The perpetuum mobile of yesterday's blog post is spoiled by the large gravity of the pressure in the fluid. See the note which we added to the top of the post.
In a rubber sheet model, incompressible fluid in the vessel is able to keep the spatial metric stretched. Then the mass M, which originally produced the spatial stretch and the depression in the rubber, can be moved away.
In Einstein's gravity, pressure cannot keep the spatial metric stretched. Moving of the mass M is prevented by the infinite gravity which the infinite pressure would cause in the fluid.
In a rubber model, we might simulate the Einstein gravity by attaching strings to the rubber sheet and the mass M so that moving M would necessarily straighten the rubber sheet. If the sheet cannot straighten => M cannot move.
Even though we have so far failed to construct a perpetuum mobile in the Einstein gravity, we still face the strange strictness of Birkhoff's theorem: it requires the matter field lagrangian L_M to conserve energy. A rubber sheet model would allow L_M to create or lose energy - the rubber sheet is not a dictator to other fields.
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