Inverse-Square-Law Experiments

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An inverse-square-law experiment does not measure an exponent in isolation. It measures a displacement, force, or torque from extended bodies and asks whether its separation and modulation dependence is explained by Newtonian gravity after the instrument and all nongravitational forces are modeled. The strongest design uses geometry to move the Newtonian signal and a possible short-range signal into known harmonics while providing configurations in which one of them is suppressed.

Required background. Weak-Field Gravity supplies the Newtonian limit and the distinction between potential and observable; Probability and Statistical Inference supplies likelihood, covariance, nuisance-parameter, and interval reasoning. Helpful background. Classical Field Theory supplies Green-function and source-distribution methods used for finite bodies.

Force laws become geometric harmonics

For point masses, a common test coordinate augments the Newtonian potential by a Yukawa term,

V(r)=-\frac{Gm_1m_2}{r}\left(1+\alpha e^{-r/\lambda}\right).

Here $\alpha$ is dimensionless and $\lambda$ is a length. The checks are immediate: $\alpha=0$ recovers Newtonian gravity, $\lambda\to 0$ suppresses the extra force at fixed nonzero separation, and $\lambda\to\infty$ rescales the Newtonian interaction rather than creating a new distance law. A real apparatus requires the density integral

U(\varphi,s)= -G\!\int_{V_1}\!d^3x\!\int_{V_2}\!d^3x'\, \frac{\rho_1(\mathbf x)\rho_2(\mathbf x')} {\lvert\mathbf x-\mathbf x'\rvert} \left[1+\alpha e^{-\lvert\mathbf x-\mathbf x'\rvert/\lambda}\right], \qquad N=-\frac{\partial U}{\partial\varphi},

where $s$ is the measured surface separation and $\varphi$ is the attractor angle. The dimensions are $[U]=$ energy and $[N]=$ energy per radian, hence torque. Coatings, glue, holes, density variations, tilt, and edge shapes belong inside $\rho_i$ and the integration domains; replacing them by point masses is generally wrong when $\lambda$ is comparable to those dimensions.

Patterned source and detector bodies give a Fourier series $N(\varphi,s)=\sum_n N_n(s)\cos[n\varphi+\phi_n]$ . Choosing two different azimuthal symmetries tests two effective length scales in the same instrument. The Eöt-Wash experiment, for example, used 18-fold and 120-fold patterns and read the corresponding torque harmonics; its geometry, calibration, displacement metrology, and harmonic fit are documented in Lee et al. 2020, Eq. (1) and Figs. 1–5. A useful null pattern cancels the leading Newtonian multipole while leaving a finite-range interaction incompletely canceled. That is much more discriminating than simply looking for a nonzero force.

Why parallel plates do not follow an inverse-square distance law

“Inverse square” describes the pairwise point-mass kernel, not every apparatus signal. An infinite Newtonian slab of density $\rho_1$ and thickness $t_1$ produces the exterior acceleration $2\pi G\rho_1t_1$ , independent of distance. A second slab of areal density $\rho_2t_2$ therefore feels pressure $-2\pi G\rho_1\rho_2t_1t_2$ , also independent of the gap. A measured plate–plate force cannot be tested by fitting it to $s^{-2}$ . Finite edges and patterning restore geometric variation that must be computed from the density integral. The corresponding Yukawa slab response instead contains $e^{-s/\lambda}$ , making the contrast particularly useful.

From twist data to a gravity likelihood

The evidence basis used here is current through 2026-08-19. Any numerical conclusion must identify the primary data version or article, geometry and density files, calibration version, retrieval date, and any superseding product. A complete measurement chain is:

record twist angle, attractor encoder angle, capacitance or interferometric separation, temperatures, tilts, magnetic monitors, and timestamps;
calibrate angle-to-torque transfer, torsional resonance, filtering phase, absolute torque scale, and the three-dimensional body displacement;
apply a declared selection and quality cut for transients, dust contacts, drive loss, excessive tilt, temperature excursions, or contaminated frequency bands, and report the retained exposure rather than calling the selected sample complete;
fit harmonic amplitudes without using the science harmonics to tune the background model; and
compare Newtonian and finite-range templates while marginalizing the shared calibration and geometry variables.

