† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11575160 and 11805074) and the Postdoctoral Science Foundation of China (Grant Nos. 2017M620308 and 2018T110750).
In the measurement of the Newtonian gravitational constant G with the time-of-swing method, the influence of the Earth’s rotation has been roughly estimated before, which is far beyond the current experimental precision. Here, we present a more complete theoretical modeling and assessment process. To figure out this effect, we use the relativistic Lagrangian expression to derive the motion equations of the torsion pendulum. With the correlation method and typical parameters, we estimate that the influence of the Earth’s rotation on G measurement is far less than 1 ppm, which may need to be considered in the future high-accuracy experiments of determining the gravitational constant G.
The Newtonian gravitational constant G is one of the most fundamental and universal constants, which is closely related to theoretical physics, astrophysics, and geophysics, while its precision is the lowest so far.[1–5] Although the precision of G has been improved over the past few decades, the values in CODATA 2014 are still in poor agreement because of the extreme weakness and nonshieldability of gravity, which indicates that there may be some systematic errors that have not been discovered or correctly understood in the experiments.[6,7] At present, the best results of G measurement have been given by a group in Huazhong University of Science and Technology (HUST), who reported two independent values of G using torsion pendulum experiments with the time-of-swing method[4,8–10] and the angular acceleration method.[1,11,12] The G values given by these two experiments are 6.674184 × 10−11 m3 · kg−1 · s−2 and 6.674484 × 10−11 m3 · kg−1 · s−2, with relative standard uncertainties of 11.64 ppm and 11.61 ppm, respectively.[13]
The effect of the Earth’s rotation[6,14,15] on G measurement has been roughly estimated to be very small before, but it is necessary to conduct a comprehensive and detailed modeling for it, which may be helpful for the G measurement with higher accuracy in the future. In the G measurement with the angular acceleration method, the effect of the Earth’s rotation has been fully analyzed, but the analysis of this effect in the time-of-swing method is still missing. This work is to solve this problem. In this paper, we present the derivation of the Lagrangian of the torsion pendulum in the general relativistic frame. Based on this, the motion equations of the torsion pendulum can be obtained. With the correlation method[16–18] and the numerical simulation in MATLAB, we extract the periods of the torsion pendulum before and after adding the perturbation brought by the Earth’s rotation, respectively. From the difference of the periods, the effect of the Earth’s rotation on measuring G can be obtained. This study shows that the influence of the Earth’s rotation mainly contributes to the G measurement by coupling itself with the pendulum motion of the torsion pendulum and if the amplitude of the pendulum motion is controlled at the milliradian level, the Earth’s rotation only contributes an uncertainty about 10−3 ppm to the G value.
The outline of this paper is as follows. Section
Let us have a simple review on the principle of the time-of-swing method. The time-of-swing method was proposed by Braun in the 1890s and developed by Heyl, Cohen, and Taylor later,[19–22] which has been widely used to measure G now. In this method, a torsion pendulum is suspended by a very thin fiber, and two source masses are placed on opposite sides of the pendulum, as shown in Fig.
In vacuum, when the source masses are placed around, the motion equation of the torsion pendulum can be written as[23,24]
Here, we consider the influence of the Earth’s rotation on G measurement with the time-of-swing method. This section focuses on obtaining the motion equations of the torsion pendulum containing the Earth’s rotation. In order to achieve this, the Lagrangian expression of the torsion pendulum should be obtained first. We start from the relativistic expression of the Lagrangian density in any curvilinear coordinate system
When we only consider the angle deflection θ, the position vector of the infinitesimal mass dm can be expressed as (x, y, z) = (Lp cos θ, Lp sin θ, −l) in the suspension point frame, in which Lp is the distance between an arbitrary point on the torsion pendulum and the center of the pendulum, and l is the length of the torsion fiber. However, in real experiments, due to the vibrational noise from seismicity or control systems, the torsion pendulum has a pendulum motion in addition to the horizontal rotation. Therefore, the position vector of dm will change. For simplicity, we set δφx(t) and δφy(t) as the angular displacements of the fiber from the vertical position in y and x directions, respectively. With the coordinate transform shown below
Based on the Lagrangian expression in Eq. (
Substituting Eqs. (
To extract the effective oscillation frequency of the motion described by Eq. (
At present, the highest precision of G value is given by the angular acceleration method and the time-of-swing method. However, there are still some systematic errors that need detailed modeling and analysis, such as the Earth’s rotation effect. For the angular acceleration method, the relevant analysis of the Earth’s rotation effect has been completed, but in experiments of the time-of-swing method, we only roughly estimated the magnitude of the effect before. Therefore, we present a more complete analysis and assessment process of this effect here. We derive the motion equations of the torsion pendulum with the Lagrangian expression in the general relativistic frame. After the calculation and simulation, we find that the main effect of the Earth’s rotation contributes to G measurement by coupling itself with the pendulum motion of the torsion pendulum. This effect is far less than 1 ppm, as long as we control the amplitude of the pendulum motion at the level of milliradian. The model we put forward is applicable to other similar gravitational experiments with torsion pendulum, in which the influence of the Earth’s rotation may need to be carefully considered.
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