Built on Facts

Spherical Waves and the Hairy Ball Theorem

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Boy howdy do we love spheres in physics. Sure we might tell you that the reason involves deep truths in topology, and symmetry, and group theory, and all that mathematical arcana, and in fact there's a lot of truth to that. But if we're completely honest, or at least if I'm completely honest, I have to admit that I love spheres because they're easy. All that lovely deep symmetry tends to produce enormous simplifications in whatever actual calculations we happen to do involving spheres.

Hence, our love for pretending everything is a sphere, or at least close enough for govermnemt work. There's even a name for taking this particular approximation method to an extreme - the spherical cow model.

In optics, we work with spherical waves and isotropic radiators and various other means of pretending that light waves are cleanly and coherently originating from a point source. Of course you wouldn't expect such an idealization to perfectly describe any physical device in the laboratory, but in theory it's fine. Right?

Let's say you have an electric field which is spherically symmetric. That is, the field is of the form:

I.e., the magnitude of the field is uniform at any distance r from the center, and is directed radially outward. Faraday's law states that:

The curl of a radially-symmetric vector field is zero, so the right-hand-side of that equation is zero. Thus a radially symmetric electric field necessarily implies that the magnetic field B is static. You can go through exactly this same argument for the B field using Ampere's law, and you'll find that a radially-symmetric magnetic field necessarily implies a static E field. Thus if you want both fields to have radial symmetry, they both have to be static. That means Maxwell's theory limits our spherical cow models to situations which are constant in time, which is a rather substantial limitation.

Ok, but what about spherically propagating waves? Light waves propagate perpendicular to the E and B fields, so perhaps we can have a light wave propagating radially outward. The E and B fields would be tangent to the outgoing spherical waves, skirting the problem that you can't have non-static E and B fields both pointing radially outward.

Unfortunately we run into a brick wall there too. There's a theorem in mathematics which states given a vector field defined on a sphere such that every vector is tangent to that sphere, the vector field must be zero on at least one point on that sphere. This theorem is called the hairy ball theorem. I'm not making this up. Basically if you have a basketball covered in fur, there's no way to comb the entire thing smoothly. You'll always have at least one cowlick.

But if either one of the E or B fields is zero at some point, the Poynting vector ExB will be zero too, meaning no light is being radiated from that point. Thus a spherical light wave is impossible too, not just in practice but in theory as well. Sometimes that isn't a problem. We might be interested only in some region in which the assumption of spherical symmetry works just fine. Still, we are constrained and if we want to maintain connection with physical reality we can't let nice spherical cow assumptions get the best of us.

We'll finish this off with some comments on some possible counterexamples:

First, what about a steel ball-bearing which has been heated red hot, or a spherical star for that matter? Sure enough, both systems have spherically-symmetric time-averaged intensity distributions. But that doesn't make them sources of spherical waves. Every point on the surface of those objects is an independent source of radiation, and the E and B fields vary wildly at any given point near the surface. This wild variation will give a uniform time-averaged Poynting vector, but the fields and instantaneous intensity are far from uniform.

Second, what about a single atom in free space which has been raised to an excited state via a laser? Eventually it'll radiate by spontaneous emission, and since spontaneous emission is completely isotropic the field generated by the radiating atom has to be spherically symmetric. But since we're dealing with a single atom, quantum mechanics indicates that the atom emits a single photon, and that photon will assuredly be detected at exactly one point on a detecting sphere which surrounds the atom, breaking the symmetry.*

The moral of the story? How about this: even if you buy the (spherical) cow, the milk isn't necessarily free.

*I tend more and more toward the Willis Lamb view that the photon-as-a-particle picture is badly misleading, so we should really treat this glib explanation with more care. But in the spirit of the spherical cow, this handwaving argument is probably good enough for the moment.

Posted by Matt Springer at 2:24 PM • 8 Comments • 0 TrackBacks

Sunday Function

Category: Sunday Function

A reader asked me about the hyperbolic trig functions, sinh(x) and cosh(x). What are they for, and do they have an intuitive interpretation in physics?

