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To my knowledge, some famous physicists such as Dirac and Dyson (perhaps among others) have advocated for the possibility that Newton's gravitational constant $G_N$ might be "time-dependent".

I have not dug deep into their proposals, but this line-of-thought always appeared awkward to me for several reasons, among which...

1) if we think of G fundamentally as the "proportionality function" in the Einstein field equation $G_{\mu\nu}=\frac{8\pi G_N}{3}T_{\mu\nu}$, then covariant conservation of $G_{\mu\nu}$ and $T_{\mu\nu}$ forces $G_N$ to obey the equation $(\partial_{\mu} G_N) T^{\mu\nu}\equiv 0$, which under a weak energy condition would imply that $G_N$ is constant anyway (At least I think so. Certainly in an FLRW metric with positive density $\rho$, this holds true).

2) More basically, the fact that the sentence "Could $G_N$ depend on time?" involves talking about dependencies on a certain co-ordinate function gives the question a "non-covariant" flavour from the onset. It seems that if the proportionality function $G_N$ from the previous point is not constant, then the most descriptive, comprehenive way of summarizing the law governing its evolution must involve covariant objects and/or mechanism that ought to be expressed covariantly.

3) It is not clear which equivalence principle or other tenets of classical GR are rejected or assailed in these theories.

Hence my question: Can you say where the research program of "time-dependent $G_N$" is roughly situated (in such a way that answers my complaints summarized above)? Is "dependence on time" the most comprehensive way to summarize that research program?

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    $\begingroup$ I think they had in mind something like Brans-Dicke theory, and when they said that $G_N$ might vary with time, they really meant that the value of the scalar field $\phi$ might change over time. $\endgroup$ – tparker Mar 31 '19 at 21:45
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    $\begingroup$ You can also describe Brans-Dicke gravity as a theory in which the units of measurement depend on position. In this form, it's a metric theory rather than a tensor-scalar theory, and the Einstein field equations hold. See Dicke, "Mach's principle and invariance under transformation of units," Phys Rev 125 (1962) 2163 . I have a discussion of units in relativity in section 9.6 of my SR book, which is free online: lightandmatter.com/sr $\endgroup$ – Ben Crowell Mar 31 '19 at 22:01
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Most proposals of "changing constants of physics" actually involve changing dimensionless numbers like $\alpha$ - as you say, merely changing a single constant of proportionality will not actually change physics since it can also be seen as a rescaling of coordinates in a time-dependent manner.

But generally speaking, the changing constants idea does not seem to have got popular traction because there is both a lack of convincing empirical evidence and it does introduce a lot of awkwardness as you point out. It seems that it is generally recognized that varying constants do break the equivalence principle. Some people are willing to bite the bullet and say that if we find that $\alpha$ varies we will have to throw out the equivalence principle, although I suspect there is a quiet majority who just trust the principle on conceptual grounds and hence give a very low prior probability to constants changing.

I do not think there is any unified "time dependent constants" research program. Dirac proposed his idea for very different reasons from the people trying to explain dark matter, do thermodynamic gravity, get rid of inflation, or just poke at the standard model to see how it holds up.

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