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Here is a recording of Paul Dirac, talking about dimensionless constants and their significance. He gives some examples of such constants(ratio of the masses of an electron to a proton, the fine-structure constant) and then touches upon the subject of the relative strength of electromagnetic force compared with that of gravity. He says, the ratio of the electromagnetic force to that of gravity is $10^{39}$. The age of the universe(which according to the estimation of their time is the "false" $18$ billion years), When expressed in atomic units of time, is also $10^{39}$. He believes this is more than a coincidence, and hence he developed a theory in which $G$ and the age of the universe are related, where $G$ is decreasing with time, so it's not a constant.

I also read in The Feynman lectures, Feynman talking about the same subject(the relation between $G$ and the age of the universe) but the only difference was that the ratio was about $10^{42}$ not $10^{39}$.

Has any progress been made in working out the relation between this constant and the age of the universe? or Has it been discredited or falsified?

these two questions are related to mine, but I think they're different. (link one, link two).

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Good question. I don't know the answer myself, but I am aware that some so-called constants aren't actually constant, and I can understand why Dirac thought the force of gravity could be reducing in an expanding universe - think rubber sheets and shallower slopes. But IMHO complications arise because the expansion is not uniform, as per the raisin-cake analogy. Space expands between the galaxies, but not within. – John Duffield Jul 12 '15 at 20:29
Related: – Qmechanic Jul 12 '15 at 20:34
The biggest problem with the argument is that gravity is not actually a force, so the ratio of the electromagnetic force and the acceleration of gravity is not truly dimensionless, even though an endless amount of theoretical nonsense has been the result of starting with that false premise. – CuriousOne Jul 12 '15 at 21:48
@CuriousOne What do you mean it's not a force? – Omar Nagib Jul 12 '15 at 22:07
a 2014 paper focused on the fine-structure constant : Planck intermediate results. XXIV. Constraints on variation of fundamental constants – igael Jul 12 '15 at 22:13

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