That's a pretty good question. After all, most of the time you first meet the hyperbolic trig functions in intro calculus, where their rather odd definitions are presented and then used as test beds for blindly applying newly-learned differentiation rules. Ok, great. But what are they really?

To answer the question, we should start off with Euler's identity, which relates the exponential function with the regular trig functions. Proving this identity would take us a little far afield for this post, so for now we'll just take it for granted:

Now, replace x with -x and write down the equation again. But remember that cos(-x) = cos(x) and sin(-x) = -sin(x) because of the even/odd properties of those functions. With that in mind, Euler's identity is just as well written:

Now we have two equations, and by adding the first equation to the second we can cancel out the sin(x) terms. Or by subtracting the second equation from the first we can cancel out the cos(x) terms. We might as well do both, an we end up with:

and

This is kind of neat - we've taken functions that have their origins in ancient people studying triangles and we've written them in terms of the modern language of exponential functions and imaginary numbers. Pythagoras and crew would have no idea at all what something like cos(iπ) would be, but now we're in position to answer those sorts of questions. In the equations we've just derived, substitute ix in place of just x. Since i*i = -1 by definition, the complex exponentials become purely real, like this:

and

Thus if we want to know what cos(iπ) is, we just plug in (1/2)*(e^π + e^-π) into our calculators, and it turns out to be about 11.592.

Which brings us to the hyperbolic trig functions. Instead of the strange ex cathedra definition in intro calc books, we see that they're simply defined as the regular old trig functions when you plug imaginary numbers into them:

and

Which is actually sort of a nice little connection. For completeness, here's their graphs:

sinh(x):

cosh(x):

Now what's the physical interpretation of sinh and cosh? To be honest there isn't much of one - really they're just sort of a change of basis, and anything you can write in terms of sinh and cosh can usually be written more clearly in terms of exponential or ordinary trig functions. They do crop up in various differential equations, including the Laplace equation in classical E&M, and potential steps in the Schrodinger equation. But at least now we know where they come from.

Posted by Matt Springer at 2:09 PM • 21 Comments • 0 TrackBacks

Popular Science and Time Travel Shenanigans

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Our department here at Texas A&M has a student chapter of the Optical Society of America, and each week a student or professor gives a talk about something interesting while the rest of us eat pizza. I've been working on and off on a talk I'm going to give, tentatively titled "Just what the @#$% is a photon anyway?". The more I dig into the subject, the more I start to think that (like the rubber-sheet analogy in GR) the "particles of light" view that tends to be the common impression tends to cause more confusion than enlightenment. I have some good company here - E.T. Jaynes wrote a famous paper expressing his own problems with the concept, and I have to say he's pretty convincing.

The BBC article percolating aroud the web reenforces my suspicions. "Time travel: Light speed results cast fresh doubts" Its intro sentence:

Physicists have confirmed the ultimate speed limit for the packets of light called photons - making time travel even less likely than thought.

Honestly the article isn't that bad. I'm used to much worse in the popular press. It does get across the point relatively intact. The headline is a little sensationalist - as I told an emailer, really it should be something like "Scientists measure speed of a light photon in rubidium vapor, turns out to at travel speed of light". It is, after all, just one (important) measurement in one (very interesting) physical system. It doesn't prove that the result holds in all times and places, indeed no experiment can. The paper is here, if you're curious. It's a elegant experiment and I congratulate the authors on a fine job.

Here's the sketchy BBC paragraphs that take the "photon as particle" view too literally and run into trouble:

While the limit in vacuum is a fixed number - some 300,000km per second - the speed of light can vary widely in different materials.

These differences explain everything from why a straw looks bent in a glass of water to experiments in cold gases of atoms in which light's speed is actively manipulated.

Some of those experiments showed "superluminal" behaviour, in which photons travelled faster than the speed of light in a given medium.

It remained, however, to determine whether or not individual photons could exceed the vacuum limit.

