The mass that would make Gauss's law leak

Counting what comes out found that the flux of the electric field out through any closed surface equals the charge inside divided by $\varepsilon_0$, whatever the surface’s shape and size, and that this is a restatement of the inverse-square law. The shape decides the falloff found that the same law makes a line charge’s field fall as $1/r$ and a sheet’s not at all, and the number of dimensions a planet can orbit in that the exponent 2 is the dimension of space written as a power. The inside of a conductor found the law’s most practical consequence — the interior of a closed conducting shell is field-free — and noted that this is also its most sensitive test, since confirming a zero needs no calibration.

That test is usually described as a measurement of the exponent: if the force fell as $1/r^{2+\epsilon}$, the inside of a charged shell would carry a small field, and none is found. But there is a second way Coulomb’s law could fail, which is not a change of exponent at all, and it behaves so differently that the tests which best constrain it are not laboratory experiments. It is the possibility that the photon has a mass. This essay follows what a mass would do to Gauss’s law, why the two kinds of failure need opposite kinds of apparatus, and what “the photon has a mass” would mean in a world where light already acts as if it had one inside every plasma and every superconductor.

What a mass does to a field #

A field carried by a particle of mass $m$ does not reach arbitrarily far. The static potential of a point charge, which for a massless photon is $1/r$, becomes Yukawa’s

$$\phi(r) \propto \frac{e^{-\mu r}}{r}, \qquad \mu = \frac{mc}{\hbar},$$

and dies off exponentially beyond the length $1/\mu = \hbar/mc$, the particle’s reduced Compton wavelength. This is the form Hideki Yukawa proposed in 1935 for the nuclear force, whose short range he explained by a carrier with a mass, and the pion he predicted is that carrier. Alexandru Proca had written down the equations for a massive vector field the year before. For electromagnetism the change is that Gauss’s law acquires an extra term,

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} - \mu^2 \phi,$$

and the new term is the entire content of the modification: the divergence of the field is no longer set by the charge alone but by the charge and the local potential together.

The figure computes the consequence for the flux. Round a point charge the flux through a sphere is no longer the same at every radius. It is $(1 + \mu r)e^{-\mu r}$ of the Gauss value: nearly all of it at a tenth of the Compton length, three-quarters at one length, a fifth at three. The flux no longer counts the charge; it counts the charge minus a contribution from the potential spread through the volume inside the surface, which acts like a smeared-out charge of the opposite sign. Everything counting what comes out built on the surface-independence of the flux would fail at distances comparable with $\hbar/mc$, and would fail slightly at every distance.

The inside of a charged shell #

The shell test is where the failure shows most directly.

For a massless photon the interior of a closed conducting shell is an equipotential, whatever charge the shell carries, because a field-free interior is the only solution of Laplace’s equation that matches a constant potential on the boundary. With a mass the equation inside becomes $(\nabla^2 - \mu^2)\phi = 0$, whose solution regular at the centre is $\sinh(\mu r)/r$, and the potential sags towards the middle. The figure plots it for three sizes of shell measured against the Compton length. At $\mu R = 0.5$ the centre sits four per cent below the shell; at $\mu R = 2$, forty-five per cent. For a small mass the sag is

$$\frac{\Delta\phi}{V_0} \approx \frac{(\mu R)^2}{6},$$

and the structure of that formula is the whole of this essay. The effect of a mass is proportional to the square of the size of the apparatus measured in units of the Compton length. A shell twice as large shows four times the effect.

Two ways to be wrong #

A departure in the exponent behaves differently, and the difference is a matter of what lengths the law contains.

Suppose the force between two charges fell as $1/r^{2+\epsilon}$ with $\epsilon$ small. That law, like Coulomb’s, contains no length: it looks the same at every scale, magnified or shrunk. A shell of radius one metre and a shell of radius one centimetre, charged to the same potential, must then have interiors that look identical once distances are measured as fractions of the radius. The interior is no longer exactly equipotential — the argument that made it so needed the exponent to be exactly 2 — and the calculation gives a sag at the centre of $\epsilon(1 - \ln 2)$, about $0.31\epsilon$, whatever the shell’s size.

The figure plots the two side by side. A wrong exponent gives a flat line: making the apparatus larger or smaller changes nothing, so the only way to improve the test is to improve the detector. A mass gives a steep line, rising as the square of the size: making the apparatus ten times larger improves the test a hundredfold, with the same detector. A departure with no length in it is best caught by the most sensitive instrument; a departure with a length in it is best caught by the largest.

