The paradox of incompressible spatial metric - does it make a perpetuum mobile possible?

UPDATE July 10, 2019: The pressurized vessel perpetuum mobile does not work - energy is conserved. The extra pressure in the vessel, when the mass is removed from the vessel, causes a gravitational pull which offsets the gain from the extra elastic energy E in the fluid. The gravitation from the mass-energy E is small, but the gravitation from the positive pressure in the vessel is large.

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Our previous blog post observed that the interior Schwarzschild metric has the spatial part of the metric independent of the pressure.

The temporal part of the metric does change with pressure, but that does not help if we are trying to squeeze more incompressible fluid into a spherical vessel.

The thought experiments of our previous posts have tacitly assumed that using enough pressure we can change the spatial metric of spacetime. If that is not possible, many strange things may occur, including a perpetuum mobile.

Suppose that we have a dense body of mass M attached to a very strong frame which surrounds a considerable volume of space around M. The mass M has deformed the spatial metric like in the Schwarzschild exterior solution. Let us build a perfectly rigid grid G which models the local spatial metric some distance s away from M. The grid is not Minkowskian (cartesian) but slightly distorted. We attach G and M to the strong frame.

Now, use a strong force to move M suddenly. The metric of spacetime can only update at the speed of light. At first, the metric stays constant at G. There should be no problem in moving M. But when the spatial metric tries to update at G, it cannot do so because pressure cannot change the metric. The force on the grid G grows infinite. We can use this infinite force to generate mechanical energy as much as we wish.

A pressurized spherical fluid vessel with a uniform extra mass m of weights in the fluid

Suppose that we have a vessel whose wall is very rigid, and the fluid inside is almost incompressible. We add a uniform density of weights, with a total mass m to stretch the spatial metric inside the vessel and fit more fluid into it.

Let the locally measured mass-energy of the fluid be L at the start of the experiment. L contains rest mass as well as some elastic energy from the pressure that gravity exerts on the fluid.

https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric

Let r_g be the global Schwarzschild coordinate radius of the sphere. The Schwarzschild radius is r_s.

According to Wikipedia, the Gaussian curvature radius R of a two-dimensional spatial slice fulfills

       R^-2 = r_s / r_g^3.

The larger the mass, the shorter the curvature radius and the more the volume within a radius r_g in the global Schwarzschild metric. The pressure does not affect the curvature at all.

We then slowly remove the extra mass of weights m, keeping the locally measured mass-energy density within the vessel always uniform throughout the volume.

The volume of the vessel decreases.

The rigid wall starts exerting a pressure on the fluid inside. The fluid contracts to fit in the vessel, and the fluid contains at the end of the process a considerable extra amount E of elastic energy.

The metric inside the vessel is at all times close to the Schwarzschild interior metric. The mass m is slowly replaced by E, but E can be much less than m.

The extra pressure P inside the vessel causes extra gravity, according to the Komar mass formula.

We gained an amount V of potential energy when we lowered the weights m into the liquid. How much extra work W we now need to spend to lift the weights out? We need to win the extra gravity exerted on m by E and by the extra pressure P. 

We assume that r_g is much larger than r_s. The work W for m to win the gravity of E and P is much less than E.

Note July 10, 2019: The extra pressure can make a much larger gravitational field than the elastic energy E itself. This is what spoils the perpetuum mobile in this case.

Is it a surprise that general relativity might allow a perpetuum mobile?

ADM and several others have proved that the mass-energy of a closed system cannot change if it lives in an asymptotic Minkowski space.

https://arxiv.org/pdf/1010.5557

Hans C. Ohanian remarked in his paper that ADM and others do not handle "non-minimal" couplings between gravity and other force fields.

If there is a complex relationship between the value of the lagrangian L_M and the spacetime metric in the Einstein-Hilbert action, then the coupling is non-minimal, and energy may not be conserved.

Birkhoff's theorem proves that the gravitational mass of a closed spherical system cannot change. We have to check what assumptions Birkhoff uses about the couplings.

Rubber sheet models of general relativity conserve energy. We need to study what is wrong with the Einstein-Hilbert action. It is very unlikely that nature would break conservation of energy in an asymptotic Minkowski space.

In rubber sheet models, pressure does affect the spatial metric inside a fluid sphere, at least in some configurations.