Index photos courtesy ETH-Bibliothek,
Zurich Bildarchiv

Why I'm No Longer a Christian

Who Was Hermann Weyl?

Wheeler's Tribute to Weyl (PDF)

Old Stuff
2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

email:bill@weylmann.com

Math Tools
Weyl's Spinor and Dirac's Equation
Weyl's Conformal Tensor
Weyl Conformal Gravity
Weyl's 1918 Theory
Weyl's 1918 Theory Revisited
Weyl v. Schrodinger
Why Did Weyl's Theory Fail?
Weyl and the Aharonov-Bohm Effect
The Bianchi Identities in Weyl Space
Weyl's Scale Factor
Weyl's Spin Connection
Weyl and Higgs Theory
Weyl & Schrodinger - Two Geometries
Lorentz Transformation of Weyl Spinors
Riemannian Vectors in Weyl Space
Introduction to Quantum Field Theory
Electron Spin
Clebsch-Gordan Calculator
Bell's Inequality
The Four-Frequency of Light
There Must Be a Magnetic Field!
Non-Metricity and the RC Tensor
Curvature Tensor Components
Kaluza-Klein Theory
A Brief Look at Gaussian Integrals
Particle Chart

Uncommon Valor

Sophie did not forget Jesus!
Hans: "Long live freedom!"

 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.

Shortly after Einstein announced his theory of general relativity (gravitation) in November 1915, Weyl began an intensive study of the theory's mathematics and was soon publishing related scientific papers dealing with its physical applications. In 1918 Weyl published his book Raum-Zeit-Materie (Space-Time-Matter), which provided the first fully comprehensive analysis of the geometric aspects of the theory and its relationship with spacetime physics. One of the topics covered in the book was Weyl's idea that gravity and electromagnetism might both be derivable from a generalization of Riemannian geometry, the mathematical basis from which Einstein had developed his relativity theory. Weyl's idea was based on a new mathematical symmetry that he called gauge invariance.

I came across Weyl's book in 1975, but it didn't impress me very much because I didn't know general relativity. However, in the summer of that year I stumbled across Misner-Thorne-Wheeler's massive Gravitation during a one-week work assignment in the microscopic rural town of Lone Pine, California (which then had a population of perhaps 500 people). Miraculously, the town's tiny public library somehow had this book, which is now regarded as a classic graduate text on 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. (I spoke with co-author Kip Thorne about this in 1994, and he was quite amused to learn where one of his books had ended up.)

But Gravitation is not an easy read, and I had to look for more introductory texts. I soon came across Adler/Bazin/Schiffer's Introduction to General Relativity, which besides being easier had a chapter on unified field theory, including Weyl's 1918 theory of the combined gravitational-electromagnetic field. For whatever reason, the theory's mathematical beauty absolutely fascinated me. I had known about local and global phase invariance from my studies of quantum mechanics, but I was not aware that Weyl's theory was the origin of this powerful symmetry principle in quantum physics.

I have since read all of Weyl's books and many of his papers. Although today I believe that my interest is now based more on an appreciation of modern gauge theory (easily the most profound and beautiful concept of physics), I credit Weyl for having initiated the idea in 1918 and for his subsequent (1929) seminal application of the idea to the then still-developing quantum theory.

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.

Weyl was an exceptionally gifted mathematician and physicist, but he was also a highly cultured man in the classical German tradition. He studied and wrote extensively on philosophy and was a serious student of German poetry and literature. His mathematical writing style could be exceedingly obtuse, but his other writings reveal a genuinely warm person who truly understood the human condition. Weyl was also very human himself; he could be overly thoughtful and cautious, often to the point of being unable to take action or make even basic decisions, and sometimes with the result that he became physically incapacitated. He was a devoted and loving husband and father, yet he could also be persuaded to stray, in accordance with the surprisingly liberal attitudes of post-World War I Weimar society.

This website is my feeble attempt to document (and in many cases expand on) 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 (I'm even of the opinion that much of Weyl's work can be understood and appreciated at the high school/beginning university level). 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.

