The Implications of the Sun’s Dragging Effect on Gravitational Experiments

Experimental determinations of Newton’s gravitational constant, Big G, have increased, in number and precision, during the last 30 years. There is, however, a persistent discrepancy between various authors. After examining some literature proposing that the differences in Big G might be a function of the length of the day along the years, this paper proposes an alternative hypothesis in which the periodicity of said variation is a function of the relative periodicity of the Sun-Earth distance. The hypothesis introduced here becomes a direct application of the Kerr Metric that describes a massive rotating star. The Kerr solution for the equations of the General Theory of Relativity of Albert Einstein fits well with this relative periodicity and adequately predicts the arrangement of the experimental G values reported by sixteen different laboratories. Also, the author explains how the Sun disturbs gravity on the surface of the Earth.


Introduction
One of the oldest constants in Physics, the constant G (commonly called Big G) that appears in Newton's Law of Universal Gravitation, is the least precise value reported in official tables. In academia, this situation is justified by stating that the gravitational force is weak. The previous argument, however, is not justified if we consider the very precise experiments that had determined G in the last 30 years. Those results, however, show bar errors depicting a gap in error between scholars thus indicating that one, or more, systematic errors might be present. In 2015, Anderson, et al. [1] proposed a possible correlation between the observed experimental variations of G and the length of the day, which is known to have a period of 5.9 years as shown by the 2013 work of Holme and Viron [2]. However, in 2015, M. Pitkin [3] applying a Bayesian model, concluded that the previous period of around six years is not admissible when a complete table reported by Schlamminger et al. [4], is included in the analysis. Pitkin ran a function with an age ranging from two to eight years (logically covering the six-year periodicity) and showed no strong correlation between data and hypothesis. This letter introduces a simple model to solve this apparent variability in the experimental determination of the universal gravitational constant.
The author assumes that gravitational interactions do not obey the superposition principle. That is, a third body can perturb the interaction between two bodies (or parts of an extended body). This idea is not new in physics. Hawking and Rayner express in [5] that per the Mach's principle, the "Local physical laws are determined by the large-scale structure of the universe". In addition, Pugach [6] reports extensive experimental work indicating that torsion balances are disturbed by the Sun's activity. The work reported in this paper will be limited to create the connection between a well-established theory and the experimental data available. Also, the author will make an evaluation of the possible perturbative parameters that disturbs the determination of Big G.

Model
Let's separate the experimental value of gravitational constant G [G(exp)] into the universal constant Big G and a perturbative effect δG(r) that is a function of the distance to the perturbative source. In other words, G(exp.) = G + δG(r).
Considering that the Sun produces the next biggest gravitational field on Earth (only superseded by the Earth's field) we can redefine the perturbation parameter δG(r) as expressed in Equation (1) where a is the average separation between the Earth and the Sun, r S is the instantaneous Earth-Sun distance, and dG is a gravitational range that must be founded. A combination of theory and experimental data will help to set the number n that is expected to be an integer power (with more probable values  (2) International Journal of Astronomy and Astrophysics Equation (2) was made using astronomical data, and consequently, its parameters are independent of our choice. Parameters in Equation (2) are a unit base of 1, an amplitude of 0.017 that is the Earth's orbital eccentricity e, a period of 1.0 year, and a negative initial phase of 3.2 rad.
The total perturbative influence of the planets and the Moon in G(exp) are too low in comparison with the contribution of the Sun. So, they were ignored in the calculations. However, to get seven significant figures in Big G, it will be necessary to include at least Jupiter.

Direct Observations of G
Over the last 30 years, a good number of experimental points with very precise results defining Big G's value with six significant figures have been accumulated.

Influence of Big G on Terrestrial Gravity
An additional correlation between the author's model and other parallel gravitational measurements connected indirectly with Big G is available. For example, the work done by Nicolas et al. [23] in Grasse, France shows the correlation.

