“Stop the presses! The Earth’s core stopped spinning! In fact it is now spinning backwards!”

Well, that’s pretty much how much of the popular press handled a recent article, published in Nature Geoscience, under the far less pretentious title, “Multidecadal variation of the Earth’s inner-core rotation”.

And indeed, the first half-sentence in the abstract says it all (emphasis mine): “Differential rotation of Earth’s inner core relative to the mantle“.

It’s not like the core stopped spinning. It’s just that the core is sometimes spinning slightly faster, sometimes spinning slightly slower than the mantle, an oscillatory pattern that has to do with the complex interaction between the two.

How much faster/slower? Don’t expect anything dramatic. At most a few degrees a year, but more likely, just a small fraction of a degree a year. So even if the ~70-year cycle (deduced by the authors of the recent article — there are other estimates) is valid, the core would only get ahead, or behind, the mantle by just a few degrees before it slows down or catches up again.

And this is what supposedly happened: the core was slowing down until a few years ago, its rotation came to be in sync with that of the mantle. Slowing down further, it’s now falling ever so slightly behind, only to catch up again, presumably, a few decades from now.

The way it is misleadingly presented in the media and the degree to which it is sensationalized demonstrate that we live in the era of hype.

The National Ignition Facility has achieved a net power gain in its experimental fusion reactor. This is heralded as a major breakthrough.

Does this mean that in 50 years, we will have practical nuclear fusion power our world?

Oh wait. We were told exactly that some 50 years ago:

At the beginning of the 1950s, it seemed that success is not far away. But later, difficulties arose one after another […]

Unfortunately today there are still gigantic difficulties in the path towards utilizing this fabulously rich supply of energy […]

In fourteen countries of the world, more than two thousand engineers and scientists are laboring on working out different types of fusion devices.

To date, more than a hundred different models have been devised […]

Let us introduce only one group of these: the Soviet Tokamak devices, because around the world, these are the ones in which researchers have the most faith, viewing them as prototypes of future fusion power plants.

A year and a half ago, in an experiment carried out in collaboration between Soviet and English physicists, they directly measured the temperature and density of the plasma of Tokamak-3, and it became clear that the results were even better than indicated by prior measurements. To date, no other device could produce plasma of such quality.

When will the first fusion power plants be realized, when will the investigation of controlled nuclear fusion exit the constraints of laboratory experiments? According to Professor Igor Golovin, the world-renowned expert on thermonuclear research, it will be possible to develop Tokamak devices into electricity-producing equipment by the last decade of our century. L. Hirsch, one of the leading physicists of the American Atomic Energy Commission is a little more cautious. According to him the path from the first experiments to the worldwide spread of fusion power plants is longer, and we’re lucky if they will enter the world’s energy production market in fifty years.

These are all quotes (my translations) from a 1972 Hungarian-language educational children’s publication, “Boys’ Almanac 1973”.

As I express my (probably uninformed) skepticism concerning practical fusion power generation, I note that in the deep interior of the Sun, under gravitational confinement due to the combined mass of more than 300,000 Earths, fusion progresses at the leisurely rate of a few hundred watts per cubic meter. (The power output of a well-maintained industrial compost pile.) For practical power generation, we need something that is at least a million times that, a few hundred megawatts per cubic meter… and we don’t have 300,000 Earths for gravitational confinement.

Of course I’d be delighted if they proved me wrong.

Every so often, I am presented with questions about physics that go beyond physics: philosophical questions of an existential nature, such as the reasons why the universe has certain properties, or the meaning of existence in light of the far future.

I usually evade such questions by pointing out that they represent the domain of priests or philosophers, not physicists. I do not mean this disparagingly; rather, it is a recognition of the fact that physics is about how the universe works, not why, nor what it all means for us humans.

Yesterday, I came across a wonderful 1915 painting by Russian avant-garde painter Lyubov Popova, entitled Portrait of a Philosopher:

What can I say? This painting sums up how I feel perfectly.

Today is November 5, 2022. Here we are in Ottawa, supposedly the second coldest capital city in the world after Ulan Bator. It is 9:15 PM.

And it is 21 degrees (centigrade — 70 F, for my American friends.)

