How did Eötvös measure gravity at sub 100 micron length scales by drilling holes in larger objects?

Question

Space.com is not the right place to read about fundamental physics, but nonetheless I just saw Pioneering gravity research snags $3 million physics Breakthrough Prize which says:

Measurements by Adelberger, Gundlach, Heckel and their colleagues recently showed that the inverse square law holds even for objects separated by a mere 52 microns (0.002 inches), "establishing that any extra dimension must be curled up with a radius less than 1/3 the diameter of a human hair," Breakthrough Prize representatives wrote in today's award announcement.

The abstract of the linked arXiv preprint says:

We tested the gravitational 1/r² law using a stationary torsion-balance detector and a rotating attractor containing test bodies with both 18-fold and 120-fold azimuthal symmetries that simultaneously tests the 1/r² law at two different length scales. We took data at detector-attractor separations between 52 µm and 3.0 mm. Newtonian gravity gave an excellent fit to our data, limiting with 95% confidence any gravitational-strength Yukawa interactions to ranges < 38.6 µm.

To push the range below 100 microns it's clear that they didn't build a 100 micron wide torsional pendulum, but instead did something to the shape of a larger pendulum.

Question: Is it possible to explain in a simple and straightforward way how the apparatus was able to explore deviations from 1/r² at distances below 100 microns using larger attractors and pendulums and putting holes in them?

Answer 1

In your favorite electromagnetism text, there is a set of exercises proving that a $1/r$ potential for a point charge leads to a $\ln r$ potential ($1/r$ field) for an infinite line charge, and to a linear potential (constant field) for an infinite plane charge. In the cases of a finite line charge or a finite plans charge, a good textbook (or a good instructor) will show how the infinite-line and infinite-plane results emerge in the limit where the interaction distance is much smaller than the length scale which characterizes the finite object.

Gravity and electromagnetism are both $1/r$ potentials, so all of these arguments apply equally well to mass distributions. A famous example which you have used yourself is that the infinite-plane/constant-field result is a really good approximation for the gravitational field near Earth's surface, as long as the distances involved are small with respect to Earth's radius.

In the Eöt-Wash gravitational torsion pendula, the test masses are uniform-density and -thickness plates with precision hole patterns drilled in them. The plates are several centimeters across, and are separated by hundreds of micrometers (or less). That ratio of interaction length to overall scale is small enough that the infinite-plane approximation is useful. The test masses are substantially larger than 100 microns, but they are everywhere less than 100 microns apart.

If I remember correctly, the analysis treats the plates-with-holes as uniform positive-mass plates superimposed with negative-mass discs with the geometry of the holes --- another transfer from electromagnetism, where superposition of opposite charges has a stronger phenomenological motivation than superposition of opposite gravitational masses, but the arithmetic is identical. (The negative-mass holes are attracted to each other.) Test mass plates whose hole patterns have different symmetries therefore probe interactions over a range of length scales; the analysis for extracting these made for interesting reading.