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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1$\begingroup$ What is a “black hole toy”? $\endgroup$– rob ♦Commented Sep 5, 2022 at 14:03
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1$\begingroup$ Why would having a black hole help? $\endgroup$– PM 2RingCommented Sep 5, 2022 at 14:04
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$\begingroup$ black hole toy is just a black hole $\endgroup$– Tentacles3587Commented Sep 5, 2022 at 14:05
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2$\begingroup$ What exactly is your method for using a black hole to measure the gravitational constant ? $\endgroup$– gandalf61Commented Sep 5, 2022 at 14:30
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$\begingroup$ Humanity currently cannot make black holes at all, and has not yet found any smaller than 3.8 solar masses, or nearer than 5200 light-years. $\endgroup$– notovnyCommented Sep 5, 2022 at 14:44
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3 Answers
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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.
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$\begingroup$ +1. Could you have a planet and the black hole orbit their mutual center of mass, and infer the black hole mass from the radius of its orbit? $\endgroup$ Commented Sep 5, 2022 at 16:02
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$\begingroup$ @mmesser314 How, then, would you measure the mass of the planet? $\endgroup$ Commented Sep 5, 2022 at 16:57
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1$\begingroup$ @mmesser314 No, because there is no analytic solution for a two-body system in GR. So the result would not be of high accuracy. $\endgroup$ Commented Sep 5, 2022 at 20:45
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2$\begingroup$ @safesphere No analytic solution, does not imply no highly accurate solution. Some aspects of orbital dynamics around a black hole have been calculated with thousands (!) of digits of accuracy. $\endgroup$– TimRiasCommented Sep 6, 2022 at 9:56
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1$\begingroup$ @safesphere How much accuracy can be achieved (and through what method) depends the orbital separation of the binary, the mass-ratio, compactness of the object orbitting the black hole. For example, of the object is compact and small enough, its internal composition would not be relevant for high accuracy calculation. All this is moot though, since no matter how accurate the simulation data, you would still not be able to measure G. $\endgroup$– TimRiasCommented Sep 6, 2022 at 16:15
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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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$\begingroup$ I disagree that it's fundamentally impossible, but it's surely not possible in practice. You "simply" need to measure $r_s$ of the BH (eg by determining the photon sphere radius via photon deflection), drop a chunk of matter of known mass into the BH, then measure the new $r_s$. Of course, other sources of gravitational potential must be negligible, and to convert the observer's measurements to Schwarzschild coordinates you have to account for the spacetime curvature in their vicinity (and their velocity). $\endgroup$– PM 2RingCommented Sep 6, 2022 at 17:06
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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"?.
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$\begingroup$ While yes, energy loss due to gravitational waves is definitely proportional in magnitude to $G$, we can barely measure it as is, so I doubt you'll be able to beat cavendish-style experiments that way. $\endgroup$ Commented Sep 6, 2022 at 15:31
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$\begingroup$ I have rather thought of Planck force (~$c^4/G$) measurement. In principle it should be possible to determine $\kappa$ in the strong gravitation limit. Imagine Einstein living at the center of star on the verge to become a black hole. He would develop theory of spacetime (GR) without need or use of Newton's constant. $\endgroup$– JanGCommented Sep 6, 2022 at 18:44