Electromagnetism

The imbalance of charge that would outweigh gravity

Gravity and electricity both fall as the inverse square, so Gauss's law treats them identically: the field of any round body counts what is inside it. If every nucleon in ordinary matter carried a tiny unbalanced charge, the body's charge would follow its mass everywhere, and its electric repulsion would be a fixed fraction of its gravity at every size from a grain of sand to the universe. Below about one part in a billion billion of an electron's charge, gravity always wins; above it, repulsion always does. In 1959 two astronomers proposed that the universe expands because the true imbalance is just above that line.

18 min read 5 figures The same equation again What stays the same

Assumes: Counting what comes out, and never looking inside · The pull that grows on the way down

Counting what comes out, and never looking inside introduced Gauss’s law as a statement about electric charge: the field crossing any closed surface counts the charge inside it, and cares nothing about where inside or how it is arranged. The pull that grows on the way down used the same law for gravity, because Newton’s gravity is also an inverse-square field, and found the Earth’s pull increasing as one descends through the mantle towards the dense core. The two laws are one law with two different charges. One is electric charge, which comes in both signs and cancels; the other is mass, which comes in one sign and accumulates.

Put them together and an old question becomes sharp. Ordinary matter is neutral because each atom has as many electrons as protons and the electron’s charge is exactly the negative of the proton’s. Suppose it were not quite exact. Suppose that every nucleon in matter, together with its share of electrons, carried some small leftover charge, a fraction $\delta$ of the electron’s. Then every body would carry a charge proportional to its mass, and Gauss’s law would apply to the charge and the mass at the same time, with the same geometry. The consequence is not that small bodies are nearly neutral and large ones heavily charged. It is that electric repulsion and gravitational attraction would stand in a fixed ratio at every scale.

A charge that follows the mass falls off exactly as gravity does. Inside and outside a planet of the Earth's size with a dense core (11,000 kg/m³ to 55 per cent of its radius) and a lighter mantle (4,500), against distance from its centre in planetary radii: the pull of gravity on each kilogram (solid), and the electric push each kilogram would feel if every nucleon carried an unbalanced charge of 4.47 × 10⁻¹⁹ e, half the critical value (dashed), both from Gauss's law with the enclosed mass computed layer by layer. Gravity reaches 9.94 m/s² at the surface; the push is a quarter of it there, and a quarter of it at every radius — through the core, across the step at its edge and out into space — because the enclosed charge is a fixed multiple of the enclosed mass everywhere. At the critical imbalance, 8.93 × 10⁻¹⁹ e per nucleon, the two would be equal everywhere and the planet would weigh nothing. — Fig. 1 Inside and outside a planet of the Earth’s size with a dense core (11,000 kg/m³ to 55 per cent of its radius) and a lighter mantle (4,500 kg/m³): gravity on each kilogram (solid) and the electric push each kilogram would feel if every nucleon carried $4.47\times10^{-19}$ of an electron’s charge, half the critical value (dashed), both from Gauss’s law with the enclosed mass computed layer by layer. Gravity reaches 9.94 m/s² at the surface; the push is a quarter of it there, and at every radius, through the core, across its edge and out into space.

The charge follows the mass, and so does the field

The first figure works it through for a layered planet. Gravity at a radius $r$ inside it is set, by Gauss’s law, by the mass enclosed within $r$ , divided by $r^2$ ; it rises through the dense core, dips slightly in the mantle, as the pull that grows on the way down found for the real Earth, and falls as the inverse square outside. If each nucleon carries a charge $\delta e$ , the charge enclosed within $r$ is $\delta e/m_u$ times the mass enclosed, where $m_u$ is the mass per nucleon. The electric field is therefore the same shape exactly, scaled, and the outward push it gives each kilogram of the planet’s own matter is a fixed multiple of the inward pull of gravity.

The multiple is

\frac{\text{repulsion}}{\text{gravity}} = \frac{(\delta e/m_u)^2}{4\pi\varepsilon_0 G} = \left(\frac{\delta}{\delta_c}\right)^2, \qquad \delta_c = \sqrt{4\pi\varepsilon_0 G}\;\frac{m_u}{e} = 8.9\times10^{-19}.

