stretched out into these extra dimensions). This is the reason people have recently decided to hype up the hope that mini-black-holes might appear at CERN -- string theory says there is a larger R_s, so it's easier to cram sufficient mass into the region. (This idea was sketched in a CERN press release a few years back.) But of course, there is zero empirical evidence for this (and all) string theory. So your empiricist scruples set you free here -- mini-black-holes at CERN are little more than a fancy speculation. Maybe there's monsters in the closet too, but I'm not holding my breath.

Whew. Do you feel liberated? I feel liberated.

For more information, download the expert safety reports at the CERN website.