 Weyl and Grandson — Posted Monday, 2 May Here's a nice photo of Hermann Weyl with his grandson Peter in 1949. Weyl was probably 63 at the time, four years younger than I am now. God, I feel old. (From the 2009 book Mind and Nature: Selected Writings on Philosophy, Mathematics and Physics by Hermann Weyl with the express permission of the University of Princeton Press. Please respect the copyright when sharing)
 Logic is Fun? — Posted Thursday, 28 April "Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever." — N. H. Abel Speaking of logic, readers may remember that back in 2014 there was a YouTube video that got millions of views because it claimed to prove the seemingly ridiculous expression $$(1) \quad S = \sum_{k=1}^{\infty} k = 1 + 2 + 3 + 4 + \ldots = - \frac{1}{12}$$ This is in fact a famous expression that has been investigated by the likes of Euler and other great mathematicians over the past 200 years and, while it appears to be ridiculous, it is by no means trivial, having several important applications in mathematics (e.g., the Riemann $$\zeta$$-function) and physics. Still, the very notion that this obviously diverging expression does not go to infinity seems not worth bothering with. Surely, it's completely wrong. My first encounter with this equation was in 1998, when I picked up UC Santa Barbara physicist Joe Polchinski's two-volume textbook String Theory. When I got to Eq. (1.3.32) on Page 22 of the first volume (which is also cited in the YouTube video), Polchinski uses the above identity not only to regularize the Hamiltonian of a bosonic string (using Weyl rescaling) but to derive the dimensionality of spacetime in the theory (it's 26 — one time dimension and 25 space dimensions). Then M-theory came along and reduced that number to just 11 (boy, that's a relief). For all you masochists out there, the first 60 pages of Polchinski's text can be read for free over at Google Books. To get a better feel for what's going on here, let's split up (1) and write it as the infinite sum of odd and even integers: $$S = 1 + 3 + 5 + 7 \ldots + 2 + 4 + 6 + 8 \ldots$$ or $$(2) \quad S = 1 + 3 + 5 + 7 \ldots + 2(1 + 2 + 3 + 4 \ldots )$$ But the last term is just $$2 S$$ so, after some rearranging, we can write (1) as $$S = - 1 - 3 - 5 - 7 \ldots$$ so that the infinite sum of all positive integers is also equal to the infinite sum of all negative odd integers (alternatively, $$\infty = - \infty$$). This may not be as concise as (1), but the logic is solid, even though it's just as crazy. What's going on? First, follow the video beginning at around the 1:52 mark, where the presenters evaluate the "infinite" sum $$S_1$$ used in the derivation of (1). It's either $$0$$ or $$1$$, depending on whether you truncate the sum at $$1$$ or $$-1$$, so our heroes simply average the sum and get 1/2 as the answer. The rest of the derivation follows easily from there. Now, any mathematician will tell you that their evaluation of $$S_1$$ is just plain nonsense; the "sum" is either $$0$$ or $$1$$, and using the "average" is totally meaningless. Nevertheless, there are better and more solid derivations of (1) available, and they're not so easily dismissed. So again, what's going on? The answer is, at least as I see it, intimately tied to our notion of infinity. We tend to think of infinity as a number, and a kind of "finite" number at that, since we have a nice symbol for it, but we also allow ourselves to think of $$\infty$$ and $$\infty - 1$$ as being the same thing. That makes perfect sense, except when we're dealing with diverging series, because in that case where you decide to truncate the series (and that's where the "$$\ldots$$" comes into play) makes all the difference. The " $$\ldots$$ " is a convenience we use to keep things simple, but sometimes one needs to consider the penultimate term $$\infty - 1$$ instead. By writing $$S$$ as $$S = \sum_{k=1}^N k$$ and paying close attention to the sum and its terms as $$N$$ approaches infinity, it is a simple matter to see that Eq. (2) is completely invalid. For the same reason, Eq. (1) is also completely wrong. But now comes the magic part. Mathematicians and physicists are not stupid, and they know damn well that the above expressions are bonkers. In quantum field theory, quantities commonly arise in certain calculations that give nonsensical answers, such as negative probabilities, vectors having negative lengths, or quantities that should be finite but are instead infinite. To deal with such problems they have devised clever ways of effectively "truncating" infinity to produce sensible results for physically meaningful quantities, such as the half-life of an unstable particle. Their calculations also tend to produce quantities that they are not interested in. These quantities are typically infinite, and they are used to "suck up" the unwanted infinities — in the words of Caltech physicist Richard Feynman, these unwanted quantities are simply "swept under the rug" and forgotten. What's truly amazing is that the physically meaningful parts of the calculations typically agree with experiment to fantastic accuracy, often to 10 or 12 decimal places. It turns out that Eq. (1) is fully applicable to string theory, but its seeming craziness is masked by careful manipulation of the underlying mathematics. The great (greatest?) mathematical physicist Paul Dirac always hated this technique of "sweeping away" the infinities (called renormalization), and he believed that quantum field theory was incomplete in the sense that the ultimate theory would not involve infinities to begin with. For this reason, many physicists have steered away from string theory, hoping that a more formal and logical approach to quantum field theory is needed, particularly since string theory mathematics is so damned hard to learn.
 Logic is Fun! (Well, Maybe) — Posted Thursday, 28 April David Marans is Professor of Logic at St. Thomas University in Miami, Florida. I don't know him personally, but he's a fellow Doctors Without Borders (DWB) supporter, professional piano player, world traveler and all-around good guy. We occasionally exchange emails on topics that are usually far over my head. David wrote and occasionally updates his book Logic Gallery, which you can either buy here (a portion of which goes to DWB) or view online for free here. It's literally a gallery of nearly 200 famous logicians, mathematicians and scientists, replete with photos, short articles, quotes, tidbits and links about people like Descartes and even Hermann Weyl. Disclosure: Although I did very well on my chemistry, physics and engineering Graduate Record Exams, my scores in Logic were absolutely pitiful, somewhere below the 40th percentile. Non-mathematical logic simply escapes me, and Logic's poor cousin, Philosophy, finds me equally at a loss. But Logic Gallery is fun to read, and I encourage anyone who is logically impaired like me to check it out.
 Off On Another Tangent — Posted Wednesday, 27 April Turner Classic Movies recently aired the restored version of German director Fritz Lang's 1922 silent masterpiece Dr. Mabuse, der Spieler, starring the great Rudolf Klein-Rogge (Metropolis, 1927). "Spieler" in German generally means "gambler," but here it's more like "puppeteer." Mabuse is a brilliant psychoanalyst whose hypnotic powers can induce people to do his will, usually for monetary, political or sexual gain. At nearly five hours, it requires some effort to get through, but the film is worth it. Midway through the film Mabuse encounters the beautiful Countess Dusy Told, whom he sets out to seduce by first destroying her husband. The philosophy of the monomaniacal Mabuse is summarized in the following scene. The Countess is a good woman, but she's bored with her husband and seeks fun at gambling parties and other "sensational" outlets. But she learns that virtuous love (Liebe) is far nobler than anything the senses can provide, a knowledge she relates to the lustful Mabuse. But his response is "There is no love, only passion! There is no chance, only the will to power!" — Klein-Rogge, whose lifespan (1885-1955) coincided with Hermann Weyl's, is deliciously evil in the film, and its popularity in the Weimar Germany of the 1920s led Lang to direct two successful sequels. However, the Nazis disliked the films, mainly because they portrayed the vain-glorious, all-powerful madman Mabuse in an unfavorable light. Today we would view Mabuse not as a madman but as a neoliberal, a term whose definition has varied widely since its introduction in 1938. Regardless of its original meaning, today's neoliberals are characterized as being disdainful of government and governmental regulation, while praising privatization of all public services in deference to a universal, rule-free market economy. Like Mabuse, neoliberals constantly seek influence and power through competition, viewing cooperation and other mutually-beneficial actions as destructive to free-market principles. Consequently, public services such as social security, welfare and governmental health programs are viewed by neoliberals as social evils whose elimination would result in increased profits to private enterprise. In short, neoliberals, like the good Herr Doktor Mabuse, are in reality evil sociopaths interested only in their own personal gain. And that brings us inescapably to the likes of Republican presidential candidate Donald Trump. Although Trump's sweeping victories in all five state primaries last night mean that he will almost certainly be the Republican nominee, I can now only hope that he is defeated in the general election by the presumptive Democratic nominee, Hillary Clinton. I am not surprised that Trump's victories are due in no small part to white evangelicals, whose earlier hesitant support for Trump has now been replaced with near-overwhelming acceptance. But I am surprised that roughly 50% of Americans can be taken in by this strutting, billionaire egomaniac whose lifestyle and philosophies go against anything remotely appealing to America's "Christians." Perhaps human beings are really nothing more than intelligent but savage apes, grabbing everything they can however they can.
