Torsion-Balance Free-Fall Tests

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A rotating torsion balance turns a composition-dependent horizontal acceleration into a sinusoidal torque at a known phase. Its strength is not merely sensitivity: material interchange and attractor modulation make a genuine equivalence-principle signal transform differently from tilt, thermal drift, magnetic pickup, and gravity-gradient backgrounds. The result is a differential-acceleration test, not an absolute measurement of $G$ .

Required background. Equivalence Principles and Gravitational Measurement supplies free fall and the operational scope of WEP. Probability and Statistical Inference supplies likelihoods and nuisance parameters; the Gaussian time-series model below is the bounded bridge used here.

Helpful background. Weak-Field Gravity connects a weak potential to acceleration, while Tests of Gravity gives the cross-regime interpretation.

Composition dipoles and torque harmonics

Let test bodies $A$ and $B$ occupy alternating positions on a pendulum with lever arms $\mathbf r_i$ . A horizontal differential acceleration $\Delta\mathbf a=\mathbf a_A-\mathbf a_B$ produces a torque about the suspension axis,

\tau_z(t)=\sum_i\left[\mathbf r_i(t)\times m_i\mathbf a_i\right]_z.

If the turntable angle is $\phi(t)=\omega_t t+\phi_0$ , the WEP component appears in a known combination of $\sin\phi$ and $\cos\phi$ . The observed twist is filtered by the complex torsional response $H(\omega_t)$ , so amplitude and phase calibration are both essential. The target is commonly reported as

\eta_{AB}=\frac{2(a_A-a_B)}{a_A+a_B}\simeq \frac{\Delta a}{g_{\rm source}},

with the source direction, projection, material order, and sign stated. Rotation moves the signal away from zero frequency, where drift is worst, but it also places turntable-synchronous systematics exactly where the signal lives.

A minimal mechanical model makes the calibration explicit. With moment of inertia $I_p$ , torsional spring constant $\kappa$ , and damping coefficient $b$ , the twist obeys $I_p\ddot\Theta+b\dot\Theta+\kappa\Theta=\tau_z$ . For the site’s $e^{-i\omega t}$ convention,

H(\omega)\equiv\frac{\Theta(\omega)}{\tau_z(\omega)} =\frac{1}{\kappa-I_p\omega^2-i b\omega}.

Thus the measured twist amplitude alone does not fix torque: the response phase changes through resonance, and the low-frequency limit is $H(0)=1/\kappa$ . This viscous-damping model is illustrative; a precision fibre can require measured anelastic loss instead.

The Eöt-Wash Be–Ti implementation used a fourfold-symmetric composition dipole, continuous rotation, an autocollimator, and periodic material interchange. Its apparatus, environmental controls, segmentation, harmonic fit, and correction budget are described in Schlamminger et al. 2008, Fig. 1, Table I, and pp. 1–3, PDF. Those published products provide summary values and model details, not a complete public time-stamped torque archive; an independent numerical reproduction must therefore use released data from the experiment under study or a clearly labeled synthetic benchmark.

Calibration, selection, and likelihood

The calibrated observation vector contains twist angle, turntable angle, timestamps, tilt sensors, thermometry, magnetic monitors, and a gravity-gradient model. A minimal linear model for segment $s$ is

\boldsymbol\Theta_s =X_{\rm EP,s}\boldsymbol\beta_{\rm EP} +X_{\rm harm,s}\boldsymbol\beta_{\rm harm} +X_{\rm env,s}\boldsymbol\nu +\boldsymbol n_s,

where $X_{\rm EP}$ fixes the predicted signal phase, $X_{\rm harm}$ contains other rotation harmonics and drifts, and $X_{\rm env}$ projects monitored nuisance channels. With declared covariance $C_s$ , the normalized Gaussian likelihood includes $\det C_s$ as well as the quadratic residual. Priors on calibration coefficients should come from independent exaggerated-response measurements, not from forcing the science channel toward zero.

Selection is part of the measurement. The 2008 analysis divided the record into two-turntable-revolution segments, fit nine harmonics plus drift terms, and excluded about seven percent of segments for specified ion-pump spikes or abrupt vertical-axis changes Schlamminger et al. 2008, data-analysis paragraph and Fig. 2, PDF. A modern analysis must publish every quality cut, the live-time loss, and whether cuts were fixed before unblinding. Completeness here means completeness in usable observing time and orientation coverage, not a population-selection function.

Covariance must follow shared calibrations. A single tilt matrix or temperature coefficient applied to many segments creates correlated uncertainty; adding its error independently to every segment would make the combined result spuriously precise. Posterior-predictive or frequentist residual checks should examine unused harmonics, opposite composition orientations, and time splits.

Backgrounds transform differently

Gravity gradients couple ambient multipoles to imperfect pendulum mass moments. Deliberately rotating compensator masses measures that coupling. Tilt converts motion of the fiber attachment into apparent pendulum rotation; imposed tilts determine the response matrix. Applied magnetic fields and deliberately exaggerated thermal gradients bound magnetic and thermal transfer coefficients. Electrostatic effects require conducting shields, charge control, and reversal tests.

The decisive operation is composition reversal. A true $A-B$ signal changes sign when the materials exchange positions while an offset fixed to the pendulum frame does not. Attractor or source reversal supplies another phase rule. The 2008 experiment’s separated means before and after interchange illustrate this logic in Fig. 2, while Table I keeps statistical acceleration, gravity-gradient, tilt, magnetic, and temperature terms distinct Schlamminger et al. 2008, Fig. 2 and Table I, PDF.

An independent benchmark should inject a blinded harmonic into raw or realistically sampled synthetic twist data, propagate it through $H(\omega)$ , and recover its amplitude and phase. Repeat after composition reversal, source reversal, and removal of each environmental regressor. Failure to recover the injection, or recovery in a null harmonic, blocks a gravity interpretation.

What the results establish

Evidence assessed through 2026-08-19 supports rotating torsion balances as high-sensitivity WEP and long-range-force instruments, with null results in the tested material/source domains. It does not establish universality for untested composition charges, interaction ranges, spin couplings, or gravitational self-energy. A current numerical bound belongs with its exact dataset, calibration, interval convention, and source model rather than as an undated fact on this method page.

The end-to-end record needed to quote a result is: named dataset or primary record; instrument and transfer calibration; all segment and quality selections; covariance and priors; correction table; material and source model; software version or machine-readable fit recipe; and the result of reversal and injection controls. A superseding time-series release or materially changed tilt, thermal, or gravity-gradient model requires reanalysis.

Exercise

A pendulum-fixed thermal offset has fitted amplitude $b$ at the turntable frequency. A WEP signal has amplitude $s$ . Write the measured amplitudes before and after reversing the composition dipole, and show how to isolate $s$ .

Solution

Before reversal the fitted amplitude is $A_+=b+s$ , because the thermal term and composition signal project with the same chosen phase. Reversing the materials changes the composition dipole but leaves a pendulum-fixed thermal coupling unchanged, so $A_-=b-s$ . Therefore $(A_+-A_-)/2=s$ , whereas $(A_++A_-)/2=b$ . This separation works only if the response calibration and environmental state are stable or their changes are included as correlated nuisance parameters.

References

Schlamminger, Stephan, Ki-Young Choi, Todd A. Wagner, Jens H. Gundlach, and Eric G. Adelberger. “Test of the Equivalence Principle Using a Rotating Torsion Balance.” Physical Review Letters 100 (2008): 041101. DOI; Open PDF.