My work always tried to unite the
Truth with the Beautiful, but when I
had to choose one or the other, I usually chose the Beautiful.

 Hermann Weyl
 

I died for Beauty, but was scarce
Adjusted in the tomb,
When one who died for Truth was lain
In an adjoining room
Emily Dickinson

Hermann Klaus Hugo Weyl (1885-1955). German mathematical physicist. In 1918, proposed an early form of gauge symmetry in an attempt to unify electrodynamics and gravitation. Subsequently applied a similar approach to quantum physics and discovered what is today considered one of the most profound and beautiful concepts in modern physics ― the principle of gauge invariance.

I discovered Hermann Weyl (pronounced vile) and his ideas on gauge invariance in 1975 after stumbling across Misner-Thorne-Wheeler's massive Gravitation during a one-week work assignment in Lone Pine, California (population maybe 1,000). Miraculously, the town's miniscule public library somehow had this book, which is now regarded as a classic graduate text on geometrodynamics, also known as general relativity. I checked out the book and brought it back to the hotel room to read in the off-hours. The book took immediate and total possession of me, and motivated me to learn everything I could about general relativity.

But Gravitation is not an easy read, and I was forced to seek out simpler texts. Fortunately, I came across another book, Adler/Bazin/Schiffer's Introduction to General Relativity, which still stands out in my mind as the best relativity text of its kind. Even better, it had a chapter on unified field theory, including Weyl's 1918 theory of the unified electrodynamic/gravitational field. For whatever reason, the theory's mathematical beauty absolutely fascinated me. I had known about local and non-local phase invariance from my studies of quantum mechanics, but I was not aware that Weyl's theory was the origin of this powerful symmetry in quantum physics. I have since read all of Weyl's books and many of his papers, and my fascination with the man and his theories continues to grow.

In his 2002 biographical memoirs, the great contemporary mathematician Sir Michael F. Atiyah praised Weyl as the discoverer of the gauge concept and as the driving force behind the current emphasis of gauge theories on modern theoretical physics:

The past 25 years have seen the rise of gauge theories--Kaluza-Klein models of high dimensions, string theories, and now M-theory, as physicists grapple with the challenge of combining all the basic forces of nature into one all embracing theory. This requires sophisticated mathematics involving Lie groups, manifolds, differential operators, all of which are part of Weyl's inheritance. There is no doubt that he would have been an enthusiastic supporter and admirer of this fusion of mathematics and physics. No other mathematician could claim to have initiated more of the theories that are now being explored. His vision has stood the test of time.

This website is my feeble attempt to document Weyl's ideas and thoughts on gauge symmetry in a manner that will be accessible to anyone with a basic understanding of calculus. Not a lot has been written about the original theory's underlying mathematics, and I wanted to provide a fairly detailed and complete mathematical description for those who want to learn about Weyl's ideas and to appreciate the beauty of his gauge theory. As this site progresses, I will also include discussions of other topics in mathematical physics (as well as some related scientific philosophy) which exhibit a similar mathematical beauty and elegance.
 
 
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Wilder and Wilder -- Posted by wostraub on Wednesday, March 10 2010
The journal New Scientist reports that renowned Oxford mathematician Roger Penrose has a new book coming out—with another one coming out after that. Amazing, considering Penrose is pushing 80 now.


Sir Roger Penrose (born 1931)

As I think I've mentioned earlier on this site, Penrose is the author of the Weyl curvature hypothesis, which attempts to explain the relationship between entropy, the arrow of time and cosmological gravitation. It is really nothing more than the observation that entropy, which was next to zero at the time of the Big Bang, coincided with a vanishingly small Weyl curvature tensor, which describes gravitational tidal forces. Naturally, the tidal force at the universe's birth would have been miniscule; over the intervening 13.7 billion years, entropy has grown, while Weyl tidal forces have increased because of cosmological gravitation. Penrose believes that Weyl curvature will may even become infinite in the far distant future of our universe.

