In 1984 Braginsky, Polnarev and Thorne proposed measuring the Earth's gravitomagnetic (Lense–Thirring) field — the dragging of inertial frames by the Earth's rotation — by monitoring the precession of the plane of swing of a Foucault pendulum at the South Pole, referenced by an astrometric telescope to the azimuth of a fixed star, and concluded that with great effort a 10% experiment in a measurement time of several months might be achieved, conditional on laboratory development they explicitly identified as unresolved: control and subtraction of frictional anisotropy; characterization and subtraction of the Pippard precession (about 10310^3 times the signal) with a 10410^4-dynamic-range readout; compensation of pendulum frequency anisotropy to below 10−810^{-8}; antiseismic isolation at factors 10310^3–10510^5 with dual-pendulum differencing; and few-milliarcsecond multi-week astrometry (two-star tracking, two-colour refractometry, telescope thermal stability of order 0.01 K). Whether such an apparatus can be developed — whether the several effects that compete with or exceed the signal can be controlled so that the cumulative error budget permits the 10% measurement — is the feasibility question posed here.

Background

In general relativity, a rotating mass drags the local inertial frames around with it — the Lense–Thirring effect, also called frame dragging. In the weak-field, slow-motion limit, the metric components g0jg^{0j} (those coupling space and time) act like a vector potential whose curl behaves as a magnetic-like field, called the gravitomagnetic (GM) field HH. In 1984 this component of the Earth's gravitational field had never been directly detected, although its analogues play a central role in astrophysical models of rotating black holes. A Foucault pendulum — a mass on a long fiber whose plane of swing behaves as a gyroscope with two degrees of freedom — normally precesses relative to the ground at a rate set by the Earth's rotation (equal to Ω⊕sin⁡λ\Omega_\oplus\sin\lambda at latitude λ\lambda, where Ω⊕\Omega_\oplus is the Earth's angular velocity); relative to the distant stars, a laboratory at colatitude α\alpha (the angle between the local vertical and the Earth's rotation axis) produces a residual classical precession Ω⊕(1−cos⁡α)\Omega_\oplus(1-\cos\alpha), which vanishes exactly at the geographic pole. There the support of a pendulum at rest on the Earth's surface has zero velocity in the Earth-centered frame, so the competing relativistic precessions of a gyroscope — the geodetic precession (from motion through the Earth's curved spacetime) and the Thomas precession (from the rotation of the support's acceleration) — vanish as well, and the swing plane precesses relative to a star-referenced azimuth at the GM rate alone: H/2cH/2c, printed by Braginsky, Polnarev and Thorne as 0.281 arcsec/yr and described by them as five times the frame-dragging rate in the polar orbit of the Gravity Probe B gyroscopes (GPB; later recomputations by Pippard, Pascual-Sánchez and Cartmell et al. give about 220 milliarcsec/yr for the same quantity — the value actually consistent with that factor of five). Earth-based rotating-sphere gyroscopes cannot reach this sensitivity because support-system imperfections and Newtonian gravity acting on gyro inhomogeneities produce precessions orders of magnitude larger; that is why the superconducting gyroscopes of GPB had to fly in space. Braginsky, Polnarev and Thorne therefore proposed an earth-based variant of GPB: a Foucault pendulum and an astrometric telescope in an underground vacuum chamber within a few kilometers of the South Pole, the telescope locked to the azimuth of a reference star (e.g., Canopus), the swinging fiber used as a light pipe so that the angle between the swing's principal axis and the stellar azimuth, and the swing ellipticity (the ratio of the minor to the major axis of the nearly linear swing), can be monitored optically. A 10% measurement of the accumulated GM angle (about 0.046 arcsec per 60 days) requires an integrated precision of roughly 0.004 arcsec per 60 days over a campaign of several months. The body of the 1984 paper quantifies the error sources that dominate this tiny signal. Velocity-proportional forces precess the swing plane directly: besides the GM force to be measured, these are magnetic forces on residual charge (suppressed by metallizing fiber and bob) and anisotropic frictional damping, which with the support fixed to the Earth produces a precession sinusoidal with a 24-hour period that must remain stable enough over months to be subtracted; and the Pippard precession — because an Earth-fixed support forces the bob to carry spin angular momentum mk2Ω⊕mk^2\Omega_\oplus (mm the bob mass, kk its radius of gyration), whose direction changes as the pendulum swings, a gyroscopic force acts and produces an accumulated angle δϕP=(Ω⊕/2)(k2/l2)τ^\delta\phi_P=(\Omega_\oplus/2)(k^2/l^2)\hat{\tau} for fiber length ll and duration τ^\hat{\tau}, about 10310^3 times the GM signal, so (k/l)2(k/l)^2 must be calibrated to 10−410^{-4} and the readout must span a dynamic range of 10410^4. Position-dependent forces (gravity gradients, static light pressure) instead drive growth of the ellipticity, which must be held below 10−710^{-7} in relative size by gravitational feedback; ellipticity precesses the swing only through a frequency anisotropy between the two horizontal swing directions, which for a finite-amplitude fiber pendulum is intrinsically 10−310^{-3} to 10−510^{-5} (finite-amplitude nonlinearity plus fiber anisotropy) and must therefore be compensated to below 10−810^{-8}, by gravitational balancing masses, by electrostatic plates, or by a mass magnetically levitated over a superconducting surface shaped to cancel the leading nonlinearity. Seismic support motion (from World-Wide Standard Seismograph Network data at the South Pole) requires antiseismic isolation factors of 3×1033\times10^3 to 9×1049\times10^4 depending on pendulum length and frequency, with residual seismic precessions removed by differencing two pendula mounted on one support and swinging out of phase. Finally, the astrometric reference itself is limited by azimuthal atmospheric refraction — the best 30-day combined astrometric accuracy of the era was about 0.006 arcsec — and by telescope distortion and tilt, to be countered by tracking two stars on opposite sides of the sky, two-colour refractometry (using the wavelength dependence of atmospheric dispersion to correct refraction), telescope thermal stability of order 0.01 K, and tilt feedback. The paper's conclusion is that the experiment is very difficult but might be doable, and that determining just how difficult (or impossible) it is requires laboratory development in several areas: frictional anisotropy, Pippard precession, frequency compensation and antiseismic isolation for the pendulum; atmospheric refraction and physical distortion for the telescope; and dynamic range for the readout system.