 What's It All About, Alfie? — Posted Tuesday, 26 April The long-wished-for "Theory of Everything" will be so simple that we will be able to summarize it in a single equation on a t-shirt. —Every Living Physicist I already quoted Einstein in my previous post, so I won't repeat it here, but it actually has a more appropriate application regarding what follows below. Dutch theoretical physicist Martinus Veltman was the co-recipient (with fellow Dutch physicist Gerald 't Hooft) of the 1999 Nobel Prize for his work in particle physics. He has written several books, at both the lay and advanced level, and one of his technical but still somewhat readable texts is his 1994 book Diagrammatica: The Path to Feynman Diagrams. In Appendix E Veltman details the full Lagrangian for the Standard Model of physics, consisting of seven parts: Just one of those parts, the weak-force Lagrangian $$\mathcal{L}_w$$, goes like If you happen to be intimately familiar with all seven parts, including the Feynman rules required for their application, you'd be in pretty good shape (don't look at me; I surely am not). But I think you get the picture — although the Standard Model has never failed, it's not quite the theory one would like to display on a t-shirt, much less carry around in one's head. Even worse, the SM doesn't include the gravitational interaction, because gravity just refuses to be put into the box. University of Oxford physicist Vlatko Vedral has a fascinating article in this week's online Aeon magazine that basically asks the question: "With the exception of gravity, the Standard Model describes the underlying physics of all inanimate matter about as well as we could ever hope. Why can't living matter be similarly described?" This is really a very profound question, because although every living thing is composed of inanimate atoms and molecules, self-replicating assemblages of nucleons and elementary particles simply defy mathematical description. In short, physics and mathematics do not explain life; there is no Lagrangian for living systems. If there were, it would surely be infinitely more complicated than the mess displayed above. In a somewhat related Aeon article, British writer Stephen Poole addresses the problem of teleology, which has to do with the meaning or purpose of things. Teleology is really nothing more than the ultimate philosophical question, although it is perhaps the most fundamental issue humankind could ever ponder. Admittedly, physical laws allow for "islands of complexity" in the universe that conserve energy and momentum and do not violate the Second Law of thermodynamics, but why should "complexification" even be allowed in a universe supposedly bound by simple (!) physical and mathematical laws? Religious types might respond to that question with some kind of unsupportable dogma involving God (or gods), but even that does not answer the question, because one could then ask what God's underlying purpose is. To them, the whole issue then becomes a matter of argumentum ad ignorantiam: "We don't really know the answer, but it's as we say, regardless." Believe me, as one gets older these questions become important. My younger son got his PhD in molecular biology, immunology and genetics at UCLA in 2010 and is now a scientist at the Centers for Disease Control and Prevention in Atlanta. Before leaving home, he and I used to have endless discussions on these very topics. We were never able to resolve anything, of course, but oh, God, how I do miss those days now.
 Back to Weyl — Posted Saturday, 23 April Hence, after the development of modern quantum theory, Hermann Weyl interpreted the ideas of gauge invariance and the corresponding mathematical formalism as connected with transplanting the state vector of a quantum-theoretical system. Be this as it may, there seems to be a very suggestive and potentially significant content to [Weyl's 1918 theory]. Physical reasoning led Weyl to a model of differential geometry which is of great theoretical interest and aesthetic appeal. — Adler, Bazin & Schiffer, Introduction to General Relativity, 1975 Physics should be made as simple as possible, but not any simpler. — Einstein Although Weyl's 1918 metric gauge theory is now almost 100 years old it is still regularly resurrected in the literature, usually in the hope that something relevant to modern physics will spin out of it. This is undoubtedly due to the unmistakable beauty of the theory, which today arguably remains the only potentially viable route to a practical non-Riemannian theory of gravity. In the last lecture of Benjamin Schumacher's excellent Great Courses video series Black Holes, Tides and Curved Spacetime: Understanding Gravity, the Kenyon College professor talks about the next revolution in physics, which will undoubtedly be the unification of general relativity and quantum mechanics. He briefly describes string theory, M-theory and loop quantum gravity, all relatively new ideas that are mathematically beautiful and intriguing but totally lacking in any empirical evidence or experimental data to back them up. But he then mentions the very new idea of entropic gravity, which ties together quantum information theory and gravitation via a simple rubber band argument! It's fascinating, and I encourage you to borrow this great (but expensive) video series from your library (as I did). But back to Weyl. In a short and very readable paper released just last month, University of Connecticut physics professor Philip D. Mannheim shows by a clever argument that the gauge freedom of the massless Dirac Lagrangian, metric tensor, Dirac vierbein and electromagnetic potential can be "balanced out," effectively producing a purely Riemannian theory consistent with a local conformal invariance that comes out of the formalism for "free." Symmetry breaking of the Lagrangian can then be used to introduce a mass term. In Mannheim's view, demanding that gravity be conformal at the outset provides the basis for a very natural approach to both gravity and quantum physics. It was Weyl himself who in 1922 introduced the notion of conformal gravity with the conformally-invariant Lagrangian $$\sqrt{-g} \, C_{\mu\nu\alpha\beta} \,C^{\mu\nu\alpha\beta}$$ (where $$C$$ is the Weyl conformal tensor), which itself is the basis for this and much of Mannheim's other work. [Mannheim posted a more detailed version of the paper last year, which made explicit use of the imaginary Weyl potential $$iA_\mu$$ rather than $$A_\mu$$. Adler, Bazin and Schiffer noted in their 1975 book that while this simple substitution automatically explains the quantization conditions for electron orbits in the hydrogen atom, it also implies an imaginary fine structure constant $$\alpha$$. This problem can be tied to the seemingly unphysical variance of vector length in Weyl's original theory, which was convincingly overturned by Einstein himself.]
 Earth Day: Two Versions — Posted Friday, 22 April The planet earth from way up there is beautiful and blue And floating softly through a rainbow But when you touch down, things look different here $$\ldots$$ — Electric Light Orchestra, Mission (A World Record) On April 22, 1970 I was finishing my junior year at California State University Long Beach when I wandered into the Music Center auditorium to hear Jacques Cousteau talk about "Earth Day." As a chemistry major and ocean enthusiast (i.e., the beach), I thought I would attend. I was amazed to find thousands of fellow students there. Today Earth Day celebrates its 46th anniversary, and since that first day there has been a sea change (so to speak) in our knowledge of what human activities are doing to the planet's ecosystems (if you're a Republican you can stop reading now). Cousteau himself could not have imagined that rising atmospheric CO$$_2$$ levels could have such a devastating effect on the oceans he loved, particularly with regard to the effects of acidification and warming on coral reefs. Meanwhile, the amount of plastic and other floating debris (much of it indestructible) in our oceans represents a global disgrace — an alien civilization visiting our planet and scanning the amount of atmospheric, water and soil pollution would be amazed at how stupidly an otherwise supposedly advanced civilization could contaminate its own world. And it doesn't end there — there are roughly 200 million pieces of non-functioning space junk now in orbit around the planet, requiring careful consideration every time a rocket, shuttle or satellite is sent into Earth orbit, lest it get impacted and destroyed, thus becoming yet more space junk. Einstein once remarked that both the universe and human stupidity are infinite, and that the definition of insanity is doing the same thing over and over while expecting a different result. Today we're a lot more knowledgeable regarding the impacts our activities are having on Planet Earth, but our efforts to reverse, minimize or nullify those impacts have been relatively sparse and ineffective. Conversely, the world's humans seem to care a lot less about the impending death of their home planet than they do about the passing of admittedly talented but behaviorally insane and inconsequential cultural icons. For that reason, I am posting this rather more appropriate Earth Day graphic:
 Bottle and Beam — Posted Tuesday, 12 April For those of you who, like me, have wondered just exactly what the half-life of a free neutron is, this month's Scientific American has an interesting article describing two separate attempts to definitively nail down that aspect of this most basic and important particle. Undergraduate physics students learn early on that the half-life of the elementary muon has been very precisely measured to be $$2.1969811(22)\times 10^{-6}$$ s, mainly because the muon is often used in the unfolding of the time dilation effect in special relativity. But the neutron, present in the nuclei of all elements except elementary hydrogen, is almost always assumed to be stable and eternal. Well, it is stable in its bound state, but when isolated the neutron is actually unstable, decaying into a proton, an electron and an anti-electron neutrino: Most textbooks that even deal with neutron decay tend to report that its half-life is around 10 to 15 minutes, an uncertainty that is rather embarrassing to physicists, considering how important and fundamental the neutron is. The authors of the Scientific American article admit this embarrassment and, with each participating in two separate experiments, they have attempted to define the half-life as precisely as possible. The results of these experiments are most interesting. One approach is to put a bunch of neutrons in a bottle, wait for a while, and then count the ones that have not decayed. While the concept is simple and straightforward, the experiment is fraught with difficulties, mainly because neutrons, being electrically neutral, can pass through the bottle's walls. But those difficulties can largely be overcome with some ingenuity. The authors' other approach is to basically monitor a beam of neutrons along a conduit, then count the protons produced in their decay. Protons, being positively charged, cannot easily escape the conduit walls and are easily counted, but again there are difficulties in the experiment, requiring ingenuity in its design. After the scientists performed numerous experiments as carefully as possible, they found to their surprise that the results differed significantly. The difference in the measured half-life of the neutrons was about 9 seconds — the bottle approach gave a half-life of $$879.6\pm0.6$$ seconds, while the beam method gave a figure of $$888.0\pm2.1$$ seconds. This is actually a huge difference but, while the scientists were dismayed, they also considered the possibility that neutron decay, which takes place via the weak nuclear force, may actually involve some exotic (and exciting) new physical phenomenon not yet known to science. Still, of all the basic fundamental constants of Nature, none comes close to that of gravity in terms of measuring frustration. In spite of the fact that it's the most basic of Nature's forces, the gravitational constant $$G$$ has only been measured to the relatively imprecise value of $$6.67408(31)\times10^{-11}$$ N-m$$^2$$-kg$$^{-2}$$ (by comparison, Planck's constant is known to better than 9 decimal places). Even worse, the basic measurement approach for $$G$$ has not changed in over two hundred years — placing two heavy masses suspended by a thin fiber and measuring the tiny amount of rotation that the fiber undergoes when another heavy mass is brought next to one of the masses. For 21st-century physics, this is barely more sophisticated than what you might find in a high school science lab.
