A force that can hide at any one distance

A planet’s orbit closes to within a part in many millions, and the inverse-square law holds so well between the Sun and the planets that a closed ellipse is often taken as proof of it. The natural conclusion is that the exponent is extremely well known. That is right for the distances between planets. It says rather little about any other distance, and the distances on which gravity has actually been measured are a small fraction of those on which it might be doing something else.

The usual way to state the question is to add a second term to the potential between two masses. If a new force were carried by a particle with a small mass, its range would be finite, as is the nuclear force carried by a pion:

$$V(r) = -\frac{GMm}{r}\left(1+\alpha\, e^{-r/\lambda}\right).$$

Here $\alpha$ says how strong it is compared with gravity and $\lambda$ how far it reaches. It is a parameterisation and is not a theory. But most theories of modified gravity, extra dimensions and light scalar fields reduce to it, at least locally. Two numbers define a possible new force, and the map of what has been excluded is a plot in those two numbers. The shape of that map says what the experiment is actually doing.

Two constants in one law #

The force from that potential is

$$F(r) = \frac{GMm}{r^2}\left[1+\alpha\left(1+\frac{r}{\lambda}\right)e^{-r/\lambda}\right].$$

At a separation much larger than $\lambda$ the bracket is 1, and at a separation much less than $\lambda$ it is $1+\alpha$. So what the term does is make the gravitational constant take two different values at two different distances.

This one picture implies most of what follows. An experiment at one separation measures $G$ at that separation, and the value of $G$ has no theory behind it to compare with. It takes a measurement of the ratio of the force at two distances to detect the term, and the two distances must lie on either side of $\lambda$. If both are inside it, or both outside, the ratio is what the inverse-square law says.

The history of measuring $G$ has a bearing here that is rarely mentioned. Laboratory measurements of the constant disagree among themselves by far more than their stated errors, at the level of hundreds of parts per million. Could a short-range force account for this? The apparatuses vary in size only over a factor of a few, and they measure at about the same distance. The scatter does not correlate with the distance in any obvious way, and limits from elsewhere are much tighter at those ranges than the disagreement. So the disagreement is thought to be in the instruments rather than the force. It is striking that the least accurately known constant in physics is also one whose absolute value has hardly any bearing on this kind of test.

A theorem that fails for this force #

There is a further wrinkle that shapes every experiment. The famous reason a planet can be treated as a point is the shell theorem, which holds exactly for an inverse-square force and for a force linear in the distance, and for no other power law. A Yukawa term has no such theorem. A spherical source of radius $R$ and uniform density produces, outside itself, the same form as a point source multiplied by a form factor,

$$\Phi(x) = \frac{3\left(x\cosh x-\sinh x\right)}{x^3},\qquad x=R/\lambda .$$

For a small sphere, $\Phi\to 1$. For a sphere much larger than the range, $\Phi$ becomes exponentially large, and that large factor is cancelled at the surface by the exponential of the distance. What remains at the surface is a fall-off:

A physical account of this is simple. Only the matter within one range of the test mass can reach it, and in a large sphere that is a small fraction of the whole. For this reason, the Earth is a poor source for any force shorter than about its own size: all that its enormous mass supplies is a thin layer of rock near the surface. A lake works as well as a planet. For that reason experiments in the 1980s took place in mines, down boreholes in ice, and on tall towers, comparing the measured change of gravity with the local rock density. A local geological density variation became a major source of error, and in several of these there seemed to be a signal for a while, until the mapping of density improved.

The same fact governs the design of short-range experiments. To look for a force of range $\lambda$, both masses and their separation must be of about that size. Force grows with mass, so high density helps, and the Newtonian background must be cancelled by design.

Every orbit is a filter #

Orbits have a similar limitation. A near-circular orbit under an added Yukawa term has an apsidal advance, per revolution, of

$$\Delta\varpi = \pi\alpha\left(\frac{a}{\lambda}\right)^{2}e^{-a/\lambda}.$$

This has the same form for every planet. Plotted against $\lambda/a$ it has a peak.

So a planet is a narrow-band detector. A force much shorter than the orbit’s radius is exponentially small at that radius. And a force much longer than the radius is simply absorbed into the fitted value of $GM$ for the Sun, which is known only as a product from the orbit itself. Only near a range comparable to the orbit is there any effect.

