AfterMath

Perhaps you've seen this trick: You have a penny-size round hole in a piece of paper and are asked to push a quarter coin through the hole without tearing the paper. How is it possible? I did it, and it's easy. It's based on the difference between two dimensions and three dimensions. Who only knows what the 4-dimensional spacetime we live in is capable of when it's warped (like in gravity)?

Here's how the penny trick is done.

Now that you know how the trick is done, show that a coin of diameter \(D\) can pass a coin of diameter \(\sqrt{2}\,D\) without tearing the paper.

Remarkably, the trick is similar to that of geometric angular deficit and its relationship to gravity. If someone eats a slice out of a flat (2-D) pizza, then joining the two open pieces requires bending the pizza into a cone, which is a 3-D object. As far as we know, only gravity can warp space like this!

"Why are you f**king with our lives?" — Vincent D'Onofrio's simulated character in The Thirteenth Floor

I watched the vastly underrated 1999 sci-fi film The Thirteen Floor again this morning. What draws me to it every time is some new feeling that its premise reflects a reality we're actually living in, not exactly a simulation (like in the movie) but something very close to it.

Forgive me if I sound nuts (and I probably am), but when Jesus Christ arrived and then somehow magically disappeared from the world seems similar to what is portrayed in the film—that there is a higher reality (perhaps even higher multiple realities, as the movie suggests) that exists beyond our knowledge or understanding, a quantum reality of some kind that we cannot explain. In the movie the higher (or highest) reality is the true reality that we all hope to comprehend.

Consider the video games Grand Theft Auto and L.A. Noire (which is far more realistic). As a video game player you really don't care about the characters who suffer and die because you know it's just a game. But in a more realistic simulation you might actually care to some extent, and you might attempt to intervene. That's beginning to reflect how I see the world today.

Have you ever wondered why and how we seem to be the only sentient players in the universe? That's the so-called Fermi Paradox, and it's very possible that we are.

Those of you who love the Japanese puzzle Sudoku will be glad to know that there's a limited number of possible unique solutions, calculated to be only 6,670,903,752,021,072,936,960 (about \(6.67 \times 10^{21}\)). I can solve an expert or extreme Sudoku puzzle in about 20 minutes 95% of the time without guessing, but my hopes of memorizing all the combinations to beat the puzzle are likely dashed.

Alas, by comparison the number of possible \(3\times3\times 3\) Rubik's cube combinations is only a "small" fraction of this number (43,252,003,274,489,856,000), but after learning the solution trick many years ago I can no longer solve the damned thing. Amazingly, there's a number of people who can solve a random Rubik's cube behind their backs in a matter of seconds. A pox upon them.

So why aren't such people working on more important matters, like climate change, nuclear fusion, cancer and quantum gravity?

Consider one of the two most successful equations of fundamental physics (the other being in quantum field theory): $$ R^{\mu\nu} - \frac{1}{2}\, g^{\mu\nu} R + g^{\mu\nu} \Lambda = \frac{8 \pi G}{c^4}\, T^{\mu\nu} \tag{1} $$ which represents Einstein's gravitational field equations (both sides vanish under covariant divergence, which kinda implies a conservation law, but that law doesn't mean that mass-energy is conserved). The \(\Lambda\) term is the famous cosmological constant, and it's believed to be a true constant throughout all time and space (if it isn't, then (1) is invalid). \(\Lambda\) appears to be positive, but exceedingly small and non-zero, and it's responsible for the accelerating expansion rate of the universe, which I touched on in my previous post. It sorta represents the energy of the vacuum of space, rather like the zero-point energy of spacetime. It exerts a repulsive gravitational effect, so as the universe gets bigger the universe tends to expand at an increasing rate.

If \(\Lambda\) is a true constant, then the universe will eventually expand at the speed of light in the distant future. It would also rupture all matter down to the sub-atomic level, resulting in a universe we can't even imagine. But some astrophysicists are beginning to think that the cosmological constant is getting weaker with time, which would presumably spare the universe from complete destruction.

Other astrophysicists believe that \(\Lambda\) is identically zero, and that the apparent acceleration of universal expansion is wrong. I also suggested something along that line in my previous post. If true, then the universe would continue to expand, perhaps even linearly, but the end result would still be an enormous void of nothingness—a very depressing thought.

There are also some scientists who think that the cosmological constant might flip from positive to negative at some time, meaning that universal expansion would ultimately halt and eventually collapse back in on itself. Recent Nobel prize winner Roger Penrose's theory of an eternally expanding and collapsing (cyclical) universe might be what's in store for the world.

Quantum field theory has its own version of the possible fate of the universe. It doesn't involve a cosmological constant, but considers whether or not the vacuum of the universe is in a ground state or an excited state. If the latter, then it's just a matter of time before the vacuum collapses down to the true ground state, with an unbelievably enormous release of energy. That energy would totally rip apart the universe down to the sub-atomic level, annihilating everything—yet another depressing thought.

I once had an idea for a short story in which a slightly less-than-omnipotent God creates billions of sentient human beings (at present, we're up to about 110 billion humans who have ever existed) for the purpose of helping Him find a solution to the ultimate annihilation of the slightly less-than-perfect universe He created and then got stuck with. When I suggested this idea many years ago to a church pastor, he said it sounded like blasphemy, so I never pursued the story. Still, I wonder.

I want to do a little opining here regarding the so-called Hubble tension that keeps popping up in the science news. I'll assume that all the readers of this site are familiar with it, since over the years I've talked about it many times.

Let's say that you want to measure the width of a room. You have two measuring devices; one is a 25-foot metal tape measure, and the other is a 1 inch-long measuring stick. Which would you choose? Obviously, using the stick is going to introduce some systematic error, since you'll have to overlap each 1-inch measurement with another, and there's bound to be some error in each measurement. For a 10-foot wide room, you'll need to do 120 measurements with the stick, and the overall error will likely be cumulative. By comparison, using the tape measure requires a single measurement, and the error will be negligible.

I liken the stick approach to the cosmic distance ladder (CDL) method, which starts out using relatively accurate stellar parallax distance measurements, followed by variable Cepheid star observations (and there's several types of Cepheid stars), followed by Type 1a supernova observations. All three measurements have their own errors, and stitching them together creates an unavoidable large measurement error. Furthermore, absolute Type 1a supernova magnitudes (assumed to be identical) can nevertheless vary according to issues like stellar metallicity, intergalactic dust and other environmental factors.

Meanwhile, I liken the tape measure method to distance calculations based on the cosmic microwave background (CMB). The approach is based on Einstein's gravity theory, extended throughout the universe's 13.8 billion year lifetime, and it provides a single measurement technique lacking the cumulative error issue.

Here is a graph of the data derived from the distance ladder approach. The \(z\) abscissa denotes increasing distance from us in certain units, while the ordinate is related to observed stellar magnitude. The trend away from a constant expansion rate (fainter dotted line, while the upper very faint line appears to be a least-squares fit of the data) at high \(z\) is apparent, but the high degree of scattering detracts from the claim that the expansion rate is really accelerating. You decide if this claim is strictly valid.

Why is all this a big deal? Because estimations of the rate of expansion of the universe vary with respect to which observational method is used. Currently, the calculated rate of expansion differs by around 8% depending on which of the two methods is used. It would be really nice to know which approach is the correct one.

But so what? The universe is currently expanding at around 22 kilometers per second per lightyear (or 70 km/s per megaparsec), and by itself that may be interesting but is otherwise of no greater scientific value. However, since 1998 the cosmic distance ladder method seems to indicate that the rate of expansion is actually accelerating, as implied by the graph. If true, it raises the question of just exactly what is causing the apparent acceleration and what the ultimate fate of the universe will be if it continues.

As the linked article notes, recent data from the James Webb Space Telescope is not making the Hubble tension situation any clearer. As the article also claims, something's wrong with either the CMB or CDL approach (and possibly both are wrong).

Tyranny is government for people who want things done NOW.
— Peter Dinklage, How to Become a Tyrant

"Get out and vote, just this time, because you won't have to do it any more. Four more years, and you know what? It'll be fixed, it'll be fine, you won't have to vote any more, my beautiful Christians!" — Donald Trump, July 29, 2024

I don't agree with everything that Nova Scotia's Anglican priest Edward Trevors has to say, but he is spot on regarding the irrationality of Christian support of the wannabe Hitlerian maniac Donald Trump. In Trevors' latest video, he suggests that conservative MAGA Christians are simply impatient with the return (advent) of Jesus Christ so, by forcing Trump onto the world stage as the Antichrist, they believe Trump will prompt Christ's return within their lifetimes. That also means they would not have to die, so their irrationality may boil down to the fear of death.

I generally agree with this, although I also believe MAGA Christians are just plain ignorant, with many being simply stupid. Ignorance is not a vice, as it can be fixed by education and enlightenment, but stupidity is forever. It's also akin to willful ignorance, which explains why Trump supporters do not believe in facts, hard evidence and truth when it conflicts with their adoration and worship of Trump.

It's a short video, and well worth watching.

In the Old Testament Book of 1 Kings, the value of the transcendental number \(\pi\) is given as 3. That was good enough when constructing water wells, I suppose, but later it was expressed more accurately as 22/7 and 355/113. Today we know \(\pi\) to trillions of decimal places. In principle, the number can be calculatd to any number of digits.

Similarly, in 1928 the gyromagnetic ratio of the electron was thought to be exactly 2, until quantum field theory and experiment showed it was closer to 2.002. Now it's known to twelve decimal places, one of the most precise predictions of quantum theory.

Then came the discrepancy between the calculated and experimental values of the magnetic moment of the muon, a heavier but otherwise identical twin of the electron. For several decades its observed magnetic moment differed slightly but significantly from the calculated value (admittedly a very difficult computational task), giving hope that new physics was involved. Alas, it turns out that the calculated value was off a bit, and now revised calculations agree with the observed value, thus erasing the discrepancy.

One of the greatest remaining physical discrepancies today is the so-called Hubble tension, which arises due to conflicting measurements of the expnsion rate of the universe. One measurement is based on observations of the cosmic microwave background, giving a value of 67 in standard units, while the other (based on stellar distance measurements) is 73. Both measurements are precise, with non-overlapping error bars, so something is definitely amiss. My guess is that the distance method is off, but a usable value of 70 can be used until the problem's resolved.

