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Weyl's Spinor and Dirac's Equation
Weyl's Conformal Tensor
Weyl Conformal Gravity
Weyl's 1918 Theory
Weyl's 1918 Theory Revisited
Weyl v. Schrodinger
Why Did Weyl's Theory Fail?
Did Weyl Screw Up?
Weyl and the Aharonov-Bohm Effect
The Bianchi Identities in Weyl Space
Conformal, Parameter-Free Riemannian Gravity
Gravity Wave Tutorial
Conformal Kerr-de Sitter Gravity
A Child's Guide to Spinors
Levi-Civita Rhymes with Lolita
Weyl's Scale Factor
Weyl's Spin Connection
Weyl and Higgs Theory
Weyl & Schrodinger - Two Geometries
Lorentz Transformation of Weyl Spinors
Riemannian Vectors in Weyl Space
Introduction to Quantum Field Theory
A Children's Primer on Quantum Entanglement
Veblen and Weyl
Graphing the Lorentz Transformation
FLRW in de Sitter Spacetime
FLRW with a Constant Curvature Scalar
Is There a Flaw in the FLRW Metric?
On the Mannheim-Kazanas Spacetime
Electron Spin
Clebsch-Gordan Calculator
Bell's Inequality
The Four-Frequency of Light
There Must Be a Magnetic Field!
Non-Metricity and the RC Tensor
Curvature Tensor Components
Kaluza-Klein Theory
The Divergence Myth in Gauss-Bonnet Gravity
Schrodinger Geometry
A Brief Look at Gaussian Integrals
Differential Forms for Physics Students
Particle Chart
Einstein's 1931 Pasadena Home Today

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2006 Archives

The Conformal Tensor and Weyl's Gauge Theory -- Posted by wostraub on Tuesday, December 19 2006
Some time ago I wrote about Weyl’s conformal tensor. It has some neat properties, but it usually crops up only in a gravitation/cosmology context, and hardly ever in differential geometry. But it was in that sense that the conformal tensor was used by Einstein to get around his primary objection to Weyl’s 1918 gauge theory, which was that the line element ds is not invariant with respect to a metric gauge transformation (also known as a conformal transformation of the metric).

Recall that an infinitesimal local gauge transformation of the metric gμν → (1 + ε π) gμν regauges the lengths or magnitudes of vectors under physical transport, where π(x) is the gauge parameter. Consequently, the line element ds2 = gμν dxμdxν is also regauged in accordance with ds → ½ ε π ds.

Einstein’s argument was that ds can represent time as well as distance, so time-independent processes such as the spacings of atomic spectral lines can be invariant only if the line element is gauge invariant. Since it is not, Einstein thought Weyl’s theory had to be wrong.

But later, Einstein took up the problem once more. He felt that ds could be made gauge invariant if the line element were revised to ds2 = J(x) gμν dxμdxν, where J is a scalar function of the coordinates whose gauge variation goes like δ J = -ε π J (that is, J must be of gauge weight -1). This would cancel out the gauge change in the metric tensor and leave the line element invariant.

Try as he could, Einstein could not come up with an appropriate scalar. Finally, he noticed that the Weyl conformal tensor Cαμνβ was exactly what he needed, for the combination √ Cαμνβ Cαμνβ is of gauge weight -1 in a Riemannian space.

Unfortunately, the Weyl conformal tensor vanishes in the absence of a gravitational source, leaving a null line element (ds = 0) whose gauge invariance is now trivial. Furthermore, the counterpart of the foregoing expression in a Weyl space is unknown.

What Einstein apparently overlooked is the scale factor from the Weyl theory itself, which considerably simplifies things. Consider the integral quantity k ∫ φμ dxμ, where k is a constant and φμ is the Weyl vector (which he identified as the electromagnetic four-potential). Under a metric gauge transformation, the Weyl vector varies in accordance with δ φμ = λ ε ∂μπ, where λ is another constant. Gauge-transforming the above integral puts the gradient μπ under the integral, which is easily integrated. We can now set the Weyl scale factor to J via J = ek ∫ φμ dxμ, which, by appropriate selection of the constant k, will have gauge weight -1.

This seems like a better approach than that provided by the conformal tensor, because in the absence of the electromagnetic potential φμ the exponential term is identically 1. Thus, the line element can be made gauge invariant only in a Weyl space containing a non-zero electromagnetic field!

I haven’t found any evidence that Weyl resorted to this counterargument to Einstein’s objection, but by that time Weyl had moved on, anyway. In 1929, Weyl applied the gauge concept to quantum theory, which was a huge success. One has to assume that he never looked back.
Hermann Weyl and Dimensional Reduction -- Posted by wostraub on Monday, December 18 2006
In his neat little book The Dawning of Gauge Theory, Dublin physicist Lochlainn O’Raifeartaigh writes
The procedure by which higher-dimensional systems are reduced to lower-dimensional ones is called dimensional reduction. The reason that dimensional reduction is so powerful from the point of view of gauge theory is that it converts coordinate transformations in the full space into gauge transformations in the subspace.
Historically, the most famous example of this statement comes from Kaluza-Klein theory. In 1919, the German physicist Theodor Kaluza postulated the existence of a fifth dimension which was hidden from observation because it was too small to be seen. Kaluza thought that the electromagnetic four-potential of Maxwell’s electrodynamics resided in this dimension, but that its effects were observable only in the more familiar four-dimensional world we humans reside in.

Kaluza assumed that the true metric tensor gμν(x) was five-dimensional. Viewed as a 5x5 symmetric matrix, it has a 4x4 subblock representing ordinary four-dimensional spacetime, while the g "boundary" elements include the potential Aμ by way of the identifications g = g55Aμ (μ = 0,1,2,3) and g55 is a constant. Thus, the four-potential Aμ lives in the fifth dimension.

The potential is brought down into our world via dimensional reduction. Kaluza took as his action quantity the integral

√ –g R d5x

where the metric determinant g and the Ricci scalar R are the old familiar ones, but now in five-dimensional form. Using Kaluza’s above formulas for the g quantities, this five-dimensional integral can be reduced to four-dimensional form, which is

√ –g (R – FμνFμν ) d4x

This, amazingly, is the familiar expression for the combined gravitational-electrodynamic action! (Physicist Ian Lawrie considers this result a minor miracle. It isn't, because God just made it that way!) I find it remarkable that Kaluza was able to deduce this way back in 1920, because the calculation (while straightforward) is not trivial.

(Kaluza excitedly sent his paper to Einstein in 1919 to get a recommendation for publication. Einstein, though quite impressed, was nevertheless uncomfortable with a five-dimensional world, and so suppressed publication until 1921. Kaluza was not particularly happy about this!)

The Swedish physicist Oskar Klein published a subsequent paper in 1926 that made numerous important improvements to Kaluza’s idea in the context of the then-emerging quantum theory. Hence the theory's present Kaluza-Klein moniker.

Interestingly, in 1953 the great Austrian physicist Wolfgang Pauli took Kaluza-Klein theory one step further -- that is, one dimension further, to n = 6. This resulted in the very first non-abelian approach to non-gravitational (particle) physics. Several years later, using a similar approach, Yang and Mills developed the first consistent theory for the strong interaction.

You might note that, in accordance with O’Raifeartaigh’s assertion, the coordinate-invariant form of Kaluza’s five-dimensional action results in a fully gauge-invariant term (√ -g FμνFμν) following dimensional reduction, while the original action is not gauge invariant at all. We got a gauge-invariant term by reducing the dimension by just one; imagine the possibilities if one started with, say, an eleven-dimensional action! This is the so-called M-theory of string physics, which promises great things (but has delivered nothing to date except beautiful mathematics). Note, however, that Kaluza-Klein theory, while interesting, eventually lapsed into obscurity because it did not predict any new observable phenomena – it was just a pretty theory. String theory is now finding itself in the same boat, and if the legions of brilliant physicists now grinding away (and maybe wasting their precious talents) at this theory cannot produce anything predictive from it (like explaining the magnitudes of the gravitational and electromagnetic coupling constants), it may also be forgotten.

Did Hermann Weyl play around with dimensional reduction? Did he ever consider the possibilities of a higher-dimensional gauge theory? I’ve seen no evidence that he ever did. By dying in 1955, Weyl missed Yang-Mills and a lot of other neat stuff he would have undoubtedly contributed to.

Weyl was taken from us too soon.
Louise Brooks: Lulu Forever -- Posted by wostraub on Saturday, December 16 2006
Peter Cowie's new book Louise Brooks:Lulu Forever is out, and at long last. Finally we have a large-format book with hundreds of rarely-seen photos, motion-picture production stills and first-person accounts of 1920s actress-flapper Louise Brooks, who would have turned 100 years old last month (she passed away in 1985).

I probably would not care so much for this actress if it were not for the fact that I first saw her signature film Pandora's Box (filmed in Germany as Die Büchse der Pandora) as an impressionable young college student in 1970. At the same time, I was taking an elective course in literature (very odd for a chemistry major), where I was also reading Vladimir Nabokov's irreducible masterpiece Lolita for the first time. It was in Chapter 6 of the novel that I encountered Monique, Professor Humbert's French girl-whore, the predecessor of one Ms. Dolores Haze. To me, Louise and Monique were one and the same at the time, and I have forgotten neither in all these years.

Of course, as a Christian I have mixed feelings about all this now, but literature is literature, and life itself isn't squeaky clean. Humbert, Monique and Lulu all paid dearly for their shortcomings (as did Louise Brooks), and so I will let it go at that.

Cowie's book can be purchased from Amazon for about $35. If you're interested, you might also consider buying Lolita* which, in my humble opinion, is the third greatest book ever written (right behind Hamlet and the New Testament). Exceedingly well-written, hilarious, disturbing and heart-breaking at the same time, it's all the more amazing that it was written by a Russian who picked up the English language later in life (much like Joseph Conrad, who also ranks right up there).

* Five points to the person who figures out the identity of John Ray, Jr. PhD, credited as co-author of the book
Bombs Bursting in Air -- Posted by wostraub on Tuesday, December 12 2006
Several weeks ago, I was driving through San Raphael near San Francisco and happened to stop by Autodesk, the company founded by AutoCAD's creator, John Walker.

Coincidentally, Walker's name popped up on an Internet search with that of John von Neumann, the great mathematical physicist and close friend/colleague of Hermann Weyl (see my December 9 post). Von Neumann worked on the Manhattan Project, where (among many other things) he discovered that an atomic bomb would be much more destructive if detonated high above the target area (something involving shock wave pressures, which I know nothing about).

It turns out that John Walker is also interested in such things, if only academically (unlike me, he is extremely wealthy and has even more time on his hands). He has a website that explains the effects of nuclear weapons on human populations, something we should all get familiar with as long as President Bush is running the world.

Anyway, Walker's site includes print-out materials and instructions for making a nuclear effects calculator. It's basically a circular slide rule that will allow you to ponder (in a very quantitative way) the death and destruction that a nuclear device can have on your least favorite city (Crawford, Texas, for example). Well, I made one, and it's very neat. It's one way to personally experience the practical aspects of the complicated science that folks like von Neumann, Oppenheimer and Teller turned into godless, immoral sin.

[Note: Optimum burst height = maximum resultant death and destruction]

Walker warns that his calculator won't be of much use in a post-nuclear war world. But that may not be that far off -- I'll bet you anything that one of the alternatives Mr. Bush is considering for the New Way Forward© in Iraq is to nuke Iran, in which case all bets are off.
Weyl and von Neumann -- Posted by wostraub on Saturday, December 9 2006
From the recollections of mathematician Herman Goldstine, friend of Hermann Weyl and the great mathematical physicist John von Neumann, and (with von Neumann) one of the developers of the early ENIAC computer:

Hermann Heine Goldstine, 1913-2004
"I always was struck by the difference between Weyl and Johnny von Neumann. There are jokes, one of which Johnny always swore was false. That's the story that, I don't know, Hermann Weyl was going to prove some theorem, a very deep and profound theorem, let's say it was the Riemann-Roch theorem. I don't know if it was the Reimann-Roch theorem, but that was one I always have trouble with, so let's say that was the theorem. And Weyl gave a lecture on why this is a very deep, profound result, and he gave a very complicated proof. And the apocryphal story goes that at the end of the lecture there's this kid who is supposed to have raised his hand at the back of the class and said, 'Professor Weyl, may I show you a proof?' And goes up to the board and goes zip, zip, zip, zip, and in about 15 lines has a brilliant proof of this thing.

"I asked Johnny about it, and he said no, that wasn't true. But it is true, if you talk to Natasha Brunswick, who was in those days Natasha Artin. Natasha says that there was always Johnny with these tight pants on. All of Johnny's life, whatever size suit he bought, he always ate too much, and the suit was always one size smaller than Johnny. Even as a student in Göttingen, his behind was always ready to bulge out of his pants. I guess Natasha and everybody in the class were always charmed.

"But Joachim, who was one of Hermann's children, told me that when Hermann used to work in his house on Mercer Street, in the study in there, you would hear groans coming out of the study. That Weyl worked at things in sort of anguish, that it was hard for him, that he delivered his theorems practically like a woman giving birth to a child. That's so different from Johnny, because when he and I would be working at something, when we'd get stuck, he'd say, 'Okay, that's it, ' and pack it up. It might be that he'd phone at two in the morning to say, 'This is how the proof goes.' But it might be three weeks, a month or so later, or it might even be I who would come in a month or so later and say, 'This is, maybe, how to go.' But he never struggled with something. When he got stuck, he filed it somehow, and it just came out easily. I suspect that Weyl was probably the deeper of the two mathematicians."
Louise Brooks at 100 -- Posted by wostraub on Sunday, December 3 2006
[Follow-up to my October 17 post.] While visiting San Francisco recently, I stopped by the city's Main Public Library, which is featuring an exhibit on the American actress Louise Brooks. Brooks, who passed away in 1985, would have turned 100 on November 14.

Astonishingly beautiful, Brooks created the movies' "bobbed" hairstyle look around 1926. I still remember the recollections of my late mother who, as a lovely teenager herself in the late 1920s, begged and begged her parents to get her hair bobbed á la Brooks. But as strict Southern Baptists, such a hairstyle (not to mention even going to the movies!) was absolutely verboten.

Back from SF, I was happy to have finally received Criterion Collection's two-disc DVD set of Brooks' early 1929 psycho-sexual drama Pandora's Box, the actress' signature film (filmed in Berlin by the great German director, Georg Wilhem Pabst). Regarded as one of the top ten greatest silent movies of all time, Criterion's digitally remastered version of the Munich Museum's restored film includes four different musical scores, Lulu in Berlin (a rare filmed interview with the actress, produced in 1984), Looking for Lulu (the one-hour, 1998 Hugh Neeley documentary on Brooks' life), the book Reflections on Pandora's Box, and assorted essays, audio commentaries, interviews and stills. If you're into this actress, this is a must-have DVD.

Brooks at age 64, in a rare 1971 interview with British filmmaker/
film essayist (and Harvard physics graduate!) Richard Leacock

Tragically, parental neglect and childhood sexual abuse (at the age of nine) most likely destroyed Brooks' life, and she went on to become the same kind of woman she portrayed in Pandora's Box and Diary of a Lost Girl (also 1929). An ultra-liberal, chain-smoking, alcoholic, partying sexual abandonee and iconoclastic loner until very late in life, to her enduring credit she renewed her Catholic roots, took up writing and turned herself around. She died of emphysema at the age of 78. May God save her soul.
My candle burns at both ends;
It will not last the night;
But ah, my foes, and oh, my friends--
It gives a lovely light!
-- Edna St. Vincent Millay (1920)
[More pics of the exhibit can be found here]
Another Hermann -- Posted by wostraub on Monday, November 27 2006
If you’ve read my articles on Weyl spinors, Dirac’s equation and quantum field theory (or been bored by them), then you’ve probably wondered why no mention was made of Grassmann numbers.