What does this mean? Penrose thinks that for the second law of thermodynamics to exist (and to have a universe resembling the one we now see), God (or Whoever or Whatever) would have had to choose a single cell of phase space out of the approximately 1010123 cells that make up the total volume of the observable universe (the total phase space is essentially the number of coordinates needed to describe all the particles in the universe). Anyone who has ever studied statistical mechanics will be blown away by such preposterous odds. If Penrose is right, it says to me that the likelihood that our universe came into being by statistical chance alone is practically nil. [On the other hand, an infinite multiverse could easily give rise to an initially low-entropy universe (actually, an infinite number of them!) so it cannot be taken as a proof of the existence of God.]

Penrose presented his hypothesis in two of his books, The Emperor's New Mind and (one of my favorites) The Road to Reality. The first can be read by anyone, while the second requires earnest curiosity and patience from non-experts.

As New Scientist observes, The Road to Reality was wildly popular in spite of the fact that it consists of more than 1,000 pages of fairly dense mathematics and convoluted diagrams. Perhaps it was popular because it is one of those books that are good for one's soul. Folks, the issues that Penrose addresses involve the very meaning of existence, and so should be important to everyone.

Penrose's upcoming book is Cycles of Time: An Extraordinary New View of the Universe. According to New Scientist, Penrose believes he has found a way around the cosmological heat death issue, which is thought to result from dark energy expanding the universe to the point of boring near-nothingness. Penrose thinks that the only particles in existence at that far-off time will be massless, traveling at the speed of light. If he is right, then the flow of time will cease to exist (think about it). According to Penrose, this will then allow for more creation events like the Big Bang—unending universes, again from nothingness.

And Weyl curvature—will it then somehow revert from being near-infinite back to zero? Or do we just start with new universes? Surely Penrose will address these questions.

Amazon doesn't have the book yet, and I've seen no publication date. But I'll be one of the first to get it when it does come out. Update: Found it.

Today's Tom Tomorrow -- Posted by wostraub on Tuesday, March 9 2010


I always thought the Supreme Leader's spatial coordinates of origin described hawaiiglox, but kenyaglox is a distinct possibility.

While I don't normally watch Glox News, I admit to having a crush on the pretty blond glox with the enormous orbs.

Meitner and Noether -- Posted by wostraub on Tuesday, March 9 2010
Speaking of Lise Meitner, here is the July 12, 1933 testimonial that Hermann Weyl wrote on behalf of another prominent Jewish researcher, the noted mathematician Emmy Noether:
Emmy Noether has attained a prominent position in current mathematical research – by virtue of her unusual deep-rooted prolific power, and of the central importance of the problems she is working on together with their interrelationships. Her research and the promising nature of the material she teaches enabled her in Göttingen to attract the largest group of students. When I compare her with the two woman mathematicians whose names have gone down in history, Sophie Germain and Sonja Kowalewska, she towers over them due to the originality and intensity of her scientific achievements. The name Emmy Noether is as important and respected in the field of mathematics as Lise Meitner is in physics.

She represents above all "Abstract Algebra." The word "abstract" in this context in no way implies that this branch of mathematics is of no practical use. The prevailing tendency is to solve problems using suitable visualizations, i.e. appropriate formation of concepts, rather than blind calculations. Fräulein Noether is in this respect the legitimate successor of the great German theorist R. Dedekind. In addition, Quantum Theory has made Abstract Algebra the area of mathematics most closely related to physics.

In this field, in which mathematics is currently experiencing its most active progress, Emmy Noether is the recognized leader, both nationally and internationally.
It is interesting to note that Weyl's testimonial, which was subsequently submitted to the Nazi Reichsministerium in Berlin, failed in its attempt to save Noether from dismissal from her university teaching job by the Nazis (in April 1933, they initiated a the summary firing of all Jewish teachers and public service workers in Germany).

Meitner and Noether were both secular, non-practicing Jews, but of course it was their race alone that motivated Nazi hatred against them and their kind. But what made Noether's tenure even more unstable involved her reputation as an anti-war pacifist.