Lastly there's the dark matter problem, which is not so much a measurement discrepancy problem but a fall-back option based on ignorance. As far as anyone knows, dark matter particles don't interact with anything—including themselves—but only feel the effects of gravity. Since particle physics relies on the observed effects of interactions (particle collisions), some physicists now believe that dark matter may never be detected, at least based on standard interaction methods. If so, then dark matter theory must be relegated to the domain of religious faith, not science.

But wait! Perhaps dark matter can "interact" with a black hole, assuming it's on a collision course with the hole. It's not really an interaction, it's just that nothing can get out of a black hole once it falls in. In her latest video, German physicist Sabine Hossenfelder believes that the unexplainably fast growth of galactic supermassive black holes might be due to their gobbling up dark matter. But if so, I assert, that would deplete the supposed dark matter halos around galaxies, rendering distant stellar rotation curves subject to plain old Newtonian physics. Instead we observe flat stellar rotation velocities, assumed to be the effect of dark matter. So what gives?

Today marks the 5th anniversary of my dear wife Munira's passing. It was also on Sunday, July 28, and I'm still deeply mourning her death.

She bravely struggled with metastatic breast cancer, diabetes amd a number of other illnesses, but she never complained, accepting the illnesses through her faith in Jesus Christ. Throughout our marriage and until her death I was something of a Christian skeptic, interested primarily in the history of the early church.

Munira was a Coptic Orthodox Christian, having been baptized in her native city of Cairo, Egypt. It's an ancient church that arose in the city of Alexandria, Egypt, founded by the evangelist Saint Mark, the author of what is considered to be the first Gospel to be penned, written around 65-70 AD. The Coptic Orthodox Church represents some ten million Copts in Egypt today, with many millions more around the world.

Munira and I along with our two sons were infrequent attendees at the Coptic Church in Los Angles, where our sons were baptized. She came to America in August 1970 with her brother and attended Saint Mark's Church in West Los Angeles. She arrived with a Bachelor's Degree in Chemical Engineering, later graduating with a Master's Degree with honors in the same field.

After her passing, I joined the Coptic Church in 2019 where I was also baptized, later becoming a deacon. Admittedly, I originally joined out of reverence for my wife's memory, but today I see the Church as the only true Christian Church in existence. Today I view Jesus Christ as my only hope for salvation after seven decades of sin, self-centered interest, ignorance and skepticism.

At 75, I hope to join my wife someday, which will likely not be too far off. I suffer from severe Meniere's disease and major depressive disorder, being mentally cogent less and less each month. But like Munira I view my problems as gifts from God, since suffering is a great way to focus one's mind on things that are of true importance. After all, the Church teaches us that we are all just sojourners in this world, awaiting a new and better life with our Lord and Savior Jesus Christ.

For historical interest I'm posting a translation of the January 13, 1916 paper that German physicist Karl Schwarzschild submitted for publication. It was an exact solution of Einstein's general theory of gravitation that was published only two months before, and is still of great theoretical cosmological importance.

Schwarzschild wrote the paper while serving as a German soldier on the front in World War I. Suffering from an autoimmune disease, he left the German army in March 1916 and died of the disease in May of that year. He was only 42.

While his paper demonstrates something of a hodgepodge of mathematical thinking, he arrived at the correct answer. While reading the paper shortly after Schwarzschild's death, Einstein was amazed at how quickly his fellow German physicist had arrived at an exact solution.

Cosmologists are at a loss to explain several strange observations of stellar galactic rotation curves, gravitational lensing and galactic clustering without invoking the existence of dark matter, a hypothetical particle that only interacts with gravity, making observable methods based on light, electromagnetic and particle-particle interactions inapplicable. Dark matter has never been detected despite decades of heroic and costly experimental efforts, and the dark matter problem remains as perhaps the most perplexing problem in physics today.

Current and past efforts to detect hypothetical dark matter particles have relied heavily on the use of vats of pure liquid xenon (an inert, stable gaseous element) in the hope that a passing dark matter particle would interact with a xenon nucleus, resulting in a detectable ionization or scintillation event. Despite several decades of research using this technique, no discernable particles have been detected— until this month. However, this turns out to be bad news for dark matter.

It has always been generally surmised that neutrino interactions with liquid xenon would be non-existent, or could be eliminated as stray events. Unfortunately, the results of the July 10 paper show that neutrinos can and do interact, frustrating the direct detection of uniquely dark matter particles and making such detection a seeming impossibility.

If neutrino-xenon interactions cannot be reliably accounted for and eliminated from detection data, it would seem that xenon and other heavy-nucleus techniques now have to be abandoned. This is too darned bad, because I estimate that hundreds of millions or even billions of dollars have been expended to date on xenon detector experiments. Whatever new techniques are possible is now a debatable question.

This leaves open even greater theoretical efforts to explain the cosmological effects of dark matter by revising Einstein's general theory of relativity (gravitation). Numerous theories have been proposed to date that can account for most of the effects of hypothetical dark matter, and it's hoped that some new Einstein will come along to find the theory that works.

Thanks to Donald Trump, you could soon be scrubbing your own damned toilets, mowing your own damned lawns and doing your own childcare and housekeeping, America.

At the Republican Convention last week, Trump supporters rallied behind the former president's claim that immigrants are "poisoning the blood" of Americans, requiring the mass deportation and/or incarceration of some 20 million undocumented or casual residents, with "Mass Deportation Now" (MDN) signs displayed everywhere at the Convention.


The immigrant and race issues are sure to be major points of contention in the GOP's efforts to reelect Trump, particularly now with the pending Democratic nomination of Vice President Kamala Harris for the Executive Office, whose black-Asian-female background will likely induce rabid conservative accusations of the unwanted "other," despite her being an American.

The rise of Donald Trump (the original surname is from the German Trumpe) is eerily and sadly reminiscent of Adolf Hitler's rise in early 1930s Germany, which resulted in mass deportations (later followed by the murder of six million Jews and millions of other unwanted "others," including Poles, Gypsies, the mentally impaired and gays), the subjugation of German women as Nazi baby-makers, and Hitler's "Make Germany Great Again" appeal to the general German public (not an official German slogan, but applicable).

If you want the likes of Russia's Vladimir Putin, China's Xi Jinping, North Korea's Kim Jong Un and other authoritarian dictators invited to the White House by Trump to sleep in the Lincoln Bedroom and perhaps establish permanent military bases in America, you'll be voting for Trump. Otherwise, it's got to be Kamala Harris for President.

Our three primary equations of physical reality

Over the years, experts have contributed papers and videos on the Internet far surpassing anything I've tried to do. One such expert is UC Santa Barbara physicist Richard Behiel, whose insightful, educational and entertaining YouTube videos go from beginner to advanced. One of the better ones is his talk on Deriving the Dirac relativistic electron equation.

As Behiel notes, the Dirac equation automatically produces antimatter and particle spin, while also introducing the spinor, a mathematical object which he accurately describes as a "horror." The spinor is something I never truly understood, despite my best efforts (see here and here). However, the spinor describes all of the ordinary matter in the universe, so it can't be ignored.

But Behiel also notes that the equation will change your life, as it did mine so many years ago. Enjoy.

A short list of papers I have written over the years can be found here. A complete list can be found at my old website weylmann.com. If interested, read or download while you can, because if Donald Trump wins in November I'm shutting down for good. I have nothing much left to say anyway.

Today, the New York Times Editorial Board recommended that President Biden leave the 2024 presidential race for the good of the country and hand the reins over to a more capable candidate who has a better chance of defeating Donald Trump (the Antichrist himself) in November.

As the article notes, Biden is simply not the man he was in 2020 when he defeated Trump in what was even then a tight contest. Biden now seems incapable of understanding the enormous stakes involved, which is nothing less than a choice between democracy or authoritarian dictatorship.

Biden comes across now as a doddering old man pushing the bounds of senility. Despite colleagues' urging to go after Trump's uncountable lies, falsehoods and venal crimes in last night's presidential "debate," Biden seemed to be completely unaware of Trump's evil history and current plans to remold America in his own evil image. Biden's overall performance was a disaster not only for the country but for enlightened Christian progressivism itself.

The dark matter/modified gravity argument goes on, and currently it looks like modified gravity (MOG) theory is winning out.

The flat stellar galactic curve problem has been known since the 1970s, when the existence of hypothetical dark matter (DM) appeared to be the only logical explanation. DM was presumed to surround galaxies in a roughly spherical halo, extending well beyond the influence of classical (Einsteinian) gravity. It was always assumed, however, that the density of the DM halo would diminish with distance from the galactic center, eventually going to zero at a distance of maybe a few hundred thousand lightyears. Any greater distance didn't make any sense, otherwise DM would be essentially omnipresent over all the universe with little density change or clumping occurring.

Just how far out the effect of DM on stellar velocities might persist has been an open question. But now recent observations appear to show that flat stellar velocity curves extend well beyond the possible range of DM, and the only reasonable answer lies in MOG theory. I first read about this issue at Triton Station, a website created and maintained by Case Western Reserve University astronomy professor Stacy McGaugh, one of the principle investigators. The paper reporting their work can be found here.

Noted German physicist Sabine Hossenfelder has frequently commented on the problem, switching sides from DM to MOG just as often. Now she's back on the MOG side:

U.S. Surgeon General Dr. Vivek Murthy has now declared gun violence to be a public health crisis, with mass shootings accounting for the greatest percentage of deaths in children, above disease, accidents and other forms of murder.

But former president Donald Trump has a simple GOP-approved solution: If reelected, he'll get rid of Dr. Murthy (like he did in 2017) and encourage Americans to buy more guns.

Meanwhile, on a slightly more positive (if tragic) note, children are now being trained how to stop a fellow gun-shot schoolmate from bleeding out.

No doubt Mr. Trump will end such training as well, since it would increase public school funding costs.

The Empire Theater in Quincy, Illinois opened its doors in 1893, where it hosted vaudeville acts and (later) silent films. It was briefly renamed the Hippodrome (as depicted in the photo), but was subsequently renamed the Empire.

