email:wostraub@gmail.com

AfterMath

Index photos courtesy ETH-Bibliothek,
Zurich Bildarchiv

Who Was Hermann Weyl?

Wheeler's Tribute to Weyl (PDF)

Old Stuff
2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

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

Uncommon Valor

She did not forget Jesus!
"Long live freedom!"

Visitors since 11-4-2004:

# 2014 Archives

 Ignore the Trolls — Posted Monday, December 8 2014 Years ago I would often go to Caltech's Millikan Library to read its sets of collected papers by Hermann Weyl and Albert Einstein, which included numerous letters of correspondence between the two scientists. But they were in the reference section, so I could never check them out and take them home. Now The Princeton University Press has released 5,000 papers written by Einstein over the years, all available for viewing free online, in the original German and in their English translations. These papers are interesting because they include personal notes and love letters written by Einstein along with other non-science stuff. The letters he wrote to Weyl are all from 1918 on, usually on the topic of gravitation and unified field theory. Einstein also wrote a glowing review of Weyl's 1918 book Space-Time-Matter, which represented the best explanation of Einstein's 1915 general theory of relativity available at the time. Also of interest is a letter Einstein wrote to Marie Curie in 1911, encouraging her to ignore the "trolls" and the "reptiles" who were attacking the brilliant chemist-physicist simply because she was a woman. The (colorized) photo above is the famous group picture from the October 1927 Solvay Conference on Physics in Brussels, showing no fewer than 17 Nobel prize winners, with Curie (a two-time Nobelist, physics and chemistry) the only woman in the field (looking to his right, the ever-iconoclastic 1945 physics Nobelist Wolfgang Pauli seems distracted, perhaps by a pretty girl walking by; a second group photo shows him also looking away). In the exact center of the group (seated behind Einstein's right shoulder) is Paul Adrien Maurice Dirac, who had just turned 25 at the time. While probably an accident, his central position in the photo is justified — as the discoverer of the relativistic electron equation in 1928 and too many other achievements to be listed here, he is in my opinion the greatest physicist who ever lived. " $$\ldots$$ Because God made it that way." — Engraved on Dirac's Tallahassee, Florida gravestone
 Are the Stars Aligned? — Posted Saturday, November 29 2014 You may have read of a recent discovery made at the Very Large Telescope, a 4-mirror visible/infrared instrument (each mirror 8.2 meters in diameter) operated by the European Southern Observatory on Cerro Paranal in the Atacama Desert of northern Chile, where a team of Belgian and German astronomers found a highly unusual alignment in the spin directions of very distant ($$z \gt 1.3$$) supermassive black holes. The initial paper is posted here at arXiv.org. Although the black holes cannot be seen directly, their spin directions can be deduced from the polarization of light being given off from the centers of the galaxies they reside in. Out of the 93 quasars (very active black holes) the scientists studied, 19 exhibit significant alignment with themselves and with the large-scale host structures they inhabit. The scientists estimate that the chance that the observed alignment is purely random is only about 1%. People are already making claims that the alignment is proof that the universe is the result of intelligent design, or that the observed filamentous distribution of matter is actually a vast system of connected, living neurons. I think such speculation is not only highly premature, but almost certainly wrong. For example, because gravitational energy is negative, the energy content of the entire universe is almost certainly exactly zero, as is its total angular momentum. The observed alignment of black holes may be a consequence of this 'zeroing out' of angular momentum. It could also be a coincidence, and it's also possible that the observational data have been misinterpreted. While I have often mused about the possibility that our universe is a computer simulation conducted by future, highly-evolved human programmers for whatever reason, it is doubtful that either they or God or any intelligent designer for that matter would make their existence known in so obvious a manner. However this recent discovery spins out, I still think that scientific observation and deduction makes a lot more sense than relying on the ancient blatherings of a bunch of illiterate semi-nomads for guidance and inspiration.
 Mathematics and the Perils of Human Logic — Posted Saturday, November 22 2014 In a famous episode from the original Star Trek series, Captain Kirk and his crew defeat an overbearing android with the statement "I am lying." The android, whose very existence relies on perfect reasoning ability and mathematical logic, cannot comprehend the sentence's impenetrable circular self-contradiction, and so blows a circuit. The crew of the Enterprise is saved, at least for another season's renewal. The "liar's paradox" is in fact a Kindergarten version of Kurt Gödel's 1931 incompleteness theorems, which proved (much to the amazement and consternation of the world's leading mathematicians), that logical, axiomatic mathematical deduction is fundamentally incapable of resolving certain problems, somewhat like the way Heisenberg's uncertainty principle $$\Delta x \Delta p \ge \hbar/2$$ prevents absolute knowledge of a particle's position and momentum. In his 1980 book Mathematics: The Loss of Certainty, New York University's Morris Kline sees Gödel's work as one of the "disasters" that befell mathematics in the 19th and 20th centuries (non-Euclidean geometry was another). Kline doesn't necessarily conclude that mathematics is itself flawed, only that its underpinnings — human logic and reasoning — are themselves flawed, with the result that mathematics is grounded on a foundation made of sand. The above quote by Hermann Weyl (one of my favorites of his) does not contradict Kline's claim since, like hygiene, human logic is never perfect. Indeed, in 1940 Weyl himself noted that "In spite, or because, of our disposed critical insight, we are today less sure than at any previous time of the ultimate foundations on which mathematics rests." He later added "The question of the ultimate foundations and the ultimate meaning of mathematics remains open: we do not know in what direction it will find its final solution or whether a final objective answer can be expected at all. 'Mathematizing' may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalization." The above Weyl quote is in fact a screen capture I made from one of Bruce H. Edward's (University of Florida) excellent Teaching Company lectures called Prove It: The Art of Mathematical Argument. In one lecture he notes that, having proved a mathematical theorem, we know it to be true now and for all time, and it is likewise true everywhere in the universe. I have always believed this, so it is disconcerting to me to imagine that certain (even simple) mathematical proofs might not be true in some sense. But what I think bails out mathematical truth is the possibility that human logic and reasoning, being imperfect, make it entirely probable that absolute mathematical truth does in fact exist. So what brought all this on? Yet another, alas unresolved, friendly argument with a family member about what is real and what is not. Noting that another member of the family sports a magnetic bracelet for health reasons, I remarked that its use is akin to mere superstition, the health benefits of magnetic fields having never been demonstrated scientifically. She disagreed, arguing that if it makes the wearer feel good, then there must be something going on, even if it's only a perceived benefit occurring in the mind (dopamine release) or the central nervous system (serotonin release), and that it does no harm. I didn't dispute the possibility of pharmacological/mental feedback effects, but I did argue that this is much the same logic of a child who, in obeying the childhood rule "step on a crack, break your mother's back," also feels good when she returns to the house to find that her mother's spinal column is indeed still in one piece. My argument is that (most) people grow out of such silly beliefs, because scientific evidence has shown what's really going on. As for it doing no harm, I strongly disagreed. The ignorance of a child, if not corrected, becomes the evil of ignorant adults who lead us into war and all manner of other wrongs; after all, George W. Bush, as he once proudly admitted, earnestly but ignorantly believed that God had personally directed him to invade Iraq. I then noted that as-yet unresolved phenomena such as magnet "therapy" is nothing but the "God of the Gaps" argument all over again. Thousands of years ago people saw God's handiwork everywhere, since science as a formal discipline did not exist. But when science began to explain things in a more rational manner, the number of scientifically-unexplained phenomena decreased to the point where only a few gaps in our understanding remained, and God and superstitious agents as explanations for things got pushed into these gaps. Today, those gaps are now minuscule in size, but they still offer a way to believe in superstition and the supernatural — "Science doesn't explain that!" goes the argument, thus preserving a multitude of illogical religious and non-religious belief systems. (Several years ago, conservative cable news host Bill O'Reilly famously embarrassed himself when he claimed that nobody could explain the phenomena known as ocean tides without invoking the magical power of God.) So the gaps have indeed diminished, but the power of irrational belief, superstition and ignorance sadly has not. Notwithstanding the mathematical "disasters" that Kline spoke about, I do not think it irrational to believe that science and mathematics are our best tools for uncovering the truth — whatever it may be.
 No Country For This Old Man — Posted Monday, November 10 2014 What we learn from history is that we do not learn from history. — Hegel Those who do not remember the past are condemned to repeat it. — Santayana Both the great Spanish philosopher George Santayana (1863-1952) and the even greater German philosopher Georg Wilhelm Friedrich Hegel (1770-1831) certainly saw the sad applicability of this sentiment in their own day, but it is doubtful that they could have imagined its even greater applicability in today's world of instant worldwide communication and the availability of an almost infinite amount of knowledge, all free for the taking. After many long discussions over the years on politics and religion (but rarely math or science), a friend of mine said to me, the day after the November 4 mid-term election bloodbath, "Why not give the Republicans a chance?" So I asked him if he remembered the George W. Bush years, you know, the treasonous lies about Iraq having WMDs and all that, the 4,500 American troop deaths, all for nothing, not to mention the hundreds of thousands of innocent civilians, also killed for nothing. "Well, maybe it will be different this time" he replied. Left speechless by this incomprehensible remark, I could only walk away. It was the same day that a close family member, suffering from an advanced neurological disease, told me that what kept her going was her Christian faith and her patriotism. Patriotism? Patriotism?! Out of compassion (or cowardice) I didn't challenge her, but I said to myself "What the fuck does patriotism have to do with anything?" At long last, I have given up. The GOP takeover of the Senate will introduce a whole new level of anti-science and anti-environmental doctrines, corporate greed, militarism and blind religious dogma to the country, not to mention the historical revisionism that my friend seems to have embraced. I remind myself that people will believe what they want to believe, and no amount of facts, logic or truth will have any effect whatsoever on those beliefs. So I've de-registered as a voter, having finally realized that voting is a meaningless exercise ("Should I vote for this rich, corrupt, self-interested bastard or that rich, corrupt, self-interested bastard?"), and I'm canceling my cable subscription, having also finally realized that cable news outlets like CNN, Fox and MSNBC — in reality, any source of pre-washed media "information" — is not only useless, but actually detrimental to mental health. My one saving grace is the thought that perhaps the entirety of the human experience is some kind of a millenia-long experiment, conducted by a god, Nature, fate or chance, or the 4D digital output of some nebulous, external computer simulation. The United States, I've come to believe, will simply run out its course, whatever that course may be, and its ultimate fate doesn't look at all hopeful, but I can't change that course, and I will certainly pass from the scene long before that course becomes apparent to anyone, in particular the inhuman monsters who seem to be running everything. But I'm not really blaming any one person in particular, Bush, Cheney or even the entire GOP, for that matter. One can endlessly demonize them, as I have done, but their actions cannot explain why we keep clinging to the same immoral, ruinous political and military policies that have plagued the country for the past fifty years. The fault lies with the American people themselves, whom I suspect don't really think at all but just accept things the way they are, neither expecting nor demanding reasons for why things are so fucked up. But this insane way of living is wholly unacceptable to me. Why any species with the brains to develop theories like quantum mechanics and general relativity (which have never failed a single test) would at the same time behave so irrationally, inhumanely and destructively is to me the key riddle of our existence. I haven't found an answer, but that hasn't stopped me from thinking that the root cause of our irrational behavior is our animalistic nature which, unlike other aspects of our anatomy (like cranial capacity), has sadly not evolved over the millenia. It has been said countless times that the stock market is based on two core human emotions, fear and greed, and that it is these two emotions that drive our species. But greed is itself just a consequence of fear: the need to acquire as much as one can to allay or ease the fear of suffering is not really a human emotion but a consequential behavioral trait. That fear is the underlying reason for all our political, religious and cultural problems seems to hold up quite well, considering that issues such as denialism and its fellow travelers cognitive dissonance and pseudoscience have to have a basis somewhere when considering irrational human thought and behavior, while the emerging theory of agnotology would seem to explain fear as the result of a love of willful ignorance. Whatever is wrong with us, whether or not some kind of animalistic, primal fear lies at its core, it is obvious that we're not going to do anything about it, much less even acknowledge that a problem exists. That is a great pity; it's still a beautiful world, while the entire universe, with its manifest physical laws and mathematical symmetries, points to an undeniable beauty that may or may not be of intentional design or purpose. Too bad the human species is unlikely to survive long enough to truly appreciate it.
 Do Black Holes Really Exist? — Posted Wednesday, October 22 2014 Contrary to conventional wisdom, some physicists retain doubts about the actual existence of black holes, maintaining, as Einstein did, that Nature would never allow a star to collapse to a mathematical point object of zero volume and infinite density. Einstein's view was more or less the accepted norm until 1939, when J.R. Oppenheimer and H. Snyder published a now-famous paper in Physical Review that appeared to solidify the case for black hole formation. Their analysis, which ignored the possibility of any radiation or matter escaping from a massive object under runaway gravitational contraction, concluded that no known force or process could halt collapse — the object would continue to contract until it literally "winked out" of existence, leaving behind a point singularity behind an event horizon, which is the standard definition of a black hole. The Oppenheimer-Snyder analysis is neatly described in Chapter 14 of Adler-Bazin-Schiffer's 1975 book Introduction to General Relativity, and the interested reader is referred to that book for details of the analysis. Nevertheless, the concept of a physical black hole has remained in question since that time. The latest attack comes from University of North Carolina-Chapel Hill's Laura Mersini-Houghton (pictured above) and her colleague, who argue against black holes in a relatively accessible paper (September 2014) that takes into account the effects of Hawking radiation as an object collapses. Mersini-Houghton's approach is very similar to that of Oppenheimer-Snyder with the exception that the stress-energy tensor $$T^{\mu\nu}$$ includes a term for Hawking radiation as well as the four-velocity of infalling matter (as could be expected, the Hawking term is proportional to the null four-vector of light, $$p^\mu$$). The resulting model is therefore a generalization of the Tolman-Oppenheimer-Volkov equation of hydrodynamic equilibrium, which you undoubtedly learned about in grade school. Anyway, Mersini-Houghton's analysis shows that stellar collapse follows the Oppenheimer-Snyder model until the Schwarzschild radius is approached, at which time Hawking radiation takes over and in effect blows away most of the collapsing star's mass, preventing the formation of a black hole. The analysis includes detailed numerical examples for stars of varying masses, all of which essentially "bounce" at the point of collapse at the Schwarzschild radius. The only argument I can come up with against Mersini-Houghton's analysis is that her stars invariably explode away before black hole formation, leaving nothing — but many intra- and intergalactic objects have been observed to date in which stars and gas are seen falling into or orbiting unseen objects that could only be black holes (the well-studied 4-million-solar-mass object at the center of our own Milky Way galaxy is perhaps the most notable example). In addition, although no one doubts the existence of Hawking radiation, it has never been observed, and its precise expression in the stress-energy term remains open. Furthermore, exactly how the radiation term comes to dominate the physics in the final stages of collapse also seems open to interpretation. Nevertheless, a very interesting paper!
 The Ten Commandments Found! (Sorta) — Posted Saturday, October 18 2014 In 1923, noted silent film director Cecil B. DeMille created a lavish wood and plaster set for his epic film The Ten Commandments on the central California coast near Nipomo. After filming was completed, the set, covering acres and consisting of numerous gigantic replicas of ancient Egyptian statues and monuments, was deemed too expensive to relocate or demolish, so the entire thing was carted off to the nearby Guadalupe Sand Dunes and buried. Now archaeologists are digging up the site, assisted by film historians and interested locals who want to restore some of the relics and put them on display. The movie itself juxtaposes the ancient story of the Exodus from Egypt with a modern tale of morality set in San Francisco. As a silent film buff, I was never too impressed with the movie's all-too obvious moralizing, its naive plot or the theatrical over-acting of its stars (though all were typical of films in those simple times), but the film's scale and the set itself are awesome, and the movie is still worth watching today. Amazingly, the film involved the work of thousands of carpenters, artisans, sculptors and extras but was brought in under its allotted $1.8 million budget (today's films cost a lot more, but the artisans are all gone, replaced by computer animators whose special-effect creations nevertheless look a lot better than they did in deMille's day). I once visited a factory in Lincoln, California that manufactures replica fixtures for old buildings in San Francisco whose damaged facades, constructed by artisans in the early part of the 20th century, can no longer be made either because of cost constraints or because artisans capable of doing the work no longer exist. But what truly fascinates me about the "Lost City of DeMille" story is that the Exodus story itself, still taken at face value by billions of Christians, Jews and Muslims today, never actually happened. The ancient Egyptians, notable for their record keeping (albeit often accompanied by exaggerated claims of military prowess), say nothing about enslaved Israelites or their exodus from bondage. The entire population of Egypt in the latter part of the Bronze Age (roughly 1200 to 1400 BCE) is estimated to have been about 4 to 5 million people, while the Old Testament book of Numbers gives the exact Israelite slave population as 603,550 men of military age (20 and older), from which a total population of roughly 2 to 2.5 million can be extrapolated if we include the men's wives, children, parents and other family members. The idea that Egypt's entire contingent of slave laborers, making up some 40-50% of the country's total population, simply took off one day is patently ridiculous — not only would there be a record of the event, but Egypt's entire economy and social structure would have been destroyed at a time when Egypt was at the peak of its power and influence in the region (the Bible also tells us that the Israelites took tons of gold and livestock with them, which would have further decimated the Egyptian economy). And unlike the remains being uncovered in the Guadalupe Sand Dunes, not a single trace of archaeological evidence exists for even hundreds (not to mention millions) of people traipsing through the Sinai or the Levant in the 42 encampment locations documented in the Old Testament — not a single pot sherd, campfire, piece of bone, leather sandal, grave, cartwheel or metal/wooden object has ever been found, despite many decades of intense archaeological excavation. Indeed, in 2001 noted Los Angeles rabbi David Wolpe announced to his congregation that it was time to give up the idea of an historical Exodus event since there is no physical evidence that it ever occurred. But belief in the Exodus story persists in the minds of most people of faith today, since much of Judaism and Christianity is based on the epic tale of God's people being delivered from bondage (Jesus himself spoke of it). I would also mention that the date of the Exodus cannot be determined with any precision, as the Bible does not identify which pharaoh was ruling at the time. The Old Testament says that the first Temple was dedicated exactly 480 years after the Exodus, and since we know that Solomon constructed it around 950 BCE, this would place the Exodus at around 1430 BCE, the time of 18th dynasty pharaoh Thutmose III. But Egypt was near the peak of its regional power at the time; there could have been no mass exodus of slaves, not to mention the fact that Egypt controlled all of the Levant, including the "Promised Land", making its supposed conquest by Joshua and his armies an impossibility. Alternatively, some biblical historians consider the Exodus to have taken place later, around 1250 BCE, but this was during the rule of the great 19th dynasty pharaoh Ramses II, who reigned with unquestioned, documented power over an even more powerful Egypt, making the Exodus even more impossible. Most authorities believe today that the Exodus story, which was composed by Jewish scribes around 600 BCE at the earliest, was probably confused with Egypt's expulsion of the Hyksos Canaanites, who conquered Egypt around 1700 BCE and ruled it until they were expelled around 1550 BCE. This theory is supported by the fact that the Bible's knowledge and understanding of Egypt is not only often historically wrong, but consistent with a later, post-Solomonic Egypt, which by then had lost much of its earlier power and dominance thanks to repeated invasions by the Assyrians, Nubians, Libyans and others. Head of the mummy of the 17th dynasty pharaoh Seqenenre, showing well-defined (and fatal) battle wounds, who most likely died leading the battle against the Hyksos around 1550 BCE. By comparison, where are the mummies of Moses, or Aaron, or Joshua, and where are the skeletons of the tens of thousands of Israelites who certainly died during the 40-year wandering in the Sinai or were killed in the supposed conquest of Canaan? My answer: They never existed — the story of the Exodus is a myth, a fabrication, a lie of epic proportions. Amusingly, in his autobiography DeMille spoke of a fear he had that archaeologists unearthing his film's remains thousands of years in the future would conclude that Egypt's ancient empire once extended all the way to California! That is highly unlikely, although the myth of the Exodus story will undoubtedly still be around. Pop Quiz, Hotshots: How many plagues were visited upon Egypt according to the Book of Exodus? If you answered "ten" then you're wrong, and should read your Bible more carefully.  Weyl Gravity Research — Posted Thursday, October 16 2014 Here is an interesting YouTube video announcing two grants that California State University at Fresno's Douglas Singleton received to study Weyl gravity abroad. It's a year old, and I haven't yet seen an update, but I'm posting it here as a reminder that Weyl's gravity theory is being actively investigated as a possible explanation for the galactic rotation problem (at least as an alternative for dark matter, which continues to elude scientists).  Long-Lost Ship is the HMS Erebus — Posted Saturday, October 4 2014 On April 21 and September 10 I made some comments about the ill-fated Franklin Expedition (a topic of some interest to me), which left England in 1845 in a search for the Northwest Passage. Led by famed British explorer Sir John Franklin with a complement of 134 men, the expedition's two ships, the HMS Terror and the HMS Erebus, got trapped in ice during the unusually cold summer of that same year. The men abandoned the ships and went ashore, where they all soon died of starvation, disease, exposure and lead poisoning (all the canned food tins were soldered with the toxic metal). This September Canadian scientists found one of the ships in about 35 feet of freezing water, but were unable to identify which one. Several days ago they confirmed through visual inspection that it is in fact the Erebus, Franklin's personal transport. Remarkably well preserved in the cold water, the scientists hope to be able to determine if Franklin himself abandoned the ship or died on it. Fascinating.  