The box of light that weighs something

A photon has no mass. Two of them have one.

The arithmetic, which is one line #

The mass of anything — a particle, a nucleus, a box, a galaxy — is defined by

$$m^2c^4 = E^2 - p^2c^2,$$

with $E$ and $\mathbf{p}$ the total energy and total momentum of whatever is being weighed. For one photon $E = pc$ and the difference is zero, which is what “massless” means. For two photons of energy $E$ each, at an angle $\theta$, the total energy is $2E$ and the total momentum has magnitude $(E/c)\sqrt{2 + 2\cos\theta}$, and the difference is

$$m^2c^4 = 4E^2 - 2E^2(1 + \cos\theta) = 2E^2(1 - \cos\theta),$$

so $mc^2 = 2E\sin(\theta/2)$.

There is no new physics in that. It is the same expression the rung below uses for a single particle, applied to a total four-momentum rather than to one particle’s — and the reason the answer is surprising is that the expression is not linear, so the mass of a sum is not the sum of the masses.

The non-additivity is geometry rather than an accident. The length of a four-vector is not the sum of the lengths of the vectors added to make it — two null vectors pointing different ways add to a timelike one with a length neither of them has. Mass is that length. So a system can have a mass its parts do not, without anything having been created, in the same way that two sides of a triangle can be longer than the third.

The box #

Now put the light in a container. A mirrored box with $U$ of radiation energy in it, sealed, at rest.

The photons inside are flying in all directions, so their momenta cancel and the total momentum is zero. The total energy is the box’s own rest energy plus $U$. The invariant mass is therefore

$$M = M_{\text{box}} + \frac{U}{c^2}.$$

The box is heavier by the energy of the light inside it. Not by a fraction of it, not by an amount depending on the mirrors: by exactly $U/c^2$.

Light carries momentum, so it pushes on the walls of the box that holds it; accelerate the box and the radiation resists, because the light bouncing off the receding wall arrives redder and the light off the approaching wall arrives bluer, and the imbalance is a backward push. That resistance is inertia, measured the way inertia is always measured — by how hard the thing is to accelerate.

And an accelerating box cannot be told from a gravitational field, so the same energy weighs as well as resisting. That is not an extra assumption bolted on: it is the equivalence principle applied to a box whose contents happen to be massless, and the conclusion is that a box of light sits on a scale and registers.

Three ways to weigh the box, which had better agree #

The claim that a box of light is heavier is worth checking against every meaning of “heavier” separately, because they are logically independent and it is a real result that they coincide.

By inertia. Push the box. The light inside strikes the rear wall more often and harder than the front wall, because the box is moving toward one and away from the other, and the imbalance opposes the push. Working the Doppler shifts through gives an extra inertia of exactly $U/c^2$ — the calculation Einstein published in 1905 in the paper that has the mass–energy relation in its title, and it is done for a body emitting light rather than containing it.

By weight. Hold the box in a gravitational field. A photon travelling upward inside it loses energy and one travelling downward gains it, at the rate the gravitational redshift gives, so the downward-travelling light pushes harder on the floor than the upward-travelling light pushes on the ceiling. The net downward force is $Ug/c^2$.

By the invariant. Compute $E^2 - p^2c^2$ for the contents, as above, and take the square root.

Three different questions — how hard is it to accelerate, how hard does it press on a scale, and what is the length of its four-momentum — and one answer. That they agree is the content of the equivalence principle plus special relativity, and none of the three arguments assumes either of the others.

Einstein’s argument, and why it needs two frames #

The inertia calculation above is quoted as Einstein’s and is worth setting out, because the striking thing about it is that one frame is not enough.

Take a body at rest that emits two equal pulses of light in opposite directions, each of energy $E/2$. Its momentum is unchanged, since the two pulses carry opposite momenta, and it stays at rest. Its energy has fallen by $E$. In that frame alone nothing can be concluded: energy left, and there is no reason yet to say anything about mass.

Now watch the same event from a frame moving at $v$. Each pulse is Doppler-shifted — one up, one down — and the shifts do not cancel in energy, because the Doppler factors are not reciprocal in the way that would be needed. Working it through, the total energy carried off in the moving frame is $\gamma E$, larger than in the rest frame.

So the body has lost $E$ in one frame and $\gamma E$ in the other. The difference, $(\gamma - 1)E$, is kinetic energy — the body’s kinetic energy in the moving frame has fallen, while its speed has not changed at all. The only quantity in $(\gamma-1)mc^2$ left to change is $m$, and matching the two accounts requires

$$\Delta m = \frac{E}{c^2}.$$

The argument is two paragraphs long and it needs both frames. In one, energy simply left. In the other, kinetic energy left too, and a body whose kinetic energy fell at constant speed has less mass. Neither observation alone forces the conclusion, and the comparison does.

That is worth noticing as a method rather than as history. A quantity that changes in one frame and not in another is a component; a discrepancy between two frames’ accounts of the same event is where an invariant is hiding, and the discrepancy here is the whole of the derivation.

What everything is actually made of #

If this were a curiosity about photons it would be a footnote. It is the ordinary case.