 Interestingness — Posted Sunday, 10 April UC Berkeley astronomy professor Alex Filippenko is a phenomenal scientist and lecturer, and his 50-hour, 96-part Great Courses video lecture series is well worth wading through (it took me several months to get through it). The last few lectures are the most enlightening, when he talks about the expansion of the universe, the role gravity played in its formation, and how tiny (1 part per several hundred thousand) density differences in the otherwise perfectly uniform structure of the very early universe ultimately led to the formation of stars and galaxies (and us). Indeed, without gravity (which we still do not fully understand) the universe would have been very boring — a homogenous soup of mostly hydrogen and helium gas, existing forever without doing anything. While I'm not a great fan of the series Through the Wormhole hosted by actor Morgan Freeman, one episode talked about the purpose of the universe and the meaning of it all. Even if you're devoutly religious, you have to admit that the teleology of existence cannot be easily explained. I consider answers like "The purpose of our lives is to become one with God and live in His countenance and love forever" to be naive, meaningless feel-good gibberish, and it's surprising how many people seem to think it's an answer at all. But the question of what exactly's going on here is perfectly justifiable. Contrary to what a lot of fundamentalists believe, the Second Law of Thermodynamics is not violated simply because complex systems come into existence. Those systems effectively borrow energy and entropy from elsewhere, creating "islands" of complexity that would otherwise appear to defy explanation, while overall the entire system strictly obeys $$dS \ge dE/T$$. Over the years I've thought about this many times, and the only answer I've been able to come up with is that, for some reason, the universe likes things to be interesting. But "interestingness" is a very subjective, human concept, and I'd be challenged if I had to quantify it in terms of some teleological purpose. It also gives "agency" to the universe, like a god or super-human computer programmer, and I'm uncomfortable with that. I can accept evolution, natural selection and random change without difficulty, but that hardly addresses the question of "why." Here's a clip from Lecture 93 of Filippenko's talks, showing how super-computer simulation allows us to understand how the observed filamentous structure of the universe probably occurred over the 13.8 billion years following the Big Bang. The "nodules" that also form are the galaxies and super-clusters of galaxies, the individual stars being too small to model. In 2014 a team of scientists at Los Alamos and Stanford completed the largest such effort to date, involving some 1.07 trillion particles in the computer simulation. All of these efforts perfectly match our observations of the universe at the largest scales. Filippenko notes at the end of the clip that the universe is probably not rotating (a good question would be "rotating with respect to what?) As a gift to Einstein on his 70th birthday in 1949, colleague Kurt Gödel found an exact solution to Einstein's gravitational field equations showing that time travel to the past was possible in a rotating universe. Again, interesting!
 Is It Ever Over? — Posted Friday, 8 April Astrophysicists have discovered a galactic black hole that's 17 billion times the mass of the Sun. Located a mere 200 million light-years from Earth, NGC 1600 (New General Catalog item 1600) is a bright but otherwise ordinary galaxy whose heart contains an extraordinarily massive black hole. Its event horizon is larger than our solar system, so an observer falling past the point of no return would not notice notice anything particularly odd except for the appearance of the stars in the sky, which would seem to be collected into a single bright point. At 17 billion $$M_{\odot}$$, the observer would not feel the crushing tidal forces of a much smaller black hole, and her journey to the singularity would be long enough to write a scientific paper about the trip. But could that paper ever be read by others on the outside? Until relatively recently, the answer was no — all information in a black hole was assumed to be destroyed forever. That was Stephen Hawking's position until 2004, when he admitted that the ideas of eternal information preservation espoused by Leonard Susskind and others were correct. Here's a brief video overview of Hawking's current thoughts on the matter:
 Quantum Immortality? — Posted Thursday, 31 March We observe the universe, and in doing so we create the very universe we observe. — Wheeler Frankfurt physicist Sabine Hossenfelder has an article in today's Aeon Magazine on the so-called information paradox that exists between quantum mechanics and Einstein's general theory of relativity. Basically, general relativity says that if a one-kilogram rock is tossed down a black hole, it just increases the hole's mass, while tossing a one-kilogram book or set of videos does exactly the same thing — the rock's "information content" is essentially nil, while the book and videos contain lots of information, but that information is lost forever in the black hole. On the other hand, the principle of quantum unitarity requires that a packet of mixed quantum states remains mixed forever unless acted upon by the unitarity operator $$e^{-iHt/\hbar}$$, meaning that the information content of books, videos and human beings cannot be lost from the universe when they fall into a black hole. Information-wise, a 1-kg rock and a 1-kg book are intrinsically distinct, and their information content cannot be destroyed by a black hole. Hossenfelder's article is interesting, but she notes that the information "conundrum" is based on general relativity's tendency to destroy information. Since we don't have a workable theory of quantum gravity, I believe that statement is either wrong or premature. A number of physicists believe that the information of an in-falling body is somehow preserved in outgoing Hawking radiation, or even transferred to another universe. At the same time, we don't really know how the unitarity operator affects a quantum system when wave function collapse occurs; it is spontaneous (meaning that the operator itself becomes meaningless during collapse) or is it merely rapid, in which case one has to wonder how a simple measurement by a conscious observer can possibly affect the operator. It's really neat how black holes today are increasingly being viewed as the one viable link between quantum physics and gravitation. Hossenfelder points out recent studies indicating that a black hole is essentially a Bose-Einstein condensate consisting of a soup of gravitons, the totality of which serves to preserve information forever. And just exactly where does information originate? I believe it's somehow tied to human consciousness and the myriad, near-inifinite number of observations we make, the decisions and actions we take based on those observations, and how these actions might affect the rest of the universe. Einstein once jokingly asked how the observations of a mouse could possibly induce the very existence of the moon, an inquiry that to him justified the Newtonian determinism of the universe while refuting the notion of quantum indeterminacy. But a mouse is not a sentient being, while humans are (I leave out Republicans, who are neither sentient nor human). By comparison, the late physicist John Archibald Wheeler once suggested that the universe exists so that observers could be created, and that those observers in turn made the universe possible — the ultimate circular logic, to be sure, but who's to say he was wrong? Hossenfelder goes on to posit the possibility that the universe is a gigantic quantum computer, inputting, processing and outputting information on a continuous basis. It's not an original idea, but it seems plausible. Imagine that you're a computer programmer, simulating the flow of water in a waterfall or the motion of several million particles. You may be using a supercomputer to do the calculations, and that computer may take days or months to complete a single time-step computation. But Nature does the same thing instantaneously and seemingly effortlessly. Exactly how does Nature do that? Perhaps that kind of thing is what the universe has been doing all along.
 Weyl and Hilbert — Posted Monday, 28 March Here's an exceedingly rare film of Hermann Weyl with his PhD advisor, mentor and close friend, the great German mathematician David Hilbert. At a conference in Paris in 1900, Hilbert famously proposed 23 mathematical problems that he hoped would be solved in the then-new century. A few of these problems were indeed solved, but only relatively recently. This video clip is taken from the PBS program celebrating the life of the American mathematician Julia Robinson, who spent the bulk of her professional career trying to prove Hilbert's tenth problem. She didn't quite make it, but her work paved the way for others. Hilbert's tenth problem basically asks whether it is possible to prove that, by any finite algorithm, a solution to a given $$n$$-order polynomial expression $$f(x_1,x_2, \ldots x_n) = 0$$ (with integer coefficients) can be expressed in terms of integers. The answer, which was finally proved in 1970, is no. The film was most likely made in or around 1930, when Weyl was appointed the chair of mathematics at the University of Göttingen to replace Hilbert, who retired that year.
 Weyl in Spain — Posted Monday, 21 March Here's a nice photo of Hermann and Hella Weyl taken in the summer of 1922 while they were visiting the ancient Moorish palace at the Alhambra in Granada, Spain. Weyl, already being considered the mathematical heir to his advisor and mentor, the great German mathematician David Hilbert, traveled frequently throughout Europe to lecture, attend conferences and to vacation with his wife, and by the late 1920s he was also visiting America to lecture. He once traveled as far west as Colorado, but the staff at the Einstein Papers Project at Caltech has no record of Weyl ever coming to Pasadena. (From the 2009 book Mind and Nature: Selected Writings on Philosophy, Mathematics and Physics by Hermann Weyl with the express permission of the University of Princeton Press. Please respect the copyright when sharing)
 Dark Energy — Posted Saturday, 12 March The standard form of Einstein's field equations is $$G^{\mu\nu} = R^{\mu\nu} - \frac{1}{2} g^{\mu\nu} R + \Lambda g^{\mu\nu} = \frac{8 \pi G}{c^4} T^{\mu\nu}$$ where the $$\Lambda$$ term is Einstein's famous cosmological constant, thought by many physicists to be entirely or at least partly responsible for the phenomenon known as dark energy, a mysterious force that appears to be causing the observed acceleration of the expansion of the universe. Interestingly, the cosmological constant is a true fudge factor that Einstein included in his field equations to actually halt the expansion and keep it a constant size, an assumption he subsequently called the greatest blunder in his life when Edwin Hubble discovered in 1929 that the universe was truly undergoing expansion. But with the proper sign $$\Lambda$$ acts as an accelerant, causing the universe to expand at an ever-increasing rate. In this month's issue of Scientific American, astrophysicists Adam Riess (co-recipient of the 2011 Nobel Prize in physics for the observed confirmation of the accelerating universe) and Mario Livio write about the "Puzzle of Dark Energy," it being a puzzle because no one really knows what the hell it is, what it's made of or where it comes from, although it appears to account for roughly 70% of all the mass-energy in the universe. The authors suggest three alternatives for dark energy: 1) the cosmological constant, which is believed to be very small but is nevertheless a significant contribution to the Einstein field equations; 2) quintessence, a kind of field that can change with time and position in space that could also result in the ultimate collapse of the universe; and 3) there is no dark energy at all, but simply an aspect of gravity that we don't yet really understand. In any of these alternatives, vacuum fluctuations due to quantum effects (particle creation and annihilation) could also be contributing to what is being observed. [The above graph summarizes the basis for Riess's Nobel Prize discovery. For a larger view, see my post dated 11 November 2013.] The only reason Einstein could stick a cosmological constant into his equations is due to the fact that both sides of the above equation must be have a vanishing divergence; that is, $$G_{\,\,\,||\nu}^{\mu\nu} = 0$$ where the double-bar subscript notation refers to covariant differentiation. In Riemannian geometry (which Einstein's theory is based upon), the metric tensor $$g^{\mu\nu}(x)$$ acts as a pure constant under covariant differentiation, a property that allowed Einstein to introduce $$\Lambda$$ in the first place. But this is not the case in most non-Riemannian geometries; for example, the geometry in Hermann Weyl's 1918 theory includes a metric tensor that does not have this property. On the other hand, the vanishing divergence of Einstein's equations is assumed to mimic the classical conservation condition for mass-energy, but true energy conservation in gravitation has always been an ambiguous issue — it can be shown to be conserved in some coordinate systems but not all. It would then appear that the zero-divergence property of the gravitational field is really not a motivating issue, especially considering the likelihood that quantum fluctuations don't obey anything but Heisenberg's uncertainty principle (which is not a classical conservation principle). Riess and Livio note that the next few years are likely to be a pivotal time for dark energy research, although what is really needed is a consistent theory of quantum gravity, a problem that is now approaching the one-hundred year mark, and one we don't seem to be getting any closer to.