Wheeler, Everett and Nuclear War — Posted Wednesday, October 1 2014 Peter Byrne's 2010 book The Many Worlds of Hugh Everett III devotes an entire chapter to John Archibald Wheeler (1911-2008), the noted Manhattan Project nuclear physicist who served as the advisor to Richard Feynman at Princeton University and the somewhat lesser known Hugh Everett, father of the multiverse theory. I always admired Wheeler for his efforts to resurrect general relativity in the 1950s (when it was languishing as a "classical" theory and taking a back seat to quantum field theory) and for his work on black holes a decade later (Wheeler in fact coined the term "black hole"). At the same time I was aware of his conservative tendencies (he rarely appeared anywhere without a suit and tie), but it wasn't until I read Byrne's book that I learned just how conservative Wheeler really was. Wheeler was extraordinarily brilliant, obtaining his physics PhD from Johns Hopkins University at the tender age of 22 (he didn't even pause to get a BS or MS degree, but went straight on through). He then went to Copenhagen to do post-doc work for the great Niels Bohr, whose influence remained with Wheeler his entire life. There he also met and collaborated with Werner Heisenberg, whose Nazi sympathies were based on German patriotism and an admiration for Hitler. Just how this influenced Wheeler is subject to debate, but upon his return as a professor at Princeton he was, as Byrne notes, an unreserved "might-is-right" Nazi sympathizer himself (he admired Hitler's steely resolve), a characteristic that was not lost on Wheeler's fellow physics professors, many of whom were Jewish. Following his stint with the Manhattan Project, where he worked to amass a quantity of plutonium sufficient to annihilate Nagasaki, Wheeler became a doyen of the military and corporate world, consulting for the Department of Defense on the hydrogen bomb program. As Byrne describes in the chapter, Wheeler became an almost fanatical opponent to communism, "cheerfully serving" the Defense Department warloads and consulting with contractors like Du Pont, General Atomics, Convair, Lockheed and General Electric, all of which had lucrative deals with the Pentagon. During this time, the persecution of many of Wheeler's colleagues (notably J.R. Oppenheimer and David Bohm) by the House Un-American Activities Committee's paranoid McCarthyites did not faze him in the least. As a prototypical "cold warrior" like his friend and fellow H-bomb developer Edward Teller, in 1959 Wheeler joined forces with none other than Henry Kissinger (then a Pentagon consultant) to write, along with fellow Pentagon employee Hugh Everett, "A Doctrine for Limited War," which openly advocated the use of tactical nuclear weapons in small ground wars. I picked up Byrne's book to learn more about Hugh Everett III, who was Wheeler's PhD student at Princeton in the early 1950s, and to learn more details about Everett's multiverse theory, which Wheeler initially championed. But I learned a lot more. Wheeler's hero Niels Bohr panned Everett's idea of multiple universes, which resulted in Wheeler's subsequent tepid support of the theory, leading to Everett's disillusionment with academia. After receiving his PhD in 1956, Everett left Princeton and went to work as a highly-paid war operations strategist with the Pentagon, where he spent the bulk of his brilliant career. A heavy smoker, drinker and skirt-chasing hedonist, he died at age 51 from a massive heart attack. Truth be told, I was shocked at how gifted scientists like Wheeler and Everett were so easily taken in by the lure of jingoistic patriotism and lucrative military contracts, while physicists like Oppenheimer and Bohm were fired and shuttled off to relative obscurity by the pro-war fanaticism that dominated the 1950s. Although Byrne's biography of Everett talks in detail about the multiverse theory, it unabashedly condemns the prostitution of science for military purposes. I cannot help but look back in horror at the efforts of these men, who eagerly plotted the deaths of hundreds of millions of human beings. And after reading the book, I wonder how the planet ever survived. One of my prized possessions is a copy of the obscure 1965 book "Gravitation Theory and Gravitational Collapse," written by Wheeler and co-author Kip Thorne of Caltech (who kindly autographed the book for me many years ago). But, having read Byrne's book, Wheeler's name will always leave a bad taste in my mouth from here on out. Mr. President, I'm not saying we wouldn't get our hair mussed. But I do say no more than ten to twenty million killed, tops! Uh, depending on the breaks. — General "Buck" Turgidson (George C. Scott), Dr. Strangelove (Everett's favorite film)  Happy Anniversary — Posted Saturday, September 27 2014 I recall one day in the early 1990s when one of my sons, who was taking precalculus in high school, was looking at some of my old textbooks. He asked "Dad, why do these books have all these tables of logarithms and trig functions?" but I'm certain he knew that in my school days we didn't have calculators or computers (though we did have slide rules), so a large portion of every math and science textbook had to have tables of the damned things. We were even taught how to laboriously interpolate their values between the tabulated ones to get more accurate answers. As science writer Tom Siegfried notes, this year is the 400th anniversary of the discovery of logarithms by the Scottish landowner and (oddly enough) theologian John Napier. The article also notes that the discovery extended the productivity of scientists in Napier's day by at least 100%, since many lengthy hand calculations were reduced to simply looking numbers up in tables. (Curiously, Napier's discovery, coupled with the fact that his lawn was always greener than those of his neighbors, gave rise to the belief that he was a sorcerer. The article also relates how it was believed Napier owned a magical black rooster. Perhaps his sideline as a theologian saved him from being burned at the stake.) Siegfried also notes that 2014 is the anniversary of many notable discoveries, including quarks (1964), the cosmic microwave background (also 1964) and the notion of atomic number, the number of protons in an atomic nucleus (1914). Next year, of course, will mark the 100th anniversary of perhaps the greatest achievement of the human mind, Einstein's general theory of relativity. Hopefully, our increasingly fractured world will pause to take notice.  Self-Correcting Science — Posted Friday, September 26 2014 In March of this year the Background Imaging of Cosmic Extragalactic Polarization ( BICEP) detector, operated by numerous universities and located in Antarctica, reported evidence of the polarization of the faint radiation associated with the cosmic microwave background (CMB). This was initially hailed as a major discovery, as it provided direct evidence for primordial gravitational waves and Guth's inflation theory. But subsequent data obtained from the European Space Agency's orbiting Planck Space Telescope cast doubt on the BICEP data, showing that intragalactic dust could mirror the polarization effects seen by the BICEP team. Although the BICEP analysis took into account dust in the Milky Way Galaxy, it apparently undercorrected for the effect. The Planck team is scheduled to release additional data in October that is expected to either confirm or refute the BICEP analysis. Coincidentally, the October issue of Scientific American has an article by noted Arizona State University astrophysicist Lawrence Krauss on the BICEP and Planck data. It may be my imagination, but my take on the article is that it was written not long after the March BICEP data appeared and was intended to praise the results, as confirmation of inflationary theory leads to strengthened support of the multiverse theory and the notion that gravity must be quantum in nature. However, Krauss does include a detailed discussion of the refuting Planck data, thus bringing the article in line with the most recent (and depressing) aspects of the dust issue. I think all this is very interesting. Like Krauss and many others, I would love to see confirming evidence of the multiverse. However, what we're witnessing is a triumph of science irregardless of how all this spins out — in spite of the subjective preferences, hopes and even wishful thinking of physicists, the self-correcting foundation of the scientific method invariably bends towards objective truth. Can anyone in politics or religion say the same? For a readable account of the issue and for numerous useful references, see the New Scientist article here.  Superheavy Einstein — Posted Saturday, September 20 2014 It surprises many students when they learn that the familiar neutron is actually an unstable particle, decaying into a proton, electron and antielectron neutrino with a half life of about 10 minutes. Fortunately, when bound into the nucleus of an atom the neutron is completely stable, with an expected half-life equivalent to that of the proton (measured to be $$\gt 10^{32}$$ years). Such is not the case, however, for superheavy elements whose nuclei are packed with so many neutrons and protons that radioactive instability is the norm. The artificially-produced element seaborgium ($$^{271}$$Sg, atomic weight 106) has a half-life of only 2 minutes, although some transfermium elements (aw $$\gt$$ 100) can hang around for hours or days. But studying atoms associated with these nuclei (say, Sg with a full complement of 106 electrons) is difficult because of their short life spans. The speed of a single ground-state electron circling a superheavy nucleus is so great that relativistic calculations are necessary, which is not the case for garden-variety atoms in which the Schrödinger equation suffices. Earlier this year, that all changed when relativistic effects were seen for seaborgium-106 (bonded to six carbon monoxide molecules) at the Helmholtz Centre for Heavy Ion Research, an accelerator laboratory located in Darmstadt, Germany, whose observations confirmed the relativistic prediction. This represents not only a triumph for superheavy element chemistry, but yet another confirmation of Einstein's special theory of relativity (not that there has been any doubt). There is a really neat inverse way of looking at all this. Instead of having a superheavy isotopic nucleus orbited by an ordinary electron, one might consider instead an ordinary lightweight nucleus orbited by a superheavy electron. Such an electron does exist — it's the muon, a variant of the electron with a mass approximately 207 times that of its lightweight cousin (there's also the tau, an even heavier variant). Both the muon and the tau exhibit properties identical to that of the ordinary electron, except for the mass aspect. What this means is that if the electron in an ordinary hydrogen atom (proton plus electron) were replaced by a muon, the muon would orbit the proton 207 times closer to the nucleus. If a muonic form of the hydrogen molecule H-H is made (which has been done), the two protons could get much closer, allowing for nuclear fusion at much lower initiating energies. Unfortunately, the muon is itself unstable, disintegrating into a muon neutrino, an electron and an antielectron neutrino in 2.2 microseconds (that's too bad for the human race — if muonic hydrogen nuclear fusion could be made practical, it would solve all our energy problems, and almost without any attendant pollution). But could the muon or tau be stabilized in such a way to prevent or postpone their decay until fusion can be made possible? Doubtful, but one can always hope. After reading the New Scientist article linked above, I immediately thought of Wikipedia's evil twin, Conservapedia (whose article on Einsteinian relativity appears to have been hacked!) Conservapedia, the braindead offspring of noted Christian fundamentalist Phyllis Schlafly and her equally incoherent son Andy, denounces relativity as a false theory because it counters the action at a distance notions of God, which are required to take place instantaneously. Willful ignorance, indeed.  Sleepwalking into War — Posted Friday, September 19 2014 Some readers of Christopher Clark's bestselling 2014 book The Sleepwalkers: How Europe Went to War in 1914 will be disappointed in the 700-page book's lack of battle photographs and diagrams, but what it lacks in graphics it more than makes up for in content. By restricting itself to the how and why of World War I rather than the battles, Sleepwalkers documents how Europe (if not the entire world) was primed for a "good war," not that there was any paucity of conflict and bloodshed in the early part of the 20th century. Still, it seems strange today that events taking place in far-off Sarajevo (which most Americans had never heard of at the time) should play so pivotal a role in setting off the first major world conflict. As Clark explains it (and as many others have also noted, notably Prof. Vejas Liulevicius of the University of Tennessee), the world's people at the time appeared primed for war, not simply because of rising nationistic fervor and political tensions, but out of a strange kind of mass psychology that gave rise to a baffling feeling of societal unity and oneness and an unquenchable desire for personal recognition, heroism and vainglory. After reading the book, I can only conclude that World War I took place simply because people felt that it was time modern society had itself a good fight, and that civilization needed to let off some steam. As Clark notes, Germany's two-front Schlieffen Plan, designed to bring a rapid and conclusive victory to that nation, failed because of a combination of military missteps and the modernity of armaments (like America's Civil War, artillery and rapid-fire weapons technology made conventional combat, like mounted cavalry and mass attacks, obsolete). In a way, World War I was already over by late 1914, when mutually-outgunned armies found themselves hopelessly slogged down in the endless filth and disease of trench warfare. The 100-year anniversary of the assassinations of July 1914 and the start of The War to End All Wars has come and gone, but mankind's desire to settle its differences through war has not changed. I find it ironic that the profoundly beautiful revolutions in science that Einstein and others had begun in relativity and quantum physics during WWI made little real difference to a world intent on destroying itself. I find it similarly ironic that the same situation exists today, in spite of quantum-leap advances in instant, world-wide communication, medicine and technology. Coincidentally, I recently discovered a WWI poem that my father had published at the tender age of 13. It appeared in the May 9, 1918 issue of The Quincy (Illinois) Daily Journal, and sadly was a paean to the militaristic jingoism that we still see today. Funny, but most of Quincy's citizens at the time were German immigrants or their offspring, so maybe all the patriotic talk was only so much apple-polishing. (Nice try, Dad. You didn't know any better.)  Ubi Ego Vadis? — Posted Thursday, September 18 2014 I will probably shut this site down in November (its tenth anniversary), as it has become less of a science and math blog and more of a political/religious sounding box. The reason for this has been my endless frustration over the refusal of my country to listen to reason, preferring instead the comforting balm of religious dogma and exceptionalist (USA! USA!) thinking, regardless of how wrong or insane those mindsets are, as exemplified by my country's decision to once again go to war over the oil resources of another nation based on fear and greed couched in sanctimony. [I know, I know — insert whiny sound here.] While my opinions and comments have not been confined to this site but many others as well (example), I realize I've just been singing to the choir of like-minded people who also feel the world has gone mad. I'm now 65 years old and need to move on. That, plus I've got my first grandchild to think about now; maybe that will help me refocus my priorities.  Two Views: Sarah and Sy — Posted Tuesday, September 16 2014 I love that smell of the emissions. — Sarah Palin I had just finished reading Alex Bellos' clever new book The Grapes of Math: How Life Reflects Numbers and Numbers Reflect Life this morning when Naomi Klein's latest book This Changes Everything: Capitalism vs. The Climate arrived on my doorstep. At only 330 pages, Bellos' book is one every person should read, while Klein's 600-pager is also required reading, but with a bit more dedication. Bellos spends an entire chapter on the transcendental exponential number $$e$$ and how it is used to describe (appropriately) exponential growth. He starts the chapter with an aside about the late University of Colorado at Boulder physics professor Albert Bartlett, the noted population control advocate whose talk Arithmetic, Population and Energy was given nearly 2,000 times to hundreds of thousands of students and concerned citizens (you can catch the first of eight YouTube videos of the talk here). I met Bartlett in Colorado sometime in 1995, and found his thesis to be both alarming and spot-on: for whatever reason (perhaps simple stupidity), the human race has been either unwilling or unable to understand the concept of exponential growth. A favorite analogy that Bartlett used is this: Imagine a test tube containing a single microorganism and (for all intents and purposes) an unlimited supply of food and an unrestricted method for waste disposal (germs poop too, you know). The bacteria can multiply once a minute on average, taking into account the deaths of loved ones and fatal accidents. The cell is introduced into the test tube at precisely 11:00 am, and bacterial reproduction begins. This goes on for awhile, but at 11:55 am a very intelligent bacterium, who has measured the volume of the test tube and calculated the average volume of a single living cell, figures that by noon the test tube will be full. She informs her fellow bacteria, but they are skeptical that any problem exists. "Our test tube home is only 3% full," they reply, "and there's plenty of room for continued growth." A minute goes by (which is a along time for a bacterial cell), and the test tube is now 6% full. After another minute, it's 12% full. "Still plenty of time for us to do something, if it should come to that," they tell her. Now it's 11:58 am, and the tube is 25% full. "Let's talk about it later, no need to get everyone worried," they say. At 11:59 am the tube is half full. Somewhat alarmed, the bacteria convene a committee to discuss the problem, but at noon the tube is full, and citizens start falling to their deaths over the lip of the test tube. It's finally too late to do anything. At the current annual growth rate of just 1.2%, the world's population will double about every 57.7 years, adding a population equivalent of ten New York Citys every year. Plenty of time to fix all our problems, right? But look at what we've been doing — finding ways to make agricultural yields greater to feed more people, advancing our medical technology to keep people alive longer and with fewer deaths by disease, and increasing our ability to exploit and consume the planet's resources to accommodate the desires of business and its demands for ever-continuing growth. That's a fine plan if you've got an infinite planet, but the Earth is sadly all too finite. Perhaps more importantly, its ability to deal with the pollutants we're spewing out is also limited. I haven't read Klein's book yet, of course, but I've read the reviews and seen the interviews. I did read her earlier book The Shock Doctrine: The Rise of Disaster Capitalism, which was a disturbingly prescient view of how natural disasters and wars allow governments and wealthy individuals to scare citizens into supporting greedy and even inhuman actions (like the Iraq War). Her current book likewise talks about the connection between global climate change and capitalism and how we're destroying the planet's ecosystems for short-term profit. I promise not to post a book report here once I've finished it, but instead encourage you to read it for yourselves. So, are humans smarter than bacteria? Apparently not. This is perhaps the greatest reason why I often believe the human race (and maybe the entire universe) is a computer simulation, perhaps designed to find out what difference intelligence makes when confronted with irrational fear, greed and cognitive dissonance, all of which may be common shortcomings of any sentient civilization. We're fucked. — Sy Hersh  Data Patterns — Posted Tuesday, September 16 2014 In early 1981 I received my company's first IBM PC, a floppy-disk driven thing with a clunky monochrome screen and a single piece of software (Lotus 1-2-3) to develop our inital program base. There were no software compilers for PCs in those days, and ours was shipped with Microsoft's BASIC Interpreter (the assembly-language application that made Bill Gates his first million). One day during lunch I wrote a simple program to gradually fill the screen with dots generated by random $$x-y$$ points (I didn't want to use the interpreter's built-in random number generator, thinking it wasn't truly random, so I made my own). I was surprised to see that the program produced clearly recognizable patterns of dots, a completely different pattern being generated each time I ran the program. Many of them were shockingly beautiful, and I kept wondering how random numbers could produce such things. I didn't know it then, but this was my first rudimentary experience with the mathematical subject known as chaos theory. I found another odd pattern in a paper I wrote in 1999, in which I noted that pairs of galaxies appear to be lognormally distributed in terms of their distances from one another: But this is nothing compared to the distributions cosmologists are finding today, where clusters and superclusters of galaxies are exhibiting patterns that apparently defy understanding: The Nature article on which this video is based talks of The Great Attractor, a kind of cosmologically-sized gravitational sink where galaxies and clusters of galaxies appear to be headed. I haven't read the article, but the notion of an attractor is a fundamental aspect of chaos theory, leading me to conclude that the universe, as inconceivably huge as it is, nevertheless obeys a sort of structural algorithm or program that defies rationalization. (On the other hand, the video notes that the effects of dark energy in the simulation have been ignored, so maybe it's not all it seems.)  Perfect — Posted Tuesday, September 16 2014 Here's a World War I political cartoon I first saw in my freshman high school history book. I never forgot it, but couldn't find it until today. It still rings true:  How Very Christian of Us — Posted Sunday, September 14 2014 Having lived in Pasadena for many years I can recall driving to Old Town to attend a play, shop for clothes, see a movie or have a meal while parking my car just about anywhere for free. Then about 10 years ago the City and its businesses decided they would meter every and all parking spaces, in fact just about any area that would take a metering pole or assigned-slot meter box — streets, parking lots, parking structures, alleys, abandoned or empty lots, you name it. Even large private parking lots reserved for stores like Tower Records and The Good Guys fell to the metering god (parking at Target is still free, but who shops a Target?) Meanwhile, parking at the City's premier mall, The Paseo Colorado, is also metered (and isn't cheap, either), though shoppers don't seem to mind (the City's previous mall, unkindly dubbed "The Corpse on Colorado," featured free parking but was not successful; the whole thing was torn down at enormous cost and replaced with trendy open-air shops overlooked by expensive private apartments and even more expensive condos). Old Town Pasadena is an interesting place, but on weekends it's horrendously crowded and a bit too trendy (and too expensive for retirees like myself). Funny how back in 1972, when The Sting was being filmed on locations there, Old Town was really just that — an embarrassingly run-down remnant of the 1920s featuring cheap garment shops, low-class eateries and panhandlers. But the City Fathers knew an opportunity when they saw it, and used the enormous popularity of the film to rehabilitate Old Town into the swanky place it is today. The garment shops and Joe's Eats are now all gone, replaced with overpriced clothing outlets and ambiance-drenched restaurants and breakfast nooks (where you can get your basic pancakes and eggs for just$20), but the panhandlers are still there, providing a nostalgic glimpse of the Old Town that used to be (I'm still amazed how people can walk out of the Apple Store on Colorado Boulevard, having dropped several grand on the latest MacBook Pro, and having to step over homeless people sleeping on the sidewalk right outside without a twinge of guilt). Anyway, I'm digressing again. The City Fathers (maybe there's some Mothers now, too) have seen the plight of the Pasadena homeless and down-and-out crowd, and in the sheer goodness of their hearts are setting aside 14 parking meters whose income will be used to benefit the less fortunate. They're not clear on how exactly the money will be distributed (as the article notes, homeless people with rotten teeth don't need granola bars), and the City admits that the program is being implemented mainly to promote awareness from the more fortunate. But we're already aware of the problem, and acutely so — the homeless are now present at every freeway underpass, on many street corners, in the public libraries, outside supermarkets and fast-food restaurants, while every early Tuesday morning in my neighborhood they're out with their carts and bags, scavenging from the recycling containers. The self-operated glass and plastic recycling centers are busier than ever, taking in the pickings from the homeless and the not quite destitute, dispensing redeemable coupons and food script for the local Ralph's, Von's and Whole Foods. It is amusing to note that the Old Testament required crop harvesters to leave a remnant for the poor (Leviticus 19:9) while also requiring that people be put to death for the most trivial reasons (the recent beheadings in Iraq are a drop in the bucket compared to the Old Testament). But the notion of leaving something for the poor and hungry makes some sense; the gleanings weren't much, but the poor did have to work for it. But the piddly populations of Davidic or Solomonic Jerusalem (about 2,000 people) were nothing compared to any of America's major cities today, which together represent over 600,000 homeless people. They can't survive on gleanings, as much as our Republican conservatives would like you to believe, which is why we have welfare. They need jobs, but there simply aren't enough to go around (one long-time homeless acquaintance of mine who frequents my local public library was a $90,000-a-year IT consultant until he got laid off at age 45. Unable to find a job, he lost everything). Meanwhile we're going back to war in Iraq (because freedom™), and again my head is spinning over the billions we'll spend there, the civilians who will die, and the little good it will do. And the roughly 20% of the federal taxes I pay every year will continue to go to war and the preparation for war, making the occasional 20 bucks you and I hand out to our local homeless the greatest moral travesty I can think of.  Monty Hall, Once More — Posted Friday, September 12 2014 Many thanks to the dozen or so people who emailed me regarding my 5 September post on the Monty Hall problem. The answer still seems counterintuitive, but I finally understand it now. One person sent me this simple probability table, which I wish I had known about earlier. An interesting story behind the puzzle is told in Gary Smith's book Standard Deviations, which relates how Marilyn vos Savant, the American columnist who holds the world's record for the highest confirmed IQ (228, measured at the age of 22 years 10 months), instantly solved the puzzle and was immediately challenged by numerous noted academics and mathematicians who insisted she was dead wrong. But computer simulations proved her right, resulting in many letters of apology and much crow eating. (Not a bad-tasting bird, actually, as I've certainly eaten my fair share.)  