The nature of Nature

One of the fundamental questions for any freshman studying physics is the following: if an electron orbits around a nucleus, the various accelerations and decelerations should cause it to emit electromagnetic radiation. Hence, it should gradually lose energy and, therefore, the orbital cannot be stable. Therefore, the atom cannot be stable. So we have a problem. 🙂

A related but different question is: why don’t the electron and the nucleus simply crash into each other? They attract each other, very strongly, right? Well… Yes. But, as I said, that’s a related but different question. Let me first try to handle the first one – as good as I can. 🙂

So… Well… It’s a simple question but – as you know by now – the science of physics seldom gives us simple answers to simple questions. Worse, I’ve studied physics for many years now – admittedly, in my own stubborn, critical and generally limited way 🙂 – and I feel the answer I am getting is not only complicated but also not very real. So… Well… We might want to think we probably do not quite understand what is going on really.

This lack of understanding is nothing to be ashamed of, as great physicists such as Richard Feynman (and others) acknowledge: “Atomic behavior appears peculiar and mysterious to everyone—both to the novice and to the experienced physicist. Even the experts do not understand it the way they would like to.” So… Well… If you’d be in need of a rather spectacular acknowledgement of the shortcomings of physics as a science, there you have it: physicists don’t understand their own science, it seems. 🙂 But let’s go beyond that. Let’s talk about the wavefunction because… Well… You know it’s supposed to describe the electron, right? So what is it?

Well… Unfortunately, physics textbooks won’t tell you what the wavefunction is. They’ll tell you it’s a mathematical construct. A solution to some differential equation (Schrödinger’s equation), to be precise. 😦 However, they will you – from time to time, at least – tell you what it isn’t. For example, Feynman’s most precise description of the model of an electron – or an electron orbital, I should say – might be the one he offers when, while deriving the electron orbitals from Schrödinger’s equation, he says what the wavefunction is surely not:

“The wave function Ψ(r) for an electron in an atom does not describe a smeared-out electron with a smooth charge density. The electron is either here, or there, or somewhere else, but wherever it is, it is a point charge.” (Feynman’s Lectures, Vol. III, p. 21-6)

So… Well… That’s not too bad as an explanation. 🙂 But… Well… While fairly precise, I’d think we can improve on Feynman’s language. For starters, we should distinguish the concept of an electron and the concept of its charge. When the electron is in some stable configuration – i.e. in an orbital as described by its wavefunction Ψ(r) – the idea of the electron combines both the orbital and the point charge. Let’s be precise here:

  1. The charge is what, when probing, we’ll effectively find “here, there, or somewhere else” in the space that is being described by our wavefunction Ψ(r).
  2. As for the electron… Well… We know that – by applying operators to the wavefunction – we’ll not only get information about its position, but also about its linear or angular momentum, its energy, and whatever other characteristic of the electron that we’re describing. In that sense, we might say that the wavefunction completely describes the electron and that, therefore, the electron is not the point charge itself, but the orbital, as described by the wavefunction, with its point charge somewhere.

In short, for all practical purposes, we might say that the electron is the wavefunction, and vice versa. 🙂 Indeed, when studying quantum mechanics, one effectively does end up equating the particle with its wavefunction, not with its charge. And rightly so ! An elementary particle – be it an electron or a quark – is more than just its charge: it has energy, momentum (linear or angular), occupies some space and – in the case of quarks – has a color too ! 🙂

But that still doesn’t answer the simple question I started out with: the electrons – or the point charges in those orbitals – don’t emit radiation. Why not? Well… If I’d be your professor, and you’d be sitting for an exam in front of me, then I’d expect you to start talking about the Uncertainty Principle, wavefunctions, energy states and what have you. But I am not your professor (I am not a professor at all, in fact), and so I don’t want hear that answer. To be precise, I don’t like that answer because, just like Feynman, I don’t quite understand it the way I would like to understand it! So… What other answer can we think of? Can we think of something that is, perhaps, more intuitive?

I think we can. I, for one, am thinking, once more, of that profound statement that Einstein made back in 1916, when explaining his relativity theory to a broader audience:

“Physical objects are not in space, but these objects are spatially extended. In this way, the concept “empty space” loses its meaning.”

In fact, I’d go one step further and say: objects create their own space.

