When we remove the mass, the space deformation in the vessel is produced by pressure.
Let us modify the experiment in such a way that the mass is originally in a thin shell of radius r. The mass is then lowered down in small amounts, so that at the end we have a spherical mass of a radius r, and of a constant density. That is, we build from a thin shell a solid sphere of a constant density. We can collect a "binding energy" E when we lower the pieces of the mass down.
The volume of the sphere of a radius r is then slightly larger than the corresponding sphere in the Minkowski geometry. We then put the rigid vessel around the sphere and fill it with weightless incompressible fluid.
After that we start lifting the mass gradually back to its original position in the thin shell. We may assume that the geometry of space inside the vessel stays roughly constant through the process, because the volume of the vessel stays constant.
We have to do a work E' in lifting the mass. A very rough estimate is that E' = 2E. Then the energy E'' of the pressured vessel is roughly the same as E, the "binding energy" of the original spherical mass.
How do we interpret this? While we lowered the mass, we were able to harvest an amount E of the potential energy. At the same time, an equal amount E of potential energy flowed into the deformation of space. When we lift the mass back up, we have to pay back the energy 2E.
Now we have a rough guess of what is the energy of a deformation of 3D space. It is of the same order of magnitude as the gravitational binding energy for a mass which produces a similar deformation.
Most of the energy in a normal mass is in the mass itself. The deformation of space around it carries a very small amount of energy unless we are dealing with a black hole.
This is not much different from an electric charge. If we assume that all the mass-energy of an electron lies in its electric field, then almost all mass is contained within a few classical electron radii, where the classical radius is 2.8 * 10^-15 meters.
If we have a spherical shell made of metal, and which contains some reasonable amount of charge, then the energy of the electric field outside the shell is much less than the mass of the shell. The energy in that field is at a macroscopic distance from the shell, in contrast to the field of a single electron.
Harvesting kinetic energy of a mass through a gravitational wave antenna
Any elastic solid object acts as a gravitational wave antenna. It resists a change in the geometry of the 3D space inside it. A change in geometry will, in general, produce vibrations.
If we have a spherical mass moving by, it distorts the geometry of space with its Schwarzschild metric. It is like a "gravitational wave" which moves at a slow speed.
The Einstein-Hilbert action
The action is
S = the integral over the whole spacetime
(1 / (2κ) R + L_M) sqrt(-g) d^4x,
where R is the scalar Ricci curvature of the metric and L_M is the lagrangian density of matter and other fields. The system tries to find a minimum of the action. The first term with R might be interpreted as some kind of energy of the deformation of space.
In the Schwarzschild solution, S is zero outside the gravitating mass. If we model curved space with a rubber membrane, then obviously the deformation does contain positive energy also outside the mass, since the membrane is deformed there. The first term above cannot be the deformation energy density. It can be the correct total energy over the whole space, though.
If a spherical mass moves by us, we can harvest energy from the outskirts of the gravitational field through the deformation it causes in a flexible object. How do we describe that with the Einstein-Hilbert action? The energy L_M increases in the flexible object in the outskirts of the field. It has to be balanced by a change in the kinetic energy of the mass. It is better to keep the spherical mass static. The flexible object flies by and starts to vibrate. The energy came from the kinetic energy of the flexible object. The object did not "harvest" any energy from the deformation but converted some of its own kinetic energy into vibrations.
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