Trespassing, Indeed — Posted Wednesday, September 10 2014 I must be getting really cranky in my old age. Back on August 14 I wrote about astrophysicist Katherine Freese's new book The Cosmic Cocktail, finding it to be a self-serving, mostly autobiographical account posing as a physics book. Then yesterday I found Amanda Gefter's Trespassing on Einstein's Lawn: A Father, a Daughter, the Meaning of Nothing, and the Beginning of Everything on my library's new book shelf. Gefter is a science writer whose New Scientist articles I read regularly. I'd always assumed she had a degree in math or physics, as she knows how to explain technical stuff pretty well (better than I can), so I checked out the book and read it last night. Gefter admits that she started out in the business of science writing by fraudulently posing as a science journalist, thus gaining access to noted physics luminaries like John Archibald Wheeler. Being young and attractive, and financially enabled by her medical doctor father to travel wherever and whenever, she was soon being invited to physics conferences, where she got the opportunity to interview the likes of Alan Guth, Leonard Susskind, Joe Polchinski, Gerard 't Hooft, Martin Rees, several other Nobel laureates, and even the Holy Grail of interviewers, Stephen Hawking (she confesses that she wanted to sit on his lap). At one point she found herself actually moderating a talk on inflationary theory with Guth and Susskind! I later determined that Gefter does not have a science degree, although she does have an MA in science history from the London School of Economics (?) But this still does not excuse the author's overall giggly, excited demeanor about being able to hobnob with the greats of science, nor her tendency (like Freese) to present group photographs of noted physicists with her own image prominently circled. Lay persons will find the book informative, but only at a very shallow level. It has no index (an old author's trick for getting the reader to read the whole thing to find a specific topic), while the glossary is comprehensive but dumbed-down (Gefter's definition of gauge theory is woefully inadequate, as are many other definitions). The book's major shortcoming is the author's tendency to write excitedly about things whose details she obviously knows little about ("I went to bed but could not sleep, because my mind was spinning over things like multidimensional brane-worlds and quantum entanglement") or just hanging out with cool guys (tooling around UC Davis with Timothy Ferris in his Porsche). Meanwhile, at 400+ pages the book is really too long, and at the end the reader will feel like he's been at a gala geek party, introduced to hundreds of people he's now forgotten. Lastly, those who read the book's back cover will find that noted Caltech physicist Sean Carroll has given Gefter a nice tribute (he's married to a non-scientist science writer, so perhaps he's better qualified to comment on the book than I am). True story: Years ago I worked with a female engineer who had a friend whose sole goal in life was to marry a brilliant scientist. I met "Bambi" (I swear to God that was her name) at a party, and found her to be vacuous but beautiful, with all the requisite physical assets needed to land a Poindexter. Last I heard, she'd married a mathematician from the Jet Propulsion Laboratory here in Pasadena (nowadays she'd be trolling in the Silicon Valley, not JPL). But, lest the reader think I'm being overly critical of women, I'll add that of all the really smart people I've met in my life, the smartest have all been women (I once tutored a 14-year-old Black-Asian girl, then in her senior year of high school, whose knowledge and understanding of math and physics was equivalent to a junior at university. I wasn't able to teach her much). Women seem wicked, when you're unwanted.* — The Doors (When You're Strange) * (And when you're not as bright as they are!)  Franklin Expedition Boat Found — Posted Wednesday, September 10 2014 Back on April 21 I posted a story about the ill-fated Franklin Expedition, which started out from England in 1845 to find the Northwest Passage from the Atlantic to the Pacific via the Arctic above Canada. Now one of the expedition's two boats — either the Erasmus or the Terror — has been found, lying in freezing water only 35 feet below the surface. Enabling the discovery was the thawing effect of global warming, which is hitting the Arctic hard, accompanied by the desire of several nations (including Canada) to claim the Arctic Ocean for themselves for oil exploration purposes. (Yeah, go figure.)  Take My Money. Please! — Posted Wednesday, September 10 2014 A guy comes up to you and says he's got a great new way for you to pay your bills immediately, and all it will cost you is$400. Would you buy it? Of course you would. It's Apple Pay, the exciting new way for Apple to get your money without your even thinking about it. After all, you're a freedom-loving American whose food preferences tend toward the Texas Thickburger (Six slices of bacon! Three slices of cheese!), so you're obviously a highly educated, discriminating consumer who demands only the best. You may recall the Tom Cruise movie Minority Report (I can't stand Tom Cruise, but he's not going away, so there you are), whose plot line is incomprehensible (but then it's a Tom Cruise movie). At one point he enters a department store, where an automated imaging system scans his retina, checks a database for his previous buying preferences, and instantly informs him of the store's great new bargains for the day. I don't know about you, but I say fuck this! You may love Apple (the Macbook Pro is tops with me), but exactly how Apple Pay represents anything but the continued commoditization of human beings is beyond me. Has instant gratification and consumerist narcissism reached the point where self-imposed feudalization is considered cool?
 Damn That Monty Hall! — Posted Friday, September 5 2014 If you've heard about the famous Monty Hall Problem then you can stop reading this right now, as you won't learn anything. On a previous post I freely admitted that I stink at general logic. And even though I know the answer, the Monty Hall Problem still gets my goat, literally. But it popped up again last night, so I challenged my family (who've never heard of the problem) to think about it, adding that if they get it right, I'll consider them to be geniuses and will forever defer to their opinions in the future. Monty Hall was the host of the old TV game show Let's Make a Deal (which I never watched). On one segment of the show, he'd bring a contestant up to three closed doors and invite them to choose a door (rather reminiscent of the old Frank Stockton short story, The Lady, or the Tiger?). Anyway, behind one of the doors was the grand prize, and behind each of the others was a goat. Consequently, the contestant had an even one-third chance of picking the door with the grand prize. Okay? But after having picked a door, Mr. Hall would then invariably open one of the doors, revealing a goat. He'd then ask the contestant if she'd like to stick to the door she picked, or select the other closed door. What would you do? Most people (like me and, as it turns out, my equally illogical family) would say that each of the two remaining closed doors now has a 50% chance of being the door with the grand prize, so you might as well stick with your first choice. But the logic behind this is wrong. This is how the logic is supposed to go: Your initial choice does indeed have a 1/3 chance of being the winner, but you also know that there are goats behind two of the doors. After Mr. Hall reveals one of the goats to you, you haven't really learned anything about the door you picked, as you already knew that two of the doors had goats. Consequently, Mr. Hall hasn't given you any useful information — your choice still has only a 1/3 chance of being the winner. However, the one remaining closed door must now have a 2/3 chance of being the winning door, so you should switch your choice to that door! This is in fact the correct logic, and has been proven by innumerable computer simulations that have tested the logic using many thousands of random Monty Hill problems. So, you should always switch your choice to the other door! I get that, but my brain always goes on to tell me that the other door's probability of being the winner is only 50%, since there are now only two doors, so the total probability (after the smirking Mr. Hall gives you that final option) is only $$\frac{1}{3} + \frac{1}{2} = \frac{5}{6}$$ and not exactly one, as I would expect. So there's a piece missing*, and it can only be resolved by assuming that the choice is 50%, since $$\frac{1}{2} + \frac{1}{2} = 1$$. So I still don't get it. Go figure. You know, getting the goat doesn't sound half bad. And yet, that awful tiger, those shrieks, that blood! — The Lady, or the Tiger? * Again, my logic is wrong. Computer simulation shows that the one-third chance of the chosen door persists, meaning that the other door's chances of winning are increased to two-thirds, making it twice as likely to be the winner. Like quantum mechanics, it doesn't lend itself to common sense, but there you are.
 Weyl and Dark Energy — Posted Thursday, September 4 2014 The cosmic microwave background (CMB), the extremely weak radiation left over from the Big Bang, is just ordinary light, and light can exhibit polarization. In March of this year researchers reported a degree of polarization that seemed to confirm the inflationary hypothesis of the very early (the first $$10^{-30}$$ second or so) universe. The "discovery" was hailed as a milestone of cosmological research, at least until later this year, when it was realized that intergalactic dust may have skewed the data. The jury is still out, but further data is due out this November. The observed expansion of the universe could be at least partly the result of this early expansion, but in 1998 it was discovered that the expansion is actually speeding up, not slowing down (as one would expect from the continuous pull of gravity). I commented on this discovery (which won the researchers the Nobel Prize) back on 11 November 2013, in which I noted the rather extreme scatter of the data. Nevertheless, the acceleration of the expansion appears to be real, and many researchers are wondering whether inflation or dark energy is the primary cause. Today the journal New Scientist carried an informative article on the subject. One of the questions asked in the article is whether dark energy is real, and if so, whether it is constant or changing with cosmological time. You may recall that Einstein's original gravitational field equations $$R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = -\frac{8\pi G}{c^4} T_{\mu\nu}$$ were modified by the great scientist to $$R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R + \Lambda g_{\mu\nu}= -\frac{8\pi G}{c^4} T_{\mu\nu}$$ where $$\Lambda$$ is the cosmological constant (Einstein later referred to $$\Lambda$$ as his greatest blunder, but he may have been right all along). While the term cannot be derived, and does not come out of the field equations as an "integration constant" or anything like that, it is a perfectly legitimate addition to the field equations because its divergence (like the equations themselves) is zero. The cosmological constant $$\Lambda$$ serves as a valid explanation for dark energy, as it can account for the observed accelerated expansion effect. It is sometimes invoked in the following manner. The effect of $$\Lambda$$ can exist only when space exists, so it is zero "outside" the expanding universe. Expansion literally creates space, which in turn creates dark energy. But matter (and possibly dark matter) dominated the early universe, so the attractive force of gravity overwhelmed dark energy's "antigravity" effect. However, as the universe expands the more or less constant effect of gravity loses out to ever-increasing levels of dark energy, so the expansion accelerates. Well, it's plausible, I guess. The only other way to view dark energy is to propose a new kind of matter or energy field that no one has ever detected. Currently, dark energy is estimated to represent about 68% of the universe's total energy, so we're sure as hell missing something. A fairly recent (2009) article on dark energy, and how Weyl conformal geometry enables it, is On Dark Energy, Weyl Geometry and Brans-Dicke-Jordan Scalar Field. It's short and readable, and very typical of the kinds of tensor-scalar field theories that have dominated the "classical" approach to relativity and the dark energy/dark matter problems.
 Piles of It — Posted Thursday, September 4 2014 A seagull, fully sated with the gorgings of a day's worth of inland urban scavengings, flies over a crowded summer time beach packed with sunbathers. He's headed for home, but first he needs to relieve himself, and one very unlucky sunbather is the recipient. Scraping the white mess off her hair, shoulders and bikini top, she moans "Why me?!" As regular Scientific American contributor Michael Shermer notes, the probability of an event can be exceedingly small, but once it happens, the probability is exactly 100%. Not only that, but given the right circumstances the event can be unavoidable: on a beach with many thousands of sunbathers, your chances of being splat upon by an insentient avian are pretty small, but the chances of someone getting it are almost certain. This in essence is the inverse of survivor's bias, which normally accompanies something that's better than being shat upon. Invariably, the person who gets the bird's load or wins the lottery is seen by others as someone either damned or damned charmed. In Shirley Jackson's famous short story The Lottery, the winner finds herself surrounded by people picking up stones off the ground, while Steve Jobs or Bill Gates get to go live in 50,000-square-foot mansions, sleeping on big piles of money with many beautiful ladies (as The Simpsons' Rainier Wolfcastle would put it). Many if not most people, their minds unable to comprehend such fantastic circumstances, tend to attribute such gargantuan luck or misfortune to some divine entity, but to a great extent it's all just a matter of chance. Many physicists today believe that we live in one universe out of perhaps an infinite number of possible universes, all of which differ to varying extent in terms of physical constants, initial conditions and other random parameters. Our universe does indeed appear to be "fine tuned" regarding its suitability for life, and for this reason many people believe God created the universe solely for our benefit. But if a suitable universe were to pop out of some random quantum vacuum fluctuation, its inhabitants would undoubtedly also come to believe it was "intended" just for them. And that's the bias Shermer is talking about. Another example: an airliner crashes and burns on takeoff, killing all aboard. All, that is, except for one grieviously injured child, who is rushed to the hospital. Amazingly, following emergency treatment, the doctors' prognosis is full recovery. "It's a miracle! Thank God!" we all say. As Shermer points out, survivor's bias always ignores the ones who didn't make it. Suppose that I am dealt five playing cards: the three of clubs, eight of clubs, eight of diamonds, queen of hearts, and ace of spades. Isn't that amazing? The chances of being dealt those cards is about one in three million and, yet, there it is! — Gary Smith, Standard Deviations
 Quo Imus? — Posted Wednesday, September 3 2014 Vectors are great for describing the motion of a particle. But now suppose you need to analyze something more complicated, where multiple magnitudes and directions are involved. Perhaps you're an engineer calculating stresses and strains in an elastic material. Or a neuroscientist tracing the changing forces on water flow near nerve cells. Or a physicist attempting to describe gravity in the cosmos. For all that, you need tensors. And they might even help you unify gravitational theory with quantum physics. — Tom Siegfried, Science News The problem of quantum gravity is arguably the greatest problem in physics today. Even before the advent of modern quantum theory, physicists have tried valiently to derive a theory that explains both gravitation and basic quantum physics under one mathematical roof — indeed, tried and failed miserably. The likes of Einstein, Weyl, Nordström, Schrödinger, Pauli, Kaluza, Klein and many others have expended prodigious amounts of time and mental energy on the problem, to no avail. In recent decades, literally thousand of theoretical physicists have plumbed the mathematical depths of string theory, loop quantum gravity, extra dimensions, parallel universes and other approaches, with little to show for their efforts. There is not a single experimental observation or piece of empirical evidence whatsoever to show that any of these ideas is on the right track. In his 2013 book Bankrupting Physics: How Today's Top Scientists are Gambling Away Their Credibility, German physicist Alexander Unzicker details how nearly all of these concepts have resulted in an almost religious faith among their brainy adherents despite the total absence of experimental data. I suspect that he is right, despite his obvious cynicism. Tom Siegfried, a science writer with Science News, reports on yet another attempt to reconcile quantum physics and gravity using a new idea championed by Roman Orus of Germany's Johannes Gutenberg University called tensor networks. This theory assumes that the quantum entanglement of many particles can be expressed by a kind of tensor formalism (this is good news, since everyone knows that tensors are easy while quantum physics is hard). The Orus paper claims to be introductory, yet at 51 pages with many complicated diagrams it is rather difficult to follow (the diagrams remind me of Roger Penrose's work, in which he describes things like the Riemann-Christoffel tensor as a diagrammatic machine with inputs and outputs). After perusing the paper, I find Siegfried's claim in the quote above to be a bit over the top, but, since tensor networks involve profound things like quantum information, density matrices and entanglement, maybe there's something to the theory after all. A consistent quantum gravity theory seems to be shaping up to be the ultimate intellectual quest of mankind, while the hope that such a theory does indeed exist borders on religious faith. I keep wondering what will happen if CERN's Large Hadron Collider, scheduled to go back online at full strength in 2015, finds nothing in the way of supersymmetric particles, extra dimensions, strings or any of the other hoped-for discoveries that scientists have anticipated using the world's largest and most powerful accelerator. Undoubtedly, if nothing is found then all will not be lost; scientists will simply go back to their blackboards to develop new explanations for the null results or, almost certainly, come up with new theories requiring even greater collider energies. Perhaps the $10 billion cost of the LHC was just a drop in the bucket. While we're hardly at the endpoint of physics, I can't help but wonder if all this isn't part of some kind of global or universal experiment (or joke) being conducted on us humans, by Fate or God or the Giant Computer Simulator in the Sky or Whoever or Whatever. The solution to the problem of quantum gravity may represent such an endpoint, but it's possible that the solution doesn't exist, which by itself is also an endpoint. What we do then might actually represent the purpose of such a universal experiment. Indeed, what then?  Logical Fallacies — Posted Monday, September 1 2014 Many years ago, on three separate occasions, I took the Graduate Record Exam (GRE) in chemistry, physics and engineering. My verbal and math scores were high, but my "logic" scores were invariably poor. Having never studied mathematical logic, I brushed it off as simply ignorance on my part, but in fact the GRE does not deal with mathematical logic; it tests general logic. So, as it turns out, I'm not a very logical person, which you've probably already deduced from my posts. However, I recently discovered a wonderful website called Fallacy Files, which has helped me deal with certain types of illogical thinking, at least as it applies to debate and argumentative discussion. The site includes a detailed taxonomy of various types of illogical thinking that is particularly enlightening. I encourage you to check this list out to see how easily it is to fall into the various logical traps we humans are prone to, in spite of our supposedly enormous intellect. I do not think I'm being illogical when I claim that Americans of a certain type (religious fundamentalists and conservative Republicans) are practically the definition of illogical thinking. So go to the site and see how many you can pick out.  You Can Learn Anything — Posted Monday, August 25 2014 A little rusty on your math and physics, or maybe your biology or your statistics or your economics or your art history or your European aristocracy or your SAT preparedness? The free tutorial website Khan Academy has been around since 2010, providing highly detailed videos of a variety of subjects, mostly for the high school and undergraduate crowd. Former hedge fund analyst turned educator Sal Khan teaches most of these subjects himself (how does he do it?!), but he has an entire tutoring staff on hand to help you out on just about any topic. Best of all, by watching the videos and testing yourself on the site's interactive tracking utility, you can learn at your own pace (I used it to brush up on my understanding of Green's theorem this morning, and I'm an old man!) And if you've got an extra 30 GB or so available on your hard drive, you can download all 5,000 videos (most are from 10 to 15 minutes long) legally and for free. How does Khan make any money off this? He doesn't — although he has a number of individual and corporate sponsors that fund the site, Khan Academy is a labor of love. And while the site has a small "donation" link, there's absolutely no advertising anywhere. Stick that in your bun, Burger King!  Go Figure — Posted Monday, August 25 2014 In 1918 Hermann Weyl came up with the idea of "purely infinitesimal geometry," the notion of continuous local transformations that leave certain quantities invariant with respect to the transformations. His theory of the combined gravitational-electromagnetic field was invariant with respect to the local change of scale $$g_{\mu\nu} \rightarrow \lambda(x) g_{\mu\nu}$$, as was the associated Weyl connection term $$\Gamma_{\mu\nu}^\alpha$$, which is symmetric with respect to interchange of the lower indices. The connection measures the amount of rotation a vector undergoes as it is moved about, and is itself a measure of spacetime curvature. In flat Minkowski spacetime there is no curvature, the vector does not rotate, and thus it remains the same everywhere. Exactly ten years after Weyl published his theory, Einstein came up with the idea of distant parallelism or teleparallelism, a rather odd theory in which all curvature vanishes, vectors remain unchanged everywhere, but the space is not flat or Riemannian. Instead, Einstein presumed nonsymmetry in the connection, and the antisymmetric part in effect takes the place of curvature. Einstein believed that he could unify gravity and electromagnetism with this new theory, and following several papers he published in 1928 the media somehow got the idea that Einstein was about to revolutionize physics yet again. On November 3, 1928 a persistent reporter from the Berlin bureau of the New York Times, one Paul Miller, showed up at Einstein's apartment and somehow made his way past Einstein's wife, Elsa, and into Einstein's study, where the physicist was so busy calculating he did not notice the reporter's presence at first. Einstein was pissed at the intrusion, but following some heated discussion managed to get rid of the guy by telling him he was "treading on the edge of a great scientific discovery, one that will startle the world far more than relativity." The next day the New York Times front page carried the article Einstein on Verge of Great Discovery; Resents Intrusion. World famous since the 1919 experimental confirmation of Einsteinian light deflection by the Sun, Einstein was again big news. Selfridge's, a fashionable department store in London, pasted the pages of one of Einstein's Fernparallelismus* (distant parallelism) papers on its windows, attracting hordes of sidewalk gawkers. The streets around Einstein's apartment in Berlin were flooded with people and news hounds, all seeking a glimpse of the great scientist who was about to astound the world with a fantastic new theory of the universe. But the theory went nowhere quickly; it turned out to be just another one of Einstein's crackpot theories. And it was not even new: the French mathematician Élie Cartan had already independently discovered Einstein's "new" geometry years before. But there was a theory published in 1928 that did indeed change the world, although the world took little notice. This was Dirac's theory of the relativistic electron, which successfully joined quantum theory with special relativity for the first time. The theory also ushered in the age of modern quantum physics, including quantum field theory, which today stands as the most successful theory mankind has ever devised. I find it amusing to note that in the Old Testament, 1 Kings 7:23 gives the value of $$\pi$$ as 3 while, thanks to Dirac, modern quantum theory routinely agrees with experiment to 10 or 12 decimal places. Yet more people today know of King Solomon and Hiram of Tyre than they do of Paul Adrien Maurice Dirac (pictured above), the British physicist who received the 1933 Nobel Prize for his work. Go figure. * English translations of Einstein's papers on distant parallelism are hard to find, but German physicist Alexander Unzicker has posted these papers on his website in both English and German for those of you who are interested (and who presumably have no life, like me). Unzicker's 2013 book Bankrupting Physics: How Today's Top Scientists are Gambling Away Their Credibility was panned by several notable physicists, but it's actually a pretty good book.  Thinking and Climate — Posted Monday, August 18 2014 A psychologist could barely dream up a better scenario for paralysis. — Daniel Gilbert, Harvard University I've never heard of George Marshall, the author of the upcoming book Don't Even Think About It: Why Our Brains Are Wired to Ignore Climate Change, but from the review over at New Scientist I'm inclined to believe Marshall's spot-on as to why climate change is not being taken seriously. (Google Books has a preview of the book's first five chapters here.) The review includes some insights by 2002 Nobel economist Daniel Kahneman, whose own bestselling 2011 book Thinking, Fast and Slow brilliantly explains how we think about issues that require little consideration (2 plus 2) versus how more complex issues are handled in the brain. I read the book when it came out, and I know I posted something on it here, but all I can remember is that things like deliberation, complex thought, nuance and consideration are avoided by some people, who prefer the shoot-now-and-ask-questions-later approach, which offers them immediacy, certainty and concreteness of action (I'm also certain that I must have tied such people to the Republican Party and to fundamentalist religious belief systems, but I'll forgo that discussion for now). Kahneman's book also explains why we're not addressing the climate change problem. I believe climate change is definitely occurring, that the bulk of it is due to human activity, and that things are going to get really bad. That's the conclusion of 97% of the world's climatic and atmospheric scientists, who've now studied the problem in detail for three decades, although it's only been seriously addressed for the past ten or fifteen years. I recall attending a conference on climate change in San Diego in 1989, and thinking that those early warnings were hardly worth worrying about. I know better now. I've lived in southern California most of my life, and I've never seen it so dry. The past three years have seen us receive roughly one-third the rainfall we normally get, but the central part of the state is far worse off. The bulk of the nation's orchard trees (fruits and nuts, mostly) are located there, and the water needed to maintain them is just not there. Surface water supplies have dried up or are being brutally rationed, while groundwater levels are dropping faster than new wells can get to them — today's Los Angeles Times and others report that in some areas groundwater levels have dropped a hundred feet in a matter of weeks. Pasadena has implemented fine-based restrictions on wasteage and overwatering ($500), while 90% of the entire state is experiencing between severe and "exceptional" drought conditions with similar or harsher restrictions (thanks to a shut-off valve I installed, I limit my showers to five gallons). A fourth dry year is expected to bring disaster. [Update: LA Times] So what to do? "Adapt" seems to be the official policy, because it's probably too late to do anything else, at least in the near term. But not to worry: if things do get worse, the Republican Party will blame it on a devil-worshiping President Obama, or on God punishing the nation for its lackadaisical attitudes toward abortion, gay marriage and women's rights (the party and its acolytes also strongly recommend this). Yes, once the stonings, hangings, burnings, prayers and self-flagellations get going, the problem will no doubt go away.
