In 1946, Albert Einstein was in sombre mood. Gone was the euphoria of his *annus mirabilis* forty years earlier, when his paper on the equivalence of mass and energy was first published. Now, with Hiroshima and Nagasaki fresh in his memory, Einstein, too, found himself “the destroyer of worlds”, as Oppenheimer put it. That is why, when he was approached by the publishers of Science Illustrated in the US to write an article on “what takes place in the operation of his law”, he entitled it:

E=MC^{2} – the most urgent problem of our time.

Like his publishers, Einstein wanted to give the general public some basic understanding of the science involved in nuclear power and its terrible potential. In the process, from our modern point of view, he manages to clarify a few basic issues with the formula itself. Oddly, considering it is probably the most famous equation in the world, there is still considerable dispute as to what it actually means. What, for example, is equivalence? Does it mean that mass and energy are the same thing, or two equivalent but different things? Is one just convertible into the other? What is the role of c^{2} in the equation? And so on.

The Stanford definition1is quite specific:

According to Einstein’s famous equation *E* = *mc*^{2}, the energy *E* of a physical system is **numerically equal** (my emphasis) to the product of its mass *m* and the speed of light *c* squared.

In the article, being forced to write for a readership of non-mathematicians, Einstein’s use of plain language makes his own understanding of the concepts behind the symbols quite clear:

“E is the energy that is contained in a stationary body; *m* is its mass. The energy that belongs to the mass *m* is equal to this mass, multiplied by the square of the enormous speed of light – which is to say, a vast amount of energy for every unit of mass.”

The first major clue is his use of the word “enormous”. The speed of light is constant; it does not change.2However, if it is to “multiply” mass, the speed of light must be a number, and quite a high one, too. In the article, Einstein uses 186,000. That is the speed of light in miles per second, which would be familiar to his US audience, but he is just using it as a number that gives him a square of 34 billion, which is large enough to account for Hiroshima.

The important point is that, for bombs – or, more usefully, nuclear power stations – to work, *some* actual number is required. With bombs, you might well wonder what the real mass was, given that it is now spread across several square miles of former city, but nuclear power stations offer a much more controlled environment, where *E* = *mc*^{2} yields fixed values that are a reliable part of the fuel purchasing calculation.

But how can this be? The speed of light can be virtually any number, from one (light years per year) to just under a billion (feet per second), anything in between, or even far greater (millimetres per hour, anyone?). The answer lies in the fact, as is clear from the article, that Einstein views the equivalence of energy and mass as a fixed ratio. How could it be otherwise? It does, of course, vary from measurement system to measurement system, from SI Units to Imperial Units to US Customary Units, and there are elaborate conversion3rules and ratios to convert from one to the other, but it has to be borne in mind that these are competing systems yielding different results for the same formula. However, it is a logical impossibility for the actual ratio to vary, or the bombs wouldn’t work, or at least would have wildly varying impact.

For instance, he suggests an experiment to verify the relationship:

“I can easily supply energy to the mass –for instance, if I heat it by 10 degrees. So why not measure the mass increase, or weight increase, connected with this change? The trouble here is that in the mass increase, the enormous factor *c*^{2} occurs in the denominator of the fraction. In such a case the increase is too small to be measured directly; even with the most sensitive balance.”

It is obvious from this that, to his mind, the energy implicit in mass and the energy released in, say, nuclear fission – the ‘bound’ and the ‘unbound’ energies, to coin a phrase – are in a true ratio, i.e. between two states of the same property of nature. If he is right – and who am I to quibble? – then the factor *c*^{2}, being a true ratio, is dimensionless, and the choice of the correct “speed of light” is central to the validity of the formula.

The SI Units (or metric system, as it is known) has a value for *c*^{2} of approximately 90 quadrillion, while the other two have values of 34 billion in miles per second, or almost a quintillion in feet per second. No kind of conversion ratio can make those compatible; they are either right or wrong. Einstein knew that. He just didn’t know which was which.

Could Einstein’s formula be tested experimentally? To be valid, we would need values for mass and energy that can be established through direct measurement in physical experiments. The ratios between these values should be judged either directly if the units are compatible, or, as a last resort, through established convention.

In the World Year of Physics 2005, as a centennial celebration of Albert Einstein’s achievements in 1905, just such a test was performed4, essentially the one that Einstein proposed for himself, except that we do now have a sufficiently “sensitive balance”: “the loss of mass to an atom and a neutron, as a result of the capture of the neutron and the production of a gamma ray, has been used to test mass–energy equivalence to high precision, as the energy of the gamma ray may be compared with the mass defect after capture. In 2005, these were found to agree to 0.00004%, the most precise test of the equivalence of mass and energy to date.” That’s four myriadths of a per cent. Four myriadths of a per cent is amazingly precise – four parts per ten million – as well as being a tribute to the sophistication of the measurement technology to which it bears witness. We are the first generation(s) with the power to *know* this.