I'm guessing, but what probably "saved" Meitner (if that's the right word) was the fact that she was a noted nuclear physicist and close friend and colleague of the brilliant radiochemist Otto Hahn, who also spoke to the Nazis on her behalf. Meitner remained in Berlin until 1938, when even the great Hahn could no longer protect her.


Emmy Noether (1882-1935) shortly before her death

In view of Weyl's mention of Meitner and his references to the implication that abstract algebra is not "practical," it seems also probable that the Nazis at the time simply valued nuclear physics more than mathematics, although in 1933 no one could have known about the fissile properties of uranium 235. That was not elucidated until 1939, and even then it is doubtful that Germany recognized its awesome potential as bomb material. It is ironic that, had the Nazis retained the services of the many brilliant Jewish scientists and mathematicians that they kicked out, Germany's atomic experiments might have had a much more frightening outcome.

I find it also ironic that Meitner routinely worked with radioactive materials but lived to be almost 90, unlike Marie Curie, whose exposure to radium was almost certainly responsible for her death by anemia at the age of 66. By comparison, Noether died at the relatively young age of 53, felled by infection following routine surgical removal of an ovarian cyst. In view of this, and speaking as a non-mathematician myself, I'm tempted to believe that mathematics is indeed more hazardous than physics!

Lise Meitner Book -- Posted by wostraub on Friday, February 26 2010

Lise Meitner as a PhD student, about 1905

Austrian-born Jewish nuclear physicist Lise Meitner was perhaps second only to Marie Curie in terms of sheer experimental brilliance. Author Ruth Sime has written an excellent biography of the scientist in her wonderful book Lise Meitner: A Life in Physics, and it's recommended reading.



If you don't have the time to watch the entire video, here's a little bit about Meitner's life—

After receiving her PhD in physics in 1906 at the University of Vienna (only the second woman to do so), Meitner went to Berlin, where she eventually went to work with the gifted German radiological chemist, Otto Hahn. When Hitler assumed power in Germany in January 1933, Meitner was doing cutting-edge nuclear research under Hahn who, though a Christian German patriot, protected Meitner as best he could against the summary firing of Jewish scientists by the Nazis. She was advised to leave Germany by many emigrating physicists, including Hermann Weyl, who notified her of a junior professorship in America at Swarthmore College. But the great German physicist Max Planck assured Meitner that the Nazi storm would soon blow over, so she remained in Berlin.

But Planck was wrong. By 1938, emigration of all remaining scientists and intellectuals was banned, and Meitner, lacking the exit visa she could have easily obtained earlier, and facing increasingly rabid anti-Jewish persecution by the Nazis, found herself trapped in Germany. At great personal risk, Meitner managed to flee to still-unoccupied Holland by bribing some border guards.

Just before Christmas 1938, Meitner went hiking in the snow with Otto Frisch, a fellow physicist who was also her beloved nephew. They had kept up with the work of Meitner's prior supervisor Otto Hahn, who was having difficulty understanding some of his lab results. In a brilliant flash of mutual insight, Meitner and Frisch did some calculations on a few scraps of paper and discovered what what actually happening in Hahn's lab. These calculations, which utilized Einstein's famous E = mc2 for the first time as an experimental tool, showed that Hahn's research had resulted in nuclear fission—the splitting of the uranium atom.

Meitner was subsequently invited to work on the Manhattan Project in Los Alamos, New Mexico. But she was vehemently opposed to the military application of nuclear fission, which she feared had the capacity to ultimately destroy the world. "I will have nothing to do with a bomb!" she declared.

Inexplicably, Hahn alone received the Nobel Prize in Chemistry for his work in 1944. Meitner's contributions, which were fully equal to Hahn's, were not even mentioned. Despite this enormous professional slight, Meitner remained a close friend with Hahn. Traveling to the United States after the war, Meitner was greeted as a hero, receiving numerous awards, citations and honorary degrees. This reception, to a great extent, helped her overcome the disappointment she felt for having been robbed of a Nobel prize.

