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There are no established relationships between the Newtonian gravitational constant $G$ and other fundamental constants, so there is no way to derive its value from other quantities. Its value must be measured experimentally, but there have been some surprises. The results obtained in recent years are sometimes in tension to each other (a nice plot can be found in this Nature article). This inconsistency between the results has even caused some speculations and discussions [1],[2],[3].

However, several modern quantum gravitational theories produce expressions that would in principle allow predictions of $G$. I would be interested in the numerical values of $G$ that these theories predict. It would be good to know how these values fit into the landscape of experimental results. And in the future, when better experimental values for $G$ are available, they would help to confirm or refute such theories, which could otherwise be difficult to test.

Are there any numerical predictions for $G$ from modern theories?

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  • $\begingroup$ Why should quantum theory be able to predict this constant? Gravity seems to be a theory of spacetime, while quantum theory starts with the assumption that spacetime exists. If anything, quantum theory and gravity have to follow from a root theory that we don't have, yet, and maybe that theory can predict numerical values, but structurally it is unlikely that quantum theory can, on its own, be used for that purpose. $\endgroup$ – CuriousOne Jun 29 '16 at 1:10
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    $\begingroup$ The two main ones are Loop Quantum Gravity and String/M Theory. The first has not even been able to find yet a classical correspondence limit to general relativity, much less Newton. The second has more free variables than anything you have ever seen, a landscape of variables. Don't think either one predict G. $\endgroup$ – Bob Bee Jun 29 '16 at 2:44
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Are there any numerical predictions for G from modern theories

What are physics theories? They are mathematical models which fit data, measurements in a laboratory or observations with real numbers. If they only fit data, then they are just mathematical maps of reality. To be a theory they also have to predict new setups, they have to be predictive to be a theory.

For a mathematical model to correspond to reality, postulates(laws) have to be posited as strong as axioms , which connect measured values to the mathematics output. Take Newton's laws where G is defined to be measured, and then used to predict new measurements.

newtongravit

You are asking if an esoteric gravitational theory from the microcosm of the constituents can predict a value for G, instead of having it as a measured input.

If a theory of everything is ever discovered that does not depend on measured parameters, like the velocity of light and h_bar then gravity will be part of it and G should come out. At the moment this is a science fiction scenario.

The only candidate for a theory of everything which will include the standard model of particle physics and quantization of gravity at the moment seems to be a string theory. String theories can embed the standard model which has a plethora of constants taken from measurements; in addition there are thousands of possible vacua . Any picked specific model will be full of constants taken from measurements. Thus at the moment there is no predictive power in the proposed theories that could give G . At most G might come out as a function of the other numerous constants, but that is not really a prediction, but a reshuffling of the constants.

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This may or may not answer your question, but it's worth looking at: https://en.wikipedia.org/wiki/Planck_units. By one point of view, units we choose, such as meters, kilograms, etc. are arbitrary, and thus we need to use empirical evidence to adjust our arbitrarily defined units to universal constants such as G.

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  • $\begingroup$ Even if the units we use to prescribe a value to $G$ are arbitrary, it does have some constant value, and the question is asking if we can somehow derive its value rather than just measure it $\endgroup$ – Pawr Jun 21 '17 at 23:22

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