Huh? Yes. Think of the planets – including Earth – going around the Sun. Einstein’s general relativity theory tells us they are in their own space. Indeed, Einstein told us we should not think of gravitation as a force: the masses involved just curve the space-time fabric around them in such a way that these planets just keep going and going around and around. They are basically moving in free space: their path just happens to be curved from our perspective. If we wouldn’t impose this abstract (or mathematical, I should say) rectangular Cartesian coordinate space on our description of them, but accept this system of large objects creates its own space, we’d understand why it’s stable: it doesn’t need any energy from the outside to keep going. Nor does it radiate anything out.

Let me emphasize this point: they are in their own space because they don’t radiate anything out. And, I should add, nor do they absorb any energy from the outside. Of course, you’ve heard about gravitational waves – and most notably the one detected by the LIGO Lab last year – but note that gravitational wave was created when two black holes spectacularly merged. That’s because black holes do emit radiation, as a result of which do lose mass and, therefore, this system of large objects became unstable. Of course, if we’d detonate all of the atomic bombs we’ve built, we might also cause our planetary system to become unstable, but you’ll understand that’s a different discussion altogether. 🙂

So… Well… I like to think a wavefunction for an orbital represents the same: we’re looking at a charge that moves around in its own space. In our Cartesian reference frame, this looks like a terribly complicated oscillation. In fact, the oscillation is not only complicated but also – literally – complex, because we’re keeping track of two dimensions simultaneously: the real and imaginary component of the wavefunction. Both are equally real, of course, in a physical sense (and we can argue about what that means, exactly, but not about the statement itself). But so… Well… It’s just a spacetime blob. The charge itself just moves around along a geodesic in its spacetime, and that’s why it doesn’t emit or absorb any energy from the outside. 🙂

Of course, the question now becomes: if an electron orbital is nothing but a weird blob of curved spacetime – in which our charge moves around like a planet moves around in a planetary system – then what’s causing the curvature of space? For our planetary system, we know it’s mass.

So… Well… What can I say? Well… What’s mass? Energy has an equivalent mass, and mass has an equivalent energy. In my previous posts, I look at mass as an oscillation itself and, as I show in one of my papers, that might allow us to interpret Schrödinger’s wave equation as an energy diffusion equation, and the wavefunction itself as a self-contained and self-sustaining gravitational wave. So… Well… If the wavefunction represents a blob of energy – some two-dimensional oscillation – then… Well… Then it could create its own space, right? Just like our Sun and the planets create their own space, in which they move without absorbing or radiating any energy away. In other words, they move in a straight line in their own space. I am tempted to think our pointlike charge must also be moving in a straight line in its own space because… Well… It would, effectively, be emitting radiation otherwise. 🙂

So what’s the nature of Nature, then? Well… All is movement, it seems. Panta rhei ! 🙂 And… Well… I’ll let you do the philosophy now. For example, if objects create their own space, how should we think of their interactions? 🙂


Mass as a two-dimensional oscillation (2)

This post basically further develops my speculative thoughts about the real meaning of the E = m·c2 formula. However, I’ll use the relativistically correct formulas for the calculations this time, so it may look somewhat more complicated. However, I think you should be able to digest it relatively easily, as none of the math is exceedingly difficult.

My previous post explored the similarity between the formula for the energy of a harmonic oscillator and the E = m·c2 formula. Now, there is another formula that sort of resembles it: the E = m·v2/2 formula for the kinetic energy. Could we relate them somehow and – in the process – gain a better understanding of Einstein’s famous formula? I think we can, and I want to show you how. In fact, in this post, I will try to relate all three.

We should first note that the E = m·v2/2 is a non-relativistic formula. It is only correct if we assume the mass – defined as a measure of inertia, remember? – to be constant, which we know isn’t true. As an object accelerates and gains kinetic energy, its effective mass will increase. In fact, the relativistically correct formula for the kinetic energy just calculates it as the difference between (1) the total energy (which is given by the E = m·c2 formula, always) and (2) its rest energy, so we write:

K.E. = E − E0 = mv·c2 − m0·c2 = m0·γ·c2 − m0·c2 = m0·c2·(γ − 1)

