We cannot define "the curved metric of spacetime" for a complex system?

UPDATE October 12, 2021: We added that one can define pressure by varying the spatial metric, and toned down our claims.

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Consider again the pressurized vessel example from yesterday. We have a test particle inside the pressurized vessel. This time the particle carries both an electric charge q and a mass M.

              _______________

             |                            |

             |    ● M, q             |

             |______________|

              pressurized vessel

        charge q ---- electromagnetic interaction ---- atoms in the liquid

        mass M ---- gravity ---- atoms in the liquid

The particle interacts with the atoms in the liquid.

We assume that both the electric charge q and the mass M affect the pressure in the vessel.

We can determine the force F on the particle by varying the position of the particle and calculating the total energy of the system at different positions. There is an effective potential for the particle inside the vessel.

Can we determine which part of the force is due to gravity and which part due to the charge q? 

Usually not. The potential typically is not linear on both M and q. If it is not linear, there is no meaningful way to divide what part of F is due to gravity and what part due to electromagnetism.

One could claim that the contribution of gravity to F is what the Einstein field equations calculate from the pressure, and the rest is due to electromagnetism. This claim sets electromagnetism inferior to gravity.

A complex system under gravity and electromagnetism

         •       •       •          system S

         •       •       •          of many particles

         •       •       •

                    ● test particle M, q

Let us then consider another experiment. There are many particles bound to each other with various forces. Let us call this the system S.

We define that in a volume element, the pressure in the x direction is obtained by varying the spatial metric in the x direction, and calculating the change in the energy of the system.

If we move a test particle with a mass M and a charge q close to the system, then S is deformed, and the energy of the system changes. There is an effective potential on the test particle, and a force F.

Can we this time determine what part of the force is due to gravity?

Again the potential probably is not linear on M and q, and there is no meaningful way to divide the force F between the effects of gravity of M and the charge q.

The Einstein field equations calculate the metric from the pressure, and other parameters. If the potential from pressure varies with q, then the equations say that the gravitational force is different for particles with the same mass M, but a different electric charge q. This breaks an equivalence principle which says that the gravitational force is the same for all particles which have the same mass.

What about setting q to zero but keeping the other system as is, varying the position of the test particle, and claiming that the calculated force on the particle is the force of gravity? This is unsatisfactory because then we ignore possible gravitational effects of the electric field of q.

What if we remove pressure and shear stresses from the stress-energy tensor?

Then the metric of spacetime would be defined solely based on the location and the movement of mass-energy. It is like the electromagnetic field, which is determined by the charges and their movement.

Could this work?

The metric is then different from the metric of general relativity, though, since pressure and shear stresses no longer have an influence on the metric.

Does this affect the Schwarzschild solution outside the central mass? Yes. The metric must be matched with the interior solution, and the interior solution is affected by pressure.

We conclude that it is not a good idea to drop pressure and shear stresses from the stress-energy tensor.

Conclusions

For a complex system we cannot define "the curved metric of spacetime" in a good way. The hamiltonian or lagrangian of the system couples gravity and other forces, and we cannot split the total energy between different fields. We cannot say what is the separate effect of gravity alone.

If we treat gravity as an ordinary field under the flat Minkowski metric, this problem does not arise. There is no need to split the combined effect between different fields.

Under most conditions, gravity successfully imitates a curved metric of spacetime. This is why people in the last century were mislead to believe that gravity is a curved metric of spacetime. The example of the system S shows that the geometric interpretation is not very good for a complex system.