Meitner traveled to Cambridge, England, where she died in 1968, a week before her 90th birthday. On her gravestone is engraved
LISE MEITNER

1878-1968

A PHYSICIST WHO NEVER LOST HER HUMANITY

What Time Is It? -- Posted by wostraub on Friday, February 26 2010
The objective world simply is; it does not happen. Only to the gaze of my consciousness, crawling upward along the life line of my body, does a section of this world come to life as a fleeting image in space which continuously changes in time. — Hermann Weyl
What Weyl meant in this famous quote is that time, which is necessary for all change to occur, may only be a phenomenon of the mind. Weyl thought that perhaps the external universe is a fixed tableau of space, energy and matter that only appears to evolve as our mental processes progress through it. In this sense, time is a purely human construct with no physical existence of its own.

Caltech physicist Sean Carroll is the latest scientist to tackle the question of time. In his new book From Eternity to Here: The Quest for the Ultimate Theory of Time (which I have not yet read), Carroll addresses the question of time and its relationship with entropy, which involves the tendency for disorder to increase with time.

String theory aside, physicists believe we live in a 4-dimensional world composed of ordinary 3-space and time. But unlike the familiar up, down, sideways and back and forth of space, time as a dimension escapes understanding, except as a mathematical construct. St. Augustine of Hippo was perhaps the first modern person to address the puzzle of time, when he noted (and I paraphrase) "I know what time is when I don't think about it, but when I think about it I don't know what it is."

Well, whatever time is, it may in fact prove to be something along the lines indicated in Weyl's quote. It would appear that sentience is related to time in a very intimate way, so it is possible if not probable that a mouse or a leopard does not truly perceive time in any meaningful sense—all time is pretty much the present as far as unconscious entities are concerned. Recall the final words of Robert Burn's poem To a Mouse:
Still you are blessed, compared with me!
The present only touches thee:
But oh! I backward cast my eye,
On prospects drear!
And forward, though I cannot see,
I guess and fear!
So it may be that time is a purely human phenomenon which, like much of physics, cannot truly be understood.

But I feel one thing is certain: when we die, we no longer exist, and whatever consciousness we may retain (if any) is instantly whisked to what the physicist Frank Tipler calls the omega point, the final destination of everything and everyone. I liken this to the Lorentz transformation taken to its ultimate extent, where v = c and the concepts of space and proper/coordinate time truly end or becomes meaningless. Perhaps eternity itself is nothing more than a space-time inhabited only by entities traveling on null geodesics.

If this point of view is anywhere near the truth, then it's no wonder we remain flummoxed by the nature of time. And, at the same time (!), it might also explain the essential ineffability of God.

Weyl's Cones -- Posted by wostraub on Monday, February 1 2010
Spinors are mathematical quantities that represent the middle ground between the vectors and tensors of ordinary geometry. Unlike the rotation of vectors, spinors are characterized by the fact that they require a rotation of 720o (4π radians) before they return to their original position; a rotation of 360o only reverses their direction, so that Ψ(θ) → − &Psi(θ+2π). In mathematical physics, spinors are used to describe particles with intrinsic spin; for example, electrons, protons, and neutrons are all spin-½ objects.

This odd behavior has surprising analogs in the ordinary world. For example, lay a cup in the palm of your right hand, with your right arm stretched in front of you. Now rotate your hand and arm toward your body and then away, twisting your hand back to the front again, keeping the palm up (your elbow is now also pointing up). The cup has undergone a rotation of 360o (2π radians), but your right elbow is now pointing painfully up as well. Now, sweep your arm counterclockwise over your head, and continue the rotation until your hand and arm return to their starting point. Getting things back to normal took a rotation of exactly 720o (4π radians).



There are other examples, but around 1929 Hermann Weyl came up with a far more interesting (and less painful) analog.