The distinction has a general form worth keeping. A modification of a law that introduces a new length shows itself in proportion to how the experiment’s size compares with that length, and the best test is the one whose size is closest to it. Tests of Newton’s inverse square at short range, which the number of dimensions a planet can orbit in described, look for a Yukawa correction with a length below a millimetre and are therefore built as small as possible — torsion balances with masses a few tens of micrometres apart. A photon mass, if it exists, is known to have a length far larger than any laboratory, and the tests that bound it are therefore built as large as possible, which means they are not built at all but found.

Fields the size of the Solar System #

The laboratory shell test was pushed to its limit in 1971 by Edwin Williams, James Faller and Henry Hill, with concentric shells about a metre and a half across, charged by a ten-kilovolt alternating voltage and monitored for any potential difference between them with a lock-in detector. They found none: the exponent’s departure from 2 came out consistent with zero to within about $3 \times 10^{-16}$, and translated the null into a bound on the photon’s mass of about $10^{-14}$ electronvolts. That is a Compton length of twenty thousand kilometres — more than ten million times the size of the apparatus, reached because the detector could see one part in $10^{16}$.

Larger fields do better, and the figure lists them against their sizes. A massive photon changes the shape of a planet’s magnetic field as well as a charge’s electric field: a dipole field would fall off faster than $1/r^3$ at distances comparable with the Compton length, and extra field components would appear. When Pioneer 10 flew past Jupiter in 1973 and mapped its field out to many planetary radii, the absence of any such distortion bounded the mass at a few times $10^{-16}$ electronvolts. The magnetic field carried outward by the solar wind is larger still. Its structure — the spiral wound up by the Sun’s rotation, measured by spacecraft all the way out past Pluto — would be distorted by a photon mass in ways that depend on the field’s own size, which is tens of astronomical units, and its observed shape bounds the mass at $10^{-18}$ electronvolts, a Compton length of about two hundred million kilometres, more than the distance from the Earth to the Sun. That is the bound usually quoted.

The Galaxy’s magnetic field, coherent over thousands of light years, would give far stronger bounds, down to $10^{-27}$ electronvolts, and such bounds have been claimed. They rest on assumptions about how that field is sustained — whether a massive photon’s field could be maintained by currents in the galactic plasma in the way the argument supposes — that have been challenged, and they are not generally accepted. Every point in the figure sits far below the dashed line where the Compton length equals the apparatus: every test reaches much further than its own size, because each is sensitive to the square of a small ratio.

There is a third kind of test, independent of the others. A massive photon would travel more slowly at low frequencies, arriving later than high frequencies from the same event, in the way the photons that raced for seven billion years tested for a different speed at high energy. Short radio bursts from other galaxies, whose arrival times at different frequencies are measured to milliseconds, give bounds comparable with the laboratory’s — weaker than the solar wind’s, but with no model of a magnetic field behind them.

Where light already has a mass #

There is a reason to take the question seriously beyond curiosity, and it is that light does behave as if it had a mass whenever it moves through matter that can respond to it.

In a plasma, transverse waves obey $\omega^2 = \omega_p^2 + c^2k^2$. That is exactly the relation between energy and momentum for a particle of mass $\hbar\omega_p/c^2$, and the consequence is the one the frequency below which nothing gets in followed: waves below the plasma frequency cannot propagate and are reflected, which is how the ionosphere bounces short-wave radio round the Earth. In a superconductor, the field that is pushed out found that a static magnetic field dies away over the penetration depth $\lambda$, exponentially — exactly the behaviour of a field carried by a particle with Compton length $\lambda$. For niobium that is a mass of about five electronvolts. In the theory of superconductivity this is not an analogy but an identity: the photon inside a superconductor acquires a mass through the pairs’ coherent response, and the mechanism is the same one, lifted into particle physics, by which the carriers of the weak force acquire their masses from the Higgs field.

What distinguishes these masses from a mass in empty space is where the missing flux goes. In a plasma the field of a charge is screened by the cloud of opposite charge that gathers round it — the long-range force that does not reach followed that cloud — and in a superconductor by the currents in its surface. Include those charges and currents in the count, and Gauss’s law holds exactly: the flux through a large sphere is small because the charge inside it is small, the original charge and its screening cloud nearly cancelling. A photon mass in empty space would give the same exponential falloff with nothing to account for it. The flux would fall short of the charge, and there would be no charge anywhere to make up the difference.

