Mass is a form of energy, which is not the same as a source of it

Push something with a constant force for long enough and Newtonian mechanics says it goes arbitrarily fast. The work done is force times distance, the kinetic energy is $\tfrac12 mv^2$, and there is no speed at which the arithmetic runs into trouble.

Relativity says there is, and the way the trouble appears is worth drawing rather than asserting.

The correct expression is

$$E = \gamma mc^2, \qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}},$$

with the same $\gamma$ that slows a moving clock. As $v \to c$ the factor diverges, so accelerating a massive body to the speed of light would take infinite energy — which is the quantitative form of the statement that nothing with mass gets there, and a considerably more informative form than the bare prohibition.

The term that does not vanish #

Expanding $\gamma$ for small speeds gives

$$E = mc^2 + \tfrac12 mv^2 + \tfrac{3}{8}\frac{mv^4}{c^2} + \dots$$

The second term is the Newtonian kinetic energy, recovered exactly, which is what a correct generalisation must do. The third is the first relativistic correction, and it is why the two curves in the figure separate the way they do.

The first term is the argument of this rung. It does not contain the speed. It is there when the body is at rest, when nothing is happening to it, and it does not go away.

That is the content of $E = mc^2$, and it is a stranger claim than the popular reading. The popular reading is that mass can be converted into energy under special circumstances — in a reactor, in a bomb, in the sun. The equation says something stricter: a mass is an energy, in the same way that a length in metres is a length in feet. There is no conversion event. A kilogram of anything, sitting on a table doing nothing, is $9\times10^{16}$ joules, and the number is a restatement of the mass rather than a prediction about its future.

Why almost nothing releases it #

If every kilogram is that many joules, the immediate question is why the world does not look like it.

Burning a kilogram of coal releases 32.8 megajoules, and dividing by $c^2$ gives a mass change of 3.6 parts in $10^{10}$ — 0.36 micrograms, out of a kilogram. The equation applies perfectly well to the chemistry. The products of the fire weigh very slightly less than the reactants, exactly as predicted, and the difference is a hundred times too small to weigh on the best balance ever built.

That is the whole reason mass–energy equivalence was not discovered by chemists. Conservation of mass and conservation of energy looked like two separate exact laws for two centuries because every process available to test them converted a part in a billion or less, and a part in a billion was invisible.

Nuclear processes are different by six orders of magnitude, not because different physics applies but because nuclear binding energies are millions of times larger than chemical ones. Fission of uranium-235 converts about 0.09 per cent of the mass; deuterium–tritium fusion about 0.38 per cent. Those are still small fractions — a fusion reactor does not annihilate its fuel — and they are enormous compared with anything a fire can do.

What “the mass of a system” then means #

The equation runs in both directions, and the direction that gets less attention is the more surprising one: adding energy to a system increases its mass.

A compressed spring weighs more than a relaxed one. A hot brick weighs more than a cold one. A box of gas weighs more than the sum of the masses of its molecules, by their total kinetic energy divided by $c^2$. None of these is measurable, and all of them are exactly true.

Where it becomes measurable is the nucleus. A helium nucleus weighs 0.7 per cent less than two protons and two neutrons weighed separately, and the deficit is the binding energy — the energy that would have to be supplied to take it apart. The mass of a composite object is not the sum of the masses of its parts. It is the one quantity every observer agrees about: the sum of the parts’ masses plus their kinetic energies plus their potential energies, all divided by $c^2$, and for a nucleus the potential term is large and negative.

The proton itself is the extreme case. Its three valence quarks have rest masses summing to about one per cent of the proton’s mass; the other ninety-nine per cent is the energy of the gluon field and the motion of the quarks inside. Nearly all of the mass of ordinary matter — the mass of a reader, a planet, a star — is not the mass of anything. It is binding energy and kinetic energy, counted as mass because that is what mass is.

The collision that gains weight #

The cleanest argument for the whole business is a collision, and this site already has the figure for it.

Take two bodies of mass $m$ approaching at the same speed and colliding perfectly inelastically. The total momentum is zero before and after, so the result is at rest. The total energy before was $2\gamma mc^2$; energy is conserved, so the object at rest has energy $2\gamma mc^2$; and an object at rest with energy $E$ has mass $E/c^2$.

Its mass is therefore $2\gamma m$, which is greater than $2m$. The kinetic energy has not disappeared into heat as a separate accounting category — it has become mass, and the heat is that mass. Weighing the fragments after a perfectly inelastic collision would give a larger answer than weighing them before, by exactly the kinetic energy divided by $c^2$.

That argument uses only conservation of energy, conservation of momentum, and the requirement that both hold in every frame. It does not require the light clock, the postulates, or any of the machinery of the spacetime diagram — which is why Einstein could present it in a three-page paper as a consequence rather than as a new assumption.