Take two identical cones with a vertex angle of φ and fix their tips together with a flexible cord so that the upper cone is free to roll without slipping on the other. After a full rotation, how many degrees has the base of the upper cone undergone?

Naively, one would think 360o. But this overlooks the fact that every point on the upper cone's surface undergoes both a rotation about its own axis and a translation about the fixed cone. To answer the question, we have to consider the movement of the base of the moving cone, which is located a distance r + R = R[1 + cos(φ)] = 2R cos2(½φ) from the center of the fixed cone's base. The ratio of the circumference of this radius to the cone's radius is therefore 2 cos2(½φ), which is also proportional to the actual circumference that the cone traverses. Note the presence of a half angle, which is a signature of spinors.

When φ = 90o, the upper cone's base is perpendicular to that of the fixed cone, and the cone rotates exactly 360o. But when φ approaches zero, the cones, which now resemble thin needles, describe a rotation approaching 2(360) = 720o. This is the spinor case.

Weyl also noted that when φ approaches 180o, cos (½ φ) → 0, and the cones are so flattened that the upper cone just wobbles atop the other, and a full rotation approaches zero degrees!

The noted British mathematical physicist Roger Penrose tells an abbreviated version of this story on Page 41 of Hermann Weyl, 1885-1985: Centenary Lectures. Another good reference is Christoph Schiller's mammoth (164 MB, 2,000 pages and counting) and free online physics text, Motion Mountain. If you're ever stranded on an island, this is definitely a book you'll want to have.

My article Weyl's Spinor and Dirac's Electron Equation describes the mathematics of spinors in much more detail.

Another Hermann -- Posted by wostraub on Saturday, January 23 2010
Recall your earliest days when you studied calculus in college (it could also have been high school). When you got to the subject of double integrals, you probably did not note that the quantity dx dy in the integrand was not the same as dy dx. As an engineering student, it sure as hell didn't get my attention, and I'll bet that few of my professors were aware of it, either. About as far as the professors would go was to say that we could write it either way, but we should always avoid writing dx dx and dy dy, as they were either meaningless or vanishingly small.

However, if you went to a really good school, your professor would have told you something along these lines: The quantity dx dy represents an infinitesimal area, and area is directional—that is, there is a unit vector n1 that points away from one side of the area, and there is another vector n2 that points away from the other side. Clearly, n1 = - n2, so it makes sense to think that dx dy = - dy dx. But can this be proved mathematically?

Born in 1809, Hermann Günter Grassmann was the son of a German minister who taught mathematics and physics (isn't it neat that enlightened people like that existed once?) In due course, Grassmann became an accomplished philosopher, linguist, physicist, humanist, and all-around scholar. He did not get interested in mathematics until his early 20s and, though he was brilliant in the field, his convoluted writing style prevented his ideas from being given much credence during his lifetime. Anyway, in 1844 Grassmann wrote his magnum opus, The Theory of Linear Extension, a New Branch of Mathematics. It introduced to the world a lot of new stuff, including 3+n-dimensional manifolds, tensors, and the concept of a vector space, which Hermann Weyl would later recall as an "epoch-making" endeavor. But it also introduced a new kind of number, which today we call a Grassmann number. Grassmann numbers are unusual in the sense that they are very simple to learn and use but are unlike anything you would have ever dreamed up yourself.


Hermann Grassmann, 1809-1877. Not just another pretty face!

Simply stated, Grassmann numbers are anticommutative under multiplication; that is, if A and B are Grassmann numbers, then AB = - BA (and consequently AA = 0). Perhaps you're saying that this is no big deal, since matrices can behave the same way. But it is a big deal, because while matrices may or may not anticommute, Grassmann numbers always anticommute. And strangest of all, they are just numbers, not matrices.