 Gravity, Entropy and Information — Posted Monday, August 18 2014 Information theorist Claude Shannon (1916-2001) I spent a good part of this past weekend watching the 12-hour video series Black Holes, Tides and Curved Spacetime: Understanding Gravity, by Kenyon College physics professor Benjamin Schumacher. It's pretty standard, high school-level stuff, but there's a few gems. In the last lecture, Schumacher addresses quantum gravity, and briefly mentions the present status of efforts to solve the problem, arguably the greatest problem of physics today. In addition to string theory, M-theory and loop quantum gravity, Schumacher, who's also an expert on quantum information theory (he invented the term qubit), suggests that some combination of gravity and entropy might provide a solution to the problem. Alas, he doesn't elaborate. I've always been fascinated by the subject of teleology, which deals with the ultimate purpose of things and the universe's apparent drive toward a purposeful finality. The key question for me is the driving force behind everything, which totally escapes me. In physics, chemistry and thermodynamics, the driving force tends to be the separation of a system from the equilibrium state; in Newton's law of cooling, for example, a hot object cools at a rate proportional to the difference between the object's temperature and its surroundings. This is really nothing but the inviolate tendency of entropy to increase with time — entropy remains stationary only when equilibrium is achieved. But this only begs the question of why Nature wants to maximize entropy. To what end? In his lectures Schumacher addresses the oft-asked issue of how gravity seems to violate the entropic principle $$dS/dt \ge 0$$. A quantity of some gas introduced into a room will expand and mix with its surroundings, thus producing a more disordered system, but in space gravity acts instead to compress the gas into a seemingly more ordered arrangement; the ultimate compressed object, a black hole, would seem to represent the most ordered system possible. But Schumacher, drawing on the work of physicists like Stephen Hawking and Jacob Bekenstein, points out the fact that black holes actually are systems of maximal entropy. Consequently, gravity produces orderly systems like stars, galaxies and black holes, but it does not violate the entropic principle. In his groundbreaking 1948 paper on information theory, Bell Labs mathematician and theorist Claude Shannon (a student of Hermann Weyl) established the definitive link between entropy and information. Shannon showed that entropy is proportional to the sum of probabilities $$- \Sigma p_i \log p_i$$ associated with the observation of some physical system, a notion that was carried over to quantum mechanics, where entropy is related to the quantity $$Tr (\rho \log \rho)$$, where $$\rho$$ is the density matrix of the system (density matrices are indispensible in the understanding of quantum entanglement). This leads to the interesting notion that the universe is attempting to maximize the amount of information contained in it. But again, if information is just entropy, what's the use of doing that? In his work, the noted British mathematical physicist Roger Penrose has considered what the ultimate state of the universe is almost certain to be: a sparse gas of random, high-entropy photons produced by the leftover detritus of burned-out stars and evaporated black holes. In this state, the universe will have presumably achieved maximum entropy. If that is indeed the ultimate fate of the universe, we might rightly ask: What good is that?! Penrose has ruminated on the possibility that, at this ultimate stage in the evolution of the universe, the universe simply "forgets" everything, resets itself, and is then reborn in a new Big Bang. If true, that would be many billions, if not trillions, of years from now (assuming time itself still exists in some form). The purpose of life is to survive so that it can produce new life. According to Penrose, the universe, faced with the ultimate heat death scenario just described, would also seem to be striving for a future in which life of some form is possible forever through an endless sequence of heat deaths and Big Bangs. But again, what is the ultimate purpose of that? Perhaps, as nihilists would say, there is no purpose to the universe. But it sure as hell seems to be striving for something. An interesting take on the question of information, entropy and the meaning of it all can be found here.
 GA Again — Posted Monday, August 18 2014 On July 24 I spoke about geometric algebra, spacetime algebra in particular, and how it was rediscovered by Arizona State University's David Hestenes. As I noted then, the concept actually goes back to the mid-19th century, with Hermann Grassmann and William Clifford, including some notable contributions from William Hamilton, but then got waylaid for over half a century by the standard vector formalism of J. Willard Gibbs. Hestenes and many others have tried valiantly to reintroduce the GA formalism into the standard curricula of mathematics and physics from high school through university, with some (but not enough) success. I've been playing with GA for awhile now, and it has convinced me that, whatever approach extraterrestrials are using regarding the mathematics of geometry and physics, it certainly involves GA or some variation of it. It is powerful, intuitive and simple, and it lends itself to all branches of higher physics theories, including general relativity, quantum mechanics and quantum field theory, as well as more mundane stuff like video game animation. Many years ago in my physics thesis I took the square root of the line element $$ds^2 = g_{\mu\nu} dx^\mu dx^\nu$$ and got $$ds = dx^\mu \gamma_\mu$$, where I assumed the $$\gamma_\mu$$ were Dirac's gamma matrices. I was bothered by the fact that these four quantities look and behave like vectors (they're just matrices), and I had to dance all over the place trying to fit them into a variant of Hermann Weyl's 1918 theory. Well, in GA they really are vectors, and my life at the time would have been much easier had I only known about Hestenes' work. Wanting to see Hestenes' "Ur-work" on the subject, I recently borrowed his 1966 book Space-Time Algebra from UC Berkeley. Compared with his other books, it's very slim (only about 90 pages), but what a revelation of clarity and power! Already he had translated Maxwell's equations into GA (all four equations are contained in $$\Box F = J$$, where $$\Box$$ is the d'Alembertian operator $$\gamma^\mu \partial_\mu$$), along with Dirac's relativistic electron equation and Einstein's gravitational field equations, and expressed the Lorentz transformation and the two- and four-dimensional spinor notations into GA formalism. My God, Hestenes' bivectors can do just about anything, and the underlying formalism is so simple a high school kid can pick it up. And, perhaps the most amazing thing of all, GA is all completely real — there are no imaginary or complex numbers in the formalism at all. Why is this not being taught?
 Some Conformal Stuff — Posted Saturday, August 16 2014 Hermann Weyl's 1918 gauge theory was a conformal theory, meaning that the physics is invariant with respect to a local rescaling of the metric tensor $$g_{\mu\nu} \rightarrow f(x) g_{\mu\nu}$$, where $$f(x)$$ is an arbitrary function. Conformally-invariant theories are all the rage now, from the microscales of quantum mechanics to the cosmological scales of the universe. German physicist Hans Kastrup (professor emeritus, RWTH Aachen University) has published extensively on conformal physics, with many of his papers available on his website. His interest in the topic goes back to his doctoral thesis of 1962, which is also posted on his website. I'm mentioning him here only because in 2008 he wrote a particularly clear paper on the historical development of conformal theories that includes a nice overview of the friendly "spat" between Einstein and Weyl on the physical reality of Weyl's 1918 theory. It goes on to discuss more modern developments, such as the theory of "AdS/CFT correspondence," a now rather famous 1997 theory by Juan Maldecena which posits that our (supposedly conformal) 4-dimensional universe is really the compactified boundary of a 5-dimensional universe. (You will forgive me here for trying to appear as if I really understand the theory. I've read Maldecena's paper numerous times, but I still can't follow most of it.) Fortunately, Kastrup's paper is accessible to the interested undergraduate. It's 82 pages, but there's some 30 pages of references, so it's not so bad.
 Amazing — Posted Friday, August 15 2014 Harvard post-doc Mike Rubenstein (and fellow University of Southern California PhD) has built an army of 1,024 tiny motorized robots and programmed them to self-assemble into predetermined 2D arrangements. As discussed in today's New Scientist article, the potential applications in structural engineering, medicine and computer simulation are endless: By giving the robots any number of instructions, you could have them play the game of Life from a random starting arrangement, determine how genetic molecules could correct for random coding errors, use them to solve the traveling salesman problem, or have them design and construct a bridge or other structure using a minimal amount of starting material (themselves!) Rubenstein says one of the next steps is to use his roving robots in real-life 3D applications (zero-g space construction would be one obvious application), while miniaturized, nanoscale bots could be used to clear plaque from congested arterial walls and other tissues.
 Does the CMB Hold Fossil Evidence of a 5D Universe? — Posted Friday, August 1 2014 Superstring theory requires ten spacetime dimensions, while M-theory posits the existence of eleven dimensions. No wonder there is still so much interest in the far more acceptable five dimensions of Kaluza-Klein theory! But 5D spacetime has other appealing characteristics which are still being explored. The spacetime known as anti-de Sitter space is currently of much interest in cosmology, for the simple reason that our ordinary 4D universe appears as a bounded hyperplane or "slice" in AdS space. This effectively "captures" the gravitational force (which would otherwise be of infinite extent), making the goal of quantum gravity and its unwieldy infinities perhaps a little more achievable. A 5D AdS space was also used to notable effect in 1999 by Harvard's Lisa Randall (then at Princeton) and Boston University's Raman Sundrum, who proposed that the unseen fifth dimension might not be as small as once thought (their 1999 article is still one of the most cited papers in physics). The cover article of this month's Scientific American also involves 5D spacetime in an interesting way. A massive 3D star can collapse to a 3D black hole, but the event horizon surrounding the hole is an ordinary 2D spherical surface. Susskind's theory of the holographic universe assumes that all the information that falls into a black hole is not lost but maintained on the hole's event horizon. The authors of the Scientific American article take this reasoning all the way back to the Big Bang. Since a black hole's singularity is cloaked by a 2D event horizon, the presumed singularity of the Big Bang might also be hidden behind a 3D surface, which the authors argue is the universe itself. The Big Bang itself would therefore have been born with four spacial dimensions which, coupled with time, becomes the 5D early universe of the article. Crazy?The idea may sound crazy, but there are several ways one might be able to test it. One way is by studying the cosmic microwave background [CMB] radiation. Outside of our three-brane [3D universe], we would expect there to be some extra four-dimensional bulk matter — something pulled close by the gravitational pull of the black hole. We can show that thermal fluctuations in this extra matter will create fluctuations on the three-brane that in turn distort the CMB by small but potentially measurable amounts. Plato's cave figures prominently in the article, which I encourage you to read.
 "Are They Your Men?" — Posted Friday, August 1 2014 My aging brain often coughs up the dim memory of some story I read long ago, but nevertheless retained because it moved me in some way. Unfortunately, I usually can't remember who wrote it, or when. This morning I remembered an old Jack London story I read in high school, and I spent the better part of an hour trying to find it on the Internet. All I could remember was the story's very last word. Not much to go on, but I finally found it, only to also find that London didn't write the story at all — it was Saki (H.H. Munro), the famous British author who died young from a sniper's bullet in World War I. Tragically ironic, the story may remind you of an ancient conflict now playing itself out (once again) in the Middle East. It's also very short, and I hope you enjoy it as much as I did.
 … But Pigeons Are Not Electrons — Posted Thursday, July 31 2014 If you have three pigeons and two boxes in which to hold them, logic says that at least two of the pigeons will end up in the same box. But quantum mechanics says otherwise. Yakir Aharonov is a famous Israeli physicist who, with the late and also great physicist David Bohm, proposed the Aharonov-Bohm effect in 1959. This month Aharonov, now with Chapman University in California (where my daughter graduated law school, yay), published an interesting paper called The Quantum Pigeonhole Principle and the Nature of Quantum Correlations. Aharonov and his colleagues use a neat twist on the Mach-Zehnder interferometer experiment (see my post of June 21, 2011 for the math) to show that three electrons can indeed exhibit the "pigeonhole principle." It's a short, fun paper that isn't hard to follow. As a kid I used to raise pigeons, but I never saw the principle at work. Perhaps I was just not paying attention.
 Solitary, Poor, Nasty, Brutish and Short (or Infinitesimal) — Posted Wednesday, July 30 2014 If men are left to their own devices, according to Hobbes, fear of one's neighbor leads to war, war leads to more fear, which in turn leads to more war. Under such conditions there is no point in investing in the future, and life is a misery. — Amir Alexander One of Zeno's famous Paradoxes involved a race between Hercules and a tortoise. Being obviously slower, the tortoise is given a head start, and at the shout of "Go!" (or whatever they yelled in those times) both the tortoise and Hercules race for the finish line. Zeno reasoned that it would take some finite time for Hercules to reach the half-way point of wherever the turtle might be during the race, then one-fourth, then one-eighth, then one-sixteenth, etc. Assuming that this halving of the distance could go on forever, Zeno concluded that Hercules could never reach the tortoise, much less win the race, since the process of taking one-half the distance could go on for an infinite number of steps. But Hercules won the race (and later enjoyed turtle soup for dinner). But how did he do it? Did Zeno not prove mathematically that a Herculean win was impossible? The basic idea of elementary differential calculus is that one can take smaller and smaller increments of distance, time or whatever while preserving the idea of a finite result when all is said and done. After all, the finite result of Zeno's paradox is that Hercules won the race! But the notion of letting things get smaller and smaller without end was a real problem for some people a long time ago, and their refusal to believe such a thing resembles something akin to what we see in this country today. Let's take the slope of some curve $$f(x)$$, which is defined as $$f^\prime (x) = \lim_{dx \to \, 0} \frac{f(x+dx) - f(x)}{dx}$$ Four hundred years ago the focus would have been solely on the denominator: "Division by zero is impossible!" But the numerator also goes to zero in the limit, making it possible to have a finite result. Thus, the slope of the parabola $$f(x) = x^2$$ at any point $$x$$ is just $$f^\prime(x) = 2x$$. But four hundred years ago, the Church might have had you burned at the stake for advocating such heresy. In his new book Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, mathematician Amir Alexander describes how the battle over infinitesimal quantities such as $$dx$$ raged in the times of Galileo, Louis XIV and Thomas Hobbes. In 1632, Rome's influential Society of Jesus, an order of supposedly learned men espousing many of the beliefs held by the Jesuits and other religious and doctrinal groups of the day, declared the idea of infinitesimals "repugnant", "improbable" and therefore "forbidden in our society." Their reasoning? The universe was ruled by divine order of God, and the notion of infinitesimal quantities such as microscopic distances, areas, volumes and "atoms" violated God's order. I can only imagine how many scientists would have been burned at the stake if by some miracle quantum mechanics had been discovered in those days. As Kelly Bundy would say, "The mind wobbles." Meanwhile, the concept of "infinitesimal" is taking on a new meaning in Williamstown, Kentucky where, in conjunction with the Answers in Genesis folks and their magical Creation Museum, are currently constructing a full-scale "replica"of Noah's ark. In all fairness, the builders are not using "gopher wood" as God demanded of Noah (the wood does not and never did exist), but the dimensions of the ark will comply with biblical specifications: 300 cubits in length (450 ft), 50 cubits in width (75 ft), and 30 cubits high (45 ft). If one neglects interior volume taken up by bulkheads, flooring and miscellaneous storage areas, that gives the ark something like 1.5 million cubic feet of volume, into which Noah and his small family of eight had to cram up to 9 million species of terrestrial creatures, either two at a time ("unclean" animals) or seven ("clean" animals). And, don't forget, this had to include some pretty large critters: since the Answers in Genesis folks claim that the Earth and universe are only 6,000 years old, herds of dinosaurs and huge herbivorous megafauna like Paraceratherium would have also been housed on the ark, not to mention fresh-water crocodilians and all manner of fish and birds. In the 2014 movie Noah these animals were all put to sleep with a kind of aromatic sedative to make them more docile, but it sadly did nothing about the overcrowding problem. Many apologetic Jews and Christians believe God had a magical "shrinking ray" that squeezed the animals down to shoe-box or smaller size, thus explaining the storage issue. But most believers are willing to just "leave it to the heretics to figure it all out." A lot of people haven't progressed much over the past 400 years. Isn't it wonderful to be stupid?