The γ in this formula is, of course, the ubiquitous Lorentz factor. Hence, the correct formula for the kinetic energy is m0·c2·(γ − 1). We shouldn’t use that m·v2/2 formula. Still, the two formulas are remarkably similar: there is a squared velocity (v2 and c2) and some factor (1/2 versus γ − 1). Why the squared velocity? That’s child play, right? Yep, I effectively wrote a post on that for my kids. We have a force that acts on some object over some time and over some distance, and so that force is going to do some work. While it’s child play, we’re calculating a path or line integral here:

KEChild play? Perhaps, but many kids don’t know what a vector dot product is (the F·dx), and they also don’t realize we can only solve this because we assume the mass m to be constant (i.e. not a function of the velocity v). So… Well… In our flywheel model of an electron, we’ve been using a non-relativistic formula, but we’ve calculated the tangential speed as being equal to c. A recipe for disaster, right? 🙂 Can we re-do the calculations? We can. You can google a zillion publications on relativistic harmonic oscillators but I took the derivation below from a fairly simple one I’d recommend. The only correction we’ll do here is to use the relativistically correct expression of Newton’s force law: the force equals the time rate of change of the (relativistic) momentum p = mvv = γm0v. So we write:

F = dp/dt = F = –kx with p = mvv = γm0v

Multiplying both sides with = dx/dt yields the following expression: relativistic spring energyNow, when we combine two oscillators – think of the metaphor of a frictionless V-twin engine, as illustrated below 🙂 – then we know that – because of the 90° angle between the two cylinders, the motion of one piston will be equal to x = a∙cos(ω∙t), while the
motion of the other is given by y = a∙cos(ω∙t–π/2) = a∙sin(ω∙t).V-2 engineNow how do we calculate the total energy in this system? Should we distinguish the x– and y– components of the total momentum p? We can easily do that. Look at the animation below, and you’ll immediately understand that we can easily calculate the velocities in the x– and a y-direction: vx = dx/dt = −a·ω·sin(ω∙t) and vy = dy/dta·ω·cos(ω∙t). The sum of the square of both then gives us the tangential velocity vv2 a2∙ω2∙sin2(ω∙t) + a2∙ω2∙cos2(ω∙t) = a2∙ω2 ⇔ va∙ω.  Circle_cos_sinBut how do we add energies here? It’s a tricky question: we have potential energy in one oscillator, and then in the other, and these energies are being transferred from one to another through the flywheel, so to speak. So there is kinetic energy there. Can we just add it all? Let us think about our perpetuum mobile once more, noting that the equilibrium position for the piston is at the center of each cylinder. When it goes past that position, extra pressure will build up and eventually push the piston back. When it is below that position, pressure is below equilibrium and will, therefore, also want to pull the piston back. The dynamics are as follows:

  • When θ is zero, the air in cylinder 1 is fully compressed, and the piston will return to the equilibrium position (x = 0) as θ goes to 90°. The flywheel will transfer energy to cylinder 2, where the piston goes from the equilibrium position to full compression. Cylinder 2 borrows energy, and will want to return to its equilibrium position.
  • When θ is 90°, the air in cylinder 2 is fully compressed, and the piston will return to the equilibrium position (y = 0) as θ goes to 180°. The flywheel will transfer energy back to cylinder 1, where the piston goes past the equilibrium position to create a vacuum. The piston in cylinder 1 borrows energy, and will want to return to its equilibrium position.
  • When θ is 180°, the piston in cylinder 1 is fully extended, and will want to return to equilibrium because the pressure is lower than when in equilibrium. As θ goes from 180° to 270°, the piston in cylinder 1 does effectively return to equilibrium and, through the flywheel, pushes the piston in cylinder 2 past the equilibrium to create vacuum. The piston in cylinder 2 borrows energy, and will want to return to equilibrium.
  • Finally, between 270° and 360°, the piston in cylinder 2 returns to equilibrium and, through the flywheel, causes the piston in cylinder 1 to compress air. The piston in cylinder 2 borrows energy, and will want to return to equilibrium.

