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 g0μ "boundary" elements include the potential Aμ by way of the identifications g0μ = 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 g0μ 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]
 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 where 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.
 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.
 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. Article
 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
 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?
 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.
 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!
 "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 abλ 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.