Spinor Info

Mathematical mysteries

Mar 202009

In the Futurama movie, The Beast with a Billion Backs, one scene features a blackboard with two different proofs of the Goldbach conjecture. The Goldback conjecture is one of the oldest unsolved problems in mathematics. One of those problems, like Fermat’s last theorem or the 4-color problem of maps that is deceivingly easy to state and fiendishly hard to prove: that every even number greater than 4 can be expressed as the sum of two primes.

Just how intriguing this problem is, it’s well illustrated by the following plot that shows the number of ways an even number between 4 and 1 million can be split into a sum of two primes:

Golbach partitioning to 1000000

This plot, taken from Wikipedia, clearly shows that the results cluster along curves (asymptotes? attractors?) that follow some kind of a power law with an exponent between ~0.68 and ~0.77, and there may also be some fractal splitting involved, too. This plot is known as Goldbach’s comet. All I have to do is look at it to understand why many people find number theory endlessly fascinating.

Killer minister

Aviation, Politics No Responses »

Mar 182009

In 2002, a tragic accident occurred over the skies of Europe, as a Russian passenger liner and a DHL cargo plane collided, causing the deaths of some 70 people, including the family of a certain Vitaly Kaloyev, a Russian architect working in Barcelona at the time.

Two years later Kaloyev killed Peter Nielsen, the air traffic controller that he believed was responsible for the death of his daughter. He was duly convicted and spent some time in prison. He was eventually released in late 2007 after winning an appeal on the grounds that his mental state at the time of the killing was not taken properly into account. He returned to Russia where many greeted him as a hero.

This is where things turn bizarre. Not long after Kaloyev’s return, Russia went to war with Georgia. One outcome of this war was the declaration of independence by the state of South Ossetia. Nationalists feelings were high on both sides of the intra-Ossetian border. And Vitaly Kaloyev was named deputy minister of housing in North Ossetia.

I can understand Kaloyev’s feelings. I can even understand why he killed Nielsen, even though Nielsen was himself a victim of incompetent management and bad organization. What I don’t understand is how a convicted killer can be named to such a high-profile public position. I think it speaks volumes about the politics of the region.

Phony outrage

Economy, Media, Politics No Responses »

Mar 172009

This phony, populist outrage over the AIG bonuses is really becoming ridiculous. They make it sound as if AIG’s top executives just took the bailout money as bonuses and ran with it. But that’s nonsense. First of all, the bonuses in question amount to barely 0.1% of the bailout that AIG received. Second, there are managers and there are managers… the ones getting these bonuses, are they the top level managers responsible for AIG’s demise, or are they the managers running, say, a successful branch office of AIG insurance?

Of course answering these questions might require some investigative journalism, some thinking, some hard explaining… not the kind of stuff the soundbyte journalism of the cable news universe likes. Glad I am not going to be home tonight, as I will not even accidentally watch the populist tripe of Lou Dobbs on CNN as he, no doubt, will take his turn at expressing outrage. Now if instead of spewing indignation, he actually took the trouble of locating a typical AIG executive who received a bonus and sat him or her down for a meaningful interview… of course I am quite willing to bet that this is not going to happen, not on CNN nor anywhere else… with the possible exception being BBC News.

The right way to think about the Pioneer anomaly

Physics, Space No Responses »

Mar 152009

I often get questions about the Pioneer anomaly and our on-going research. All too often, the questions boil down to this: what percentage of the anomaly can <insert theory here> account for?

This is a very bad way to think about the anomaly. It completely misses the fact that the Pioneer anomaly is not an observed sunward acceleration of the Pioneer spacecraft. The DATA is not a measured acceleration; what is measured is the frequency of the spacecraft’s radio signal.

I already capitalized the word DATA in the previous paragraph; let me also capitalize the words MODEL and RESIDUAL, as these are the right terms to use when thinking about the Pioneer anomaly.

As I said above, the DATA is the Doppler measurement of the spacecraft’s radio frequency.

The MODEL is a model of all forces acting on the spacecraft including gravity, on-board forces, solar pressure, etc; all effects acting on the spacecraft’s radio signal, including the Shapiro delay, solar plasma, the Earth’s atmosphere; and all effects governing the motion of the ground stations participating in the communication.

The RESIDUAL is the error, the difference between the MODEL’s prediction of the Doppler measurement vs. the actual measurement. This RESIDUAL basically appears as noise, but with characteristic signatures (a diurnal and an annual sinusoid along with discontinuous jumps at the time of maneuvers) that suggest mismodeling.