It is a funny thing. Where is the energy in this system? Energy is not supposed to be thought as being directional but, here, direction clearly matters! We need to think about averages here (kinetic energy is a non-directional (scalar) quantity but it’s a function of velocity, and velocity is directional. If we have two directions only (x and y), then we can write: 〈vx2〉 = 〈vy2〉 = [〈vx2〉 + 〈vy2〉]/2 = 〈v2〉/2. So this gives us a clue, but we won’t make things to complicated here. Think of it like this. While transferring energy from one piston to the other, the crankshaft will rotate with a constant angular velocity: linear motion becomes circular motion, and vice versa. So what is the total energy in the system? What if we would want to use it? What can we take from it? You’d agree we would have to take it from the flywheel, right? The usable energy is in the flywheel. Let’s have a look at that energy conservation law we derived above: conservation lawThe usable energy in the flywheel is the E = m·c2 term. This, and my previous post, suggests we may interpret the mass of an electron as a two-dimensional oscillation. In fact, I think my previous post is an easier read because I use the classical (non-relativistic) formulas there. This post, hopefully, demonstrates that a relativistically correct mathematical treatment doesn’t alter the picture that I’ve been trying to offer. 🙂

Of course, the more difficult thing is to go beyond this metaphor and explain how exactly this motion from borrowing and returning energy to space would actually work. 🙂 So that would be a proper ‘ring theory’ of matter. 🙂

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 mass?), I am not going to talk about the Higgs field or other terribly complicated stuff. In fact, I’ll be talking about relativity using formulas that I shouldn’t be using. 🙂 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 and E = m·a2·ω2/2 formulas. The E = mc2 formula is very well known but also very mysterious. In contrast, the E = m·a2·ω2/2 formula is not so well known but not mysterious at all! It is the formula for the 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 a·ω identity that comes out of this, of course – but I will get there in a moment. 🙂

We can get rid of that 1/2 factor by combining two oscillators. Think of two frictionless pistons (or springs) in a 90° 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° degree angle explains why a Ducati is more efficient than a Harley-Davidson, whose cylinders are at an angle of 45°. The 45° angle makes for great sound but… Well… Not so efficient. 🙂

V-2 engine

The in the E = m·a2·ω2/2 formula is the magnitude of the oscillation, and the motion of these pistons (or of a mass on a spring) will be described by an x = a·cos(ω·t + Δ) function. The x is just the displacement from the center, and the Δ is just a phase factor which defines our t = 0 point. The ω is the angular frequency of our oscillator: it is defined by the time that is needed for a complete cycle, which is referred to as the period of the oscillation. We denote the period as t0, and it is easy to understand that t0 must be such that ω·t0 = 2π. Hence, t0 = 2π/ω. Note that, because of the 90° angle between the two cylinders, the phase factor Δ would be zero for one oscillator, and –π/2 for the other. Hence, the motion of one piston is given by x = a·cos(ω·t), while the motion of the other is given by x = a·cos(ω·t–π/2) = a·sin(ω·t).

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


Now think of an electron as a charge moving about some center, so that’s the green dot in the animation above. We can then also analyze its movement in terms of two perpendicular oscillations, i.e. the sine and cosine 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 write:

ψ(θ) = a·ei∙θ = a·ei∙E·t/ħ = a·cos[(E/ħ)∙t]  i·a·sin[(E/ħ)∙t]

Remembering that multiplication by the imaginary unit (i) amounts to a rotation by 90°. To be precise, because of the minus sign in front of the sine, we have a rotation by minus 90° here, but that doesn’t change the analysis: Nature doesn’t care about our convention for 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 ± ħ/2 for an electron). But let’s further develop our analogy by getting back to our oscillators. The kinetic and potential energy of one oscillator – think of one piston or one spring only – can be calculated as:

  1. K.E. = T = m·v2/2 = (1/2)·m·ω2·a2·sin2(ω·t + Δ)
  2. P.E. = U = k·x2/2 = (1/2)·k·a2·cos2(ω·t + Δ)

The coefficient k in the potential energy formula characterizes the restoring force: F = −k·x. From the dynamics involved, it is obvious that k must be equal to m·ω2. Hence, the total energy—for one piston (or one spring) only—is equal to:

E = T + U = (1/2)· m·ω2·a2·[sin2(ω·t + Δ) + cos2(ω·t + Δ)] = m·a2·ω2/2

That is the formula we started out with and, yes, if we would add the energy of the two oscillators, we’d effectively have a perpetuum mobile storing an energy that is equal to twice this amount: E = m·a2·ω2.

