The pull that has to equal the pull back

The floor that cannot be told from gravity stated the equivalence principle and used it. The fall that does not depend on what is falling followed three centuries of tests showing that gravity accelerates every material equally, now to a part in $10^{15}$. The binding energy that has to fall too extended the test to a body’s own gravitational energy, using the Moon’s orbit, and the radiation a difference in falling would make listened for the dipole a violation would add to a binary pulsar.

All of those tests share a direction. Each takes two bodies, puts them in a field made by something else — the Earth, the Sun, the galaxy — and asks whether they respond equally. The mass being compared is the one that gravity pulls on. But mass appears in the law of gravitation twice, and the second appearance is the one that makes the field in the first place. Whether that mass is the same for every material is a separate question, and it turns out to be the question of whether Newton’s third law holds for gravity. It has been tested far less often and far less precisely, and the best test uses the shape of the Moon.

Three masses in one law #

Newton’s law of gravitation, written with care, contains three different masses. A body has an inertial mass, which resists being accelerated and appears in $F = ma$. It has a passive gravitational mass, which measures how hard a given field pulls on it. And it has an active gravitational mass, which measures how strong a field it makes. The force of body 1 on body 2 is

$$F_{12} = \frac{G\, m_{A1}\, m_{P2}}{r^2},$$

the active mass of the source times the passive mass of the body being pulled.

The torsion balances and the satellite MICROSCOPE compare passive with inertial: two test masses in the Earth’s field, and whether they fall at the same rate. The equality they test is what makes free fall universal, and it is known to a part in $10^{15}$. The active mass never enters, because the field is the Earth’s and the Earth is the same for both test masses.

Now take two bodies of different materials and let each be the source for the other. Body 2 pulls body 1 with $G m_{A2} m_{P1}/r^2$, and body 1 pulls body 2 with $G m_{A1} m_{P2}/r^2$. These are equal and opposite only if $m_{A1}/m_{P1} = m_{A2}/m_{P2}$. Write each active mass as $m_A = m_P(1 + \varepsilon)$, with $\varepsilon$ a property of the material. The two pulls then differ by

$$\Delta F = \frac{G\, m_{P1}\, m_{P2}\,(\varepsilon_2 - \varepsilon_1)}{r^2},$$

and that difference is a net force on the pair from nothing outside it. The figure draws the arrows for a difference of 30 per cent, so that it can be seen: the unusual body pulls harder than it is pulled, and the pair moves off towards the ordinary body with no external cause.

A third law that could have been false #

That is a startling consequence, and it is worth being precise about what it violates. It violates Newton’s third law, which says that the forces two bodies exert on each other are equal and opposite. It therefore violates the conservation of momentum, since an isolated system would change its momentum. Collisions are easier than forces built the whole of collision mechanics on that conservation, and the point that keeps moving derived from it that a system’s centre of mass moves uniformly whatever its parts do. A universe with different ratios of active to passive mass would have a centre of mass that could accelerate itself, and a spacecraft built of two suitable materials would need no propellant — the thing the push that needs nothing to push against rules out for a rocket.

Newton did not assume the third law for gravity; he argued for it. In the Principia he considered the Earth divided into two parts by a plane and noted that if the parts did not attract each other equally, the whole Earth would accelerate for ever towards one side — “which is absurd”. The absurdity is an appeal to experience rather than to logic. Nothing in the inverse-square law requires the active and passive masses of every material to be proportional; the claim that they are is an empirical statement about gravity, of exactly the same kind as the claim that every material falls at the same rate. In general relativity it is built in, because the field equations conserve energy and momentum locally. That is an argument for general relativity if the equality holds, not a reason for it to hold.

Why a laboratory cannot simply watch #

The direct test would be to build a dumbbell of two materials, isolate it from every other force and watch whether it moves.

The figure computes how far such a dumbbell would travel in a year. The force between two half-kilogram spheres ten centimetres apart is about $1.7\times10^{-9}$ newtons, the weight of a speck of dust. If the materials’ ratios differed by a part in a thousand, the dumbbell’s self-acceleration would be $1.7\times10^{-12}$ metres per second squared, and in a year it would carry itself 831 metres — a surprising distance for so small a push, because a year is $3\times10^{7}$ seconds and the distance grows as its square. At a difference of $10^{-12}$ the same dumbbell moves less than a micrometre.

The catch is the year. No object on the Earth can be kept free of every other influence for a second, let alone a year. Its own weight is $10^{10}$ times the self-force, and the tiniest stray gradient, radiation pressure or electrostatic patch would overwhelm the effect before it had moved anything. The experiment is conceptually trivial and practically impossible, which is why the question was asked in a completely different way — by measuring how hard a piece of one material pulls, with the other material arranged so that it could not contribute.