You may have been taught that dx is just a differential, but in a space of more than one dimension it magically becomes a Grassmann number. I will give you the simplest demonstration of how this comes about. Consider a coordinate transformation from polar to cartesian, where we have the familiar relationships x = r cos θ, y = r sin θ. Differentiating, we get

dx = cos θ dr - r sin θ dθ
dy = sin θ dr + r cos θ dθ

Now multiply dx times dy, keeping all the differential terms in their exact order. You get

dx dy = sin θ cos θ dr2 + r cos2θ dr dθ - r sin2θ dθ dr - r2 sin θ cos θ dθ2

Now, if all the polar differentials are taken as Grassmann numbers, we will have dr2 = dθ2 = 0 and dr dθ = - dθ dr, which leaves us with

dx dy = r dr dθ

which is the correct expression for differential area in cartesian coordinates. And this is the real reason why we don't put things like dx dx under the integral sign!

Because of their anticommuting nature, functions of Grassmann numbers are simple. For example, the exponential function exp(aB), where a is an ordinary number and B is Grassmannian, is just exp(aB) = 1 + aB. And any function of two Grassmann numbers can be written as f(A,B) = a0 + a1A + a2B + a3AB. They don't get any simpler than that!

If you're wondering where these wonderful numbers come from, don't ask me—I'm just as bewildered as you are. Maybe God got bored one day, and said "What the hell, I'll make those, too." But they're the basis of what is called external algebra, and they play an important role in the algebra of differential forms. Grassmann numbers also lie at the basis of fermionic quantum field theory, an important application that Grassmann himself could never have imagined.

I won't go into the details here (you can get them in my write-up on Gaussian integrals), but where else can you find an algebra where the integral is an operator, equal to its own derivative:

∫ dθ = ∂/∂θ

I was impressed when I first studied complex analysis, which shows that the imaginary number i = (-1)1/2 has as much to do with reality (and perhaps more) as any real number. But I think Grassmann numbers have it beat hands down.

Reinventing Gravity -- Posted by wostraub on Wednesday, January 13 2010
In 1918, Hermann Weyl modified Einstein's then-new general relativity theory in a failed effort to unite it with electromagnetism. Then Kaluza and Klein tried it, then Schrödinger, then Pauli, and then, ironically, Einstein himself, whose failed effort spanned the final 30 years of his life. Along the way, an untold number of lesser physicists also tried their hands at the problem.

In the 1980s, quantum theorists noticed that their evolving string theories actually demanded a spin-2 particle, which they obligingly identified as the graviton, the hypothetical quantum of the gravitational field.

More recently, University of Toronto gravity theorist John Moffat has also tackled Einstein's theory, albeit with a more modest (if that's the right word) goal, which is not to embed electromagnetism into the theory, but to explain the so-called "Pioneer anomaly" and maybe even the apparent acceleration of the expansion of the universe.

In his 2008 book Reinventing Gravity: A Physicist Goes Beyond Einstein, Moffat describes his effort to derive a variant of Einstein's theory by way of an action Lagrangian that includes a variety of scalar, vector and tensor terms alongside the usual Einstein-Hilbert term. Judging by the theory's seeming agreement with observation, it looks like a pretty good theory (see the August 2009 paper written by Moffat and co-author Viktor Toth). Perhaps best of all, the theory dispenses with the need for "dark matter."

While the Moffat-Toth paper is clear and readable (any undergraduate can follow it), in my opinion it succeeds by throwing everything into the Lagrangian. Dirac tried a similar approach, which included a scalar field term in the action, but in Moffat's theory we have it all: scalars, vectors and tensors, along with mass terms for the scalar fields. Even the gravitational "constant" G is a field, as is the mass μ of the vector field! By comparison, the current action for the standard model of quantum field theory has something like 40 terms in the Lagrangian, so perhaps Moffat's theory isn't so complicated after all. But still ...


And this is only part of it!

I can't help but think about the old practice of curve fitting, which involves fitting experimental data to empirical mathematical expressions that often have no theoretical basis (for example, one can describe the relationship between water vapor pressure and temperature fairly accurately with a simple parabolic curve). But curve fitting requires parameters that have to be adjusted by hand to fit the data. (In his book, Moffat states that his theory requires no parameters at all, so maybe he's on to something.)