Let us now think this through. If E and m are the energy and mass of an electron, then the E = m·a2·ω2 and E = m·c2 equations tell us that a·ω. What are a and ω here? Well… The de Broglie relations suggest we should equate ω to E/ħ. As for a, we could take the Compton scattering radius of the electron, which is equal to ħ/(m·c). So we write:

  a·ω = [ħ/(m·c)]·[E/ħ] = E/(m·c) = m·c2/(m·c) = 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·a2·ω2 = m·c2 equation might (let me put in italics: might) make sense. 🙂

Let me show you something else. If this flywheel model makes sense, then we can, obviously, also calculate a tangential velocity for our charge. The tangential velocity is the product of the radius and the angular velocity: vr·ω = a·ω = c. In our previous post, we wrote that we should think of the 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 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? Well… From our a·ω = c, we get that:

ω = c/= c/[ ħ/(m·c)] = (c/ħ)·(m·c) = m·c2/ħ = E/ħ

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. The natural frequency of a spring (ω) depends on (1) the mass on that spring (m) and (2) the restoring force, which is equal to F = −k·x. The k factor here 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 from m and ω. We wrote: k = m·ω2. 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 not some actual mass going up and down and back and forth. In fact, if the tangential velocity of our charge would be equal to c – which it is in our model – then the charge itself should have zero (rest) mass ! Hence, the stiffness would be equal to k = 0·ω2 = 0! 

Let me offer another calculation instead. If this flywheel model makes sense, then the electron will have some angular momentum, right? The angular momentum is equal to L = ω·I, so that’s the product of the angular velocity (ω) 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 ± ħ/2. So now we can calculate the moment of inertia as I = L/ω = (ħ/2)/(E/ħ) = ħ2/(2·E). Now, substituting E for E = m·c2 and remembering that = ħ/(m·c), we can write this as:

I = ħ2/(2·m·c2) = (ħ2·m)/(2·m2·c2) = (1/2)·m·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. 🙂

So… Well… That’s it! 🙂



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

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 that this analogy may represent something real. Why? Because it would allow us to associate the wavefunction with something real and, therefore (see my paper on this), it would also allow us to think of Schrödinger’s equation as representing something real. To be precise, it would allow us to think of Schrödinger’s equation as an energy diffusion equation, but… Well… That is somewhat more difficult to explain than what I explained above. 😦

The essential question, of course, is: what gives that pointlike charge that circular motion? What is the origin of what Schrödinger himself referred to (I admit: he did so in a very different context) as a 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 equivalent 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 consistent 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·a2·ω2/2 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 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. 🙂 Our model assumes it is, effectively, the charge of the electron that is oscillating. Hence, why wouldn’t it be the same?

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

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·v2/2 = (1/2)·m·ω2·a2·sin2(ω·t + Δ) formula, right? 🙂 The relativistically correct definition of kinetic energy is slightly different than the T = m·v2/2 formula. It may – or may not – make a difference. 🙂 I’ll talk about that in my next post.

E = mc2

Isn’t it remarkable that the best-known formula in physics, E = mc2, is actually one that we cannot really prove? As such, this formula is like a physical law: we think such laws are universally true, but we cannot be sure. Why? Because the experiments and observations since Einstein came up with this formula 115 years ago (for the context, you can check the Wikipedia history section on it) only suggest it is true. Of course, these experiments and observations very strongly suggest the E = mc2 formula is true but… Well… Karl Popper (and – more importantly – Albert Einstein himself) told us that we should always be skeptical: some more advanced alien might visit Earth one day and demonstrate – through some spectacular experiment – that our E = mc2 formula (and all of the other associated laws and lemmas) is, in fact, not quite correct.

That would be even more dramatic, I guess, then the 1887 Michelson-Morley experiment – that ‘most famous failed experiment in history’ which, instead of confirming what everyone thought to be true at the time, told us that Galilean (or Newtonian) relativity was, in fact, not true. It took Einstein and a whole lot of other bright guys (the likes of James Clerk Maxwell and Hendrik Antoon Lorentz) almost 20 years to fix the problem (Einstein published his special relativity theory in 1905), so… Well… We can only hope that this alien will be friendly and, immediately after blowing up our cherished beliefs, will also show us how to fix our formulas. 🙂

Of course, you will say one can find lots of proofs of E = mc2 when surfing the web, but these ‘proofs’ are actually just simple illustrations of the law: they only ‘prove’ that the E = mc2 formula is consistent with other statements and laws. For example, a lot of these so-called proofs will show you that the E = mc2 is consistent with special relativity. They will show you, for example, that the E = mc2 formula is consistent with the relativistic mass formula m = mv = γm0, or that radiation has, effectively, some equivalent mass. But then… Well… That consistency doesn’t prove the E = mc2 formula.