That is why a photon mass is not merely a matter of degree. Ordinary electromagnetism leaves the potentials partly arbitrary — the zero of the electric potential, and the gauge freedom the field with no ends met in the magnetic case — because only the fields they produce can be measured. The extra term $-\mu^2\phi$ makes the potential itself observable, so a photon mass destroys that freedom. The theory remains consistent — Proca’s equations conserve charge and are Lorentz-invariant — but it is a different theory, with three polarisations of light instead of two, and the bounds in the figure are bounds on how far from ordinary electromagnetism the world could be.

The third way of shaking, and why it would hardly show #

A massless photon can shake in two directions, both across its direction of travel, which is why light has two polarisations and why the curve that would not come down counted two modes for every wavevector in Planck’s cavity. A massive vector particle has three: the two transverse ones and a longitudinal one, shaking along the direction of travel. Counting modes naively, a photon of any mass at all, however small, would give black-body radiation half as much energy again as Planck’s law — a discontinuous jump in one of the best-measured spectra in physics, produced by a mass too small to measure any other way.

That cannot be right, and the resolution is instructive. The longitudinal mode couples to charges with a strength proportional to $(mc^2/\hbar\omega)^2$: as the mass goes to zero it does not merely become lighter, it decouples, and ordinary matter can neither emit nor absorb it except at a vanishing rate. Alfred Bass and Erwin Schrödinger worked out in 1955 how long the walls of a cavity would take to bring the longitudinal mode into equilibrium with the other two, for a photon mass anywhere near the bounds of the time, and found it far longer than any experiment and, for small enough masses, longer than the age of the universe. A cavity with a massive photon in it would show Planck’s two-mode spectrum for any practical length of time, with the third mode empty.

This is a case where a limit that looks discontinuous on paper is continuous in practice, because the extra degree of freedom that appears the moment the mass is not exactly zero also becomes inaccessible in proportion to how small the mass is. The same structure appears whenever a symmetry is broken by a small amount: the states the symmetry forbade become allowed, and they become allowed only weakly. It is also why the black-body spectrum, though measured to parts in a hundred thousand in the cosmic microwave background, gives no useful bound on the photon’s mass at all.

Where the calculation stops #

The figures take the simplest form of each failure: an exponent that is constant at every distance, and a mass that enters only through Proca’s equations. Other departures are possible — a force law that changes form between scales, a coupling of the photon to some other field that mimics a mass only in certain conditions — and each needs its own analysis. The magnetic-field bounds depend on models of the fields involved: Jupiter’s and the solar wind’s are well measured and well understood, which is why those bounds are trusted; the Galaxy’s is neither, which is why its bound is not. And the shell calculations assume perfect spheres and perfect conductors; the real experiments were differential measurements designed so that imperfections in either cancel to first order, and their quoted limits include the residual uncertainty from that cancellation.

What the pictures cannot show #

The figures show a static field and a static shell. The actual laboratory test charged its outer shell with an alternating voltage and looked for an alternating response on the inner one, because an alternating signal can be picked out of noise by a lock-in detector and a static one cannot; the analysis for a massive photon at finite frequency adds terms the static picture omits. The figures cannot show the third polarisation a massive photon would have — a longitudinal wave that couples very weakly to ordinary sources and would add a small extra contribution to the thermal radiation of a black body. And the bounds figure places each test at a single size, when each actually measures a field over a range of distances and the bound comes from how its shape changes across that range.

Still open: whether the photon’s mass is exactly zero #

No experiment can show that a quantity is exactly zero; it can only push the bound lower. The standard model of particle physics sets the photon’s mass to zero because the symmetry that gives electromagnetism its gauge freedom forbids a mass term, and a massless photon is the simplest theory consistent with everything measured. But the same symmetry, in the weak interaction, is broken by the Higgs field, and there is no theorem that forbids some much weaker mechanism from giving the photon a mass far below every bound. The bounds reach a Compton length of more than an astronomical unit from the solar wind, and many light years if the galactic arguments are accepted. Whether they will ever meet a nonzero value, and whether the question of a mass below all conceivable tests has physical meaning at all, are open.

The habit worth carrying away is to ask whether a proposed failure of a law has a length in it. A departure with no length — a wrong exponent — shows equally at every scale and is caught by the most sensitive detector; a departure with a length — a mass — shows as the square of the apparatus measured against that length, and is caught by the largest field that can be found. The inside of a charged shell tests the exponent to one part in 10¹⁶; the magnetic field the solar wind carries past Pluto tests the mass a ten-thousand-fold better.