Our A/C ran several times today, especially while we were baking something.

This is beyond incredible. I’ve seen snowstorms in this town in October. I’ve never seen summer-like weather in November.

If this is global warming… well, if folks who will likely be swept away by the sea in places like Florida don’t care, who am I to complain?

Still… weather like this in November is a bit creepy.

[And yes, I still use Windows gadgets, with the help of third-party software. What can I say? I like them.]

There are only about six days left of the month of October and I have not yet written anything in this blog of mine this month. I wonder why.

Ran out of topics? Not really, but…

… When it comes to politics, what can I say that hasn’t been said before? That the murderous mess in Ukraine remains as horrifying as ever, carrying with it the threat of escalation each and every day? That it may already be the opening battle of WW3?

Or should I lament how the new American radical right — masquerading as conservatives, but in reality anti-democratic, illiberal authoritarianists who are busy dismantling the core institutions of the American republic — is on the verge of gaining control of both houses of Congress?

Do I feel like commenting on what has been a foregone conclusion for months, Xi “Winnie-the-pooh” Jinping anointing himself dictator for life in the Middle Kingdom, ruining the chances of continuing liberalization in that great country, also gravely harming their flourishing economy?

Or should I comment on the fact that prevalent climate denialism notwithstanding, for the first time in the 35 years that I’ve lived in Ottawa, Canada, our air conditioner came online in the last week of October because the house was getting too hot in this near summerlike heat wave?

Naw. I should stick to physics. Trouble is, apart from the fact that I still feel quite unproductive, having battled a cold/flu/COVID (frankly, I don’t care what it was, I just want to recover fully) my physics time is still consumed with wrapping up a few lose ends of our Solar Gravitational Lens study, now that the NIAC Phase III effort has formally come to a close.

Still, there are a few physics topics that I am eager to revisit. And it’s a nice form of escapism from the “real” world, which is becoming more surreal each and every day.

I don’t always agree with Sabine Hossenfelder but every once in a while, she hits the nail spot on.

Case in point: Her article, published in The Guardian on September 26, about the state of particle physics.

Imagine going to a zoology conference, she says, where a researcher discusses a hypothesis (complete with a computer-generated 3D model) of a 12-legged purple spider living in the Arctic. Probably doesn’t exist but still, how about proposing a mission to the Arctic to search for one? After all, a null result also contains valuable data. Or how about a flying earthworm that lives in caves? Martian octopuses, anyone?

Zoology conferences do not usually discuss such imaginary monsters but, Sabine argues (and she is spot on) this is pretty much what particle physics conferences are like: “invent new particles for which there is no evidence, publish papers about them, write more papers about these particles’ properties, and demand the hypothesis be experimentally tested”. Worse yet, real money is being spent (wasted might be a better word) on carrying out such experiments.

She points out that while it is true that good science is falsifiable, the opposite isn’t always the case: Just because something is falsifiable does not make it good science.

And not just particle physics, I hasten to add. How about cosmology and gravitation? Discussions about what may or may not have happened during the Planck epoch? Exploring exotic spacetime topologies, often in dimensions other than four? And let me not even mention quantum computing or fusion energy…

Perhaps I am a born skeptic lacking imagination, but to me, these are all 12-legged purple Arctic spiders. The science we actually know and have the ability to confirm are general relativity in a spacetime that is by and large the perturbed Minkowski metric; and the Standard Model of particle physics, extended with a neutrino mass mixing matrix. These are the things that work. Not perfectly, mind you. General relativity needs “dark matter” (name aside, we don’t know what it is except that it has a dust equation of state) and “dark energy” (again, it has a name but beyond that, we don’t know what it is beyond its equation of state) to account for galaxy dynamics and cosmic evolution. The Hubble tension, the discrepancy between values of the Hubble parameter measured using different methods, is real. Observations by the James Webb space telescope suggest that we do not understand the “dark ages”, the first few hundred million years after the surface of last scattering (i.e., the epoch when the cosmic microwave background radiation was produced), well. Massive neutrinos invite the question about the apparent absence right-handed neutrinos.