 What We Should All Know About String Theory — Posted Friday, 11 March In June of last year Edward Witten of the Institute for Advanced Study gave a talk at the International Centre for Theoretical Sciences on an elementary approach to quantum gravity. Beginning with a toy theory in one dimension, Witten showed that by considering the metric tensor $$g_{\mu\nu}$$ to be a $$1\times 1$$ matrix the massive Klein-Gordon equation could be derived for both Euclidean and curved space, which represents kind of a first step toward quantum field theory. This one-dimensional approach is analogous to Feynman's diagrams, which consist of 1-D stick figures glued together at vertices, and Witten shows that this is essentially the basis for the unwieldy divergences that plague QFT. But in two dimensions the stick figures become tubes, which merge smoothly together at their vertices and thus avoid the infinities. This is the beginning of string theory. Witten shows, however, that this 2-D approach necessitates a $$2\times 2$$ metric tensor that must be conformally invariant, meaning that a rescaling transformation of the form $$g_{\mu\nu} \rightarrow \lambda(x) g_{\mu\nu}$$ cannot alter any of the underlying physics. Witten refers to these rescaling as Weyl transformations, which formed the basis for Hermann Weyl's 1918 unified field theory. Witten is considered by most physicists today to be the smartest man in the world (although that overused phrase tends to demean the man's vast contributions to both physics and mathematics) and the heir apparent to Einstein. His talk is posted on YouTube under the title "What every physicist should know about string theory." It's suitable for undergraduates, and is well worth watching (but use the full-screen option if you want to actually see the equations) :
 New Superconductor — Posted Tuesday, 8 March Here's an interesting compound: Sn$$_6$$Sb$$_6$$Ba$$_2$$MnCu$$_{13}$$O$$_{26}$$, a metallic ceramic that happens to become superconducting at 110 degrees Celsius, ten degrees higher than the boiling point of water. Are room-temperature superconductors, allowing for the everyday application of lossless power lines and magnetically levitated trains, not too very far in the future? I never took a class in solid-state physics (condensed-matter physics), so I can't offer any intelligible comments on this finding. But thanks anyway to Professor Golden Nyambuya of the National University of Science & Technology Department of Applied Physics for notifying me of this neat discovery.
 An All-Red America? I Sure Hope Not — Posted Tuesday, 8 March As we all know the Republican Party's establishment acolytes are ramping up pressure against front-runner Donald Trump, if only for the purpose of avoiding Trump's outrageous political positions from sticking to the GOP brand. But they generally dislike Ted Cruz, while Marco Rubio is barely hanging on by his political fingernails. John Kasich seems to be the most sane of all the candidates, but Republican voters aren't buying into him. So what's gonna happen? Without question, the GOP establishment is doing what it can to prevent Trump from accumulating the 1,237 delegates he needs to win the Republican nomination. He may indeed be the candidate with the most delegates, but if he doesn't get to 1,237 then all bets are off — the party elites are no longer beholden to him and can nominate anyone they want. That would be the brokered convention everyone's talking about, and while it's unconventional (excuse the pun) it's been done before. In this case, it's almost guaranteed if, by convention time, Hillary Clinton (or maybe Bernie Sanders) is considered all but ensured of winning the general election in November against Trump (or Cruz). If the GOP is successful in preventing Trump from getting the necessary number of delegates, I believe they'll almost certainly nominate former Wisconsin senator and current House Speaker Paul D. Ryan. Yes, he's said on numerous occasions that he would never accept the Speaker position, and yes, he's stated at least once that he would not pursue the presidency at this time, most likely preferring to defer it until 2020 or even 2024, when he would still be only 52. But with the Executive Office and a conservative Supreme Court makeup at risk and with the entire Congress already now in Republican hands, I think Ryan's forced nomination is all but guaranteed. And, like the house speakership, Ryan would almost certainly fall on his sword and accept being drafted come July 18-21, 2016. Ryan's primary objection to accepting the House speakership was based on his stated preference to spend more time with his family, a claim that may well have been legitimate, but a President Ryan would have his family with him in the White House, negating that issue. Also, George W. Bush managed to spend 533 vacation days away from the White House clearing brush on his ranch when he was president, an activity Ryan certainly wouldn't engage in. What's even scarier is that, with a united Republican Party at his back and little serious troublesome political baggage, he would most likely be elected in November.
 Einstein Papers — Posted Tuesday, 1 March Here's a neat site where you can read the papers published by Einstein in the German journal Annalen der Physik (Annals of Physics). Presented by the Max Planck Institute for the History of Science in conjunction with Wiley-VCH, the papers are all in German but the math is readable and there are many notations in English that describe them (the site includes a search function but it's not working yet). Even better is the set of collected works (with English translation supplements) available for free online from the Princeton University Press. Volume 7 covers the years 1918-1921, when Einstein and Hermann Weyl communicated extensively on a range of subjects, including gravitation, cosmology, unified field theory and even philosophy.
 Linger Yet, Thou Art So Beautiful! — Posted Tuesday, 1 March My favorite classical opera was composed by Arrigo Boito, who wrote Mefistofele in 1868. It didn't do well at first (mainly because it was over five hours long), but Boito trimmed it down and it soon became a hit. It's rarely performed today, but in the 1950s and 1960s it was fairly popular. I just finished reading writer Brian Morgan's 2006 book Strange Child of Chaos, a biography of the great bass-baritone Norman Treigle (pronounced Tray-gull). Treigle began performing the part of the opera's devil in 1954 (along with the same part in the 1859 opera Faust by Charles Gounod) and his brilliant singing and acting quickly made him the part's signature player. But what I found most interesting to learn was that Treigle, who was born in New Orleans in 1927, was a devout Southern Baptist who also sang Christian songs and hymns throughout his life, yet was married twice, was something of a ladies' man and smoked four packs of cigarettes and a quart of scotch every day. A lifelong insomniac, he regularly abused sleeping tablets until he died of an accidental overdose in 1975 in London at the age of 47. Nearly six feet tall, he never weighed more than 138 pounds. But his voice was nothing short of miraculous. The book's title is taken from Dr. Faust's line Strano figlio del Caos, which he utters in response to Mefistofele's great "Whistle Aria." Every song in the opera is a gem, and the 1973 New York performance (directed by Julius Rudel and starring the great tenor Plácido Domingo) features the incomparable voice of Monserrat Caballé, perhaps the finest soprano of all time. But Treigle's Mefistofele steals the show. Here is the opera's final aria, All'erta!, in which the damned soul of Faust is saved by grace, accompanied by a deafening chorus of celestial beings, with the defeated Satan taking it all in stride but screaming Trionfa il Signor, ma il reprobo fischia! ("The Lord has triumphed, but the reprobate whistles!"), which he does to the very end.
 Do Not Attempt This At Home On Your Commodore 64 — Posted Tuesday, 16 February Two journals published by the American Physical Society, Physical Review (established in 1893) and Physical Review Letters (established in 1958), are the most prestigious physics journals in the world — every serious physicist wants to get published in them, but only a relative few are accepted. While abstracts of published papers are posted regularly by these journals, the full papers generally are not — unless the papers are of historic significance. Last week, Physical Review Letters posted the entire paper "Observation of Gravitational Waves from a Binary Black Hole Merger," which announced the now-famous detection of gravitational waves (you can download a copy of the paper here or go online, where you can find it just about everywhere now). While every major newspaper on the planet covered the discovery, I think what's nearly as significant as the detection effort itself is the computer programming that simulated the gravitational waves coming off a binary black hole merger. Most students of general relativity rarely get beyond the derivation of the Schwarzschild model, and maybe a few can actually understand the Kerr model of a single rotating black hole. But to take Einstein's equations and apply them to two spinning, co-orbiting black holes is a challenge I cannot even imagine. Yet, the scientists at Caltech and the other participating universities did just that. While detecting a tiny signal and confirming it with a predictive mathematical model is a pretty neat accomplishment in itself, the fact that the project's two detectors saw the exact same pattern separated by a time lapse consistent with the light speed of the passing gravitational wave (the Livingston, Louisiana detector and the Washington State detector in Hanford) just blows my mind: As the black holes converged (and this was 1.3 billion years ago!), the detected gravitational wave pattern was relatively quiet, and the corresponding computer model showed a bit less correlation. But as the holes merged, all three — the computer prediction and the two signals themselves — precisely agreed with one another. That final extended blip at the right end of the graphs is the "chirp" you've been reading about, but the simulated audio version simply doesn't do the science justice. Principal investigator B.P. Abbott from Caltech and his hundreds of colleagues (see the citation list at the end of the paper) are to be congratulated, and I cannot imagine the 2016 Nobel Prize in physics bypassing these guys.