Experimental Approach
The possible variation of the gravitational interaction between any two masses with the distance to the Sun is something relatively easy to corroborate in comparison with the mathematical evaluation of that interaction. Besides that, the variation will not be easy to observe because those values could be many times smaller than the ambient noise if the apparatus is not carefully prepared.
The author's setting included three variations to the Cavendish's balance [24].
The main one is the introduction of two strings versus the one used by Cavendish. The nine-millimeter separation of the two strings works like a mechanical filter that reduces the effect of weak perturbations with low duration of time.
The second variation was the introduction of a microscope because the microscopic displacement of the balance's iteration is stable enough to be observed. The experimental balance has a permanent state of vibration. The technique for this kind of situation is to accumulate enough data to find the average position and its standard deviation. In this case, that method is not efficient. A better method is to record the two points that are farthest from the equilibrium point daily and represent both points in the graph. Of course, the size of the error bar of every point will fit inside the two points that were recorded. Figure 3 has the observations from December the 2 nd to July the 4 th . The negative slope on the graph is matching well with the results reported over the last 30 years.
A bonus from Figure 3 is that the curve shows the superposition of two permanent signals with an average repetition of around ten and six days respectively. Both periodicities can be explained if the nucleus of the Sun is a plasma torus that rotates with an angular momentum of value J. We can associate the ten days with a plasma torus precession of 20 days' period and the six days with its nutation. This idea is in correspondence with the visual information that the Sun's equator is rotating with a period of approximately 25 days.
The General Theory of Relativity (GTR) of Albert Einstein [25] is the necessary tool to understand this problem. The solution of the GTR found by Kerr [26], for a non-charged rotating body, discloses the consequences that can be used to explain all the problems analyzed in this paper. Kerr where the mass of the rotating body is M, and J is its angular momentum. The functions A, B, C, and D were introduced in (3)  The qualitative correspondence between theory and observation moves the author to do a quantitative analysis. Per the Kerr's Metric, a gyroscope spinning inside the field of a rotating mass has a precession at the frequency [27], where G is the gravitational constant, J is the total angular moment of the rotation mass, c is the speed of light, and ρ is the distance to the dragging source.
Starting from Equation (4) where R is the Sun's plasma torus major radius, and α is the ratio of minor radius to the major radius (the value α comes close to 0.01 from the relativistic condition that the velocity of the plasma torus's faster points must be smaller than the speed of light), M is the mass of the Sun, and ρ is the Earth-Sun dis-   The author, correlating its experimental points with the result from [8]- [22], deduces that the Sun's plasma torus plane of rotation makes an angle of 80 degrees with the zenith of the solar system (nutation cover only 2 degrees). Applying that angle in Equation (6), the gravity on Earth should oscillate annually between 100 × 10 −8 m/s 2 and 92 × 10 −8 m/s 2 . The difference between those numbers (8 × 10 −8 m/s 2 ) is close to the 9 × 10 −8 m/s 2 annual variation of g in France, reported by Nicolas et al. [23]. By now, the reader will know that the power of the inverse of the distance in our initial hypothesis becomes n = 3 per Equation (4).
That inverse power is the same that appears in the tidal equation and that maybe is not a numerical coincidence. The analysis of that coincidence will also be discussed in a following paper.
where day N represent the number of the day from January first to July fourth, x is a parameter without units that represent the distance to the Sun ( )

Conclusions
The most influential labs in the world always turn toward the values of the fundamental constants in physics. Not just for trustworthiness but also for safe The author founded the mathematical explanation that covers all the points on the determination of Big G in the General Theory of Relativity under the assumption that the core of the Sun has the form and condition of a relativistic rotating plasma torus. The Sun's core is massive enough to produce visible dragging effects here on Earth assuming its spin suffers precession and nutation.
The experimental work introduced here clearly shows how others failed to null the dragging effects from the Sun by making, half of the time, the gravitational force to be right-handed and, the other half of the time, left-handed because those effects are not symmetric in space and cannot be neutralized. The remaining difference showed up in every result of Big G and if displayed on an annual graph, will create a pattern as an inverse relationship with the distance to the Sun. Modern gravitational theory dictates that the connection between the experimental value of Big G and the distance to the Sun, follows a negative third power law. One of the questions that arises from this research, "Is the dragging effect changing along the years, forcing in that way the variation of the length of the day"? This question serves as an exigency for the opening of labs to make diary controls of the dragging variation. The relativistic nature of this effect could potentially be dramatically unstable. Furthermore, while one might suggest the duration of the day could be an explanation it could also be the case that the inverse result is true. The author will show experimental evidence on a new paper that this dragging function is responsible for the Sun's tidal effect, meaning they are not correlative but rather one in the same.