And yes, we are very much in the blind concerning these issues. Nature has not yet provided hints and we are not smart enough to figure out the answers entirely on our own. But how is that an excuse for inventing 12-legged spiders?

I think it isn’t.

Putting aside Trumpism, woke-ism, the politics of the day, populism, the whole kaboodle, here’s something to contemplate.

Tonight, Russia is continuing its efforts to subjugate the independent nation of Ukraine, not refraining from committing serious, intentional, criminal acts against the country’s civilian population to further its goals.

Also tonight, the space agency of the United States, NASA, is conducting a ground-breaking experiment, the first of its kind testing a method that might one day avert a global disaster, protecting the entire planet from an asteroid impact.

DART: View of the asteroid Dimophos 20 minutes to impact

I think it speaks volumes about the different ways in which these countries see their respective roles in the world.

I almost long for Soviet times. The regime was assuredly brutal, but at least it professed to seek noble goals. Not anymore, I guess.

Oops. It’s past midnight already, so technically it was yesterday but to me it is still today, September 12.

The sixtieth anniversary of John F. Kennedy’s famous “we choose to go to the Moon” speech. How many more years before another human sets foot on the Moon?

Oh, and it was thirty years ago that Ildiko and I became married.

Yup, that’s us; 1979 vs. 2019.

A few days ago I had a silly thought about the metric tensor of general relativity.

This tensor is usually assumed to be symmetric, on account of the fact that even if it has an antisymmetric part, $$g_{[\mu\nu]}dx^\mu dx^\nu$$ will be identically zero anyway.

But then, nothing constrains $$g_{\mu\nu}$$ to be symmetric. Such a constraint should normally appear, in the Lagrangian formalism of the theory, as a Lagrange-multiplier. What if we add just such a Lagrange-multiplier to the Einstein-Hilbert Lagrangian of general relativity?

That is, let’s write the action of general relativity in the form,

$$S_{\rm G} = \int~d^4x\sqrt{-g}(R – 2\Lambda + \lambda^{[\mu\nu]}g_{\mu\nu}),$$

where we introduced the Lagrange-multiplier $$\lambda^{[\mu\nu]}$$ in the form of a fully antisymmetric tensor. We know that

$$\lambda^{[\mu\nu]}g_{\mu\nu}=\lambda^{[\mu\nu]}(g_{(\mu\nu)}+g_{[\mu\nu]})=\lambda^{[\mu\nu]}g_{[\mu\nu]},$$

since the product of an antisymmetric and a symmetric tensor is identically zero. Therefore, variation with respect to $$\lambda^{[\mu\nu]}$$ yields $$g_{[\mu\nu]}=0,$$ which is what we want.

But what about variation with respect to $$g_{\mu\nu}?$$ The Lagrange-multipliers represent new (non-dynamic) degrees of freedom. Indeed, in the corresponding Euler-Lagrange equation, we end up with new terms:

$$\frac{\partial}{\partial g_{\alpha\beta}}(\sqrt{-g}\lambda^{[\mu\nu]}g_{[\mu\nu]})= \frac{1}{2}g^{\alpha\beta}\sqrt{-g}\lambda^{[\mu\nu]}g_{[\mu\nu]}+\sqrt{-g}\lambda^{[\mu\nu]}(\delta^\alpha_\mu\delta^\beta_\nu-\delta^\alpha_\nu\delta^\beta_\mu)=2\sqrt{-g}\lambda^{[\mu\nu]}=0.$$

But this just leads to the trivial equation, $$\lambda^{[\mu\nu]}=0,$$ for the Lagrange-multipliers. In other words, we get back General Relativity, just the way we were supposed to.

So in the end, we gain nothing. My silly thought was just that, a silly exercise in pedantry that added nothing to the theory, just showed what we already knew, namely that the antisymmetric part of the metric tensor contributes nothing.

Now if we were to add a dynamical term involving the antisymmetric part, that would be different of course. Then we’d end up with either Einstein’s attempt at a unified field theory (with the antisymmetric part corresponding to electromagnetism) or Moffat’s nonsymmetric gravitational theory. But that’s a whole different game.

I’ve been wanting to write about this all the way back in April, when folks became rather upset after Florida rejected some school math textbooks. A variety of reasons were cited, including references to critical race theory and things like social-emotional learning.