One way to read this is as a method for getting the most out of the Solar System. The Moon’s distance is known by laser ranging to about a centimetre, and the rate of its perigee to better than a part in a hundred thousand of its total, so it gives the strongest limit anywhere, at $\alpha$ of about $2\times10^{-11}$, at $\lambda$ of about $4\times10^{8}$ metres. Mercury’s perihelion rate, the one that gave 43 arcseconds for relativity, probes a range of about $3\times10^{10}$ metres. Mars, whose position has been measured by landers and orbiters to a few metres, probes a range of about $10^{11}$ metres. These are different instruments, and each only sees its own band.

Reading the map #

Assemble all these windows and they make a shape.

The most striking feature is how uneven it is. At the Moon’s distance, a new force at $10^{-11}$ of gravity is ruled out. At a metre, it might be as large as $10^{-3}$. It is a surprise to many people to learn that around the size of a building, the law of gravity is tested eight orders of magnitude less precisely than it is at the size of the Moon’s orbit. Nothing about a metre makes that hard in principle. It is that a pair of masses a metre apart gives a very small force compared with any other that might act on them, and there is no vacuum at that scale in which to isolate the pair.

The valley’s left wall is the most thought-about part of the plot. At short range, the Newtonian force between two small masses is dominated by electromagnetic forces: Casimir forces, residual charges, and patch potentials on surfaces. A group at the University of Washington, under Adelberger, Heckel and Hoyle, has built a sequence of torsion pendulums that have pushed the limit down. Their instruments rotate a pair of disks with matching patterns of holes, which modulates any force at a known harmonic and which is nearly blind to anything else. By 2007 they had shown that the inverse-square law holds at $\alpha=1$ down to $\lambda$ of 56 micrometres, and later work has taken this a little further. That matters because large extra dimensions had been proposed in 1998, for which gravity would become stronger below a certain distance, possibly a millimetre. That whole region has since been closed, in part because it was tested by an experiment on a laboratory table.

The right wall is unknown for a different reason, though a change of $G$ with time is limited separately. Beyond about a thousand astronomical units there are no objects with measured orbits, and there is no longer a test at a level comparable to the rest. The hints of change on cosmological scales, such as the acceleration of the expansion, are in that range or far beyond it. But a test of gravity there is a different kind of measurement, and it is usually expressed in other parameters.

The fifth force that came and went #

The whole subject had a remarkably public episode. In January 1986, Ephraim Fischbach and his colleagues published a reanalysis of the data from Loránd Eötvös’s torsion balance measurements, published in 1922. Eötvös had looked for a difference in the rate of fall of different materials, and found none within about a part in $10^8$, and his small residuals had been regarded as noise. Fischbach showed that, when plotted against baryon number per unit mass, these residuals fell on a line, and the slope implied a new force of strength about one per cent of gravity and range of a few hundred metres. It was at about this time that Frank Stacey’s mine measurements also suggested an anomaly in $G$.

The claim drew a lot of attention, and in the following few years it was tested many times, partly with far greater sensitivity. Most experiments found nothing, a few found marginal signals. By about 1990, it had been effectively ruled out, with the best limits now five orders of magnitude below what was claimed. The reason for the original correlation in Eötvös’s data remains uncertain. It may have been due to thermal gradients that happened to be correlated with the samples’ mass. In any case the Eötvös experiment’s samples were few and it would take very little for a pattern to appear.

Those few years made the plot above. Modern torsion-balance methods were largely developed in order to test this claim, and the concept of a composition-dependent force at a finite range became an organised programme. It is a useful example of a wrong claim that left behind much better instruments and a clear way of thinking about the problem.

What makes two materials different #

The Eötvös claim also brought to light the question of how a new force might couple to matter. Gravity couples to mass, which includes binding energy. A new force need not. A natural guess is that it couples to the number of baryons, a conserved quantity that is not quite proportional to mass. If it does, the test object’s charge per unit mass, the quantity that decides how it falls, is its baryon number divided by its mass.

This figure is simply the nuclear binding curve seen from a different direction. Iron has the most tightly bound nucleus and so has the least mass per nucleon, and the largest baryon number per unit mass. It means a composition-sensitive experiment is in a curious position. It can only compare objects that differ in that small amount. Modern tests use beryllium and titanium, since they have a large difference and can be machined to high precision. The best limits are from the Eöt-Wash group, and by the MICROSCOPE satellite at about a part in $10^{15}$, comparing titanium and platinum in orbit for over a year.

In the same way, the Earth and Moon falling towards the Sun form an enormous composition test, because the Earth has an iron core and the Moon hardly any. And that is what lets lunar laser ranging test both kinds of departure from general relativity at once.

