Alan Guth gives a thought experiment to show that a gravitational field has negative energy. (See the picture below.) Consider a thin spherical shell of elastic, compressible matter, of radius $R_o$. Inside the shell, there is no gravitational field. Outside the shell, the gravitational field is the same as if all the mass were concentrated at the center.
$\quad\quad$ Attach ropes to points on the shell, connected to turbine-generators. Allow the shell to slowly collapse under gravitational attraction, doing work on the turbines. When the shell reaches a smaller radius $R_i$, there are three regions:
$\quad\quad(1)\,\,$ In the region $r<R_i$ the gravitational field is unchanged (zero).
$\quad\quad(2)\,\,$ In the region $r>R_o$, the gravitational field is unchanged.
$\quad\quad(3)\,\,$ In the region $R_i < r < R_o$ there is a new gravitational field.
Hence, the gravitational field in the spherical annulus has negative energy, equal to the work done on the turbines. This is not potential energy, but actual (absolute) negative energy.
QUESTION: The energy density at radius $r$ can easily be calculated to be $-GM^2/4\pi r^4$, where $M$ is the mass of the shell. This is a Newtonian gravity calculation. In general relativity, however, there is no such concept as conservation of energy-momentum, since we do not have a conservation law $T^{\mu\nu}_{\,\,\,\,,\nu}=0$ but rather $T^{\mu\nu}_{\,\,\,\,;\nu}=0$.
$\quad\quad$What happens to this thought experiment in general relativity? Presumably, one must work with the Schwarzschild metric.