In QFT, scalar particles and fields can be described by a path integral involving infinite-dimensional products (123…dφinfinity ) under the integrals. But for fermions (electrons, quarks and the like), whose fields are actually operators, the fields ψ(x) obey instead an anticommuting algebra. Thus, ψ1ψ2 = - ψ2ψ1, which needless to say complicates fermionic QFT. The first thing I thought when I saw this was “well, matrices and differential operators can anticommute, so these fields are just matrices or differentials.” No, I was wrong -- there are plain old ordinary numbers out there that can anticommute. They are called Grassmann numbers.

Hermann Grassmann was born in Stettin, Germany in 1809. He loved math and physics, but was also drawn to theology, chemistry, Latin, philosophy, linguistics and neohumanism, so much so that he ultimately went on to teach all of these subjects. Amazingly, he never took any formal classes in mathematics or physics, yet he excelled in these subjects to the extent that famous mathematicians of the day (including Möbius and Kummer) considered him their equal. But because he was not formally educated, Grassmann was not recognized in his day for his genius.

Grassmann was apparently the first researcher to realize that linear vector spaces need not be limited to three dimensions. His work on infinite-dimensional vector spaces predated by many years the work of Hermann Weyl, Elie Cartan and other mathematicians, but intriguingly it also provided the mathematical basis for fermionic QFT.

When Einstein tackled gravity in 1911, he found he needed a type of mathematics that described gravitational physics that did not depend on any particular coordinate system. He was advised by friends to study tensor calculus, which had been worked out fifty years earlier by the likes of Riemann, Christoffel and Ricci. I find it remarkable that quantum electrodynamics would similarly be worked out in the 1940s using a mathematics that had been discovered by Grassmann almost one hundred years earlier.

The books I have on QFT explain only the merest fundamentals of Grassmann algebra, while math books I have seen on the subject go far over my head. Still, I am amazed that God could come up with something so strange and counterintuitive -- and useful. In fact, since fermions make up all of the ordinary matter in the universe (including you and me), God must have had Grassmann numbers in mind from the very beginning. What a Creator!

One of the oddest things about Grassmann calculus is its underlying simplicity. For example, the most complicated single-variable math formula f(x) you can think of can be expanded as a Taylor series, which in Grassmann algebra consists of just two terms: a + bx, where a and b are constants. Thus, the basics of Grassmann's discovery can be grasped by anyone in about five minutes.

You can read more about Hermann Grassmann here:
Units -- Posted by wostraub on Thursday, November 16 2006
One of the more appealing aspects of Hermann Weyl's metrical gauge theory deals with the concept of "units." Humans measure length in terms of meters and feet and, in ancient times, cubits -- different, but all interchangeable, and therefore the same thing. But in the presence of a strong gravitational field (or when dealing with velocities approaching the speed of light), the lengths of physical objects can become ambiguous -- the length of a physical measuring rod, for example, can depend on the observer.

In Weyl's original gauge theory, length can be continuously redefined as one goes from one point in spacetime to another.

The basic units of length, time, mass etc. used to be based upon physical objects or anthropological effects (all called "artifacts") that were explicitly defined to represent the units they stood for. For example, the meter used to be defined as 1/10,000,000 of the distance from the equator to the North Pole (via Paris). Similarly, the second was once defined as 1/86,400th of a day. In these examples, the physical earth was a measurement artifact.

All of these artifact-based units (except the unit for mass) have since been replaced by non-anthropological representations. For example, the meter is now defined by a specified number of wavelengths of the emission spectrum of a certain cesium isotope. Similarly, the speed of light in vacuo is now fixed at exactly 299,729,458 meters per second. The second itself has a specific definition based on isotopic transitions. But to date the kilogram has resisted all such conversions. Officially, it is still defined as the mass of this platinum-iridium alloy cylinder having equal dimensions of length and diameter (39 mm) maintained near Paris:

But this object is not entirely stable. It has been observed to change on the order of 50 parts per billion per year (Corrosion? Sublimation? Old age?). Now scientists are attempting to revise the definition of a kilogram to a non-artifact basis. But it has been a difficult road.

The December issue of Scientific American describes the most recent attempt. It is based on a nearly perfect, ultra-pure sphere of crystalline silicon-28 having a number of atoms that is very nearly that of Avogadro's number (roughly 6 x 1023), which is defined itself as the number of anything in one mole of a pure elemental substance.

But to my mind, this just replaces one artifact with another. Furthermore, Avogadro's number is another "unit" having an anthropological basis. Is there no way to define the unit of mass that is free of some kind of human subjectivity?

Quantum physicists long ago realized that their equations could be greatly simplified by setting Planck's constant and the speed of light to unity. But this is really nothing more than a convenience, as these simplifications only establish yet another set of units that is no better than any other now in use.

My suggestion? Define the kilogram as the mass of one atom of hydrogen and be done with it.
Weyl and Einstein, Again -- Posted by wostraub on Friday, November 10 2006
Taken aback by Hermann Weyl's insistence that his gauge theory was valid despite the physical evidence, Einstein wrote to his friend on 1 May 1918 with this remarkable correspondence:
Could one really charge the Lord with inconsequence for not seizing the opportunity you have found to harmonize the physical world? I think not. If He had made the world according to you, you see, Weyl II would have come along to address Him reproachfully thus:

"Dear Lord, if it did not suit Thy way to give objective meaning to the congruency of infinitesimal rigid bodies, so that when they are at a distance from one another one cannot know whether or not they were congruent, then why didst Thou, Inscrutable One, not decline to leave this property to the angle or to the similarity? If two infinitely small, initially congruent bodies K, K' are no longer able to be brought into congruency after K' has made a round trip through space, why should the similarity between K and K' remain intact during this round trip? So it does not seem more natural for the transformation of K' relative to K to be more general than affine."

But because the Lord had already noticed, long before the development of theoretical physics, that He cannot do justice to the opinions of mankind, He simply does as He sees fit.
You may not always agree with Einstein, but he just nails it here.
Stupid Notation -- Posted by wostraub on Friday, November 10 2006
From September 1918 until late November of that year, Hermann Weyl and Einstein corresponded repeatedly, with the main topic being Weyl's geometrical gauge theory. Einstein loved the basic idea, but was upset over the fact that the line element

in the theory was not gauge invariant. This unsettling little fact ultimately doomed Weyl's idea.

But the two also bickered over Weyl's expression for the equation of the geodesics, which is obtained by extremalizing the related integral expression
Weyl's result was


Einstein vehemently stated that this was wrong. Weyl disagreed, and for three months the issue came up again and again. The two men never resolved it, and Weyl persisted in using his expression in all five editions of his book Space-Time-matter. So who was correct?

Well, Einstein was right after all, but the whole thing was trivial, and the two great scientists should have known better. The correct expression is

What's the difference? It's in the partial derivative term for the metric tensor: Weyl used a covariant term for x in the denominator when he should have used the contravariant term.

It's no big deal, but it serves to show how important it is to maintain consistency in your tensor notation. Few areas of mathematics have displayed such a wide and bewildering range of notation as has tensor calculus in its 150-year history. In the years immediately following Einstein's general relativity theory, it seems that everyone was using a different notation (even Einstein). Contravariant and covariant indices were constantly being intermixed, and that is really what lies at the bottom of this little Weyl-Einstein disagreement.
Spin and the Early Universe -- Posted by wostraub on Thursday, October 19 2006
Abraham Loeb, Professor of Astronomy at Harvard University, has an interesting article in the November issue of Scientific American that deals with the so-called "Dark Ages" of the universe.

According to current cosmological thought, about 380,000 years after the Big Bang the universe had cooled enough for neutral (non-ionized) hydrogen atoms to form. This prevented the microwave background radiation from continuing to interact with electrons and ionized hydrogen via Thomson scattering, so it began to leak out into the expanding universe. Because stars had not yet started to form, there was no source of light, and the universe went almost completely dark -- the start of the Dark Ages.

During this darkness, which is believed to have lasted about a billion years, gravitation gradually coalesced matter into stars and galaxies. Light radiation from the resulting fusion reactions reionized most of the hydrogen in the still-expanding universe, and light returned to the cosmos.

As a consequence, the farthest astronomers can see with their telescopes is about one billion light years. Whatever occurred prior to that cannot be detected.

Or can it? Loeb believes that the universe preserves an imprint of the Dark Ages through what he calls a menage a trois between the backround radiation, the kinetic energy of neutral atoms, and a type of energy called hydrogen spin energy. By modeling how these energies must have interacted, astronomers can compare theoretical calculations with observations of the sky at long radio wavelengths and test it out.

In the article, Loeb describes how a neutral hydrogen atom, consisting of a proton and electron, can exist in its ground state in two distinct energy states -- one with the spins of the proton and electron aligned, and another in which the spins oppose each other. The energy difference is minute, but becomes important when the background radiation level or kinetic energy is smaller. Loeb believes that as the dark universe expanded, the spin energy, radiation energy and kinetic energy took turns being top dog, and that the echoes of this cosmic energy dance are still detectable in the night sky.

If Loeb is right, then humans will be able to see the unseen, and perhaps get an even better glimpse of the hand and mind of God.
Pandora's Box on IFC -- Posted by wostraub on Tuesday, October 17 2006
Shot in Berlin in the waning years of Germany's Weimar Republic, the late silent film classic, "Pandora's Box" (Die Büchse der Pandora) is airing on the Independent Film Channel (IFC) at midnight tonight and tomorrow at 7:45 am PST (October 18). The film is shown uncut and uninterrupted, with both its original German and English subtitles.

German filmmaker Georg Wilhelm Pabst's 1929 classic stars the hauntingly beautiful American actress Louise Brooks (1906-85) as the libertine but curiously innocent dancer/vamp Lulu. Now widely regarded as a cinematic masterpiece, the film received surprisingly scathing reviews because of its (then) shocking sexuality (but there's no nudity, parents).

Sickened by the excess and amorality of Hollywood (though hardly an ingenue herself), and stuck in a series of profitable but brainless "flapper" roles, Louise Brooks left to further her career in Germany, where she starred as Lulu in "Pandora" and Thymiane in "Diary of a Lost Girl" (Das Tagebuch einer Verlorenen, also 1929). Following another starring role in the 1930 French film Prix de Beauté ("Beauty Prize"), Brooks returned to the states. She grudgingly made several more films in the 1930s, but she was essentially blacklisted by the film industry because she refused to play by its rules. She left Hollywood for good in 1939 and went to New York, where she lived a rather impoverished, hand-to-mouth existence as best she could until her death in 1985.

A victim of childhood sexual abuse and gross parental neglect, Brooks ironically and tragically became a hedonistic abandonee herself, and by the mid 1950s was, in her own words, "a questionable East Side dame." But about that time she started writing about her life and the many stars she had known personally (often very personally) during her acting days. While her work was not prolific, her writing demonstrates a remarkable talent for intelligent exposition. Her 1982 book, Lulu in Hollywood, reflects a truly brilliant mind.

A chronic drinker and smoker, Brooks succumbed to emphysema on August 8, 1985 after a long struggle with the disease.

Brooks led an absolutely amazing life, which is chronicled in Barry Paris' excellent 1989 book, Louise Brooks: A Biography.

Silent film fans around the world will celebrate Louise's 100th birthday next month (on November 14), at which time Criterion Collection Films will release a digitally remastered DVD of Pandora with many extras, including a rare filmed interview of the actress from 1979.
Melvin Schwartz Dead at 73 -- Posted by wostraub on Wednesday, August 30 2006
1988 Nobel Prize winner Melvin Schwartz has died at 73. He shared the prize with Leon Lederman (The God Particle) and Jack Steinberger for their work on the weak interaction and their discovery that neutrinos come in different flavors. But what appealed most to me about Schwartz was his approach to electromagnetism.

Like many other befuddled graduate students in the 1970s, I had the great misfortune of being forced to learn electrodynamics from J. Jackson's Classical Electrodynamics, arguably the most difficult text on the subject (the third edition was presumably "dumbed down" in the 1990s in belated response). It's a shame that Schwartz' Principles of Electrodynamics, first published in 1972, didn't achieve the same (inexplicable) popularity as Jackson's book, because Schwartz' approach is much clearer. It's even entertaining -- he starts it off with the statement Electrodynamic theory is beautiful! What a wonderful way to start a book!

Schwartz was one of the few physicists who insisted that electric and magnetic fields, which are essentially the same thing, share the same units. This in itself represents a tremendous simplication of the subject, as the "units problem" in electrodynamics has caused no end of troubles for students.

He similarly simplifies the understanding and calculation of the Lienard-Wiechert potentials, another chronic stumbling block for mediocre students like myself. The subject is laid bare in a wonderful chapter entitled Let There Be Light!, in which the author unashamedly shares his enthusiam for and appreciation of God's scientific and mathematical wisdom. Indeed, Schwartz' writing style is peppered with statements like At this point when the laws were being written, God had to make a decision ... God naturally chose the antisymmetric tensor as His medium of expression (Chapter 3). I love it!

Fortunately, Schwartz' book is available as a Dover reprint and can be had for about $10, so you have no excuse for not buying it. No physics library should be without it.
Dark Matter Discovered? -- Posted by wostraub on Tuesday, August 22 2006
On August 15, a group of astrophysicists announced they had seen indirect evidence for the existence of dark matter. What does this mean?

For decades, astronomers have noticed that the rate of rotation of galaxies does not jive with the amount of matter contained in them – that is, there is not enough gravity contained in the galaxies to keep them from flying apart. Astronomers therefore believe that there must be a form of matter unlike normal matter that keeps the galaxies together. This matter have been given the name dark matter. It is optically invisible because it does not interact with ordinary matter.

Scientists have no idea what dark matter is composed of. Since it cannot be made of ordinary stuff like protons and electrons, other, more exotic forms of matter have been proposed (axions, anyons, etc.). But to date, all this conjecturing has been purely theoretical.

Now a team of scientists (including members from the Stanford Linear Accelerator Center and the University of Arizona) have announced indirect evidence of dark matter in the Bullet Cluster, two groups of over one thousand small galaxies that collided about 100 million years ago in the constellation Carina, forming the shock wave shown in the above photo (which is a composite of visible and x-ray photographs). As the galaxies collided, the ordinary matter slowed down as one would expect in any physical collision. However, the dark matter component, which is immune to any kind of physical (mainly electromagnetic) interaction, kept right on going. The scientists were able to deduce this by measuring the amount of gravitational lensing caused by the dark matter on more distant galaxies seen in the photo's background (dark matter may not interact directly with ordinary matter, but it can still affect it gravitationally). Thus, a cosmic collision event can serve as a means of "filtering out" dark matter from its ordinary counterpart. Indeed, there is speculation that past events have generated "dark matter galaxies," whose presence can only be deduced by gravitational lensing effects.

[Interesting Question: Is intelligent dark matter "life" possible, or does it require the usual quarks and leptons? Maybe God is not quarkic/leptonic at all!]

Who cares, you might be tempted to say. But astrophysicists have estimated that dark matter makes up about 25% of the total matter in the universe, whereas ordinary matter accounts for only about 5%. The remaining 70% is thought to consist of dark energy, a hypothetical energy field (called quintessence by some scientists) that permeates the entire universe. Thus, the visible universe you and I know and love accounts for only 5% of physical reality. This concept is truly mind-boggling.