The last sentence directly contradicts the one before it, because in so-called superluminal experiments nothing actually propagates faster than the speed of light. The pulse of light looks like it exceeds c, but only because the material has been "pre-prepared" in such a way as to amplify the leading edge of the pulse which makes the pulse peak appear to shift forward.

Now, Shengwang Du and colleagues at the Hong Kong University of Science and Technology have measured what is known as an optical precursor.

Like the wind that moves ahead of a speeding train, optical precursors are the waves that precede photons in a material; before now, such optical precursors have never been directly observed for single photons.

By passing pairs of photons through a vapour of atoms held at just 100 millionths of a degree above absolute zero - the Universe's ultimate low-temperature limit - the team showed that the optical precursor and the photon that caused it are indeed limited to the vacuum speed of light.

"By showing that single photons cannot travel faster than the speed of light, our results bring a closure to the debate on the true speed of information carried by a single photon," said Professor Du.

"The waves the preced photons in a material?" Oh dear. Light is light. Waves are made of photons, and individual photons (contra their particle-like popular image) express wave-like behavior. But however you look at it, light never travels faster than c according to both classical E&M and modern QED.

I'd also quibble a tiny bit with professor Du's quote. He has shown that that particular precursor in that particular material travels at c. Of course we would all be stunned if any material turned out to be an exception, but as careful scientists we shouldn't state that any single experiment proves a universal truth.

Still, not a bad article. Now if I can figure out exactly what photons are, I'll let you know. But it's murky waters...

Posted by Matt Springer at 10:04 PM • 10 Comments • 0 TrackBacks

Inventing Relativity, 1860s style

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By the 1860s, the classical theory of electricity and magnetism was on a very solid theoretical footing. Maxwell's equations describing the interplay of charges and currents with electric and magnetic fields were on paper by 1862, and with some changes in notation they're the exact same today. Relativity wouldn't be invented for another half-century or so, and that makes it all the more remarkable that Maxwell's equations don't actually need to be modified at all to work in a relativistic framework. Lorentz covariance is built right in, though it's a bit hidden.

But Maxwell and Faraday and Ampere and the rest didn't know that. There were some tantalizing hints though, and in fact it was the exploration of classical electrodynamics that led Einstein to the theory of special relativity. It's entertaining to take a look at some of those hints, which are lurking right there in second-semester intro physics.

Consider a uniformly charged wire alongside a particle of charge q:

(Apologies for the sloppy PowerPoint graphic, but it probably gets the gist across.) We know from freshman physics that (via Gauss' law), the electric field generated by that charged wire is:

The force experienced by the charge q in the field is:

We'll say the particle and the wire are both positively charged, so the force is repulsive and pointing radially outward from the while. For simplicity, we won't bother with vector notation in this post, but do keep in mind that forces and fields are vectors and do have a direction that we have to pay attention to.

Now let's start moving the wire and the particle to the right at a constant velocity v. Or equivalently, move ourselves to the left at a constant velocity v, leaving the wire and particle stationary in the lab frame. Physically, they are the same situation, and this ends up being a key part of relativity. A moving charged wire is a current carrying wire, since an electric current is just moving charge by definition. The current I is given by I = λv, since charge/time is the same thing as (charge/length)*(length/time). A current produces a magnetic field which wraps radially around the wire with a magnitude of

So now we have electric and magnetic fields, like so:

Now the force on a charged particle moving in a magnetic field is F = qvB, and in this case it'll be directed radially inward, toward the wire.

Now hold on - when everything was standing still, the net force was qE, pointed away from the wire. When we changed nothing at all except sliding our own chair in the lab to the left at speed v, suddenly the net force is F = qE - qvB, which is something completely different. Substituting the expressions for E and B in, the net force is:

What the heck? The force is an objectively measurable thing which gives the particle a specific acceleration. It can't possibly be different depending on whether we're moving or not. If classical electromagnetism makes such a prediction, surely the theory is wrong. Right?

Right - if we assume that all these charges and fields and currents and lengths are all the same in both frames. And that simply isn't the case. You need relativity.