It contains no radius, no density and no structure. It is the ratio of the electric force between two nucleons, each carrying $\delta e$ , to the gravitational force between them, and Gauss’s law guarantees that the ratio survives being added up over any distribution of matter whatever, because charge and mass are added up the same way. The figure’s planet has a core and a mantle; the ratio is the same through both and outside.

Two inverse squares, and why only one adds up

The comparison is worth making explicit, because the two forces are so alike in their geometry and so unlike in their effects. Both fall as the inverse square, both obey Gauss’s law, and the shape decides the falloff found that both give the same field patterns for the same arrangements of their sources — a sphere acting as a point, a sheet giving a field that does not fall with distance. The difference is entirely in the sources. Electric charge comes in two signs, and ordinary matter carries both in equal amounts, so its field outside cancels and the enormous strength of the electric force never shows on large scales. Mass comes in one sign, so it only accumulates, and a force $10^{36}$ times weaker per particle ends up governing everything larger than a mountain.

Gauss’s law is what makes the cancellation complete. The field outside that cannot find the core found that the field outside a sphere depends only on the total enclosed, so a body whose charges sum to zero has no electric field outside it at all, however its charges are arranged within. Any leftover in that sum adds up, like mass, over every atom in the body. A leftover per nucleon is therefore just another source that accumulates with the amount of matter, and it is no surprise that it competes with gravity in a fixed proportion. The surprise is only in how small the leftover would need to be.

An imbalance below the line never matters, and one above it always does

How much a tiny imbalance of charge would take off a body's weight. The electric repulsion between the parts of any body, as a fraction of their mutual gravity, if every nucleon carried an unbalanced charge δ (in units of the electron's), on logarithmic axes. It is (δ/δc)², with δc = 8.93 × 10⁻¹⁹, for a grain of sand, a planet or a galaxy alike. At the imbalance Lyttleton and Bondi proposed, 2 × 10⁻¹⁸, repulsion would beat gravity 5.0 times over, everywhere — their mechanism for an expanding universe. Experiments with gas let out of tanks had bounded the imbalance near 10⁻²⁰ within a year, where the ratio is 1.3·10⁻⁴, and modern ones near 10⁻²¹, 1.3·10⁻⁶. — Fig. 2 The electric repulsion between the parts of any body as a fraction of their mutual gravity, $(\delta/\delta_c)^2$ with $\delta_c = 8.93\times10^{-19}$ , against the unbalanced charge per nucleon, on logarithmic axes. Lyttleton and Bondi’s $2\times10^{-18}$ would make repulsion five times gravity everywhere; the bound near $10^{-20}$ set within a year leaves a ratio of $1.3\times10^{-4}$ , and today’s near $10^{-21}$ , $1.3\times10^{-6}$ .

The second figure draws the ratio against the imbalance. Because it does not depend on size, there are only two regimes. With $\delta$ below $\delta_c$ , gravity wins at every scale: planets hold together, stars hold together, galaxies and clusters of galaxies hold together, each only slightly less tightly than with no charge at all. With $\delta$ above $\delta_c$ , repulsion wins at every scale, and nothing held together by gravity — no planet, no star, no galaxy — could exist; only bodies held by other forces, rocks and people, could.

The threshold is a remarkable number. It says that the electric force is about $10^{36}$ times stronger than gravity between two protons, so a leftover charge of one part in $10^{18}$ is enough to undo gravity entirely. It also says that the neutrality of matter is not a curiosity but a precondition for astronomy. Every gravitationally bound object in the sky is evidence that the leftover charge, if there is one, is below the line.

