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Makes a lot of sense.

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Right now physicists are struggling with the 'firewall paradox' - a problem in our theory of black holes.  But this is far from the first time physicists have been stuck with an annoying 'paradox'.

Back in the late 1800s, physicists noticed that an electron should get mass from its electric field.  Nowadays we'd say this is obvious.  The electric field has energy, and E = mc², so it contributes to the mass of the electron.  But this was before special relativity!

How did they figure it out?  They were very clever.  They used Newton's F = ma.  When you push on something with a force, you can figure out its mass by seeing how much it accelerates!

So, they did some calculations.   When you push on an electron with a force, you also affect its electric field.   It's like the electron has a cloud around it, that follows wherever the electron goes.  This makes it harder to accelerate the electron.  So, it effectively increases the electron's mass.  They calculated this extra mass.

They also did an easier calculation: how much energy this electric field has!  

Say m is the extra mass due to the electric field surrounding the electron, and E is the energy of this electric field.  Then they discovered that

E = ¾mc²


Had they made an algebra mistake?  Not really. 

Some really smart people all got the same answer!   Oliver Heaviside got it in 1889 - he was one of the world's smartest electrical engineers.  J.J. Thomson got it in 1893 - he's the guy who discovered the electron!   Hendrik Lorentz kept getting the same answer, even as late as 1904  - and he's one of the people who paved the way for relativity! 

But in 1905, Einstein wrote his paper showing that E = mc² is the only possible answer that makes sense.

So what went wrong? 

All those guys were assuming the electron was a little sphere of charge.   Why?  In their calculations, if was a point, the energy in its electric field would be infinite, because the electric field gets extremely strong near that point.  The mass contributed by this field would also be infinite.

If the electron were a tiny sphere, they could avoid those infinite answers.

But then they ran into this E = ¾mc² problem.  Why?  Because electrical charges of the same sign repel each other.  So a tiny sphere of charge would explode if something weren't holding it together.  And that something - whatever it is - might have energy.  But their calculation ignored that extra energy.

In short, their picture of the electron as a tiny sphere of charge, with nothing holding it together, was incomplete.   And their calculation showing E = ¾mc², together with special relativity saying E = mc², shows this incomplete picture is inconsistent.

So in the end, it's not a case of people being stupid.  It's a case of people discovering something interesting... by taking a plausible idea and showing it can't work.

Quit reading here if your brain is tired.


If you want the details, read what Feynman has to say:

• Richard Feynman, Electromagnetic mass,

He does all the calculations that explain this problem.  I've been reading his books since high school, but never really understood this part until now.  I'm thinking about problems with infinity in physics.

Here's a bit of what he says:

The discrepancy between the two formulas for the electromagnetic mass is especially annoying, because we have carefully proved that the theory of electrodynamics is consistent with the principle of relativity.  [...] So we are in some kind of trouble; we must have made a mistake. We did not make an algebraic mistake in our calculations, but we have left something out.

In deriving our equations for energy and momentum, we assumed the conservation laws. We assumed that all forces were taken into account and that any work done and any momentum carried by other “nonelectrical” machinery was included. Now if we have a sphere of charge, the electrical forces are all repulsive and an electron would tend to fly apart. Because the system has unbalanced forces, we can get all kinds of errors in the laws relating energy and momentum. To get a consistent picture, we must imagine that something holds the electron together. The charges must be held to the sphere by some kind of rubber bands—something that keeps the charges from flying off. It was first pointed out by Poincaré that the rubber bands—or whatever it is that holds the electron together—must be included in the energy and momentum calculations. For this reason the extra nonelectrical forces are also known by the more elegant name “the Poincaré stresses.” If the extra forces are included in the calculations, the masses obtained in two ways are changed (in a way that depends on the detailed assumptions). And the results are consistent with relativity; i.e., the mass that comes out from the momentum calculation is the same as the one that comes from the energy calculation. However, both of them contain two contributions: an electromagnetic mass and contribution from the Poincaré stresses. Only when the two are added together do we get a consistent theory.

