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The force near a black hole (outside event horizon $r=3r_s/2$) onto a mass $m$ can be calculated by General Relativity:

$$F=\frac{GMm}{r^2}\frac{1}{\sqrt{1-\frac{2GM}{c^2r}}}.$$

However, there must be a distance $r$ where the black hole's gravity basically becomes Newtonian only:

$$F=\frac{GMm}{r^2}.$$

At which distance $r$ does this happen?

Marcus
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    Related post by OP: https://physics.stackexchange.com/q/584522/2451 – Qmechanic Oct 07 '20 at 18:02
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    Expand F in inverse powers of r. Show your thoughts. – Cosmas Zachos Oct 07 '20 at 18:06
  • @Cosmas Zachos: I'm not sure what you mean by this...clearly for a big $r$ the second fraction becomes = 1 and the force becomes Newtonian, but where is the threshold? – Marcus Oct 07 '20 at 18:15
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    Ipso facto, this is what I mean by this: how did you get it into your head there is threshold behavior at all? – Cosmas Zachos Oct 07 '20 at 18:17
  • I mean, when plotting it I get $r = 3r_s$ where the functions tend to overlap. Clearly, $3r_s$ is the minimal radius where circular orbit is possible, but I don't know how to mathematically derive this value. – Marcus Oct 07 '20 at 18:21
  • @A.V.S.: Got that corrected, thx. The question works with static case as well... – Marcus Oct 07 '20 at 19:38
  • @Cosmas Zachos: Not "overlap", but "tend to overlap" = almost overlap, so as an approximation...looking at the function diagram they become very similar indeed. Still, they are around 22.47% off of each other, but it doesn't look like much on the diagram since both values are pretty small compared to the big values at $r < 2r_s$. – Marcus Oct 07 '20 at 21:13

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From your formula, you can see that $F \approx F_\text{Newtonian}$ when $2GM/c^2r \ll 1 $, or if you rearrange, $2GM/c^2 = r_s \ll r$. In other words, the farther from the Schwarzschild radius, the closer you get to Newtonian gravity.

We can take the ratio $F/F_N$ to get an idea of how far off we are from Newtonian gravity. At $r = 10 r_s$, it is $1/\sqrt{9/10}$ which is about $1.054$, so this is 5.4% off from Newtonian gravity.

At $r = 100 r_s$ it is $1/\sqrt{99/100}$ which is about $1.00504$, so this is 0.504% off from Newtonian gravity.

UrsaCalli79
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  • +1, but could you also discuss $3r_s$ as the minimal radius where a circular orbit around a black hole is possible, and why this might be? – Marcus Oct 07 '20 at 19:06
  • You're right to ask why F (not F_N) does not suggest unstable orbits for r<3r_s. I haven't questioned where the formula comes from, but I can only assume that this formula is already an approximation only valid within a certain regime, likely already far from r_s. Even if this is not the case, the force you feel says nothing about the shape of geodesics. At this point, we have to turn to GR in full and we can't rely on approximations. – UrsaCalli79 Oct 07 '20 at 19:09
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Although UrsaCalli79's answer is very effective at explaining this mathematically, I will attempt to explain it in layman terms.

You do not leave either Newtonian theory nor Relativistic theory, it is only that one serves better to mathematically predict the forces. This can be seen in the similarity of the equations. Newtonian theory can predict the forces with a fair amount accuracy, and Relativistic theory with more accuracy.

What UrsaCalli79 mathematically demonstrated was that as you take distance from the Schwarzschild radius $-$ which also can be said to be the start of black hole's singularity $-$ the Newtonian equation can more accurately describe the forces.

To end, this essentially means there is a difference of how accurately one theory is versus the other, this difference in accuracy decreases as $r$ increases.

Broken Admin
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  • But how come that these two functions tend to overlap at $r = 3r_s$? Because this would mean that only when General Relativity becomes Newtonian, circular orbits are possible...and this would actually also prove that Newton mechanics includes only circular orbits - celestial bodies if you will... – Marcus Oct 07 '20 at 18:37
  • ...and this in the other hand allows us to prove that Newton mechanics does not apply within an $r = 3r_s$. So we have a place in space and time which common physical laws do not apply. This is like quantum mechanics, but large scale ;-) – Marcus Oct 07 '20 at 18:43
  • Overlapping between these functions can be looked at similarly as the overlapping of other equations of contrasting theories which attempt to predict forces. – Broken Admin Oct 07 '20 at 18:44
  • Newtonian theory does not accurately predict within $r < 3r_s$, although it can still be used to predict less-than-accurate force values within $3r_s$. It applies, just not as accurately as Relativistic theories. – Broken Admin Oct 07 '20 at 18:47
  • Looking at the data, I don't believe that a reliable prediction can be made with Newton mechanics for $r < 3r_s$, it's rather similar to quantum mechanics. Interestingly enough, we are currently struggling to understand atoms and their quantum mechanics, simply because we can't measure anything anymore. But yet there's an huge object in space that is far away, but Newtonian mechanics won't (really) work with that. So why not start to think differently and look at atoms as black holes, or look at atom cores like at black holes, and start from there...TOE, you know. – Marcus Oct 07 '20 at 19:09
  • Well, a reliable and precise prediction can be made, but it may not be as accurate as the currently accepted theory, that specifically being Relativistic gravitation. Although for $r < r_s$ we don't have a currently accepted theory of quantum gravitation as we do not have a TOE. – Broken Admin Oct 07 '20 at 19:43
  • I'm not exactly sure why you touched on atoms, as we do have a theory concerning their elementary particles $-$ see particle physics and the Standard Model for more $-$ as for the lack of measurements, I assume you are referring to the Uncertainty Principal, which concerns itself with the elementary particles of the Standard Model. – Broken Admin Oct 07 '20 at 19:43
  • Currently we look at atoms as consisting of said elementary particles, and at the forefront there is superstring or string theory which is concerned with the possibility that the elementary particles of atoms are created by said superstrings. – Broken Admin Oct 07 '20 at 19:46
  • The reason why I touched on atoms, is because Bohr used Newtonian mechanics within solar systems to explain the electron trajectory in atoms...or at least the absorption spectrum of atoms, and this worked out very well (btw Bohr was very well aware of this model's drawbacks, unlike some may believe today). So why as well not using relativity in order to explain the workings inside atoms better? Or similar to what Bohr did, using black holes as a model for atom cores, hence whole atom = galaxy? Just some food for thought...especially when it comes to the dark matter question ;-) – Marcus Oct 07 '20 at 20:04
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the orbit will be never closed (remember, the system emits gravitational waves), but will spiral towards the center. The smaller $r$ the longer it takes.

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The general relativity equations have the term $r_{s}/r$.

When $r_{s}/r \approx 0$, then the equations reduce to Newton's law of gravitation.

A six solar mass black hole has $r_{s} \approx$ 10.875 miles (about 17,400 meters) At 93 million miles, $r_{s}/r \approx 1.1694*10^{-7}$, which makes Newton's equations an extremely good approximation at this distance.

Michael Ejercito
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