A cylinder that weighs nothing and pulls anyway #

In 1968 Lloyd Kreuzer, a graduate student at Princeton, found the arrangement.

He immersed a cylinder of Teflon in a liquid — a mixture of trichloroethylene and dibromomethane — whose proportions he adjusted until the cylinder floated in neutral balance, neither rising nor sinking. At that point the cylinder and the liquid it displaced have exactly the same passive mass. He then moved the cylinder back and forth inside the liquid near the test mass of a torsion balance and watched for a change in the balance’s pull. If active mass is proportional to passive mass for both materials, moving the cylinder changes nothing: it simply trades places with an equal passive mass of liquid, which by assumption pulls identically. If the two materials’ ratios differ, the cylinder pulls slightly harder or slightly more weakly than the liquid it displaces, and the balance sees a pull that follows the cylinder about.

The figure draws what the balance would see. As the cylinder passes, the change in pull traces the shape of its own inverse-square field at the test mass, scaled by the difference $\Delta\varepsilon$. Kreuzer’s apparatus could resolve a change of five parts in a hundred thousand of the pull the cylinder would exert in empty space, and he saw nothing. Teflon is three-quarters fluorine by mass, and the liquid was dominated by bromine and chlorine, so the null result says that the active mass per unit passive mass of those elements agrees to $5\times10^{-5}$.

The design has the same elegance as a block the water does not lift run backwards: buoyancy is used to cancel exactly the quantity that is not being tested. The passive masses are matched by flotation, so the only thing left free to show up is the active mass. It is also why no better laboratory test followed. A laboratory is a place where objects are held, and every holding force is ten orders of magnitude larger than the effect. To go further, a body was needed that had been left alone for a very long time and was made of two materials arranged unevenly.

The Moon’s heavier near side #

The Moon is not uniform. Its crust, made largely of light aluminium-rich feldspar, is thicker on the side facing away from the Earth than on the near side — around sixty kilometres against thirty, as measured by the gravity-mapping spacecraft GRAIL. The denser interior, rich in iron-bearing minerals, therefore bulges slightly towards the Earth, and the Moon’s centre of mass sits about 1.9 kilometres closer to the Earth than the centre of the sphere that its surface defines.

In 1986 David Bartlett and Dave van Buren saw what that offset meant. The crust and the interior are two bodies of different composition, arranged off-centre with respect to each other. If aluminium’s ratio of active to passive mass differed from iron’s, the crust’s pull on the interior and the interior’s pull on the crust would not cancel, and the Moon would feel a net self-force along the line joining the two centres. In the model drawn here, the self-acceleration comes out at about $\Delta\varepsilon \times 1.8\times10^{-3}$ metres per second squared.

What makes the Moon far better than any dumbbell is not only its size but its tidal lock. The Moon keeps the same face towards the Earth, so the offset, and any force along it, always points the same way relative to the Earth–Moon line. A force fixed in that direction does not average to zero over an orbit, as a force fixed in space would; it acts like a steady change in the Earth’s attraction and alters the Moon’s orbit in a way that laser ranging, already measuring the lunar distance to millimetres, can look for. Bartlett and van Buren found no such effect and concluded that the active-to-passive ratios of aluminium and iron agree to four parts in $10^{12}$ — an improvement of seven orders of magnitude on Kreuzer. A reanalysis with modern ranging data in 2023 lowered the bound to about four parts in $10^{14}$.

Why a steady push turns an orbit #

It is worth seeing why a push that never changes direction relative to the Earth is something ranging can find, rather than something absorbed into the numbers the orbit is fitted with. A steady outward push along the Earth–Moon line looks, at first sight, exactly like the Earth being slightly lighter: the Moon would settle a little further out, or go round a little more slowly, and nobody knows the Earth’s mass to better than the orbit itself says. If that were the whole of it, the effect could hide in the value of $GM$ for ever.

It is not the whole of it, because the push does not fall off with distance as the Earth’s pull does. The Moon’s orbit is an ellipse with an eccentricity of 0.055, and its distance from the Earth changes by about eleven per cent over a month. An extra force that is the same at perigee and apogee is, in effect, a different force law added to the inverse square, and the arrow that says which way the orbit points showed what that does: only the pure inverse square keeps the ellipse from turning, because only it conserves the vector pointing at the perigee. Add a force with any other dependence on distance and the ellipse rotates, as the orbit that does not come back to itself drew for a general central force. A self-push would therefore show up as an extra, steady rotation of the Moon’s perigee, on top of the large rotation the Sun already causes, and of a size fixed by the push’s strength and the orbit’s eccentricity.

