What is the best way to amend General Relativity to include a variable gravitational 'constant' $G$, that depends on the positions of all other masses?
That is, if the amount of 'bending of space-time' caused by a mass $m$ depends on the proximity of all other masses $m_i$ and has an individual $G$ to keep
$$mc^2 = \sum_{i=1}^nG\frac{mm_i}{r_i}\tag1$$ always true.
Where the sum is over all other particles in the visible universe, up to a horizon $c/H$, where $H$ is the Hubble parameter.
The advantages of such a theory are that it naturally explains the flatness problem of cosmology and the 'coincidence' of the 'Large Number Hypothesis'.
It allows conservation of energy for a change of length scales as $$mc^2 - \frac{GMm}{R} = 0 \tag2 $$ where $M$ and $R$ represent the mass and radius of the universe.
Some links describing the motivation are at the bottom.
There seem to be two approaches, can the first be used?
- In
$$G_{\mu\nu} + g_{\mu\nu}\Lambda =\frac{8\pi G}{c^4} T_{\mu\nu}\tag3$$
change $G$ to $G(r)$, defined for each position $r$ by 1), so 3) becomes
$$G_{\mu\nu} + g_{\mu\nu}\Lambda =\frac{8\pi G(r)}{c^4} T_{\mu\nu}\tag4$$
What's the proper way to express the idea in tensor language?
Or 2.
Is a Scalar Tensor Theory approach needed, using
$$S=\frac1c\int d^4x\sqrt{-g}\frac1{2\mu}\times\left[\Phi R-\frac{\omega(\Phi)}{\Phi}(\partial_{\sigma}\Phi)^2-V(\Phi)+2\mu\mathcal L_m(g_{\mu\nu},\Psi)\right]\tag5$$
from Scalar Tensor Theory?
Any advice on this would be appreciated.
Links:
Reduction in the strength of gravity