Many physicists have expressed their hope of one day having a unified theory of everything, a theory that's conceptually so simple that it will fit into a single line of mathematics that can be worn on a T-shirt. Moffat's theory, if correct, would require a mighty big T-shirt (I take 42 large), whereas string theory will take one the size of Nevada.

By the way, I've communicated with Viktor Toth a number of times, and he's a really neat guy. He's a Hungarian-Canadian computer/software expert and author whose website reveals the same love of physics I have (be sure to check out his short physics and math articles).

Cracking the Code -- Posted by wostraub on Wednesday, January 6 2010
The subject of Fulvio Melia's 2009 book Cracking the Einstein Code: Relativity and the Birth of Black Hole Physics is really New Zealand physicist Roy Kerr, who in 1963 found an exact solution to Einstein's gravitational field equations for a massive spinning object. Mostly non-technical, the book's 150 pages can be read in a few hours, and it's worth the time and effort. (For related information, see my 10 October 2009 post.)

Melia's run-up to the subject of Kerr and black holes includes some truly fascinating history on the first tests that were performed on Einstein's theory, which was published in November 1915. The very first test was not a test at all, but an explanation for an astronomical puzzle that had vexed astronomers since Newton's day. We tend to visualize these early observers with their clunky, primitive reflecting telescopes, whose primary mirrors were made of polished speculum metal, but that view is wrong. The accuracy of the equipment and the veracity of the astronomer's calculations (laboriously done by hand) almost defy description: astronomers had to take great pains in making their observations and correcting for the gravitational effects of all the planets on one another to determine the true orbital motions. But by the mid-1700s, the Keplerian ellipticity of the orbits of the planets was amply confirmed, and the calculated positions of the planets from day to day and from year to year exactly matched observation. The Newtonian "clockwork universe" seemed to be assured.

But then, when improved observational equipment became available in the early 1800s, it was noticed that the observed position of the planet Mercury (which is closest to the Sun) did not quite match the calculations. Again, the astronomers checked their orbital calculations and compared the result to what they saw in the telescopes. By extrapolating the minute discrepancy between Mercury's observed and calculated position for a period of one hundred years, Le Verrier found that the planet's orbit was 35 arc-seconds off. Decades later, Newcomb's refined calculations showed a discrepancy of 43 arc-seconds. By comparison, the Moon subtends an angle of about 0.5 degree in the sky, or 30 arc-minutes, about 40 times Mercury's orbital discrepancy for an entire century!

Continued observation conclusively confirmed the 43 arc-second figure, and astronomers were at a total loss to explain it. Some postulated the existence of an unseen planet ("Planet X" or "Vulcan") that orbited in sync with Earth but always behind the Sun, perturbing Mercury's orbit but otherwise eluding detection. Others suggested that asteroids or dilute solar-system dust might provide the explanation for the perturbation of Mercury's orbit. But nobody dared think that there might be something amiss with Newton's physics.

Within a few months of Einstein's 1915 announcement of general relativity, the German physicist Karl Schwarzschild solved the Einstein equations exactly for the simple two-body orbital case, and discovered that planets did not rotate about the Sun in perfect ellipses, but in slightly precessed elliptical orbits. That is, after completing one orbit the planets' positions would be slightly advanced. Schwarzschild easily calculated the advancement for Mercury, and found it to be 43 arc-seconds. He had found the answer to Mercury's anomalous orbital behavior! When he communicated this finding to Einstein, Einstein was so giddy with joy and excitement that he could not sleep for several days. [Some say that Einstein himself, armed with Schwarzschild's metric, calculated Mercury's orbit. Either way, it's a relatively simple calculation, so I'm giving Schwarzschild the credit.]