The limits that the map assumes #

The curve is a rounded envelope. It smooths a set of published limits with different assumptions, confidence levels and treatment of source geometry. For a detailed use, the original bounds would be necessary. It should be read as giving the shape and the rough heights.

The Yukawa form is not the only form. A force that varies as a power law of distance gives a different plot, and some models have forces which depend on local density and hide in dense environments, a behaviour known as screening. Such a force would evade every test in the Solar System by construction, and there would be no point on the map at which it could be detected. The map says nothing against it.

The coupling is assumed the same for all matter. A force that distinguishes composition would show up in equivalence tests, at a far higher sensitivity than in inverse-square tests at the same range. The map drawn here is for a composition-independent force, or else for the part of a coupling that survives the comparison.

$\alpha$ may have either sign. A repulsive force has the same effect in most experiments except for a change in sign of the signal. The vertical axis takes the absolute value, and limits are somewhat different for the two signs near the steep parts of the curve.

A map drawn by people chasing different things #

The chain’s most curious feature is that its links were not forged for this purpose. Lunar ranging was justified as a test of general relativity and as a way to measure the Moon’s interior. Satellite geodesy was intended to measure the Earth’s gravity field for missile guidance and for oceanography, and the LAGEOS satellites were built to measure continental drift. Planetary ranging is a by-product of navigating spacecraft to Mars. Only the torsion balances are aimed at the inverse-square law directly.

So the map is a fair picture of what kinds of measurement people happen to be doing. Its deep notches lie where a measurement of very high quality was needed for another reason. This is also why its ridge, between a few centimetres and a few kilometres, is where it is: nobody has needed to know gravity between a metre-scale mass and a test mass better than about a part in a thousand for anything else. There are suggestions for space experiments, in which free-falling masses have much better isolation from seismic noise, at ranges up to a few metres. None has been funded so far. This range is uninteresting to most theories, and the absence of a limit there has not caused much concern.

How a torsion balance sees one part in a hundred thousand #

The instrument at the left of the map is worth describing, because the difficulty in measuring gravity on a tabletop is not the one that would be expected. A pair of masses each a few grams, a millimetre apart, attract with a force of about $10^{-10}$ newtons or less, which is small but not absurdly so. The problem is everything else that acts on them. A stray electric charge of a few thousand electrons, or a thin film of contamination, gives a much larger force.

The method that won out is to avoid measuring a force at all. The detector is a pendulum hanging from a thin fibre, with a ring of holes bored in its rim, facing a similar plate with a matching pattern. The plate rotates slowly and continuously beneath it. Each time the holes come into line the gravitational attraction differs slightly, giving the pendulum a small twist at a frequency equal to the rotation rate times the number of holes. A thin stretched conducting membrane between the two blocks electrostatic forces. And because a signal at a high harmonic of the rotation has no simple cause other than the pattern of holes, the method rejects nearly everything else that might twist the fibre. Any tilt or temperature cycle shows up at the rotation frequency instead.

The trick is that the comparison is built into the geometry. A second ring of holes is placed beneath the first, offset by half a period, with dimensions chosen so that an exact inverse-square force nearly cancels, and only a departure from that law is left over. So the instrument is, by design, nearly blind to Newtonian gravity. It looks at the ratio that the first figure said is the only thing to look at, without any calibration of $G$. Indeed the same fibre method goes back to Cavendish and to Michell, and its precision has never helped with the value of the constant itself, since the value and the ratio are separate questions.

Still open: whether something sits below the micrometre #

The short-range wall may be the most interesting place to look. If dark energy has a characteristic length, the natural scale is about $(\hbar c/\rho_\Lambda)^{1/4}$, which turns out to be near 85 micrometres. There is nothing in the established physics that picks out this distance, but it is about the current limit of laboratory tests, and a change of gravity there would be astonishing. The best tests are already within a factor of about two of it, at a coupling of order one, though not at the smaller couplings that some theories would allow.

Beyond that limit, the Casimir force between two surfaces becomes larger than the Newtonian force by orders of magnitude, and getting below it will need cancelling or calibrating a much larger force of a known but only partly understood kind. Below a micrometre, the limit on $\alpha$ rises steeply, and at a nanometre a new force could be many billions of times stronger than gravity with no measurement yet able to detect it. The inverse-square law is often called the most thoroughly tested in physics. Over most of the range where forces can act, it is hardly tested at all.