A proton weighs 938 MeV. Its three valence quarks weigh, between them, about 9 MeV — one part in a hundred. The other 929 come from the kinetic energy of the quarks and the energy of the gluon field that binds them, both of which are energies of massless or nearly massless things arranged in a confined region.

So the claim that mass is not a property of stuff is not a philosophical position. It is an accounting fact about the objects on any table: the mass of ordinary matter is dominated by the energy of the arrangement, and only about one per cent of it is the mass of the constituents.

The numbers, on things that can be weighed #

It is worth seeing how large the effect is in cases where the answer is known to many figures, because the size ranges over forty orders of magnitude and the physics does not change anywhere in it.

A helium nucleus weighs 28.3 MeV less than two protons and two neutrons — seven parts in a thousand of its own mass, and the largest fractional binding of any light nucleus. That deficit is measurable on a mass spectrometer and has been since 1920.

A hydrogen atom weighs 13.6 eV less than a proton and an electron, which is fifteen parts in a thousand million. Nobody has weighed that directly and it is not in doubt.

A cup of boiling water weighs about $4\times10^{-12}$ kilograms more than the same water at room temperature. That is four picograms, which is at the edge of what a good balance can do and is swamped by buoyancy corrections.

A wound clock spring carries perhaps a joule, and so about $10^{-17}$ kilograms.

A proton is the extreme case in the other direction: 99 per cent of its mass is arrangement, and the constituents are the correction.

The ordering is not accidental. The fraction of a system’s mass that is binding or kinetic energy is a measure of how strongly it is bound relative to the rest energy of its parts — parts per thousand for the strong force acting on nucleons, parts per thousand million for the electromagnetic force acting on electrons, and essentially all of it for quarks, whose own masses are small compared with the energy scale that confines them.

The measurement that tests it to a part in a million #

The relation is quoted so often that it is easy to forget it has been measured, and the sharpest test is a nuclear one that compares two quantities determined by entirely different apparatus.

Take a nucleus and let it capture a neutron. The product is heavier by one neutron and lighter by the binding energy released, which comes out as a gamma ray. So there are two independent numbers: the difference in mass between the initial and final nuclei, and the energy of the gamma ray. The relation says the first times $c^2$ equals the second.

The mass difference is measured in a Penning trap, by comparing the cyclotron frequencies of single ions held in a magnetic field — a frequency measurement, ultimately, and good to a part in $10^{11}$. The gamma energy is measured by diffracting it from a silicon crystal whose lattice spacing is known, which turns an energy into an angle and a length. Nothing about the two techniques is shared: one is electromagnetic and frequency-based, the other is a wavelength measured against a crystal.

Compared on silicon and sulphur, the two agree to within about four parts in ten million. That is the most precise direct check the relation has, and its value is in the independence of the two sides rather than in the number of digits: an error in the mass spectrometry and an error in the crystal diffraction would have to conspire to hide a discrepancy.

It is also a test of exactly the statement this essay is about. What is compared is a change in a system’s mass against an energy that left it — not a conversion of one into the other, but the bookkeeping identity that a system lighter by $\Delta m$ is a system that has emitted $\Delta m c^2$.

Why colliders collide #

The non-additivity has an engineering consequence large enough to have decided how every high-energy laboratory in the world is built.

What matters for making a heavy particle is the invariant mass of the collision — the total four-momentum’s length, since that is what is available in the frame where the products can be made at rest. For two equal beams meeting head-on, the momenta cancel and the invariant mass is simply the sum of the energies: two beams of 7 TeV give 14 TeV to work with.

Fire the same beam at a stationary target instead and the arithmetic collapses. The total momentum is now the beam’s, undiminished, so most of the energy is locked up in the motion of the products and is unavailable. Working the invariant out gives roughly $\sqrt{2mE}$, with $m$ the target’s mass — a square root rather than a sum. A 7 TeV proton striking a stationary proton delivers about 115 GeV of usable energy, against the collider’s 14,000.

A factor of a hundred and twenty, from geometry alone. And the scaling is worse than the ratio suggests: doubling a collider’s beam energy doubles what is available, while doubling a fixed-target beam’s energy multiplies it by only $\sqrt2$. There is no beam energy at which a fixed target catches up.

The price is that two beams are enormously harder to collide than one beam is to aim at a block of metal, because a target is dense and a counter-rotating beam is not. Every collider is that trade — a hundredfold in useful energy against a collision rate many orders of magnitude lower — and the trade is forced by the fact that mass is a length rather than a sum.

The reactor, which is where the standard story goes wrong #

The sentence mass is converted into energy is in every popular account and every school textbook, and it describes something that does not happen.

Take a sealed reactor: fuel, moderator, walls, nothing entering or leaving. Run it for a year. Its invariant mass at the end is exactly what it was at the start, because the total energy inside is what it was and the total momentum is still zero. Nothing was converted, and nothing weighs less.

What has happened is a redistribution: some rest energy has become kinetic energy and radiation, and those are inside the box. Open a valve, let the heat out, and now the mass falls — by the energy that left, over $c^2$.