Hermann Weyl and others postulated that what we today call dark energy is nothing more than a mathematical artifact of Einstein’s general theory of relativity called the cosmological constant (I tend to agree, as "quintessence" sounds a tad like the old "ether" idea of the early 1900s). The cosmological constant is simply a term in Einstein’s gravitational field equations which, depending on its sign, can either act with or against the usual attractive force of gravity. Many scientists believe that a non-zero, repulsive cosmological constant exists and is responsible for the observed large-scale repulsion effect that is forcing the universe to expand at ever-greater velocities. If true, the universe will eventually expand at the speed of light, resulting in a rather bleak future for all existence.

The relationship between dark matter and dark energy has not been established. If the human race can keep from blowing itself up over petty tribal conflicts (which I find highly doubtful), we may have a chance at someday understanding the fantastic universe that God has made for us.
String Theory Unraveling? -- Posted by wostraub on Tuesday, August 15 2006
The great Austrian physicist Wolfgang Pauli once remarked "What God hath put asunder, let no man join." He was referring to the seemingly-intractable problem of unifying quantum theory with general relativity, two theories that work just fine by themselves but which have, since Hermann Weyl's time, resisted all attempts at unification. It's the one great open problem of physics, and every physicist worth his salt worries about it.

Why bother with unification when the theories work fine on their own? Because only through unification can we simplify our world and begin to grasp the mind of God. Recall James Clerk Maxwell, who in the 1860s discovered that the electric field and magnetic field are the exact same thing. Instead of a hodgepodge of unconnected, complicated vector equations, we have the four Maxwell equations expressing the unified electromagnetic field (which, I might add, are among the most beautiful mathematical expressions mankind has ever gotten its hands on).

The unification problem can be traced to the fact that the general theory of relativity is not renormalizable (which means that infinite probabilities invariably arise during the perturbation step), so efforts to describe gravitation using a quantum-perturbative approach fail miserably.

Thirty-odd years ago, string theory promised a way around this problem. The theory's early developers noted that it demands the existence of a massive, spin-two particle, and it was assumed that the as-yet undetected graviton feld would fit the bill.

Initially, it looked good on paper. But the most advanced version of string theory requires that spacetime consist of 11 dimensions (1 time dimension and 10 spacial dimensions) and not the usual 4. In order to demonstrate the existence of all these extra dimensions, physicists would need access to energies that are many magnitudes beyond those currently produced in the most powerful particle accelerators. Indeed, these energies rival that of the Big Bang itself, making it almost a certainty that string theory can never be tested.

The theory's critics insist that any untestable theory is unscientific, and therefore has no place in science. Some have even gone so far as to say that it is akin to religious faith. Anti-evolutionists (and Republicans) might be crazy about this, but not physicists.

My personal problem with string theory is far simpler -- I just can't follow the mathematics. Back when Einstein first announced his general theory of relativity in November 1915, it was said that only a dozen people in the world could understand it. That was simply not true -- relativity is pretty straightforward, and while the math at the time was unfamiliar, it wasn't difficult at all. Same thing with Heisenberg's matrix mechanics in 1925 -- physicists just weren't all that familiar with matrices, which even today's middle school kids can understand. String theory, by comparison, is nothing less than a convoluted maze of unbelievably complicated mathematics that seems beautiful only to the relatively few people who can work with it. And in their own words, even they don't really understand what they're doing!

So now we get the August 21 issue of Time magazine, which has an article entitled The Unraveling of String Theory. It reports that two new books by respectable physicists (Lee Smolin and Peter Woit) are heralding a renewed criticism of string theory that might just catch on.

The criticism advances the now decades-long suspicion that string theory, which provides absolutely no testable predictions, may be nothing but mathematics after all. If this can somehow be demonstrated, it would serve to free up the minds of some pretty smart people (like Ed Witten at the Institute for Advanced Study at Princeton) who currently are obsessively pursuing M-theory, which is the 11-dimensional version of strings I mentioned earlier.

In my mind, it's entirely possible that God considered basing physical reality on string theory, but gave it up because it gave even him headaches -- and a theory with headaches lacks beauty, and God's way of thinking always involves beauty. But if not strings, then what? Is there no way to unify gravity with quantum mechanics? Was Pauli's admonition correct after all?

If string theory bombs, then we're back to where Weyl, Einstein, Pauli and many others were 80 and 90 years ago. To be sure, we know a lot more than those folks did, but one thing remains the same -- our intellectual curiosity is simply not matched by our intellectual ability.
Straumann Again -- Posted by wostraub on Thursday, August 3 2006
Here's a new article from Norbert Straumann (University of Zürich), which was the basis of a talk he gave in 2005. Some new stuff on Hermann Weyl and early gauge theory, along with some reflections on the gauge principle in quantum electrodynamics.

Persistence -- Posted by wostraub on Monday, July 31 2006
Edison once said that discovery is 1% inspiration and 99% perspiration. Einstein asserted that persistence trumps intelligence. Weyl's efforts to bail out his 1918 metrical gauge theory certainly represents a classic example of persistence in the face of withering criticism. Weyl persisted because he believed he was in possession of the truth.

Recall that Mr Einstein rejected Hermann Weyl’s original gauge theory on the basis that it did not preserve the invariance of the line element ds under a gauge transformation. In spite of the simplicity of Einstein’s argument, Weyl tied himself in knots desperately looking for a way out. As far as I know, he tried three escape routes.

One was to assume that the ds of measurement was not the same as the mathematical ds. That is, what we measure as ds is a true invariant, whereas the mathematical version is not. This almost metaphysical option was quickly dismissed by Einstein, Pauli, Eddington and others.

Weyl then moved on to a line element that replaced the metric tensor gμν with the Ricci tensor Rμν, a quantity that is a true gauge invariant in Weyl’s geometry. This was an interesting dodge, but it too was thrown out.

Weyl’s last gasp was to make ds invariant by multiplying the metric tensor with a scalar J(x) of gauge weight –1, so that the line element now goes like ds2 = J gμν dxμdxν. After considerable thought, Weyl realized that the only reasonable J-quantity had to be the square root of Cμναβ Cμναβ, where the C-quantity is the Weyl conformal tensor (see my pdf article on this tensor on the menu to the left). This time Einstein was impressed, though to this day no one knows if Weyl’s J-quantity has any relevance in physics.

It is straightforward, if rather tedious, to calculate the equations of the geodesics associated with Weyl’s J-invariant. I did the calculation many years ago, and found that they’re completely nonsensical. I’m sure Weyl did the same calculation, and maybe that’s when he finally tossed in the towel.
Riemannian Vectors in a Weyl Space -- Posted by wostraub on Sunday, July 16 2006
I've posted the final write-up on Riemannian Vectors in a Weyl Space, which tries to address a mathematical inconsistency in the original Weyl theory (and which has nothing to do with the conformal aspects of the theory). Fixing the inconsistency leads to simple derivations of the Klein-Gordon and Dirac equations. I've also included lots of other junk as food for thought.

In this paper I've tried to include all the reasons why I think Weyl was really close to a unified theory of the combined gravitational-electrodynamic field, but please believe me when I say I have no illusions that this will ever be rigorously demonstrated -- certainly not by my overly-simplistic treatment. Feel free to criticize.

Riemannian Vectors in a Weyl Space
Houston, We Have a Problem -- Posted by wostraub on Saturday, July 15 2006
The disturbing events in the Middle East and the recent hoopla over the space shuttle mission made me think about that old science fiction movie in which astronauts take off for the moon or Mars or someplace only to learn that a world war has destroyed earth's population (along with everything else) and they have nothing to return to. I'm also reminded of the scene in Planet of the Apes (the 1960s version, that is) in which one of the returning astronauts plants a little American flag in the lifeless soil, while Charleton Heston laughs hysterically.

Now that the shuttle missions have been reduced to meaningless public-relations trips designed to see whether the ship's insulation is still intact, I am again forced to unveil the true stupidity behind America's "space travel" experience. Here it is:

1. The shuttle orbits at an altitude of about 210 miles. At that height, the force of gravity is still more than 90% of what it is here on earth's surface. It's just like an astronautic flea who "soars" above the surface of an onion by jumping onto the nearest outer skin layer. The astronauts are not in "outer space."

2. The weightlessness of the astronauts is induced solely by the orbital centripetal force. If they increased the shuttle's speed by only 2.5%, they could orbit the earth at a height of one foot.

3. Shuttle missions are incredibly boring. There's little for the astronauts to do other than maintain their life-support equipment. All meaningful on-board experiments have been done to death, including observing how crystals grow and ants propagate in microgravity. In their free time, the astronauts can look out and see the curvature of the earth. That's about it.

4. Without the protection of earth's atmosphere, dangerous levels of cosmic rays constantly permeate the shuttle and its inhabitants. Do you recall the "flashes of light" reported by Armstrong, Aldrin and Collins during their interplanetary trip on Apollo 11? Those flashes were caused by "Z particles" (cosmic ions with atomic weight around that of carbon) piercing the astronauts' corneas. Meanwhile, microgravity induces rapid loss of bone density and muscle tone.

5. There are literally thousands of pieces of space junk now orbiting the earth, from grain-size ejecta particles to car-sized failed satellites. And it's all flying around at many thousands of miles per hour -- many times faster than a speeding bullet. NASA mission planners have to keep track of every known chunk to avoid a catastrophic collision with the shuttle. One of these days they'll lose track of something, with disastrous results. All it takes is a particle the size of a grain of salt.

6. In spite of all the high-tech you see in the orbiter, its boosters and the tracking equipment, the shuttle is really nothing more than a fancy Chinese rocket utilizing 2,000-year-old chemical technology. Chlorine- and nitrogen-containing pollutants spewed out from each launch measurably impact the earth's biosphere (chlorine is about the worst thing you can imagine for the ozone layer).

7. President Bush says we're going to Mars! The round trip will take years, cost more than $100 billion, and if cosmic rays don't kill the astronauts the boredom will drive them insane. But it's a trip Americans may have to buy into whether they like it or not, as it's rumored that Mars has weapons of mass destruction. We can't just stand here and wait to be killed! They don't call it the Red Planet for nothing!

The real reason why Americans support the "space" program is that they don't know a solar system from a galaxy, a mile from a megaparsec. I constantly hear people say things like "Our brave astronauts are out there among the stars and galaxies, blazing the trail to discovery." No, they're 200 miles above your head, idiot. And if I ever hear that poem again about "touching the face of God" (in a nuclear-armed fighter jet, yet), I'll scream. And "spin-offs"? Please, did we have to spend a trillion bucks for Tang and freeze-dried ice cream?

All this nonsense, and for what? A trifling trillion dollars or so to date, and counting! The real purpose of these missions today? That's a national secret, but I can tell you that it involves looking down on you and me and everyone else, and not looking out on this wonderful universe God made for us.

My suggestion? Let's first get our planet in order, make our resources and human, animal and plant life sustainable, find a way to deal with our aggressions, then reach for the heavens. Wouldn't this please God more than what we're doing now?
Weyl and Dirac -- Posted by wostraub on Monday, June 26 2006
Someone asked me for a copy of a 1973 paper by P.A.M. Dirac today. I got it out of the garage to email, and read it again for the first time in years. In it, Dirac uses Weyl's gauge theory in an attempt to explain why the gravitational constant G should be decreasing with time. In the paper, Dirac reveals a fondness (if that's the right word) for Weyl that I had missed earlier. He even provides a counter-argument for Einstein's famous objection against Weyl's theory.

But then he goes on to describe how a non-moving charged particle in a Weyl field can be used to break charge symmetry while maintaining CPT symmetry. Dirac's argument is simple: a vector associated with a particle in a Weyl field changes magnitude according to dL = φμ L dxμ. If the particle is at rest, vector length still changes with the flow of time according to dL = φ0 L dx0, where φ0 is the Coulomb potential and dx0 = cdt. If the change in length is positive with increasing time, then it must shrink with decreasing time (and vice versa). Regardless of the convention you choose, the change in length is effected by the sign of the particle's charge. Thus, symmetry is broken between positive and negative charge. It's so beautiful.

Dirac, who won the 1933 Nobel Prize in Physics at the age of 31, was once asked if there was anyone who was so smart even he couldn't understand. "Weyl" was Dirac's answer.
Hermann Weyl Resources Online -- Posted by wostraub on Sunday, June 25 2006
Among the papers, books and articles I have collected on Hermann Weyl are a number of contributions made by the German mathematical historian Erhard Scholz of the University of Wupperthal. Scholz has written extensively about Weyl's mathematics (from a primarily historical perspective), although I find his English difficult to follow for some reason. Nevertheless, his online materials are well worth acquiring. Just Google "erhard scholz, weyl" and you'll finds lots of stuff, mostly in pdf format.

You might also want to Google "john l. bell, weyl" (presumably no relation to the John S. Bell of Bell's Theorem fame) regarding several online papers he's written on Weyl and his philosophical leanings. I really don't "get" philosophy, but it's worth checking out.

Another resource that I have not yet acquired is "Hermann Weyl -- Mathematics and Physics, 1900-1927," a 1991 Harvard University PhD dissertation by Skuli Sigurdsson. I haven't found it on any of the online dissertation libraries, so it's probably not out there. I suppose I'll have to get it directly from Harvard for much more than I care to pay. I'll let you know if I find it.

[God bless the Pasadena, California Library! It acquired a set of Einstein's collected writings (German and English translations) after a loan request I made several months ago. The collection includes many references to Weyl and his gauge theory and is just plain fun to read.]
Albert, Mileva and the Noble Engineering Profession -- Posted by wostraub on Thursday, June 22 2006
I've been reading the letters that Einstein and his first wife, Mileva, wrote to each other in the period 1914-19. This was a period of increasing estrangement between the two of them following their split-up around 1913, and the correspondence ranges from cordial to angry.

The letters take on a decidedly monetary tone after 1916, when it became apparent that Einstein would eventually win the Nobel Prize. Mileva was constantly asking for money, and Einstein provided it, often grudgingly. Indeed, the letters from 1916 to 1919 seem to be nothing but arguments over money. But Mileva was hard up, unemployed, and looking after two young children, while Einstein, not yet famous, was himself just getting by. (Einstein got the Nobel in 1921, and all the prize money, as he promised, went to Mileva. It amounted to 121,572 Swedish krona. Worth roughly $20,000 back then, it's not much today, and it wasn't that much even in 1921. Nowadays, the prize is about $1 million.)

Mileva seems to have used their two boys, Hans Albert and Eduard (nicknamed "Tete"), as a means of coercing funds from her estranged husband, but the real villain of the story is Einstein himself, who was never really cut out to be a husband or father. In the letters, Einstein frequently apologizes for having to cancel out on planned visits and such, and he seems content to simply blow kisses at them while coolly blowing off Mileva's demands for money.

Later in the decade, we see letters to and from Einstein and his soon-to-be second wife, Elsa. It's almost disgusting to experience Einstein's kissy-kissy attitude with Elsa in these correspondences, especially when one knows that this marriage was also a colossal failure. Mileva was no beauty queen, but I could never understand Einstein's attraction to that pudding of a woman, Elsa.

Anyway, I got mildly ticked off when I read Einstein's letter to Mileva dated November 9, 1918 (also Weyl's 33rd birthday!), in which he impuned all us noble engineers:
I am glad that Hans has an intense interest in something. On just what it is directed is less important to me, even if it is engineering, by God!
The nerve of the guy!

PS: Einstein's insult to the engineering profession backfired on him. Hans Albert Einstein went on to become a noted professor of civil engineering at UC Berkeley. Ha!