 The Old Dog, New Trick Conundrum — Posted Thursday, July 24 2014 Geometry is the gate of science, and the gate is so low and small that one can only enter it as a child. — William K. Clifford Much has been said about Maxwell's equations since the brilliant Scottish mathematical physicist Jame Clerk Maxwell first wrote them down in 1861. What is not said is that Maxwell's original vector notation was nearly incomprehensible, and it wasn't until Oliver Heaviside and Josiah Williard Gibbs came along in the latter part of the 19th century to put the equations in the vector form we all know and love (or hate) today, along with their divergences, gradients, curls and other vector whatnots. The equations can also be cast in beautiful covariant form, which I presented in my post dated March 6. However, there is a sad twist to the whole vector notation story, which goes all the way back to the German mathematician Hermann Grassmann (1809-1877) who in 1844 discovered (or invented) an algebra involving mathematical quantities that anticommute under multiplication ($$ab = - ba$$). His work was largely ignored, but 100 years later anticommuting Grassmann numbers became the basis of fermionic quantum field theory. About the same time (1840s), the British mathematician William Rowan Hamilton sought to extend the idea of a complex number to three dimensions. He came up with the quaternions, which brilliantly anticipated the Pauli and Dirac matrix formalism of quantum mechanics in the 1920s. Hamilton's quaternions were also pretty much ignored. Then came the British mathematician William Kingdon Clifford (1845-1879), whose Clifford algebra resurrected many of the ideas of Grassmann and Hamilton. It has been said that if Clifford had known about quantum mechanics, he might have beat out Paul Dirac in his 1928 discovery of the relativistic electron equation (which, IMHO, is the most profound achievement of the human mind). Indeed, the Dirac or gamma matrices $$\gamma^\mu$$ of quantum field theory encapsulate much of Clifford's algebra. So what has all this dry math history to do with vector notation? As authors Venzio de Sabbata and Bidyut Kumar Datta describe in their slim and very readable 2007 book Geometric Algebra and Applications to Physics, what the likes of Grassmann, Hamilton and Clifford discovered was geometric algebra, which describes geometry and physics in terms of a purely coordinate-free, geometrical approach. It is ironic indeed that Gibbs' vector notation caught on, because not only is it vastly inferior to geometric algebra, it nevertheless became the standard notation for generations of students and scientists since he first developed it. (It's also ironic that most of Gibbs' scientific papers were famously grammatically obtuse. Outside of Gibbs' free energy, how many people have ever heard of the guy?) The sad twist to the story, at least for me*, is that trying to learn geometric algebra (in spite of its intuitive, powerful and very natural notation), requires simultaneously unlearning the standard vector notation that we've all used for the past 120 years. Consider the fact that the cross product of two vectors $$\mathbf{a} \times \mathbf{b}$$ or the curl of a vector $$\nabla \times \mathbf{v}$$ in the standard notation only makes sense in three dimensions. In GA (written as $$a \wedge b$$ and $$\nabla \wedge v$$ ), they make sense in any dimension. Not only that, but all vector quantities in GA are tied fundamentally to the geometry of points, lines, areas, volumes and higher-dimensional objects. Best of all, the notation is coordinate-free (and nearly index-free). The real guts of GA lies in the simple vector product $$ab$$. In two dimensions, a vector $$a$$ can be expanded in terms of its components $$a_i$$ and the orthogonal direction base vectors $$e_i$$, or $$a = a_1 e_1 + a_2 e_2$$. The base vectors as expected obey $$e_1 e_1 = e_2 e_2 = 1$$ and $$e_1 e_2 = - e_2 e_1$$, so that $$ab$$ can be written simply as $$ab = a \cdot b + (a_1 b_2 - a_2 b_1) e_1 e_2$$ or $$ab = a \cdot b + a \wedge b$$ Note that in GA, a vector product is the sum of a scalar ($$a \cdot b$$) and a bivector ($$a \wedge b$$), an unheard-of concept in ordinary vector algebra. Note also that the square of $$e_1 e_2$$ is $$(e_1 e_2) (e_1 e_2) = e_1 e_2 e_1 e_2 = -1$$ so that $$e_1 e_2$$ behaves as the imaginary number $$i$$, a characteristic that persists in any dimension $$n$$ ($$e_1 e_2 e_3 .. e_n$$). We can then write $$ab = |a| |b| \cos \theta + i |a| |b| \sin \theta$$ or $$ab = |a| |b| e^{i \theta}$$ If $$a,b$$ are unit vectors, the product $$ab$$ can be used as a vector rotation operator, a property of GA that immediately lends itself to countless applications in physics, including quantum theory. For almost fifty years, the leading proponent of GA has been Arizona State University's David Hestenes, who has heroically championed the overturn of standard vector analysis and calculus in favor of the GA approach. Today, many schools (from high school to Cambridge University) routinely teach GA to students. But will it take another 100 years for it to finally catch on? I found that the Doran and Lasenby book on GA is one of the best to learn the subject. You can also download a sample of the book here. It's worth looking into; give it a try. * I read several books on GA two years ago, and was surprised at how difficult it was for me to unlearn the notation I've used for many years. But I finally got it figured out. Then yesterday, turning again to GA to understand some of its applications to computer animation, I realized I had forgotten it all. I'm chalking it up to age, but you don't have to!
 Let's Get Small — Posted Friday, July 18 2014 It has been known since the 1930s that ordinary matter can be compressed to fantastic densities in stars, with the compressing force being provided either by impact or gravity. The primary forces opposing compression in a star's interior are generally the ordinary hydrostatic and electrostatic forces that accompany heat and outgoing photon pressure. But for very massive stars gravity can overcome these forces and squeeze matter until electron degeneracy (that it, the resistance associated with electrons being pushed into disallowed shared quantum states, thus violating the Pauli exclusion principle) takes over and halts continued contraction of the star; a white dwarf star is a common and well-studied example. However, if the star's mass is truly enormous gravity can overcome the degeneracy pressure and squeeze the matter down to where electrons and protons join, forming neutrons. This produces an even stronger kind of degeneracy pressure that characterizes a neutron star. Modern thermodynamic equations of state accurately describe the densities of even these stars, where matter is compressed to the order of $$10^{17}$$ kilograms per cubic meter. Ouch. But if a star has sufficient mass, gravitational collapse can continue unabated until the physical star itself literally "winks out" of existence, producing a black hole. In classical general relativity there is no known force or process that can prevent the total collapse of matter to a mathematical point of zero volume and infinite matter density. An important assumption in this classical point of view is that spacetime is smooth and continuous all the way down to zero volume. This week's issue of Nature reports on a theory in which spacetime is considered to be granular or foamy at distances near the Planck length ($$10^{-35}$$ meter); that is, the theory posits that spacetime is quantized at some sufficiently tiny scale, so that spacetime volumes below this scale do not exist. The underlying theory, known as loop quantum gravity, has actually been around for about as long as modern string theory. But the Nature article raises the possibility that gravity cannot "scrunch" matter down to zero volume because such a volume simply does not exist. A collapsing star would resemble a classical black hole only until the central portion reaches the fundamental "Planckish" volume limit, with the result that the matter rebounds off this volume. The black hole then explodes, producing a white hole. Interestingly, the arXiv paper on which the article is based reports that the time frame of this rebounding process could be many billions of years, making it unlikely that even a single event could ever be witnessed. Carlo Rovelli, one of the paper's authors, is a leading theorist in quantum gravity. He is publishing a new book called Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory (which appeals to me mainly because it has the word "elementary" in its title!) The book is scheduled to come out in November, but if I can get my hands on a preview copy I'll let you know if it's worth reading. Update: I found what appears to be a legally-downloadable preview draft of Rovelli's book here at the Aix Marseille Université-Centre de Physique Théorique website (it's only 277 pages, but at 9 MB it may take a while to download). The book has numerous spelling and grammatical errors, none of which are critical to the subject matter (hopefully all the math is correct). I've only skimmed through it so far, but it does look to be fairly readable. Have a nice weekend.
 "Something Funny is Happening to the Sunset" — Posted Thursday, July 17 2014 Lemmings leaping to their deaths by the thousands. Whales beaching themselves in droves. Worldwide colony collapse disorder in bees. I don't believe in intentional animal suicide; it's not their nature. When the food runs out, or when there are few females to mate with, sure, somebody's gonna get hurt, maybe even die. But suicide? No. It takes higher brain functioning to even contemplate that. It takes the ability to actually think. "Momma, when I am prezdint I want a nucular war this big!" Okay, yes, it's a cheap shot, posting Sarah Palin's brain-damaged child Trig here, but I'm trying to make a point. It not only takes the ability to think to commit suicide, but you also have to be insane. And that seems to describe the world today. Israel initiated a ground campaign in Gaza this morning, a passenger jet carrying 295 people was shot down over Ukraine, Australia has abandoned its global warming mitigation efforts, extreme drought in America is expanding, the entire legislative body of the United States is dysfunctional, the Supreme Court thinks corporations are people, America and Russia are at each other's throats again, European countries are going hard right wing, and the Museum of Creation in Kentucky is building a full-sized replica of Noah's Ark to proselytize the notion that the Earth is only 6,000 years old and that humans lived with dinosaurs. If this isn't insanity, I don't know what is. I recall my older son Kris suggesting to me years ago the idea that while homosexuality has always existed, its apparent expansion in recent years may be Nature's way of trying to cope with 7 billion human beings on a planet designed for far fewer. In hindsight today, I think he was on to something. But it's also possible that it was no accident that Nature allowed a single species to develop the ability to annihilate itself. (BTW, my son and his wife will make me a grandfather for the first time next month, making me wonder what kind of world the kid will be coming into.) Many years ago I visited the birthplace and home of science fiction writer Robert A. Heinlein in Butler, Missouri, one of those tiny, nondescript, fundamentalist Christian burgs in western Missouri ("The Electric City," it calls itself) along the eastern edge of Tornado Alley. Heinlein's 1952 short story The Year of the Jackpot sticks out in my mind as a particularly prescient tale of what we seem to be witnessing today, although in that story Nature was a bit more direct in Her dealings with us. Trig Palin for President in 2040! Sorry, Trig, but I think someone will get to the planet long before you do.
 Amen — Posted Thursday, July 17 2014 Dear Devoted Reader: Is your corporation or 501(c)(3) washed in the Blood of the Lamb? How can you know for sure that your business is going to Heaven? Just send $19.95 cash, check or money order to William O. Straub care of this website to learn how you can spend eternity with your beloved corporation or for-profit tax exempt charity! Hurry, and God bless you! Update: Hey folks, I'm being sarcastic here! Enough with the stupid emails!  Einstein and Unification — Posted Friday, July 11 2014 Jeroen van Dongen is an assistant professor at Utrecht University in the Netherlands and an editor at the Einstein Papers Project here at Caltech in Pasadena. I'm currently reading his 2010 book Einstein's Unification, which chronicles Einstein's efforts to find a common geometrical basis for gravitation and electromagnetism, which was essentially his life's work from 1925 until 1955 when, as biographer Abraham Pais eloquently put it, he laid down his pen and died. The book has a rather lengthy section on Einstein's involvement with Kaluza-Klein theory, which he worked off and on with as late as 1938, along with many other of his own theories, including teleparallelism, semivectors, spinors and his final work on non-symmetric unified field theory. The book includes numerous references to Hermann Weyl and his related work on unification, Einstein's attempts to reconcile his distain for modern quantum theory (inlcuding quantum entanglement), and the collaborations that Einstein enjoyed (and sometimes merely tolerated) with the likes of Weyl, Schrödinger, Heisenberg, Planck and Bohr, along with Wolfgang Pauli, who often served as devil's advocate for many of Einstein's more hairbrained ideas. Readers of Einstein have no doubt noted that many young physicists of the time (perhaps anxious to further their careers or just move on to more productive fields of research) seemed to exhibit a true disliking of Einstein, whom many viewed as a relic of an earlier, classical age despite Einstein's truly momentous contributions to quantum theory and relativity. And while von Dongen doesn't address the matter, I for one am puzzled by the fact that, during their twenty years together at Princeton's Institute for Advanced Study, Einstein and Weyl never collaborated on anything. At nearly$100 on Amazon, van Dongen's slim (213 pages) book is probably not worth purchasing, but it's definitely worth reading (my copy was a loaner from the California Lutheran University, of all places), as it sheds a little more light on why the otherwise brilliant Einstein was so reluctant to give up the unification idea, while others (notably Weyl) saw the writing on the wall and moved on to more productive endeavors.
 Kaluza-Klein at 100 — Posted Thursday, June 26 2014 Finnish relativist Gunnar Nordström (1881-1923). In 1919 the German physicist Theodor Kaluza proposed a novel generalization of Einstein's 1915 gravity theory that posited the existence of a fifth dimension, in addition to time and the usual 3-space we all know and love. More importantly, Kaluza believed that all of Maxwell's electrodynamics lived in this fifth dimension. He communicated a paper he wrote on the idea to Einstein, who was enthralled. But when Kaluza asked Einstein to recommend the paper for publication, for some reason Einstein sat on it for two years. It didn't see the light of day until 1921, but it caused quite a stir. In 1926 the Swedish mathematician Oskar Klein made important improvements to Kaluza's theory, which then became known as Kaluza-Klein theory (in my opinion, it should have been called Klein-Kaluza theory). The beauty of KK theory is that it automatically undergoes dimensional reduction to give both Einstein's gravitational field equations and Maxwell's equations. All that by simply adding another space dimension! Unfortunately, that dimension is unimaginably small, on the order of the Planck length, which is why you don't back over it when you leave for work in the morning. The basic idea is extended by superstring theory, which posits no fewer than 10 space dimensions, of which 6 are, like the KK fifth dimension, unimaginably tiny. But the idea of a fifth dimension seems to have originated with the Finnish physicist Gunnar Nordström in 1914 (I knew a German kid named Gunnar in high school). But, according to the Wikipedia article, the idea was either ignored or forgotten. However, Nordström was not forgotten — he went on to discover (with Hans Reissner) the metric for an electrically charged black hole. But that's another story. In his wonderful book Einstein Gravity in a Nutshell, Physicist Anthony Zee beautifully expresses his hope that a modern version of Kaluza-Klein theory will someday answer all our physics problems. I hope he gets his wish.
 More History of Unified Field Theories — Posted Tuesday, June 24 2014 The dream of unifying all fundamental interactions in a single theory by one common represention still captures the mind of many a theoretical physicist. — Hubert Goenner Hubert F.M. Goenner, Professor of Physics Emeritus at Germany's Göttingen University, has written extensively on general relativity and on early attempts to unify it with electromagnetism. Years ago I posted a link to a 2004 article he wrote (it's more like an online book) entitled On the History of Early Unified Field Theories, a concise and very readable introduction into the work of Hermann Weyl, Theodor Kaluza, Oskar Klein, Arthur Eddington, Oswald Veblen and others, who all tried their hand at generalizing Einstein's 1915 gravity theory in the hope that other forces of Nature were somehow also describable in terms of pure geometry. (Although a German, Goenner's English is superb, far better than mine, and he thankfully provides translations for many of the original German scientists' quotes into English.) Yesterday Goenner posted a much longer article on the same subject, this time covering the period 1930 to 1965. I've only just started reading it, but it includes an extensive discussion of Einstein's last unification efforts and attempts made by both physicists and mathematicians to connect quantum theory with general relativity. Fascinating stuff, and the best part is that the interested physics or mathematics undergraduate should have no trouble following most of the material.
 Free Relativity Book — Posted Saturday, May 31 2014 Not much is known about science author Kevin S. Brown. He doesn't show up on the Internet, and no one I've talked to has any idea who he is. But he has a fantastic (and at over 700 pages, very comprehensive) book that you can read online for free over at Mathpages. It's an enormous collection of articles that Brown has written and also collected into book form called Reflections on Relativity. While the online book is free, the math typsetting — though clean — is a tad awkward. I'd recommend buying the much better version at Amazon, but at $40 I'll stick with the online version. Furthermore, while Brown makes numerous references to Hermann Weyl and his theories, the online book is not indexed, making it somewhat difficult to find specific topics. The book's many quotes and stories remind me of Anthony Zee's Einstein Gravity in a Nutshell, which is probably the best book on general relativity ever written. However, Brown has a number of calculations you won't find in Zee's text, and it's just as much fun to read. Enjoy.  Einstein and Eddington — Posted Thursday, May 29 2014 Do not Bodies act upon Light at a distance, and by their action bend its Rays, and is not this action strongest at the least distance? — Isaac Newton I just finished watching Einstein and Eddington, a 2008 film dramatizing the mutual personal, political, religious and scientific struggles that Germany's Einstein and England's Arthur Stanley Eddington endured while the two countries fought one another in World War I. Einstein was desperately trying to finalize his general theory of relativity, which held little interest for the proponents of the German war machine, while the devoutly religious Eddington, a Quaker, was troubled over the potential religious aspects of the theory's possible overturning of Newton's law of gravitation and its perfect "clockwork" description of God's universe. The film is decent as a dramatization, but its historical accuracy leaves much to be desired. For one, Einstein wasn't quite the doting father or the brash, wisecracking know-it-all he's portrayed in the movie, while the film's treatment of Newton's law of gravitation would lead one to believe that starlight is gravitationally bent only in Einstein's theory. (In truth, Newtonian physics predicts one-half the relativistic deflection; by a neat coincidence, the Newtonian deflection in radians is the Sun's Schwarzschild radius divided by its actual radius.) The film also glosses over the significant technical problems Eddington experienced when trying to quantify the deflection of starlight by the Sun. He had to wait for a solar eclipse in May 1919 to do the experiment, which he personally supervised on a months-long journey to Principe, West Africa. Almost miraculously, the cloud cover broke just minutes before totality, but damaged photographic equipment allowed him to get only two decent plates of the eclipsed Sun and the star field surrounding it. And even then, the few stars visible on the plates were significantly displaced from the solar disk, decreasing the amount of observable relativistic deflection (the path a light ray takes skimming past the Sun is an exceeding flat hyperbola). Nevertheless, Eddington was able to prove that deflection had occurred in accordance with Einstein's prediction. Here is one of the plates, which is overlaid with the same star field in the absence of the Sun, which allowed Eddington to measure the deflection: The results of Eddington's Principe expedition literally made Einstein a scientific superstar, and within just days of the news ordinary people around the world were talking about the warping of spacetime. The movie portrays Einstein's achievement in light of the political differences between Germany and England at the time — England hoping that his theory was wrong, in deference to Newtonian (and British) science, and the Germans hoping that Einstein was right, as a means of exemplifying German scientific genius, but in fact it didn't play out that way. Germany lost the war, thus nullifying a lot of its scientific pride, while Einstein became a reviled Jew in his own country. German right-wing attacks began against Einstein and "Jewish physics" almost immediately, intensifying with the rise of the Nazis in the early 1930s. Indeed, right-wing hatred of legitimate science continues to this day in our own country. As we approach the 100th anniversary of Einstein's 1915 gravity theory, I see a world becoming more and more like it was in the days of Einstein and Eddington. Militarism, nationalistic pride, racial bigotry, greed, anti-gay hatred and willful scientific ignorance abound, as evidenced by the recent disturbing resurgence of extreme right-wing politics in Europe. Something seems to be brewing, perhaps the unconscious acknowledgment of catastrophic global climate change, the coming end of the age of cheap oil and food, or perhaps it's just a kind of growing mental vacuity or spiritual nihilism on the part of the planet's increasingly restless 7.2 billion inhabitants. It would be a shame if the centennial of the theory of general relativity is met with indifference or, if the Republicans win big in November, outright rejection of science.  Where is that Damned Dark Matter? — Posted Thursday, May 29 2014 The stars at the outer portions of most spiral galaxies are moving too fast, and still nobody knows why. The observed matter density of these galaxies is too low to hold the outer stars in place, even when only Newtonian physics is assumed. Most astrophysicists believe there is an unseen type of exotic matter called dark matter responsible for this anomaly, but what it's composed of no one knows, as all detection efforts so far have failed. First noticed in the 1930s, it's now an 80-year-old mystery. Lisa Grossman at New Scientist writes that it's becoming do-or-die time with respect to WIMPs, the hypothetical weakly interacting massive particles many believe constitute dark matter. Numerous highly sensitive detection experiments have been conducted over the past decade, but to date nothing has been seen, not counting a few candidate events, but these outliers have been dismissed as spurious. By comparison, the masses of the three known flavors of neutrino are too small to account for the anomaly, spurring ideas that there might be fourth kind of neutrino (the sterile neutrino), but this is just another kind of hypothetical matter. (Time is also running out on supersymmetry, the theory whose particles were deemed certain to be spotted at the Large Hadron Collider, but were not.) An increasing number of astrophysicists believe that the missing dark matter isn't there at all, and that stellar velocities can be explained by assuming that Newton's law of gravity needs to be modified. Modified Newtonian Dynamics (MOND) is the theory that, in addition to the usual Newtonian gravitational acceleration $$GM/r^2$$ acting on a massive body, there is a much smaller fixed acceleration term $$a_0$$ that has to be added to the calculations. This has been done, and the results agree with observation, but exactly what is the justification for assuming $$a_0$$, other than that it makes theory fit observation? I've mentioned more than once on this site the theory known as conformal gravitation, which is based on Hermann Weyl's conformal tensor $$C_{\mu\nu\alpha\beta}$$. This quantity replaces the Ricci scalar $$g^{\mu\nu}R_{\mu\nu}$$ in the Einstein-Hilbert Lagrangian with $$C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}$$. As numerous investigators have discovered (notably Mannheim and Kazanas), the equations of motion associated with such a Lagrangian explains the apparent hypervelocity of outer galactic stars quite well (alas, the inclusion of the mass-energy tensor in the theory makes it quite complicated, but at first glance it can be ignored). A more recent paper by Mannheim and O'Brien can be read here; it shows the velocity-distance curves for dozens of galaxies, including the one for NGC 3198 depicted above. This is Mannheim's stellar velocity-distance curve for the galaxy NGC 3198; the abcissa is the distance from the galactic core in kiloparsecs, and the ordinate is the star velocity in km/sec. Dots are observations, and the heavy dashed line is the expected data based on Newtonian gravitation. The solid line is calculated from conformal gravitational theory, which matches observation fairly well. The other two curves are lower-order contributions to the full theory and can be ignored here. Will dark matter turn out to be just another erroneous aether theory, phlogiston, or a Cheshire Cat grin? Time will tell, hopefully.  