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Let d be the distance (in some coordinate system) from a particle to the event horizon of a black hole. For a particle starting at distance d>0 above the event horizon to rise away to a very large (infinite distance) would require an energy E. As the initial distance d shrinks to zero, E rises without limit (to ∞). At d=0, the particle is at the event horizon. Alright, reverse this sequence.....the particle falls back toward the event horizon from a large distance. It would seem that the limit of kinetic energy E acquired by the particle would be infinite as d shrunk to zero. Since this is clearly not possible, the particle should not be able to cross the event horizon (in either direction). The issue remains even if the acquired kinetic energy is radiated away or converted to mass, since the potential well is still infinitely large.

Where is the flaw in this scenario?

Tom McFarland
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  • From the infinity observer's point of view the particle never crosses the event horizon and, since energy is observer dependent, from a falling observer's point of view, the particle energy never becomes infinite. End of giant foam hand show. – CuriousOne Jul 11 '16 at 09:34
  • Fine.....but Curious One, we are not the falling observer. We are looking at these black holes from a large distance. From our viewpoint, the falling object is being gifted with potentially unlimited energy (or mass?) How does this gift affect the parts of the universe at large distances from the event horizon? – Tom McFarland Jul 11 '16 at 19:33
  • If I remember correctly, the most energy one can extract from mass falling into a black hole was a fraction of its rest mass and not even the entire rest mass energy, so your argument that the energy is infinite is not true, even though I don't know what the theoretical calculation of that looks like. I also don't find arguments of the form "It would seem" particularly convincing. They indicate that your intuition on the matter is limited, but that doesn't mean anything for nature. – CuriousOne Jul 11 '16 at 19:54
  • Correct: my intuition is limited, which is why I ask question; I am only a mathematician with interest (but no training) in cosmology. Am I wasting your time? – Tom McFarland Jul 11 '16 at 20:43
  • I also recall a finite limit on extracted energy. That said, however, if only a finite energy can be converted by a fall toward the event horizon, then what prevents the reversal of this scenario? And if a remote observer of this fall never sees the particle crossing the event horizon, then from that observer's perspective (our perspective), are all the black holes in the universe nothing but the original collapsed objects surrounded by in-falling matter just outside the event horizon? Tom McFarland – Tom McFarland Jul 11 '16 at 21:24
  • You are not wasting my time, but you will have to start thinking about general relativity beyond the language of Newtonian mechanics. Even in that case one would have to calculate the integral of Fds, which may be finite, even if F is not. I am kind of surprised that John Rennie hasn't cleared this up with the proper calculation, already. – CuriousOne Jul 11 '16 at 22:32
  • The question of energy exchange in a black hole potential well is not yet answered, but the following discussion on this forum goes a long way to clarify the related question of whether a particle can "fall into" a BH. – Tom McFarland Jul 12 '16 at 08:45
  • (continuation of above comment) I am satisfied with this discussion: http://physics.stackexchange.com/questions/21319/how-can-anything-ever-fall-into-a-black-hole-as-seen-from-an-outside-observer – Tom McFarland Jul 12 '16 at 08:53

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