The construction the factor comes from is a clock carried past at speed. Its light has further to go, so it ticks slow, and the same factor multiplying the rest energy is what makes the energy diverge as the speed approaches $c$. One geometric fact, two consequences that look unrelated until they are written down — and it is why a result about clocks turns out to constrain what a reactor can do.

Relativistic mass, and why it is avoided #

An older presentation writes $E = mc^2$ with a mass that grows with speed, $m_{\text{rel}} = \gamma m_0$, so that the formula stays superficially Newtonian. It is not wrong arithmetically and it is a poor idea, for three reasons worth stating.

It suggests that a fast object becomes intrinsically harder to accelerate in all directions equally, which is false — the resistance to a force along the motion and across it differ by a factor of $\gamma^2$, so no single number called “mass” describes both.

It suggests that a fast object would collapse into a black hole if it went fast enough, which is false, and is a paradox that dissolves the moment mass is understood as frame-independent.

And it wastes the word. The useful quantity is the one every observer agrees on: the rest mass, which is an invariant in exactly the way the spacetime interval is. The relation

$$E^2 = (pc)^2 + (mc^2)^2$$

makes the structure explicit — energy and momentum are the components that depend on the observer, and $m$ is the length of the vector they form, the same for everyone. Written this way the equation also covers particles with no mass at all: setting $m = 0$ gives $E = pc$, which is the correct statement for light and which the $\gamma m c^2$ form cannot express, since $\gamma$ is infinite and $m$ is zero.

That one function governs the energy, the time and the length is the whole economy of special relativity. There is a single factor, it depends on speed and on nothing else, and every kinematic quantity carries it or its reciprocal. Learning the subject is largely a matter of finding out which.

Where the energy for a star comes from #

The sun radiates $3.8\times10^{26}$ watts, and dividing by $c^2$ gives a mass loss of 4.3 million tonnes per second. That is a striking number and it is worth putting beside the sun’s mass of $2\times10^{30}$ kilogrammes: the loss over the sun’s entire main-sequence lifetime is about a thousandth of it.

The mechanism is fusion of hydrogen to helium, which converts 0.7 per cent of the mass of the hydrogen involved. Before this equation, the energy source of the sun was an open problem with no acceptable answer: chemical burning would exhaust the sun in a few thousand years, and gravitational contraction — the best available proposal, from Kelvin and Helmholtz — gives about 30 million years, which was already in flat contradiction with the geological evidence for an Earth far older.

That contradiction was one of the sharpest in nineteenth-century science, with Kelvin’s physics on one side and Darwin’s and Lyell’s geology on the other. It was settled by a term in an expansion, and the settlement is a good illustration of what a hidden constant term can do: nothing in the observation changed, and the energy budget of the solar system was rewritten.

What the curve cannot show #

The hero figure plots energy against speed, and both axes conceal something.

The horizontal axis stops before it gets interesting. It runs to $0.98c$, where $\gamma$ is 5. Modern accelerators run at $\gamma$ in the tens of thousands, at which the speed differs from $c$ by parts in a billion and every point sits on the same pixel. A speed axis is simply the wrong coordinate above about $0.99c$ — the useful variable there is the energy itself, or the rapidity, and the figure’s asymptote is the visual reason why.

The vertical axis is in units of the rest energy. That makes every body’s curve identical, which is the point, and it hides how enormous the vertical offset is. Drawn to a scale on which the rest energy of a gram were one millimetre, the kinetic energy of a rifle bullet would be under a nanometre. The whole familiar range of mechanical energies lives in the thickness of the line at $E/mc^2 = 1$.

Nothing on the curve is a process. A body does not travel along it. The curve is a relation between two properties of a state, and the work needed to move from one speed to another is a difference of heights on it — which is the only sense in which the divergence at $c$ is a statement about what can be done.

The letter, and the sentence that ends it #

The equation did not appear in the paper that founded special relativity. It appeared in a three-page afterthought published later the same year, titled as a question — Does the inertia of a body depend upon its energy content? — and its argument is the collision argument in reverse: a body emitting light in two opposite directions loses energy, must be left at rest by symmetry, and is found to have lost mass.

Einstein’s own summary at the end is worth quoting for what it concedes: the result would be testable, he wrote, with bodies whose energy content is variable to a high degree, such as radium salts. He did not have a way of measuring it. The equation was published as a consequence with a suggested experiment attached, and it took until the 1930s and the mass spectrometer for the arithmetic to be checked directly — Cockcroft and Walton’s lithium splitting in 1932 being the first case where the masses in, the masses out and the energy released were all measured and agreed.

That sequence — consequence first, measurement decades later — is worth registering against the popular telling in which the equation arrives with a mushroom cloud attached. For thirty years it was a curious corollary about the inertia of hot bodies.

The most direct test, and the unit that followed #

The equation was checked indirectly for a century — every nuclear reaction’s energy balance is a test of it — and the sharpest direct comparison was made in 2005 by measuring both sides on the same nucleus.