PPS: The letter, sent by Einstein without a return address, was opened and read by a Berlin government censor, who threatened to withhold future deliveries unless the address was clearly marked. Sounds very similar to what's going on in this country today.
Looking for Lulu -- Posted by wostraub on Tuesday, June 20 2006

The other night Turner Classic Movies reaired the 1999 documentary Looking for Lulu, a great one-hour overview of the life and works of American silent film actor Louise Brooks (1906-1985), whose character Lulu in the 1928 German classic Die Büchse der Pandora (Pandora's Box) is said to have enraptured Adolf Hitler long before Marlene Dietrich or Eva Braun came along. I'll bet anything Hermann Weyl and Albert Einstein for once agreed with Hitler on something (however, Hitler subsequently denounced the film itself as "degenerate art").

I saw the film years ago at the old Vagabond Theatre in Los Angeles and fell head over heels for this lady, whom I consider to be easily the most beautiful film actor of all time. But she wasn't just a pretty face -- she was a child prodigy, educated in classical literature from an early age, and a gifted classical dancer with an equally brilliant mind. In her early films she played a typical 1920s "flapper," but soon left for Germany to seek more demanding roles. In Germany she was known as Schwarze Sturzhelm (Black Helmet) because of her unusual coiffure.

Amazon sells the documentary DVD for $90. I burned it on DVD from the TCM airing and will send it out for a nominal fee if you're interested, provided I don't get inundated by hundreds of requests. Drop me a line.

PS: Pandora's Box is currently unavailable in Region 1 (USA) DVD format, and Kino Video does not plan to release it anytime in the near future. If you live in the UCLA area, you can attend a free screening of the film at the Armand Hammer Museum at 8 pm on July 7, 2006.

Update: The Criterion Collection will release the digitally remastered Pandora's Box on American region DVD on November 10, 2006. It will include four different musical soundtracks, the Looking for Lulu documentary, stills, an interview with Brooks, and other extras.
Feynman's Wheels -- Posted by wostraub on Tuesday, June 20 2006
While purging files from my Powerbook today, I came across a couple of pictures having to do with Caltech physicist Richard Feynman (I don't remember where I got them, but they must be fairly old, as the guy died in 1988). Anyway, here is his license plate (I would have gotten quanta for my plate, but we can excuse Feynman for the bad spelling).

This next shot of Feynman's van is interesting because it was obviously taken while parked at the Dorothy Chandler Pavilion in downtown Los Angeles. How do I know that? Because the tiered building in the background is the home of my old employer, the Department of Water & Power!

It doesn't show up very well, but Feynman's van is covered with paintings of (appropriately enough) Feynman diagrams. I wonder which auto detail shop in Pasadena did that (I might get something Weylian for my Prius).

Today's Factoid: The DWP has a really neat engineering library on the fifth floor, and I looked out from that vantage point one day many years ago to see then-Mayor Tom Bradley standing with Queen Elizabeth in the Pavilion right across the street. You don't see that every day!
Fourth Order -- Posted by wostraub on Friday, June 9 2006
One of Einstein's objections to Weyl's theory of the combined gravitational/electrodynamic field was that Weyl's field equations were of fourth, not second, order in the metric tensor gμν and its derivatives. However, variation of the fourth-order Weyl action

with respect to the metric tensor for empty spacetime gives

where the subscripted bar and double bar notation indicates partial and covariant differentiation, respectively. It is relatively easy to show that this differential equation has a non-trivial solution only when the Ricci scalar R is a non-zero constant; the second term then vanishes, and R can be divided out of the remainder, leaving a term of second order. The surviving term can be solved (it's almost the same as Schwarzschild's solution), giving the familiar expressions for the advance of Mercury's perihelion, the deflection of starlight, etc., provided R is taken as a small constant.

The great Austrian physicist Wolfgang Pauli was aware of this calculation as far back as 1921 (when he was just a 21-year-old kid), and noted that Weyl's theory was just as capable of explaining the perihelion shift and light deflection as was Einstein's theory.

The Schwarzschild-like solution includes a small repulsion term (proportional to R) that might have something to do with the observed accelerated expansion of the universe. Numerous researchers have linked this term to the cosmological constant.

It is also interesting that Weyl's theory gives an Einstein tensor with a 1/4 term (rather than 1/2). This makes it automatically traceless, a desirable feature that Einstein himself searched for in vain. No wonder Weyl thought he was really on to something!
Ray Davis Dead at 91 -- Posted by wostraub on Saturday, June 3 2006
Raymond Davis, the Brookhaven/University of Pennsylvania physical chemist and 2002 Nobel Laureate who devised accurate neutrino detection and counting methods, has died at his Blue Point, New York home. He was 91.

Davis, whom I wrote about on this site a few months ago, showed conclusively that the number of neutrinos (elementary particles first anticipated by the work of Hermann Weyl) reaching Earth from the sun is only one-third the number predicted by the standard solar model developed by the late astrophysicist John Bahcall (and a close friend of Davis). It was learned in the 1980s that the three types of neutrino can morph into one another, so out of 100 solar neutrinos emitted by the sun, only one-third will still be solar neutrinos by the time they reach Earth. In an amazing case of theoretical/experimental jousting, both scientists were proved to be right!

At the time of Davis' Nobel Prize in Physics, Bahcall said of his friend
Ray is not only an extraordinary scientific person, but also an extraordinary human being. Ray treats the janitor in the laboratory with the same friendliness and respect that he does the most senior scientist. And for that, he is loved by his colleagues.
Davis is survived by his wife of 57 years, Anna.
Hermann Weyl and Imaginary Length -- Posted by wostraub on Saturday, June 3 2006
Mathematical symmetries, like Hermann Weyl's gauge symmetry, are essentially undetectable aspects of action Lagrangians. This is the essence of all mathematical symmetries. For example, the electromagnetic four-potential Aμ has no absolute value -- an arbitrary gradient can be added to it without changing Maxwell's equations. Before the advent of the gauge revolution in physics, it was thought that the four-potential therefore has no intrinsic meaning, and that the electric and magnetic fields E(x) and B(x) represent the only true reality. Nowadays we know better; E(x) and B(x) are themselves composed of various derivatives of Aμ which, though "undetectable" in a real physical sense, is the true underlying reality. To paraphrase Columbia University's Brian Greene, trying to determine the absolute value of Aμ is tantamount to trying to figure out if the number 9 is happy.

Those of you who studied complex analysis in school may recall the theory of residues, which provides a means for evaluating certain improper integrals by integrating around the singular pole in the Argand plane. Probably the first problem you solved involved the "single pole" integral

where z is a complex quantity and i is the imaginary number (-1)1/2. It is interesting to note that Einstein's objection to Weyl's gauge theory can be avoided by an appeal to this pathetically simple equation if we identify z with the (variable) length of a vector L under parallel transplantation in a Weyl manifold. In fact, the German mathematical physicist Fritz London used this equation in 1927 to derive the quantized radii of orbital electrons for the Bohr atom in a Weyl space.

The only downside is that quantities such as vector length L and the four-potential itself become essentially imaginary quantities in Weyl spacetime. This observation has interesting consequences, and perhaps the most profound consequence is that Weyl's theory has validity only in quantum mechanics (where imaginary quantities are de rigeur), not geometry. If you have followed this site at all, then you already know that in 1929 Weyl successfully applied his gauge concept to quantum theory, where it now represents one of the most profound ideas in all modern physics.

But there are some researchers (and they keep emailing me!) who insist that imaginary vector length is ok provided the square of the length L2 always comes out real (reminiscent of the probability interpretation of the square of the wave function Ψ2, which is real). Well, I still don't know about all this, but I keep thinking about it. If one always gets L2 when doing a physical measurement, its complex or imaginary aspects are totally hidden from us because we always just take the square root, thinking that it, too, is real. For example, the square root of the real quantity |z|2 is not +/- z, but a +/- ib, where a and b are real numbers.

I'm an idiot, it's true, but you have to admit it keeps one's mind off the moronic (and criminal) antics of President Bush, whose mind (and legitimacy as a human being) are pure imaginary but whose crimes are all too tragically real.
Hermann Weyl and Steve Martin -- Posted by wostraub on Saturday, June 3 2006
The comedian Steve Martin, who was a philosophy major at California State University at Long Beach (my undergraduate school!), once said that he learned just enough philosophy there to screw him up for the rest of his life. I was luckier than he was -- not only have I never taken a class in the subject, it wouldn't have made any difference anyway, because I just don't get philosophy at all.

Stanford University Professor of Philosophy Thomas Ryckman does get it and, more importantly, one of his specialties is the relationship between the development of general relativity and the state of German philosophy in the early 20th century. He has written a book on the subject, The Reign of Relativity, in which both Einstein and Weyl play prominent roles. Einstein himself was an armchair philosopher, but Weyl was much more active on the subject. He was an early adherent of the great German philosopher Edmund Husserl, and in fact married one of Husserl's students, Helene Joseph.

Both Weyl and his wife were not only good philosophers, they were gifted linguists. In the preface to his seminal book The Classical Groups: Their Invariants and Representations, Weyl tells us
The gods have imposed upon my writing the yoke of a foreign tongue [English] that was not sung at my cradle.
(Weyl wrote this in English, not German, and it has always been one of my favorite quotes of his.)

Anyway, back to Ryckman, who in January 2001 gave a lecture at Berkeley on the influence of Husserl on Weyl's gauge idea. I will not pretend that I understand the philosophical part, as my brain is not really wired for it (and it's not a chronic "senior moment" thing for me, either; like pure math and mathematical logic, it just plain escapes me). But Ryckman's talk did provide a pretty good introduction to Weyl's gauge principle, and you just might understand the rest of it as well, especially if you have ever studied transcendental phenomenology or logical empiricism (whatever the hell they are). Here is Ryckman's lecture in Microsoft Word format: Article
Absolute Truth in an Age of Lies -- Posted by wostraub on Tuesday, May 30 2006
In a letter to Einstein dated 19 May 1918, Hermann Weyl asserted
As a mathematician, I must absolutely insist that my geometry is the true, local geometry [reine Nahegeometrie]; the fact that Riemann posited only the special case Fμ ν = 0 has no substantive legitimacy other than a merely historical one ... If in the end your views are correct concerning the actual world, then I would regret having to accuse the dear Lord of a mathematical inconsistency.
Einstein himself once stated that if his theory of general relativity (gravitation) was not correct, he would have pitied the Lord for having overlooked such a beautiful idea. This is what sheer truth and beauty does to a person -- it is so compelling that it takes on almost divine qualities, even to scientists who are otherwise devoid of any religious faith. In the purest of examples, it is completely objective, overriding any issues of ego or self-righteousness.

Another case in point: I am rereading The Physics of Immortality: Modern Cosmology, God and the Resurrection of the Dead by the noted astrophysicist Frank Tipler (he's the same guy who proved that an infinitely-long rotating cylinder could be used as a time-travel device). I am looking at it again only for the mathematics, which may or may not be relevant to the author's central thesis -- that religion is actually a branch of physics, and that we will all be resurrected to eternal life by God when the universe reaches the so-called Omega Point some umpteen zillion years from now. As a newly-minted PhD in 1976, Tipler was a diehard atheist until he experienced an epiphany of sorts while playing with Einstein's gravitational field equations.

Whether one completely agrees with Tipler or not is beside the question (as a Christian, I do not, but the stuff's interesting nevertheless). The main point is that mathematical and physical truth has a beauty to it that transcends much of what one experiences in day-to-day living. Part of that truth (at least for me) is the realization that God exists and had a purpose for putting us here in the first place (either as Adam and Eve or as a couple of enlightened Australopithecines). I'm not always sure he did the right thing, considering the mess we've made of the world, but that's another story.

Weyl's own Road to Damascus occurred in 1918, when the concept of gauge symmetry sprang into his mind. Einstein's was in the period 1913-15, when he realized that another symmetry -- spacetime invariance -- could be used to develop a theory of gravity. Both men were absolutely convinced that they were in possession of the truth, and it changed their lives forever.

I often ask myself what inspires or moves other people. Is it absolute truth, or what they themselves believe to be the truth based on what others have told them? How can we recognize absolute truth, and not be fooled by others (or ourselves) that that truth is not in fact a lie? To me, the only path is math and science, in combination with the teachings of Jesus Christ, because these things cannot lie. But not everyone finds math and science to be very interesting. Can truth be found in accounting, economics, politics, American Idol or auto mechanics? Can truth be found in the New Testament if mathematics and physics are ignored? The answer is very clear to me, but who am I to impose my beliefs on others?
Gravitational Lensing of a Quasar -- Posted by wostraub on Thursday, May 25 2006
This amazing photograph, taken by the Hubble Space Telescope, shows a cluster of galaxies (about 7 billion light-years distant) splitting the image of a single very distant quasar (about 10 billion LY away) into no fewer than five images (the bright bluish-white points of light near the photo's center). The galaxies act as a gravitational lens that imperfectly reproduces the quasar's image in a circular arc about the galactic field of view.

The photo also shows distorted images of galaxies near those responsible for the lensing. Quasars (quasi-stellar objects) are themselves the cores of galactic-sized objects containing super-massive black holes. The extreme luminosity of a quasar is powered by matter being accreted into the hole; as it spirals in, friction from the accretion heats the matter up to the point where intense x-ray and gamma radiation comes pouring out. Quasars were originally a great mystery to astrophysicists because their great luminosities didn't seem to agree with their extreme distances.

Another example of God's miracle universe. Sadly, the Bush Administration, in its hatred and fear of legitimate science, has cut funding for the Hubble Space Telescope, whose orbit will eventually decay until it burns up in Earth's atmosphere. On the plus side, the money saved will be available to help fund new wars of aggression for oil and other dwindling resources, but in the name of truth and justice and liberty and Christian goodness. But hey, whaddya want, America -- buck-fifty gasoline or a geeky orbiting science project?
Reality v. Formalism -- Posted by wostraub on Sunday, May 21 2006
In the preface to his 1917 book The Continuum, Hermann Weyl tells us
It is not the purpose of this work to conceal the bedrock on which the house of analysis is founded with a fake wooden structure of formalism -- a structure which can fool the reader and, ultimately, the author himself into believing that [formalism] is the true foundation. Rather, I shall show that this house is, to a great extent, built on sand.
Weyl goes on to say that the then-popular use of arithmetic and irrational number theory to solve the problem of the continuum should be stamped (sarcastically) with the title Pythagoras. Therein lies the seed of Weyl's thesis: much of real analysis at the time was based on circular logic; it was non-rigorous, and therefore corrupted by false or meaningless formalism. In his book, Weyl set out to make things right.

Today we have a few people around who are brilliant in both modern mathematics and physics; Witten, Penrose and Baez come immediately to mind. But Weyl was the first of this kind to come upon the scene. Trained initially as a mathematician, he immediately ventured out into physics, where he made many profound and fundamental discoveries.

Weyl was not afraid to attack what he perceived as either unintentional misrepresentation or outright lies. In his book he attempts to set straight issues that at the time contradicted unquestioned mathematical thinking that greats like Dedikind had established decades earlier. Later, when Weyl proposed his theory of the combined gravitational-electromagnetic field, he did not back down even when his theory was questioned by the great Einstein. Weyl's persistence, which was based on a firm conviction that what he had proposed was based in absolute truth and beauty, paid off when he applied his theory to the then-emerging field of quantum mechanics. Weyl's gauge principle stands today as one of the most profound tenets of quantum physics.

