E = mc² and mass-energy

Mass and energy are not two different things. They are the same thing, measured in different units, like miles and kilometres. Einstein's famous equation, E = mc², is just the recipe for swapping one for the other.

The c in that equation is the speed of light, which is enormous, and the equation asks you to multiply by it twice. So the "exchange rate" from mass into energy is gigantic. Even a tiny scrap of mass, if you could fully unlock it, holds a staggering amount of energy. A single paperclip's worth would run a home for years.

This is the secret behind the Sun and behind nuclear power. Both of them get their energy by turning a very small amount of mass into a very large amount of energy. The Sun does it by squashing hydrogen together in its core. A nuclear power station does it by splitting heavy atoms apart. In each case only a whisper of mass disappears, but because you multiply by that huge number twice, the payoff is immense.

So mass is really a kind of frozen energy. Most of the time it stays locked up and does nothing. But nature has a few ways to prise a little of it loose, and when it does, the results power the stars.

The full sentence behind the slogan is this: an object that is sitting still still has energy, just because it has mass. That is its rest energy, and it is exactly

\[ E = m c^2. \]

The reason the numbers get so large is the \(c^2\). The speed of light is about \(3 \times 10^8\) metres per second, so \(c^2\) is roughly \(9 \times 10^{16}\) in metric units. One kilogram of anything, converted completely, would release about \(9 \times 10^{16}\) joules, which is a few weeks of the entire electricity output of a large country. That is why one gram fully converted is comparable to a small nuclear bomb.

The catch is that we almost never convert all of it. What actually happens in a reactor or a star is subtler, and it comes down to binding energy. When protons and neutrons lock together into a nucleus, the finished nucleus weighs slightly less than the loose parts you started with. That missing mass is called the mass defect, and it left as energy when the pieces bound together. Break the nucleus up again, or build a tighter one, and you can collect that difference.

Two routes exploit this. In fission, a heavy nucleus like uranium splits into lighter pieces that are more tightly bound, and about \(0.1\%\) of the mass is released. In fusion, light nuclei like hydrogen merge into helium, which is bound even more tightly, and about \(0.7\%\) of the mass is released. That sounds tiny, but with the \(c^2\) exchange rate it is colossal. The Sun runs on fusion, converting roughly four million tonnes of mass into energy every single second, and it has done so for over four billion years without running low.

Try both routes on the converter above. Even at a tenth of a percent efficiency, a paperclip's mass carries the energy of tens of tonnes of TNT.

Rest energy is only half the equation. \(E = mc^2\) is the special case for an object at rest. The full relation from special relativity ties energy, momentum and mass together:

\[ E^2 = (pc)^2 + (mc^2)^2. \]

Set the momentum \(p\) to zero and you recover \(E = mc^2\). Set the mass \(m\) to zero, as for a photon, and you get \(E = pc\): a massless particle still carries energy and momentum, and always moves at \(c\). The \(m\) here is the rest mass, or invariant mass, and it is genuinely invariant. Every observer, whatever their speed, agrees on it, because it is the length of the energy-momentum four-vector.

Forget "relativistic mass." Older texts describe a mass that grows with speed, \(m_{\text{rel}} = \gamma m\), so that a fast object gets "heavier." It is a bookkeeping device, not a good idea, and modern practice drops it. Mass is the invariant \(m\); what actually increases with speed is energy and momentum. Keeping mass fixed and letting energy carry the \(\gamma\) avoids a pile of confusions, including the false notion that a fast particle could collapse into a black hole.

Most of your mass is not the Higgs. Here is the part that surprises people. The mass of a proton is about \(938\) MeV/\(c^2\), but the three valence quarks inside it, whose masses come from the Higgs field, add up to only a few MeV. The rest, well over \(90\%\), is the energy of the gluon field and the kinetic energy of the quarks swirling inside, all counted as mass through \(E = mc^2\). So the mass of ordinary matter, and therefore most of your own weight, is confined energy, not the intrinsic mass the Higgs hands to the quarks. Binding energy is not a footnote to mass; for the proton it is almost the whole story.

The clean case is annihilation. Fission and fusion free a fraction of a percent because they only rearrange binding energy. When a particle meets its antiparticle, the entire rest mass of both converts, usually into photons: an efficiency of \(100\%\). An electron meeting a positron yields two \(511\) keV gamma rays, each carrying the full \(mc^2\) of one electron. That is the ceiling the converter's annihilation setting shows, and nothing chemical or nuclear comes close to it.