We have a beautiful theory that puts each of nature's forces into a single, neat package. The whole of it can be summed up in a single line of very compact—and for most, including me, incomprehensible—mathematics. At least, that is what we would like to be able to say, but this beauty is marred. Imagine the Mona Lisa with an eyepatch drawn in using crayon.

That is modern physics. The eyepatch is gravity.

There are many ideas about how to remove the crayon eyepatch from the masterpiece of modern physics and create a single, unified theory, but there's little evidence to support any of them. Among the ideas are theories involving extra dimensions (like string theory). And for nearly 10 years, physicists have been fruitlessly searching for evidence for these hidden dimensions.

Now, one of the most sensitive experiments yet has reported another null result. But it's a very cool experiment nonetheless.

Why do quantum mechanics and gravity always argue, Mom?

Here is the sort of conflict facing physicists: quantum mechanics takes place on the fixed stage of space and time; in gravity, space and time are actors. In quantum mechanics, the actors are not always continuous, often making sharp jumps. Unfortunately, gravity requires that space and time be smooth and continuous. These conflicting requirements are, so far, unresolved.

One approach to resolving this problem is for space and time to have more than four dimensions. I can't pretend to have a great understanding of this subject, but my impression is that all the jumpiness of space time will get absorbed into the dimensions that we don't observe. Why don't we observe these dimensions? Because they are curled up into tiny volumes.

Yet particles should still feel them. Imagine that you are a particle. You have some mass, and therefore, you bend space and time such that nearby objects feel a force attracting them to you—even better, it is a mutual attraction, the best sort of attraction.

But that force falls off in a very specific way. For three spatial dimensions, if you double the distance between you and the particle you are trying to attract, the force between you reduces by a factor of four. This is called the inverse square law, and it has been calculated and verified by numerous experiments going right back to Newton. However, if there are more than three spatial dimensions, the force will reduce even faster than predicted by the inverse square law.

That is something that we can measure. By placing two masses in proximity to each other and measuring the force of attraction between them, we can attempt to measure any deviation in the inverse square law.

Hidden dimensions and 18th century pendulums

The basic idea is to use a pendulum and observe the amplitude and phase of its swinging due to some nearby mass. Usually the idea is to use two identical pendulums and observe how they influence each other. The problem with this approach is that the drive frequency—we have to supply some energy to the pendulum to keep it in motion—is the same as the frequency where the signal should be.

To overcome this problem, researchers from China have come up with a new design for the venerable pendulum experiment. In their version, the (torsional) pendulum swings next to a disk. The disc has eight masses and rotates at the same frequency as the swinging motion pendulum. As each mass passes the end of the pendulum it perturbs its motion, meaning that the signal is at eight times the oscillation frequency of the pendulum. This makes it very easy to sense, because for natural harmonic motion, there is no detectable component at that frequency.

Another advantage of the design is that the pendulum can be placed as close as 0.3mm from the disk, allowing the force of gravity to be tested over quite short distances. At this distance, there are all sorts of other problems that come into play. For instance, the electrons in the disk start to feel the electrons in the pendulum. Unless steps are taken, they will arrange themselves to generate a mutually attractive force.

It's really an awesome setup. What we have are a few precision milled masses, glued to very low thermal expansion glasses, coated with a highly conducting metal, and separated by an additional conducting barrier. Add in a few control electronics, a vacuum chamber, and some test masses for calibration, and you have a relatively simple but highly sensitive experiment. It's amazing how sensitive it is given the starting materials.

How sensitive? On the order of 10-17Nm (that's the torque on the pendulum, the actual force is on the order of 10-19N, but torque is what is actually measured). This is about the equivalent to the force of 10,000 water molecules falling onto a surface (a single millilitre of water has over 1022).

And, after all of that work... the inverse square law is still intact down to a length scale of about 50 micrometers.

So, where will things go from here? It should be possible to improve the sensitivity of the current experiment and to get the pendulum and the disk closer together to test even smaller separations. The biggest hole in the experimental data is in separations of 10 micrometer or less, which I don't think this experiment will ever get close to. The problem with going to closer distances is that the forces due to the electrons and protons in the masses will dominate gravity. To avoid that, entirely new approaches (probably based on MEMs, or microelectromechanical devices) will be required.

But we don't have to look for hidden dimensions using the force of gravity, as electrostatic charges also obey the inverse square law. That leads to a final thought: we already know from the way matter behaves that electrostatic forces obey the inverse square law to a distance of at least 10-10m. So, I guess the argument is that gravity might use these extra dimensions, but photons—the force carrier for electromagnetism—don't. That seems a bit on the unlikely side.

Physical Review Letters, 2016, DOI: 10.1103/PhysRevLett.116.131101