Who today has the courage to go up against established authority? Today we are told lie after lie by our political leaders, and we swallow every one of them, hook, line and sinker. Who among us has the courage to tell Bush and Company that they are liars and false prophets?

Weyl denounced Hitler, but had to flee his beloved Germany in 1933 because neither he nor all his brilliant contempories could stem the tide of nationalistic insanity that was sweeping the country. Not long afterward, magazines and posters were displayed with Einstein's photo and Noch ungehängt! (not yet hanged) all over Germany. Not surprisingly, Einstein too fled the country. A dedicated Nazi effort to discredit the theories of special and general relativity soon had the German people thinking they had been tricked into believing a lot of intellectual hokum. Books were burned, ideas themselves were banned, and the great edifice that was once Deutsche Mathematik und Physik was destroyed in favor of ignorant, arrogant nationalism.

Sadly, this is America today. But the danger is heightened infinitely by America's possession of 10,000 nuclear bombs, spy satellites that can watch and record everything we do and say, a hatred of legitimate science and truth, and a crazed, fanatical, nationalistic Christian millennialism that wants desperately to hasten Armageddon through unilateral, preemptive war.

It is ironic that today Weyl and Einstein would almost certainly be forced to return to Germany to escape the fascist madness that has overtaken our country.

This is "circular logic" of a most disturbing kind. God save us all.
Einstein -- Collected Papers -- Posted by wostraub on Monday, May 8 2006
I spent several hours at Caltech today perusing its copy of the Collected Papers of Einstein (writings and correspondence, nine volumes, with a few English translation versions). I went there to copy Einstein's correspondence with Hermann Weyl, only to realize that I already have most of it.

But I was unprepared for the sheer volume of Einstein's correspondence with other notables of the time. Letters in those days (this was 90 years ago) was the email of their time, and Einstein must have spent a fair amount of his free time just writing letters.

Particularly interesting are the letters to Mileva (his ex-wife) and son Hans Albert, all of which show varying degrees of the man's emotions, including warmth, concern, impatience, intractitude, and even a little hostility. Though he genuinely cared for his two sons (in 1903 he and Mileva had an out-of-wedlock daughter, Liserl, who was given up for adoption), Einstein was not a family man, and his boys must have suffered for it. Hans went on to become a professor of civil engineering at Berkeley (he developed the Einstein bed load function in sedimentation theory), while Eduard had mental problems all his life and died at an early age. The fate of little Liserl is a mystery.

The Collected Papers abounds with correspondence between Einstein and hundreds of notable scientists, mathematicians, philosophers and political scientists. It's well-organized and makes fascinating reading, if you've got the time. It's also available for purchase, but each volume runs around $100, which is far beyond my pocketbook.

As for Weyl, Einstein and my favorite mathematical physicist wrote to each other dozens of times, discussing many different topics, including Weyl's gauge theory and related/unrelated mathematics, gravitation theory, philosophy, German politics, the war, and what kind of salaries professors should be given. Weyl was also the frequent subject of Einstein's correspondence with others. It very much takes you back to a time when it appeared that Einstein's relativistic theories (and generalizations) would eventually solve all the standing problems in physics (this was before quantum theory, of course).
Wilczek on Weyl -- Posted by wostraub on Friday, April 28 2006
In October 2005, MIT's Frank Wilczek, the winner of the 2004 Nobel Prize in Physics (for discovering the principle of quark asymptotic freedom), wrote a nice tribute to Hermann Weyl. Here it is in .PDF format.
Einstein v. Weyl -- Posted by wostraub on Wednesday, April 26 2006
Partly as a means of ridding my mind of the preposterously immoral state of this country and the criminal actions of the Bush Administration, I'm writing a brief synopsis of the argument that Einstein and Weyl had regarding Weyl's early metric gauge theory.

Several people have written in, asking what was behind Einstein's objection to the theory, was it valid, did they remain friends, etc. Hardly the appropriate subject matter for the general public, but the story itself is fairly interesting, and I hope I can do it justice. I'm getting ready for a trip, but I'll try to have it up in the next few days.
Derivation of the Weyl Conformal Tensor -- Posted by wostraub on Wednesday, April 12 2006
Some time ago I mentioned the Weyl conformal tensor, which is fundamental to the understanding of gravitational tidal effects. Whereas Einstein's equation (which involves only the Ricci tensor and scalar) describes gravitational compression and compaction of matter (volume reduction via gravitational attraction), the Weyl tensor is responsible for the deformation of matter, with the initial volume of matter remaining intact. For example, if you ever happen to fall into a black hole, your body's volume will be retained but you'll be increasingly squished sideways and elongated in the direction of the hole. This rather unpleasant phenomenon, known to black hole afficionados and the cognoscenti as spaghettification, is due to the Weyl conformal tensor. Why God allows black holes to exist is anybody's guess (maybe just because they're fascinating).

How did Weyl discover this tensor? I could never find out. He seems to have simply written it down (he was that good).

Numerous people have asked me how the tensor can be derived. Since I've never seen the derivation, I'd never done the calculation, at least until now. It's much simpler than you might think.

The file conformal.pdf is on the menu to the left.
Weyl and the Question of Asymmetric Time -- Posted by wostraub on Thursday, April 6 2006
What really interests me is whether God had any choice in the creation of the world. -- Albert Einstein
In the early 1920s, Hermann Weyl discovered a new tensor quantity (the Weyl conformal tensor) which is basically the Riemann-Christoffel curvature tensor with the contracted pieces (the Ricci tensor and scalar) removed. The resulting tensor is conformal (angle preserving) as well as metric gauge invariant. Weyl must have come across the tensor while investigating the consequences of his 1918 gauge theory and its presumed (but wrong) unification of gravitation and electrodynamics, but I have been unable to confirm this.

The Weyl curvature tensor is zero for flat spacetime, but for curved manifolds it is non-zero, even in the absence of matter. The tensor is responsible for gravitational tidal effects, in which (say) a spherical collection of particles is contorted into a prolate ellipsoidal shape (although the tensor preserves the initial volume). In fact, Weyl curvature is responsible for the tidal bulge in the Earth's oceans caused by the moon's gravitational pull. By contrast, the Ricci curvature terms deform matter by gravitational compression, and volume is not preserved.

In 1979, the British mathematical physicist Roger Penrose (also a gifted science writer) announced the Weyl Curvature Hypothesis, which essentially states that the Weyl tensor was precisely zero when the Big Bang occurred and will become infinite if and when the controversial Big Crunch occurs. On the basis of this hypothesis, Penrose believes that time must be asymmetrical; that is, time proceeds from the Big Bang to the Big Crunch in only one direction. This contradicts the CPT theorem, which basically states that physics is also valid for reversed time (that is, all equations remain valid if we replace t with -t ). The laws of physics may be time direction-invariant, but on a universal scale this might not be the case. Penrose believes that a consistent quantum-gravity theory (assuming we ever come into possession of it) will demonstrate that the direction of time is really only one-way.

Whether the universe will end in a Big Crunch is debatable (current data indicate that the universe will continue expanding forever), but what is certain is that much of the matter in the late universe will coalesce into black holes. Spacetime curvature near the event horizon of a black hole is highly Weylian, so even if Penrose is wrong the totality of Weyl curvature in the late universe will undoubtedly be extremely high if not infinite.

I've mentioned Penrose before. He has two excellent (and very readable) books out: The Emperor's New Mind (Penguin Books, 1989) and The Road to Reality -- A Complete Guide to the Laws of the Universe (Knopf, 2004). The latter book is a life-altering text that should be read by everyone who has even the slightest interest in physics, the universe, and God's role in it all. Buy this book, read it carefully, and then place it next to the Bible and Hamlet on your bookshelf; you will then be able to call yourself an enlightened member of the human race.

The Weyl Curvature Hypothesis provides a direction for time's arrow, and is therefore intimately connected with the increase of entropy in the universe (as demanded by the second law of thermodynamics). Indeed, Stephen Hawking and others have proved mathematically that the surface area of a black hole's event horizon is proportional to the hole's entropy. Thus, in the late universe, the level of entropy contained in black holes will be enormous. By comparison, the entropy of spacetime at the time of the Big Bang was very low, if not exactly zero. Thus, the Big Bang and Big Crunch are distinctly different events. This calls into question the reality of a "cyclic universe," that is, one that comes into existence and then recollapses over and over.

I think Weyl would have been pleased that his curvature tensor is today profoundly associated with the fate of the universe and related unsolved problems in modern physics.
Inflation Theory Getting Dissed -- Posted by wostraub on Sunday, March 19 2006
After the Washington Post's article on cosmic inflationary theory came out a few days ago (and discussed by me below), a veritable raft of conservative idiots have come out of the woodwork proclaiming that it's just nonsense:
I just hate it when the media reports carefully vetted scientific data as fact and not as just one of many valid points of view. I'm not asking for them to ignore the opinions of these so-called scientists, but they really should report the fact there's a lot of controversy about whether this kind of evidence is valid. LIke, were you there, huh, Mr. Hotshot Washington Post? As if this ludicrous nonsense - a marble blows up like a baloon [sic] to become the entire universe in a trillionth of a second - is more plausible than Genesis? Give me a break!
To be fair, it doesn't help things when the WaPo reports that the "universe expanded from the size of a marble to the expanse of the entire cosmos in a trillionth of a second." This is just plain wrong, but I think conservatives would have a problem with the article even if it had been reported accurately.

Alan Guth's inflation theory says that about 10-35 seconds after the Big Bang, the universe began to expand at an exponential rate. This increased rate of expansion lasted until about 10-30 seconds after the Bang (a total duration of very nearly 10-30 seconds). During this expansion, the volume of the still-infant universe increased by a factor of some 1050. Expand something the size of a marble by that factor, and you get a marble the size of a globular cluster -- big, but still quite tiny compared to the size of the universe today.

But I think conservative young-Earth creationists still have problems with this. To them, there was nothing until God created it a scant 5,000 years or so ago. Adam and Eve rode to work on dinosaurs (they actually believe this), and all animals were herbivorous (even T. rex, who needed those fangs and claws to get through the notoriously chewy plants that grew back then). Then came the Fall of Mankind, when some animals literally became evil and began to eat their more peaceful cousins. What utter, puerile nonsense! "Where were you when the Big Bang occurred?" seems to be their standard question to cosmologists these days. Well, you creationists weren't around either -- so there, idiots.

I think a good part of the problem can be traced to the use of terms like "millionths of a millionth of a trillionth of a second." These terms are almost meaningless when taken out of context. I recall when my high school physics teacher told us about Planck's constant, which is roughly 6 x 10-34 joule-sec. My first thought was "Hell, that's practically NOTHING. Anything that small is really no different from ZERO." But I couldn't have been more wrong! The problem seems to be related to the fact that humans cannot imagine something as small as a decimal point followed by 33 zeros and the number 6.

Similarly, many people have a hard time imagining even a trillionth of a second, which is much larger. "How could anything happen in a millionth of a millionth of a second?" they ask. Well, an unstable particle that has an average lifetime of that long is what particle physicists call a very long-lived particle. Most unstable particles exist for much shorter periods. Some are around for only 10-20 seconds.

The 10-35 second following the Big Bang is on the same level of smallness as Planck's constant. If you want, we can sit here and talk about tiny numbers all day long -- how about 10-1000 seconds? Me, I can't even begin to imagine such small numbers. And I'm stupid at both ends -- I can't imagine what infinity looks like, either. But I know they exist.

What do I believe? If anybody really gives a damn, I believe that at t = 0, God said "Let's have a quantum fluctuation take place in this boring, expanseless nothing of a universe and get something interesting going." BANG. Later, he created creatures with wonderful minds that could actually calculate what the early universe MIGHT have looked like.

And, for better or for worse, God also created creatures that did not want to use their minds. These he called "Republicans," although in the early days they went by the names Pharisees and Sadducees.
Inflation Theory Verified? -- Posted by wostraub on Friday, March 17 2006
New data from the Wilkinson Microwave Anisotropy Probe (WMAP), a satellite launched in 2001, has provided scientists "smoking gun" proof that the so-called Inflationary Universe, first proposed by astrophysicist Alan Guth in 1981, is correct.

The data provide much finer details of the distribution of the cosmic microwave background (CMB), which is an "echo" of the Big Bang. The CMB was first detected in 1965 as a uniform, isotropic radiation representing a universal thermal background of about 2.73o Kelvin, or 2.73 degrees above absolute zero. [Note: the "degree" label o for Kelvin temperatures is always suppressed, so it's just 2.73 K.]

The microwave radiation "echo" actually occurred about 270,000 years after the Big Bang. Why is that? It's because prior to that, the expanding ball of ionized plasma and intense radiation was so opaque ("thick") that light from within couldn't get out. At around t = 270,000 years the fireball had cooled to the point where the plasma became transparent, and THERE WAS LIGHT!

Later data and refinements showed that the background radiation is not quite uniform, but "granular." This graininess was predicted by Guth's theory, but proof of the theory had eluded scientists until now. The figure below is WMAP's map of space showing cooler (blue) and hotter (red) regions (the temperature gradients are truly small, on the order of a fraction of a Kelvin). Numerous scientists have remarked that looking at the map is akin to looking at the face of God.

The inflationary theory states that the Big Bang, which occurred very nearly 13.7 billion years ago, began as a uniform expansion of spacetime which then accelerated briefly (and by briefly, I mean on the order of 10-30 seconds). This brief expansion, now almost jokingly called the "Inflationary Epoch," was due to fluctuations in what is believed to have been the false vacuum of the early stages of the Big Bang.

Columbia University's Dr. Brian Greene, a renowned astrophysicist, has called the findings "spectacular and "stunning," while Dr. Michael Turner, assistant director for mathematics and physical sciences at the National Science Foundation, called the data "absolutely amazing."

Meanwhile, concerned Republicans are scrambling to figure out how the universe could be 13.7 billion years old, when everybody knows that God created it only 5,000 years ago. Earlier today, there were indications that Senate Majority Leader Bill Frist would propose a bill that would nullify the WMAP data and imprison, torture and execute the study's research scientists as godless, freedom-hating terrorists. But President George W. Bush, who subsequently inquired about the cosmic microwave background himself, remarked "If that there kozmick microwave is a new kind of meat cooker, then I'm all fer it."
The Fly in the Cathedral -- Posted by wostraub on Wednesday, March 15 2006
I just finished reading Brian Cathcart's excellent 2004 book The Fly in the Cathedral (Farrar, Straus and Giroux, publishers), which describes in detail the discoveries of Rutherford (the atomic nucleus), Chadwick (the neutron), and Cockcroft and Walton (induced atomic fission). The book's title refers to a comment made by Rutherford, whose original atomic "plum pudding" model gave way to the correct view of a tiny, lone nucleus sitting in the vast empty space of the atom -- like a fly in a cathedral.

But the bulk of Cathcart's book is taken up by the story of John Cockcroft and Ernest Walton, who in early 1932 bombarded lithium metal with accelerated protons. It is an intriguing tale of frustration, dashed hopes, personal tragedy, and ultimate victory. The scientists' apparatus, Neanderthal by today's standards, continually broke down, adding months to their efforts (and always overshadowed by lack of funds in those early days of worldwide depression). But their efforts were repaid many times over -- they ultimately found to their utter amazement and delight that protons could split lithium-7 -- a stable element -- into two helium atoms. The Cockcroft-Walton experiment was the very first experimental observation of man-made atomic fission, the transmutation of one element into another, the first splitting of the atom.

