It is extremely difficult to fetch accurate value of gravitational constant. How about we have a black hole toy and laboratory for it? I am (currently) not a university student and need help for it.
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No. To measure G, you must have independent measurements of mass and gravitational field. The mass of the black hole can only be inferred from its gravity.
John Doty
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This is fundamentally impossible.
In the metric for the black hole, $G$ only ever appears in the combination $GM$. Consequently, there is no way of interacting with a black hole that will allow you to independently measure the mass $M$ and $G$.
The same is true for the dynamics of a black hole binary. G will always be paired with one of the masses (and vice versa).
Not coincidentally, the same is true in Newtonian gravitational physics, which is why we know $GM_{\odot}$ very accurately, but not G.
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I think yes. In general relativity $G$ can be derived from the knowledge of Einstein's gravitational constant $\kappa$ and $c$, the propagation velocity of gravitation waves. See my answer to the question Is the Planck force a truly "Planck unit"?.
JanG
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