Mass as a two-dimensional oscillation

This post is actually not about Einstein or anything he wrote. It is just for fun. It really is because… Well… While it’s about one of the most fundamental questions in physics (what is聽mass?), I am not聽going to talk about the Higgs field聽or other terribly complicated stuff. In fact, I’ll be talking about relativity using formulas that I shouldn’t be using. 馃檪 But you should find this very straightforward and thought-provoking. Or so I hope, at least. So see if you like it and – if you do or you don’t, whatever – please let me know by posting a comment.

In my previous post, I noted the structural similarity between the E = mc2聽and E = m路a2路蠅2/2 formulas. The聽E = mc2聽formula is very well known but also聽very mysterious. In contrast, the聽E = m路a2路蠅2/2 formula is not so well known but not mysterious at all! It is the formula for the聽energy of a harmonic oscillator: think of an oscillating spring, for example. The only difference between the two formulas is the 1/2 factor and… Well… We would also need some interpretation of the c聽=聽a路蠅 identity that comes out of this, of course – but I will get there in a moment. 馃檪

We can get rid of that 1/2 factor by combining聽two聽oscillators. Think of two frictionless pistons (or springs) in a 90掳 degree angle, as shown below. Because their motion is perpendicular to each other, their motion is independent, and so we can effectively add the power (and energy) of both. In fact, the 90掳 degree angle explains why a Ducati is more efficient than a Harley-Davidson, whose cylinders are at an angle of 45掳. The 45掳 angle makes for great sound but… Well… Not so efficient. 馃檪

V-2 engine

The a聽in the聽E = m路a2路蠅2/2 formula is the聽magnitude of the oscillation, and the聽motion of these pistons (or of a mass on a spring) will be described by an聽x聽=聽a路cos(蠅路t + 螖) function. The x is just the displacement from the center, and the 螖 is just a phase factor which defines our聽t聽= 0 point. The 蠅 is the聽angular聽frequency of our oscillator: it is defined by the time that is needed for a complete聽cycle, which is referred to as the periodof the oscillation. We denote the period as t0, and it is easy to understand that聽t0聽must be such that聽蠅路t0 = 2蟺. Hence, t0 = 2蟺/蠅. Note that, because of the 90掳 angle between the two cylinders, the phase factor 螖 would be zero for one oscillator, and 鈥撓/2 for the other. Hence, the motion of one piston is given by聽x聽=聽a路cos(蠅路t), while the motion of the other is given by聽x聽=聽a路cos(蠅路t鈥撓/2) =聽a路sin(蠅路t).

The animation below abstracts away from pistons, springs or whatever other聽physical聽oscillation we might think of – but it represents the same. We just denote the聽phase itself聽as聽胃 =聽蠅路t.


Now think of an electron as a聽charge moving about some center, so that’s the聽green dot in the animation above. We can then also analyze its movement in terms of two perpendicular oscillations, i.e. the聽sine聽and聽cosine聽functions shown above. Now, you may or may not know that an elementary wavefunction consists of the same: a sine and as cosine. Indeed, using Euler’s notation, we聽write:

蠄(胃) = a路ei鈭櫸聽=聽a路ei鈭橢路t/魔= a路cos[(E/魔)鈭檛] 聽i路a路sin[(E/魔)鈭檛]

Remembering that multiplication by the imaginary unit (i) amounts to a rotation by 90掳. To be precise, because of the聽minus聽sign in front of the sine, we have a rotation聽by minus聽90掳here, but that doesn’t change the analysis:聽Nature doesn’t care about our convention for聽i, or for our convention for measuring angles clockwise or counter-clockwise, and it does allow the angular momentum to be either positive or negative (it is 卤 魔/2 for an electron). But let’s further develop our analogy by getting back聽to our oscillators.聽The聽kinetic and potential energy of聽one聽oscillator 鈥 think of one piston or one spring only 鈥 can be calculated as:

  1. K.E. = T = m路v2/2 =聽(1/2)路m路蠅2a2路sin2(蠅路t + 螖)
  2. P.E. = U = k路x2/2 = (1/2)路k路a2路cos2(蠅路t + 螖)

The coefficient k in the potential energy formula characterizes the restoring force: F = 鈭択路x. From the dynamics involved, it is obvious that k must be equal to m路蠅2. Hence, the total energy鈥攆or聽one聽piston (or one spring) only鈥攊s equal to:

E = T + U = (1/2)路 m路蠅2a2路[sin2(蠅路t + 螖) + cos2(蠅路t + 螖)] = m路a2路蠅2/2

That is the formula we started out with and, yes, if we would add the energy of the聽two聽oscillators, we’d effectively have a聽perpetuum mobile聽storing an energy that is equal to聽twice聽this amount: E = m路a2路蠅2.

Let us now think this through. If E and m are the energy and mass of an electron, then the E =聽m路a2路蠅2聽and E =聽m路c2聽equations tell us that聽c聽=聽a路蠅.聽What are聽a聽and聽蠅 here? Well… The聽de Broglie聽relations聽suggest we should equate聽蠅 to E/魔. As for聽a,聽we could take the聽Compton scattering radius聽of the electron, which is equal to聽魔/(m路c). So we write:

聽聽a路蠅 = [魔/(m路c)]路[E/魔] = E/(m路c) =聽m路c2/(m路c) =聽c

Did we prove anything here? No. We don’t prove anything in this post. We’re just having fun. We only showed that our E = m路a2路蠅2聽= m路c2聽equation might (let me put in italics: might) make sense. 馃檪

Let me show you something else. If this聽flywheel聽model聽makes sense, then we can, obviously, also calculate a聽tangential velocity for our charge. The tangential velocity is the product of the聽radius聽and the angular聽velocity:聽v =聽r路蠅 =聽a路蠅 =聽c. In our previous post, we wrote that we should think of the聽speed of light as the resonant frequency of the spacetime fabric, but we should probably take that back. The speed of light emerges as the speed of the charge聽in what I’ll now officially refer to as my flywheel model of an electron.

