Isn’t it remarkable that the best-known formula in physics, E = m*c*^{2}, 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 = m*c*^{2} 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 = m*c*^{2} 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 = m*c*^{2} when surfing the web, but these ‘proofs’ are actually just simple *illustrations *of the law: they only ‘prove’ that the E = m*c*^{2} formula is *consistent *with other statements and laws. For example, a lot of these so-called proofs will show you that the E = m*c*^{2} is *consistent *with special relativity. They will show you, for example, that the E = m*c*^{2} formula is consistent with the relativistic mass formula m = m* _{v}* = γm

_{0}, or that

*radiation*has, effectively, some equivalent mass. But then… Well… That

*consistency*doesn’t

*prove*the E = m

*c*

^{2}formula.

The E = m*c*^{2} 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 m_{0}. 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 = m_{0}*c*^{2}. 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·s^{2}/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 = m*c*^{2} 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 = m*c*^{2} 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

*meta*physics 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:The *d*T/*d*t is the time rate of change of the kinetic energy, and we use bold letters (e.g. **F**, ** v** or

**) to denote vectors, so they are directional numbers, so to speak: they do not only have a magnitude but a**

*s**direction*as well. The product between two vectors (e.g.

**F**·

**) is a vector dot product (so it’s commutative, unlike a vector**

*v**cross*product). OK. Onwards. You should note that the formula above is fully relativistically correct. Why? Because the formula for the momentum

**p**= m

**= m**

*v*_{v}

**uses the relativistic mass concept, so it recognizes the mass of an object increases as it gains speed according to the Lorentz correction:**

*v**is the*

**Onwards!**T*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 = m

*c*

^{2}formula is correct, we can write the following:Note that we substituted

**F**for

*d*

**p**/

*d*t = d(m

**)/dt. This too is relativistically correct:**

*v**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**=

*d*

**p**/

*d*t = d(m

**)/dt = m**

*v**/*

_{v}d**v***d*t = m

_{v}**. All we do is substitute the mass factor m for the**

*a**relativistic*mass m = m

*. Now, it takes a few tricks (e.g. multiply both sides by 2*

_{v}*m*) to check that this equation is equivalent to this:In 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:

*m*

^{2}

*c*

^{2}=

*m*

^{2}

*v*

^{2}+

*C*.

*What is the constant*

*C*? The formula must be valid for all

*v*, so let us choose

*v*= 0. We get:

*m*

_{0}

^{2}

*c*

^{2}= 0 +

*C*=

*C*. Substitution then gives us this:

*m*

^{2}

*c*

^{2}=

*m*

^{2}

*v*

^{2}+

*m*

_{0}

^{2}

*c*

^{2}. Finally, dividing by

*c*

^{2}and rearranging the terms gives us the relativistic mass formula:Isn’t this amazing? We cannot

*prove*the E = m

*c*

^{2}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 = m*

*c*

^{2}formula is true,

*then*the m = m

*= γm*

_{v}_{0}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:

- E = m
*c*^{2} - E = mω
^{2}/2

In fact, I should write the second formula as E = m·*a*^{2}·ω^{2}/2: the *a *is the amplitude of the oscillation, which may or (more likely) may *not* be equal to one. The point is: the *c *and the ω in these two formulas both describe a velocity – linear or, in the case of E = mω^{2}/2 – *angular*. Of course, there is the 1/2 factor in the E = mω^{2}/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∙ mω^{2}/2 = mω^{2}. Indeed, Einstein’s E = m*c*^{2} equation implies the ratio between the energy and the mass of any particle is always the same:If you have ever read anything about oscillators – mechanical or electrical – this should remind you of the ω^{2} = C^{−1}/L or ω^{2} = k/m formulas for electric and mechanical oscillators respectively. The key difference is that the ω^{2} = C^{−1}/L (electric circuit) and ω^{2} = k/m (mechanical spring) formulas introduce two (or more) degrees of freedom. In contrast, c^{2} = 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:

- The quantum-mechanical wavefunction as a gravitational wave; and
- Further reflections on the reality of the wavefunction.

Have fun ! 🙂