The goal is to make this RESIDUAL “vanish”; by that, we mean that only random noise remains, any diurnal, annual, or maneuver-related signatures are reduced to the level of background noise.

The RESIDUAL can be made to vanish (or at least, can be greatly reduced) by incorporating new contributions into the MODEL. These contributions may or may not be rooted in physics; indeed, orbit determination codes typically have the ability to add “unmodeled” effects (basically, mathematical formulae, such as a term that is a quadratic or exponential function of time) to the MODEL, without regard to the physical origin (if any) of these effects.

Anderson et al. found that if they add an unmodeled constant sunward acceleration to the MODEL, they can make the RESIDUAL vanish. This is the result that has been published as the Pioneer anomaly.

If one has a physical theory that predicts a constant sunward acceleration, it is meaningful to talk in terms of percentages. For instance, one may have a physical theory that predicts a constant sunward acceleration with magnitude cH where c is the speed of light and H is Hubble’s constant at the present epoch; it then makes sense to say that, “using the widely accepted value of H ~= 71 km/s/Mpc, the theory explains 79% of the Pioneer anomaly,” since we’re comparing two numbers that represent the same physical quantity, a constant sunward acceleration.

However, note (very important!) that the fact that a constant sunward acceleration fits the data does not exclude alternatives with forces that are not constant or sunward pointing; the DATA admits many different MODELs.

Now let’s talk about the thermal recoil force. It is NOT constant and it is NOT sunward pointing. As we recompute this force, incorporating the best thermal model that we can compute into the MODEL and re-evaluate it, we obtain a new RESIDUAL. There are, then, the following possibilities:

Suppose that the new RESIDUAL is as free of a mismodeling signature as the constant acceleration model and that its magnitude cannot be reduced by adding any unmodeled effects (i.e., we reached the level of our basic measurement noise.) Does it then make sense to speak of percentages? OK, so the thermal recoil force is 30%, 70%, 130%, you name it, of the constant sunward acceleration. But the thermal recoil force is neither constant nor sunward, and by incorporating it into the MODEL, we got a different trajectory than the constant sunward acceleration cas. Yet the RESIDUAL vanishes, so the MODEL fits the DATA just as well.
Suppose that the new RESIDUAL is half the original RESIDUAL at least insofar as the apparent mismodeling is concerned. What does this mean? Does this mean that the thermal recoil force and the resulting acceleration is half that of the constant sunward value? Most certainly not. Say it’s 65%. Now did we explain 50% of the anomaly (by reducing the RESIDUAL to 50%) or did we explain 65% of the anomaly (by producing a thermal recoil acceleration that’s 65% of the published constant sunward value?)

Instead of playing with percentages, it makes a lot more sense to do this: after applying our best present understanding insofar as thermal recoil forces are concerned, we re-evaluate the MODEL. We compute the RESIDUAL. We check if this residual contains any signatures of mismodeling. If it doesn’t, we have no anomaly. If it does, we characterize this mismodeling by applying various unmodeled effects (e.g., a constant sunward force, exponential decay, etc.) to check if any of these can characterize the RESIDUAL. We then report on the existence of a (revised) anomaly with the formula for the unmodeled effect as a means to consisely characterize the RESIDUAL. If this revised anomaly is still well described by a constant sunward term, we may use a percentage figure to describe it… otherwise, it’s probably not helpful to do so.

The stuff sci-fi fans like

Film, Pop culture No Responses »

Mar 152009

I’m watching a horrible movie (“Recon 2020“) and in an attempt to find out what it is about, I happened upon a Web page where I read this:

“Action packed lots and lots of monster killing. You know all the stuff sci-fi fans love.”

No. That’s not what sci-fi fans like. Or, at least, that’s not what this here sci-fi fan likes.

I grew up on the sci-fi of the likes of Asimov, Clarke, the Strugatsky brothers, Lem, or for that matter, Verne and Wells. Science fiction that contemplated the future of humanity, the role of science and technology, our destiny, the dangers we face, our chance of survival, our responsibilities. THAT’s what science fiction means to me. I don’t mind action and monsters, if well done it can even be fun, but no action or monster killing can make up for the absence of a credible plot and a meaningful story.

Einstein and Silberstein, Part 2

Physics No Responses »

Mar 102009

Einstein was right and Silberstein was wrong, but it’s beautifully subtle why.