Two New Quantum Interpretations — Posted Monday, May 26 2014 Two new papers are out by physics Nobelists Steven Weinberg (1979) and Gerardus 't Hooft (1999) dealing with quantum measurement, wave function collapse and entanglement, problems which have defied solution since quantum theory's earliest days. Weinberg's paper, which proposes dumping the state vector in favor of the density matrix, is 28 pages long but is relatively readable. 't Hooft's paper is 202 pages long and appears less accessible, but I haven't read the whole thing yet so maybe I'm wrong. You can read an elementary introduction to the papers here, which includes links to the papers on arXiv.org that you can download for later reading. Richard Feynman once famously remarked that nobody really understands quantum mechanics. But we can still hope.  Are We Being Had? — Posted Monday, May 26 2014 It is hard to say to someone face to face: Your paper is rubbish! — Max Planck If you see fraud and don't shout fraud, you are a fraud. — Nassim Taleb I have several books by City College of New York professor of physics Michio Kaku, a highly respected theoretical physicist with a solid background in string and quantum field theory. But his primary occupation in the last decade seems to be publishing a lot of tripe about hyperuniverses and such, while making appearances on speculative whoosh-bang science programs of the purely infotainment variety ("If you see a black hole coming, watch out!") Nowadays, when he shows up on one of these programs I just turn the station. I read two recently-published books over the weekend, each dealing with what many have called "fairy-tale physics." No, one of them is not the ridiculous The Physics of Star Trek (although I'm an admirer of its author, Arizona State University physicist Lawrence Krauss). Instead, the books I read involve very speculative and untested physics, the kind that is being peddled today as mostly silly CGI entertainment on cable television. The books also talk about how over-attention to these theories is corrupting the field of physics itself. Science writer Jim Baggott's Farewell to Reality: How Modern Physics Has Betrayed the Search for Scientific Truth is arguably the better of the two. It challenges many of modern physics' latest concepts, such as brane worlds, supersymmetry, string theory and inflationary cosmology, primarily from a set of six principles that the author asserts are necessary to establish the validity of any physical concept: the expectation of reality, not fantasy; the acquisition of facts; the creation of a sensible theory; the testability of the theory; the theory's veracity; and something that Baggott calls the Copernican principle, which is basically the requirement that a theory not rely on any kind of anthropogenic authority. Taken together, these principles essentially constitute the scientific method, albeit in a rather longwinded manner. Then there's Alexander Unzicher, a German theoretical physicist who, with co-writer Sheilla Jones (acting mainly as his German-to-English translator), is the author of Bankrupting Physics: How Today's Top Scientists Are Gambling Away Their Credibility. The entire book is basically one long rant against the silliness of fairy-tale physics, but the book's far more pithy, interesting and entertaining than Baggotts' rather dry critique of modern trends. Filled with personal experiences (as a high school physics teacher, yet), spot-on quotes by famous theorists and a wonderfully sardonic sense of humor, I couldn't put the book down (even though he appears to have a caustic attitude toward black holes, one of my favorite subjects). Unzicher has just released another book called The Higgs Fake: How Particle Physicists Fooled the Nobel Committee. Based on initial reviews of the book it would appear that Unzicher may have gone too far with his criticisms, but when it shows up at my local library I'll give it a look. Regardless of the reviews, these books made me rethink my own attitudes towards theories involving parallel universes, string theory, extra dimensions, supersymmetry and brane worlds. Both authors bemoan the fact that, for the past 30 years or so, string theory has dominated every other field of physics despite having provided no evidence whatsoever to show that it mght be true. Indeed, if it is necessary that we be able to probe Nature at the Planck scale ($$10^{-35}$$ meter) to test the theory (which will almost certainly be forever impossible), then string theory and its ilk might end up being a kind of religious faith among physicists who simply refuse to let go of it.  Memorial Day — Posted Monday, May 26 2014  Mister Feynman's Neighborhood — Posted Wednesday, May 21 2014 This will sound presumptuous, but intellectually I have one thing in common with the late Caltech physicist and 1965 Nobel Prize winner Richard Feynman: a need to understand things for myself. I can't just read about physics, I have to actually do the calculations myself before I can believe that something's true. That probably explains why popularized "physics for the layperson" books don't do much for me. At best they're just entertainment, and at worst they're just meaningless handwaving. I was thinking about Feynman this morning for a reason I will get to in a moment. I recently came across some old 8mm home movies in the garage that my father had shot in Duarte, California back in 1955-56, and I can't believe what a different world that was (I transferred them to DVD, and will send my sister a copy, as it includes views of the Wayfarer's Chapel in Palos Verdes, where she got married in 1955). We moved to Duarte in 1949 just after I was born in Altadena, California, and for some reason I decided to drive by the old house in Altadena today. It's still there, but it hasn't changed much since 1946: (That's my mother and sisters, but the boy isn't me, as I hadn't arrived yet. Also, in the original photo I can see my Dad taking the photo in the car's reflection. Like me, he didn't like having his picture taken.) Anyway, the house is a stone's throw from the Mountain View Cemetery, where Feynman and his third wife Gweneth were buried in 1988 and 1989, respectively. Here's the shot I took this morning: I don't get reverential very often, but I couldn't help but reflect on the fact that the guy six feet below my shoes had changed the world with his ideas and discoveries. He was a friend, colleague and contemporary of the likes of Einstein, Weyl, Dirac, Pauli, von Neumann, Bethe and so many others, yet here he lies with his wife in a grave as modest as those of all the ordinary people laid out around him. Dying of stomach cancer in 1988, Feynman's last words were "I'd hate to die twice — it's so boring." He got his wish. Adieu, Professor Feynman.  A Miracle — Posted Wednesday, May 21 2014 I've been reading Anthony Zee's Einstein Gravity in a Nutshell again, whose nearly 900 pages of dense text will likely take me the rest of my life to work through. Published in 2013, it's already a classic on general relativity, easily eclipsing all the other texts out there. Zee's presentation of just about everything involving Einstein's 1915 theory (including its most recent applications) is also replete with the author's many clever insights, jokes, puzzles, witticisms and personal stories. The last chapter of the book describes Zee's take on Kaluza-Klein theory, along with his sincere hope that some modern form of the theory will eventually lead to a complete description of reality. In my own previous take on the theory I expressed the opinion that the original 5-dimensional theory's dimensional reduction to the usual 4-dimensional Einstein-Hilbert-Maxwell lagrangian was only a fortuitious coincidence of the mathematics. The calculation, which both Kaluza and Klein had to do by hand in the 1920s, is complicated to say the least, and some years back I did the calculation with the help of some computer software. Maybe that's why I was not so impressed, since by taking a computer shortcut I missed out on all the (gory) details. Tonight I could not sleep, so I decided to do it by hand myself. It took three hours, making me question my sanity in my old age. Those of you familiar with tensor calculus will know that such calculations produce many terms, but reduction (and much relabeling of dummy indices) invariably eliminates nearly everything through cancellation, leaving a nice, simple result. The Kaluza-Klein problem is that kind of calculation, but on steroids. The initial expansion of the 5-D Ricci scalar $$\tilde{g}^{AB} \tilde{R}_{AB}$$ results in about two hundred terms, which then slowly collapse down to just two, the famous $$R + 1/4\, F_{\alpha\beta}F^{\alpha\beta}$$, $$F$$ being the electromagnetic tensor. I now stand chastened by the experience — it's a magical result, and Zee's remark that he would be sorely disappointed if Nature didn't somehow involve higher dimensions truly hits home. Like Hermann Weyl, Theodor Kaluza was also a German mathematician who happened to be born on the same date as Weyl, 9 November 1885. I'm not aware of any correspondence beteen the two (although they must have written each other), but it's inevitable that Weyl would have seriously considered the possibility that the world was, as Kaluza was presumably the first to suggest, five dimensional. As others have noted, Weyl was a lifelong prolific mathematical physicist, while Kaluza was your typical one-hit wonder (though he is said to have been fluent in more than a dozen languages). Even then, Kaluza's theory would have probably been stillborn without the added insight of the Swedish mathematical physicist Oskar Klein, who considerably improved the theory while showing that it might also have application to quantum physics. Zee's book expresses the Kaluza-Klein 5-D metric tensor in a slightly more compact form than the one I used years earlier, but it still has the superfluous constant $$k$$ (which I suppose could simply be absorbed into the identification of the electromagnetic four-potential $$A_\mu$$). Zee uses $$\tilde{g}_{AB}= \begin{bmatrix} g_{00}+k^2A_{0}A_{0} & g_{01}+k^2A_{0}A_{1} & g_{02}+k^2A_{0}A_{2} & g_{03}+k^2A_{0}A_{3} & kA_{0} \\ g_{01}+k^2A_{0}A_{1} & g_{11}+k^2A_{1}A_{1} & g_{12}+k^2A_{1}A_{2} & g_{13}+k^2A_{1}A_{3} & kA_{1} \\ g_{02}+k^2A_{0}A_{2} & g_{12}+k^2A_{1}A_{2} & g_{22}+k^2A_{2}A_{2} & g_{23}+k^2A_{2}A_{3} & kA_{2} \\ g_{03}+k^2A_{0}A_{3} & g_{13}+k^2A_{1}A_{3} & g_{23}+k^2A_{2}A_{3} & g_{33}+k^2A_{3}A_{3} & kA_{3} \\ kA_{0} & kA_{1} & kA_{2} & kA_{3} & 1 \end{bmatrix}$$ or, in $$2\times2$$ shorthand notation, $$\tilde{g}_{AB} = \begin{bmatrix} g_{\mu\nu}+k^2A_\mu A_\nu & kA_{\mu} \\ kA_\nu & 1 \end{bmatrix}$$ from which we see that the metric determinant in 5-D is the same as it is in 4-D, or $$\sqrt{-\tilde{g}} = \sqrt{-g}$$ (I had to use Mathematica to prove this, as I wasn't about to do the $$5\times5$$ determinant by hand). It should be easy to see that the inverse 5-D metric is just $$\tilde{g}^{AB} = \begin{bmatrix} g^{\mu\nu} & -kA^{\mu} \\ -kA^\nu & 1+k^2 A_\mu A^\mu \end{bmatrix}$$ The Ricci scalar $$\tilde{R} = \tilde{g}^{AB} \tilde{R}_{AB}$$ is then $$\tilde{R} = g^{\mu\nu} \tilde{R}_{\mu\nu} + 2 g^{\mu5} \tilde{R}_{\mu5} + g^{55} R_{55}$$ (Computationally, the last two terms are a snap. It's the first term that's the troublemaker.) The Kaluza-Klein action then reduces to $$\int d^5x \sqrt{-\tilde{g}} \tilde{R} = \int dx^5 \int d^4x \sqrt{-g} \left( R + \frac{1}{4} k^2 F_{\mu \nu}F^{\mu\nu} \right)$$ Remarkably, in Zee's earlier text Quantum Field Theory in a Nutshell he states that the entire calculation can be avoided using basic symmetry arguments. If I only had a brain like that!  The Money Pit Syndrome, or It's Time to Grow Up — Posted Tuesday, May 20 2014 Oak Island off the southern coast of Nova Scotia is the location of the infamous Money Pit, which supposedly contains a treasure that was buried there over two centuries ago. Over the years, treasure hunters have discovered tantalizing traces of jewelry and other objects buried at increasingly deeper levels in the pit, which is naturally flooded at its deepest level by seawater. Numerous searchers have lost their lives looking for valuables, while vastly more treasure has been sunk into the hole by disappointed searchers than has been taken out. Still, there seems to be no end of treasure hunters and wealthy investors who are convinced that a large treasure awaits anyone who can successfully overcome the pit's constant cave-ins, floodings and accidents. Somewhat more inaccessible are carbon-rich planets believed to harbor fantastic quantities of pure diamond, and even further out may be entire cold, dead stars (carbonized white dwarfs) consisting of single crystals of pure diamond (but I think de Beers has already staked a claim on them). Closer to home are the asteroids that orbit the Sun between Mars and Jupiter, many (if not most) of which are made primarily of primordial iron and nickel, though there's no reason not to think they may also have huge amounts of gold, platinum and other precious metals. All of the gold ever produced on Earth would fit into a cube 50 feet on each side, and a single gold asteroid could easily surpass that amount. We can forget about diamond planets, as the logistics of finding and exploiting them are surely insurmountable. But while the Money Pit is probably only a myth or elaborate hoax, it is nevertheless enticing. Also enticing is the thought of somehow capturing a precious-metal asteroid. But why? I believe it has to do with the fact that we have the technologies to get at them, and so we tend to think of them as free resources waiting to be plundered. That is certainly true of asteroid mining, a subject that is becoming more and more of interest as precious metal resources here on Earth are diminished. The idea of space mining is not new, but a recent article in New Republic addresses the recent heightened interest in the subject, particularly by wealthy entrepreneurs who believe that the not-inconsiderable logistical problems can be overcome with existing technologies. And what are those technologies? Current mining practices and rocket propulsion, both of which are ancient, inefficent and environmentally destructive. That's about it. And that's what's driving these entrepreneurs to think about space mining in the first place — the hope of using these rudimentary technologies to make even greater fortunes for themselves. Even someone as normally level-headed as astrophysicist Neil deGrasse Tyson is excited about space mining, as the article notes. Tyson asserts that an asteroid (or meteorite) the size of a house might contain more platinum than has ever been mined on Earth. And even if that asteroid contained little platinum, the iron and nickel metal alone would still be worth billions of dollars. But getting either an asteroid or the metal it contains down to Earth is horrifically problematic, and that's where I see these optimistic entrepreneurs getting the cart before the horse. Take the case of Tyson's house-sized asteroid. Space miners wouldn't be interested in rocky or stony asteroids; they'd certainly focus on metallic asteroids, which have an average density of around 7 or 8 grams per cubic centimeter. A house-sized object would therefore have a mass of roughly $$10^{7}$$ kilograms, or about 25 million pounds. But you're not likely to find one of these just hovering out in space, available for the plucking; they typically travel at about 25,000 meters/second, giving our house-sized object a kinetic energy of about $$10^{16}$$ joules. That's the energy equivalent of some 250 million gallons of gasoline, which is what you'd have to burn with 100% efficiency to bring the asteroid to a halt. And assuming you could stop it in its tracks 10,000 miles from Earth, you'd have to expend the equivalent of millions more gallons to provide the associated gravitational braking energy to bring it to Earth's surface safely. (It has been suggested that refined materials could be delivered back to Earth using space parachutes to avoid the problems of braking. But if that's such a good idea, why aren't parachutes used today?) Space-mining apologists counter such arguments with tragically broken logic. You don't have to stop an asteroid, they claim, just hop onto it with your mining equipment, mine it at your leisure, and ship the refined resources down to Earth via space shuttles. But think about it — the energy requirements I mentioned above would be exactly the same. Even more problematic would be getting the mining equipment up to the asteroid in the first place. The current cost to transport one pound of equipment into space via space shuttle is roughly$10,000. Apologists have suggested that we shoot the equipment up instead with giant Earth-based space cannons, but cost issues associated with propulsion energy and air friction are essentially the same as with shuttle transport, while simply capturing a payload at the mining site would also require energy. Furthermore, since gasoline doesn't burn in space, the miners would have to use storage batteries or nuclear power to run their operations. I can promise you that those heavy materials won't be sent into space for a long time. Space-mining entrepreneurs talk a lot about how mankind's future home is in the stars, while the need for metals will only accelerate in the here and now, so we might as well get started. Unfortunately, they're only interested in making themselves richer, not advancing the condition or welfare of mankind. To date, you can hop on a Russian rocket for a trip to the International Space Station if you have the requisite $25 million, while Richard Branson and his ilk are promising more local trips via hypersonic aircraft to the same wealthy travelers. But these technologies are all still based on rocket propulsion, a low-tech and wasteful technique invented by the Chinese a thousand years ago, and they will never be made available to the average person. Well then, they say, how about antigravity propulsion, powered by matter-antimatter annihilation? Or maybe dilithium crystals? Yeah, how about that? As long as we're driving, flying and riding around in fossil-fuel powered cars, planes and trains, the prospects for such futuristic technologies will likely remain with the magic elixir that turns a tankful of water into gasoline.  Weyl (Early 1940s?) — Posted Tuesday, April 29 2014 My computer crashed recently, and while recovering some files I came across this photo. It's undated, but I had labeled it as "Fuld Hall," so it must have been taken at the Institute for Advanced Study (IAS) in Princeton. My guess is that it was taken around 1945 or a little earlier. I can recognize four individuals: the first person on the left is mathematician James Alexander (a topologist); don't know the next guy; then Einstein; don't know the next guy; then Hermann Weyl (mathematician); and the lanky guy on the far right is Oswald Veblen (mathematician). Weyl, Einstein, Veblen and John von Neumann were the very first to sign on with the IAS when it opened in 1933. (Did Einstein ever wear a suit and tie?)  Annemarie Schrödinger, 1886-1965 — Posted Tuesday, April 29 2014 Erwin and Annemarie Schrödinger wedding photo, March 1920. In April 1963 the noted American physicist and philosopher of science Thomas Kuhn traveled to Vienna to interview Annemarie (Anny) Schrödinger, the widow of 1933 Nobel laureate Erwin Schrödinger (1887-1961). The American Institute of Physics recently posted a transcript of the interview (in English), which you can read here. In the interview the then 67-year-old Anny shared many memories of her husband and his colleagues, which included Hermann Weyl. She doesn't go into any details, but she and her late husband shared a notoriously open marriage. Erwin had many girlfriends, and he took one with him on a Christmas tryst to the Alps in December 1925, where the seemingly indefatigable Austrian physicist somehow also found the energy to discover wave mechanics. A close friend, Weyl would later characterize Schrödinger's discovery as the intellectual result of "a late erotic outburst" in the Nobelist's life (he was only 38 at the time, but I guess that was considered past one's prime in those days, and not just in physics). I was surprised to learn some years ago that Anny had a fling of her own, and that her lover was none other than Weyl himself. Schrödinger knew about the affair, but the free-swinging physicist didn't mind, and he and Weyl remained close lifelong friends. I was disappointed that Kuhn's interview didn't reveal any details of the Annemarie-Weyl affair, but then 1963 was a different time as well. As Mark Twain once remarked in The Adventures of Tom Sawyer, "Let us draw the curtain of charity over the rest of this scene."  The Unified Field Theory of Inequality* — Posted Friday, April 25 2014 Inexplicably, French economist Thomas Piketty's book Capital in the Twenty-First Century, at nearly 700 pages of dense economic prose, is a runaway best seller. The Los Angeles Times reports that Piketty has hit an ideological nerve in America, which partly accounts for its popularity. Liberals love it because it brings attention to the growing problem of wealth inequality, while conservatives hate it for the same reason. Amazon's bi-modal book reviews seem to verify the love-hate aspect of Piketty's work. Stranger still is the fact that Piketty's thick French accent, exhibited in the numerous television interviews I've seen, would ordinarily stanch any interest in his book. My e-book copy clocks in at only 642 pages, though I'm only halfway through it. I'm not particularly adept at economics, but I can understand the data and his analyses well enough to say that Piketty's assertions are spot-on. But we're in big trouble, because not even Piketty can see a way out of the mess we're in. His basic conclusion is this: if $$r$$ is the long-term private rate of return on capital and $$g$$ is the rate of income and output growth, then The inequality r > g implies that wealth accumulated in the past grows more rapidly than output and wages. This inequality expresses a fundamental logical contradiction. The entrepreneur inevitably tends to become a rentier, more and more dominant over those who own nothing but their labor. Once constituted, capital reproduces itself faster than output increases. The past devours the future.Piketty goes on to note that capital growth averaging 4 or 5 percent per year will always outstrip income growth, leading to ever greater inequality. Historically, wealth inequality was countered somewhat by two world wars, whose emphasis on production far exceeded private wealth accumulation. But those wars were primarily the result of geopolitical instability, not proletarian worker discontent over income. Today, both of those conditions exist. (By the way, if you can't or won't read the book, at least read Nobel economist Robert Solow's review, which will bring you up to speed on the book's main points.) I have not finished the book, but so far I feel that Piketty has missed an important aspect of wealth inequality, at least as far as this country is concerned. That is the idea that people today have become so overly arrogant when issues of financial success and prosperity are raised that they're willing to tolerate any level of inequality provided either that they are wealthy or are able to maintain the expectation that they will become wealthy. In addition, many of the wealthy today are not only greedy but feel entitled to their wealth, and are not ashamed to express their pride and haughtiness about it. Worse are the sanctimonious conservatives who believe their prosperity is God's gift to them for being so good, and whose revilement of the poor stems from a core belief that poverty is God's punishment of the poor's unworthiness. You may recall around the time of the 2012 presidential election that the issue of "Who built my business" became a conservative rallying cry. Obama rightly if naively asserted that no entrepreneur becomes successful on his or her own — the country and its people as a whole contibute to their success through public education, infrastructure and other "village" amenities. Mitt Romney and Paul Ryan stoked conservative hubris by claiming that succesful people have earned their prosperity all by themselves. It worked, but it didn't win them the election. A few days ago on my way to the gym I pulled up next to a large motorhome pulling an SUV. The large, white stenciled letters on the back window read "The government didn't build my business. I BUILT IT!" I rolled down my window and said something like "Did you really get no help from anyone else?" to the driver. Instantly annoyed, he responded to my query with the bird. I suppose I deserved that. * What Nobel economist Paul Krugman calls Piketty's book  Quantum Mechanics, Susskind/Friedman — Posted Thursday, April 24 2014 I friend loaned me this book, wanting to know what I thought about it. I have the first of Lenny's "Theoretical Minimum" books (classical physics) and wasn't too impressed with it, mainly because of the book's awful typeset equations (although the contents are pretty good). This new one's on quantum mechanics, and overall it's a distinct improvement. If you managed to get through all of Susskind's YouTube video lectures on the subject you'll find this a great reference. The sections on quantum entanglement are especially clear and informative, requiring only a little familiarity with complex numbers and linear algebra. It even manages to get into what's known as the density matrix which, besides being a fascinating subject all its own, is arguably the best way to approach the topic of quantum entanglement, which a number of notable physicists have called "the only mystery." Like the lecture series, Susskind intended this book for those who majored in science or engineering in school and managed to maintain an active curiosity of the physical world, even if they didn't go into physics as a career. His video lectures were videotaped before an informal Stanford University audience comprised of mostly aging computer scientists, software/hardware engineers and other geek types who kind of remember the math and want to satisfy their intellectual cravings concerning the Higgs field and all the new stuff that's going on. BTW, co-author Art Friedman is the owner of School Year Data (whatever that is), which is located in the San Francisco Bay Area. He's a software engineer who's also taught high school math and science. His CV includes the above book, which he considers suitable "for mathematically literate nonphysicists." If that describes you, invest the$17, read the book, and become an enlightened human being.
