[This piece first appeared at Double X Science. Thanks to Emily Willingham for her ninja editing prowess.]

The association is strong in our minds: Albert Einstein. Genius. Crazy hair. E = m c². Maybe many people don’t know what else Einstein did, but they know about the hair and that equation. They may think he flunked math in school (wrong, though he did have conflicts with some teachers), that he was a ladies’ man (true, he had numerous affairs during both of his marriages), and that he was the smartest man who ever lived (debatable, though he certainly is one of the central figures in 20th century physics). Rarely, people will remember that he was a passionate antiracist and advocate for world government as a way of bringing peace.

Obviously whole books have been written about Einstein and E = m c², but a blog post at io9 caught my attention recently. The post (by George Dvorsky) itself looked back to a scholarly paper by David Topper and Dwight Vincent [1], which reconstructed a public lecture Einstein gave in 1934. (All numbers in square brackets [#] are citations to the references at the end of this post.) This lecture was one of many Einstein presented over the decades, but as Topper and Vincent wrote, “As far as we know [the photograph] is the only extant picture with Einstein and his famous equation.”

Well, kind of. The photograph is really blurry, and the authors had to reconstruct what was written because you can’t actually see any of the equations clearly. Even in the reconstructed version (reproduced below)…there’s no E = m c². Instead, as I highlighted in the image, the equation is E₀ = m. Einstein set the speed of light – usually written as a very large number like 300 million meters per second, or 186,000 miles per second – equal to 1 in his chalkboard talk.

What’s the meaning of this?

It is customary to express the equivalence of mass and energy (though somewhat inexactly) by the formula E = mc², in which c represents the velocity of light, about 186,000 miles per second. 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. –Albert Einstein [2]

Before I explain why it isn’t a big deal to modify an equation the way Einstein did, it’s good to remember what E = m c² means. The symbols are simple, but they encode some deep knowledge. E is energy; while colloquially that term gets used for a lot of different things, in physics it’s a measure of the ability of a system to do things. High energy means fast motion, or the ability to make things move fast, or the ability to punch through barriers. Mass m, on the other hand, is a measure of inertia: how hard it is to change an object’s motion. If you kick a rock on the Moon, it will fly farther than it would on Earth, but it’ll hurt your foot just as much – it has the same mass and therefore inertia both places. Finally, c is the speed of light, a fundamental constant of nature. The speed of light is the same for an object of any mass, moving at any velocity.

Mass and energy aren’t independent, even without relativity involved. If you have a heavy car and a light car driving at the same speed, the more massive vehicle carries more energy, in addition to taking more oomph to start or stop it moving. However, E = m c² means that even if a mass isn’t moving, it has an irreducible amount of energy. Because the speed of light is a big number, and the square of a big number is huge, even a small amount of mass possesses a lot of energy.

The implications of E = m c² are far-reaching. When a particle of matter and its antimatter partner meet – say, an electron and a positron – they mutually annihilate, turning all of their mass into energy in the form of gamma rays. The process also works in reverse: under certain circumstances, if you have enough excess energy in a collision, you can create new particle-antiparticle pairs. For this reason, physicists often write the mass of a particle in units of energy: the minimum energy required to make it. That’s why we say the Higgs boson mass is 126 GeV – 126 billion electron-volts, where 1 electron-volt is the energy gained by an electron moved by 1 volt of electricity. For comparison, an electron’s mass is about 511 thousand electron-volts, and a proton is 938 million electron-volts.

In our ordinary units the velocity of light is not unity, and a rather artificial distinction between mass and energy is introduced. They are measured by different units, and energy E has a mass E/C² where C is the velocity of light in the units used. But it seems very probable that mass and energy are two ways of measuring what is essentially the same thing, in the same sense that the parallax and distance of a star are two ways of expressing the same property of location. –Arthur Eddington [3]

Another side of the equation E = m c² appears when we probe the structure of atomic nuclei. An atomic nucleus is built of protons and neutrons, but the total nuclear mass is different than the sum of the masses of the constituent particles: part of the mass is converted into binding energy to hold everything together. The case is even more dramatic for protons and neutrons themselves, which are made of smaller particles knowns as quarks – but the total mass of the quarks is much smaller than the proton or neutron mass. The extra mass comes from the strong nuclear force gluing the particles together. (In fact, the binding particles are known as gluons for that reason, but that’s a story for another day.)

A brief history of an idea

The E₀ = m version of the equation Einstein used in his chalk-talk might seem like it’s a completely different thing. You might be surprised to know that he almost never used the famous form of his own discovery: He preferred either the chalkboard version or the form m = E/c². In fact, in his first scientific paper on the subject (which was also his second paper on relativity), he wrote [4]:

If a body gives off the energy L in the form of radiation, its mass diminishes by L/c². The fact that the energy withdrawn from the body becomes energy of radiation evidently makes no difference, so that we are led to the more general conclusion that … the mass of a body is a measure of its energy-content …

In other words, he originally used L for energy instead of E. However, it’s equally obvious that the meaning of E = m c² is present in the paper. Equations, like sentences in English, can often be written in many different ways and still convey the same meaning. By 1911 (possibly earlier), Einstein was using E for energy [5], but we can use E or L or U for energy, as long as we make it clear that’s what we’re doing.

The same idea goes for setting c equal to one. Many of us are familiar with the concept of space-time: that time is joined with space (thanks to the fact that the speed of light is the same, no matter who measures it). We see the blurring of the boundary between space and time when astronomers speak of light-years: the distance light travels in one year. Because c – and therefore c² – is a fixed number, it means the difference between mass and energy is more like the difference between pounds and kilograms: one is reachable from the other by a simple calculation. Many physicists, including me, love to use c = 1 because it makes equations much easier to write.

In fact, physicists (including Einstein) rarely use E = m c² or even m = E/c² directly. When you study relativity, you find those equations are specific forms of more general expressions and concepts. To wit: The energy of a particle is only proportional to its mass if you take the measurement while moving at the same speed as the particle. Physical quantities in relativity are measured relative to their state of motion – hence the name.

That’s the reason I don’t care that we don’t have a photo of Einstein with his most famous equation, or that he didn’t write it in its familiar form in the chalk-talk. The meaning of the equation doesn’t depend on its form; its usefulness doesn’t derive from Einstein’s way of writing it, or even from Einstein writing it.

References

1. David Topper and Dwight Vincent, Einstein’s 1934 two-blackboard derivation of energy-mass equivalence. American Journal of Physics75 (2007), 978. DOI: 10.1119/1.2772277 . Also available freely in PDF format.
2. Albert Einstein, E = mc². Science Illustrated (April 1946). Republished in Ideas and Opinions (Bonanza, 1954).
3. Arthur Eddington, Space, Time, and Gravitation (Cambridge University Press, 1920).
4. Albert Einstein, Does the inertia of a body depend upon its energy-content? (translated from Ist die Trägheit eines Körpers von seinem Energiegehalt abhängig?). Annalen der Physic17 (1905). Republished in the collection of papers titled The Principle of Relativity (Dover Books, 1953).
5. Albert Einstein, On the influence of gravitation on the propagation of light (translated from Über den Einfluss der Schwerkraft auf die Ausbreitung des Lichtes). Annalen der Physic35 (1911). Republished in The Principle of Relativity.
6. Albert Einstein, Relativity: The Special and the General Theory (1916; English translation published by Crown Books, 1961).