So, which was it to be? SI, Imperial, US? Given the almost unlimited number of possible results, the most likely outcome of the Direct Test was none of the official systems. There was no design constraint that would force a choice between the three main measurement systems. And yet, what was this precise value? 89,875,517,873,681,800.00, otherwise known as the SI value for *c*^{2}.

This was an astonishing result, not just for its precision, but because of where we got the number to which we’re comparing it (See A Short History of *c*). Almost any other number from the huge range of possible speeds of light was infinitely more likely. Yet we have this one, and with it a strong suspicion that it is dimensionless, which reinforces the probability that energy and mass really are two states of the same property of nature.

Not that it should come as a surprise. We have been running nuclear power stations for decades, and this has been the value for the mass/energy ratio for all that time. Nonetheless, it has never been academically verified until now. You may, of course, still be entertaining doubts about the directness of the test, but you don’t have to take it from me; feel free to ponder the legitimacy or otherwise of this experiment. However, remember that this test convinced Stephen Hawking, and all other physicists of our time, of the validity of Einstein’s formula, so answers on a postcard please, to . . .

Again, what does this all mean?

Well, for a start, it means that, thanks to all the people involved, we now have an extremely precise numerical value for *c*, the speed of light. Why is this important? Because *c* is a universal physical constant, and it shows up in calculations for almost all the other physical constants. Take the natural units, for example:

“Physical constant.” *Wikipedia, The Free Encyclopedia*. Retrieved 30 Jul. 2017

The point is, we humans wrote all those, and all the formulae that use them. Bear in mind that the value we have given *c ^{2}* is around ninety quadrillion, so its position above or below the line makes an enormous difference, as Einstein pointed out in his article. We use these constants all the time, and we’ve measured the results. They are correct. They work.

So, Rainville et al have established that the measured ratio between the binding energy in a unit of mass and that energy once released is 89,875,517,873,681,800.00, which just happens to be the very same number that the French and the Babylonians conspired to set for *c ^{2}*. So, is this the coincidence of the title?

Yes. On the one hand we now have scientific proof that Einstein’s theory and its mathematical predictions were correct, including the value for *c ^{2}*, and on the other, the historical fact that it was the Babylonians who, 4,000 years ago, chose to divvy up into units of time the rotation of what we are assured – by Carl Sagan and Stephen Hawking among other luminaries – is merely “an insignificant planet of a humdrum star lost in a galaxy tucked away in some forgotten corner of a universe” – which, in combination with the arbitrary and inaccurate measurement of the surface of that same planet by the French, gave us the 299,792,458.00 value for c in the first place. I cannot begin to imagine what possible causal connection there could be between their activities and the empirically confirmed fundamental constants of the universe. On the other hand, you’ve got to admit, as coincidences go, it’s a Duesy5, because, thanks to Rainville et al, we now

*know*it to be so.

If it’s *not* a coincidence, and there *is* a causal connection, that would mean that we on this planet, *because* we’re on this planet and no other6, have been able to define *celeritas* or *constant*, or whatever you think the *c* stands for, for the entire universe, and thus all formulae with *c* in them. Apart from the Planck units, that includes most of the universal constants, and therefore almost all of modern physics. This is completely artificial, in the original sense of the word. There is no equivalent in nature to one 86,400^{th} of the averaged and corrected time it takes our “insignificant planet” to rotate on its own axis, even if you take a year’s worth7, just as there is nothing natural, despite the best efforts of Méchain and Delambre, about an incorrect measurement of a quarter arc of that same planet. But it is very human8. All the faffing about, rounding off and shrugging, not to mention the warfare and terror. Imagine that the English and French had got along, and collaborated through Greenwich instead. Or worse, that the English had gone it alone with yards per second; *c* would be 327,853,032.0688, and *c ^{2}* would be 107,487,610,636,706,0009, or nearly 20% greater than the number we have now; no four myriadths10of a per cent there. I’m not even going to mention

*miles*per second11.

So it has to be just the most amazing coincidence.

Unless, of course, it isn’t, but be careful with that. That way madness lies12.

[i] Adams, C.F. (editor) (1850–56), *The works of John Adams, second president of the United States*, vol. VIII, pp. 255–257, quoted in Ayling, p. 323 and Hibbert, p. 165, retrieved from Wikipedia *George III*, 9/9/14

[ii] Alder, Ken (2002). *The Measure of all Things—The Seven-Year-Odyssey that Transformed the World*. London: Abacus. ISBN 0-349-11507-9.