Better rubber sheet model of gravity: prevent horizontal movement of rubber

We have a tense horizontal rubber membrane. We can put weights on it, and springs between weights. The springs model pressure.

    ---|-------|-------|-------|----  tense rubber sheet

 

      steel spikes prevent

      horizontal movement

The new feature is that we have a steel spikes which prevent any horizontal movement of the rubber sheet. Only a vertical movement is allowed.

We may imagine that there are holes in the rubber, reinforced with steel rings. The spikes go through these rings.

The spikes prevent longitudinal waves in the rubber sheet. The model is now closer to general relativity. Only the vertical elevation of the sheet matters.

Birkhoff's theorem now holds? No

 

        ----___          ___----  rubber sheet

                   • • • • 

              particles in 

        circular formation

Recall an example from November 2023. We suddenly increase the pressure inside a circular mass on the rubber sheet.

The pressure starts to stretch the rubber sheet horizontally within the circle, and succeeds in that, too. It pushes the outer parts of the circle upward, trying to make the rubber area within the circle as large as possible.

The rubber sheet is lifted up outside the circle. Birkhoff's theorem fails because the vertical shape of the rubber sheet changes outside the circle.

Can we decouple the metric of space from the metric of time?

The model differs from general relativity in that that the metric of time (vertical elevation) determines the metric of space uniquely. Can we decouple these?

In our previous blog post we have serious problems varying the metric of time and space separately in general relativity. Is it so that they cannot be decoupled?

Pressurized vessel: removing suddenly the negative pressure in the skin of the vessel

Let us have a ring of particles lying on the rubber sheet. We put squeezed springs between them inside the circle, modeling a positive pressure there. Stretched springs between the particles balance the forces and make the system static.

                         particle

        -------•                •--------  rubber sheet

                  \______/

            positive pressure

                   in a "pit"

   

There is a sharp bend in the rubber at the skin of the vessel. In the diagram, the bend is at the particle. The pressure inside the ring pushes the rubber sheet a little down inside the ring. There is a pit.

We suddenly remove the stretched springs, i.e., we remove the negative pressure. The positive pressure starts to accelerate the particles horizontally outward, and also vertically up.

If the particles are very heavy, they accelerate slowly. The pit and the bend may linger there for a long time.

In general relativity, pressure is a "charge". It affects the metric. A sudden loss of a charge confuses the Einstein field equations. They may fail to have a solution.

In our rubber model, a sudden loss of a pressure is no problem. We can calculate the development of the system without any problem.

The negative pressure between the particles simply acted as a force which keeps them static. It was not any "charge" which would affect the shape of the rubber sheet instantaneously.

If we suddenly remove the pressure inside the ring, what happens?

If there are particles also inside the ring, then their movement prevents any abrupt movement of the rubber.

In the rubber sheet model, pressure is not a "charge" which creates a dent in the rubber sheet. The process is more complex.

A "charge" is something which cannot be created or destroyed. The mass of the weights in the rubber sheet model is a charge.

Conclusions

We have worked very hard to find a rubber model which would reasonably approximate general relativity – in vain.

In the previous blog post we were unable to get the Einstein-Hilbert action to work on the Schwarzschild interior metric. This casts a doubt on the correctness of the action.

Our own Minkowski & newtonian model claims that spacetime cannot be "bent" or stretched. The effects of gravity are due to newtonian gravity and the inertia changes of the test mass (or photon) in the gravity field. Then it is unlikely that a rubber model can imitate gravity.

In our own gravity model, the lagrangian density is quite complicated because it has to take into account the changes in inertia, and the kinetic energy stored into the field when the inertia of a test mass m grows.