 "Duck Soup, My Ass" — Posted Monday, April 21 2014 Speaking of lead poisoning, the latest episode of Cosmos tells the story of how Caltech scientist C.C. Patterson discovered the true age of the Earth (4.55 billion years) in 1956 by analyzing the relative concentrations of lead and uranium in zircon crystals. Patterson was also the guy who discovered the dangers of lead in the environment caused by the anti-knock gasoline additive tetraethyl lead. The discovery almost destroyed his career, since the chemical industry (primarily the Ethyl Corporation) was making billions with the additive. They tried hard to shut him up, but Patterson persisted, and by analyzing Arctic snow and ocean water samples he finally conviced the EPA (in 1973) to ban lead compounds in gasoline. (Modern radiometric dating methods give the age of the Earth as 4.54 $$\pm$$ .05 billion years, nearly identical to Patterson's finding.) Besides showing some nice shots of Pasadena and Caltech, the Cosmos episode also relates how, for hundreds of years, the "official" age of the Earth was the one determined by Bishop James Ussher in the 1600s. By carefully studying all the "begats" in the genealogies of the Old and New Testaments, Ussher calculated that the Earth was created on Saturday night on 22 October, 4004 BC (and at 9 pm, by golly). Incredibly, there are still many Americans today, perhaps 20% of the population, who still believe the Earth is only about 6,000 years old. And that percentage is much higher on the Republican side of the House of Representatives. An interesting story, and one showing that lead, which featured prominently in the fall of the ancient Roman Empire and the demise of the Franklin Expedition, can't be blamed for the insanity running amok in our country today.
 Some Things Never Change — Posted Tuesday, March 25 2014 Hermann Weyl and Paul Dirac both considered the possibility that the Newtonian gravitational constant G might be changing with time, as this would support theories they developed to explain why a certain large number ratio seems to appear again and again in Nature (for example, the ratio of the electromagnetic to gravitational force is about $$10^{40}$$, as is the ratio of the radius of the observable universe to that of the electron). This idea became known as the Large Number Hypothesis, which assumed that G is getting slightly smaller over cosmic time periods. The notion of a non-constant G has also been used by young-Earth creationists to argue that radioactive decay rates are getting smaller, which would (not really) explain why radiometric dating techniques appear to indicate a much older Earth than that indicated in the Bible. But now new research has shown that, at least for the last 9 billion of the universe's 13.8-billion-year history, the gravitational constant has not varied more than (at most) one part in a billion. After an exhaustive study of 580 observed supernovae events, professor Jeremy Mould and his PhD student Syed Uddin at the Swinburne Centre for Astrophysics and Supercomputing and the ARC Centre of Excellence for All-Sky Astrophysics showed that the Newtonian constant G has not changed appreciably over cosmic time. Their research, which focused on Type 1a supernovae, demonstrated a constant G within an upper bound of $$\dot{G}/G$$ of $$10^{-10}$$. The legitimacy of Type 1a supernovae studies was demonstrated in 1998 when three astrophysicists, Saul Perlmutter, Adam Riess and fellow Australian astronomer Brian Schmidt, used Type 1a supernovae to show that the universe is expanding at an accelerated rate, thus proving the existence of dark energy (see my post dated November 11, 2013 for more information). In 2011, the three scientists were awarded the Nobel Prize in phyics in recognition of their work. A Type 1a supernova occurs when a white dwarf accretes sufficient matter from a companion star to attain critical mass. It then explodes, and the light given off by the explosion is used as a standard candle to measure the supernova's distance from the Earth. This critical mass is the same for all Type 1a events, and it depends on the value of G in the relativistic calculations used in its determination. Observations of billion-year-old Type 1a events demonstrated the constancy of G to high precision. A less technical overview of the Swinburne study can be read here. An absolutely fascinating discovery.
 Black Friday — Posted Friday, March 21 2014 While I don't expect newscasters to be astrophysicists, I don't expect them to be complete idiots, either. CNN anchor Don Lemon, a nice enough guy, unfortunately represented the bulk of clueless humanity when he seriously asked a guest if the ill-fated Malaysian Flight 370 might have been the victim of a black hole. Also unfortunate was the fact that his guest 1) did not break out in hysterical laughter, and 2) reminded Mr. Lemon that a black hole would "suck up the entire universe." I guess she didn't know that the universe's countless black holes have somehow failed to do that just yet. And while I understand that reality shows like Life After People and Strip the City might be entertaining in a fun if impossible sort of way, I'm waiting for someone at CNN to seriously suggest that if we have the technology to Drain the Ocean, then we should use that technology to find Flight 370 (and Captain Kidd's treasure). But then if Animal Planet and even National Geographic can seriously suggest that mermaids exist, perhaps humankind has already reached the end of its rope.
 Primal Fear — Posted Friday, March 21 2014 Religion is the great salve that protects believers from death and the great unknown, right? If you believe that, then watch this short (2:42), award-winning video, Lights Out. And watch it ALONE, AND IN THE DARK. (Duct tape helped get George W. Bush reelected. But it ain't gonna save you, lady.) Sweet dreams.C'est une croix qui de l'enfer nous garde. — Gounod, Faust
 Improbable v. Impossible — Posted Tuesday, March 18 2014 Here's a highly improbable story, fictional but one that I absolutely guarantee will play out some day, and exactly as I've written it. In the not too-distant future, computers will be able to calculate the transcendental number $$\pi$$ out to previously unimaginable accuracy — to be precise, let's say $$10^{1,000,000}$$ decimal places — in a matter of only a few weeks or so. Mathematicians, being a curious sort, will concurrently develop sophisticated programs to determine if there are any unusual or weird numerical patterns in $$\pi$$, like the not too-unusual $$999999$$ sequence around the 762nd decimal you see here: Then, using the trivial code $$A=1, B=2, C=3$$ etc., at some unbelievably-distant decimal place in $$\pi$$ they discover the phrase I AM THE LORD YOUR GOD YOU HAVE AT LAST DISCOVERED PROOF OF MY EXISTENCE AND MY POWER TO MANIPULATE ALL TRANSCENDENTAL THINGS INCLUDING PI BEHOLD MY WORKS YE MIGHTY AND DESPAIRUpon publication of this discovery there is a momentous worldwide resurgence in religious belief, based on the apparent statistical impossibility of finding such a detailed, prescient message in the seemingly random infinite number $$\pi$$. But upon further investigations at an even more distant decimal place, mathematicians find the message JUST KIDDING THE BIBLE AND THE TORAH AND ALL THAT ARE JUST NONSENSICAL GIBBERISH CREATED BY FEARFUL HUMANS I DONT EVEN EXIST YOU ARE TRULY ALONEUpon publication of this numerical discovery, the human race goes into a long period of self-reflection and doubt, not knowing what to think or believe anymore. In 1985 I traveled to New Zealand on a two-week business trip. I caught a taxi at Auckland Airport and directed the driver to my hotel. It was a long drive, and along the way the cab driver and I struck up a conversation about various things, and at one point I asked him how long he'd been driving a taxi. He told me he only moonlighted as a cab driver to make ends meet, as the taxes in New Zealand, a socialist country, were pretty steep. He then told me that he worked as a water resources engineer at his day job. I informed him that I also worked in water resources, and was in New Zealand to inspect some water treatment equipment for possible purchase for my company. He replied that he had an appointment the next day to meet with a guy from Los Angeles regarding water quality issues and a potential sales contract. Upon exchanging names, we realized that he and I had been talking to each other by phone for the previous six months about water quality issues, the drought in nearby Melbourne, Australia, and life in general. We had a good laugh over this coincidence, which seemed pretty amazing at the time. But the next day in his office he introduced me to a consulting engineer for his firm, and I experienced another improbability: the consultant and I recognized each other immediately, as we had both majored in chemistry at California State University at Long Beach, graduating in 1971. He too had left chemistry and had gone into engineering. I've experienced a number of coincidences like these, but they're insignificant compared to some of the stories that Imperial College mathematics professor David J. Hand relates in his new book The Improbability Principle: Why Coincidences, Miracles, and Rare Events Happen Every Day. He opens the book with an unbelievable incident that actor Anthony Hopkins experienced in London while preparing for a role. But Hand notes that, given the billions of human beings that have ever lived and the countless experiences they have shared, unbelievable incidents are simply par for the course, mathematically speaking. In previous posts I've talked about science writer Michael Shermer and his thoughts on patternicity and agenticity. Humans are hardwired to see patterns in nature, Shermer notes, probably as a survival mechanism, even when there are no patterns. The tendency for humans to see unusual or improbable patterns and associate them with an agent (like God) is also probably hardwired into our brains. But, as Hand points out, given an infinity of alternatives and enormous spans of time for which these alternatives to play out in, and there is absolutely nothing unusual about highly improbable events. On the most recent episode of Cosmos, astrophysicist and host Neil deGrasse Tyson talked about evolution and intelligent design. He raised a favorite topic of the IDers, namely the human eye and the notion that it's too complex to have come into existence via evolution — it simply had to have an Intelligent Designer behind its construction. But Tyson notes that innumerable numbers of ocean-borne molecules (simple and complex), coupled with billions of years of random interactions, made the eye and even life itself not only probable but a certainty. We do not know if there's a Great Designer behind all things, and we'll likely not know until we die. In the meantime, I would suggest that we put our biases aside, along with our fears of death and the unknown, and try to put this universe and all its inconceivable wonders into a context that leaves us simply in awe, and not afraid. PS: I can't remember who said "It's difficult to convince someone of a factual truth when their faith or their salary depends on their not believing it." That's what we mean by bias. Here's a cute cartoon demonstrating the concept:
 Gravity Waves Detected? — Posted Monday, March 17 2014 Years ago it was noted that the observed cosmic microwave background (CMB) temperature of the universe appears to be far too uniform. If the Big Bang was literally the beginning of everything (including time and space), the expansion of the universe, however rapid it might have been, would be expected to produce some non-uniformity in the observed CMB temperature pattern (it averages only about 2.7 Kelvin, close to absolute zero). To explain this uniformity, Alan Guth proposed the theory of inflation, which conjectured that within the first $$10^{-35}$$ second or so after the Big Bang the universe experienced a brief but extremely rapid expansion, so rapid that any non-uniformities would have been effectively smoothed out. Inflation has since become the leading theory of how the universe got to be so uniform, although direct experimental evidence has been lacking. If inflation (and general relativity) is correct, then gravitational waves resulting from the Big Bang would have been produced copiously and with unimaginably small (and immeasurable) wavelengths, but within a short time inflation would have stretched the waves out to more reasonable size, making them (at least indirectly) observable. The results of the latest research appears to show that this is correct. The preliminary papers are out on arXiv.org (links here). I haven't read them yet, but notables like Lawrence Krauss, Alan Guth and Andrei Linde are saying that if the data holds up this will be the biggest thing in cosmology in the last 30 years, and certain to earn at least one Nobel Prize.
 Weyl as Art — Posted Thursday, March 13 2014 This is interesting — a beautiful interpretation of Hermann Weyl by artist Sarah Kaiser, which she created for the cover of the June 3, 2010 issue of Nature magazine.
 Multiverse — Posted Saturday, March 8 2014 After reading an interview with Wesleyan University professor of religion Mary-Jane Rubenstein on the Religion Dispatches website, I picked up her new book Worlds Without End: The Many Lives of the Multiverse, in which she tries to come to terms with the commonalities and differences between science and religion as envisioned by her in the multiverse theory, which is currently enjoying a vogue with the lay public right now. Imagine flipping a coin five times and getting heads with each flip. No big deal, you say, it's common enough. If you flipped it ten times and got heads on each flip, you'd probably think it was kind of remarkable, but still no big deal. However, if you flipped it twenty-five times and got heads each time, you'd likely think the coin was flawed, or rigged in some way, perhaps heavily weighted on the tails side. But you could still entertain the possibility that it was still only a statistical fluke, albeit a very unusual one. The odds of getting heads 25 times in a row on an honest coin are only about 0.000003%, but that's still far from zero. You say to yourself, hey, it could happen. You now flip the coin $$10^{100}$$ times (you couldn't live long enough to do it for real, but a computer might simulate the flips in a few million years or so). If the results were all heads, and you were absolutely positive that the coin (or the computer) was not flawed or rigged in any way, and you had absolutely eliminated the possibility of an outside physical force or other influence acting on the coin (electromagnetic induction, gusts of wind, losing your mind, etc.), then you would almost certainly ascribe the result to some supernatural cause or entity. In a nutshell, that's the situation between religion and the multiverse theory today. A religious person would say that it's absolutely impossible for the universe to exhibit the multitude of precise and apparently inviolable physical laws and biological processes we observe by chance alone. But a physicist or mathematician would say that the odds, while inconceivably small, are not zero. And if you have a theory in which the number of possible universes is infinite, or if the amount of time available is essentially unlimited, then the universe we live in is not just a fluke, they say, but a statistical certainty. We just happen to be living in it. If we weren't, we wouldn't be wondering about all this. That is what Stephen Hawking meant when he claimed that the universe does not need God (which isn't the same as saying that God doesn't exist). Noted science writer and author Michael Shermer (who lives a few blocks from me in nearby Altadena) has asserted the idea that humans are endowed with two attributes that together conspire to create and enforce religious or superstitious belief. The first is a survival instinct that he calls patternicity. Humans tend to see patterns in things, even when there are no actual patterns. To one person, a random collection of data points is just a meaningless scatter plot, while another person might see a definite or meaningful pattern or trend in the data. A more basic form of patternicity that humans developed long ago involves imagining dangerous animals (or rival tribe members) lurking in bushes and forests. If you see a nearby shrub shaking unexpectedly, most likely it's just the wind blowing the leaves around. But why take a chance? You run like hell! This instinct gave rise to the concept of false positives and false negatives — better to run from both if you want to survive. If it's just the wind, then all you've lost is a little energy. If it's a hungry Smilodon or cave bear, then you've improved your survival chances by hightailing it out of there. And it worked great for thousands, maybe even millions of years of human existence. The second attribute is what Shermer calls agenticity, which is the tendency to ascribe supernatural intent or force to things we do not understand or are afraid of. This also provided a sense of comfort and security and control over the natural world to early humans, which for much of our existence was totally beyond control. Agenticity also provided humans with a sense of purpose and meaning; belief in supernatural forces gave rise to acknowledgment over their power and influence, which then led to ritualized worship, obedience and even sacrifice, actions that provided a measure of purpose in human affairs. Ritual practices also made supernatural belief more "believable," even sensible. That is why atheists, agnostics and "nones" even today are shunned, if not persecuted outright by believers — non-belief on anyone's part serves to induce doubt in the believers, and they don't like that. In addition, just the act of tolerance on the part of believers induces the fear that their god will be angry with them for not getting rid of the non-believers. The Old Testament books of Exodus, Numbers and Deuteronomy are classic examples of extreme religious intolerance, while today American fundamentalist Christians fear God's wrath if they do not expunge the country from the evil-doers in their midst. But what the multiverse scientists cannot adequately explain is why physical and mathematical laws are so beautiful. And "beautiful" here is not simply a subjective description, but an undeniable attribute of our universe that chance alone would seem to have no business in creating. I suppose it's fair to say that if there is truly an infinite number of possible universes, then there is also an infinite number (or at least a large number) of worlds in which non-subjective "beautiful" physical laws and their underlying mathematical symmetries exist. But to date, I haven't been able to accept that. If there's anything interesting in Rubenstein's book worth sharing, I'll come back to it.
 Hidden Variables — Posted Thursday, March 6 2014 I'm laying hardwood flooring and, having removed all the carpeting, I carefully covered all the tack strips so I wouldn't step on them (those little nails are sharp). Well, I didn't step on any, at least until I took my shoes off for bed. I then immediately stepped on one I somehow forgot to cover. Now aching from a tetanus shot and a pierced foot (not to mention a sore back, because at 65 I'm too damned old to do this anymore), I have little to do but lay here and talk about gauge theory, one of my favorite topics. The Weyl gauge in electrodynamics is an exceptionally simple prescription that gives the electric potential $$\Phi$$ rather directly; in fact, it's just $$\Phi = 0$$. Why this prescription is attributed to Weyl escapes me, but in certain situations it's obviously very handy. But even then, the electric field $$E$$ itself remains largely undetermined. It has always amazed me that electric and magnetic fields are so easily detected and measured (an EMF meter, which is actually a kind of antenna that detects and quantifies deflections, like the jump of the needle in a voltmeter, is one such device), while the underlying four-potential $$A^\mu = (\Phi, \vec{A})$$ that $$E$$ and $$B$$ are made from is essentially undetectable and immeasurable. There is no device that can tell us in a straightforward manner that a "bare" electric potential $$\Phi$$ is nearby, or that a non-zero vector potential $$\vec{A}$$ is lurking about. And, for that matter, no device can measure their intensity. The reason for this has to do with the gauge freedom of the four-potential. As is well-known, Maxwell's equation are unchanged under the pair of gauge transformations $$\vec{A} \rightarrow \vec{A} - \vec{\nabla} \lambda, \quad \Phi \rightarrow \Phi + \frac{1}{c} \frac{\partial \lambda}{\partial t},$$ where $$\lambda(x,t)$$ is a completely arbitrary scalar function of space and time. Consequently, there is no such thing as the four-potential $$A^\mu$$ because $$\lambda$$ can have any value. Nevertheless, the gauge parameter $$\lambda$$ can be specified in such a way that makes Maxwell''s equations easier to solve. (In fact, the potentials were originally viewed as just a mathematical convenience with no physical validity, while gauge invariance was seen as a mere happenstance of the formalism.) Recall that the electric and magnetic fields can be expressed in closed form by considering the homogeneous set of Maxwell's equations $$\vec{\nabla} \times \vec{E} + \frac{1}{c} \frac{\partial \vec{B}}{\partial t} = 0, \quad \vec{\nabla} \cdot \vec{B} = 0,$$ which can be solved using simple vector identities to give $$\vec{E} = -\vec{\nabla} \Phi - \frac{1}{c} \frac{\partial \vec{B}}{\partial t}, \quad \vec{B} = \vec{\nabla} \times \vec{A}$$ As is easily shown, these physical quantities do not change under the transformations given above. But what about the inhomogeneous Maxwell's equations, which specify the sources? They are $$\vec{\nabla} \cdot \vec{E} = 4\pi \rho, \quad \vec{\nabla} \times \vec{B} - \frac{1}{c} \frac{\partial \vec{E}}{\partial t} = 4\pi \vec{j}$$ Using the identities for $$E$$ and $$B$$ above, these go over to $$\nabla^2 \Phi + \frac{1}{c} \frac{\partial (\vec{\nabla}\cdot\vec{A})}{\partial t} = -4\pi \rho$$ and $$\nabla^2 \vec{A} - \frac{1}{c^2} \frac{\partial^2 \vec{A}}{\partial t^2} - \vec{\nabla} \left( \frac{1}{c}\frac{\partial \Phi}{\partial t} + \vec{\nabla}\cdot\vec{A} \right) = - 4\pi \vec{j}$$ As many have noted, these last two expressions are ugly as hell, not to mention the fact that they're inextricably coupled in $$\Phi$$ and $$\vec{A}$$. But they are both invariant with regard to a gauge transformation, and we can use that fact to simplify them. Consider the scalar quantity $$S = \frac{1}{c}\frac{\partial \Phi}{\partial t} + \vec{\nabla}\cdot\vec{A}$$ A gauge transformation then gives $$S \rightarrow S + \frac{1}{c^2} \frac{\partial ^2 \lambda}{\partial t^2} - \nabla^2 \lambda$$ Thus, $$S$$ can be made gauge invariant if the gauge parameter satisfies the wave equation of light, which is $$\Box^2 \lambda = \frac{1}{c^2} \frac{\partial ^2 \lambda}{\partial t^2} - \nabla^2 \lambda = 0$$ Selecting a gauge parameter $$\lambda$$ whose d'Alembertian $$\Box^2 \lambda$$ vanishes (which we can always do, since it's arbitrary) thus turns $$S$$ into an arbitrary gauge invariant scalar, and for simplicity we may as well set it to zero: $$\frac{1}{c}\frac{\partial \Phi}{\partial t} + \vec{\nabla}\cdot\vec{A} = \partial_\mu A^\mu = 0$$ This is called the Lorenz gauge, and its primary value is that it uncouples the above equations in $$\Phi$$ and $$\vec{A}$$ to give the beautifully symmetric expressions $$\frac{1}{c^2} \frac{\partial^2 \Phi}{\partial t^2} - \nabla^2 \Phi = 4\pi\rho, \quad \frac{1}{c^2} \frac{\partial^2 \vec{A}}{\partial t^2} - \nabla^2 \vec{A} = 4\pi \vec{j}$$ or, in lovely covariant language, $$\Box^2 A^\mu = 4\pi j^\mu$$ where $$j^\mu = (\rho, \vec{j})$$. In principle, if you can solve one of these equations, then you can solve the other using the same approach. I've always had problems with the Lorenz gauge. For one thing, I always confused Ludwig Lorenz with the far more famous Hendrik Lorentz of Lorentz contraction fame. They're not the same guy. For another, setting $$S = 0$$ requires that the gauge parameter $$\lambda$$ satisfy the wave equation, but the opposite is not necessarily true. The only sure way of justifying the Lorenz gauge is by appealing to a rather nasty theorem in vector calculus called Helmholtz's theorem, which states that any well-behaved vector function can always be expressed as the sum of a divergence and a curl. For the vector $$\vec{A}$$, the curl is already specified in terms of the magnetic field via $$\vec{B} = \vec{\nabla} \times \vec{A}$$. But the divergence of $$\vec{A}$$ is undetermined, so we can select any arbitrary value for it. That's what I was taught, but I still don't get it. I know it should have something to do with gauge transformations, but I'll be darned if I know how. At any rate, to me the four-potential is something like God — it never makes its existence known, and is a total and profound mystery, yet it's somehow there, and can be deduced mathematically and by physical reasoning. I cover some of these thoughts in my elementary write-up on the Aharonov-Bohm effect, which explains how the physical existence of the four-potential was finally demonstrated by a very clever (and beautiful) quantum-mechanical experiment. By the way, noted UC Berkeley physics professor J.D. Jackson has written a lengthy paper detailing many useful types of gauge transformations. But be warned — Jackson is also the author of many a grad student's greatest nightmare, the seemingly impenetrable textbook Classical Electrodynamics.