More than four decades later the experiment has still never been performed. The proposal was re-analyzed only descriptively (Pascual-Sánchez's TELEPENSOUTH study, 2002–2003; Iorio, 2003), was judged pessimistically by Pippard (1988), and is being pursued by an ongoing program (Cartmell, Faller, Lockerbie et al., 2017–) whose instrument was still in build and test in 2025 and which estimates a needed resolution gain of order 10910^9. Satellite experiments (Gravity Probe B; LAGEOS/LAGEOS-2/LARES) have meanwhile measured frame dragging directly, so the open problem is specifically the terrestrial pendulum measurement and its systematic-error control, not the detection of frame dragging itself.

Problem Statement

Determine whether the South Pole Foucault-pendulum experiment proposed by Braginsky, Polnarev and Thorne is feasible: whether a pendulum-and-telescope apparatus can be developed whose cumulative error budget permits a measurement of the gravitomagnetic precession of the pendulum's swing plane, of order 0.2–0.3 arcsec/yr, with about 10% accuracy in a campaign of several months, with the apparatus located within a few kilometers of a geographic pole (local colatitude known to about 0.1 arcsec, ice motion accounted for). This requires resolving, through laboratory development and demonstrated performance of the relevant components, the error sources the authors identified as decisive, each of which is comparable to or exceeds the signal: (i) a suspension whose frictional anisotropy is stable enough over months that its induced 24-hour-periodic precession can be subtracted at the level of about 0.004 arcsec per 60 days; (ii) calibration of (k/l)2(k/l)^2 to about 10−410^{-4} together with an optical readout of dynamic range about 10410^4, so that the Pippard precession (about 10310^3 times the signal) can be subtracted; (iii) compensation of frequency anisotropy to below 10−810^{-8} — by gravitational balancing masses, electrostatic compensation, or alternative suspensions such as a mass magnetically levitated over a superconducting surface — while holding the swing ellipticity below 10−710^{-7} under feedback and without introducing new perturbations such as compensator jitter; (iv) active antiseismic isolation at factors of order 10310^3–10510^5 (set by the chosen pendulum length and frequency), together with demonstrated rejection of residual seismic precession, e.g. by differencing two pendula on a common support swinging out of phase; (v) astrometric referencing — two-star tracking, two-colour refractometry, telescope thermal stability of order 0.01 K, and tilt feedback — sufficient to hold atmospheric-refraction and telescope errors at the few-milliarcsecond level integrated over multi-week runs; and (vi) an integrated apparatus and error budget establishing that these controls jointly support the 10% measurement, or a quantitative demonstration that they cannot. Feasibility fails if some identified decisive error source cannot be stably modeled, calibrated, and subtracted (or otherwise compensated) to the residual precision required for the 10% measurement — about 0.004 arcsec per 60 days — in any admissible design. Admissible designs are the pendulum-and-star-referenced-telescope configurations the 1984 paper contemplates (“Many experimental setups are conceivable”), including the alternatives it lists — gravitational balancing masses, electrostatic compensation plates, a mass magnetically levitated over a superconducting surface, and two pendula on one common support swinging out of phase — sited within a few kilometers of a geographic pole with the local colatitude known to about 0.1 arcsec. The source treats systematics larger than the gravitomagnetic signal — the Pippard precession (about 10310^3 times the signal) and the 24-hour frictional precession (of the order of the signal) — as stable, calibratable and subtractable from the data, so the feasibility question is whether such terms can be controlled to the required residual precision, not whether they exceed the gravitomagnetic signal.

References

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