Re: Double counting gravitational potential energy

After some helpful discussions on the Physics Forums, I think I now
have a much simpler illustration of how the paradox arises, involving
only one body, so it can be matched with the Schwarzschild solution.

Consider assembling a thin shell of mass m and radius r by lowering
the mass bit by bit from infinity (in a spherically symmetrical way),
extracting the potential energy in the process. The work done is the
integral of Gm/r dm which is Gm^2/2r. When this is complete, the whole
shell is at a red-shift of the potential (1-Gm/rc^2) so the effective
energy differs from the original energy by Gm^2/r, which is twice as
much. This seems to mean that the shell has lost twice as much energy
as was extracted, hence the paradox.

My suggestion is that the missing energy must have gone "into the
field", which seemed a meaningful thing to suggest in a semi-Newtonian
model, and the amount exactly matches the amount for a similar Coulomb
model except for the sign, so it suggests that the mathematics of this
"field" should be similar to that for electrostatics.

However, the Schwarzschild solution seems to deny the existence of
this energy, in that Einstein's vacuum equations say there is nowhere
it could be outside the shell. So where did it go?

.

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