Einstein’s point was that in a region free of singularities, the circumference of an infinitesimal circle should be 2π times its radius. He then showed that an infinitesimal circle perpendicular to, and centered around, the line connecting the two mass points in Silberstein’s solution has the wrong radius, hence the line connecting the two masses must be singular.

Silberstein countered by pointing out an error in Einstein’s derivation and then showing that a particular quantity rigorously vanishes along this line, implying that yes, the circumference of the infinitesimal circles in question do, in fact, have the right radii. This solution appeared in his paper in Physical Review in 1936.

Einstein then, in his published Letter that he wrote in reply to Silberstein’s paper, provided a technical argument discussing square roots and derivatives that, while correct, is not very enlightening. (It’s one of those comments that make perfect sense once you know what the devil he is talking about, but there’s no possible way you can actually understand what he’s talking about from the comment alone. Not that I haven’t been guilty of committing the same sin, even writing things that I myself wasn’t able to understand a few months later without going back to my notes or calculations. But I digress.)

By solving for the metric and obtaining an explicit formula, I was able to better understand this mystery.

First, it should be noted that the axis connecting the two mass points is also the origin of the radial coordinate, so the coordinate system itself is singular here even if there is no physical singularity. To verify that there is no physical singularity at this location, we can draw tiny, infinitesimal circles centered around this axis and check if they obey the rules of Euclidean geometry (one fundamental rule in curved spacetime is that in a small enough region, things always look Euclidean.) This was the basis of Einstein’s argument.

We can express the ratio of the angular and radial components of the metric in the form of an exponential expression, say exp(N) where N is some expression. Clearly, we must have N(r → 0) = 0 in order for the circumference of an infinitesimal circle centered around the axis to be 2π its radius.

As it turns out, N appears in the form [] + C where [] is just some complicated expression. It turns out that this part [] is constant along the axis, but its constant value is different between the two mass points vs. outside the mass points. I can draw the axis and the two mass points like this:

––––––––––––––––X––––––––––––––––X––––––––––––––––

The bracketed expression is constant along this line (not constant outside the line, such as above or below it but that’s not relevant now) but has one constant value between the two points (the two X’s) and another constant value on the two sides. The two constant values are not the same.

We can choose the constant C to make N vanish either between the two mass points or outside the two mass points but not both at the same time. When N vanishes outside, but not between, the two mass points, the singularity that remains serves as a “strut” holding the two mass points apart:

––––––––––––––––X================X––––––––––––––––

Or, when N vanishes outside, it’s as if the two mass points were attached by “ropes” to the “sphere at infinity”, hanging from there in static equilibrium as gravity pulls them together:

================X––––––––––––––––X================

So how come Silberstein was able to dismiss Einstein’s criticism? He was able to do so by making a particular choice of the sign of a square root that made N vanish both between and outside the two mass points. But this is where it helps to write N in the form that I obtained, symbolically as [] + C, with the bracketed part having one constant value along the axis between the two masses and another value outside. Sure we can make N vanish everywhere along the axis… by allowing C to have different values between vs. outside the masses.

This is fine along the axis… the masses themselves are singularities, so they represent a discontinuity anyway, so we are free to choose different integration constants on two sides of a discontinuity. This is indeed what Silberstein has done by choosing a particular sign of a square root in his formulation of the metric.

But, as Einstein pointed out, such a choice of sign leads to a discontinuity in derivatives. More explicitly, what happens is that the function N is defined not just along the axis but everywhere in spacetime… to get from a point on the axis between the two masses to a point outside, we need not go along the axis, we can leave the axis, go around the mass, and then return to the axis. As we do this, we encounter no singularity but the value of C must jump from one constant to another at some point. In other words, by removing the “strut” or “rope” singularity from the axis, we introduced a much worse singular “membrane” that separates regions of space.

One morale of this story is that when we use a coordinate system like polar coordinates that is not well defined at the origin, we must be ultra careful about that spot… since the coordinate system is singular here, this is where things can go wrong even if they appear perfect everywhere else.

Einstein and Silberstein

Physics No Responses »

Mar 102009

I’m reading about the debate in the 1930s between Einstein and Silberstein about the (non-)existence of a static two-body vacuum solution of the Einstein field equations.

Silberstein claimed to have found just such a solution as a special case of Weyl’s metric. However, he then concluded that the existence of an unphysical solution implies that Einstein’s gravitational theory has to be modified.