An imbalance gravity could never see

There is a strange corollary to the fixed ratio. Below the line, a uniform imbalance would weaken gravity between all bodies by the same factor, $1 - (\delta/\delta_c)^2$ , at every distance and for every material. That is indistinguishable from a slightly smaller gravitational constant. The value of $G$ measured with torsion balances in a laboratory is the attraction between lumps of metal, which would carry the same imbalance as everything else, so the measured $G$ would already include the reduction; the orbits of planets, the structure of stars — which the ball of gas that heats up as it cools found to be fixed by gravity against pressure — and the dynamics of galaxies would all be computed with the reduced value and would all agree with each other. No gravitational observation could detect a universal imbalance of charge, however large, as long as it stayed below the line.

That is why the question had to be settled by measuring atoms one kind at a time rather than by measuring the heavens. And it is why the fixed ratio matters for a different reason: it depends on both forces being exactly inverse-square. The mass that would make Gauss’s law leak found that if the photon had a mass, the electric field of a charge would die away exponentially beyond a length set by that mass, so the repulsion of a charge imbalance would fall short of gravity on scales larger than that length, and the ratio would stop being independent of size. With the photon’s mass bounded as tightly as it is, the electric force reaches at least as far as the Solar System, and on all scales where the argument has been applied, the two laws still add up the same way.

A universe pushed apart by its own charge

In 1959 Raymond Lyttleton, at Cambridge, and Hermann Bondi, at King’s College London, proposed that the leftover charge is not zero but just above the line. The universe was known to be expanding; in the steady-state theory that Bondi had helped found, matter was continuously created to keep its density constant, and something had to drive the expansion. They suggested that the proton’s charge exceeds the electron’s in magnitude by about two parts in $10^{18}$ , so that a hydrogen atom carries a tiny net positive charge. On the scale of the whole universe, which cannot exchange charge with anything outside itself, that charge would push everything apart, and with the figure’s ratio the repulsion would exceed gravity about fivefold, everywhere. The newly created matter, carrying the same imbalance, would supply a charge density that grew with the volume, and the expansion would be driven exactly as observed.

It was a testable idea, and it was tested within months. Its prediction is that every atom and molecule carries a definite, measurable charge, proportional to its number of nucleons. The one quantity a boost leaves alone described the experiment that does it: a sealed metal vessel full of gas is let empty through a valve, and the charge that the escaping molecules carry away appears on the vessel and can be read with an electrometer. With $10^{22}$ molecules leaving, a per-molecule charge of the size Lyttleton and Bondi needed would be millions of electrons’ worth. A. M. Hillas and T. E. Cranshaw, at Harwell, and John King, at MIT, did the experiment with hydrogen, helium and other gases in 1959 and 1960, and found nothing, to a precision of about $10^{-20}$ of an electron’s charge per nucleon — two hundred times smaller than proposed, so that the repulsion it could provide was smaller than needed by a factor of tens of thousands. The proposal did not survive. Later experiments, with molecular beams deflected by electric fields and acoustic resonators driven by oscillating ones, have pushed the bound to about $10^{-21}$ .

Why the Earth’s charge says nothing about it

The voltage an imbalance below every bound would still produce. The voltage a rocky sphere (3,000 kg/m³) would reach if every nucleon in it carried 10⁻²¹ of an electron's charge — below every experimental bound — against its radius, on logarithmic axes (solid), with the field at its surface (dashed, in V/m). By Gauss's law the charge grows as the volume and the voltage as the square of the radius: 10⁻⁵ V for a grain of sand, 11 V for a boulder, 4·10¹⁴ V for a body the size of the Earth. Beyond a radius of about 275 km the surface field would pass the three megavolts per metre that breaks down air, and at the Earth's size it would be 7·10⁷ V/m; any real body in contact with a gas or a plasma would draw opposite charge from it and neutralise itself long before its charge mattered. Only the universe as a whole has no surroundings to borrow charge from, which is why the cosmological version of the idea was the one worth proposing. — Fig. 3 The voltage of a rocky sphere (3,000 kg/m³) if every nucleon carried $10^{-21}$ of an electron’s charge, below every bound, against its radius on logarithmic axes (solid), with the field at its surface (dashed, V/m). The voltage grows as the square of the radius: $10^{-5}$ V for a grain of sand, 11 V for a boulder, $4\times10^{14}$ V for a body the size of the Earth. Beyond about 275 km the surface field passes the 3 MV/m that breaks down air, and at the Earth’s size it is $7\times10^7$ V/m.