 Quadratic Equations — Posted Tuesday, 16 February Here's a bit of nostalgic silliness. While disposing of a bunch of video files today I found this old episode of Dr. Science I recorded on VHS from the mid-1980s. Airing Saturdays on the old Fox channel, there were only about a dozen episodes, but my kids loved it — my older boy once even dressed up as Dr. Science for a school Halloween pageant. My kids always claimed that Dr. Science and I were spitting images of one another, but of course I was far better looking. Shot on the cheap at the campus of California State University at Dominguez Hills, Dr. Science enjoyed a short-lived vogue for a time. You can still watch the most popular episode The Dr. Science National Science Test on YouTube (originally broadcast on PBS), but the show is now largely forgotten by all except for idiots like me. Some years ago I tried contacting the show's originator, Dan Coffey, by email several times but he never answered. But apparently the Dr. Science radio show is still being broadcast somewhere back east, and there's even a website where you can submit questions and find out useful stuff like why bathroom water and kitchen water taste different to little kids in the middle of the night. (I just posted the episode Dr. Science Meets His Match (from which the above clip was taken) on YouTube.)
 Einstein and Unification — Posted Sunday, 14 February The late physicist Abraham Pais was a friend and colleague of Albert Einstein at the Institute for Advanced Study in Princeton and the author of numerous books. Born in Amsterdam in 1918, he received his PhD in physics from the University of Utrecht in 1941. A Jew, he was forced to hide from the Nazis until nearly the end of the war, when he was captured and sent to a Gestapo prison camp. Somehow he managed to avoid execution until VE Day, when he and the other surviving prisoners were released by the Allies. In Pais' 1982 book Subtle is the Lord: The Science and the Life of Albert Einstein, arguably the finest scientific biography of Einstein ever written, he writes: Then it came to pass that physics veered toward a different course, neither led nor followed by Einstein. First quantum mechanics and then quantum field theory took center stage. New forces had to be introduced. New particles were proposed and discovered. Amid all these developments, Einstein stayed with the unification of gravitation and electromagnetism, the final task he set himself. This insistence brought the ultimate degree of apartness to his life. [It] was not he but others who in the end ushered in the new physics. So it was to remain in the next decade, and the next and the next, until he laid down his pen and died. His work on unification was probably all in vain, but he had to pursue what seemed centrally important to him, and he was never afraid to do so. That was his destiny. Einstein's first "unification" paper appeared in 1922, but as Pais notes in the book he was certainly on the unification track far earlier, perhaps as early as 1917. Back then only two forces of nature were known, gravitation and electromagnetism, and the beauty of Einstein's 1915 general theory of relativity was so profound that many physicists simply knew in their hearts that the two forces could somehow be unified through the same mathematical formalism (similarly, the inverse square laws for both forces are essentially identical, and for many years before Einstein it was believed they must share a common basis). Hermann Weyl's 1918 unification theory was the first unified theory to appear, at least the first that appeared to really work. But, as I've noted here repeatedly, when the theory was shown to be unworkable (by Einstein, no less) he abandoned it and turned to other things. But Einstein either couldn't or wouldn't move on — perhaps his 1915 theory impressed him so much that he was certain it would yield even greater discoveries, and that he would be the one to do it. On 19 June 1945 Einstein, then 66 years of age, submitted a paper to the Annals of Mathematics on a new approach to unification that in retrospect seems to have had the same elements he stuck with until he died 10 years later. I am posting this nearly 71-year-old paper here because it introduces an interesting new aspect to the geometry of spacetime that contains at least the seeds of a formalism that could legitimately be tied to quantum mechanics. In it Einstein introduced a new metric tensor consisting of the sum of the ordinary symmetric metric tensor $$g_{\mu\nu}$$ and the antisymmetric electromagnetic tensor $$F_{\mu\nu}$$, $$\hat{g}_{\mu\nu} = g_{\mu\nu} + i F_{\mu\nu}$$ This new tensor is not symmetric but hermitian, which Einstein might have believed was somehow associated with quantum theory, and he proceeded to derive a set of field equations based on this new quantity. Unfortunately, Einstein's approach necessarily resulted in a non-symmetric connection term $$\Gamma_{\mu\nu}^\alpha$$ that he had to dance around mathematically to get something he deemed physically meaningful (in the end it failed, of course, but to Einstein there were really no failures, only ideas that didn't work). Still, he was sufficiently impressed with the basic idea that the final edition of his 1922 book The Meaning of Relativity, published posthumously in 1956, included an appendix he wrote entitled "Relativistic Theory of the Non-Symmetric Field." Perhaps even more telling is that on his 70th birthday in 1949 Einstein was presented a birthday cake (prepared by his long-time assistant Helen Dukas) decorated in icing with several equations taken from this non-symmetric theory (you can see a bigger photo of the cake on my posting dated 29 July 2015). [For reasons of completeness you might also want to read the sequel to this paper, which Einstein published in early 1946 with colleague Ernst Straus.] Although Einstein's notion of a hermitian metric tensor probably resulted simply from a desire to generalize his 1915 theory any way he could, there is a fundamentally good reason why it might actually have applicability in general relativity. I'm thinking I'll write this up myself, as I need something right now to take my mind off the fact that a leading Republican contender for the presidency (and an avowed devout Christian, natch) wants to carpet-bomb the Middle East and make the sands over there glow.
 Carroll on Gravity Waves and Locality — Posted Friday, 12 February Noted Caltech physicist Sean Carroll has an article in today's Atlantic where he talks about gravitational waves while also musing on the notion of "locality" in general relativity and quantum mechanics. I mentioned earlier Isaac Newton's dissatisfaction with the idea of "action at a distance," of which Carroll writes Newton himself never liked this feature of his own theory. It wasn't primarily the instantaneous speed of gravity, but the fact that the force seemed to propagate through empty space, rather than through a medium like the air. He found this "action at a distance" distasteful, but he left its ultimate solution "to the Consideration of my readers."Newton's inverse square force "law" of gravity does indeed imply instantaneous action but in 1687 Newton had no better understanding of gravity, so he was stuck with it. It's a testament to his genius that he was able to see action at a distance as a flaw that would ultimately be resolved by a better theory. That theory had to wait until 1915, when Einstein demonstrated that gravitation really was a "local" phenomenon that propagated at the finite speed of light — fast, yes, but not instantaneous. However, Carroll goes on to note that quantum mechanics is actually a non-local theory, at least with respect to quantum entanglement, implying that this non-locality really is a form of action at a distance (why the stupid religious fundamentalists over at Conservapedia don't latch onto that aspect of relativity escapes me, but then they're a bunch of idiots anyway). Carroll puts off any discussion on the matter for a later time, but could the infamous incompatibility between general relativity and quantum physics be tied in some way to entanglement? People like Leonard Susskind think so, and it may be no coincidence that black holes today are looking to be the gateway to a unified theory of gravitation and quantum mechanics. Indeed, I think it's highly suggestive that yesterday's announcement of the detection of gravitational radiation involves the violent merger of two black holes some 1.3 billion years ago, a finding that, if indeed true, seems to also make things like time and spacial separation vague if not meaningless concepts. Quantum entanglement simply means that if you make a measurement on one entangled particle the wave function of its partner also collapses, since they're both in the exact same quantum system. The "problem" of non-locality arises only when one considers that the particles might be in different galaxies, separated by billions of light-years, which inescapably invokes our human notions of simultaneity of measurement across space and the erroneous view that one particle must somehow send out an instantaneous signal to the other when it's observed. Yes, it's completely nonintuitive, but we humans aren't nearly as smart as we need to be to understand this stuff. I believe this explains why Hermann Weyl's ideas on the possible irrelevance of spacial scales and distances impressed me so much back in the 1970s. And if the concept of distance is irrelevant, then the concept of time itself must also be irrelevant. A physical theory that does not involve spacetime at all might be the right theory, but who the hell can even imagine what that entails? Gravity is a wave, Rodney, and I'm going to surf that wave. — Dr. Science (Dan Coffey)
 The First — Posted Friday, 12 February Ann Mayes Rutledge (1813-1835) is generally acknowledged as Abraham Lincoln's first (and perhaps only) true love, her life tragically cut short when the young woman died of the "milk sick" at the age of only 22. Lincoln's friends said he never really got over it, and they attributed his lifelong melancholy not to the travails of the Civil War or his battles over wife Mary Todd Lincoln's spending but to the death of someone near and dear to him 30 years earlier. Lincoln's love life is a fascinating story of fits and starts, replete with tales of indecision, awkwardness, shyness and even hints of homosexuality. A year after Ann's death Lincoln found himself engaged to Mary Owens, a woman he barely knew and clearly did not even like. It was apparently a set-up situation through a mutual female friend, who introduced Owens to Lincoln in 1833. For whatever reason Lincoln agreed to marry her, but after an extended, long-distance mail correspondence with the woman Lincoln practically begged to be released from his promise. When he saw her in the flesh again some years later, she had grown so fat and unattractive that Lincoln remarked in an 1836 letter to a friend "I knew she was over-size, but she now appeared a fair match for [Shakespeare's] Falstaff $$\ldots$$ when I beheld her, I could not for my life avoid thinking of my mother; and this, not from withered features, for her skin was too full of fat to permit its contracting into wrinkles, but from her want of teeth, weather-beaten appearance in general, and from a kind of notion that ran in my head that nothing could have commenced at the age of infancy and reached its present bulk in less than thirty-five or forty years; and, in short, I was not at all pleased with her." (I nearly always die laughing when I read this). A similar situation presented itself in the period 1840-1842, when Lincoln met and then married Mary Ann Todd, though Lincoln's apprehensions were less pronounced, if only because this Mary was from a wealthy family and she was even better educated than he at the time. Still, another close friend described Lincoln's relationship with