But it's 1860, and we haven't got relativity yet. How might we go about groping in the dark toward an answer? Well, we might postulate that fields aren't the same in each frame. In the frame where the system is stationary, we have (say) the electric field E, while in the moving frame we have some different electric field E', given by some coefficient E = αE', where α is a function of v. If we assume the force is the same in both frames (it isn't, but we don't know that in 1860), we can look for that coefficient by solving:

Which gives after a little algebra:

If this initial groping-in-the-dark attempt at fixing classical E&M to work like our Gallilean intuition says it should is right, the E field in the moving should be slightly bigger in the moving frame than in the rest frame. Maybe the motion with respect to the ether somehow magnifies it, I dunno. In any case the correction factor is very small. If we're talking about laboratory speeds in the m/s range, the correction factor is on the order of parts-per-quadrillion.

But is it right experimentally, if we could measure it? As it would turn out, no - but it's close. It turns out that α should be the Lorentz γ factor, but at small speeds his γ and our α have the same order of magnitude (though we're still off by a factor of 2, it turns out).

In any case our first attempt at a relativity theory is wrong - but closer to right than we were without it. Don't be tempted to think that even people like Einstein had their brilliant ideas spring into being fully formed. Even the seemingly sudden great advances represent a lot of hidden hard work, tentative steps, and false starts.

Posted by Matt Springer at 1:17 PM • 13 Comments • 0 TrackBacks

The Philosophy of Science of Lord of the Rings

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Phil at Bad Astronomy opined (and it is a common opinion) that the supernatural is incoherent:

If you posit some thing that has no perceivable or measurable effect, then it may as well not exist. And as soon as you claim it does have an effect -- it can be seen, heard, recorded, felt -- then it must be in some way testable, and therefore subject to science.

Joshua was not so sure about this. The supernatural could, perhaps, interact observably with the universe at some times but not at others. Under normal circumstances the normal laws apply, under others, supernatural stuff happens. Chad weighed in on that:

The obvious rejoinder to this, leaped upon by a bunch of people in comments, is that if the supernatural doesn't behave according to known laws of nature, that just means that the known laws are incomplete, and some more complete theory would encompass the seemingly supernatural. Which is true as far as it goes, but misses a subtle point, namely the determinability of those laws.

He went on to give the difficulty of observing quantum effects on a macroscopic level as an instance of even "normal" laws of nature being difficult to completely verify.

Basically this is a long-running argument which is interesting but basically totally irrelevant to the perennial arguments between theists and atheists (of New or other varities). In practice, no one cares if God or other possible forms of the supernatural would somehow ontologically "above" the laws of nature, or whether they would simply be a part of nature obeying laws that aren't normally apparent in everyday life. Most people only care whether God exists in a way that could be empirically verified by, say, dying and waking up in Heaven.

But I'm not about to make this the bazillion and first post on ScienceBlogs to wade into that tar pit. I care about Saruman's grad students.

In Lord of the Rings, we have a universe in which magic unambiguously exists. Sure, you could argue it doesn't exist in the philosophical nitpick sense, since magic might just be part of natural laws that are incompletely known. But again, nobody really cares about that. In the practical sense, there's magic. Magic rings, ghost armies, enchanted ropes, spoken incantations, a realm of gods reachable by boat, etc.

Professor Saruman lives at the top of a tower noodling around his library while his employees work underground doing arcane biology and chemical engineering experiments. At least some of this is manifestly ordinary science, in the sense that we wouldn't consider it magic if it happened in our world. They build war machinery. Arguably they develop and deploy gunpowder weapons:

Even as they spoke there came a blare of trumpets. Then there was a crash and a flash of flame and smoke. The waters of the Deeping-stream poured out hissing and foaming: they were choked no longer, a gaping hole was blasted in the wall. A host of dark shapes poured in.

'Devilry of Saruman!' cried Aragorn. "They have crept in the culvert again, while we talked, and they have lit the fire of Orthanc beneath our feet..."

In the book this might be interpreted as pure magic, though both I and the films think gunpowder is more plausible. Sauron's grad students have managed to figure out quite a bit of actual science despite living in a world where magic exists.