For a vector $\mathbf d$ of harmonic amplitudes, one transparent likelihood is

-2\ln \mathcal L = \bigl[\mathbf d-\mathbf m(\alpha,\lambda,\boldsymbol\nu)\bigr]^{\mathsf T} C^{-1} \bigl[\mathbf d-\mathbf m(\alpha,\lambda,\boldsymbol\nu)\bigr] +\text{constant},

with $\mathbf m=\mathbf N_{\rm N}+\alpha\mathbf N_{\rm Y}+\sum_a\nu_a\mathbf B_a$ , assuming $C$ is fixed; otherwise add $\ln\det C$ . The nuisance variables $\boldsymbol\nu$ include separation zero, density and thickness, scale calibration, tilt, magnetic susceptibility, patch-potential templates, thermal drift, and vibration coupling. Their prior or auxiliary-measurement probability must be stated. Correlations belong in $C$ or in a shared nuisance model; adding quoted systematic errors in quadrature is justified only when independence and approximate Gaussianity have been established.

The confidence or credible construction must also be named. At fixed $\lambda$ , a profile-likelihood interval in $\alpha$ and a Bayesian interval with a prior uniform in $\alpha$ are different procedures. Scanning many $\lambda$ values produces a family of correlated tests, not independent discoveries. Posterior-predictive residuals should be checked by harmonic, separation, run epoch, and apparatus generation.

Controls that separate forces

Electrostatic patch forces, magnetic pickup, Casimir forces, temperature gradients, vibration, and feedback can all change with separation. A successful control changes one cause while leaving the proposed gravitational response fixed:

reverse or deliberately vary an applied electric or magnetic field;
rotate a composition-neutral dummy pattern or use a Newtonian-null geometry;
move the signal harmonic away from the torsional resonance and known line noise;
repeat with a different attractor symmetry, material, coating, or source mass;
blind the analysis with an injected Yukawa amplitude; and
withhold separations or run epochs until the nuisance model is fixed.

The decisive reproduction test is to recover the Newtonian geometric torque, then recover a blinded Yukawa injection with its correct separation and harmonic scaling, and finally show that the same pipeline returns the null in a canceling geometry. Software, quadrature tolerances, density maps, masks, calibration covariance, and machine-readable fit residuals are necessary parts of that reproduction. A residual seen only after one background subtraction, or only in one harmonic, does not by itself establish a failure of the inverse-square law.

Check your understanding

A patterned torsion balance has a Newtonian-null harmonic at the nominal alignment. A small tilt produces a background proportional to $s^{-2}$ , while a hypothetical Yukawa signal at the tested range is approximately proportional to $e^{-s/\lambda}$ . Explain a test that can distinguish them without assuming the tilt calibration is exact.

Solution

Fit several separations jointly and include the measured tilt monitor as a nuisance template, because the power law and exponential have different curvature as functions of separation. Then deliberately reverse or enlarge the tilt to calibrate its phase and amplitude, while keeping the mass pattern unchanged. A blinded Yukawa injection tests whether the fitting procedure can recover the exponential component. Finally, repeat in a second geometric harmonic: the tilt template should transform according to the apparatus geometry, whereas the injected finite-range torque follows its independently integrated density kernel. Agreement of both channels therefore supplies a stronger discriminator than a single small residual.

References

Lee, J. G., Adelberger, E. G., Cook, T. S., Fleischer, S. M., and Heckel, B. R., “New Test of the Gravitational $1/r^2$ Law at Separations down to 52 Microns,” Physical Review Letters 124 (2020) 101101, DOI.