It might seem simpler to look for the imbalance in the charge of a large body. The third figure shows why that does not work. Because charge grows as the volume while the radius grows only as its cube root, the voltage of a uniformly charged sphere grows as the square of its radius. An imbalance of $10^{-21}$ , below every bound, would put a boulder at eleven volts and an Earth-sized body at four hundred trillion volts, with a field at its surface twenty times the strength that makes air break down. No real planet could hold that. It would draw electrons or ions from the gas and plasma around it, as any charged conductor does, until its net charge was neutralised, and the inside of a conductor found how quickly that happens in matter that can carry a current.

The Earth does carry a net charge — about half a million coulombs of negative charge on its surface, maintained by thunderstorms and leaking away through the fair-weather atmosphere, the global circuit that the field that keeps Greenwich time found rising and falling with the clock of the world’s afternoon thunderstorms. But that charge is set by the atmosphere’s electricity, not by the neutrality of its atoms, and it would be the same whatever $\delta$ was. Only the universe as a whole has no surroundings to draw compensating charge from, which is why the cosmological version of the idea was the one that could not be dismissed without measuring atoms.

The charge a star is obliged to carry

$The charge gravity makes a star keep. The net positive charge of a star made of ionised hydrogen in hydrostatic equilibrium, in coulombs. Gravity pulls a proton 1,836 times harder than an electron, so the electrons would float above the protons until the field of the separated charge holds them together: (mₚ − mₑ)g/2e, a fraction of a microvolt per metre in the Sun. Gauss's law turns that field into a charge, 4πε₀R² times it, and the radius cancels: the star carries 2πε₀GM(mₚ − mₑ)/e, 77 C for each solar mass, whatever its size — the Sun 77 C, a white dwarf 46 C, a neutron star 108 C, a red giant 92 C. Per nucleon that is 4·10⁻³⁷ of an electron's charge — exactly half the square of the imbalance at which repulsion would cancel gravity, and eighteen orders of magnitude below it.$

Fig. 4 The net positive charge of a star of ionised hydrogen in hydrostatic equilibrium. Gravity pulls a proton 1,836 times harder than an electron, so the star holds the field

(m_p-m_e)g/2e

that keeps them together, a fraction of a microvolt per metre in the Sun, and Gauss’s law turns that into a charge

2\pi\varepsilon_0 GM(m_p-m_e)/e

in which the radius cancels: 77 C per solar mass — the Sun 77 C, a white dwarf 46 C, a neutron star 108 C, a red giant 92 C.

There is one place where matter is measurably charged, and Gauss’s law says how much. A star is a ball of ionised hydrogen, electrons and protons mixed. Gravity pulls each proton 1,836 times harder than each electron, so if the two were free to settle separately, the electrons, being so much lighter, would float up like a light gas over a heavy one, and the star would separate into a positive core and a negative skin. They cannot separate far, because the first slight separation creates an electric field that pulls them back together. The equilibrium is a field just strong enough to give each species the same effective weight: upward on the protons and downward on the electrons, of size $(m_p - m_e)g/2e$ . Anton Pannekoek and Svein Rosseland worked it out in the 1920s. In the Sun’s atmosphere it is a fraction of a microvolt per metre.

A field at a star’s surface implies a charge inside, by Gauss’s law, and the fourth figure computes it. The charge is $4\pi\varepsilon_0 R^2$ times the field, and since the surface gravity is $GM/R^2$ , the radius cancels: every star carries $2\pi\varepsilon_0 GM(m_p - m_e)/e$ , 77 coulombs per solar mass, whether it is a red giant half the size of the Earth’s orbit or a neutron star the size of a city. Per nucleon that is $4\times10^{-37}$ of an electron’s charge.

