Apr 21, 2005

science - E=mc^2

So I've done a little research to find out how Einstein derived E=mc^2 and I've found some interesting things.

First, it turns out the derivation isn’t quite as easy as I had hoped it might be. Relativity derivations in general are quite straight forward. That’s probably why this particular derivation never shows up in science books. I was also wrong when I stated it was derived from the General Theory of Relativity. It’s actually derived from the Special Theory of Relativity and also uses Maxwell’s equations. Maxwell’s equations describe electromagnetic radiation and while lesser known are probably more beautiful than any other set of equations.

While I’m here, another fun fact. Einstein’s General and Special theories are named that way because the Special theory applies to a special set of reference frames – namely those that move with constant velocities in relation to other frames of interest. The General theory applies to a more general set of reference frames – namely those where acceleration also can occur. Before I knew this, I could never keep the two straight.

While I’m still here, another fun fact. Einstein first started working on his relativity theories when he was a teenager. He performed his first gedanken experiment or thought experiment in his late teens. He asked what would happen if you were moving at the speed of light. What interested him was that Newton showed that if you are moving at a constant velocity you can’t actually tell that you are moving. You could be still or moving 10,000 miles per second (assuming there are no visual clues) and you would have no way to determine this movement. That’s why when you are on a subway or a train you sometimes think you are moving when it’s another train that is really moving.

You might argue that you could feel the wind on your face or the jiggling of the vehicle you were in. But these both imply some acceleration is occuring. We're talking just straight fixed velocity movement. A better way to think about it is if you were in the middle of space with no visual cues - no stars or planets. Just black. How would you know you were moving or stationary? Newton said you wouldn't.

If light travels the speed of light and you are traveling the speed of light (we’re back to the gedanken experiment) then light should never reflect off your face - the light wave equivalent of breaking the sound barrier. And if you looked in a mirror it would be black. But then you could tell you were moving forward at a constant velocity at the speed of light. But Newton said that isn’t possible. Einstein knew something had to break – either Newton’s law or the speed of light or something. I don’t know about you but when I was a teenager I was trying to find a way to get a fake ID.

Let’s look at Einstein’s strategy to get to his famous equation. He starts with a body at rest (let’s say it’s an atom). The atom emits 2 pulses of light of equal energy in opposite directions. Einstein then analyzes this ‘act of emission’ from a second frame of reference – one that is moving at a constant velocity (that’s how he’s able to use his Special theory). Einstein needs to use Maxwell’s theory of electromagnetism and his theory of relativity to calculate the physical properties of the pulses in the second frame (I'll explain later). By comparing the two descriptions of this emission event, E=mc^2 falls out.

Actually it doesn’t. Einstein never said E=mc^2. He said m=L/c^2 where L = energy = E. And he really was talking about L as a change in energy. He ultimately is saying if a particle loses energy L, then it must lose an amount of mass equal to L/c^2.

So let’s go through his theorem. It’s posted here if you are interested in reading the original. First let’s define his variables. The atom has a certain amount of energy as it sits there, E0. After it emits these 2 photons it has a new energy, E1 (it's going to lose some energy from the emission). Let’s call the energy of the 2 photons together L. So individually they have ½ L energy. Thus by the conservation of energy,

Eo = E1 + ½ L + ½ L

Now let’s look at it from a different perspective. From a perspective moving with a constant velocity, v, with respect to the fixed atom (we'll assume the movement is parallel to the photon emissions for simplicity). Here the energy of the atom has a similar form we’ll call Ho and H1. But the energy of the light waves changes. This is due to our motion in this moving frame of reference. It’s very similar to the Doppler effect you might hear as an ambulance passes by. As it approaches the ambulance siren has a certain sound, and as it passes it drops in pitch. What’s causing this is the compression of the sound waves in front and decompression behind the ambulance. Compression leads to a higher frequency and decompression leads to a lower frequency. Same thing with light.

But with light the compression has a more significant meaning. The energy of the photon is a function of the frequency. So this change in frequency, due to the fact that we’re in a moving reference frame has to be accounted for with both Maxwell’s equations and Einstein’s special relativity equations (that work is done here). This accounting introduces a square root term involving the velocity of the reference frame and the speed of light, c. The denominator in the last 2 terms of the second equation below is effectively from Einstein and the numerator is from Maxwell. If you want to see the full derivation go to that link above (down towards the end where he talks about Doppler). It’s a bit messy but here is the full set of equations.

E0 = E1 + ½L + ½L

H0 = H1 + ½L(1 - v/c cosΦ)/(1 - (v/c)^2)^½ + ½L(1 + v/c cosΦ)/(1 - (v/c)^2)^½

Simplifying H0

H0 = H1 + L/(1 - v^2/c^2)^½

First, ignore the cosΦ (for us it's 1). Einstein is just being complete here (it refers to the angle between the x axis and the light vector). And besides, once we simplify the 2nd equation the term disappears. See how the energy terms for the light waves increase in one case and decrease in the other (note the + and – sign difference in the last 2 terms of the second equation). This is the ‘Doppler Effect’.

Now Einstein subtracts the two equations to find the energy difference between the two reference frames.

H0 - E0 - (H1 - E1) = L (1/(1 - (v/c)^2)^½ - 1)

Now this bit is a little tricky. H – E again is the difference in the energies of the atom in 2 reference frames. This difference will in part be due to some kinetic and potential energy difference. In other words,

H – E = K + P

Where K and P are the kinetic and potential components. Since we are not introducing anything to change the potential energy of the system (like adding gravity), the P component is the same before and after the photon emission. Therefore,

H0 – E0 = K0 + P


H1 – E1 = K1 + P

Subtracting these two equations we get,

H0 – E0 – (H1 – E1) = K0 – K1

This is the form in the equation on the left hand side (5 equations above, the jpg picture). Substituting we get,

K0 - K1 = L (1/(1 - (v/c)^2)^½ - 1)

Now Einstein does something called a Taylor’s series on this equation. I could write a whole set of blog entries on how fucking cool Taylor’s Series are (maybe I will). It is one of the most bad ass mathematical operations around. What Mr. Brook Taylor figured out was this – any equation can be represented as an infinite power series (see here). Now I know what you’re thinking. “Sounds great Chooky. Sign me up”. Don’t get cynical on me. See that equation above? It’s a bit of a mess because we have square roots in the denominator and so forth. If we perform a Taylor’s series on this equation we can simplify it and make it easier to work with. Assume that mess above on the right hand side is G(v). In other words, some function of v. Taylor showed that G can be represented by a formula that looks like this

G = A + B v^2 + C v^3 + D v^4 ….

In many cases the latter terms contribute very little to the overall picture. And in fact Einstein throws these higher order components out. Here is what he ends up with.

K0 - K1 = ½ L (v/c)^2

Now K0-K1 is a kinetic energy term. Kinetic energy is equal to ½ mv^2. Substitute that in for the left hand side and cancel the ½ and the v^2 terms and you get

m = L/c^2 or L = mc^2 or E = mc^2

And we're done. Not quite. Does anyone notice something funny here? I’ll let you stew on that for a while. But I’ll give you a hint. Something is terribly wrong with what I’ve shown here. Or rather I should say something is terribly wrong with what Einstein showed. And it had me screwed up for a while last night as I was deciphering his paper. Luckily I've found some other information that backs up my problems.

No comments: