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In the paper "Black holes and entropy" (JD Bekenstein, Phys Rev D 7 2333, 1973), in the section on Geroch's perpetual motion* machine, I'm trying to understand why they can state "its energy as measured from infinity vanishes"?

What they mean is that the work extracted by lowering a test mass ([itex]m[/itex]), "from infinity" (on an ideal string), to the (precise) surface of a black hole event horizon, is exactly [itex]mc^2[/itex].

I'm certainly not disagreeing: in Newtonian gravity, you can obtain infinite work by lowering one point mass toward another, but the point where the work (or - gravitational potential) equals the mass-energy happens to occur at half the Schwarzschild radius, where the approximation is invalid. Intuitively, it seems reasonable as, due to time dilation, the gravitational force becomes (permanently) zero such that it is a conceivable point where "all" of the work has been extracted. Can anyone give a more concrete explanation?

*The proposed machine takes a box "of black-body radiation" from a warm bath, lowers it toward a black hole elsewhere, and allows some energy to escape into the event horizon. The box is retrieved so as to repeat the process of converting heat into work with 100% efficiency, a violation of thermodynamics. The machine fails if the box can't quite be lowered all the way onto the surface (limited say by the radiation wavelength) so the efficiency is slightly less, and in fact turns out less than the Carnot efficiency given the black hole "temperature" (defined analogously with that in classical statistical mechanics, based on the similarity of thermodynamic entropy to event horizon surface area). It's an interesting paper, I'm reading it as a lead-up to Hawking radiation.

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# Black holes and entropy: energy with respect to infinity

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