That number is not independent of the threshold. It comes out as exactly half the square of $\delta_c$ . The star’s charge is fixed by asking for the electric force on a proton to cancel half of its weight, and $\delta_c$ is the charge per nucleon at which the electric force between two nucleons cancels their gravity, so the two are tied together by the same ratio of constants. A star is charged to the square of the level that would undo gravity — which, since $\delta_c$ is a part in $10^{18}$ , is a part in $10^{36}$ .

Five imbalances on one scale

Fig. 5 Unbalanced charge per nucleon, in units of the electron’s, on a logarithmic axis: Lyttleton and Bondi’s

2\times10^{-18}

; the

8.9\times10^{-19}

at which repulsion cancels gravity; the bounds near

10^{-20}

in 1960 and

10^{-21}

now; and the

4\times10^{-37}

that equilibrium puts on a star.

The last figure sets the numbers side by side. The cosmological proposal sat just above the line at which repulsion overtakes gravity. The experiments put the real imbalance two and then three orders of magnitude below that line, so the repulsion it could produce is at most a millionth of gravity. And the one charge matter is known to carry for gravitational reasons, a star’s, sits eighteen orders of magnitude below the experimental bounds. If the electron’s and proton’s charges differ at all, the difference is smaller than any measurement has reached; most physicists expect it to be exactly zero, because in the standard model the cancellation of certain quantum inconsistencies, the gauge anomalies, ties the charges of the quarks and leptons in each generation together, and with the neutrino neutral it forces the proton’s charge to equal the electron’s.

Where the argument needs its assumptions

The fixed ratio of the first two figures needs charge to follow mass, which means the same imbalance per nucleon in every material. If instead the imbalance lay between the proton and the electron, with the neutron exactly neutral, the charge per nucleon would vary with composition — a hydrogen atom would carry a full share per nucleon, an iron nucleus less than half — and the repulsion between bodies of different composition would differ. That kind of imbalance would show up in tests of the equivalence principle, which compare how different materials fall and are now sensitive to differences of a part in $10^{15}$ , as a force that depends on what a body is made of. The neutrality experiments distinguish the cases by using gases of different composition.

The argument also treats the bodies as isolated. Any real body is embedded in surroundings that can neutralise it, and once that is allowed, a macroscopic charge measures the surroundings rather than the atoms. And the field of the third figure is for a uniform sphere, held to its charge with nothing to discharge it, which no real material would be at those fields.

Bounds drawn as points

The figures compute the fields of idealised spheres with an assumed imbalance, and the bounds they mark are drawn as points on a line, which is easy to read as measurements. They are not: no experiment has measured a nonzero imbalance, and every value drawn below the threshold is the largest one an experiment failed to see. The star’s charge is computed for pure ionised hydrogen in static equilibrium; a real star has helium and heavier elements, a magnetic field, a wind and an atmosphere that is not static, and the Pannekoek–Rosseland field is a small correction inside the larger electric structure those produce. None of that changes its order of magnitude or the cancellation of the radius.

Still open: whether the charges are exactly equal

The equality of the proton’s and electron’s charges is one of the most precisely tested facts in physics, but it is a measured fact, and the reason for it is not settled by experiment. In the standard model it follows from anomaly cancellation only within the model’s particle content; in theories that unify the strong, weak and electromagnetic forces it follows from the structure of the unified group, which also predicts that protons decay. Neither the decay nor any departure from neutrality has been seen. Experiments with levitated microspheres and with atom interferometers are being developed to push the test of neutrality further, and a nonzero result anywhere would say something about physics beyond the standard model, with or without any consequence for the universe’s expansion.

The habit worth carrying away is to ask what a law adds up. Because Gauss’s law sums charge and mass in the same way, an imbalance of δ per nucleon makes every body’s electric repulsion exactly $(\delta/\delta_c)^2$ of its gravity, with $\delta_c = 8.9\times10^{-19}$ , at every size and every radius; Lyttleton and Bondi’s $2\times10^{-18}$ would have pushed the universe apart, gas let out of a tank showed the imbalance to be below $10^{-20}$ within a year, and the only charge matter is known to carry for gravitational reasons, a star’s 77 coulombs per solar mass, sits at half of $\delta_c^2$ .