The E = mc2 formula is just something like Newton’s law of gravitation, or Maxwell’s equations: we can’t prove those either. We can just work out all of their implications and check if they are consistent with experiments and observations, and we accept them because they are. In fact, most of these so-called proofs don’t even help to understand what Einstein’s mass-energy equivalence formula actually means. They just talk about its manifestations or consequences.

For example, we all know that the equivalent mass of the binding energy between matter-particles in a nuclear fission reaction is converted into destructive heat and radiation energy. However, we also know that a nuclear explosion does not actually annihilate any elementary particles. So you might say it doesn’t really convert (rest) mass to energy. It is just binding energy that gets released – or converted into some other form of energy. As such, this oft-quoted example just illustrates that energy has an equivalent mass. So it just says what it says. Nothing more. Nothing less. This example does not tell us, for example, if and how it might work the other way around. Can we convert radiation energy back into mass? Probably not, right? Why? Because of entropy and what have you. In other words, we can surely not say that mass and energy are the same. Equivalent. Yes. But not the same.

Now, you might still be inclined to think they are, because there is actually a much better illustration of mass to energy conversion than the classical nuclear bomb: think of a positron and an electron coming together. [Just to make sure, anti-matter is just matter with an opposite electric charge. There is no such thing as negative matter.] Let’s say their rest mass is m0. So… Well… The positron and the electron will effectively annihilate each other in a flash and the  resulting radiation will have an energy that’s equal to E = m0c2. So that’s a much better illustration of how the rest mass of an elementary particle can be converted directly into energy.

Still, this equivalence between mass and energy does not imply the energy and mass concepts are, effectively, the same. For starters, their physical dimension is different. Equivalent (1 kg = 1 N·s2/m), yes, but not the same. Or “same-same but different”, as they’d say in Asia. 🙂 More importantly, this illustration of this so-called equivalence between mass and energy still doesn’t prove the formula: this experiment just adds to a zillion other observations and experiments which have turned this formula into a generally accepted statement – something that is thought of as being true. In fact, in physics, we cannot really prove something is true: everything we know is true only until someone comes along and shows us why it is not true. Experiments can only confirm what we believe is true or – if they don’t work out – they prove us that what we believed is wrong. Hence, strictly speaking, experiments and observations can only tell us what might be true, or confirm our beliefs by showing us what is definitely not true. Of course, that is more than good enough for most of us. I, for one, am convinced that the E = mc2 formula is true. So it’s my truth, for sure! [Just to make sure you know where I stand: I fully accept science! Creationism and other nonsense is definitely not my truth!]

The point is: I want to understand the formula, and that’s where most of these proofs fail miserably too: not only don’t they prove anything, but they also don’t really tell us what the E = mc2 formula really means. How should we think of the annihilation of matter and anti-matter? What happens there, really?

To answer that question, we need to answer a much more fundamental question: what is mass? And what is energy? It is not easy to define energy. It comes in many forms (e.g. chemical versus nuclear energy), and various other distinctions – such as the distinction between potential and kinetic energy – may cause even more confusion. Is it any easier to define mass? Maybe. Maybe not. Let’s try it. Newton’s laws associate two very different things with mass: it is, first, a measure of inertia (resistance to a change in motion) and, second, it is the cause of the gravitational force.

Let us briefly discuss the second aspect first: Einstein’s general relativity theory sort of explains gravitation away, by pointing out that a mass causes spacetime to curve. We no longer think of spacetime as an abstract mathematical space now, but as a physical space: it is our space now, and it’s bent. Think of Einstein’s famous remark: “Physical objects are not in space, but these objects are spatially extended. In this way, the concept “empty space” loses its meaning.” Hence, general relativity theory is not just another equivalent representation of the same thing (gravitation): the metaphysics are very different.