Conversely, not all magic in the world is available for scientific examination despite manifestly existing. The One Ring by its nature tends to spend its time on the finger of someone who's not going to make it available for peer review. No doubt knowing there was a Nobel in it, Saruman convened professional conferences on the topic, with little success. Systematic scientific methods just didn't work very well.

Ok, ok. I admit this post isn't serious. But I hope it illustrates why I can't take the whole "the supernatural and science are/aren't incompatible by definition" argument seriously either. In any practical sense, the question of the philosophical comparability of science and the supernatural is completely orthogonal to the question of whether the supernatural exists in a way that most people are likely to care about. Nor do I claim that science is irrelevant to that question; some claims about the supernatural are testable by science. By all means feel free to argue apologetics until you're blue in the face. But the particular philosophical question being batted around here is basically on the level of speculating whether we're really in the Matrix.

Which I'll admit is also fun to argue about...

Posted by Matt Springer at 10:00 AM • 21 Comments • 0 TrackBacks

Sunday Function

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Head down to Box Office Mojo and pull up the list of the top grossing films of the year thus far. Seven of the top ten have a dollar gross beginning with the number 1. Okay, that's not too weird. Big films tend to pull down somewhere between $100-200 million, while only the real monsters have high grosses. So what if we look at the inflation-adjusted all-time list, which is less likely to be fixed by the coincidental size of the film-going public and ticket prices? Again, seven of the 10 have grosses beginning with 1.

Well, maybe movies are just weird. What about cities? In the US, five of the top ten cities have a population figure which begins with a 1.

Maybe cities are just weird too. How about election results? If you rank the states of the 2008 US presidential election by Obama's vote total, zero of the top ten have Obama vote totals beginning with 1 - but then again, all the rest of the top 20 did.

Why this preponderance of numbers that happen to start with 1? Is it just an artifact of the data sets I've picked, or something more interesting. Try a thought experiment:

Pick a number, say, one million. Write it out in decimal notation and it reads 1,000,000. Its first digit is the number 1. If you increase or decrease 1,000,000 by ten percent, you get 1,100,000 or 900,000, which start with 1 and 9 respectively. If you increase or decrease 1,000,000 by twenty percent, you get 1,200,000 or 800,000, which start with 1 and 8 respectively. If you increase or decrease 1,000,000 by thirty percent, you get 1,300,000 or 700,000, which start with 1 and 7 respectively.

Continue this exercise and basically the pattern continues. Essentially the million numbers following 1,000,000 start with 1, but the million below 1,000,000 can start with just about anything, including 1.

Obviously had you started with (say) 3,000,000 the effect would be much less pronounced, but it would still be there. It's possible to rigorously analyze this sort of thing, and the result is Benford's Law, which gives the probability distribution for the first digits of random numbers:

Plotting this distribution gives:

From Benford's law, you'd expect around 30% of leading digits to be the number 1. Not every set of randomly chosen integers satisfies the conditions required to Benford's Law and its odd preponderance of 1s, but lots of them do. In the financial industry, the law has even been used to search for fraud. Humans are generally terrible at making up random numbers that act anything like actual random numbers, and as a result the figures they make up when cooking the books don't tend to satisfy laws like Benford's.

Unless you're angling to hang out with Bernie Madoff in Club Fed, you should probably use your math knowledge for good rather than evil. But if you're gonna cook your books, your recipe should probably include about 30% 1s as leading digits...

Posted by Matt Springer at 3:37 PM • 4 Comments • 0 TrackBacks

Maru the Cat does dimensional analysis

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Here is a picture of (I think) Maru the cat playing in a bag. He loves bags.

Here is the same picture of Maru, at half the size:

Now imagine that Maru is a physicist and the pictures are not pictures but instead windows into the universe he occupies, separate from ours with (possibly) its own unique set of physical laws. The only difference between the two universes is that one has the lengths of everything reduced by a factor of 2.

Can the parallel versions of Maru tell which universe they're in - the smaller or the larger? Or if you want to imagine what you might do, suppose that in some Kafkaesque way you find that you wake up one morning and everything if twice the size it was the day before, including you. Could you tell? After all, all the rulers are twice as big too.

