### Abstract

Periodic oscillations in Newton's constant G are contemporaneous with length of day data obtained from the International Earth Rotation and Reference System. Preliminary research has determined that the oscillatory period of G is ≈ 5.9 years (5.899 ± 0.062 years), which is close to one−half the principle period of solar activity. It will be shown mathematically that the variations in G are concomitant with gravitomagnetically induced torsion on the Earth's spin during the ≈ 5.9 year period.**The gravitomagnetic acceleration at the Earth's equator is in good agreement with experimental measurements and falsifiable predictions are given to test the torsion hypothesis.**

### INTRODUCTION

Measurements
of G oscillate between 6.672
× 10^{−11} and
6.675 × 10^{−11} N·(m/kg)^{2
}with
a periodicity of ≈ 5.9 years (a difference of 10^{-4
}%)
[1, 2]. Scientists studying this anomaly have found that the
variations can be predicted from length of day (LOD) data obtained
from the International
Earth Rotation and Reference System [3].

fig.
1 G/LOD synchronicity: The solid curve is a CODATA set of G
measurements and periodic oscillations in length of day (LOD)
measurements are represented by the dashed curve. The green dot, with
its one−sigma error bar, is the mean value of the G
measurements. The LENS−14 outlier conducted in 2013 by the
MAGIA collaboration was the only measurement which utilized quantum
interferometry, while the other 12 measurements were determined
macroscopically.

It was suggested by Anderson et. al. [2] that an actual increase in G “should pull the Earth into a tighter ball with an increase in angular velocity and a shorter day due to conservation of angular momentum,” which would contradict the G/LOD synchronicity in fig. 1. It can be shown mathematically, however, that this reasonable assumption is not entirely accurate.

The
gravitational field produced by a massive spinning body in Einstein's
general theory of relativity can be described by equations which have
the same form as Maxwell's equations for electromagnetism (in the
flat spacetime weak field limit). These equations are known as the
gravitoelectromagnetic (GEM) equations [4], and the gravitomagnetic
torsion **ξ**
(an effect of frame−dragging) is
equivalent to half of the Lense−Thirring precession frequency
[5, 6],

where
*c*
is the velocity of light in a vacuum and **L**
is the body's angular momentum (according to Mashoon et. al. [6] the
denominator 2 “can be traced back to the spin−2 character
of the gravitational field” as opposed to the spin−1
electromagnetic field in Maxwell's theory).

For
brevity, let us assume the measurements of G are taken at the Earth's
equator so the dot product of **L**
and **r**
vanishes in Eq. 1. By approximating the Earth as a solid ball−shaped
body of uniform density, **L**
can be expanded and we get

where
M_{0}
is the Earth's rest mass, ω is its angular velocity, and ⌀
is its diameter. Since LOD data
indicates ω varies periodically [3],
let us assume fig. 1 is accurate and G
varies with ω. The velocity of light *c*
is a “rigid” constant, so *c*^{2}
can be used as the constant of proportionality and we get

where

(we
will see in a moment that the 2/5 ratio in Eq. 3 is linked to Euler's
number *e*).
In order for *c*^{2}
to remain constant while G and ω oscillate, **ξ**
and ⌀ must also oscillate since M_{0}
is a conserved quantity. It can be deduced from Eqs. 3 and 4 that
a decrease in the Earth's angular velocity ω (an increase in
its LOD) would result in an increase in G, confirming the G/LOD
synchronicity in fig. 1. It can also be deduced that an increase in **ξ**
would pull the Earth into a tighter ball, while an increase in G
would
be contemporaneous with an increase in the Earth's oblateness.

### ANALYSIS OF THE **2/5
RATIO**

Reducing *c*^{2}
to *c* in Eq. 3 we get

where

This ratio is commonly encountered in the realm of electrical engineering with the time constant τ,

where
*e* is Euler's number,
and

where
R, C, and L are the resistance, capacitance, and inductance of a
series circuit respectively, and f_{c}
is the cutoff frequency (2πf = ω). From the relationship

where
ϵ_{0 }and µ_{0}
are the electric and magnetic constants respectively, an alternative
version of Eq. 3 can be given as

