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 ! 🙂

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3 thoughts on “E = mc2

  1. Ok, I haven’t managed to understand even your basic introduction but two thoughts come to my (little) mind:
    (1) ontology: it’s not because (two) things are equivalent (mass~energy) that they are equal or one can be dispensed with – the conversion is not just a numerical constant (it’s not like your converting inches to cm) but its dimensional (a squared speed, and actually a ‘limit’ at that) which means that, at the very least, you’re switching reference frameworks so it’s a different conceptualization.
    (2) phenomenology: it’s not because things are reducable or convertable to each other that their meaning is the same: ice, water and steam are all just H2O but saying that this means you can dispense with the concepts of ice and steam is reducing reality too far.
    But I guess I should make some time first to try understand what you’ve written and revert back to my comment later 🙂 Keep up the good work!

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    1. Feynman said: “Same equations. Same solutions.” And, yes, physical constants, as opposed two purely numerical constants, have TWO functions: (1) scaling, (2) ensuring dimensions on both sides are compatible. I am not saying mass and energy are the same. I am trying to give some MEANING to their equivalence… 😊

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