The Sun’s own perturbation turns the perigee once every 8.85 years, and that is computed to far better than the precision needed; general relativity adds about two hundredths of a second of arc a year, which is also computed. What ranging looks for is anything left over after both. The bound on $\Delta\varepsilon$ is, in the end, a bound on an unexplained rotation of an ellipse — the same kind of measurement as the forty-three seconds of arc a century that Mercury had left over, made on a body that has been kept at the same orientation to its planet since shortly after it formed.

Two equivalences, not one #

The two equalities are often folded into one statement — gravitational mass equals inertial mass — and the folding hides how differently they have been tested.

The figure sets the bounds side by side. The passive equivalence was tested to $5\times10^{-9}$ by Eötvös before the First World War and to $10^{-15}$ by a satellite a century later. The active equivalence was not tested at all until 1968, stood at $5\times10^{-5}$ for two decades and was then improved in a single step, by a factor of ten million, when somebody noticed that a planetary body with an asymmetric interior was a natural experiment. It remains about a hundred times weaker than the passive one.

The two equalities also say different things about theories. A theory in which inertial and passive masses differ would let a body’s trajectory in a given field depend on its composition, which is what most proposed violations, from new long-range forces coupling to baryon number or to lepton number, would do. A theory in which active and passive masses differ would violate momentum conservation, and almost no modern theory is willing to do that, because momentum conservation follows from the homogeneity of space — the conservation law a symmetry hands over. The active test is therefore less a search for a predicted effect than a check on a foundation, and that is why a bound improving by seven orders of magnitude in one step was news.

What a field’s source has to include #

The question of what counts as active mass does not end with the elements. In general relativity the source of gravity is not mass but energy and momentum, including the energy of fields and of binding. The binding energy that has to fall too asked whether gravitational binding energy is pulled as other energy is; the matching question for the source is whether it pulls. For the Moon and the Earth, whose binding energies are a few parts in $10^{11}$ and $10^{10}$ of their masses, lunar ranging answers both at once, because the orbit is set by both bodies acting as sources and as test masses. The electrostatic energy inside nuclei is a larger fraction — a few thousandths of the mass in heavy nuclei — and it varies from element to element, so Kreuzer’s and Bartlett’s null results also say that electrostatic energy pulls in proportion to how it is pulled, at the level of their bounds.

There is a subtler point, which the momentum of something that is not moving touched from another side. In relativity, a body under internal stress has an inertia that includes the stresses, and the source of its gravity includes them too. For a body in equilibrium the stresses integrate to zero and drop out of both, so a static body’s active mass is simply its energy. That is a theorem, not an assumption, and it is part of why general relativity guarantees the equality this essay has been testing: the field equations make the source conserved, and a conserved source cannot accelerate itself.

Where the picture stops #

The figures reduce the Moon to two components, a crust and an interior, with one number describing each composition. The real Moon has a small iron core, a mantle whose composition varies with depth, and a crust whose thickness varies from place to place. The bound from lunar ranging depends on a model of how much of the Moon’s asymmetry is carried by which minerals, and the figure’s self-acceleration of $1.8\times10^{-3}$ metres per second squared per unit $\Delta\varepsilon$ is an order-of-magnitude estimate from the offset alone, not the calculation the published bounds rest on. Nor does the picture show how the orbit responds: a steady force along the Earth–Moon line is partly indistinguishable from a small change in the Earth’s mass, and separating the two needs the orbit’s eccentricity and the Sun’s perturbations, which make the analysis long.

It also says nothing about the direction of any violation. The sign of $\Delta\varepsilon$ decides whether the Moon would push itself towards the Earth or away, and the arrow in the figure could point either way. What is bounded is the size.

Still open: whether anything pulls differently from how it is pulled #

The active equivalence is tested to a few parts in $10^{14}$, for aluminium against iron. Other pairs of materials, and other forms of energy, are tested less well or not at all. Proposals to improve the bound include ranging to asteroids with known asymmetric compositions, and analysing the self-forces in binary systems where one member is compact and the other is not; neither has yet produced a limit. Whether a violation exists at any level is not known, and almost no theory predicts one. But the passive equivalence was also tested for centuries without a theoretical prediction to guide it, and its current precision was reached by people who thought the answer was worth knowing whatever it was.

The habit worth carrying away is to count the roles a quantity plays before assuming it plays them equally. A mass that is pulled and a mass that pulls are two different properties of a body, and their equality is Newton’s third law — an experimental claim about gravity, tested to a part in 10¹⁴ by a Moon that has kept the same lopsided face towards the Earth for four thousand million years. A test that drops two things in a field someone else made can never see the second property, however precise it becomes.