The
Earth was approximated as a ball−shaped body of uniform density
in Eq. 3, but Eq.
10 suggests that the rate of change in G' and ω' are dependent
upon the composition
of M_{0}.
If this is an accurate interpretation, G' and ω' may change
dualistically by the Euler factor (1
− 1/*e*^{t}^{/τ}).
Including the Lorentz factor from Einstein's special theory of
relativity, Eq. 10 can be given as

where
ǝ_{1}
and ǝ_{2}
are the Euler factors and γ
is the Lorentz factor,

### CONCLUSION

From
Eq. 11, the gravitomagnetic acceleration **g(r)
**at
the Earth's equator is

Even
though the torsion **ξ**
is minute (≈ 1.012
× 10^{−14} Hz
at the equator) it has a hefty impact on gravity since it scales with
2*c*^{2
}(γǝ_{1}ǝ_{2}
≈ 2/5 when *v*<<*c*
so the torsion is essentially multiplied by 5*c*^{2}
for the Earth's rotation):

(14)
5*c*^{2}
= (5)299,792,458^{2}
= 4.493775893684088 × 10^{17
}m/s.

Plugging and chugging for the gravitomagnetic acceleration proximal to the equator we get

The
CODATA recommended value (2014) for the standard acceleration of
gravity is 9.806 (65) m/s^{2},
which differs from the above approximation by 0.010 (65) m/s^{2}.

As
discussed in Eq. 8, the time constant τ in the Euler factors ǝ_{1}
and ǝ_{2}
is
dependent upon a body's resistance, and superconductors have zero
resistivity [7]. This may explain why the gravitomagnetic
acceleration measured by Tajmar et. al. (2006) [8]
with
spinning superconductors (≈ 6,500 rpm) was at least one
hundred million trillion times greater than what was predicted with
general relativity (≈ 100 millionths of the acceleration due to
the Earth's gravity). As stated by Tajmar, “We ran more than
250 experiments, improved the facility over 3 years and discussed the
validity of the results for 8 months before making this announcement.
Now we are certain about the measurements.”

It is hypothetically possible that the Sun's gravitomagnetic field induces torsion on the Earth's spin during the ≈ 5.9 year G/LOD period (close to one−half the principle period of its magnetic field reversals). If this is true, the Sun's diameter ⌀ and angular velocity ω may slightly fluctuate dualistically at nearly the same rate as the G/LOD variations (hysteresis would be expected due to the Earth's inertia). It was shown by Holme R. and de Viron (2013) [3] that sudden changes in the Earth's LOD are concomitant with jerks in its magnetic field, which may support this hypothesis. The anomalously low magnetic field of Venus may also be linked to its relatively low angular velocity (≈ 243 days in retrograde rotation).

fig.
2 An equation of time graph:
Positive
time values indicate an accurate clock ticking faster than a sundial
and negative values indicate the opposite (in minutes). Annual
variations in G should be detectable with torsion balance schemes.
Taking
into account the Earth's obliquity (mauve dashed curve) and
eccentricity (blue dash−dot curve), the difference between the
annual variations in G (red curve) are
hypothesized to be greatest on the dates marked by the green dots. G
is predicted to be at maximum around 12 FEB and at minimum^{
}around
3 NOV
(assuming the measurements are made proximal to the equator).

**References**

[1]
Zyga L.
2015 “Why
do measurements of the gravitational constant vary so much?”
*Phys.org
http://phys.org/news/2015-04-gravitational-constant-vary.html*

[2]
Anderson
J., Schubert G., Trimble
V., and M. R. Feldman
2015 “Measurements
of Newton's gravitational constant and the length of day”*
EPL Europhysics Letters *Vol.
110, Num. 1

[3]
Holme
R. and de Viron 2013 “Characterization and implications of
intradecadal variations in length of day” *Nature* 499 202

[4]
Mashhoon B. 2003
“Gravitoelectromagnetism: a Brief Review”. *arXiv*: gr
qc/0311030.

[5]
Pfister H. 2007 “On the history of the so-called Lense–Thirring
effect”. *General
Relativity and Gravitation* 39 (11):
1735–1748

[6]
Mashhoon
B., Gronwald F., and Lichtenegger H.I.M. 1999 “Gravitomagnetism
and the Clock Effect” *Lect.Notes
Phys.* 562:
83–108

[7] Bardeen
J., Cooper L., Schriffer J. R. 1957 “Theory of
Superconductivity”. *Physical
Review* 8 (5):
1178.

[8]
Tajmar
M., Plesescu F., Marhold K., & de Matos C.J. 2006 “Experimental
Detection of the Gravitomagnetic London Moment”. *arXiv*:
*gr−qc/0603033v1*.

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