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Let's say that the gravitational constant changes with time. $~G~\to G(t)$. ​​​​ (It is essentially an isotropic and homogeneous scalar field)

If we were to re-derive the Einstein field equations using the Einstein-Hilbert action, how would this affect the derivation?

The action for this derivation can be taken as,
$$S =\frac {1}{16πG(t)} S_H + S_M$$ where $S_H = \int\sqrt{-g}~R~d^4x$.
In this case, $G$ is only a function of coordinate time, not of the metric, so I'm not sure how to proceed so it is properly included in the metric variation. Could it be absorbed into $S_H$ or $S_M$ somehow before the variation occurs?

I am aware the $\frac {1}{16πG}$ is put in the action so it comes up with the correct constant to match the weak-field newtonian approximation. My question is more how would we deal with a variable gravitational constant, if that were to be discovered, and how we could account for that in the fundamental derivation of general relativity.

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  • $\begingroup$ You may be interested in reading about Brans-Dicke theory. $\endgroup$
    – Andrew
    Commented Oct 13, 2021 at 1:56
  • $\begingroup$ Yes, that is very interesting. It looks like they accounted for a changing gravitational constant by adding a scalar field to the action. $\endgroup$
    – perchlorious
    Commented Oct 13, 2021 at 2:17
  • $\begingroup$ What if the scalar field they added only changed with time, and not with space? How could we continue the derivation? $\endgroup$
    – perchlorious
    Commented Oct 13, 2021 at 2:27
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    $\begingroup$ It's not really consistent with relativity to have a physical quantity only depend on time. How does a value of $G$ outside my light cone know to be exactly the same value as a point at my location? The only way would be to break locality, and therefore causality. $\endgroup$
    – Andrew
    Commented Oct 13, 2021 at 2:41
  • $\begingroup$ that's a fair point. What if we instead said that G is homogeneous and isotropic, and therefore can be treated as only a function of time? $\endgroup$
    – perchlorious
    Commented Oct 13, 2021 at 4:19

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