Many were aghast: Has the political right gone bonkers, seeing shadows even in math textbooks? And to a significant extent, they were correct: when a textbook is rejected because it uses, as an example, racial statistics in a math problem, or heaven forbid, mentions climate change as established observational fact, you can tell that it’s conservative denialism, not genuine concern about children’s education that is at work.

But was there more to these rejections than ludicrous conservative ideology? Having in the past read essays arguing that mathematics education is “white supremacist”, I certainly could not exclude the possibility. Still, it seemed unlikely. That is, until I came across pages like Mrs. Beattie’s Classroom, explaining “How to spark social-emotional learning in your math classroom“.

Holy freaking macaroni! I thought this nonsense exists only in satire, like a famous past Simpsons episode. But no. These good people think the best way to teach children how to do basic math is through questions like “How did today’s math make you feel?” — “What can you do when you feel stressed out in math class?” — “What self-talk can you use to help you persevere?” or even “How can you be a good group member?” The line between reality and satire does not seem to exist anymore.

In light of this, I cannot exactly blame Florida anymore. Conservatives may be living in a deep state of denial when it comes to certain subjects (way too many of them, from women’s health the climate change) but frankly, this nonsense is almost as freakishly crazy. If I were a parent of a school age child in the United States today, I’d be deeply concerned: Does it really boil down to a choice between schools governed by some form of Christian Taliban or wokeism gone berserk?

Doesn’t this cloud, photographed in the skies above Ottawa by my beautiful wife moments ago, look just like the USS Enterprise?

Maybe it is, doing its time-traveling thing, with a malfunctioning cloaking device.

There are a few things in life that I heard about and wish I didn’t. I’m going to mention some of them here, but without links or pictures. If you want to find them, Google them. But I am mindful of those who value their sanity.

• In a famous experiment, a researcher subjected rats to drowning. Rats that were previously rescued tried to stay afloat and took longer to die than those who weren’t. Hope changed their behavior.
• There was an old Chinese method of execution: literally cutting the condemned in half at the waist.
• Japan’s wartime bioweapons and chemical warfare research facility, the famous Unit 731, was so horrific, Auschwitz-Birkenau is probably like a happy summer camp in comparison (and not because Mengele was nice).
• Touch a tiny fraction of a milligram of dimethylmercury for more than a few seconds even while wearing a latex glove, and you will almost certainly die a horrible death months later, as your body and mind irreversibly deteriorate. (Someone once said that the very existence of something evil like Hg(CH3)2 is proof that there’s no God, or at least not a benevolent one.)

There may be a few other similarly unpleasant tidbits, but I can’t recall them right now, and that’s good. Mercifully, our human memory is imperfect so perhaps it is possible to unlearn things after all. (Or, perhaps I am hoping in vain, like those unfortunate rats.)

From time to time, I promise myself not to respond again to e-mails from strangers, asking me to comment on their research, view their paper, offer thoughts.

Yet from time to time, when the person seems respectable, the research genuine, I do respond. Most of the time, in vain.

Like the other day. Long story short, someone basically proved, as part of a lengthier derivation, that general relativity is always unimodular. This is of course manifestly untrue, but I was wondering where their seemingly reasonable derivation went awry.

Eventually I spotted it. Without getting bogged down in the details, what they did was essentially equivalent to proving that second derivatives do not exist:

$$\frac{d^2f}{dx^2} = \frac{d}{dx}\frac{df}{dx} = \frac{df}{dx}\frac{d}{df}\frac{df}{dx} = \frac{df}{dx}\frac{d}{dx}\frac{df}{df} = \frac{df}{dx}\frac{d1}{dx} = 0.$$

Of course second derivatives do exist, so you might wonder what’s happening here. The sleight of hand happens after the third equal sign: swapping differentiation with respect to two independent variables is permitted, but $$x$$ and $$f$$ are not independent and therefore, this step is illegal.

I pointed this out, and received a mildly abusive comment in response questioning the quality of my mathematics education. Oh well. Maybe I will learn some wisdom and refrain from responding to strangers in the future.