The same correction applies to the Sun, to a battery, and to a compressed spring — a wound clock spring weighs about $10^{-16}$ kilograms more than a slack one, which nobody can measure and which is nonetheless the same physics as the proton’s 929 MeV.

Where the mass goes when the angle changes #

The photon-pair figure has a feature worth stating plainly, because it is the sharpest test of whether the idea has landed. Two photons at 180° weigh $2E/c^2$. The same two photons at 0° weigh nothing. Nothing has been added, removed or converted; only the angle differs.

The geometry of a photon–electron collision does the same bookkeeping in the other direction. Total four-momentum is conserved, the photon’s mass is zero before and after, and the electron’s is unchanged — yet the photon’s energy falls and the electron’s rises. Nothing about mass is doing any of the work; the conserved quantity is the four-vector, and the masses are lengths of pieces of it.

An electron–positron pair annihilating at rest produces two photons back to back, each of 511 keV. Their invariant mass is $2 \times 511$ keV$/c^2$, which is exactly the mass of the pair that made them. Mass was conserved, not destroyed, and it moved from being the rest mass of two particles to being the arrangement of two that have none.

Light arrives in lumps of energy $hf$, and each lump carries momentum $hf/c$. That the two stand in the ratio $c$ is exactly the condition $E = pc$ which makes a photon massless — so the quantum statement and the relativistic one are the same statement, and neither is evidence for the other. They agree because they are one fact counted twice.

What it costs #

The box has to be closed. Every statement above is about an isolated system. The moment anything leaves, the accounting changes, and most of the confusion in the subject comes from applying a closed-system result to an open one.

“Relativistic mass” has been avoided deliberately. Assigning each particle a mass $\gamma m$ makes the total additive again and buys that with an observer-dependent mass — and then the mass of a system depends on who is looking, which is the one thing the invariant is for. The photon pair is the cleanest reason not to use it: two photons have no rest mass and a relativistic mass of $E/c^2$ each, and neither number gives the system’s mass at any angle but one.

The historical alternative is worth knowing about because it was abandoned rather than disproved. Longitudinal and transverse mass were introduced to keep $F = ma$ looking familiar, and they differ from each other by $\gamma^2$ — so a body had two masses depending on which way it was pushed. The modern convention keeps mass as an invariant and lets the force law carry the complication, which is not more correct but is very much easier to think with.

Where the model stops #

Gravitation has been treated as though mass were the source. It is not: the source is the stress–energy tensor, of which energy density is one component and pressure is three more. For a box of light the pressure terms are not negligible — a photon gas has $P = u/3$ — and the correct gravitational field of a box of radiation is not the field of a point mass $U/c^2$ except far away, where only the total matters.

The proton’s 929 MeV is not a calculation done here. It is the result of lattice quantum chromodynamics, and it is one of the more expensive computations in physics. What this essay establishes is that such a mass is possible; that it comes out at 938 is a separate and much harder statement.

Nothing here is about a single photon in a gravitational field. A photon does not “have a mass hf/c²” that gravity acts on; what happens to light in a field is a statement about the geometry it travels through, and the two accounts agree to first order and part company after that. The box argument works because a box has a rest frame; a photon does not.

And a system has to have a rest frame for any of this to apply. Two parallel photons have none: their total four-momentum is null, and there is no frame in which they are at rest. That is the reason the curve reaches zero at the left-hand edge rather than approaching it, and it is not a limiting case of the others — it is a different kind of object.

A clock made of light slows because $c$ is the same for everybody, and that same constancy is what makes $E^2 - p^2c^2$ an invariant at all. Everything in this essay rests on one thing being frame-independent, and it is worth ending by naming it: not the mass, not the energy, but the speed the whole construction is built out of.

What the pictures cannot show #

The hero figure plots a mass against an angle, and the angle is between two things that have no rest frame and therefore no proper way of being at an angle to each other in any single observer’s account. Every observer measures a different angle between the two photons and a different energy for each, and the mass they compute is the same. The axis, in other words, is a quantity that changes between frames plotted against one that does not, which means the whole curve is drawn in a particular frame and only its individual points are invariant.

Nor can any figure here show what a mass is. It has been used as a label on an axis, as a reading on a balance and as the length of a four-vector, and those three uses agree in every case examined — but the agreement is a theorem rather than a picture, and the drawings show consequences of it rather than it.

Where this ladder goes next #

Four rungs stand on mass-energy. The first found that mass is a form of energy rather than a source of it; the second found the invariant that survives a boost; the third found the mass a nucleus is missing. This one closes the pattern by removing the constituents entirely: a system whose parts have no mass at all has one, and that is the ordinary case rather than a special one.

The habit worth carrying away is about extensive quantities. Ask of any total whether it is a sum. Energy is; momentum is; charge is; entropy is, nearly. Mass looks as though it should be and is not, because it is a length rather than a component, and a great deal of confusion in this subject is one non-additive quantity being added up.

What is left on this ladder is where the energy actually sits. A system’s mass is its total energy in its own frame, and asking where that energy is located is a question with no frame-independent answer — the field of a moving charge carries energy that a different observer attributes elsewhere, and the same ambiguity is what makes localising gravitational energy impossible.