Physicists don't generally directly worry about questions like this, but they're actually very similar to questions about whether fundamental constants like the speed of light are truly constant. If the speed of light doubled overnight, what would change? Maybe lots of things - the famous Einstein relation E = mc^2 would seem to imply that the ratio between a fistfull of matter and the corresponding quantity of energy would change by a factor of four, which would certainly have an observable effect in things like the nuclear reactions that power the sun.

Or, maybe simultaneously everything suddenly got less massive by a compensating factor and the change in m balances the change in c. Or, maybe the rate of flow of time (whatever that might be) changed or maybe length scales changed, or who knows?

This isn't idle dorm-room philosophy. Physics needs to be able to deal with quantities that change in time, be they truly fundamental or not. There's actually some real subtlety involved, but the essentials revolve around just what we mean by units and dimension.

Units are specifically defined increments of a physical quantity. Meters, feet, angstroms, miles, astronomical units, and parsecs are all measuring the same thing in different increments. Likewise liters, gallons, and barn-megaparsecs. And teslas and gauss, and so forth. You can use whatever units are convenient, and keeping track of units is one of the first things you learn in freshman physics. In theoretical physics it's very common to pick a system of units where various fundamental constants are equal to 1. For instance, with c = 1, Einstein's mass-energy relation becomes the rather zen-like E = m.

Dimensions are the thing being measured, independent of units. Length in space, duration in time, volume, magnetic field strength, momentum, and so forth.

The crucial test of measurement in physics is the variation in ratios of numbers with identical dimensions. If everything in the universe doubles in size, the ratio of your height to the height of a ruler will stay constant. If only you double in size, the ratio of your height to the ruler will double also, leading to a possible NBA career. (As a naturalized Texan, I believe I should now proffer support for Dallas. Go Dallas!) Ratios of numbers with identical dimensions are said to be dimensionless, and they're the interesting quantities.

So we don't look for variation in the speed of light as such. We look for variation in dimensionless numbers involving the speed of light. The most famous of these numbers is probably the fine structure constant, which is the square of the electron charge divided by the product of Planck's constant and the speed of light. (Maybe with another constant in there too, depending on the system of units you're using for electric charge. The value of the constant, like all dimensionless constants, is that its numerical value is identical regardless of the system of units.)

The fine structure constant is approximately α = 0.0729735..., which happens to be a hair under 1/137. Why this particular value? Nobody knows, but as a dimensionless number appearing all over physics it's a great test subject for investigating the possible change in the laws of physics (including the speed of light) over time. So far there's no real indication that it has ever changed, which is a nice thing to know.

But if it ever did, we could learn about it along these lines without having to worry much about changes in our lab equipment.

Posted by Matt Springer at 6:31 PM • 14 Comments • 0 TrackBacks

Gauss' Law PROVED WRONG!

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Just though I'd try writing a post title in the style of a crank. Kinda fun!

Gauss' law, of course, is not wrong. But I got a question from a reader that deceptively simple and an interesting example of a theorem not quite working the way you'd expect. I've gone over Gauss' law before, so as a quick refresher I'll just say that it relates the electric charge at a location to the way the electric field lines diverge at that location. Symbolically (and in, appropriately, Gaussian units):

E is the electric field, ρ is the charge density. Draw a closed surface around the charge in question and integrate over the enclosed volume:

The integral of the charge density within the surface is of course just the total charge enclosed within the surface:

Quoting Wikipedia's article on the divergence theorem, it's true that the volume integral of the divergence is equal to the surface integral of the vector field itself:

Wikipedia's notation for the integrals is a tiny bit different, but you get the gist. Making this substitution in Gauss' law gives:

Where n is the normal vector - the direction locally outward perpendicular to the surface. This is useful because if the symmetry of the problem can guarantee that E is constant and parallel to n, E.n is just the scalar E and comes outside the integral. Which is nice, and allows such neat things as a proof of Coulomb's law for a point charge, the shell theorem, and so forth.