Is there a resonant frequency here? If so, how should we interpret it?聽Well… From our聽a路蠅 = c, we get that:

蠅 = c/a聽= c/[ 魔/(m路c)] = (c/魔)路(m路c) =聽m路c2/魔 = E/魔

So the answer is: no. No resonant frequency of spacetime. The frequency is the frequency of our electron – not of the fabric of spacetime. However, we can, perhaps, think of another analogy.聽The natural frequency of a spring (蠅) depends on (1) the mass on that spring (m) and (2) the restoring force, which is equal to F = 鈭k路x. The聽k factorhere is the stiffness of the spring. Could we, likewise, talk about the stiffness of the spacetime fabric? We know that, for a spring, we can calculate聽k聽from m and聽蠅. We wrote:聽k =聽m路蠅2. Can we do anything with that? Probably not. The mass in our flywheel model is the equivalent mass of the energy in the (two-dimensional) oscillation. It is聽not聽some actual mass going up and down and back and forth. In fact, if the tangential velocity of our charge would be equal to聽c聽– which it is in our model – then the charge itself should have聽zero (rest) mass ! Hence, the stiffness would be equal to聽k =聽0路蠅2聽= 0!

Let me offer another calculation instead. If this flywheel model makes sense, then the electron will have聽some angular momentum, right? The angular momentum is equal to L = 蠅路I, so that’s the product of the angular velocity (蠅) and the moment of inertia (I), aka the angular mass. Now, from the Stern-Gerlach experiment, we know that the angular momentum of an electron is equal to聽卤聽魔/2. So now we can calculate the moment of inertia as I = L/蠅 = (魔/2)/(E/魔) =聽魔2/(2路E). Now, substituting E for E =聽m路c2聽and remembering that聽a聽=聽魔/(m路c), we can write this as:

I =聽魔2/(2路m路c2) = (魔2路m)/(2路m2c2) = (1/2)路m路a2

Do we recognize that formula? Yes. It’s the formula for the angular mass of a solid disc, or a hoop about the diameter, as shown below. Which of the two makes most sense? I am not sure. I’ll let you think about that. 馃檪

So… Well… That’s it! 馃檪



Why am I smiling? Well… I hope this post makes you聽think about stuff yourself because… Well… That was my only objective: have fun by聽thinking聽about stuff yourself! 馃檪

Do these calculations – and the analogy itself – make any sense? My truthful answer is: I am not sure. I really don’t know. Of course, I would very much like to think聽that this analogy may represent something real. Why? Because it would allow us to associate the wavefunction with something real and, therefore (see聽my paper聽on this), it would also allow us to think of Schr枚dinger’s equation as representing something real. To be precise, it would allow us to think of聽Schr枚dinger’s equation as an energy diffusion equation, but… Well… That is somewhat more difficult to explain than what I explained above. 馃槮

The essential question, of course, is: what gives that pointlike charge that circular motion? What is the origin of what Schr枚dinger himself referred to (I admit: he did so in a very different context) as a聽Zitterbewegung?

All that I’ve written above, assumes space is not just some abstract mathematical space. It is real, somehow, and perfectly elastic. So that’s why we can advance a model that assumes an an electron is nothing but a charge (with zero rest mass) that bounces around in it. All of its mass is the聽equivalent聽mass of the energy in the oscillation itself. It is, of course, a crazy hypothesis that we cannot prove but, as far as I can see, while crazy, the hypothesis is聽consistent聽with what we know about the weird wonderful world of quantum mechanics.

The main weakness in the argument is the following: if the charge itself has zero rest mass, then our E = m路a2路蠅2/2 equation reduces to E = 0. Is the analogy still valid? And how can we possibly associate some angular mass with something that is going around but has zero rest mass? This is, effectively, a flywheel model without a flywheel. I may have explained matter as a two-dimensional oscillation, but I haven’t told you what is oscillating. Or… Well… I did. What is oscillating is the charge, with zero rest mass.

The analogy with a photon is obvious. A photon has zero rest mass too! Now, the wavefunction of a photon is the electromagnetic wave. Can we say what is oscillating there? Yes, we can! Of course: it is the electromagnetic field! But what’s the field? You will say the electromagnetic field has a聽physical dimension: newton per coulomb (N/C). This begs the next question: what’s the physical dimension of the wavefunction? Could it be the same?

My answer is: yes, or maybe. 馃檪 Our model assumes it is, effectively, the charge of the electron聽that is oscillating. Hence, why wouldn’t it be the same?

In any case, I will let you think about that for yourself. As聽you can see, in physics, an answer to one question may trigger many more. 馃檪 If you have any good feedback, please comment! 馃檪

Post scriptum: The mentioned weakness in the argument should also be related to the fact that we are using classical non-relativistic formulas. If our charge is really moving at a speed at or near light speed around some center, we should probably have another look at our K.E. = T = m路v2/2 =聽(1/2)路m路蠅2a2路sin2(蠅路t + 螖) formula, right? 馃檪 The relativistically correct definition of kinetic energy is slightly different than the T = m路v2/2 formula. It may – or may not – make a difference. 馃檪 I’ll talk about that in my next post.