This was a bummer back around 1905, because people had actually hoped all the mass of the electron was due to its electric field.  Note: this extra assumption is not required for the E = ¾mc² problem to bite you in the butt.   It's already a problem that the energy due to the electric field is ¾mc² where m is the mass due to the electric field. But the solution to the problem - extra 'rubber bands' - killed the hope that the electron could be completely understood using electromagnetism.

It is therefore impossible to get all the mass to be electromagnetic in the way we hoped. It is not a legal theory if we have nothing but electrodynamics. Something else has to be added. Whatever you call them—“rubber bands,” or “Poincaré stresses,” or something else—there have to be other forces in nature to make a consistent theory of this kind.

Quit reading here if your brain is tired.


Wikipedia has a good article on the history of this problem:

and this paper is also good:

• Michel Janssen and Matthew Mecklenburg, Electromagnetic models of the electron and the transition from classical to relativistic mechanics,

But if you want to understand what's going on, Feynman is better.

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A picture proof that √2 is irrational

This Friday I was hanging out with some philosophy professors.  This is always fun, because they think sort of like me, but different.  They seem more optimistic about our ability to solve all sorts of puzzles just by talking. 

To annoy them a bit, I said that philosophers are great at verbal  reasoning, but mathematicians should be good at three kinds of reasoning: verbal, symbolic and visual  reasoning.

In response, one of them showed me this picture proof that √2 is irrational.

We just need to show that it's impossible to have

a² = b² + b²

for whole numbers a and b.  So let's do a proof by contradiction.  We can assume a is the smallest whole number that obeys this equation for some whole number b.  We'll get a contradiction, by finding an even smaller one.

We do it by drawing a picture. 

The big square in this picture is an a × a square.  The two light blue squares, which overlap in the middle, are b × b squares.  

The area of the big square is the sum of the areas of the light blue squares.  But there are two problems.  First, the light blue squares overlap.  Second, they don't cover the whole big square!  These two problems must exactly cancel out.

So, the area of the overlap - the dark blue squares - must exactly equal the area that's not covered - the two pink squares. 

So, the area of the dark blue square is the sum of the areas of the pink squares!   But the lengths of the sides of these must be whole numbers, say c and d.  So we have

c² = d² + d²

But c is smaller than a.  So, we get a contradiction!

Actually this proof uses a mix of verbal and visual reasoning, with just a tiny touch of symbolic reasoning.  I wrote the formulas like a² = b² + b² just to speed things up a bit and reassure you that this was math.  I didn't really do anything with them.

The philosophers who told me about this are Mike Pelczar and Ben Blumson.  The picture here comes from a website Mike pointed me to:

+David Richeson,Tennenbaum’s proof of the irrationality of the square root of 2,

Richeson says:

Apparently the proof was discovered by Stanley Tennenbaum in the 1950’s but was made widely known by John Conway around 1990. The proof appeared in Conway’s chapter “The Power of Mathematics” of the book Power, which was edited by Alan F. Blackwell, David MacKay (2005).

On the other hand, Ben says John Bigelow published the proof in his book The Reality of Rumbers in 1988, without citing anyone.

We wondered if it was known to the ancient Greeks.

You can do similar proofs of the irrationality of √3, √5, √6 and √10:

• Stephen J. Miller and David Montague, Irrationality from the book,

And this particular style of proof by contradiction is famous!  It's called proof by infinite descent.  You assume you have the smallest whole number that's a counterexample to something you want to prove, and then you cook up an even smaller one.  It's really just mathematical induction in disguise, but it's more fun.  It was developed by Pierre Fermat - who, by the way, was a lawyer.

If you want to take all the fun out of the proof I just gave, you can do it like this.  

Assume a is the smallest whole number for which there's a whole number b with

a² = b² + b²


c = 2b - a


d = a - b

Then c and d are whole numbers and

c² = d² + d²

(You can do some algebra to check this.)  But c < a, so we get a contradiction.

Wikipedia shows you how to prove by infinite descent that whenever n is a whole number, either √n is a whole number or it's irrational:

Fermat did a lot more interesting stuff with this method, too!

#spnetwork arXiv:0909.4913 #geometry

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“The Weird History of the Pledge of Allegiance”

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