Ok, so here's the reader question: consider a constant ρ(x,y,z) = &rho₀;. In other words, a uniform charge that fills all of space. By symmetry, it's clear that there's no preferred direction and thus the electric field is zero everywhere. But pick a point as an origin and draw a sphere around it - inside the sphere is some positive amount of charge Q, but the symmetry requires the total flux be zero. There seems to be a contradiction in a fairly simple-seeming example of Gauss' law.

I have to issue my standard disclaimer that I'm not a mathematician (or even a theorist!), but I think I've sussed out the problem. The problem is not with Gauss' law, but with the divergence theorem itself. As you'd expect, a theorem is a mathematically proven statement about abstract objects that satisfy certain conditions. Of course in this case the vector field F has to be a well-behaved function in some sense, which basically all functions in physics are. But less well known is that F must be compact supported - ie, that it's only nonzero over some finite region. This ρ isn't. The divergence theorem fails to hold, and it's no longer possible to draw conclusions about the electric field in the usual way. You have to try something else.

Exercise: The usual undergrad method of Gaussian surfaces works just fine with lines of charge and sheets of charge, even though they don't vanish at infinity. Why? (Seriously, why? I don't know for sure - I suspect it's because the charge distributions are zero at infinity except for a set of measure zero, but I couldn't say for sure.)

Posted by Matt Springer at 9:35 AM • 32 Comments • 0 TrackBacks

Thoughts on Osama

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Some initial thoughts, on a beautiful day in a palpably better world without Osama:

1. I'm astonished he was still alive. I was certain he died from an anonymous bomb or health problems sometime between '01 and '04. He hadn't released any tapes or videos with unambiguous confirmation of when they were recorded, and the general consensus was that al-Qaida was now a fully decentralized organization that had adjusted to operating without him. The fact that he wasn't dead meant that, unlike his suicide bombers and guerrillas, he had basically abandoned his own cause to live in (very) quiet luxury. So add "coward" to "monster".

2. Weekly World News (a now-defunct humor tabloid that reported entirely made-up news) used to report that Osama was alive and hiding in Florida. They weren't that far off. Hiding in a mansion next to a military base in a major city near the capital does not give me a very high opinion of Pakistan.

3. Frankly three wars is three more than we should be in, but I wouldn't be too upset if we added a fourth to decapitate the Pakistani government if it turns out they were complicit in hiding Osama. Of course cooler heads will prevail and it won't happen, but that's my emotional instinct.

4. Most of the reaction among my friends on Facebook has been celebratory. There was only a little political sniping ("Obama did what Bush couldn't" vs. "Bush should get the credit for everything that lead up to it"). In any event that argument is pretty silly, akin to crediting or blaming the Secretary of Transportation about the pothole on your commute to work. Both presidents deserve their share of credit, and the great majority of the credit goes to the anonymous intelligence analysts, spies, and special forces who did the hard work.

5. On almost every other issue I'm an Obama critic, but he gets top marks from me on this one. Top marks plus extra credit, because he ordered the dangerous but much more effective ground assault rather than a bombing run.

6. In his speech - which was almost surreal to watch - Obama said that he ordered the mission with the goal of the death or capture of Osama bin Laden. I wonder if the actual order was to kill him, full stop. One of the serious weaknesses of the US counterterrorism effort is a nearly complete lack of a clear and uncontroversial way to try suspected terrorists in court. There is almost no possible way a trial of Osama could have been anything but a debacle, and dead men need no trial.

7. Not sure I like the burial at sea though. Obviously you don't want him stuffed at the Smithsonian, but you'd like to have some independent forensic analysis by countries whose citizens are not likely to immediately believe the White House.

8. Of course Osama's death won't make much difference in regard to preventing future terrorism. But who cares?

9. Nobody who was part of the raid will ever have to buy a beer again.

10. "Never forget" eventually became something of a punchline, but somewhere deep within even the jaded and partisan souls of modern Americans, we never did forget.

Posted by Matt Springer at 9:55 AM • 32 Comments • 0 TrackBacks