For many years on Friday evenings the Empire hosted senior graduation ceremonies for nearby Quincy High School (my father graduated from the school 100 years ago this coming week). Two years later, in 1926, the theater closed and remained vacant for many years during the Depression. The building still stands today at 111 Eighth Street in Quincy, although it has been converted into an apartment building.

My father used to tell me about the silent films he saw at the Empire, especially the Phantom of the Opera from 1925. The scene where Mary Philbin removes the mask from Lon Chaney Sr. scared the wits out of Dad, although seeing it as a child many years later on the old 1960 television show Silents Please didn't disturb me at all. Today's youth I'm sure are even more jaded.

Dad died in 1981, but he lives on at Classmates.com, where I entered his name years ago just to see who might have still been alive at the time. I got one notable response from a fellow 1924 graduate, but she died shortly thereafter. They're all gone now.

I have all of Dad's high school yearbooks except the one for 1923, which I'm still looking for. Here he is as a proud high school graduate from June 1924. God willing, I will see you again someday, Dad.



PS: Here's a photo dated March 1920 that I'd never seen before, found at the Quincy Historical Society website. Dad joined the school band and orchestra as a sophomore in 1919, when the band was founded. Dad's in the first row with his trumpet (which I still have), second to the right of the xylophone player:

"Welcome to Nazi.USA". Ironically, a German protest poster.

Today is the 80th anniversary of D-Day and the beginning of the liberation of Europe from Nazi Germany. A year later World War II ended, with over 400,000 American troops dead.

Every anniversary, battle films and documentaries are aired on television, all extolling the price America paid for freedom from the tyranny of ruthless authoritarian dictators. It's ironic that America is now becoming a dictator nation itself, with Donald Trump and his insane MAGA adherents plotting the elimination of democracy and the freedoms those 400,000 troops died for.

In 2018 Trump infamously stated that Americans who died in wars were "losers" and "suckers" for getting killed. His worshipers turned a deaf ear.

Tragically, it appears that those 400,000 men and women died for nothing.

I remember when scientific calculators first came out, around 1970. They had basic trigonometric function capability, and I can recall this joke that circulated around that time:

TRIGONOMETRY 101
Problem: How to calculate the tangent of 37.1 degrees:
1. Set calculator to degree mode.
2. Enter 37.1 and push the TAN key.
3. Read the answer: 0.75629
4. Congratulations! You now know basic trigonometry!

In his latest video, popular science and technology geek Joe Scott asks the question "Why learn anything any more?" He references the cultish 2006 Mike Judge film Idiocracy, in which an average-joe Army guy gets transported 500 years into the future via experimental military cryotechnology. Because America has grown insipidly stupid and lazy over the intervening years, he finds himself the smartest man in the world. It's an okay film, with funny if vulgar asides like the Carl's Jr. ad "Fuck You, I'm Eating," but otherwise it's more like a documentary of a future America under the likes of Donald Trump.

But most of Scott's video concerns the potential impact of near-future humanoid robots running under artificial intelligence, which pretty much renders humans obsolete and irrelevant. Scott then points out how automobiles replaced horses from 1907 to 1922, prompting his observation that "We're the horses now."

A few months ago University of Chicago computer scientist Geoffrey Hinton gave an impassioned talk entitled "Are We Doomed?", in which he spoke about the impending perils of climate change, artificial intelligence, resource depletion, nuclear annihilation and related end-of-times disasters facing a modern world of over eight billion people. Typical of a New Yorker article, it's damned overly long but can be succinctly summarized by the writer's admission "I'm glad I'm 76," meaning he doesn't expect to live long enough to see what's going to happen.

I'm 75, and my dear brother-in-law in Christ is 76, and as Christians we both are glad we're likely not going to see what's coming, which doesn't bode well for the human race or the natural world.

In the Gospel of Mark 16:28, Christ states what has been called by many (including C.S. Lewis) the most embarrassing verse in the New Testament:

Truly I tell you, there are some standing here who will not taste death before they see the Son of Man coming into His Kingdom.

Lewis, Bart D. Erhman and many other noted Christian writers have thus claimed that Jesus Christ by His own account was a failed apocalypticist, who seemed to be predicting the imminent End of the World in His own time. But shortly after Christ's claim, His apostles Peter, James and John personally witnessed the Transfiguration of Christ, and so they did indeed see the coming of the Son of Man into His Kingdom.

From a purely practical point of view, the End of the World occurs when you die. In that sense, you're always living in the Last Days, and it bears to emphasize Christ's warning to always be on the watch, for you never know when the end is coming (Matthew 24:42).

Now take a close look around you: do you really believe this world is sustainable in its present form? Oceanic fisheries are collapsing, ancient permafrost is melting at an alarming rate, releasing methane gas (one of the worst greenhouse gases), while fossil fuel production and utilization continues unabated, pushing more and more carbon dioxide into the atmosphere, while eight billion humans, all striving for developed-world living standards, will do anything to own cars, live in air-conditioned homes and feed and raise upwardly-mobile, social media-addicted children regardless of what's happening around them.

So read the article, and give it some thought.

The famous but now retired tennis pro Roger Federer (I loved his one-handed backhand stroke), hails from Basel, Switzerland, the home of an even greater pro (in mathematics), Leonhard Euler (1707-1783). It was Euler who, in 1734, solved what came to be known as the Basel Problem, a math puzzle that had stumped many notable mathematicians of the day. It involves the deceptively simple infinite series $$ S = \sum_{n=1}^\infty \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} \ldots $$ Standard series analysis says it must converge to a finite number, but the series converges very slowly, which for many years complicated its solution. Euler finally proved that \(S = \pi^2/6\), a discovery that made him famous.

There are numerous solutions to the problem known today, the simplest involving the trigonometric function \(\sin(x)\), which can easily be proved by any curious high school student. What I find interesting, however, is that the trigonometric function \(\tan(x) = \sin(x)/\cos(x)\) can also be used to derive a solution using the exact same approach that Euler used. Out of sheer boredom today, I tried it out and got the unambiguous (but ridiculously wrong) answer \(S = - \pi^2/3\). Where did I go wrong, and where the heck did that minus sign come from? But admittedly, I'm no Euler.

It reminds me of a nearly equally famous series problem from quantum string theory, which is $$ S = \sum_{n=1}^\infty n = 1 + 2 + 3 + 4 \ldots = - \frac{1}{12} $$ This is also clearly nonsense, but the \(-1/12\) factor is intriguing because it leads to all kinds of interesting and mathematically useful aspects of string theory. If you dig out my post of May 30, 2016 I show that the correct answer is actually \(S = \infty - 1/12\), which at first glance is hardly interesting at all.

Truly historic news for America, but will it make a difference? Trump may still win in November, and then he'll proceed with the executions he's promised.

My German is a little rusty, but yesterday former president Donald Trump promised a Geeintes Reich (unified rulership) in America if he wins reelection in November. Dear God, he actually used the word "Reich" in his social media blog, not rulership. Worse, he also suggested that current president Joe Biden would be executed under a 2025 Trump presidency.

Wake up, you stupid, ignorant Americans, Trump is either the Antichrist or the next Adolf Hitler!!!!!

German physicist Sabine Hossenfelder's latest video is so beautiful that I'm showing it below. She goes on at some length to describe how Einstein's attempts at unified field theory were doomed to fail, but also notes that his notion that matter somehow springs out of spacetime itself remains viable, although no one knows how to prove it.

In one of his first attempts at a unified theory, Einstein came up with this equation

which caught my eye immediately because his friend and colleague Hermann Weyl had derived it earlier in 1918, also in an attempt at a unified theory.

Hossenfelder points out that Einstein's efforts were purely classical and could never have resulted in the equations of quantum theory, the most beautiful (in my opinion) being the relativistic electron equation, derived by Paul Dirac in 1928 at the tender age of 25. Expressed in profoundly simple and beautiful relativistic notation, the equation led to enormous advances in modern quantum theory and is the cornerstone of theoretical physics today.

I recall a propaganda art exhibit I went to in Los Angeles around 1970 during the height of the Vietnam War (it may have been the Los Angeles Museum of Art) that featured large prints of Vietnam War-related posters, many produced by Russian, Red Chinese or North Korean artists. One in particular caught my attention, and for years I've looked in vain for it. It depicted a stern-faced Viet Cong soldier loading a shell into the breech of an artillery piece, with a thought balloon above his head in which he is imagining his dead wife and child. Translated, the title of the poster was "Loading His Hate Into The Gun." Oddly enough, Vietnam is now an American political and economic ally, and i've never seen any hint of revenge or reprisal since the war ended in 1975, despite America having killed two million Vietnamese in the war.

Today's online Aeon Magazine features an article entitled On Jewish Revenge. In this long article, the writer delves into the nature of hatred and revenge, and not just that involving Jews, despite the fact that Jews have been persecuted since time immemorial, likely because ancient Jews are seen as the murderers of Jesus Christ, who is the Son of God in the Christian faith.

Of course, the writer recalls the Nazi atrocities against European Jews, the horrific October 7, 2023 Hamas attack as well as other more historic pogroms against the Jews. But ironically he does not mention Psalm 137 (136 in my church) from the Old Testament, which was almost certainly written by Israelite exiles in Babylonia following that country's total destruction of Jerusalem and the beloved First Temple in 586 BC. In the Psalm, the unknown Jewish scribe laments the loss of Jerusalem's religious heritage and asks how on earth are the Jews going to continue worshiping their God while exiled in a foreign country.

But the scribe goes further—he damns the Edomites (a country just to the east of Jerusalem, which supported Babylonia's siege and who tormented the Jewish exiles as they were force-marched to the city of Babylon 800 miles away), and with particular anger he expresses a desire that Babylon's infants should be taken and "dashed against the rock," a clear expression of the desire for revenge.

In the Aeon article, the writer also talks about the nature of revenge, which is a past, present and future way of thinking and dealing with past offenses, but he also talks about the nature of forgiveness, in which past offenses are largely forgotten. Christian tradition favors forgiveness, but given particularly horrific offenses it often boggles the mind that forgiveness should be offered, especially when those offenses go unpunished.

Many of the top Nazis were punished, with the guilty being either imprisoned or hanged following the Nuremberg trials, but it is hard to see how members of Hamas (which were responsible for the October 7 attack) might be brought to justice, much less trial. Prime Minister Benyamin Netanyahu is determined to exterminate Hamas, and perhaps he has a right to feel that way, but Palestinian civilians are being lumped into the crimes of Hamas, which is completely unjustified. There's also the political issue of wiping Gaza off the map (along with everyone there) so that Israel can stake a "Jordan River to the Sea" claim on all land between the river and the sea. This would undoubtedly be done purely out of revenge (and worse, profit).

Sadly, I see no just end to this tragedy, and I fear Israel will ultimately become a pariah country that the United States, unfortunately, will be forced to support for political puposes. God help us.

Today I came across this new video entitled An Elegant Structural Overview of Modern Theoretical Physics by a guy named Preston O'Neal, who has a number of videos available on YouTube. I never heard of him, but this video is a great overview of everything you'd ever want to know about modern physics. It's over 2 hours long but expertly presented, and while it starts out easy it quickly becomes demanding.

O'Neal starts by presenting electrodynamics in three ways, from high school vectors and basic calculus to differential forms and geometric algebra. I've studied geometric algebra and think it's a far better, more fundamental and sadly much overlooked alternative to ordinary vector analysis, but I never caught on to the appeal of differential forms. For example, one can summarize all of electrodynamics with the simple equation \(dF = J\), but how in the heck can anyone calculate anything with that?

Lately I've been getting lots of email (good and bad) from a paper I wrote some years ago on differential forms, in which I posited that (with one exception) it's a waste of time for physics students to learn. Most of the responses I've gotten have been negative, asserting that differential forms lie very near to ultimate reality and should be appreciated for their beauty and elegant simplicity. Well, I still disagree.

O'Neal's video goes much deeper and should be watched to the end regardless of its mounting complexity. And while he folds into his talk the subject of Einstein's gravity, it's not an attempt at a unified theory.

During a rally speech in Pennsylvania yesterday, former president Donald Trump praised "the late, great" Hannibal Lecter—the antagonist of the 1991 movie The Silence of the Lambs—as a "wonderful man" along with other accolades. Trump seems to be unaware that Lecter is not only a fictional character but a sadistic, cannibalistic serial murderer. Trump's rally speech, which also condemned migrants being allowed into the country, resulted in screams of rabid adoration from the 80,000 rally attendees.

Trump's support base is comprised of corporatists who hope to profit from Trump's reelection plans along with white evangelical voters who view Trump as God's "Chosen One." The motivation of corporatists is understandable, as they don't give a damn about who Trump is or what he stands for as long as he makes them money. The motivation of evangelicals, however, is much more sinister. If they believe he's the next Messiah then they're just plain stupid (which is dangerous), but I fear that they also believe Trump might very well be the Antichrist. Why would they hope for the Antichrist? Because they believe the evil he would do (including instigating a global thermonuclear holocaust) will force the true Messiah—Jesus Christ—to return. This belief is insane, because Christ will return when He gets good and ready to return, and not because of the hopes of America's stupid white evangelicals.

In Romans 3:7-9, the Apostle Paul wrote

Someone might argue, "If my falsehood enhances God's truthfulness and so increases His glory, why am I still condemned as a sinner?" Why not say — as some slanderously claim that we say — "Let us do evil that good may result"? Their condemnation is just!

What Paul meant is that doing evil under any circumstances is wrong, and whatever supposed good it does will not enhance their standing as sinners in God's eyes. Evangelicals seem to believe that they will be rewarded, not punished, for telling, believing in and promoting outrageous lies. Trump is the Prince of Lies, and whether he knows it or not he has aligned himself and his followers with the Prince of Darkness.

Israel has moved ahead with its plans to militarily occupy southern Gaza and the refugee-choked Rafah crossing. This is the main point where relief supplies could get into the Gaza Strip, but that's being blocked now, guaranteeing more dead Gazan civilians, who've already suffered 35,000 mean, women and children killed by Israel.

This has happened before.

I'm a Christian, but reading the Book of Joshua in the Old Testament still bothers me. It's there that we read God ordering invading Israelites to kill and destroy every living thing, human, animal and plant in Canaan, also known as the Promised Land. The only justification I could ever imagine for this 3,200-year-old genocide was the fact that the occupying Canaanites, who worshiped other gods and practiced child sacrifice, were likely considered unredeemable or unworthy of mercy by God. However one looks at it, it's an uncomfortable bit of ancient history.

The Israelites, of course, took over the entirety of Canaan, and through King David and King Solomon established the nation of Israel, with its capital at Jerusalem.

However, when that nation (which subsequently split a century later into the two nations of northern Israel and southern Judah due to political differences) the Israelites got a taste of the same medicine—when many of their citizens began worshiping other deities, God allowed northern attacking countries to destroy Israel in 722 BC followed by Judah in 586 BC, slaughtering and exiling tens of thousands of Jews. Later, the Romans killed hundreds of thousands of Jews during the failed Jewish Revolt of 66-70 AD. As for the countless persecutions Jews suffered during the following 2,000 years and the six million Jews murdered by the Nazis in World War II, I have no answer, but I suspect that humans and not God were the ones responsible.

The October 7, 2023 Hamas attack on Israel resulted in the horrific deaths of some 1,200 Jews, along with the capture of some 200 hostages, many of whom are still being held. Israel's response was predictable, with the primary goal of exterminating Hamas, but the extent of Israel's widespread and ongoing destruction of Gaza and the deaths of 35,000 Gazan civilians to date has been nothing short of overkill. Gaza today is a wall-to-wall ruin, unfit for human habitation, its communities, hospitals and supporting infrastructure totally wiped out. Its 1.5 million Palestinian refugees have streamed south, hoping Egypt will take them in. But then where will they live and work? They sure as hell won't be going back to Gaza, both because there's nothing left there and because Israel has other plans.

Eyeing potentially lucrative real estate investments, Jared Kushner, former US president Donald Trump's Jewish son-in-law, recently suggested that Israel should move the Palestinians out of Gaza and clean it up. Yes, someone's definitely going to "clean up."

I don't watch The Simpsons anymore, mainly because the show ran out of ideas long ago, but Season 8's Episode 23 Homer's Enemy was the series' best ever. It (probably unintentionally) addressed the issue of American inequality better than any show of its kind.

Frank Grimes is a new employee at the nuclear power plant where lackadaisical Homer works. He works hard, takes night classes and strives for a better life than living in the squalid apartment below and above the bowling alleys in which he resides, but he just can't seem to get ahead. You might be able to catch the entire episode on YouTube, but here's the spoiler—thanks to Homer, Grimes dies accidentally at the plant, and as his casket is slowly lowered into the grave a drooling and mumbling Homer is seen fast asleep among the funeral attendees, who all yell out "That's our Homer!"

On the other end of the inequality spectrum is the excellent 2018 PBS documentary The Gilded Age, a two-hour presentation on the excesses of extreme wealth amid the grinding poverty and hardship of more ordinary Americans roughly over the period 1870 to 1920.

While watching the documentary, one thought constantly ran through my mind: Why do people tolerate such inequality? In the PBS show, we find that inequality is borne out of the wealth created for the few by the burgeoning railroad, oil and steel industries of the time and how industrial magnates like the Morgans, Vanderbilts, Rockefellers and Carnegies expanded their wealth through petty, illegal and even murderous schemes. Yet despite heroic populist efforts to stem the greed by labor leaders like Henry George, Mary Elizabeth Lease and Jacob Coxey and socialist writers like Upton Sincair, all these efforts failed. Making things even worse were local, state and federal governments that coddled the wealthy and ignored the plight of mainstream America.

More recently, we've seen similar Gilded Age fortunes built from the dot-com, real estate, information and social media individuals and corporations whose fortunes amazingly far eclipse those of their former ilk. At the same time, the current poverty rate in America is only about 12%, far less than it was in 1900 (80%) or 1950 (20%), although these figures are undoubtedly much higher for Americans of color. Yet today's poverty in America is unlike anything of the distant past—poor people typically have TV sets, cars, air conditioning and smart phones (a favorite argument of the Republican Party for why the poor should just stop complaining), if nothing like a realistic hope for a better life.

It may be hope for a better life that keeps the Great Unwashed fron rising up and slaughtering its fantastically wealthier cousins today, but does it explain how the truly impoverished of earlier times managed to deal with their problems? Was it hope, fawning celebrity worship (which we have today in spades), or the acceptance of poverty out of religious belief? Just asking.

In the hilarious 1982 PBS television film The Great American Fourth of July and Other Disasters, noted humorist Jean Shepherd of Christmas Story fame has teenage Ralphie going on a blind date at a movie theater. Set up by a friend, Ralphie assumes the worst, that his friend's visiting cousin will be a real troll he'll get stuck with for the evening. But Ralphie's in for a shock—the girl is drop-dead gorgeous.

When I first saw the film, I was immediately reminded of the one and only time I was on a blind date, also at a movie theater. I recall she was nice looking, but I also recall that I acted like an idiot, being socially awkward to the max and having no social skills to speak of at the age of 16. However, unlike shy me Ralphie tries the old arm-around-the-shoulder maneuver, but is immediately and harshly rebuffed. He then comes to the same epiphany I had:

But then I had one of the great searing insights of
my life — For the first time I saw myself clearly!
I saw myself for what I was — I WAS THE BLIND DATE!

A great film, not just for this scene but for many others. (Actor Matt Dillon's makeup only coincidentally resembles me at that age.)

Today is Easter in my church, the Coptic Orthodox Church. We say to one another "Christ is risen!" (Christos anesti!), while the response is "He is truly risen!" (Alithos anesti!). Church tradition has Christ's Resurrection taking place in the year 34 A.D., meaning that it occurred exactly 1990 years ago. Will a full millenium (2000 years) be eventful, especially considering the current mess the world's in? I hope so, but given God's own timeline, it's probably unlikely.

Is an object's size (or the distance between two points) an absolute characteristic of its existence, or does it depend on the observer? Einstein's special theory of relativity says it does indeed depend on the observer, and that size, length, distance or whatever you want to call it depends on the observer's frame of reference (this also applies to areas and volumes).