This morning, Google greeted me with a link in its newsstream to a Hackaday article on the Solar Gravitational Lens. The link caught my attention right away, as I recognized some of my own simulated, SGL-projected images of an exo-Earth and its reconstruction.

Reading the article I realized that it appeared in response to a brand new video by SciShow, a science-oriented YouTube channel.

Yay! I like nicely done videos presenting our work and this one is fairly good. There are a few minor inaccuracies, but nothing big enough to be even worth mentioning. And it’s very well presented.

I suppose I should offer my thanks to SciShow for choosing to feature our research with such a well-produced effort.

Saturday afternoon was stormy. The lights flickered a bit during the storm, my UPSs came online several times. But then the storm left, and everything was back to normal.

At least here in Lowertown.

I didn’t check the news, so it was not until later Sunday that I learned, from a social media post from a friend who has been without power since, just how bad things really got.

And how bad they still are.

Hydro Ottawa’s map is still mostly red. Now “only” about 130,000 customers are affected, which is certainly less than the peak of well over 170,000, but to put that into perspective, Hydro Ottawa has a total of less than 350,000 customers; that means that at one point, more than half the city was without power.

As a Hydro official said on CTV News tonight, their distribution system is crushed.

And then there are all the downed trees, destroyed traffic lights, not to mention severely damaged homes and businesses. Not quite a like a war zone (of which we had seen plenty on our TV screens, courtesy of Mr. Putin’s “special military operation” in Ukraine) but close.

And of course the damage doesn’t stop at Hydro Ottawa’s borders: Hundreds of thousands more are without power in Eastern Ontario and also Quebec.

A beautiful study was published the other day, and it received a lot of press coverage, so I get a lot of questions.

This study shows how, in principle, we could reconstruct the image of an exoplanet using the Solar Gravitational Lens (SGL) using just a single snapshot of the Einstein ring around the Sun.

The problem is, we cannot. As they say, the devil is in the details.

Here is a general statement about any conventional optical system that does not involve more exotic, nonlinear optics: whatever the system does, ultimately it maps light from picture elements, pixels, in the source plane, into pixels in the image plane.

Let me explain what this means in principle, through an extreme example. Suppose someone tells you that there is a distant planet in another galaxy, and you are allowed to ignore any contaminating sources of light. You are allowed to forget about the particle nature of light. You are allowed to forget the physical limitations of your cell phone’s camera, such as its CMOS sensor dynamic range or readout noise. You hold up your cell phone and take a snapshot. It doesn’t even matter if the camera is not well focused or if there is motion blur, so long as you have precise knowledge of how it is focused and how it moves. The map is still a linear map. So if your cellphone camera has 40 megapixels, a simple mathematical operation, inverting the so-called convolution matrix, lets you reconstruct the source in all its exquisite detail. All you need to know is a precise mathematical description, the so-called “point spread function” (PSF) of the camera (including any defocusing and motion blur). Beyond that, it just amounts to inverting a matrix, or equivalently, solving a linear system of equations. In other words, standard fare for anyone studying numerical computational methods, and easily solvable even at extreme high resolutions using appropriate computational resources. (A high-end GPU in your desktop computer is ideal for such calculations.)

Why can’t we do this in practice? Why do we worry about things like the diffraction limit of our camera or telescope?

The answer, ultimately, is noise. The random, unpredictable, or unmodelable element.

Noise comes from many sources. It can include so-called quantization noise because our camera sensor digitizes the light intensity using a finite number of bits. It can include systematic noises due to many reasons, such as differently calibrated sensor pixels or even approximations used in the mathematical description of the PSF. It can include unavoidable, random, “stochastic” noise that arises because light arrives as discrete packets of energy in the form of photons, not as a continuous wave.

When we invert the convolution matrix in the presence of all these noise sources, the noise gets amplified far more than the signal. In the end, the reconstructed, “deconvolved” image becomes useless unless we had an exceptionally high signal-to-noise ratio, or SNR, to begin with.

The authors of this beautiful study knew this. They even state it in their paper. They mention values such as 4,000, even 200,000 for the SNR.

And then there is reality. The Einstein ring does not appear in black, empty space. It appears on top of the bright solar corona. And even if we subtract the corona, we cannot eliminate the stochastic shot noise due to photons from the corona by any means other than collecting data for a longer time.