Meanwhile, Einstein dismissed Silberstein’s solution on two grounds. First, he claimed that there are additional singularities; second, he claimed that a solution that yields singularities is in any case not a proper solution of a field theory, so it certainly cannot be used to discredit that theory.

I disagree with Silberstein… just because there exist solutions that are unphysical does not unmake a theory. The equations of ballistics also yield unphysical solutions, such as cannonballs going underground or flying backwards in time… but it simply means that we chose unphysical initial conditions, not that the theory is wrong.

I also disagree with Einstein’s second argument though… field theory or not, some singularities can be quite useful and physically meaningful, be it, say, the “point mass” in Newton’s theory, the “point source” in electromagnetism, or, well, singularities in general relativity representing compact (point) masses.

But both these issues are more philosophy than physics. I am more interested in Einstein’s first argument… is it really true that Silberstein’s solution yields more than two singularities?

That is because when I actually calculate with Silberstein’s metric, I find regular behavior everywhere except at the two singular points. I see no sign whatsoever of the supposed singular line between them. What am I missing?

Oops, wrong button

Politics No Responses »

Mar 072009

During the Bush administration, we were used to dumb mistakes, but just because Bush is gone does not mean dumb mistakes have come to an end. Such as giving the Russian prime minister a gift, a button labeled “reset”… except that the mistranslated label actually carried a word that meant quite the opposite. Whatever one has to say about Condoleeza Rice, at least she spoke some Russian…

A-News at Six from the Civic Centre

Economy No Responses »

Mar 052009

That’s an ad that is running during local commercial breaks on CNN here in Ottawa, on Rogers Cable.

Too bad that as of yesterday, there is no A-News at Six. We live in interesting times…

Philip José Farmer

Books No Responses »

Mar 022009

I just saw the news… Philip José Farmer, one of the giants of science fiction, died at the age of 91. May he rest in peace.

The Sylvia Plath effect

Books No Responses »

Mar 022009

I’ve been reading Wikipedia about the so-called Sylvia Plath effect. It takes a bit of explaining.

I just finished reading Escape from Hell, the sequel to Larry Niven’s and Jerry Pournelle’s Inferno, itself a superb book, a modern rewrite of Dante’s hell. In this sequel, the principal companion of the protagonist is a woman named Sylvia Plath.

I must say I never heard of Sylvia Plath until this book. I knew nothing about her tragically short life nor about her death by suicide, a month before I was born in 1963. But that’s not why I was learning about her from Wikipedia this evening.

I finished Escape from Hell last night, less than 24 hours ago. Tonight, my wife and I watched the latest episode of The Simpsons. At the end of this (otherwise better than average) episode, there is a brief shot of Lisa holding a book in her hand. It’s the semi-autobiographical novel The Bell Jar, by Sylvia Plath.

One recurring theme in Inferno and Escape from Hell is that the protagonist meets just too many people he encountered in his earthly life for it to be a coincidence. I call such coincidences my Victrola moments… Many years ago, I played a Monty Python computer game which featured a Victrola, a word I never heard previously, but within 24 hours, I heard it in a completely different context. There were a few more instances like this in my life. Now, I can add Sylvia Plath to the list.

European socialism

Politics No Responses »

Mar 012009

What’s this “European socialist model” that Gingrich is trying to frighten American conservatives with? They make it sound bad, like, “Soviet communist model” or “German fascist model”. But last time I was there, “European socialism” was none of that. Why has Europe suddenly become a scary word in the American right’s political vocabulary? While it is reeling from the recession just like elsewhere, Europe has a strong (private sector) economy, a functioning health care system, in many ways a better infrastructure (think TGV!) than the US, a strong united currency, and last but not least, Europe has found a way for the (so far) peaceful coexistence of many peoples who speak dozens of different languages and have waged the bloodiest wars in history against each other. Is this what scares the GOP?

Personally, I’d be much more worried that under the present GOP leadership, America might end up following the Chinese corporatist model…

Sane conservatives

Politics No Responses »

Feb 282009

I take it back. There are still some sane conservatives out there in America. Some even think that right-wing talk radio is destroying conservativism. I only wish they were listened to.

How Radio Wrecks The Right

Limbaugh speaks

Politics No Responses »

Feb 282009

I am watching Rush Limbaugh on CNN and I shudder. Is this what the once mighty conservative right has become? A middle aged drug addict spewing poisonous populism? And this was once the party of Dwight D. Eisenhower, Richard Nixon, Ronald Reagan, or even, gosh, George H. W. Bush?