The experiment also offered the very first practical test of Einstein's E = mc2 formula. The observed 8.5-MeV energy of each product helium nucleus balanced the books with respect to the reactant particle energies. Like Einstein had said in 1905, mass and energy are truly equivalent! In recognition of their work, the Nobel Committee awarded Cockcroft and Walton the 1951 Nobel Prize in Physics.

The book's decription of Cockcroft/Walton's discovery is nothing short of heartwarming, but it also includes tragedy. At the time of their triumph, both men lost infants to childhood disease, tragedies that nearly destroyed the men and their wives in spite of their groundbreaking discovery.

But Cathcart saves the best for last. Ernest and Frieda Walton had a long, happy marriage, and they went on to have four more children who all pursued careers in science (three in physics!) Meanwhile, John and Elizabeth Cockcroft went on to have five more children -- a scientist, an engineer, a priest, a nurse, and a teacher.

God be praised!
The Anthropic Principle -- Posted by wostraub on Wednesday, March 15 2006
For a long time I've been planning to put up an article on the Anthropic Principle, which purports to offer some proof that an intelligent, omniscient entity (let's call it God) really did engineer the universe for our benefit. There's actually quite a bit of evidence that such a principle is scientifically valid, but, until I get around to it, here's one of the more convincing arguments.

The Big Bang, which occurred about 13.7 billion years ago, started out basically as a super-hot plasma of quarks and leptons. But within 2 minutes this plasma had cooled sufficiently to allow for the formation of protons and neutrons. Using Boltzmann's equation and the fact that neutrons decay into protons, electrons and antineutrinos (the life of an unbound neutron is about 15 minutes), it can be shown that the expanding fireball at age 2 minutes was composed of about 75% protons and 25% neutrons (with relatively minor concentrations of other stuff). Thus, three-fourths of the universe consisted of hydrogen, the basic fuel of star formation via nucleosynthesis.

Gravity gradually coalesced clouds of matter into spheres and compressed them to the point where nuclear fusion began in their cores. Almost all of this fusion involves the creation of helium nuclei from hydrogen. This is the kind of fusion that mankind is now trying to duplicate for long-term energy generation.

But after a few billion years, many stars burn up their supplies of hydrogen. A star begins to cool, contracts further under gravity, and then the core heats up again as a result of the increasing pressure. The temperatures eventually get so high that the star's helium can fuse into carbon, oxygen, neon and several other low-atomic weight elements. Some of these stars eventually explode as novas and supernovas, and their supplies of carbon, oxygen, hydrogen and other trace elements is what makes up all living things. All living things on Earth owe their existence to stardust flung out by ancient star explosions. Humans, for example, are composed of about 20% carbon and 65% oxygen, a reflection of the equilibrium battles in the sun involving these two elements.

During the helium-burning phase of a middle-aged star, there are three primary reactions going on that affect the formation of carbon. One is the triple-alpha reaction, in which three helium nuclei fuse to form a carbon-12 nucleus. The carbon formed in this process, however, can be scavenged by another process in which a helium nucleus fuses with a carbon nucleus to form oxygen-16. At the same time, oxygen-16 can also be scavenged by a helium nucleus, yielding neon-20. It turns out that the rates of these competing reactions and the physical constants that determine their equilibrium points are very finely tuned; if the excited states of carbon-12 and neon-20 nuclei were only slightly different, all middle-aged stars would evolve atmospheres that are either oxygen-rich and carbon-poor or composed predominently of neon. The physics of these processes, including a few minor ones that take place simulaneously with those I've described, have been worked out by astrophysicists over the years to a gnat's eyebrow. If the fundamental physics of the elementary particles making up carbon, oxygen, helium and neon were displaced by only a few parts per million, life could not exist anywhere in the universe. Thus, the universe as we know it must have been designed, or we wouldn't be here.

However, some physicists have argued that, if the many-worlds interpretation of quantum physics is valid, then our universe is just one of an infinite number of possible universes in which the physics happened to be "just right." In the vast majority of these other universes, the physics is "off" and life does not exist. In this way of thinking, a designer God would not be necessary because life exists here because of a statistical fluke.

But this argument is flawed. The processes I described above represent only a single example of the Anthropic Principle. There are many other processes that are also finely tuned for the existence of life, and the totality of these processes makes the statistical argument really hard to accept. I have estimated that, given the totality of "life-favorable" physical processes and constants, on average only one such universe would arise out of 1040 universes. While I suppose this is still statistically possible, it seems much easier just to admit the existence of God. In my mind, these numbers imply that the non-existence of God is statistically about one in 10-40. To me, that's essentially zero.

So get over it -- God exists! Whether he/she/it is Jehovah, Allah or Brahma is purely a matter of faith. And I'll leave it at that for now.

Post Script -- Speaking of faith, where do a lot of Americans derive it? I recall about a year and a half ago a grilled cheese sandwich that sold on eBay for $28,000. Why? Because it had what appeared to be the figure of Jesus Christ on it (the pan it was cooked in was subsequently auctioned off for $20,000). Since then, I've heard of trees, pancakes, highway overpass stains, and hotcross buns that appeared to depict Christian figures. All of these events made a big splash in the media, and literally thousands of people made pilgrimages to these "holy" sites. Yet the average American knows nothing at all about the scientific Anthropic Principle, which to me represents the only rational way of looking at physical "proof" of God's presence.

Americans truly are idiots.
Saunders MacLane and Mathematics Under the Nazis -- Posted by wostraub on Sunday, March 12 2006
While researching some notes on Hermann Weyl’s tenure at Göttingen during the period 1930-33, I came across a story involving the American mathematician Saunders MacLane of the University of Chicago. MacLane was a gifted 1930 Yale mathematics graduate who wanted to further his studies by attending the best graduate school in the world. This led him, in 1931, to the Mathematics Institute at the University of Göttingen in Germany, which was then universally regarded at the finest school of its kind. Its previous and then-current mathematics instructors included the likes of Gauss, Riemann, Dirichlet, Klein, Minkowski, Courant, Hilbert, Noether, and Weyl.

MacLane’s arrival at the university coincided with the turbulent political, economic and social scenario that was plaguing Germany at the time. It would shortly result in the demise of the Weimar Republic and give rise to Adolf Hitler and the Nazi Party.

Hitler was appointed Chancellor of Germany on 30 January 1933. A month later (27 February) the Reichstag (German parliament) building was set on fire, supposedly by a member of the German Communist Party, but most probably by a mentally ill transient (who was quickly executed). Regardless of the real cause, the Nazis used the incident to foment the fear that social and liberal radicals in the country wanted to overthrow the government.

Their first step was to enact the "Reichstag Fire Decree" (Reichstagsbrandverordnung) in March 1933, which nullified many of the key civil liberties of German citizens set forth in the country’s constitution. With Nazis already occupying prominent positions in the government, the decree was used as a legal tool to suppress publications deemed unfriendly to the Nazi Party. It also allowed the Nazis to imprison anyone considered to be in opposition to their policies and actions. The decree is generally considered to be the Nazis’ first major effort to establish a dictatorial, one-party German state.

However, the Nazis feared that the Reichstag Fire Decree would be insufficient to bring about total control of the country. Consequently, on 23 March 1933 they engineered the Reichtag's passage of the "Enabling Act" (Ermächtigungsgesetz). This act provided the Nazis the means of establishing total dictatorial control by legal mandate, because now Hitler and his Nazi-dominated cabinet could enact laws without interference by (or even participation of) the Reichstag.

The formal name of the Enabling Act was the "Law to Remedy the Distress of the People and the Reich" (Gesetz zur Behebung der Not von Volk und Reich). Cleverly worded as such, the German people were quickly brought on board Hitler’s plan to protect them through harsh (but seemingly benevolent) means.

The Nazis’ third step was to implement Article 48 of the Weimar Constitution. In reaction to the instability that arose in Germany immediately following its defeat in World War I, the government had amended the constitution to provide for emergency powers that might be needed to quell public unrest, insurgency, and the possibility of civil war. Article 48 was implemented via the "Order of the Reich President for the Protection of the People and the State" (Verordnung des Reichspräsidenten zum Schutz von Volk und Staat). Under the provisions of the article, Hitler and his cabinet acquired essentially unlimited power in Germany, all done legally and within the constraints of the constitution. This effectively ended the Weimar Republic. Germany, now officially a dictatorship, thus inaugurated the Third Reich.

Now back to MacLane who, in early 1933, was working under the guidance of Göttingen assistant professor Paul Bernays. He knew Weyl well, and on one occasion attended a dance party that Weyl and his wife gave at their apartment. Though the political strife made life uneasy, these were heady times for the young MacLane, who regularly came into contact with the greats of contemporary German mathematics. He even came into the presence of Hitler and Hermann Göring at an opera, but acknowledged later that he failed at the time to recognize the inherent evil in these men. MacLane later thought about how he might have changed history if he'd carried a pistol that night!

On 7 April 1933, following the country’s short but complete takeover just a month earlier, all Jewish professors (along with everyone not considered totally committed to the National Socialists) were summarily fired. By month’s end Courant, Noether, Neugenbauer, Bernstein, Bernays, Hertz and Lewy were drummed out of the university. MacClane, forced to seek a new advisor, went to Weyl, who took him on. MacClane subsequently received his doctorate in the summer of that year, just prior to Weyl’s decision to leave the country as well (Weyl was a Christian, but his wife had a Jewish background, so their two sons were considered to be Jewish).

So, by the summer of 1933 the world-renowned Mathematics Institute of the University of Göttingen had been decimated. A total of 18 professors had been forced to leave. A similar fate befell other universities across the country.

MacLane returned home with his wife, Dorothy, in August 1933. In his own words, he had personally witnessed “the damage done to academic and mathematical life by any subordination to populism, political pressure and proposed political principles.”

MacLane ended up at the University of Chicago, where he received many awards for his work in mathematics, especially algebraic logic. He died in 2005 at the age of 95.

I bring up this little story because I see what I consider to be an irrefutable parallel between what happened to Germany in 1933 with what is happening in my own country today. By the latter part of 2001, President George W. Bush, an unsuccessful, uncharismatic, witless, incurious, unrecovered alcoholic, was well on his way to an inglorious one-term presidency. In his first eight months in office he had accomplished nothing of any importance, and his approval ratings were well below those of his unpopular, one-term father, George Sr. Then came September 11.

The Pentagon and Twin Towers attacks, like the Reichstag fire, breathed new life into Bush’s foundering presidency. And like the Enabling Act and the enactment of Article 48, the ridiculously-named "Patriot Act" and the recently uncovered governmental domestic spying scandal have given Bush and his zealots unprecented, almost dictatorial power. And, like in Germany in 1933, the American people have bought into the lies of the Bush Adminstration because they’ve been taught to be afraid.

What we’ve seen from Bush since 9/11 is an unremitting horror of illegal war, missing WMDs, legalized killing, torture, imprisonment, secret back-alley meetings and deals to benefit the corporations, environmental destruction, astronomical deficits, wiped-out surpluses, degraded civil liberties, hatred of objective scientific inquiry, mega-capitalism and materialism, and unbelievable religious hypocrisy. The United States, thanks to Bush, is today almost univerally hated around the world.

I’m going to make a prediction before I close this overly-long diatribe: I predict that within a year we will hear about a new scandal involving illegal Bush Administration spying on the election plans of the Democratic Party. While Nixon couldn’t pull it off because what he did was deemed illegal and immoral (even by Republicans), such niceties simply do not carry weight any more under the Bush Reich.

May God save this country. Even so, come Lord Jesus.

PS - You can download a five-page PDF document of MacLane's personal reminiscences of Göttingen at
"Ich fahre nach Pasadena ..." -- Posted by wostraub on Sunday, March 5 2006
Here's a brief letter (along with its English translation) that Einstein wrote just prior to his second visit to Pasadena in 1931. In it, he lauds America's love of science and its ability to balance production and consumption.

How times have changed. Americans now prefer superstition, video games and celebrity worship to math and science, while our gluttonous material appetite threatens to consume the entire world.

Exactly where and when he wrote this (and to whom) is anybody's guess. Like it? You can have it for only $23,000.
More on Neutrino Oscillations -- Posted by wostraub on Sunday, February 26 2006
Many people have written to say that they were fascinated by last week’s PBS program on neutrinos, The Ghost Particle. It is interesting to note that Hermann Weyl also made fundamental contributions to our understanding of these particles, which may be the most numerous things in the universe. In his seminal 1929 paper, Electron and Gravitation (Zeitschrift f. Physik, 330 56), Weyl was the first to recognize that the treatment of spin-1/2 particles (like the neutrino) in a gravitational field requires a covariant derivative that is appropriate to fermionic fields.

Weyl’s development of the spin connection ω a and the associated spin covariant derivative emerged from this work, as was his recognition that the zero-mass version of the Dirac relativistic electron equation allowed for a description of particles that violate parity (this is practically the de facto definition of the neutrino!) While Weyl’s paper preceded Pauli’s 1930 neutrino hypothesis by a year (and it is doubtful that Weyl had any inkling about the existence of this particle at the time), his work nevertheless provided a sound basis for the neutrino’s subsequent mathematical elucidation.

Weyl also was totally unaware of the existence of three types of neutrino or the possibility of neutrino oscillation, which was the subject of the PBS program. Whatever the physical process behind a neutrino’s penchant for converting itself into any of the three types, it is abundantly clear that a successful description will involve the dynamics of fermionic fields against a gravitational background, and this will by necessity involve Weyl’s spin connection and derivative. Not bad for a mathematican who was once scolded by Pauli for straying into the physics community!

Several people have asked me about more advanced yet readable information on neutrino oscillation. I’m the wrong person to ask, because I know practically nothing! But there are several papers I’ve collected that have helped me understand the things a tiny bit:

Dieter Brill and John A. Wheeler, Interaction of Neutrinos and Gravitational Fields (1957). Reviews of Modern Physics, 29 465. [This is probably the first article you should seek out.]

C.Y. Cardall and George M. Fuller, Neutrino Oscillation in Curved Spacetime: A Heuristic Treatment (1997). Physical Review D, 55 No. 12, 7960.

Xin-Bing Huang, Neutrino Oscillation in de Sitter Spacetime. arXiv:hep-th/0502165 v1 (12 February 2005).

Victor M. Villalba, Exact Solutions to the Dirac Equation for Neutrinos Propagating in a Particular Vaidya Background (2001). International Journal of Theoretical Physics, 40 No. 11, 2025.
The Ghost Particle on PBS -- Posted by wostraub on Tuesday, February 21 2006
I hope you all caught The Ghost Particle on PBS tonight. The ghost particles are, of course, neutrinos, first postulated by Wolfgang Pauli in 1930. Nearly massless and traveling close to the speed of light, approximately 100 trillion pass through your body every second, and "pierce the lover and his lass," to quote the famous John Updike poem.

The program chronicles the search for the solar neutrino, and focuses on the initially-contradicting data between experiment and theory. First came the theory, proposed by John Bahcall in 1964, that electron neutrinos would be produced by the sun at a rate of X per second. Then Ray Davis and colleagues built an apparatus that could actually measure the things. Strangely, their observations indicated that the sun was producing neutrinos at the rate of only X/3. Scientists around the world couldn't figure out just who was right (if either).

The Standard Model of particle physics says that electron neutrinos are massless and travel at the speed of light. But in the 1970s and 1980s two more neutrinos showed up -- the muon neutrino and the tau neutrino. Today, the family consists of electron neutrinos, muon neutrinos and tau neutrinos, along with their antimatter counterparts. Most physicists believe that no new neutrinos will ever be found.