Let us turn to the first aspect: mass as a measure of inertia. When one or more forces act on an object with some mass, some power is being delivered to that object. I hope you’ve learned enough about physics to vaguely remember we can write that using vectors and a time derivative. Don’t worry if you can’t quite follow the mathematical argument. Just try to get the basic idea of it. If T is the kinetic energy of some object with some mass, we can write the following:F1The dT/dt is the time rate of change of the kinetic energy, and we use bold letters (e.g. F, v or s) to denote vectors, so they are directional numbers, so to speak: they do not only have a magnitude but a direction as well. The product between two vectors (e.g. F·v) is a vector dot product (so it’s commutative, unlike a vector cross product). OK. Onwards. You should note that the formula above is fully relativistically correct. Why? Because the formula for the momentum p = mv = mvv uses the relativistic mass concept, so it recognizes the mass of an object increases as it gains speed according to the Lorentz correction:F2Onwards! T is the kinetic energy. However, if kinetic energy is all that changes (the potential energy is just the equivalent energy of the rest mass here), then the time rate of change of the total energy E will be equal to the time rate of change of the kinetic energy T. If we then assume that the E = mc2 formula is correct, we can write the following:F5Note that we substituted F for dp/dt = d(mv)/dt. This too is relativistically correct: the force is the time rate of change of the momentum of an object. In fact, to correct Newton’s law for relativistic effects, we only need to re-write it like this: F = dp/dt = d(mv)/dt = mvdv/dt = mva. All we do is substitute the mass factor m for the relativistic mass m = mv. Now, it takes a few tricks (e.g. multiply both sides by 2m) to check that this equation is equivalent to this:F6In case you don’t see it, you may want to check the original story, which I got from Feynman here. Now, if the derivatives of two quantities are the same, then the quantities themselves differ by a constant only, so we write: m2c2 = m2v2 + C. What is the constant C? The formula must be valid for all v, so let us choose v = 0. We get: m02c2 = 0 + C = C. Substitution then gives us this: m2c2 = m2v2 + m02c2. Finally, dividing by c2 and rearranging the terms gives us the relativistic mass formula:F2Isn’t this amazing? We cannot prove the E = mc2 formula, but if we use it as an axiom – so if we assume it to be true – then it gives us the relativistic mass formula. So the logic is the following: if the E = mc2 formula is true, then the m = mv = γm0 is true as well. The logic does not go the other way. Why? Because the proof above uses this arrow at some place: ⇒. One way. Not an arrow like this: ⇔. 🙂

Still, the question I started out with remains: what is mass? I haven’t said anything about that so far. The truth: it is a bit complicated. In fact, I have my own little fun theory on this. It is based on the remarkable structural similarity between the relativistic energy formula and the formula for the total energy of an oscillator:

  1. E = mc2
  2. E = mω2/2

In fact, I should write the second formula as E = m·a2·ω2/2: the is the amplitude of the oscillation, which may or (more likely) may not be equal to one. The point is: the and the ω in these two formulas both describe a velocity – linear or, in the case of E = mω2/2 – angular. Of course, there is the 1/2 factor in the E = mω2/2 formula, but that is exactly the point that inspired me to explore the following question: what if we’d think of mass as some oscillation in two dimensions, so it stores an amount of energy that is equal to E = 2∙ mω2/2 = mω2. Indeed, Einstein’s E = mc2 equation implies the ratio between the energy and the mass of any particle is always the same:F7If you have ever read anything about oscillators – mechanical or electrical – this should remind you of the ω2 = C−1/L or ω2 = k/m formulas for electric and mechanical oscillators respectively. The key difference is that the ω2 = C−1/L (electric circuit) and ω2 = k/m (mechanical spring) formulas introduce two (or more) degrees of freedom. In contrast, c2 = E/m for any particle, always. But that is exactly the point: we can modulate the resistance, inductance and capacitance of electric circuits, and the stiffness of springs and the masses we put on them, but we live in one physical space only: our spacetime. Hence, the speed of light c emerges here as the defining property of spacetime – the resonant frequency, so to speak. We have no further degrees of freedom here.

This gives rise to what I refer to as a flywheel model for elementary particles. More about that later. 😊 Or… Well… If you don’t want to wait, here are the links to my two papers on this:

Have fun ! 🙂