Then only the topmost photons do not observe any pressure in the fluid. The photons at the bottom of the shell do observe the fluid already contracted. They see approximately the final, spatially flat metric of the vessel. In particular, the bottom photons see the full gravity of the pressure in the final metric of the vessel.
The photons, on the average, see half of the final gravity caused by the pressure. If we would move the massive shell very slowly upward, the result would be roughly the same.
The photons cannot escape the change of the metric which the photons themselves create.
The construction of a perpetuum mobile in the Einstein equations turns out to be hard. We need to check what types of lagrangians does the conservation of the ADM mass allow. It looks like ADM handles a pressurized vessel, after all, and the coupling between gravity and the matter fields is minimal for a pressurized vessel. What kinds of matter field lagrangians would be nonminimal?
We also need to study the differences between the Einstein model and rubber models. If we fill the pressurized vessel with incompressible fluid, then we cannot move the mass M at the center at all. Any movement would raise the pressure infinite and the infinite gravity of the pressure prevents M from moving from the center. The infinite gravity would also stop the locally measured time inside the vessel.
In a rubber model, we are able to move the center mass M. It just has to climb uphill from the depression in the rubber sheet and over the incompressible fluid. The pressure in a rubber model is able to stretch the spatial metric.
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Our earlier pressurized vessel thought experiment was based on very slow movements. The perpetuum mobile did not work because the added pressure created a strong gravitational field which we had to fight when we raised the big mass M.
But can we fool nature? Let us put a very strong pipe through the pressurized vessel.
Let us for now make the unnatural assumption that the fluid in the vessel is very lightweight.
We use a bunch of photons in a very strong box as the mass M which stretches the spatial metric. The photons are inside the pipe in the middle of the vessel.
The volume of the vessel has grown because of the mass M of the photons. We filled the vessel to the brim with the fluid which is almost incompressible.
Let us then open the sides of the photon box. The photons escape at the speed of light.
The pressure in the vessel starts to build up and grows very high. But the photons do not know anything of the changed gravitational field of the vessel because they are escaping at the speed of light. For the photons, the vessel might as well be pressureless.
The photons escape. They lose the exact same energy when climbing up from the gravitational well which they gained when we packed them into the box.
We are left with a vessel with a huge pressure in it. The pressure can do a lot of work which came out of nothing. Conservation of energy is broken.
Analysis of the requirement of "lightweight" fluid
Let us analyze the impact if the fluid has a considerable mass. What if the mass of the extra fluid spoils the perpetuum mobile?
The start configuration of the experiment is that we have a diffuse gas of photons at the infinity and some extra fluid at the infinity.
At the origin we have the vessel full of fluid and the box for the light.
We use a pulley P to lower the extra fluid down from the infinity and gain some energy E.
The newtonian approximation tells us that the photons lose the same energy E when they climb back to infinity, compared to the case where we would not have lowered the extra fluid.
The non-zero mass of the fluid does not change the fact that we have a perpetuum mobile.
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