Dark energy: a balloon model

In December 2025 we had problems finding a reasonable mathematical model for retarded gravitation in an asymptotic Minkowski space. Let us try it again, this time for a spherical balloon model. The spacetime is then topologically different from a plane.

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Can the equation of spacetime be local for a balloon model?

Let us think of the Schwarzschild solution. The solution extends through the entire asymptotically Minkowski space. The central mass M affects locations very far away.

For a spherical rubber balloon, the equation governing its stretching is local, if we assume that the pressure inside is uniform.

Friedmann equations in general relativity look local, but that is only because we assume a uniform mass distribution. Any retardation effects are masked.

A rubber string and a weight

                 rubber string

           ●----------------------------●

      point                            point

Let us draw points uniformly around a sphere, and connect them with tense rubber strings which run on the surface of the sphere. The rubber strings simulate the gravity between the mass points. If the balloon is static, or inflates or deflates at a constant rate, this model looks like an ordinary balloon: the tension in the strings is uniform along the string.

A rubber string and a weight which the string slows down

Let us consider the simplest possible model of a tense rubber string plus weight:

          

                                              force

                                         F  <------ 

         ==| ~~~~~~~~~~~~~~ ● ---> v

    wall         rubber string            M weight

We are interested in what kind of longitudinal waves form in the string when the weight moves to the right while the tension in the string pulls the weight to the left.

Free waves in the string are sine waves. Since the speed of the waves is finite, there is "retardation".

As the weight slows down, it creates new waves. The weight perturbs the string. Calculating the precise form of the wave probably requires a computer. But we are only interested in very crude estimates.

Do binary pulsars prove that there are no longitudinal gravitational waves? Are longitudinal waves confined between masses?

Binary pulsars have confirmed that transverse waves and linearized Einstein field equations explain the energy loss of a binary pulsar up to a precision of 0.1%.

In an earlier blog post we suggested that longitudinal waves must be "absorbed in" by matter quickly, because they cannot propagate in empty space. This hypothesis would explain why binary pulsars match the transverse wave model so well.

Another hypothesis: longitudinal wave effects can only exist in the field between two masses M₁ and M₂ and can never escape to empty space.

Longitudinal electromagnetic waves exist in plasma, but they do not exist in empty space. Could this be analogous to our model of retarded gravity?

***  WORK IN PROGRESS  ***