Mary Todd as "a burning, scorching hell, as terrible as death and as gloomy as the grave." But, as they said then and still say today, "Lincoln soldiered on." Though somewhat uncomfortable around women, Lincoln had a great sense of humor and was a famous teller of off-color jokes and stories, some of which would raise eyebrows even today, though I find it difficult to imagine Lincoln and Mary having much of a sex life (Abe: Brace yerself, Mary! Mary: Oh, Abe!) I mention this story because of an article I read today by Ellen McCarthy, a writer for the Washington Post, and it struck a nerve. It deals with that first real brush of adoration we all experience, usually at a post-adolescent but still impressionable age. For me it was in Mrs. Farrell's French II class in my sophomore year in high school (and no, it wasn't Mrs. Farrell herself, who I spoke of in my 1 March 2013 post, and it wasn't Shirley or Bonnie or Joanne). Still, 51 years later and counting, I still think about that girl classmate with the golden hair. "I think it's not just about the other person. It's about who we were at that time. We're relishing the image of ourselves. They give us license to be the person we were once again — young and vibrant and beautiful." — Jefferson Singer I think Connecticut College psychologist Singer's analysis is correct, although I can never recall a time when I was "young and vibrant and beautiful" — maybe just young. And, for whatever reason, when I read the article my mind went straight to Lincoln and Ann Rutledge. Ann has a new headstone now, but the original was inscribed with this poem by Edgar Lee Masters: Out of me unworthy and unknown The vibrations of deathless music: "With malice toward none, with charity toward all." Out of me the forgiveness of millions toward millions, And the beneficent face of a nation Shining with justice and truth. I am Ann Rutledge who sleep beneath these weeds. Beloved in life of Abraham Lincoln, Wedded to him, not through union, But through separation. Bloom forever, O Republic, From the dust of my bosom! Memory, especially long-term memory, can be both a blessing and a curse. In the immortal words of yet another poet, England's great Walter de la Mare: Here lies a most beautiful lady, Light of step and heart was she; I think she was the most beautiful lady That ever was in the West Country. But beauty vanishes, beauty passes; However rare — rare it be; And when I crumble, who will remember This lady of the West Country?
 Joseph Weber — Posted Thursday, 11 February In my 5 February post I mentioned an article written by UC Irvine astronomer Virginia Trimble (please see this link to the 1962 article "Behind a lovely face, a 180 I.Q.") She just happens to be the widow of Joseph Weber, largely recognized as the father of gravitational wave detection. He died in 2000, and I like to think he would have been tickled by today's news regarding the confirmed detection of gravitational waves. Weber's approach was to construct large cylinders of aluminum outfitted with sensitive strain gauges, a design that he hoped would be able to pick up the faint signal of a passing gravitational wave. That design failed, if only because today's announcement indicated that a gravitational wave would produce a deformation in spacetime smaller in size than the diameter of a proton. The LIGO experiment that successfully detected gravitational waves relied on interferometry, an insanely-sensitive approach equivalent to that used in the 1880s by Michelson and Morley to dispel the notion of the "luminiferous aether." That was also a ground-breaking discovery, as it conclusively demonstrated that the velocity of light is independent of the velocity of the observer. Einstein's 1905 theory of special relativity relied on that independence, and it quickly became a cornerstone of modern physics. I will now reiterate a comment I have made numerous times on this site: fundamentalist religion rejects all of Einstein's theories precisely because those theories contradict the notion of "action at a distance," which Christian theologians have relied upon for 2,000 years to enable the instantaneous action of God's miracles. The concept of action at a distance resulted when Isaac Newton developed his theory of gravitation, but he was reluctant to accept it because he could not understand how a physical phenomenon could occur instantaneously. I again direct my readers to the Christian website Conservapedia, a highly influential website created by conservative writer and author Phyllis Schlafly and her son Andrew that purports to overturn science (especially evolution and relativity) in favor of fundamentalist religion.
 Gravitational Wave Announcement — Posted Thursday, 11 February What happens when two black holes having masses of 29 and 36 times that of our Sun rotate about one another? They blow off energy in the form of gravitational waves en route to eventually merging into a single black hole of 62 solar masses, the mass difference having been converted into gravitational radiation. At least that's the scenario about to be reported by the LIGO (laser interferometer gravitational-wave observatory) team. Official announcement of the LIGO finding is imminent, and if it bears out then science will have a truly major discovery, one that goes far beyond mere confirmation of the existence of gravitational waves, which physicists have been pretty sure of since Einstein predicted them in 1916. Indirect (but highly precise) evidence of energy loss due to gravitational radiation in a binary pulsar system led to the 1993 Nobel Prize in physics. If the current discovery holds up, it will likely win another prize. We'll see. Update: It's been confirmed (see also today's New York Times, which has a great video explaining the discovery). When all the shouting dies down we'll find out exactly what they discovered. A great day for science!
 Gravitational Waves Discovered? — Posted Tuesday, 9 February "Tidal" effect of a passing gravitational wave. From Adler, Bazin and Schiffer, 1975. Southern California has been laboring under the apparent delusion that the "Mother of All El Niños" would be drenching us non-stop with record rains, and I admit that I was also caught up in the hope that the four-year drought here might finally be broken. But the facts appear to belie the hype, much to SoCal's great disappointment. In recent weeks I've blogged repeatedly about the tanned, wide-stanced, action-figure television weather forecasters who constantly praise SoCal's warm weather and cloudless, sunny clear skies while surreptiously avoiding the uncomfortable certainty that we're headed into a fifth year of record-setting drought. For this reason I'm hesitant to embrace recent reports that gravitational waves have finally been discovered, with a five-sigma reliability to boot. While even noted mathematical physicist John Baez is hopeful that Einstein's as-yet unproven prediction has finally been proved, I remain unconvinced. In 1922 Hermann Weyl examined the mathematics behind gravitational waves in the context of the weak-field limit ($$g_{\mu\nu} \rightarrow \eta_{\mu\nu} + \gamma_{\mu\nu}$$) of Einstein's gravitational field equations (others, including Einstein, also examined the effect). Because the field equations are highly non-linear, only an approximate solution is possible, which indicates that gravitational waves are essentially akin to electromagnetic waves propagating at the speed of light. Anyway, the weak-field limit predicts that separated test particles will experience a very slight physical displacement when a gravitational wave passes by. This displacement is exceedingly tiny, yet recent LIGO results indicate that such a displacement has indeed been detected. The result will be published this Thursday in the journal Science. The experimental search for gravitational waves has been ongoing for nearly fifty years. I hope this settles it.
 Weyl and Noether — Posted Tuesday, 9 February Here's the photo I alluded to earlier this year. It's summer, 1932, at the Gasthof Vollbrecht in the little town of Nikolausberg, a little outside of Göttingen, Germany, where Hermann Weyl found himself the chair of mathematics at the university following the retirement of his mentor, the great German mathematician David Hilbert. To the immediate left we have the very young German mathematician Ernst Witt, beside him the Swiss mathematician Paul Bernays, and then Hella and Hermann Weyl with their 17-year-old older son Joachim, who also became a noted mathematician. The chunky woman is none other than Emmy Noether (pronounced nuhr'tah), arguably the greatest female mathematician who ever lived. The attractive Mädchen on the far right is "E. Bannow", whose identity I have been unable to determine. [On 12 February I received an email from Malcolm Douglas informing me that the young lady is Erna Bannow, Ernst Witt's first doctoral student. Born in 1911, she got her PhD under Witt in 1939, and the two married in 1940. While flanking the group in the photo, I can't help wondering if they already had designs on one another! I also learned that Erna was the first to sign a 14-page petition to the German authorities to allow Noether to remain as a teacher: The last sentence reads "This is the reason why we would appreciate it if Frau Prof. Noether, [whose reputation] stands alone in Germany, was given the opportunity to continue to exercise her activities as a teacher." The petition failed, and Noether was banished from teaching in Germany.] Weyl and Noether both emigrated to the United States in 1933 to escape Hitler, although Noether dallied a bit before leaving. As noted, Noether was peremptorily fired from teaching due to her being a Jew, and she was reduced to teaching informally from her apartment until she finally left for America. Even then she could only find a half-salaried position at Bryn Mawr University in Pennsylvania, where she died unexpectedly from a malignant ovarian tumor in 1935 at the age of only 53. Einstein and Weyl presided over her funeral, where Weyl nobly accredited her mathematical genius as far superior to his. (From the 2009 book Mind and Nature: Selected Writings on Philosophy, Mathematics and Physics by Hermann Weyl with the express permission of the University of Princeton Press. Please respect the copyright when sharing)
 MOND — Posted Monday, 1 February Frankfurt theoretical physicist Sabine Hossenfelder has an interesting article in this week's online Aeon magazine that talks about the dark matter problem. As for what that problem is and why it's so important to physicists today can be gotten from the article itself, but it raises the potentially greater problem of what gravity is and how we've looked at it over the last 100 years since Einstein brought it into its modern, post-Newtonian form. Hossenfelder notes that the failure to date for experimentalists to irrefutably detect a single dark matter particle may be due to the fact that it's not a particle problem but a gravity problem. She cites a recent paper by Lasha Berezhiani and Justin Khoury of the University of Pennsylvania pointing to the possibility that dark matter is essentially a superfluid that exists thanks to the extreme coldness of interstellar space. The lengthy (44 pages) paper infers that it's undetectable because our Solar System is basically too warm for its gravitational effects to be noticed in our neighborhood, although its effects are noticeable in the dynamics of other galaxies. In this respect, Hossenfelder claims that the superfluidity of dark matter allows it to mimic gravitational effects, thus making the Berezhiani/Khoury idea a kind of modified Newtonian dynamics (MOND) theory. I've skimmed over the latter's 2015 paper, and while I haven't fully digested it yet I am confused — dark matter either exists or it doesn't, as Hossenfelder also notes, but if it does then I don't see it as being any kind of MOND theory at all. In 2009, John Moffat of the Perimeter Institute for Theoretical Physics in Canada published Reinventing Gravity: A Physicist Goes Beyond Einstein, in which he posits a true MOND theory that makes certain adjustments to Einstein's 1915 gravity theory to fit observations. However, like other theories that impose various tensor, vector, spinor and scalar components into a suitable action Lagrangian density for gravity, Moffat's theory fits the observational data well but only at the cost of introducing a lot of parameters that reduces it to an exercise in mere curve fitting. And while I haven't completely read the Berezhiani/Khoury paper, I suspect it might be something along the same lines. Conversely, I still prefer the approach first proposed some years ago by Mannheim and Kazanas, which to me represents a true MOND theory, although it utilizes Hermann Weyl's conformal tensor as the starting point. You might also want to look at this paper by the same folks (to see the entire paper, click on 'print this article', which will bring up the entire paper. I dare you to check the calculations in Appendix A, which are undoubtedly the work of some poor grad student). I hope time will tell who's right in the end.