Anyway, to make a long (but very fascinating) story short, it was later discovered that the three kinds of neutrinos can randomly oscillate from one kind into another. Thus, the number of solar electron neutrinos reaching Earth is reduced by a factor of two-thirds. Bahcall's theory was vindicated, as were Davis' experiments. Both physicists were right!

Because neutrino oscillation requires that these particles have a non-zero proper time measure, neutrinos cannot travel at the speed of light, so they must have a tiny but non-zero mass. Consequently, there was early conjecture that neutrino mass might account for the "missing mass" in the observed universe (the total number of neutrinos in the universe is almost unimaginable, so even a tiny mass would add up to something truly significant). However, it is now believed that other, more exotic forms of non-baryonic matter make up the vast bulk of the known universe's mass-energy. Oddly enough, the ordinary matter that you and I know and love (protons, neutrons, electrons, hamburgers, etc.) accounts for only about 5% of the "stuff" of the universe. The rest is "dark matter" and "dark energy." Very odd, indeed.

Although Davis (now in his 80s) is suffering from Alzheimer's disease, he was fully cognizant back in 2002 when, to the delight of his family and fellow researchers, he won the Nobel Prize in Physics for his neutrino work. He was accompanied in Stockholm by no fewer than 23 ecstatic family members -- wife, children and grandchildren. God be praised!

I recorded this excellent PBS program on DVD-R. Let me know if you'd like a copy.
Weyl's Take on the Gravitational Energy-Momentum Tensor -- Posted by wostraub on Friday, February 17 2006
Shortly after Einstein's November 1915 announcement of his general theory of relativity, Weyl attempted to derive a coordinate-invariant form of the energy-momentum tensor that expressed conservation via an invariant divergence formula. His failure to find a fully-covariant expression of this tensor puzzled many physicists at the time. And despite repeated attmepts over the years by many scientists, no one has discovered a satisfactory form of the tensor.

This is very odd, because general relativity is practically the de facto definition of invariance theory, yet something as conceptually simple as gravitational energy-momentum conservation continues to elude us.

Weyl's attempt is documented in the first edition of his 1918 book Space-Time-Matter (Raum-Zeit-Materie). It's a mess, if only because of the inconsistent index notation he used in those days. But, yes, a divergenceless energy-momentum tensor can be written down (it looks like T μν + t μν) but the quantity t μν is really only a pseudotensor -- it's not invariant with respect to a change in the coordinates, and it's not even symmetric with respect to the indices. This is very frustrating!

Long ago, I thought it might be possible to use Weyl's φ-field to derive a truly covariant form of the energy-momentum tensor. I failed in this attempt, but I'm little better than a total idiot, so it's still possible that this approach is valid. Something to think about on an otherwise cold and rainy night.
Spin Connection -- Posted by wostraub on Saturday, February 11 2006
I rewrote The Spin Connection in Weyl Space (a somewhat pretentious title, I know) and included an elementary overview of vector parallel transfer and covariant differentiation. The .pdf file is posted on the menu to the left. The typos are all fixed now (I think), but I'm washing my hands of the whole thing, as it still doesn't read the way I wanted it to. Enjoy it if you can.
DPGraph -- Posted by wostraub on Monday, January 30 2006
Years ago when I was teaching, I used a simple but powerful program for creating static and animated computer graphics called DPGraph. If you have ever attended university, chances are the program can be downloaded for free (the site has a list of hundreds of site-licensed universities, and all you have to do is click on yours [be honest, now] and the program is automatically downloaded gratis). You can also buy the thing outright for just $10. Either way, it's a bargain, and it's a blast.

Just enter a 2-D or 3-D algebraic expression on the command line, toss in a few parameters, and the program gives you fantastic graphics. It's used by countless schools (including elementary, middle and high schools), but with a little imagination you can easily create exceedingly-complex graphics and animations that look like output from a graduate institution.

The program, which only takes up about 500 KB on your hard drive, will run on any Windows OS (it also runs on my Mac Powerbook via Virtual PC, but Virtual PC is such a dog that very high-resolution graphics [especially animated graphics] take a long time to generate).

The program's documentation is minimal; there's no manual, but the HELP menu should provide all the information you'll ever need. Also, the DPGraph website has many hundreds of free downloadable graphics files that serve as useful examples of the required programming.

DPGraph is a far cry from professional programs like Wolfram's Mathematica, but that's overkill for most people, anyway (it's also a far cry from my old Keuffel & Esser sliderule, but that's another story, and I don't want to date myself too badly here).
"Now We're All Sons of Bitches" -- Posted by wostraub on Thursday, January 26 2006
These were the words of physicist Kenneth Bainbridge to J. Robert Oppenheimer, director of the Manhattan Project, immediately following the successful test of the atomic bomb at Alamogordo, New Mexico, on July 16, 1945.

Why my renewed interest in JR? I was looking over a copy of a letter Weyl had sent to Oppenheimer in 1934, and Weyl practically pleaded with "Oppie" to accept the directorship of the recently-founded Institute for Advanced Study (IAS) in Princeton. But Oppenheimer repeatedly begged off, thinking that he "would be useless" in such a place.

At the end of the war, with Germany defeated and Japan reeling under two atomic blasts and the concurrent deaths of 225,000 civilians, both Weyl and Einstein felt that Oppenheimer was no longer suitable for the IAS job, and they campaigned for Wolfgang Pauli (who also happened to win the Nobel Physics Prize that year). Nevertheless, Oppie got the job in April 1947, where he was to remain almost 20 years until his death in early 1967.

My search for a Weyl-Oppenheimer association led me to what is currently considered the definitive study of the "father of the atomic bomb" in the new book American Prometheus: The Triumph and Tragedy of J. Robert Oppenheimer by Kai Bird and Martin J. Sherwin (Alfred A. Knopf, 2005). The book has little to say about Weyl (who never worked at Los Alamos), but it gives a detailed account of the life of Oppenheimer, his early academic triumphs, his success at Los Alamos, his attempts to internationalize the science of nuclear weapons to prevent a cold war, the subsequent trials he endured as a suspected Communist sympathizer and national security threat, and his later life, which he devoted to sailing, writing, and quiet intellectual pursuits.

True to their nature, the Republicans demonized Oppenheimer as a peacenik and threat to US unilateral nuclear dominance. Eisenhower had Oppie's security clearance revoked, and the scientist was scarred by the McCarthy commie witch-hunt of 1953. It was not until the Democrats had regained power in 1960 that the US government attempted to patch up its pathetic relationship with him, and he was eventually awarded numerous honors and medals by presidents Kennedy and Johnson. It was too little, too late. Oppenheimer died on 18 February 1967 of throat cancer, much disillusioned but accepting of his fate.

I remember reading Oppenheimer's obituary in a Time magazine at my high school library. "Father of the Atomic Bomb" is about all I remember of it. Later, the noble and wise administrators at Duarte High School staged an assembly with a guest speaker who spoke to us of the benefits and charms of fascism and communism. This went on for some time, while the students all booed and hissed. Only then did the speaker reveal himself to be a true American patriot whose speech was a sham designed to expose the manifest evils behind communistic indoctrination. We all cheered. (I remember this as if it were yesterday.)

While reading the book (I have two other books on Oppenheimer, but this one's the best), I realize that it's 1953 all over again. Science is being disparaged, superstition is on the rise, crooked and corrupt politicians are again in power (perhaps permanently this time), and Americans have gone into a deep, deep sleep. The warnings of scientists like Einstein, Weyl, Oppenheimer, Bohr, Serber and many others have been totally ignored, and as a result the world is far more dangerous today than ever before. A corrupt, moronic president, in league with a corrupt but very clever and ambitious Republican Party, false Christians all, is leading us down the path to destruction. Sons of bitches, indeed.
Collapse -- Posted by wostraub on Sunday, January 22 2006
Take a test tube, fill it with sterilized nutrient broth, add a few bacteria, and incubate. The bacteria will multiply exponentially. Their waste products will also accumulate, and at some point the organisms will enter lag phase, then undergo death. The bacteria, being insensate organisms, don't know any better. As long as there's a supply of food and nutrients, they'll reproduce without any consideration of the possible consequences until either the food is gone or their wastes destroy their environment.

I've just finished reading Jared Diamond's Collapse: How Societies Choose to Fail or Succeed (Viking Press, 2005). Diamond, a professor of geography at UCLA, is a physiologist, evolutionary biologist, and a biogeographer (whatever that is). He's won numerous awards for his science and writing, including the MacArthur Foundation Fellowship. He's also the author of the Pulitzer Award-winning Guns, Germs and Steel. He is a writer to be reckoned with, and this is a fantastic book.

Collapse decribes the rise and fall of human cultures and societies like the Easter Islanders, the Maya, the Norse Greenlanders, the Icelandic Vikings, and our own Southwest American Anasazi. Each one prospered thanks to plentiful natural resources, but their populations invariably grew to the point where their respective diminished environments could no longer support them, and they quickly died off. Each tells the almost identical story of how population growth destroyed the cultures' resilience to withstand climate change and periodic drought. Without exception, each was the victim of a successive pattern of overextension and environmental destruction, followed by denial, resource wars, disillusionment, despair and even, near the very end, cannibalism.

Diamond's first case is the Eastern Island Polynesians, who came to this remote island (about 2,300 miles due west of Chile) around 900 AD. The island was small but heavily forested, including a species of vanished tree called the giant Chilean palm. Birds abounded, as did fish and shellfish, and the islanders eked out a fairly comfortable living on a diet based on limited agriculture, local birds, fish, and even porpoises. For whatever reason, the islanders began a program of competitive monument building, consisting of huge stone statues called moai and even larger stone support foundations called ahu. An extinct volcanic pit furnished an igneous rock that was easily carved, and the islanders built roads upon which they transported their massive statues by sheer muscle power. The local giant palms, with heights and diameters of up to 100 feet and 7 feet respectively, provided the construction materials needed to build the elaborate system of primitive levers, cranes and ramps required to raise the statues. Over time, many hundreds of statues were built.

Within a few hundred years, the forest was completely gone. Erosion wiped out what little arable soil the islanders had. The birds and porpoises left, and the fisheries were wiped out. The natives, desperate for food, turned to cannibalism. All this is irrefutably recorded in the island's trash heaps and burial mounds.

Ironically, in the last days the few remaining islanders overturned and/or destroyed the statues, perhaps out of rage that their priests and elders had deceived them and led them down the path to destruction.

Diamond muses over what that unnamed Easter Islander might have been thinking as he cut down the island's last tree:
Like modern loggers, did he shout "Jobs, not trees!"? Or "Technology will solve our problems, never fear, we'll find a substitute for wood"? Or "We don't have proof that there aren't palms elsewhere on Easter, we need more research, your proposed ban on logging is premature and driven by fear-mongering"?
To this Diamond might also have added "Don't worry, our gods won't fail us, the Rapture is coming, and we'll all be saved."

[BTW, for those of you who believe the Rapture is going to bail you out, you should recall the conversation Jesus had with Satan in Matthew 4:5-7.]

Diamond's book goes on to talk about modern cultures and how many are following in the exact same footsteps as the above-mentioned ancient peoples. He gives a few positive examples, like the Japanese, whose series of islands is still 94% forested, but he overlooks the fact that the Japanese are preserving their forests by cutting down everyone else's.

After checking this 570-page book out from the library, I simply could not put it down, and read it into the wee hours until I was finished. Diamond is a first-rate writer, and his narratives flow like prose. The subject material is so appropriate to the here and now that Collapse should be required reading for all high school and university students.

My only criticism of the book stems from its title, which implies that cultures can choose to fail or succeed. In my opinion, Diamond does not give an adequate description of how such decisions can be made, or even if there's evidence that today's cultures are engaged in such decision-making. As for me, the whole concept of rational decision-making lies solely in the minds of people who have been unfairly labeled liberal, God-hating pessimists. Sadly, the world is ruled by capitalist exploiters who still think that the Earth's resources are inexhaustible.

I personally see no evidence that Earth's governors are deciding anything. In spite of our sentience and ability to alter our fate, I fear we're really no different than bacteria.
Deep Down Things -- Posted by wostraub on Friday, January 20 2006
Seduced by the glowing reviews of UC Santa Cruz physicist Bruce Schumm's new book, Deep Down Things (Amazon, about $19) and, curious over the fact that the book devotes almost 80 pages to an elementary elucidation of gauge theory, I bought the thing and read it.

First off, the book is pretty much aimed at the motivated lay reader who wants to understand the non-mathematical particulars of the Standard Model of the electromagnetic and strong and weak nuclear forces (he wisely left gravity out of the loop because it's clearly beyond the scope of a book of this kind). The book includes rather extensive (though elementary) expositions on topics such as Lie groups and Lie algebra, SU(2) and SU(3) isospin and hypercharge symmetries, the weak interaction and the quark model, and by book's end I had regained much of what I invariably tend to forget about this stuff.

However, the book is inconsistent at the levels with which it treats (or should treat) complex numbers, quantum mechanical phase invariance, and related topics. For example, Schumm writes down Schrodinger's one-dimensional wave equation on numerous occasions, explaining what all the parts represent, but he doesn't feel that the reader is quite up to understanding the exponential version of complex numbers. This lack of confidence extends to his description of phase invariance, in which nary a e i θ appears in the book. This is a shame, because anyone who has even a smidgen of knowledge about z = a + ib knows that the exponential form (known as Euler's relation), which is ubiquitous in quantum mechanics, is easier to use and more intuitive. You cannot explain to someone what a unitary operator is with z! Schumm's description of gauge invariance, Weyl's brainchild, is particularly muddled. The principle of gauge symmetry is easy to understand, but not if you leave the basic math out of the discussion.

The book's last chapter, Into the Unknown, discusses a few advanced opics, along with the Higgs field and physicists' hopes to discover it with the European Large Hadron Collider, which is scheduled to go into operation in 2007. And while Schumm plays down the role of gravity in all this, he hints at the possibility that a unified theory of all four forces will radically change the way we think of everything.

The problem with Schumm's book is the same one that plagues all popularized expositions of modern physics theories -- there is precious little middle ground that these writers are willing to explore between a non-mathematical, golly-gee treatment and a higher-level textbook-like approach. In my opinion this is not Schumm's fault, but rather that of a dumbed-down reading public coupled with a rather cynical attitude of the publishers.
What a Waste -- Posted by wostraub on Monday, January 16 2006
Here's a morality play masquerading as a physics problem.

You no doubt know that controlled nuclear fusion would solve the world's energy problems for all time. Fusion is really very simple -- just get a deuterium atom and a tritium atom (these are available or easily-made isotopes of ordinary hydrogen) close enough, and they'll fuse to form helium-4 (along with a left-over neutron), with the release of lots of energy. How hard can it be to get two tiny nuclei close to one another?

Well, it's deceptive -- the Coulomb repulsion between the particles is so great that only truly enormous confinement temperatures and pressures can get them to fuse. While this has actually been done in gigantic experimental machines (like the Tokamac), the energy expended in the experiments far outweighs the energy derived from fusion. Scientists are still trying to achieve "breakeven," and as a result practical nuclear fusion is still many decades away.

But there's another way that doesn't require huge pressures and temperatures.

Take a deuterium-tritium (DT) ion with a single shared electron. Fire a muon into the ion (the muon is an elementary particle that is identical to the electron, but about 207 times as heavy). The muon knocks the electron out of the DT pair and begins to orbit the nuclei, just like its electron counterpart did. But because of its greater mass, the mean orbital radius of the muon is 207 times smaller than that of the electron. This causes the deuterium and tritium nuclei to move very close to each other. The muon's small orbital radius also effectively shields the positive Coulombic repulsion of the DT nuclei, which gets them in even closer. Within a few thousandths of a nanosecond the nuclei undergo fusion, with the release of about 17.6 MeV of energy. The muon is unharmed during the fusion event and leaves the helium-4 in search of another DT pair, where it can do its trick all over again. Because the muon comes out unscathed, this process is called muon-catalyzed fusion. It has been demonstrated many times in laboratories over the past three decades.