My PhD research focused on the size of particles and their statistical distribution. For micron-size particles, I determined that they often follow a log-normal distribution, much like the observed distribution of truly massive systems of galaxy pairs. But observations of size and distance all depend on the observer, and relativistically these have no objective meaning, This is what led me to Weyl's 1918 gravity theory, which is invariant with respect to such things. This new Scientific American article provides a little more depth into the matter, which I found relevant.

German physicist Sabine Hossenfelder has gone back and forth a few times on the issue of whether dark matter exists or if a modified version of Einstein's 1915 gravity theory more correctly explains what is perhaps the most puzzling problem in cosmology today. In her latest video, she seems to give modified gravity theory a little more credit:

Here's a puzzle for my more educated readers. If you consider the nearly-conformal action $$ S = \int\!\! \sqrt{-g}\,R^2\, d^4x $$ for free space, its extremalization provides exactly the same predictions as Einstein's original theory, but with a purely geometric nod toward an explanation of dark matter. What today is known as pure \(R^2\) (or quadratic) gravity theory was first proposed by the German mathematical physicist Hermann Weyl back in 1918 (whose story is surely known to those who've followed my website).

The paper Hossenfelder cites in her video is from Dr. John Moffat of Canada's Perimeter Institute. It's a short, readable paper, but the author's argument relies on the existence of arbitrary scalar and vector fields having nothing to do with \(R^2\) gravity theory.

Einstein's 1915 gravity theory is grounded in Riemannian geometry, a branch of differential geometry that includes non-Euclidean geometry (think of warped lengths, surfaces and volumes). It was developed in 1851 by the great German mathematician Bernhard Riemann (1826-1866), whose brilliant career was tragically cut short at the age of 39 by tuberculosis. Riemann's geometry was valid in any spacial dimension, but other than certain purely mathematical aspects the geometry remained grounded in the usual three dimensions of space. However, if Riemann had only thought of time as a fourth dimension, he might have been led to Einstein's gravity theory at least sixty years before Einstein published his monumental work in November 1915.

A key element of Riemannian geometry is that its elements are coordinate-independent, meaning that the geometry is applicable to any arbitrary coordinate system. Coordinate invariance means that non-inertial coordinate systems (accelerated systems) could be handled as well as inertial systems, an importsnt factor in Einstein's gravity theory. Einstein also recognized that time could be considered a fourth coordiinate (turning space into spactime), and the dimensionality invariance of Riemannian geometry was also key to the theory, which is four-dimensional.

Some background:

Nearly all physical theories are expressed in mathematical terms that specify a particular coordinate system. High schoolers usually learn about the \(x,y,z \,\) Cartesian coordinate system (the simplest), and slightly later they learn about the polar coordinate system \(r, \theta, \phi\). The mathematics of physical theories based in one system often look completely different in another, leading to confusion and/or disagreement over which system is the best (or easiest) one to use—it's really just a matter of preference or convenience. But physics cannot be dependent on which coordinate system is applied—that is, the predictions of a physical theory cannot change if the coordinate system is changed.

The importance of coordinate invariance was recognized by Riemann, and a number of other mathematicians set out to formalize his ideas. The Italian mathematician Gregorio Ricci (1853-1925) developed the absolute differential calculus (more commonly known as tensor calculus), which is in use today. Tensors are just mathematical quantities that don't rely on any particular coordinate system, and can be used to express any physical theory in tensor language. Concurrently, another Italian mathematician named Luigi Bianchi (1956-1926) derived fundamental mathematical relationships between certain important tensors.

When Einstein was trying to develop his general theory of relativity (gravity), the difference between inertial and non-inertial systems meant that coordinate invariance would be fundamental to the theory. But Einstein did not know how to proceed, so he turned to his close friend and colleague Marcel Grossmann, a mathematician conversant in tensor calculus. Grossmann essentially taught Einstein the formalism, and after many ups and downs over the period 1912 to 1915 Einstein finally completed his gravity theory in November of that year. It is a testament to the beauty, elegance and truth of Einstein's theory that it has not experienced a single failure in over 100 years of astronomical and cosmological observation.


However, despite the success of Einstein's gravity theory it presents several problems. For one, the theory is not conformally invariant, which means that it is fundamentally inconsistent with Maxwell's equations (although electrodynamics can be forced into theory). Another problem is that it appears to be incompatible with quantum field theory (which is conformally invariant). Still another problem is that it cannot be derived from any known fundamental principle. True, the free-space form of Einstein's theory can be deriived by extremalizing the Einstein-Hilbert action $$ S = \int\!\! \sqrt{-g}\, R\, d^4 x \tag{1} $$ where \(R\) is the twice-contracted form of the Riemann curvature tensor \(R_{\mu\nu\alpha\beta} \rightarrow R_{\mu\nu} \rightarrow R\), but then this is just an ad hoc quantity that happens to work. In addition, the action principle in physics traditionally includes a kinetic energy term and a potential energy term, and these are missing in (1). Furthermore, adding a straightforward, principle-based energy-momentum term to the action is elusive (Einstein also had issues with this), so sources of matter and energy (mass, electromagntic fields, etc.) that give rise to gravity in the first place have to be artificially tacked on.

But perhaps the biggest problem is that Einstein's gravity theory cannot be generalized in any satisfactory way that can provide for conformal invariance and mass-energy, which should somehow be incorporated into the action. The most fundamental quantity in gravity theory is the Riemann curvature tensor \(R_{\mu\nu\alpha\beta}\), which is identically zero in empty space. In non-empty space is is not, and should therefore be related to matter and energy (if it is not indeed the very source of matter and energy). But the Riemann curvature tensor is all there is, and there are some (including me) that think it should be the basis of all physical reality. By comparison, in quantum theory one has all kinds of neat things, like operators, fields and the beautiful Dirac bra-ket formalism, along with mathematically consistent ways of describing everything. In that sense, gravity is a "poor" theory, because it has to rely on so little—just the contracted Riemann curvature tensor. The only thing missing in quantum theory is gravity, which seems incapable of being jammed in—hence, the ongoing search for a quantum gravity theory, which may not even exist.

With quantum gravity, string theory and supersymmetry either going nowhere (and/or failing outright) as of today to incorporate gravity, it seems the only recourse is to generalize Einsteinian gravity. But despite ongoing efforts, including the use of scalar, vector, tensor and spinor quantities into the formalism, nothing has really worked regarding a solution to several remaining, perplexing problems like dark matter and dark energy. Meanwhile, Einstein's gravity theory remains eminently successful and unchallenged, despite its inability to be joined with quantum mechanics.

My dream of seeing a workable theory of quantum gravity in my lifetime will likely never be realized, and I'm now resigned to it.

"Look at them, Smithers: goldbrickers, layabouts, slugabeds! (Not to mention Joe Sixpacks and Susie Housecoats) — C. Montgomery Burns


I was saddened while watching this new video from German-physicist-cum-YouTube-video-wrangler Sabine Hossenfelder, whose work, books and videos I've followed and admired for years. I noticed something going on with her a year ago, and in this personally painful video "confession" she reveals all.

For many more years I've followed Cornell University's arXiv.org site, which posts many thousands of freely downloadable, pre-peer-reviewed scientific papers on dozens of academic subjects. But over the last few years I've noticed an inabiity on my part to even comprehend the paper titles on the site, much less their contents. However, my initial fear of the onset of dementia was allayed when I and many others came to the realization that 99%+ of the papers are garbage ("bullsh*t," as Hossenfelder rightly calls it). A glance at a sampling of the papers on the site's General Relativity and Quantum Cosmology section provides convincing evidence of this. As Hossenfelder notes, the vast majority of authors are post-docs, adjuncts and untenured faculty workers whose only hope for a normal, stable life and a tenured university position with decent pay and medical and retirement benefits is to crank out as many papers as possible in an unending effort to secure grants for their employers. My post-PhD son saw this early on and left academia, calling it "life sucking."

Being retired with a pension, I can write all I want about interesting stuff, but with 8 billion and counting humans on the planet now I wonder how many more unproductive wastrels and slugabeds like me will be forthcoming in a future promising far less retirement comforts.

What's the future of academia today? Would Einstein himself stand a chance in this world? Apparently not even a brilliant mind like Hossenfelder's has any hope of a decent future except to bail out and become a YouTube semi-celebrity. Watch and weep:

Although he strictly avoids the subject of the existence of God, astrophysicist Ethan Siegel's latest article talks about one of the most profound aspects of our physical world—continuous mathematical symmetries and conservation laws. These symmetries underlie perhaps the most profound reality, which is that every physical (and probably biological) law of the universe arises from consistent and inviolable mathematical rules that could not have possibly resulted from chance or a random chaotic collection of particles and fields. I urge readers of this site to read the article, and try to figure out for themselves just how such consistency, beauty and truth could have resulted by chance alone, or even the possibility of of an infinite multitude of random parallel universes, each having it own set of chaotic characteristics.

Understanding (or at least appreciating) this in the early 1970s is what initially forced me believe that God exists. Why or how He exists is a subject for theologians.

Readers might also want to read Nobel physicist Eugene Wigner's famous 1960 essay The Unreasomable Effectiveness of Mathematics in the Natural Sciences. in which he also ponders the reality of "the ultimate truth".

On April 8 (which would have been my late wife's 78th birthday), a total solar eclipse will pass over a large swath of the United States. Although many people are looking forward to it, Time Magazine is reporting its downside—major traffic jams in affected states.

The article reminded me of similar traffic problems during an eclipse from years ago, when CNN aired several interviews with people on the street. I'll never forget one woman's comment, which went something like "This eclipse thing is so stupid. I don't know why they're having it in the first place."

No, the title of this post is not from poet e.e. cummings, but from a book I read in high school entitled The Life and Times of Archy and Mehitabel, a collection of poems from 1920s writer Don Marquis and cleverly illustrated by period artist George Herriman of Krazy Kat fame. The post title comes from the "archy declares war" poem.

Archy is a large cockroach whose soul transmigrated from an undated poet, but presumably from Shakespeare's time. He lives in an office, subsisting on stale library paste, and at night when the office is vacant he laboriously loads a typwriter with paper and composes his poems by painfully leaping upon the keys. He is unable to operate the CAPS key, so all his work is in lower case.