 Being Number Two Ain't So Bad — Posted Friday, 15 January Japanese astrophysicists believe they've found the second largest black hole in our galaxy. They deduced the presence of a black hole from the range of observed velocities in the gas cloud known as CO-0.40-0.22, estimating the hole's mass to be that of about 100,000 Suns (100,000 $$M_ \odot$$). The cloud is located only about 200 light-years from the galactic center, almost a literal stone's throw from the biggest known black hole in the Milky Way (called Sagittarius A*), a monster of some 4 million $$M_\odot$$. Number Two is not in the cloud itself, but was deduced from observed extreme Doppler shifts of the emission spectra of gaseous hydrogen cyanide molecules. The data are consistent with a nearby black hole attracting the gases and then flinging them past the hole at high velocity in accordance with standard orbital "slingshot" physics. Interest in black holes has grown substantially over the past few decades. Originally dismissed by Einstein and others in the 1920s, by the 1960s it was apparent that no known physical or nuclear force could prevent their formation given enough "starting" mass. More recent observations appear to show that a massive black hole lies at the center of all galaxies, and there seems to be a very fundamental connection between black holes and the evolution of their host galaxies from nebulous gas clouds. Recent studies also indicate that there is a strong correlation between the mass of a galaxy and its central black hole, and just last month a group of astrophysicists calculated that black holes may have a theoretical limit of about 50 billion $$M_\odot$$. Earlier I posted a video link to a lecture Leonard Susskind gave recently on the connection of black holes with quantum physics, entropy and information theory. It is entirely possible that the birth, evolution and fate of the universe are tied to black holes in some fundamental way we don't yet understand. In addition, according to Susskind and others, the preservation of the principle of quantum unitarity (specifically, that information cannot be destroyed) may depend on black holes in one universe serving as portals to other universes. Fascinating.
 Waiting for Einstein — Posted Thursday, 14 January During a visit to the optometrist yesterday I took along my copy of Anthony Zee's Einstein Gravity in a Nutshell (unquestionably the best book of its kind), intending to read the chapter on Kaluza-Klein theory while waiting for the pupil-dilation drug to take effect. I was delighted to discover that the optometrist was a physics and math major in college, and he immediately recognized the book I was reading. A nice discussion followed, and we both noted that finding a kindred spirit in such an accidental manner is a rare but welcome occurrence. Anyway, in the chapter Zee discusses how Einstein reacted to the work of Hermann Weyl and Theodor Kaluza (shown) regarding their ideas on the unification of gravity and electromagnetism, and he quotes from some of the Einstein letters that Kaluza's son had preserved. April 21, 1919: Einstein writes "The idea [of unifying electromagnetism with gravity] has also frequently and persistently haunted me. The idea, however, that this can be achieved through a five dimensional cylinder-world has never occurred to me and would seem to be altogether new. I like your idea at first sight very much. From a physical point of view it appears to me more promising than the mathematically so penetrating ansatz of Weyl, because it concerns itself with the electric field and not with the, in my opinion, physically meaningless four-potential." (Zee notes that Einstein was completely wrong in this last sentence, as the four-potential is of far more fundamental importance than the electric and magnetic fields.) April 28, 1919: Einstein starts with "I have read through your paper and find it really interesting" but then adds "the arguments $$\dots$$ do not appear convincing enough." Zee goes on to note that here Einstein starts hedging, as he will ultimately (and unnecessarily) delay the publication of Kaluza's paper for two years. May 29, 1919: "It is true that I made a blunder with [some remark Einstein made in a previous letter] $$\dots$$ I see that you thought the matter over quite carefully. I have great respect for the beauty and boldness of your idea. But you will understand that I cannot take side with it as originally planned given the present factual doubts." More delays. October 14, 1921: Einstein again writes to Kaluza, saying "I am having second thoughts about having restrained you from publishing your idea on a unification of gravitation and electricity two years ago. Your approach seems in any case to have more to it than the one by H. Weyl. If you wish I shall present your paper to the academy after all, provided you send it to me." At last Einstein condescends to advise publication, but what I find remarkable is that Einstein seems to have misplaced Kaluza's original paper! Considering the gravity (no pun intended) of Kaluza's highly original and influential idea (today it's the basis of string theory and all higher-dimensional field theories), it's amazing that Kaluza, a fellow German, didn't tell Einstein to just go to hell and find someone else to promote his paper. And I find the "If you wish" part of Einstein's remark in his last letter to be particularly infuriating. What I find also remarkable is that Kaluza was born on the same day as Weyl, 9 November 1885, although he died a year earlier in 1954.
 Rambling Sunday Thoughts — Posted Sunday, 10 January People have written me to ask if I really believe in the multiverse, entangled black holes, holographic universes and all that stuff. I tell them that I try to keep up with current theories in modern physics, but at times the sheer volume of bizarre ideas that are out there today (just look at the number of papers that are being posted on arXiv.org) makes me wonder if things are getting out of hand. Twenty years ago the pinnacle of crazy, math-dense theories was considered to be M-theory, with its 11 spacetime dimensions, while until recently supersymmetry (with its bosonic-fermionic, partner-particle idea) was not only popular, but considered perhaps even true. Out of these came superstring theory and supergravity, and all the while modern physics was looking as if the mathematicians had not only taken over all of physics, but had gone mad as well. With the Large Hadron Collider scheduled for renewed power-up in March, scientists will have access to unprecedented energies (approaching 14 TeV) for particle research. While the LHC produced nearly unequivocal evidence for the Higgs boson in 2012, a series of negative results nearly hammered the last nail in supersymmetry's coffin. The collider's new power will hopefully send SUSY completely back to the mathematicians, who may then choose to make a religion out of it, but variants of string theory are likely to survive, especially if evidence for extra dimensions is found. And while I have absolutely no justification to say it, I think it's also possible that the LHC will provide nothing more than additional confirmation of the Standard Model. Beyond that may lie a virtual desert, devoid of any exciting new particles, forces and fields. If then even more powerful machines are built that produce the same non-results, would physics be essentially "finished" at that point, with nothing left for scientists to do but tack on more decimal places to the Standard Model's predictions? It's not out of the question. At the other end, perhaps at its deepest level physical reality lies at the Planck scale, where the so-called reduced Planck energy, roughly $$10^{19}$$ GeV, is needed to fully resolve what the hell's going on. But that energy is unimaginably greater than anything the LHC could ever produce (Susskind sees Planck-scale accelerators having the dimensions of the observable universe), meaning that humans will never be able to do Planck-scale physics. So, discounting a few new discoveries that the LHC may provide, there really is an ultimate limit to all this. Both string theory and loop quantum gravity (LQG) posit the existence of Planck-size vibrating strings and interconnected networks of spacetime, but again it is doubtful these ideas can ever be tested experimentally. In my opinion, LQG is more beautiful than strings, as it does away with the concept of time altogether, with space becoming granular or foamy, with quantized areas and volumes making up the resulting 3D network mesh of space. There used to be a draft version of Carlo Rovelli's 2014 book Covariant Loop Quantum Gravity available on the Internet, but it's gone now. It's hardly elementary, but the beautiful ideas it presents make me think that if I were the Creator, I might have resorted to that kind of thing. Ask your local public library if they have an interlibrary loan program that can get it for you, and check it out. And speaking of Creators, my mind goes back again to the notion of the simulation hypothesis, which is making more and more sense to me as I get older and even stupider. Unlike the "life is just a dream" idiom, it posits that in the distant future humans (assuming they're still around) will possess computer capabilities far in excess of anything we can imagine today (just extrapolate Moore's Law out a few hundred or thousand years and you'll understand what I'm talking about). Advanced humans (or what Oxford University's Nick Bostrom calls "post-humans") will be able to simulate nearly anything using computers, even computerized life forms having sentience and free will, all residing on a hard drive possessing incomprehensible capacity and speed. Bostrom believes that our own existence may in fact be a computer simulation which, if nothing else, would neatly explain the problem of theodicy. Those of you interested in this seeming lunacy fascinating idea might want to read Daniel F. Galouye's 1964 science fiction book Simulacron-3, or watch the excellent 1999 film The Thirteenth Floor based on the book. Other films, notably Inception, Dark City and World on a Wire deal with the same topic (Tom Cruise's Vanilla Sky also uses the idea, but, sorry to say, the film really stinks). From The Thirteenth Floor © Columbia Pictures and Centropolis Film Productions, 1999 Something we never could have expected?! Hell, that's the first thing I'd suspect. Perhaps the simulations go down forever, and go up forever. Perhaps their ends even link up, so that there are no simulators to begin with, and the universe we know creates itself out of nothing.