The experiment is simple to state. Let a silicon nucleus capture a neutron; it emits gamma rays whose energies can be measured to a few parts in a million by diffracting them from a perfect crystal, which turns an energy into a lattice spacing and an angle. Separately, weigh the nucleus before and after in a Penning trap, which compares cyclotron frequencies of single ions and reaches parts in $10^{11}$.

One side of the balance is an energy measured as a wavelength; the other is a mass difference measured as a frequency ratio. They agreed to four parts in ten million, which is the tightest direct confirmation there is — and the two measurements have nothing whatever in common except the nucleus they were made on.

That kind of comparison also changed the definition of the kilogram. Since 2019 the unit is fixed by declaring Planck’s constant to have an exact value, which — with $c$ also exact — makes a mass and a frequency the same quantity in two notations: $m = h\nu/c^2$. The frequency corresponding to a kilogram is absurd, some $10^{50}$ hertz, and no experiment goes near it; what is done instead is to carry the ratio down through atoms, whose masses and whose gamma-ray energies are both measurable, or across through a Kibble balance, which weighs a mass against an electrical power.

The metrological point is the one the previous section makes. Mass and energy are not two quantities related by a large number; they are one quantity, and the system of units has now been arranged to say so.

Weighing a proton that is mostly not there #

The claim that ninety-nine per cent of a proton’s mass is field energy rather than the mass of anything is the strangest sentence on this page, and it is not an estimate. It has been computed.

The theory of the strong interaction has no closed-form solution at the energies where a proton exists, so the calculation is done by replacing spacetime with a lattice of a few million points and evaluating the resulting integral numerically — a computation that took decades of method development and a supercomputer to finish. The inputs are the quark masses and one coupling strength; the outputs include the masses of the proton, the neutron and a dozen other particles.

They come out right, to a few per cent, and the proton comes out at 938 MeV with quark masses summing to about nine. Everything else is the energy of the gluon field binding them and the kinetic energy of the quarks moving inside — energy which, by the argument of this page, simply is the mass.

The sharpest way to state it is a limit. Set the up and down quark masses to zero, so that the constituents have no mass at all, and rerun the calculation: the proton still comes out at around 870 MeV. A particle built entirely from massless parts weighs almost exactly what the real one does.

So the mass of ordinary matter is not, at bottom, the mass of anything. It is confined energy, counted in kilograms because that is what the equation says to do with it.

Where the model stops #

Nothing here is about gravity. Special relativity treats flat spacetime and inertial frames. Mass–energy equivalence has consequences for gravity — energy gravitates, which is why the pressure inside a star contributes to its own weight — but that requires general relativity and is outside this rung entirely.

Rest energy has no zero. Only differences in energy are measurable in every other part of physics, so an additive constant is conventionally discarded. Here the constant is the whole claim, and what makes it more than bookkeeping is that it participates: the mass deficit in a nucleus is measured on a mass spectrometer, the energy released is measured in a calorimeter, and the two agree.

The conversions have their own costs. The budget figure gives the fraction of mass a process converts, and says nothing about whether the process can be run, contained, or made to yield net energy. Fusion converts four times the share fission does and has taken seventy years of engineering without a net-positive power plant, because the difficulty is confinement rather than arithmetic.

Antimatter is not a fuel. The hundred per cent bar in the figure is real and misleading. Antimatter does not occur naturally in usable quantities and making it costs far more energy than annihilating it returns, so it is a storage medium of terrible efficiency rather than a source. The bar shows what the equation permits, not what is available.

The constant that is a conversion factor #

One more reading of the equation is worth having, because it makes the $c^2$ look less like a physical mechanism and more like what it is.

Mass and energy are measured in different units for historical reasons — the kilogram was defined by a lump of metal and the joule by mechanical work — and $c^2$ is the number that converts between them, exactly as 1,609 converts miles to metres. In the unit systems particle physicists actually use, $c$ is set to one and the equation reads $E = m$: the same quantity, two names.

That is why the size of $c^2$ explains nothing about nuclear energy. A large conversion factor makes the number of joules per kilogram large, and it equally makes the number of kilogrammes per joule small. What decides whether a process releases usefully much is the fraction of the mass it converts, which is the budget figure above and has nothing to do with the size of $c$.

Where the ladder goes next #

The rungs from here: four-momentum, and the invariant that makes $E^2 - (pc)^2$ the natural object; massless particles and why light has momentum without mass; binding energy across the periodic table and the iron peak, which is why fusion pays below iron and fission above it; pair production and the threshold energy; the relativistic Doppler effect, where the energy of a photon becomes frame-dependent; and the equivalence of energy and inertia in general relativity, where energy of every kind becomes a source of gravity.

The claim to carry forward is the middle one. Nothing on this page is a mechanism for converting mass into energy. A mass is an energy, permanently and without any process being involved — and the reason that took until 1905 to notice is that every process anyone could run converted a part in a billion of it.