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If an object is in orbit around a star, the object has gravitational potential energy that could possibly be extracted. For example, when we perform gravitational slingshots around Jupiter, our spacecraft speeds up and Jupiter drops to a slightly lower orbit -- we convert some of its gravitational potential energy into kinetic energy which we use for our purposes. There is only so much energy that can be extracted before Jupiter would drop so low that it would get too close to the Sun for this to work (e.g. hitting the Sun's surface).
Imagine if the planet were orbiting a black hole instead of a star. In this case the process could be continued for longer. How much energy could be recovered? Can this be expressed as some fixed fraction of the lowered object's rest-mass? Is this the same as what you would get by slowly lowering the object directly into the black hole on an idealised rope connected to a turbine?
lemon gave an excellent explanation of how the gravitational potential energy difference from lowering a mass, $m$, from far away down to the event horizon is equal to half its mass-energy. This would seem to provide an upper bound for a more realistic answer. Here are a couple of additional things I'd like to see taken into account:
(1) The freed energy would get redshifted as it was brought back away from near the black hole. How does this change things?
(2) Below 1.5 times the Schwarzchild radius, there are no circular orbits (circular travel requires outward thrust). This presumably causes a lot of trouble for my method beyond that stage since I had a mass in orbit that I was slingshotting things around.
(3) Below 1.5 times the Schwarzchild radius, ballistic objects cannot escape the black hole unless they are angled away from it (e.g. light starting travelling perpendicular to the centre of the black hole just spirals in). Presumably this causes trouble for my approach since as far as I understand the slingshotting mainly involves ballistic motion roughly perpendicular to the centre of the black hole.
(4) Lower orbits are also faster, so it would seem that some gravitational potential energy goes into speeding up the lowered object, increasing its kinetic energy. This could be a very large amount if it is lowered close to the event horizon, as the orbital speed there is $c$. Thus assuming you can get all the gravitational potential energy out could be a serious over-estimate. Does anyone know how to adjust for that?
This is not a duplicate of How much energy does lowering an object into a black hole generate? as that question is not concerned at all with lowering via orbits, so for instance, my questions (1), (2), (3), and (4) don't apply there. In addition, the brief answer given there appears to be incorrect, as it contradicts the best answer here so far.