Archy has a friend, a scroungy female alley cat named Mehitabel (which means "God rejoices" in Hebrew). He is accompanied by an assortment of other odd characters whose activities he records in his poetry.

Archy also manages to visit nearby venues, including a museum and a radio facility, where he communicates with Mars. Perhaps the best poem results from his visit with an Egyptian pharaoh's mummy at the museum ("archy interviews a pharaoh"). After many millennia, the pharaoh reveals a dying thirst for beer, but Archy has to tell him "my reverend juicelessness this is a beerless country," as this is the Prohibition Era (1920-1933).

I recently found the free online book on the above link and fondly recounted how much of it I remembered verbatim after 57 years. I hope you'll enjoy it as well.

When asked what the greatest commandment is in the Law, Jesus replied:

"Love the Lord your God with all your heart and with all your soul and with all your mind." This is the first and greatest commandment. And the second is like it: "Love your neighbor as yourself." All the Law and the Prophets hang on these two commandments. — Matthew 22:37-40

That second one is the harder of the two, at least for me, and it's probably true for the entire human race. From time immemorial, people have persecuted and slaughtered one another despite enormous advances in knowledge, technology, education and communication and despite all of our collective so-called intelligence, and we're not only no better off today but poised on the brink of global self-destruction. Why? Because we allow ourselves to be deluded by deep-rooted hatreds and biases, fed by egomaniacal, power-hungry demagogues from Alexander the Great to Vladimir Putin, Donald Trump, Kim Jong Un and their morally corrupt ilk today. Recently, Trump's son-in-law advocated kicking all Palestinians out of Gaza into the Negev Desert to allow Israel to bulldoze the region so that investors like himself could make fortunes building luxury ocean front communities, not for returning Gazans but for Israeli settlers.

I'm in despair today. How am I expected to love these monsters and their insane followers, and to follow Christ's related command that we love our enemies and do good to those who hate us? (Luke 6:27)

I dare readers of this site to watch this non-religious 15-minute After Skool video on why education and intelligence do not necessarily lead to wisdom, and how they can actually feed our destructive delusions.

In German physicist Sabine Hossenfelder's latest video, she tries to explain several anomalous aspects of Sag A\(^*\), the supermassive black hole at the center of our Milky Way galaxy. She claims that the rotation rate of a black hole is limited by the speed of light, but this isn't strictly true.

For a non-rotating black hole, the first two terms in its metric (the famous Schwarzschild solution, discovered in 1916 by Karl Schwarzschild) are given by $$ ds^2 = \left( 1 - 2 m/r \right)\, c^2 dt^2 - \frac{1}{1 - 2 m/r}\, dr^2 $$ so that, at \(r = 2 m \), the second term goes to infinity. This is the Schwarzschild radius, which defines the black hole's event horizon, from which nothing can escape. (Similarly, \(r = 0\) also takes things to infinity, which defines the hole's central singularity.) However, the Schwarzschild black hole is a convenient fiction, because all black holes rotate to some extent. The same two terms in the metric of a rotating black hole (discovered by New Zealand physiciat Roy Kerr in 1963) are given by $$ ds^2 = \left( 1 - \frac{2m r}{r^2 - a^2 \cos^2\theta} \right)\, c^2 dt^2 - \left( \frac{r^2 - a^2 \cos^2\theta}{r^2 + a^2 - 2m r} \right) dr^2 $$ where \(a\) is proportional to the hole's rotational angular momentum. Oddly enough, the point \(r = 0\) is no longer a problem, but setting the denominator of the second term to zero we have the point $$ r = m \pm \sqrt{m^2 - a^2} \tag{1} $$ which takes that term to infinity. (Note that for \(a = 0\), we recover the Schwarzschild problem points \(r = 2m\) and \(r = 0\).) From (1), it's obvious that it's not possible for \(a \gt m\), so that \(a = m\) defines the maximum rotation rate of a physical black hole (no black holes observed to date have been shown to exceed this limit). I can't see how the speed of light has anything to do with this limit.

The full Kerr metric is very complicated, and its implications for distinquished regions near the black hole are nothing short of amazing. (I suppose, however, that the limiting speed of light is buried somewhere in both metrics, so Hossenfelder may be right after all.)

Really?!

This recent Quanta article talks about new research into the structure of protons.

Although protons are conventionally thought to be composed of three quarks (two up and one down) connected by gluons, it has become apparent that the internal workings are more like a zoo of quarks, gluons and virtual particles that respond to typical high-energy observations as though just three quarks existed. But by slamming protons much more forcefully with electrons, new features appear that seem to indicate that the interior of a proton is subject to truly enormous forces and pressures.

The Quanta article raises the question of how the energy-momentum tensor of these forces and pressures plays a role in proton structure and internal behavior. The energy-momentum tensor is usually reresented by a symmetric, ten-component \(4 \times 4\) matrix denoted by \(T^{\mu\nu}\), where the components represent energy density and momentum. The appearance of this tensor in particle physics studies seems very odd to me, since it usually appears only in the ten Einstein field equations of gravity expressed by $$ R^{\mu\nu} - \frac{1}{2}\, g^{\mu\nu} \, R + \Lambda g^{\mu\nu} = \frac{8 \pi G}{c^4}\, T^{\mu\nu} $$ The gravitational force is always ignored in particle physics, mainly because particle masses are so small that their gravitational effects can be ignored. However, like mass-energy, pressure and force exert their own forms of energy, and if these are as truly enormous as indicated in the Quanta article, it's possible that gravity may play an important role in the structure of protons, especially considering the tiny distances involved. Recall that gravitational force \(F\) in Newtonian physics is given by $$ F = - \frac{G\,M\,m}{r^2} $$ where \(r\) is the distance between the two masses \(M\) and \(m\). For a proton, constituent distances are likely on the order of \(10^{-20}\) meters or smaller which, coupled with massive forces and pressures, might mean that gravity does indeed play a role.

Although the Newtonian force law has been verified down to sub-millimeter distances and sub-milligram masses, it's probable that it doesn't hold within truly tiny structures like protons and neutrons.

The Quanta article raises the possibility that high-energy electron-proton collisions can interact via the appearance of a graviton, the as-yet undetected force carrier of gravity, again promoting the (highly unlikely) possibility that gravity plays a role in particle physics.

Isaac Newton once stated that if one knew the precise position and momentum of every particle in the universe, then his Newtonian physics would be capable of predicting exactly how the universe was in the past and would be in the future.

In her latest video, German physicist Sabine Hossenfelder discusses why the Butterfly Effect effectively destroys Newton's claim, because the flapping of the wings of a certain lepidopteran insect in the Amazon forest might have an effect on the likelihood of a tornado in Laredo, Texas. Hossenfelder cites the work of the late mathematician Edward Lorenz on chaos theory, which posits that the initial conditions of a physical system, unless they're known exactly, can have enormous indeterministic effects on the system at other times.

However, what Newton and Hossenfelder fail to mention is that the meaning of the word "precise" is an impossibility. The effect of a butterfly flapping its wings can be made tantamount to knowing the one-zillionth decimal in the as-yet unsolved Navier-Stokes equation, gross approximations of which are used today in weather forecasting.

Hossenfelder also notes that not only can a butterfly theoretically affect the weather many thousands of miles away, but so can the happenstance position and arrangement of simple molecules in the environment. I would go even further to say that random quantum fluctuations can have the same indeterministic effect.

In short, I'd prefer to say that if the zillionth decimal in an otherwise reliable forecasting equation is unknown, then chaos will reign and we'll never know how to forecast accurate physical systems.

Tomorrow night (March 13) Caltech will present "Einstein in Pasadena - Between Two Worlds," a talk by Diana Buchwald, Professor of History and Director of Caltech's Einstein Papers Project. The talk is sold out, but you can watch it on YouTube when it has been posted.

Einstein spent three winters here in Pasadena (1931, 1932 and 1933), following a tour given to him by Edwin Hubble at the 100-inch Mount Wilson Telescope high above Pasadena. In 1929, Hubble was the first astronomer to prove the expansion of the universe, although it had been predicted theoretically by Russian physicist Alexander Friedmann in 1922.

During his first visit, Einstein stayed in the modest home of a Pasadena couple at 707 S. Oakland Avenue, but crowds of local gawkers made the visit difficult, so later he resided at Caltech's Athenaeum (I posted a link to the house as it looks today on the menu of my website).

Years ago my next-door neighbor (then a retired professor of linguistics at USC) personally saw Einstein give a dedication talk for a new telescope facility at nearby Pasadena City College (he told me he tried to get an autograph of the famous physicist, but was unable due to the crowds). The college's telescope and building still stand.

At the time of Einstein's last visit in 1933, Caltech's then-president Robert Millikan offered Einstein a professorship at the school. But personal differences between the two (Millikan was notoriously conservative, and he privately detested Einstein's progressive views), and in late 1933 Einstein fled Hitler's Germany to take a position at the Institute of Advanced Study at Princeton in New Jersey.

By the way, the Caltech talk is being given a day before Einstein's 145th birthday. March 14 (3.14), which is also celebrated as Pi Day!

You can watch it here.

Popular German physicist Sabine Hossenfelder seems to have turned primarily to science reporting, with a new video coming out almost daily now. Of course, her videos invariably include sponsor ads at the end, which is okay because a professor's salary only goes so far nowadays.

In her latest video she talks about the non-conservation of energy, something that conflicts with what every high-schooler is taught. It seems to arise only because the universe is capable of expanding or contracting according to the laws of general relativity. For example, every photon in the universe has a specific frequency \(\omega\) that endows the photon with an energy given by \(E = \hbar \omega \), However in an expanding universe the photon's wavelength (which is inversely proportional to frequency) gets stretched, which effectively reduces the photon's energy. Where does this energy go? According to Hossenfelder and most physicists, the energy is simply lost.