So what's wrong with this picture? Nothing, but there are a few technical problems that have to be overcome. One, muons have to be created, you can't buy them in stores. They come from decaying negative pi-mesons (or pions, π-), and you need a linear accelerator to get the necessary pions. Second, the muon is itself unstable and decays into an electron, a muon neutrino and an anti-electron neutrino (a muon has a typical life of only 2 microseconds). And third, once a muon catalyzes a fusion event, it often develops the habit of hanging around the helium-4 nucleus once it has formed. In view of the muon's short lifetime, this "stickiness" of the muon wastes valuable time.

However, the first and second difficulties are not all that critical -- they can be dealt with. The most critical problem is the muon's tendency to loiter around and be unproductive. A means for making unsticky muons would represent a truly profound discovery and a wonderful gift to mankind's future welfare.

There's even another particle just like the electron and the muon called the tau (τ-), which is about 17 times heavier than the muon. Tau-catalyzed fusion might someday be demonstrated, although the tau's lifetime is about ten million times shorter than the muon's.

So what's the upshot of all this? Every year, the world's nations spend nearly $1 trillion for weapons of war (about half of this amount is spent by the United States). Recent estimates (by the 2001 Nobel prize winner in economics, no less) of the actual out-of-pocket costs of the Iraq war total about $2 trillion Article. Forgetting annual US defense expenditures, what do you think we could have done with $2 trillion? Develop practical muon-catalyzed fusion, maybe? It boggles my mind to think that the US might be able to develop non-polluting nuclear fusion energy generation if it would only get its head out of its ass!!

This is just another of President Bush's outrageous legacies -- at a time when Peak Oil is rapidly approaching, and the country is in desperate need of an alternative energy source, Bush decides that what we really need to do is monopolize (i.e., steal) the world's remaining oil resources! This cannot save us, because even if Europe, China, India and the other developing countries can be held at bay, the resulting destruction of the world's economies will also destroy ours. And my guess is that the other countries of the world wouldn't stand for it -- remember, Russia still has 6,000 nuclear weapons and the missiles to deliver them.

Make no mistake about it -- the lunatic US Emperor George W. Bush is the most dangerous man in the world, and we tolerate him at our extreme peril.
Cosmic Landscapes -- Posted by wostraub on Sunday, January 15 2006
In his new book The Cosmic Landscape: String Theory and the Illusion of Intelligent Design (Amazon, about $16), Stanford physicist Leonard Susskind suggests that the universe we inhabit is only one of a nearly infinite number of "megaverses," perhaps as many as 10500. Each of these possible universes is based on a different set of fundamental physical constants, so one universe may permit life while another does not.

Susskind, a leading string theorist, does not necessarily imply that intelligent design is wrong, it's just that in his multiverse theory there's no need for it. Given an almost infinite number of possible universes, it is inevitable that at least one universe will look just like the one we live in. And there we are!

For the same reason, Susskind feels that things like beauty and elegance are also inevitable, especially when a universe contains thinking creatures. This would seem to imply that there is no such thing as "absolute truth," which is abhorrent to me, but it's still something worth thinking about.

Is it possible to flip an "honest" coin 10500 times in a row and have it come up "heads" each time? Of course! The probability is very small, but it's not zero. If there are a similar number of universes out there, all orthogonal to the one we inhabit, the chances are good that just about any kind of otherwise implausible event or condition will be observed.

Are these universes the "many mansions" that Jesus spoke about? Susskind would probably disagree, but I argue that it's equally possible that it is.

Meanwhile, purely for your enjoyment, here's the Sombrero Galaxy (M104), located about 50 million light years away from us in the constellation Virgo. This beautiful galaxy is just one of the hundreds of billions known to exist in our one universe. God be praised!

Embracing Lies as Truth -- Posted by wostraub on Saturday, January 14 2006
I haven't read James Frey's A Million Little Pieces, and I never will, but I have to comment on Frey's appearance on Larry King Live three nights ago. The fuss centers around accusations that Frey's "redemption" is based on the many lies and half-truths contained in his book, and the fact that Oprah Winfrey had championed his book on her book club. The book sold well as a result, but the allegations are pushing sales beyond Frey's wildest dreams.

Anyway, Winfrey herself called in to King's show to defend Frey and the book (and her own reputation). Instead of admitting that she had made a mistake by trumpeting a liar's work of fiction, she made two incredible claims -- that the book's publisher was to blame for any untruths in the book, and that she and Frey had apparently created a "new genre" of legitimate literature, a pleasant blend of fact and fiction. The ancient adage about turning a sow's ear into a silk purse comes immediately to mind.

By the way, Frey also brought his mommy along to the King show, no doubt to throw added weight onto the lies he was spinning.

Anyone who believes Frey's nonsense is a boob, and I guess that includes Oprah as well as most Americans, many of whom have actively campaigned to have Winfrey run for president. Winfrey herself seems to be self-delusional, thinking that her billions are proof that she's infallible. But the real blame falls on a celebrity-intoxicated American public that cannot differentiate between truth and lies anymore.

It's no wonder Bush is president.
Teach Quantum Physics in the Churches? -- Posted by wostraub on Thursday, January 12 2006
Today, two opposing members of the Ohio Board of Education appeared on Lou Dobbs Tonight to present their respective cases for and against the teaching of intelligent design in Ohio public schools.

Although ID suffered a stinging defeat in Pennsylvania last month, its adherents are regrouping and taking their arguments to school boards in numerous states -- Ohio, Georgia, Missouri, Kansas, and even California.

Dobbs listened to both sides, and at one point actually suggested teaching quantum physics alongside evolution in the schools! He then asked if it would be proper to teach comparative religion in public school. The pro-ID guest disagreed, and implied that religious education had no place in the schools. My read on this is that ID supporters would try to quietly introduce Christian education into public schools via the teaching of ID. Apparently, IDers are sold on the idea that intelligent design belongs solely to the Christian faith; comparative religion be hanged. I have yet to see a Jew, Muslim or Buddhist demand that ID be taught as a scientific discipline in public schools.

PBS' Frontline is currently running a series (Country Boys) on the problems of rural youth -- lack of jobs, premarital sex, methamphetamine abuse, depression, etc. Last night's episode took us into a rural Kentucky high school classroom where the teacher ridiculed evolution as anti-God pseudo-science: "Did Jesus Christ look like an ape? Do you think you came from a bunch of monkeys? That's not what I believe!" or words to that effect. I really felt sorry for the school's students, who undoubtedly have enough problems in their lives. Now they can add ignorance to the list.

I really like the idea of teaching quantum mechanics (at the appropriate level) in churches, because QM is undoubtedly one of the tools God uses to run his universe. Like religion, QM is based on numerous postulates that cannot be scientifically proven, and so have to be taken on FAITH. Everyone believes in QM, because modern life could not be possible without it. Why can't the IDers look at evolution the same way? Evolution is just one of God's tools to ensure the perpetuation of the planet's species.

But, as Lou Dobbs surmised today, ID ain't going away, and it will continue to itself evolve until it is absolutely disproven as a science and banned outright or legitimized and made a mainstay in the public schools. Hopefully, the legitimization of ID, like evolution, will also take millions of years.
Fascinating Physics -- Posted by wostraub on Saturday, January 7 2006
A hydrogen atom is just an electron bound to a proton. What if you replace the proton with some other positively-charged particle?

If that particle is an antielectron (or positron), you get something called positronium, or Ps. Physics Today rightly calls it "nature's simplest atom."

Electron/positron pairs are created near charged particles, but they invariably annihilate one another, resulting in photon pairs. But under the right circumstances, and for intervals on the order of 100 nanoseconds, they join to form Ps. There are two bosonic species of Ps: ortho-Ps, in which the particle spins are aligned (spin one), and para-Ps, in which the spins cancel (spin zero).

The latest issue of Physics Today (January 2006) reports that researchers at the University of California at Riverside have found indirect evidence that two Ps atoms can join to form a diatomic molecule (Ps2). If confirmed, the researchers believe that they can form a Bose-Einstein condensate at a temperature of around 15 degrees K. This in turn might then lead to gamma-ray lasers in which each photon has an energy of about 0.5 MeV. Amazing!

I imagine even George W. Bush will be interested in this, but for another reason -- gamma-ray space lasers to zap the evil-doers (set your weapon to deep fat fry, comrade!)
To The Mall, Patriots! -- Posted by wostraub on Saturday, January 7 2006
I laughed out loud when I read Jesse Eisinger's WSJ article about neoconsumerism under the Bush Reign of Terror. He calls it zombie consumerism -- for a reason you'll need to read the article to understand.

Eisinger points out that for the first time since the Great Depression, Americans spent more money in 2005 than they earned. This negative cash flow greatly increased the nation's debt, as Americans flocked to take out home equity loans to pay for all their SUVs, gadgets and credit card installments (I guess they see this as "found money" that doesn't have to be paid back). I peeked in on the Commerce Department's website back in November, and it confirmed that Americans were indeed saving nothing but spending nevertheless.

Eisinger warns that with the flattening of the real estate market, record bankruptcy rates, usury-like credit interest rates, the spiraling differential between workers' pay and lavish CEO pay, and the very real connection between stock market performance and the availability of consumer cash, the future bodes ill for Bush's wundereconomy. WSJ Article

Can't afford food? Eat EQUITY!
Weyl on Time Travel -- Posted by wostraub on Saturday, January 7 2006
While slaving away at the gym this morning, I started to think about time travel again. This is one of my favorite topics, in spite of the fact that I really don't think it's possible, at least for any massive object.

If you check out the New Testament, you'll see numerous references to God and light. I believe this connection is more than just hyperbole, because if God is purely spiritual then he undoubtedly moves on a null geodesic, which is to say that he is free to move around just like a photon of light.

A photon lives is a very strange world, indeed. Because its line element vanishes (ds2 = gμν dxμdxν = 0), it exists everywhere in the universe at all times -- past, present and future. This fact is paradoxical to us humans, because when we snap on a light, photons are created at that instant, and when they are absorbed by our cornea, they are annihilated. Clearly, then, light can be created and destroyed in a short time interval. But a photon's own existence is much different -- to a photon, it is people whose lives are sedentary and fleeting. This is nothing more than an extreme example of Einstein's so-called twin paradox, which of course is not a paradox at all when you've understood special relativity. So the saying "God is light" is probably closer to the truth than one usually imagines.

In 1994, the noted Caltech physicist Kip S. Thorne published his wonderful book Black Holes and Time Warps -- Einstein's Outrageous Legacy. Mostly a non-mathematical look at time travel through wormholes and the like, it's a fascinating read that investigates various time travel possibilities along with their inherent problems and paradoxes. Following one of his promotional lectures (I think it was one of the Leon Pape Lectures), I got the chance to talk to Thorne about time travel, quantum field theory, relativity, and life in general. But when I asked him if he himself believed in time travel, I got an elusive answer.

[By the way, I took along a copy of Thorne's 1965 first book Gravitational Theory and Gravitational Collapse. He signed it for me, and told me that he still gets a miniscule royalty check from the publishers each year from the half-dozen or so copies that they sell.]

Thorne and his British pals and colleagues Stephen Hawking and Roger Penrose are arguably the world's foremost authorities on time travel. But many years ago, our good friend Hermann Weyl also speculated on the time-travel possibilities associated with a rotating universe:
It is possible to experience events now that will in part be an effect of my future resolves and actions. Moreover, it is not impossible for a world line (in particular, that of my body), although it has a time-like direction at every point, to return to the neighborhood of a point which it has already once passed through ... In actual fact the very considerable fluctuations of [the metric tensor gμν] that would be necessary to produce this effect do not occur in the region of the world in which we live. Although paradoxes of this kind appear, nowhere do we find any real contradiction to the facts directly presented to us in experience.
In fact, Weyl took the rather extreme view that the very concept of time is illusory and an inherent debilitation of the human mind. He once famously remarked
The objective world simply is, it does not happen. Only to the gaze of my consciousness, crawling upward along the world-line of my body, does a section of the world come to life as a fleeting image in space which continuously changes in time.
Before you dismiss these words as overly metaphysical, consider the remarks of St. Thomas Aquinas (or was it St. Augustine?), who "knew what time is except when asked what it is." What really is time? Einstein treated it as the fourth dimension, but it's obviously not just another coordinate. My guess is that we will never really know until we stand before God. But will time still exist then?

The concept of time not existing (which must have been the case "before" the Big Bang) is not so far-fetched because if time doesn't exist there can be no causal loops or other time-related paradoxes. A universe without time might not be such a bad place. At least we wouldn't get old and decrepit!

On a perhaps more realistic note, consider this. Let's say that you want to travel back to Northside Square in Bolivar, Missouri at exactly 5:00 pm CST on November 5, 1955 (like Professor Brown in Back to the Future). You get into your Way Back machine, set the dials on the flux capacitor, and hit the "go" button. An instant later, you materialize in the vacuum of empty interstellar space, where you quickly decompress and die. What went wrong?

The problem is due to the fact that Earth was not at the location you've traveled to in time. Because the Earth is rotating on its axis and moving around in the solar system (which is also moving in the Milky Way Galaxy, which is also moving relative to the Local Cluster), you don't want a time machine. What you want is a spacetime machine, a device that will take you both when and where you want to go. This means that you have to know the precise spacetime coordinates of your destination, otherwise you're likely to end up in airless space or physically embedded in a tree or mountain. To get these coordinates, you need to have the world line (4-D history) of the travel-to location. But where are the world lines of all physical objects and locations in the universe maintained?

I don't believe that UFOs are Little Green Men; I prefer to believe that (if they exist at all) they are time-traveling historians or scientists who, for whatever reason, prefer to remain undetected. If way-back time travel is at all possible, future travelers in their spaceships will know that it is far safer to approach Earth from space, where any imprecision in their spacetime coordinates won't matter (unless they happen to materialize within an asteroid, but then space travel is an adventure, isn't it?)

The best reference I've found on the subject is the 1993 book Time Machines by Paul J. Nahin, a professor of electrical engineering at the University of New Hampshire. It's still available in paperback, and it's excellent.
Math and Physics Math - There's a Big Difference -- Posted by wostraub on Wednesday, January 4 2006
Many years ago, I took a graduate course on variational calculus. I had studied the same subject in my math methods (for physicists) courses, and I thought I would try out a pure math class for the hell of it. The material was familiar stuff, so I thought it would be a snap.

Boy, did I get a surprise. The professor's chalkboard scribblings were all in formal math notation, and it was a struggle for me from day one. The professor seemed to delight in sticking it to the few physics students that took the course. I ended up with a "B" for the class, my worst math grade ever.

Over the ensuing years I have tried without success to understand pure math. This is a damn shame to me, because it makes up the vast bulk of Weyl's work. For the same reason, I've had difficulty following John Baez's excellent but otherwise unintelligible online notes on higher gauge theory (actually, anything that doesn't directly involve physics gives me fits). My pure-math stupidity seems also to be the main reason why I've had such a hard time understanding string theory.

I'm not giving up, but the time is rapidly approaching when I will do just that. I just turned 57, so not only are my little grey cells getting worn out, there's just not that many of them left now. And in not too many more years, I probably won't give a damn about even thinking anymore -- just like a Republican.