President Obama gave conservatives a chance to have their say about the stimulus package. They turned their collective backs on him. Unsurprisingly, the result is a stimulus package that serves Democratic priorities. And now they gripe. Even in the middle of a worldwide financial crisis, they’re not addressing real issues, they, Limbaugh among them, have nothing else to discuss but their partisan hatred of Democrats who, in their view, are out to destroy America.

Health care decisions

Health, Politics No Responses »

Feb 252009

Every time I hear American politicians warn their citizens about the dangers of a government-run health care system putting in the hands of government bureaucrats health care decisions that should be left to patients and doctors, I feel outraged.

I have experienced the government-run health care system in several countries. NOT ONCE was a government bureaucrat in any way consulted… on the contrary, all health care decisions were made by me and my doctor.

Indeed, the only country I know in which health care decisions are routinely scrutinized by (insurance company) bureaucrats is the United States of America, with its privately run health care system.

So I think I can tell with full confidence to my American friends that if what they are looking for is a health care system in which decisions are made by patients and doctors with no bureaucrats involved, a government-run health care system may be the very solution they’re looking for.

And the Oscars remind me of…

Film, Society No Responses »

Feb 232009

I don’t usually write about things such as the Oscars, as the topic leaves me supremely uninterested. However, it is hard to avoid it if you watch, listen to, or read the news. And I suddenly realized something, one of the reasons why I consider awards like the Oscars so irrelevant.

I probably wasn’t even in grade school yet (my guess is that it was the summer of 1968 and my family was watching the Mexico City Olympics) when I observed that there really are two types of sports. The results of sprinters, jumpers or weightlifters are measured by an objective standard: the reading from a scale, a ruler, or the face of a clock. But others, like gymnasts, are judged entirely differently: by a panel of experts who make subjective judgments of their performance. The ranking of a weightlifter doesn’t depend on which scale was used, as all are calibrated to the same standard; the ranking of a gymnast, on the other hand, depends heavily on the constitution of the panel of judges supervising the competition.

But what bothered me most is not that these differences exist, but that the world of adults did not seem to take notice. Grownups all talked about the Olympic gold of a weightlifter the same way as they did about the gold of a gymnast, as if there was no difference.

My disconnect with the world of grownups began around this time, I think. And, as my misgivings about the Oscars demonstrate, it has not ended.

That infamous stimulus bill chimp

Politics No Responses »

Feb 232009

Here’s this infamous political cartoon that everyone was so upset about:

Stimulus bill chimp

But… why? All I see is a clever political cartoon suggesting that the stimulus bill is so stupid, it must have been written by a chimp, perhaps the cimp that police shot the other day, so they need a new one to write the next bill.

Not in my wildest dreams did it occur to me to view this cartoon as racist. With all due respect, not in my wildest dreams would it occur to me that an intelligent person might compare Barack Obama to a chimp. I doubt it occurred to Barack Obama either. He doesn’t strike me as a person who takes offense when none was intended.

Socks

Cats No Responses »

Feb 212009

The White House cat during the Clinton years, Socks, died yesterday.

Socks in the White House

That infamous do-not-call list

Canada, Technology No Responses »

Feb 192009

When the Canadian do-not-call list was brought into existence, I dutifully registered my two landline telephone numbers. Little did I know that the list is available to anyone (including foreign operators who are not subject to Canadian do-not-call legislation) for a few dollars. I have not checked (stupid mistake!) only naively assumed that the list will be implemented properly, meaning that the government will offer a means for operators to submit their list of numbers for do-not-call verification. I never expected that the do-not-call list itself will be made available in its entirety. On the odd chance that I know personally some of the people behind this lousy implementation, I’ll refrain from using stronger words but… this really wasn’t a very smart thing to do, was it.

Fortunately, I never registered our cell phones. And I don’t think I will. I doubt the do-not-call list will stop fraudulent operators from trying to tell us about our vehicle warranties or Florida vacations and why help them by making our numbers readily available in the form of an easy-to-purchase, government approved list?

There are times…

Society No Responses »

Feb 172009

There are times when I wonder if moving to North Korea might be a good idea.

No, I am not altogether serious, but still… in North Korea, they may send you away for half a lifetime of hard labor for looking cross-eyed at a picture of the Dear Leader, but only in Canada can the driver of a passenger vehicle be fined for smoking while carrying a minor… even though, while the officer was writing the ticket, the minor, being a smoker, got out of the car to have a cigarette herself.

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