However, there is a kind of energy conservation law in Einstein's field equations, which are given by $$ R^{\mu\nu} - \frac{1}{2}\, g^{\mu\nu} R + \Lambda g^{\mu\nu} = \frac{8 \pi G}{c^4}\, T^{\mu\nu} $$ where \(T^{\mu\nu}\) is called the energy-momentum tensor. which is a measure of the total energy of a source of mass and/or radiation. Einstein's energy conservation law is given by the divergence quantity $$ \nabla_\nu T^{\mu\nu} = 0 $$ where \(\nabla_\nu \) stands for covariant differentiation. But strict energy conservation would be given by $$ \partial_\nu T^{\mu\nu} = 0 $$ in which \(\partial_\nu \) is the ordinary partial derivative. Try as he might, Einstein was never able to massage his gravity equations so that this latter divergence held. Consequently, energy is not conserved in our universe.

At the 5:08 point in Hossenfelder's video, she cites a recent paper by Russian researchers who try a form of modified gravity to answer the non-conservation issue. Hossenfelder is not crazy about the paper, but gives it some credit according to its innovative approach. However, I got nowhere with the paper, which you can download here. Good luck.

Meanwhile, here's Hossenfelder's informative video:

Imagine you're a librarian working in the ancient Roman resort city of Herculaneum in August of the year 70 AD. It's evening, and you've just finished rolling up scrolls of books that patrons have been reading. After placing the scrolls carefully on their shelves, you lock the place up and go home. That night you hear ominous sounds, followed by strong earth tremors. Then Mount Vesuvius explodes, sending fiery ash down on the town. You don't know whether to run or stay in your house, hoping the eruption will subside, as this has happened before. But the mountain explodes again, and this time it sends a pyroclastic flow of molten rock and fiery debris down on the town. Now you run, but you don't get far, as the air is hot and unbreathable. Your last thought is of your friends and family, and of course the fate of your beloved books. You end up under 20 feet of smoldering ash, your vaporized body becoming a hollowed-out cast.

The fate of those books was to end up as charred and blackened lumps too, but with each scroll still preserving its rolled-up shape. Modern investigators discover their remains, and a few efforts are made to unroll them to see what they have recorded. But those attempts fail, as the charred books simply fall apart into tiny bits of unreadable blackened soot.

For decades researchers have hoped that future technology would find a way to safely unwrap the scrolls, or at least provide a way of peering inside them so that they could be read. X-rays didn't work, and neither did MRI. But now machine-learning algorithms trained on the scrolls has partially revealed the Greek writing within, and artificial intelligence promises vastly greater success in virtually unwrapping the scrolls.

As reported in this new Scientific American, some several hundreds of blackened Herculaneum scrolls are being viewed as potentially important glimpses into the ancient world of Roman and Greek civilization. Perhaps some new mathematical discovery by Archimedes will be discovered, but at the present all we have is "Epicurus thinks his friend Philodemus is a know-nothing jerk." (My translation.)

We're getting record-setting tons of rain here in Southern California, so I have nothing to do but sit here and read, watch a little TV and reminisce. In my last post I mentioned how the quality of songs at live rock concert performances tended to be lousy compared with their recorded variants, but that ELO was a notible exception. Another exception was a high school performance put on by the long-extinct 1960s rock band The Bobby Fuller Four. which came to Duarte High School (California) in early 1966. Their only hit song that I recall was "I Fought the Law," a great song that still has no equal from the time as far as I'm concerned. The band performed it perfectly in our high school auditorium, and I still recall the girls (SHE in particular) going nuts over the band members, the late Bobby Fuller in particular.

The 1960s was a strange time for popular music in the sense that conventional rock music (the Rolling Stones' Get Off My Cloud, for example) could be heard on rock radio stations along with insipid niche tunes like They're Coming to Take Me Away and Baby Elephant Walk. There was no song vulgarity allowed in those days, of course, and I recall that when the Stones appeared on the Ed Sullivan Show to perform "Let's Spend the Night Together," the über-conservative Sullivan insisted that they sing it as "Let's spend some time together," a demand that the group ignored anyway, surely to Sullivan's dismay.

Here's a related photo from my 1966 high school yearbook. Upper left is Joey Paige, who sang "Goodnight My Love," followed by catcalls from the class audience to "Get a haircut!" To the right is our Senior Class President Chick Mangan welcoming Dick Biondi, a popular DJ from the radio station KRLA (he passed away in 2023). [I last saw Mangan in 1976 when he was a bailiff at a courthouse where I was a juror.] Lower left is The Deuces Wild, who performed something or other, and at the lower right is a stock publicity photo of The Bobby Fuller Four. You can read Page 9 of this old issue of KRLA Beat Magazine for more information, which indicates the bands came to Duarte High School in April 1966.

While it was nice of our high school president John McGrew to allow long-haired rock bands into the school, I never realized how preposterously conservative the school was at the time, along with most of its students and the town itself. The town's newspapers (The Duartean and The Daily News-Post) were very conservative news outlets, fully supportive of the Vietnam War, and they reported notoriously on several minor race riots the high school experienced involving attacks by whites against its few black students ("Hello" to my old friend Dumas M.!)

I suppose all that's nothing compared to what's going on in high schools today, with auditorium performances possibly involving half-naked twerking and gyrating performers. Sad.

Side Note: I just noticed the writer of the Page 9 article, one Marlyn S. Some of my fellow high school grads will definitely remember her:

In 1981 the Electric Light Orchestra released its futuristic rock music album Time, which included Here is the News. The song parodied likely news reports from a dysfunctional future Earth, all of them doom-laden. The album's songs reference the year 2095, far enough away from 1981 to lend the songs possible credence and yet portraying a frightening world sadly closer to our own time.

ELO is my favorite band, with lead singer/composer/guitarist Jeff Lynne taking top credits for the band's many hits of the 1970s through the 1990s. The Beatle's Paul McCartney was highly impressed with Lynne's innovative music, which combined rock with classical orchestral instruments, noting that had the Beatles stayed together their music would likely have pursued the same genre.

I have all of ELO's albums along with videos of the many live performances they made over the years. I was not, however, aware that music videos (remember them?) were also made of some of the lesser-known songs. "Here is the News" is one of them, featuring Lynne and his band members portraying news reporters. The lip-syncing is not great, but the song is. Enjoy.

Side Note: The music at most live rock concerts tends to sound far inferior to what you might hear on the radio or on CDs. One of the great things about ELO is the songs in their live performances sound almost exactly like their commercial recordings. I attended my first rock concert on July 5, 1968, when the Doors performed at the Hollywood Bowl, and they truly sounded awful.

"**Gasp** I ... I've become radioactive!"

In the second episode of Season 7 of The Simpsons, Bart and his pals get their hands on an extremely rare copy of Radioactive Man No. 1, which details the events leading up to the main character's becoming radioactive and subsequently a superhero. Like Peter Parker being bitten by a radioactive spider and becoming Spider Man (yet another unlikely superhero), the comics are replete with such fortunate transformations thanks to radioactivity (I can name other examples from a misspent youth of reading comic books, but I won't). But in reality, most radioactive materials are highly dangerous, having the ability to irreparably damage cellular DNA, leading to fatal cancers, not superheroes.

In her latest video, physicist Sabine Hossenfelder recalls how x-ray radiation was discovered by the German physicist Wilhelm Roentgen. The discovery didn't involve radioactive materials, but x-rays quickly found favor with merchandisers and medical quacks who figured they must be beneficial to human health. With the exception of medical imaging, x-rays were found to be also damaging to human tissue in even moderate doses. In 1898, French chemist and physicist Marie Curie and her physicist husband discovered the radioactive element radium, which also quickly garnered favor with merchandisers and medical quacks. But unlike x-rays, the unregulated public use of radium (including direct ingestion to increase "vigor") proved to have tragic consequences.

Because radium glows green in the dark due to its radioactive decay, watch makers found it useful as a means of making luminous watch dials and numerals visible in the dark. Paint infused with radium was employed for this purpose, applied by lowly-paid female factory workers who used finely-tipped brushes. To maintain the brush tip, they routinely pressed the brush between their moistened lips, which inadvertently transferred radium into their mouths. This use of radium faded within several years, when many radium girls developed oral deformalities and fatal cancers. To my knowledge, none became superheroes.

Predicably, watch factory owners and radium paint manufacturers fought vigorously for many years against legal charges that their products were causing any harm. The story is detailed in the informative and entertaining 2018 film Radium Girls.

Also interesting in the same vein was the popular 1920s practice of grafting monkey and goat glands into men's testicles to increase virility. (I understand the practice is still popular among far-right Republicans and Trump worshipers.)

My dear late Egyptian wife Munira was a devout Christian, and while she did not hold any enmity against Israel she always believed that Israel's primary goal was to annex the entire region, including Gaza and the West Bank, leaving Palestinians to either get out, starve to death or be eliminated Nazi-style. By rejecting any possibility of a future two-state solution, Israeli prime minister Benyamin Netanyahu seems to have adopted that "solution" instead.

For years since Israel's politically-based foundation in 1948, Israeli settlers have encroached onto disputed lands, kicking out generations-long Palestinian property owners and establishing militarily-protected kibbutzim and similar collective settlements in lands once owned by Palestinians. The ongoing Gaza War is now providing additional opportunities for land grabs by Israeli settlers in the West Bank, and in recognition of this President Biden has issued an Executive Order focused on violence committed by Israelis seeking to take advantage of the Gaza War.

Since Israel does not recognize any US legitimacy in the West Bank, I say good luck, Mr Biden. Meanwhile, America yawns.

German physicist Sabine Hossenfelder has apparently left the dark matter fold, admitting that she no longer believes it exists. She briefly supported the idea that some kind of universal "superfluid" was responsible, but to me the only viable option left is modified gravity. Einstein himself stated that his 1915 gravity theory (still the standard) would be shown to be only an approximation of the truth, but dark matter advocates have insisted that an undetected, unproven and hypothetical dark matter particle could be explained by other undetected, unproven and hypothetical particles, wasting billions of dollars in experimental efforts over the past 40 years. Perhaps now pen and paper will ultimately prove dark matter will go the way of the luminiferous aether, polywater, pixie dust and the unicorn.

On occasion, a stable and electrically neutral atom will emit an electron in a process known as beta decay, which occurs when a neutron in the atomic nucleus spontaneously turns into a postively charged proton. Electrical neutrality is conserved, however, with the creation and emission of the electron.

