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Dynamic measurement of gravitational coupling between resonating beams in the hertz regime

Abstract

To date, there have been few dynamic measurements of gravitation in the laboratory, and fully controlled quantitative experiments have been limited to frequencies in the millihertz regime. Here we introduce a fully characterized experiment at frequencies in the hertz regime, which allows a quantitative determination of the dynamic gravitational interaction between two parallel beams vibrating at 42 Hz in bending motion. A large amplitude vibration of the transmitter beam produces a gravitationally induced motion of the high-quality-factor resonant detector beam with amplitudes up to 10−11 m. The sub-picometre-resolution measurement is made possible by a setup that combines acoustical, mechanical and electrical isolation; a temperature-stable environment; heterodyne laser interferometry; and lock-in detection. The interaction is quantitatively modelled based on Newton’s law of gravitation. Amplitude measurements at varying beam distances follow an inverse square law and agree with theoretical predictions to within approximately three percent. Furthermore, we extract the value of the gravitational constant G and near-field gravitational energy flow. We expect our experiment to enable progress in directions where current experimental evidence for dynamic gravitation is limited, such as the dynamic determination of G, inverse square law and gravitational shielding.

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Fig. 1: Illustration of the measurement principle.
Fig. 2: Illustration of the measurement setup.
Fig. 3: Result of a measurement run.
Fig. 4: Gravitationally induced bending motion.
Fig. 5: Measurement result of the gravitational constant.

Data availability

Source data are provided with this paper. The data that support the findings of this study are available as an open access dataset37 or from the corresponding author upon reasonable request.

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Acknowledgements

We gratefully acknowledge the support of ETH Zurich, maxon motor ag and ZC Ziegler Consultants AG. We received no specific funding for this work.

Author information

Authors and Affiliations

Authors

Contributions

T.B., B.Z., F.P., J.-C.T., S.B., D.S. and J.D. designed and constructed the experiment. T.B., B.Z. and J.D. conducted the experiments, T.B., F.B. and J.D. evaluated the data and analysed the results. T.B., S.K., J.F. and J.D. derived and evaluated the theory. All the authors wrote and reviewed the manuscript.

Corresponding author

Correspondence to Jürg Dual.

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The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Andrei Metrikine and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Detector resonance frequency vs. temperature.

Detector beam resonance frequency f0 as a function of the mean temperature inside of the detector chamber. The points represent the SDOF fit result of the resonance frequency obtained from individual measurement runs (such as exemplarily shown in Fig. 3) at the mean temperature during the run (x - error bars = std). The y - error bars denote the 95% confidence interval of the SDOF fit result (not visible). The temperature of the air in the chamber and the beam’s material are assumed to be equal. A linear fit of all data points reveals a linear relation between the detector resonance frequency and its temperature with a coefficient of −0.01834(56) Hz/°C. The red shaded illustrates the 95% confidence band of the linear fit.

Source data

Extended Data Table 1 Detector and transmitter beam model parameters

Source data

Source Data Fig. 3

Data for Fig. 3.

Source Data Fig. 4

Data for Fig. 4.

Source Data Fig. 5

Data for Fig. 5.

Source Data Extended Data Fig. 1

Data for Extended Data Fig. 1.

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Brack, T., Zybach, B., Balabdaoui, F. et al. Dynamic measurement of gravitational coupling between resonating beams in the hertz regime. Nat. Phys. 18, 952–957 (2022). https://doi.org/10.1038/s41567-022-01642-8

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