 Quote of the Week — Posted Monday, March 3 2014 US Secretary of State John Kerry, on Russian warmongering in Ukraine:"You just don't invade another country on phony pretext in order to assert your interests." And that's from 2004 Democratic presidential candidate John Kerry. I can only wonder if he was intentionally including a "dog whistle" message in that statement, or if he's even aware of how monstrously hypocritical it sounds. And speaking of that erstwhile "Empire of Evil" or "Axis of Evil" country, I can remember George W. Bush (easily the stupidest and most insanely corrupt President we ever spawned) talking about Vladimir Putin not that long ago, when Bush "looked into his eyes and saw his soul." Do you remember that? Well, it seems the country's conservatives don't. But they're all over President Obama for being a total wuss for not getting tough with Putin. Fratboy George W. Bush in happier, cockier days. He's still an asshole. Yeah, let's shoot off those nuclear-tipped ICBMs and get it the fuck over with.
 Schrödinger Again — Posted Saturday, February 22 2014 Most people know Erwin Schrödinger as the father of wave mechanics and the co-recipient (with Dirac) of the 1933 Nobel Prize in physics. But he was also interested in numerous other scientific areas, including biology, genetics, general relativity and color measurement (he was also a noted womanizer, but that ain't scientific). In the 1940s his interests turned to a fundamental topic in differential geometry, that of affine connections. I just posted an online paper (suitable for undergraduates) concerning one particularly simple connection that Schrödinger presented in his short but illuminating 1950 book Space-Time Structure. I bought the book back around 1978 and still turn to it on occasion. One warning — I kind of dump on Hermann Weyl in this paper, as I believe Schrödinger's connection makes more sense than Weyl's. But whatever.
 Einstein's Cake — Posted Monday, February 17 2014 I'm getting a renewed interest in (and even appreciation of) the latter work of Erwin Schrödinger in what he referred somewhat extravagantly to as "the final laws" of gravity and electromagnetism, which he developed in the years immediately following the end of World War II. This work in many ways paralleled that of Einstein, whose interest in a unified theory of gravitation and electromagnetism continued unabated from around 1925 until his death in 1955. By 1939 Schrödinger had moved to Dublin, Ireland from his native Austria, following a long bout of political persecution from Nazi Germany, which had annexed Austria two years prior to the war. (He was not a Jew, but his progressive ideas were nevertheless annoying to the Germans. But as a co-recipient of the 1933 Nobel Prize in physics his fame fortunately outweighed his infamy in Nazi eyes, so his life was never in danger.) Schrödinger helped establish the Institute for Advanced Study in Dublin and became a naturalized citizen there in 1948. Following his retirement in 1955 he moved back to Austria, where he died in 1961. During his years in Ireland he wrote numerous papers on unified field theory (while simultaneously siring several illegitimate children with two Irish women), one series of which was titled The Final Affine Field Laws. Like Einstein, Schrödinger had decided that the symmetry of the affine connection $$\Gamma_{\mu\nu}^\lambda$$ in the lower two indices should be abandoned in order to derive a workable theory. This idea had been considered by many physicists, even as far back as 1918, but it introduces many problems. Any asymmetry in the connection of course goes unnoticed in the equations of the geodesics $$\frac{d^2x^\lambda}{ds^2} + \Gamma^\lambda_{\mu\nu} \frac{dx^\mu}{ds} \frac{dx^\nu}{ds} = 0,$$ but quantities like the Riemann-Christoffel and Ricci tensors become unwieldy, and the field equations associated with these more general tensors don't seem to produce any useful physics. Furthermore, the mathematical notation itself is messy, as one must continually work to keep the symmetric and asymmetric pieces distinct from one another throughout the derivations. In my humble opinion, it's all an exercise in futility, and I suspect Einstein and Schrödinger both feared this was indeed the case. But there are many physicists today who remain undaunted by these difficulties. Notable is physics professor Nikodem Poplawksi of the University of New Haven in Connecticut (previously with Indiana University), who has authored many papers involving general affine and asymmetric connections. His work (some of which has been featured on television programs) ranges from conventional to truly interesting, even profound to crackpot. But I don't think there are many who have investigated the connection and its possible relation to gravitation and electromagnetism as much as he has. His many papers, most of which are available on arXiv.org, have the benefit of being accessible to the motivated undergraduate, and I encourage the interested student to look into the ideas of this young New Haven professor. Near the end of his life, when he had completed his failed work on unified field theory, Einstein's personal secretary Helen Dukas had a cake baked for the great scientist in honor of his questionable "achievement." The cake was decorated with the field equations themselves, in red icing. I have a neat photo of the cake laying around here somewhere on my hard drive, and if I can locate the damned thing I'll post it here. In the meantime, you can pop over to the jstor.org academic publishing website where you can read Schrödinger's own papers (in English) on the final affine laws yourself (requires a free subscription).
 And the Survey SAYS — Posted Sunday, February 16 2014 In Arthur Conan Doyle's A Study in Scarlet, Sherlock Holmes admits to an astonished John Watson that he is not aware that the Earth orbits the Sun, nor does he care: "What the deuce is it to me?” he interrupted impatiently; “you say that we go round the Sun. If we went round the Moon it would not make a pennyworth of difference to me or to my work."By way of explanation, Holmes reveals that he is careful not to commit such "useless" information to his brain, lest it interfere with more important things, like the science of deduction. Doyle's novel was written in England in 1886, yet even then it would have been impossible to find someone who thought that the Sun revolves around the Earth. But here in 2014 America such a finding would not be unusual at all. A recent survey conducted by the National Science Foundation showed that 26% of Americans actually believe just that. Similarly, only 39% of Americans believe in the Big Bang, and only 48% express a belief in evolution. And half of Americans believe that antibiotics are effective against viruses. Not surprisingly, Europeans and Asians fared much better in the survey. Any astrophysicist will tell you that the Earth and Sun actually revolve about a common center of mass located very close to the Sun's core. I'll bet nearly 100% of Americans polled would not know that fact, but that is of course quite excusable. But to have 26% of Americans think that the Sun goes round the Earth confirms my theory that a quarter of us are certifiably insane. And they're called Republicans.
 Making the Necessary Adjustments — Posted Saturday, February 15 2014 The first-ever winner of the Hermann Weyl Mathematics Prize (2002), Edward Frenkel is a Russian-born professor of mathematics at UC Berkeley whose short NY Times article this Sunday touches on the subjectivity and objectivity of mathematics. Like Einstein's colleague Kurt Gödel, Frenkel asks whether mathematics simply "is," and is therefore subject only to discovery and analysis by humans, or if it's purely an invention, in which case it is subjective to some extent. Frenkel also addresses a favorite topic of mine, which is the question of what the ultimate reality might be. His article references a recent paper by physicists Silas Beane, Zohreh Davoudi and Martin Savage, which considers the possibility that our universe is actually a computer simulation (I posted the URL for this article two years ago, but you can also link to it from Frenkel's article). Frenkel suggests that if we are indeed living in a simulation then our mathematics might not be inherently absolute but simply a kind of artificial version handed down to us from our simulators, who decided early on that $$1+1 = 2$$ and not $$3$$, for example. The Beane et al. paper actually addresses the possibility that the proposed computer simulators' mathematics is not an invention at all, but a fixed logic like ours from which their simulation is based. What makes the paper interesting is the authors' supposition that, either due to oversight or the limitations of their technology, the simulation is "flawed" at some level, making it possible for us otherwise unwitting humans to discover that we've been had. I once suggested that far more powerful Large Hadron Collider-like machines might someday reveal such flaws — for example, we may not discover a wealth of new physics, particles and forces at all, but a barren desert representing the "pixel" limits of the simulators' impressive but ultimately constrained technology. For some reason, the article reminded me of an Amazing Stories episode from long ago (or something like it*), which featured a man who suddenly realizes that nearly everything he knows is wrong. In the end (if I remember it correctly) he has to be re-educated by his 4-year-old daughter, who reads to him from a child's reading primer. She shows him a picture of a cake, which is labeled "dinosaur" in the book. Needless to say, the man realizes he has a lot to learn (or relearn). If we are ever granted access to the true reality behind our existence, I wonder if it will be like that. But perhaps it will be like this: From The Thirteenth Floor (1999). Simulant (Vincent D'Onofrio) meets simulator (Craig Bierko) with unpleasant results. * A friend has since informed me that the episode was "Wordplay", from the newer (1985) Twilight Zone series.
 Schrödinger on Weyl — Posted Friday, February 7 2014 In 1922, Schrödinger submitted a paper to Zeitschrift für Physik that apparently represents the first attempt to tie Hermann Weyl's 1918 gauge theory to quantum mechanics. Schrödinger's On a Remarkable Property of the Quantum-Orbits of a Single Electron (Zeit. f. Phys. 12 1922, 13) notes that Weyl's proposed metric $$\hat{g}_{\mu\nu} = e^{-k\int \phi_\mu dx^\mu} g_{\mu\nu}$$ explains the energy spectrum of an electron in the hydrogen atom if the term in the exponential is an integral multiple of $$ie/\hbar c$$. Schrödinger adds that It is difficult to believe that this result is merely an accidental mathematical consequence of the quantum conditions, and has no deeper physical meaning.At the same time, he is hesistant (or unable) to expound on the role that Weyl's theory might actually play in quantum theory, which was then still in its infancy. Perhaps Schrödinger's observation that the factor was pure imaginary bothered him, since Einstein's and Weyl's theories were, after all, classical theories. Two pages from Schrödinger's notebook (mid-1925) — the genesis of his wave function concept Note that Schrödinger wrote this paper three years before his own seminal announcement of the wave equation, which for the first time fully explained the strange quantum behavior of hydrogenic electrons that Bohr originally reported on in 1913. Indeed, the then still-emerging quantum theory had not advanced appreciably beyond Bohr's work, and the paper demonstrates the kind of brilliant thinking that was to characterize Schrödinger's contributions to physics, which were finally rewarded when he shared the 1933 Nobel Prize with Dirac. One must also remember that Weyl was essentially trying to eliminate the subjective concept of scale in his theory. This has since led to conformal (scale- or length-independent) cosmological theories, which may or may not have anything to do with the problems of dark matter and dark energy. Certainly, when the Universe was born out of the Big Bang, the concept of scale or length had little if any physical meaning, since spacetime "outside" the Big Bang did not even exist! I have been unable to locate an English translation of Schrödinger's paper and, loath as I am to translate the entire (rather lengthy) article from my original German copy, am presenting here the abbreviated version reproduced in the late Lochlainn O'Raifeartaigh's indispensable 1997 book The Dawning of Gauge Theory. The Alpbach, Austria graves of Erwin and Anny Schrödinger, with perhaps the most profound physics equation of all time (and yes, it beats $$E=mc^2$$)
 Mathematical Justice? — Posted Wednesday, January 29 2014 Here's an interesting paper from last year by Alexander Afriat of the Université de Bretagne Occidentale entitled How Weyl Stumbled Across Electricity While Pursuing Mathematical Justice (see also this paper). The "justice" that Afriat talks about has to do with the equality of direction and distance in Riemannian geometry — vectors can have any direction they want, but their lengths are required to be fixed. This is in direct conflict with quantum mechanics, where the direction of a state vector $$|\psi\rangle$$ can be anything, likewise its length; multiplying the state vector by any real or complex number doesn't change the vector at all — the length of the vector is in fact essentially meaningless. But I think Afriat has confused mathematical justice with mathematical symmetry, which is what I believe Weyl was actually interested in. All of our physics appears to arise from mathematical symmetry, which is essentially the invariance of our theories with respect to coordinate change, linear and rotational translation, time translation and quantum-mechanical gauge or phase translation. These symmetries also give us the conservation laws, like those for energy, linear and angular momentum and electrical charge. To me, these symmetries are the most sublime and beautiful evidence we have that there is a Great Intelligence behind everything. And what is that Great Intelligence, you ask? I haven't the faintest idea. At any rate, Afriat's paper provides lots of neat quotes from Weyl that appear to support the contention that Weyl was indeed on a kind of philosophical or spiritual quest for the truth. I just wouldn't call it "justice."
 Entangled Systems — Posted Wednesday, January 1 2014 Although Hermann Weyl's 1918 theory of conformal invariance failed as a model for the unification of gravity and electromagnetism, it was a phenomenal success when it was applied ten years later to quantum physics. Even today, it seems remarkable that a theory that is invariant with respect to the simple phase transformation $$|\psi\rangle \rightarrow e^{i \theta} |\psi\rangle$$, where $$\theta(x)$$ is an arbitary function of the spacetime coordinates, could explain the conservation of electric charge. Renamed gauge invariance, Weyl's idea is a cornerstone of modern quantum theory. But there is another way of dealing with quantum state vectors that does not involve phase arbitrariness, and that is the density operator approach. If the state vector $$|\psi\rangle$$ contains everything we are allowed to know about the quantum state $$\Psi$$, then surely the slightly more complicated dyad operator $$|\psi\rangle\langle\psi|$$ contains the exact same information. Indeed, it not only contains the same information and is phase invariant, but it also provides a means for understanding random collections of quantum states (or mixed states), a topic usually skipped in undergraduate courses. The density operator formalism also opens the door to the branch of quantum physics known as quantum information theory, in which the mystery of quantum entanglement is explained. Although most of his texts are highly technical and written in German, University of Konstanz professor of physics Jürgen Audretsch's 2007 book Entangled Systems: New Directions in Quantum Physics is written in English at an undergraduate level that is accessible to students of physics, mathematics, chemistry and even computer science. While the book presupposes a beginner's understanding of basic quantum mechanics, it's the most accessible introduction to quantum entanglement, information and entropy I've ever seen (even better than Leonard Susskind's video lectures). The book is a great self-teaching tool, and includes many exercises. (Amazon wants seventy bucks for the book, but you can read a good portion of it for free over at Google Books.) I'm particularly impressed because for years now I've ruminated on the idea that physical reality is fundamentally rooted in the creation, propagation and annihilation of information and that, for whatever reason, the universe itself is somehow tied to the notion of "interestingness." For many years, physicists have asked the question "Why is there something rather than nothing in the universe?" The answer might simply be "Because something is more interesting than nothing." [On the downside (at least for me), Audretsch's book addresses some topics that I wanted to use in a book of my own. How can I write anything when other people keep beating me to it?] Anyway, Happy New Year!
 Is We Evolving? — Posted Tuesday, December 31 2013 To me, the most beautiful and profound aspect of physical law is that Nature invariably strives to be as efficient as possible. This is succinctly demonstrated by the fact that the mathematical quantity known as the action, which Nature is somehow intimately familiar with, is invariably extremalized (and usually minimized) in all physical interactions, from the very small (quantum physics) to the very large (gravitation). Since action is always expressed in units of momentum-displacement ($$p \cdot x$$) or energy-time ($$E \cdot t$$), Nature evidently likes to do things using the the least momentum along the shortest path, or the least energy in the least time. Indeed, the principle of least action as first developed in the 18th century was viewed as the best scientific evidence for the existence of God. When biologist Charles Darwin visited the Galápagos Islands in the 1830s he noted that there was a diverse variety of birds (notably finches) whose food preferences depended to a great extent on the size and shape of their beaks. Those who fed primarily on small seeds had small beaks, while large-beaked birds ate larger, harder seeds. Some birds, which fed mostly on hard-to-reach seeds (like those in cactus), had long, narrow beaks. While not a physicist or mathematician, Darwin saw this diversity as evidence of an evolutionary tendency that trends toward efficiency — for example, birds who feed on small seeds do not carry around heavy beaks, as this would be a waste of energy. I doubt very much if Darwin was ever aware of the principle of least action, but if he was he would probably have viewed evolution as a good example of it. True, Darwin likely saw evolution as a slow process, taking many generations of animals over huge time periods, but given the fact that environmental stressors such as climate change, disease, inter-species competition and predator population change all vary slowly with time, Darwin probably considered evolution to be a very slow but efficient process overall. [Side story — in 1971 I took an undergraduate class in biochemistry, and the professor calculated the energy efficiency of the electron-transport mechanism at 67%. He compared this with the efficiency of a car engine, which at best is only around 25%. Nature wins this contest, hands down.] But for many people at the time (and even today), Darwin's On the Origin of Species, written in 1859, went too far. It implied that humans themselves had physically evolved over time, in seeming contradiction to the Bible. Most people in Darwin's time believed that God had created man roughly 6,000 years earlier, and that man and all the other creatures were created by God in the exact same form that we see them today. Worse, Darwin hypothesized that humans had evolved from earlier primate forms related to monkeys and apes. Human ego rejected the notion that people were descended from monkeys — "If monkeys had evolved into people", the saying went, "then why are there still monkeys walking around?" Although we now know from sound genetic evidence that the genetic lines of apes/monkeys and man diverged some 5 million years ago (which is why monkeys are still around) and that physical evolution can take place startlingly fast, it should not be surprising that the theory of evolution is still being questioned today. After all, Darwin's work is rarely actually read by anyone (much less studied), and biblical genealogy from Adam and Eve on down to Jesus supports the notion that only about 6,000 years have elapsed since time began. To a person of strong Christian, Jewish or Muslim faith, evolution is not only puzzling but contradictory to religious belief as well. But what is surprising, however, is that in spite of new, ongoing and profound fossil and genetic discoveries, evolution is being increasingly rejected by people of faith in America today. Some 1,983 American adults from all 50 states were polled in a new study conducted by the Pew Research Center regarding their views on human evolution. Although 60% of Americans indicated they believe human evolution has occurred, fully 33% believed that humans have remained physically the same since they were first supposedly created by God. While marginally more Americans believe in evolution today, the percentage of white, evangelical Protestants who reject evolution has increased. Indeed, nearly two-thirds of white evangelicals polled do not believe in evolution: I don't know about you, but this scares the crap out of me. When I was teaching I used to tell my students that there are no laws in science, only theories. Electromagnetism, quantum mechanics, gravitation, chemistry, aerodynamics and germ theory are and will always be just theories, no matter how much evidence is compiled to support them — that's what the scientific method is all about. People universally believe these theories, but 33% hold that evolution is "just a theory." And I suspect that a sizable subset of that 33% is also wondering why monkeys are still walking around if there's anything at all to Darwin's ideas. Study after study now confirm that the country is becoming ever more politically and culturally polarized. I keep asking myself how otherwise intelligent people can reject facts, reason and empirical evidence in favor of dogmatic allegiance to illogical, self-contradictory religious nonsense — people who can drive a car, hold a job, program a DVR, balance a checkbook and even teach at university, but choose willful ignorance over a life of rational thought. I keep asking myself what they're afraid of, what is it that they find so frightening that they would abandon the thinking, reasoning brain that God gave them. I don't have any answers, and I have no idea what's going on around here. I can only hope that 2014 turns out better than 2013. Those who can make you believe in absurdities can make you commit atrocities. — Voltaire