When first studied in 1930, it was noticed that the outgoing electron did not have sufficient kinetic energy for the conservation of energy of the entire system. The famous physicist Niels Bohr was ready to believe that energy conservation was violated in beta decay, but the equally brilliant Wolfgang Pauli postulated that an unseen particle was also emitted with sufficient energy to maintain overall energy conservation. Pauli called this particle a "neutron," but when the actual neutron was discovered in 1932 Pauli's particle was renamed the neutrino, meaning "little neutral one."

Pauli's postulated neutrino remained undetected until 1956, but zillions of neutrinos are now routinely manufactured in linear accelerators, and they're copiously produced in the core of our Sun. Today, the neutrino is an honored member of the Standard Model of physics.

The law of energy conservation is a foundation of physics, but when Einstein announced his gravitation theory in 1915 (and when the expansion of the universe was discovered in 1929) physicists noted that both discoveries seemed to violate energy conservation. Or at least, so it seemed.

There is undeed a foundational aspect of the conservation of energy and momentum in Einstein's theory, but there is a hitch. It involves the divergence of the energy-momentum tensor \(T^{\mu\nu}\), but this divergence is a covariant one, not one involving ordinary partial differentiation. Try as he might, Einstein was not able to break down the covariant divergence into an ordinary one, and to this day it remains a key reason why a gravity-filled, expanding universe does not appear to conserve energy.

Most physicists today believe that the law of energy conservation is simply not applicable to our universe, but many beg to differ. One is the astrophysicist Ethan Siegel, who kind of waffles on the subject in the linked article. His argument is based in part on the fact that universal expansion stretches the wavelengths of photons, which decreases their energy. Where does this energy go? Siegel posits that it may power the expansion of the universe itself, but he also indicates that it might simply be lost, and that the energy budget of the universe is not conserved.

By stretching the wavelength of photons, the notion of tension (which ia kind of force) enters the picture. A stretched photon involves an increase in its "length," and tension times length is equivalent to energy. I once believed that the lost energy was taken up by the dark energy of space, but the vastness of photonic lost energy and its assumed uptake by dark energy do not seem to match, at least observationally.

This conundrum is at least partly responsible for why I believe Einstein's gravity theory needs to be modified. Einstein himself thought that his theory, likr that that of Newton's, was an approximation to the truth deserving of future modification, despite its seemingly unerring success for the past 109 years.

In 1918 the German mathematical physicist Hermann Weyl believed he had glimpsed a means of modifying Einstein's gravity theory, and in pursuing it he thought he had found a way to unify gravity and electrodynamics. I won't go into it, but it involves changing the way length, time and energy are accounted for, and it might still be a way of showing that the law of energy conservation in the universe it actually upheld. As a devotee of Weyl's work, I've struggled to find a mathematical way of resolving the problem, to no avail. Maybe some bright kid will.

Three weeks after some 1,200 Israeli civilians were murdered by Hamas in its horrific October 7, 2023 attack on Israel from Gaza, Israel's Prime Minister Benyamin Netanyahu defiantly addressed his nation. In addition to a pledge to destroy Hamas, Netanyahu asked his country to "Remember what Amalek did to you." This was a reference to Deuteronomy 25:17-19, in which Moses says

Remember what the Amalekites did to you along the way when you came out of Egypt. When you were weary and worn out, they met you on your journey and attacked all who were lagging behind; they had no fear of God. When the Lord your God gives you rest from all the enemies around you in the land he is giving you to possess as an inheritance, you shall blot out the name of Amalek from under heaven. Do not forget!

After the death of Moses, his chosen general Joshua proceeded to attack Canaan (the Promised Land). The first to fall was Jericho, with the death of all its inhabitants, and then Joshua marched against the city of Ai, in which all of its 12,000 inhabitants were killed, with only 36 Israelite fatalities. Upon completion, Israel's conquest of Canaan resulted in the deaths of perhaps 60,000 people, including all of its men, women, children, livestock and food supplies.

Israel's current reprisal war against Hamas in Gaza has so far resulted in the deaths of 24,000 Palestinian civilians. The ratio of civilian to Israeli deaths is only 20:1, far below that of the attack on Ai, but that ratio is expected to increase dramatically before Hamas is convincingly defeated as a terrorist organization.

The International Court of Justice (the Hague) is considering an accusation from member nation South Africa that Israel is committing genocide against Palestinians living in Gaza. That accusation is open to debate, but Israel's indiscriminate and widespread bombardment of Gaza is reminiscent of the total destruction that the Israelites visited upon the cities of Canaan more than three millennia ago.

Netanyahu's "Remember Amalek" remark is also chillingly reminiscent of Moses' Deuteronomy command, and Netanyahu's refusal to accept a ceasefire in Gaza appears to reflect the same determination that Joshua expressed in utterly destroying his enemies. In addition, with Gaza likely to be reduced to complete rubble, Palestinian survivors and refugees will have little recourse but to flee to Egypt's Sinai, leaving Gaza open to permanent Israeli occupation and lucrative ocean-front resorts after rebuilding.

The hoped-for "Two-State Solution" is all but dead now, having been replaced by Netanyahu as a contiguous Israel. Perhaps next to go will be the West Bank Palestinians, and then the phrase "From the River to the Sea" (i.e., from the Jordan River to the Mediterranean Sea) will have a more ominous meaning than what pro-Palestinian demonstrators were hoping for.

In my post of 4 December last year I pondered the possibility that a fuzzier or grainier aspect of the Cosmological Principle might account for the so-called Hubble tension in modern cosmology. With the possible exceptions of dark matter and quantum gravity, the Hubble tension represents cosmology's most perplexing mystery today.

The Hubble parameter is a description of the current rate of the expansion of the universe. The two main contenders for the parameter are based on measurements of the cosmic microwave background (CMB) and the cosmic distance ladder (CDL). The former gives a value of about 67 kilometers per second per megaparsec (km/s/Mpc), while the latter is about 9% higher at 73 km/s/Mpc. The uncertainties in the data do not overlap, resulting in the tension. One or both of the figures must therefore be wrong.

The cosmological principle is based on the assumption that—on average—the disribution of matter and energy in the universe is perfectly homogeneous and isotropic. This assumption is necessary if Einstein's gravitational field equations are to be capable of describing the universe, as any lumpiness (either locally or non-locally) is strictly incapable of being modeled. Nevertheless, the cosmological principle has been remarkably successful in predicting the evolution and probable fate of the universe and all manner of many astronomical observations.

In the latest episode of PBS Spacetime, astrophysicist Matt Dowd raises the possibility that non-uniformity may be the reason for the Hubble tension, at least on the local level of our galaxy. For one thing, it is known that our Milky Way Galaxy is located in a large relative void of matter, and this void might be affecting certain astronomical observations (mainly associated with Cepheid variables) giving an apparent overestimate of the Hubble parameter for the CDL approach. In one of the papers Dowd mentions, accounting for this void would lower the Hubble figure to about 69, in closer agreement with the CMB approach. On the hand, it is also known that the Milky Way is a member of a supercluster of some 100,000 neighboring galaxies called Laniakea, a Hawaiian word meaning "immense heaven." Some researchers believe this supercluster might have the oppsite effect on astronomical observations, making the actual Hubble papameter closer to the CDL approach.

The other three papers mentioned in Dowd's talk can be found here, here and here. While readable, extensive scattering of currently available astronomical data forces the papers' authors to use complicated statistical analyses that themselves are difficult to follow. Good luck.

The recreational math website MindYourDecisions posted this fascinating puzzle today:

A clever wife, tired of her husband's spending habits, gives him free access to her private ATM account pending his ability to find the pin code, which is hidden in the solution to this integral:

Like me, hubby first resorts to Wolfram Alpha, which fails\(^*\). Noticing that the denominator can be factored, he then tries a solution using partial fractions, but this too goes nowhere. He then tries to do the integral by throwing thousands of random Monte Carlo points at the plot given at 5:40 in the video, hoping that the normalized area under the curve can provide the answer. When this also fails, he then just waits for the video to provide the answer.

But to his dismay, wifey has already changed the pin code. He, like me, has wasted several hours of his life for nothing.

\(^*\) Actually, Wolfram Alpha works just fine; I simply punched it in wrong the first time:

You have to be of a certain age if you've subscribed to Classmates.com, a free social networking website founded in 1995 (has the Internet really been around that long?) that displays photos and stories from past high school graduates and attendees. I signed up sometime after retiring in 2002 mainly to see what it was all about, but there wasn't much on it back then. Today, the site has nearly 100 million subscribers.

There are several other such classmate websites, and in 2006 I saw one post from an elderly graduate of the 1924 Quincy (Illinois) High School. As best as I can remember now, she wrote "I graduated from QHS in 1924 and am now almost 100 years old. My grandson is writing this for me. Is anyone from my class still out there?" I somehow contacted the woman's family, but I never got a response. She most likely passed on shortly after her inquiry.

Struck by the pathos of her post, that same year I added my father to the Classmates site (even though he died in 1981) to find out if there was anyone still alive from his graduation class. He too graduated from QHS in 1924, and I have his rather tattered graduation yearbook, which turns 100 years old in June (I bought the yearbook from eBay in 2007). Dad was 19 years old at the time, a year older than most high school graduates of today, but very common for kids at the time (graduation from high school back then was somewhat equivalent to graduating with an undergraduate college degree today).

I occasionally get messages from Classmates saying "Hello Ogden Straub! People want to see how you look today!" (No, they do not! 😜)

My father, from his 1924 yearbook

I'm now 75, and my high school graduation took place in 1967, going on 57 years ago. The yearbook sits on my bookshelf, and I wonder where it will end up when I'm long gone.

I'm praying for a happier, more peaceful and saner New Year, but I can't help but feel pessimistic. Two major wars are ongoing, with the possibility of expanding into regional, multi-country disasters, while at home the spectre of a Trump reelection will determine if America remains a democracy or a dictatorship.

Meanwhile, I just turned 75, and while I feel okay there's no denying that my body is now three-quarters of a century old. Ugh.

I feel old, Starbuck, and bowed. As though I were Adam, staggering beneath the pile of centuries since Paradise.
— Captain Ahab, Moby-Dick