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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?

  1. 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

Cosmology: An expansion of all length scales

Solution of Einstein's equations for a cosmological model

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    $\begingroup$ There are no "position vectors" in curved spacetime. Separations are path dependent. I can think of other objections, but I think this one will be sufficient to prevent progress. $\endgroup$
    – m4r35n357
    Aug 21, 2021 at 12:35
  • $\begingroup$ @m4r35n357 So to make progress and incorporate a $G$ that varies from place to place could the minimum separation be used? For a variation with cosmological time e.g $G(t)$ is it the denominator of the RHS of equation 3) that should be changed or are you saying that there is no way to incorporate a varying $G$ in GR? $\endgroup$ Aug 21, 2021 at 12:47
  • $\begingroup$ That is all I can do I'm afraid, best wait for someone who can answer rather than comment ;) $\endgroup$
    – m4r35n357
    Aug 21, 2021 at 13:02

1 Answer 1

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(1) We can't replace $G$ with $G(r)$ in Einstein's equation, since this is not going to satisfy the 2nd Bianchi identity in general: $\nabla^aG_{ab}=0$ (this should be satisfied for all metrices), else it will impose severe restriction on possible choice of stress-energy tensor.

(2) See the discussion on development of Weyl geometry by Erhard Scholz (in https://arxiv.org/abs/1111.3220v1 ). However, there are other limitations of scalar-tensor theory, for instance we cannot get usual GR from scalar-tensor gravity by taking the limit $\omega\to\infty$ for traceless fields ($T_{\mu}^{\mu}=0$). ( https://arxiv.org/abs/gr-qc/9902083)

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  • $\begingroup$ Instead of replacing G with G(r) in Field equations, we can identify $\frac{1}{G}$ with scalar field $\Phi$ in Einstein's Hilbert action, so the action should be modified as given by eqn (5) $\endgroup$
    – KP99
    Aug 21, 2021 at 13:44
  • $\begingroup$ Thanks, can $\Phi$ vary with position? When papers talk about limits on $\omega$ they seem to treat it as a constant number, but does the kind of theory in 5) allow $G$ to vary with position? $\endgroup$ Aug 21, 2021 at 14:15
  • $\begingroup$ $\Phi$ is a scalar field which will depend on both space and time coordinates. Under some approximation the space dependence can vanish, which will depend on the model you are dealing with (for example in Mannheim's conformal cosmology, using large distance approximation) $\endgroup$
    – KP99
    Aug 21, 2021 at 15:11

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