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Can anyone suggest a way to measure or rule out a tiny cyclic variation in the earth’s gravitational constant $G_{earth}$, predicted by an alternative cosmological model? It’s an annual cyclic variation of magnitude $1.65\times 10^{-10} G$.

The cosmological model will be briefly described in part 1), in part 2) there is the derivation of the cyclic $G$ effect and in part 3) some ideas already considered to try and measure the effect.

Part 1) The Cosmological Model.

In the model the ‘expansion’ of the universe happens to all distances as in Cosmology - an expansion of all length scales , in this type of expansion all physical quantities change according to the number of length dimensions in them , $Q=Q_0e^{nHt}$.

The redshift was explained due to Planck’s constant changing with time, $h=h_0e^{2Ht}$.

This led to the conclusion that the expansion constant $H$ is half of Hubble’s constant - leading to the encouraging result that the matter density $\Omega_m$ is predicted to be observed as between 0.25 and 0.33 (although really 1.0).

The simple model led to a cosmology giving a good match to data, with no need for dark energy and a cosmological constant of zero, perhaps as Einstein said, it was his “greatest blunder”.

But does it predict anything new?

The cosmology is an apparently static universe, but a consistent cosmology must include General Relativity (or a theory very close) and the Big Bang. The Big Bang is included by the Gravitational constant reducing for dense regions of matter, allowing ejection phenomenon and the Big Bang itself.

Part 2) Trying to understand the cause of gravity.

For a mass $m$, of energy $mc^2$ according to 1) its energy would change as $(mc^2)e^{2Ht}$ and violate conservation of energy…

but gravity hasn’t been included. With gravitational potential energy included, the total energy due to the mass is

$(mc^2-\frac{GMm}{R})e^{2Ht}$

Where $R$ represents the Radius of the universe of mass $M$ and small numerical constants are omitted for simplicity.

Energy can be conserved during the expansion of the type in 1) if inertial and gravitational mass are equal and

$mc^2-\frac{GMm}{R}= 0\tag1$

$$G=\frac{Rc^2}{M}\tag2$$ this is known to be approximately true, why it’s so is known as the flatness problem and has no explanation with traditional Big Bang theory.

The interpretation of this is that gravity is caused by the expansion (of the type in part 1) in order to conserve energy as the expansion occurs.

enter image description here

For a large mass $m$ of radius $r$, with a large self-gravitational energy, (1) would be amended to (3)

$$mc^2-\frac{GMm}{R} -\frac{Gm^2}{r} = 0 \tag3$$

Leading to a reduction in $G$ for dense objects as in Does General Relativity allow a reduction in the strength of gravity?

$$G_m = \frac{c^2}{c^2/G + m/r} \tag4$$

…and for a mass near a particularly massive object, such as the earth near the Sun, $M_S$, it’s (6)

$mc^2-\frac{GMm}{R} -\frac{GM_Sm}{r_{ES}} = 0\tag5$

(where $\frac{GMm}{R}$ now includes all masses apart from the Sun)

$G_{earth} = \frac{c^2}{M/R + M_S/r_{ES}}$

$$G_{earth} = \frac{c^2}{c^2/G + M_S/r_{ES}}\tag6$$

The earth – sun distance, $r_{ES}$, varies between $1.47\times 10^{11}$m and $1.52 \times 10^{11}$m giving a predicted variation in the earth’s $G$ of $1.65\times 10^{-10} G$

Variable $G$ theories (for example Brans-Dicke theory), aren’t ruled out if the expansion is of the type in 1). With the traditional expansion, Lunar Laser Ranging seems to rule out Equation (2) as no change in $G$ has been observed over approximately 40 years, but with the expansion of type 1), no change in $R$ is observable and a continuous change in $G$ would not be measurable.

Part 3) Measuring the prediction.

Is it possible for an experiment to measure such a tiny variation to support the theory or rule it out? Limits on the variation could also restrain other variable $G$ models.

The predicted variation is an annual sinusoidal variation in $G$ of amplitude $1.65\times 10^{-10} G$.

Any comments on the ideas below would be welcome, or any other ideas.

A) Torsion pendulums.
Some of the best results for $G$ are from torsion pendulums, like Cavendish’s - but they have only achieved an accuracy of about 4 significant figures. Since it’s a variation in the product $GM_{earth}$ that matters (assuming the earth's mass is constant) and not the value of $G$ individually, is it possible to achieve the accuracy needed? Perhaps building up an oscillation with resonance over a few years - initial thoughts are that it’s unlikely to work as vibrations and temperature changes would interfere.

B) Measuring the change of a weight. Perhaps suspended by wires and using laser interferometry to measure changes in the extension. But vibrations and temperature changes would also interfere.

C) Lunar Laser Ranging. The predicted variation in the Lunar orbit would be cyclic of amplitude about 6.3cm Apparently there is an annual change in the Lunar orbit, for other reasons, but it’s much greater than the predicted change – so could the separate effects be isolated?

D) Satellite Laser Ranging such as GRACE

enter image description here

Are there any other satellite missions? Apparently there is one planned that has two satellites one ‘above’ the other instead of one following the other.

Some data from satellite missions was found in Matsuo ‘Temporal variations in earth’s gravity’,

https://www.researchgate.net/publication/268983151_Temporal_variations_in_the_Earth's_gravity_field_from_multiple_SLR_satellites_Toward_the_investigation_of_polar_ice_sheet_mass_balance

The data is being interpreted as evidence for changes in distribution of ice or water around the earth. The graph below, from the paper, shows changes in one of the earths spherical harmonics

enter image description here

It’s of the same period and similar amplitude, but can a variable $G$ effect be distinguished from other effects with satellite data?

Are there any other ideas of how to measure, or put limits on, a possible cyclic variation of $G_{earth}$?

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