Karl Schwarzschild (1873-1916). German physicist and a
tragic casualty of World War I

Schwarzschild's analysis did not include the effect of the Sun's rotation on surrounding space-time. Several researchers, notably H. Thirring and J. Lense, tried to incorporate the angular momentum of a spinning, gravitational mass into the field equations but, with the exception of a first-order approximation, none were successful.

Then in 1963, Roy Kerr, at the time a physics professor at the University of Texas in Austin, came up with an exact solution. By comparison with the Schwarzschild solution, which any undergraduate can now duplicate, the Kerr metric is devilishly more complicated. It describes space-time in the vicinity of a rotating mass, and as a result it postulates wholly-new and unprecedented insights into the nature and topology of space-time, angular momentum, and kinetic and potential energy. In particular, it can be shown that space-time itself is actually "dragged" around a rotating black hole, so that the concept of inertial rest becomes meaningless. Furthermore, it has been shown theoretically that the rotational energy of a rotating black hole can be extracted to do useful work, with the attendant depletion of the hole's mass, in exact accordance with Einstein's E = mc2 law.

Kerr's work also finally allowed the complete description of a black hole, whose only three parameters are mass, angular momentum, and electric charge (hence the adage "black holes have no hair," meaning that they're actually very simple objects*). In 1965, E. Newman and his collaborators used the Kerr metric to derive the metric of a charged rotating black hole, which predicts even more phenomena.

Kerr himself wrote the book's Afterword, where he shares some interesting anecdotes about his life in general and his discovery in particular. Highly recommended.

* It is said that when John A. Wheeler coined the term "black hole" in December 1967, French physicists were upset, believing that the term carried a sexual connotation. They were even more upset when the "no hair" phrase hit, which connoted even more sexual naughtiness. Funny, I always thought the French had the jump on everyone when it came to such matters!

The Weyl Tensor and Gravity Radio? -- Posted by wostraub on Saturday, January 2 2010
In 1831, Michael Faraday discovered his famous law of induction, which stated that wiggling a magnet in the presence of a wire will induce an electric current in the wire. As noted in my 5 December 2009 post, Faraday reasoned 20 years later that wiggling a massive object might induce a similar effect involving the object's gravitational field.

In 2003, the journal New Science reported a related effort by University of California at Berkeley physicist Raymond Chiao to detect gravitational waves using high-temperature superconductors. Chiao even constructed a home-made "gravity radio" to test his idea. While it hasn't yet been successful, some scientists think that he might be on to something.

There is a formal, if somewhat hypothetical, analogy (called gravitoelectromagnetism, or GEM) between Einstein's gravitational field equations and the Maxwell equations of electrodynamics. This analogy was unknown to Faraday but his basic idea pointed in the same direction. Chaio's initial efforts were based on calculations involving recent work on GEM but, as described in the New Science article, there was a conceptual flaw: the effect Chiao sought was of short range and hence useless as a marker for the detection of gravitational radiation. This effect is called the Lense-Thirring field phenomenon, and is related to the "dragging" of an inertial reference frame (actually, spacetime itself) by a rotating mass. But this effect is of very short range, and far too short to be of any practical use in a detector.

But is there any other gravitational "warping" effect that persists over large distances? Indeed there is, and it was discovered by Hermann Weyl around 1920. Weyl determined that the Riemann-Christoffel curvature tensor Rμνλβ could be broken up into two pieces called the Ricci term and what is today called the Weyl term. The Ricci term involves the compressive, volume-deforming effect normally associated with gravity; it is especially large very near the gravitating source. By comparison, the Weyl term involves what are known as "tidal effects," in which an object even very far from the source can be distorted in shape while the object's volume remains constant. It is the Weyl tensor that is responsible for the "spaghettification" of the unfortunate astronaut who wanders too close to a black hole.

A Berkeley colleague suggested to Chiao that the Weyl tensor might be what he was for looking for. Indeed it was, and Chiao's revised theory was accepted for publication in the prestigious journal Physical Review.

In my little article Weyl's Conformal Tensor you can read how this tensor is derived, along with some brief notes and references on its importance in general relativity.