Let's think that there is a black hole with mass $M$ and matter with mass $m$. Two objects have distance $R$. What is the gravitational potential energy between the black hole and the mass? How can I obtain the value? For the convenience of calculation, few suggestions are followed.

  1. The black hole is a Schwarzschild black hole.

  2. You can use any coordinates you want.

  • $\begingroup$ I suppose you are forcing us to use General Relativity, because the Newtonian approximation would be too easy (GMm/r). Correct? $\endgroup$ Jun 26 '20 at 19:32
  • $\begingroup$ How far apart are the two masses? In other words, what's the ratio of $\text{separation}/\text{Schwarzschild radius}$? $\endgroup$ Jun 26 '20 at 19:56
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    $\begingroup$ If you define the potential energy as the total energy minus the kinetic energy minus the rest energy you can use the definition at notizblock.yukterez.net/viewtopic.php?p=554#p554 anyway this is one way how you can solve for the escape velocity, so set the kinetic energy equal to the negative potential energy, then your particle can escape to infinity $\endgroup$
    – Gendergaga
    Jun 26 '20 at 21:04

If you're looking for a General Relativity answer, I have bad news. From the Wikipedia article on Gravitational Potential Energy, in the GR section:

In general relativity gravitational energy is extremely complex, and there is no single agreed upon definition of the concept.

amd your question would be opinion-based and non-mainstream. If you're looking for an approximate Newtonian answer, then it's

$U(R) =-G\frac{Mm}{r} $

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    $\begingroup$ The Wiki quote applies to dynamic spacetimes. In the static Schwarzschild spacetime, the potential energy is defined perfectly well. $\endgroup$
    – safesphere
    Jun 26 '20 at 20:37
  • $\begingroup$ If this is the case, then I don't know the answer: in all my GR classes, the professors always told us to be aware that anything "potential energy" in GR is a not well defined subject. Care to show the correct answer? $\endgroup$ Jun 26 '20 at 21:09

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