Let me show a plot from a paper that is work-in-progress, with the actual SNR that we can expect on pixels in a cross-sectional view of the Einstein ring that appears around the Sun:

Just look at the vertical axis. See those values there? That’s our realistic SNR, when the Einstein ring is imaged through the solar corona, using a 1-meter telescope with a 10 meter focal distance, using an image sensor pixel size of a square micron. These choices are consistent with just a tad under 5000 pixels falling within the usable area of the Einstein ring, which can be used to reconstruct, in principle, a roughly 64 by 64 pixel image of the source. As this plot shows, a typical value for the SNR would be 0.01 using 1 second of light collecting time (integration time).

What does that mean? Well, for starters it means that to collect enough light to get an SNR of 4,000, assuming everything else is absolutely, flawlessly perfect, there is no motion blur, indeed no motion at all, no sources of contamination other than the solar corona, no quantization noise, no limitations on the sensor, achieving an SNR of 4,000 would require roughly 160 billion seconds of integration time. That is roughly 5,000 years.

And that is why we are not seriously contemplating image reconstruction from a single snapshot of the Einstein ring.

Came across a question tonight: How do you construct the matrix

$$\begin{pmatrix}1&2&…&n\\n&1&…&n-1\\…\\2&3&…&1\end{pmatrix}?$$

Here’s a bit of Maxima code to make it happen:

(%i1) M(n):=apply(matrix,makelist(makelist(mod(x-k+n,n)+1,x,0,n-1),k,0,n-1))\$
(%i2) M(5);
[ 1  2  3  4  5 ]
[               ]
[ 5  1  2  3  4 ]
[               ]
(%o2)                          [ 4  5  1  2  3 ]
[               ]
[ 3  4  5  1  2 ]
[               ]
[ 2  3  4  5  1 ]


I also ended up wondering about the determinants of these matrices:

(%i3) makelist(determinant(M(i)),i,1,10);
(%o3) [1, - 3, 18, - 160, 1875, - 27216, 470596, - 9437184, 215233605, - 5500000000]


I became curious if this sequence of numbers was known, and indeed that is the case. It is sequence number A052182 in the Encyclopedia of Integer Sequences: “Determinant of n X n matrix whose rows are cyclic permutations of 1..n.” D’oh.

As it turns out, this sequence also has another name: it’s the Smarandache cyclic determinant sequence. In closed form, it is given by

$${\rm SCDNS}(n)=(-1)^{n+1}\frac{n+1}{2}n^{n-1}.$$

(%i4) SCDNS(n):=(-1)^(n+1)*(n+1)/2*n^(n-1);
n + 1
(- 1)      (n + 1)   n - 1
(%o4)               SCDNS(n) := (------------------) n
2
(%i5) makelist(determinant(SCDNS(i)),i,1,10);
(%o5) [1, - 3, 18, - 160, 1875, - 27216, 470596, - 9437184, 215233605, - 5500000000]


Surprisingly, apart from the alternating sign it shares the first several values with another sequence, A212599. But then they deviate.

Don’t let anyone tell you that math is not fun.

Move over, general relativity. Solar gravitational lens? Meh. Particle physics and the standard model? Child’s play.

Today, I had to replace the wax ring of a leaky toilet.

Thanks to this YouTube video for some useful advice, helping me avoid some trivial mistakes.

Acting as “release manager” for Maxima, the open-source computer algebra system, I am happy to announce that just minutes ago, I released version 5.46.

I am an avid Maxima user myself; I’ve used Maxima’s tensor algebra packages, in particular, extensively in the context of general relativity and modified gravity. I believe Maxima’s tensor algebra capabilities remain top notch, perhaps even unsurpassed. (What other CAS can derive Einstein’s field equations from the Einstein-Hilbert Lagrangian?)

The Maxima system has more than half a century of history: its roots go back to the 1960s, when I was still in kindergarten. I have been contributing to the project for nearly 20 years myself.

Anyhow, Maxima 5.46, here we go! I hope I made no blunders while preparing this release, but if I did, I’m sure I’ll hear about it shortly.