This paper considers the Foucault pendulum and its possible application as a measurement instrument for the relativistic effect of Lense-Thirring precession, or frame-dragging. The Foucault pendulum is considered by some to be one of the ‘top ten’ experiments of physics, first undertaken by Léon Foucault in 1851 and still of interest today [1]. This paper will summarise some recent modelling of the Foucault pendulum, under two different forms of excitation to maintain its motion over time and in the presence of dissipations [2,3]. The first of these is through parametric excitation of the length, shown to be most effective as in overcoming natural aerodynamic and frictional dissipations and also as a form of amplification of the response. When excited into principal parametric resonance the natural tendency of the motion of the Foucault pendulum to degenerate into ellipticity can also be controlled. The second form of excitation is through electromagnetic forcing, through the coupling between a strong magnet integrated into the base of the bob and an exciter coil placed directly underneath the equilibrium position of the pendulum. This excitation is shown to be just as effective as the parametric excitation, but exceeding the accuracy of ellipticity control and without the unwanted additional tendency for the bob to move vertically at the frequency of parametric excitation. >From the perspective of understanding the dynamics of the Foucault pendulum so that its performance can be reliably interpreted, and very importantly maintained over time, the paper then considers how a gyroscopic instrument of this sort may be used as the basis for measuring the tiny relativistic precessions of a test mass experiencing frame-dragging due to terrestrial gravity. The exceptional work done by the teams of scientists behind the GP-B and LAGEOS missions in measuring the Lense-Thirring precession of frame-dragging in low Earth orbit is highlighted as the base-line against which any other measurement work should refer. In the paper we develop the proposition that the Foucault pendulum may be an appropriate instrument for a terrestrial meaurement and we show that in principle at least such a measurement of frame-dragging may be possible. We start with the premise that the natural location for a frame-dragging measurement would be the South Pole [4,5,6,7] and we derive a practical expression for Lense-Thirring precession on Earth using a gravitoelectromagnetic analogy, building in a correction for terrestrial locations elsewhere than the Poles. From this point on the paper explores the major challenge in siting the measurement anywhere other than the Poles, and a summary of ideas for performing the measurement in the northerly location of Glasgow, Scotland, is presented, together with a summary of the intended implementation of the electromagnetic excitation proposal of [8]. The paper places the work done by the author, and colleagues, generally in the wider context of the literature, noting that the first serious proposal to perform a terrestrial frame-dragging test was due to [4], followed slightly less enthusiastically by [5], and others subsequently. The paper seeks to clarify the very considerable challenges involved in this measurement and as such should be regarded as a work in progress.
References
[1] https://en.wikipedia.org/wiki/Foucault_pendulum (last accessed 23/08/21).
[2] Cartmell, M.P., Faller, J.E., Lockerbie, N.A., Handous, E.: On the modelling and testing of a laboratory-scale Foucault pendulum as a precursor for the design of a high-performance measurement instrument. Proc. R. Soc. A 476, 20190680, pp. 26 (2020) http://dx.doi.org/10.1098/rspa.2019.0680.
[3] Cartmell, M.P., Lockerbie, N.A., and Faller, J.E., Towards a high-performance Foucault pendulum, Proceedings of the 2nd International Nonlinear Dynamics Conference, Sapienza University, Rome, Italy, February 16-19, 2021.
[4] Braginsky V.B., Polnarev A.G., Thorne K.S.: Foucault pendulum at the South Pole: proposal for an experiment to detect the Earth’s general relativistic gravitomagnetic field. Physical Review Letters. (9) 53, pp 863-866 (1984).
[5] Pippard, A.B.: The parametrically maintained Foucault pendulum and its perturbations. Proc. R. Soc. A 420, pp. 81-91 (1988).
[6] Ruggiero, M.L., Tartaglia, A., Gravitometric Effects, arXiv:gr-qc/0207065v2, 8 February 2008.
[7] Pascuel-Sánchez, J.F., TELEPENSOUTH project: Measurement of the Earth gravitomagnetic field in a terrestrial laboratory, arXiv:gr-qc/0207122v1, 30 July 2002.
[8] Schumacher, R., and Tarbet, B., A short Foucault pendulum free of ellipsoidal precession. arXiv: 0902.1929v3 [physics.class-ph] 15 November 2020.