An elastic rubber sheet analogy of gravity and other forces

The electric force pushes or pulls in the plane of the elastic rubber sheet

Let us have an elastic rubber sheet. We can model the pressurized vessel thought experiment by embedding a steel ring into the rubber and embedding small steel disks in the area which is surrounded by the ring. Steel is so strong that we can think of steel objects as perfectly rigid.

As we keep embedding small steel disks into the circular area, the rubber has to bulge and stretch to accommodate more.

Instead of steel disks we could embed positive electric charges. Their repulsion causes the rubber to bend and stretch within the circular area.

We see that the electric force is a direct force between charges and its direction is in the plane of the rubber sheet.

More precisely, we should model the electromagnetic field as embedded in the rubber sheet. The field pushes or pulls on charges. Then the electric force is not a direct force between the charges, but a force between the field and the charges.

Gravity is an indirect consequence of a force which pulls perpendicular to the plane of the elastic rubber sheet

Putting weights, for example, steel balls, on a horizontal rubber sheet is a well known analogy for the gravitational force. An outside force F (in this case the real gravity, not the modeled gravity) pulls them to a direction perpendicular to the sheet. The balls want to roll together because by joining their forces they can stretch the sheet more, and can settle into a lower position.

The "attraction" between the balls is an artifact caused by the interplay between the sheet and the perpendicular force which pulls the balls down. There is no "real" direct attractive force between the balls.

In this model, the electric force and the gravitational attraction are very different forces. This may explain why we cannot find a sensible definition for the energy density of the gravitational field - it is because there is no "gravitational force" in the same sense as there is an electric force.

However, we probably can find a sensible formula for the positive energy stored in the deformation of the 3D space. The total energy is the deformation energy plus the potential energy in the field of the outside force F. We can set the potential of F such that all energies are always positive. We get rid of awkward negative energies.

Cosmological models: the de Sitter space and dark energy

Let us model the expanding universe with the usual expanding rubber balloon model.

Dark energy can be explained as a positive energy of empty space, which in turn means negative pressure. The 3D space wants to contract, to turn into a negative curvature saddle shape, to reduce its energy content. In general relativity, this paradoxically leads to an accelerated expansion of the balloon.

Our planar rubber membrane model above, if generalized directly to a rubber balloon, predicts that a negative pressure would make the balloon to contract. We need to improve our model so that it explains the behavior of cosmological models.

Maybe the effect of negative pressure is not that much to contract the rubber membrane, but to make the curvature of the membrane negative. Locally, that would contract the volume enclosed into a ball of a fixed radius.

Positive pressure tries to increase the curvature of the rubber membrane. Locally, that will make the volume of a fixed radius ball bigger.

We see that locally, the effect of a positive/negative pressure is consistent with our planar rubber membrane model. It is like having an infinite balloon.

What to do to extend the model to balloons of a finite size?

In a cosmological balloon model, energy is minimized only locally, not globally

The cosmological balloon is huge and its parts get information only at the speed of light from other parts. It is possible that when each individual part tries to fall into a lower energy state, the balloon as a whole will develop into a higher energy state.

Positive pressure tends to bulge the balloon locally, that is, to increase its positive curvature. The global effect is that the balloon contracts to make the curvature bigger. A global observer outside the universe will notice that the balloon is traveling towards a higher energy state, but no part of the balloon is aware of that. Minimization of global energy might require faster-than-light communication.

Similarly, a global negative pressure in the balloon causes that each individual part wants to reduce its curvature, which in turn leads to the balloon becoming bigger, and the total energy of the whole universe keeps increasing. In a hypothetical inflation scheme in the Big Bang, the energy of the universe would grow phenomenally fast.

Physics in the Minkowski space seem to conserve energy, and try to divide energy evenly among various degrees of freedom. There is no need for faster-than-light communication to implement this. But if we allow varying geometries of the universe, the maybe it is not possible to conserve energy.

Since every concrete, everyday, rubber sheet or balloon model, which we build, lives in the Minkowski space, it conserves energy in the global view. These models are not very good at modeling the behavior of a cosmos where energy is not conserved.