Sep 262022
 

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.

 Posted by at 7:18 pm
Sep 132022
 

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.

 Posted by at 1:54 am
Sep 012022
 

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.

 Posted by at 11:40 pm
Aug 202022
 

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?

 Posted by at 10:08 pm
Jun 102022
 

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.)

 Posted by at 1:19 am
May 252022
 

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.

 Posted by at 11:46 pm
May 232022
 

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.

 Posted by at 7:22 pm
May 232022
 

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.

 Posted by at 7:13 pm
May 062022
 

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.

 Posted by at 4:01 pm
Apr 202022
 

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.

 Posted by at 1:36 am
Apr 132022
 

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.

 Posted by at 2:26 am
Mar 242022
 

Between a war launched by a mad dictator, an occupation by “freedom convoy” mad truckers, and other mad shenanigans, it’s been a while since I last blogged about pure physics.

Especially about a topic close to my heart, modified gravity. John Moffat’s modified gravity theory MOG, in particular.

Back in 2020, a paper was published arguing that MOG may not be able to account for the dynamics certain galaxies. The author studied a large, low surface brightness galaxy, Antlia II, which has very little mass, and concluded that the only way to fit MOG to this galaxy’s dynamics is by assuming outlandish values not only for the MOG theory’s parameters but also the parameter that characterizes the mass distribution in the galaxy itself.

In fact, I would argue that any galaxy this light that does not follow Newtonian physics is bad news for modified theories of gravity; these theories predict deviations from Newtonian physics for large, heavy galaxies, but a galaxy this light is comparable in size to large globular clusters (which definitely behave the Newtonian way) so why would they be subject to different rules?

But then… For many years now, John and I (maybe I should only speak for myself in my blog, but I think John would concur) have been cautiously, tentatively raising the possibility that these faint satellite galaxies are really not very good test subjects at all. They do not look like relaxed, “virialized” mechanical systems; rather, they appear tidally disrupted by the host galaxy the vicinity of which they inhabit.

We have heard arguments that this cannot be the case, that these satellites show no signs of recent interaction. And in any case, it is never a good idea for a theorist to question the data. We are not entitled to “alternative facts”.

But then, here’s a paper from just a few months ago with a very respectable list of authors on its front page, presenting new observations of two faint galaxies, one being Antlia II: “Our main result is a clear detection of a velocity gradient in Ant2 that strongly suggests it has recently experienced substantial tidal disruption.”

I find this result very encouraging. It is consistent with the basic behavior of the MOG theory: Systems that are too light to show effects due to modified gravity exhibit strictly Newtonian behavior. This distinguishes MOG from the popular MOND paradigm, which needs the somewhat ad hoc “external field effect” to account for the dynamics of diffuse objects that show no presence of dark matter or modified gravity.

 Posted by at 2:30 am
Feb 222022
 

This is the last moment until well into the 22nd century that the current time and date in UTC can be expressed using only two digits.

I can only hope that this date will not be memorable for another reason, you know, something like the start of WW3?

 Posted by at 5:22 pm
Feb 202022
 

Recently, I came across an interesting article by a Jonathan Jarry from McGill University, suggesting that the much heralded Dunning-Kruger effect is not real, but a data analysis artifact.

Here is the famous Dunning-Kruger graph:

The usual interpretation is that those in the bottom quartile significantly overestimated their ability. This is the famous Dunning-Kruger effect.

But, Jarry says, a completely random model yields a very similar-looking graph:

and thus concludes that the Dunning-Kruger effect may not be real after all.

But wait. When we compare the two graphs, there are qualitative similarities but also striking differences. Notice how, in the second graph, the two curves intersect each other at roughly the halfway point. That makes perfectly good sense: If the model is that people in all four quartiles fail to assess their abilities accurately at the same rate, those in the bottom quartile will overestimate their ability just as much as those in the top quartile underestimate theirs. This would be the effect of random noise.

However, when we look at the original Dunning-Kruger curve, this is not what we see. Those in the bottom quartile overestimate their ability to a much greater extent than those in the top quartile underestimate theirs. Even in the 3rd quartile, people tended to overestimate their abilities, though only slightly, by the same amount as those in the top quartile underestimated theirs. So what the original Dunning-Kruger curve actually appears to show is a more ore less random spread in the 3rd and top quartiles, but significant bias in the bottom and 2nd quartiles, consistent with the notion that people in these quartiles overestimate their abilities.

Of course it would be nice to see a proper statistical analysis that also evaluates the statistical significance of the finding, but a simple, qualitative comparison of the two plots seems to show is that the Dunning-Kruger effect is real, after all.

 Posted by at 2:18 am
Feb 192022
 

The 64-antenna radio telescope complex, MeerKAT, is South Africa’s contribution to the Square Kilometer Array, an international project under development to create an unprecedented radio astronomy facility.

While the SKA project is still in its infancy, MeerKAT is fully functional, and it just delivered the most detailed, most astonishing images yet of the central region of our own Milky Way. Here is, for instance, an image of the Sagittarius A region that also hosts the Milky Way’s supermassive black hole, Sgr A*:

The filamentary structure that is seen in this image is apparently poorly understood. As for the scale of this image, notice that it is marked in arc seconds; at the estimated distance to Sgr A, one arc second translates into roughly 1/8th of a light year, so the image presented here is roughly a 15 by 15 light year area.

 Posted by at 5:04 pm
Dec 242021
 

Although we are not religious, we celebrate Christmas.

And I still cannot think of a better way to celebrate Christmas than with the words of the astronauts of Apollo 8, and the sense of awe they felt when they became the first human beings ever in the history of our species to be completely cut off from Mother Earth, when their spaceship disappeared behind the Moon.

Earthrise from Apollo 8

Re-emerging, they read passages from the Book of Genesis to their audience, with Frank Borman concluding with the words:

[G]ood night, good luck, a Merry Christmas – and God bless all of you, all of you on the good Earth.

To me, this is the most beautiful Christmas message ever.

 Posted by at 2:38 am