Dear Wolfgang Konle
You asked: Do you think that it is by accident that the integral (12) ∆Wfield just has the same value if you insert -1/(8πG) for alpha?
YES. It is a Fallacy.
The point where you created your fallacy is equation 12. That is when you put together three divergent integrals into a single integral and postulate a single r_0, later to be conflated with the r in the Potential Energy calculation.
Self-Energy of Coulomb or Gravitational forces are infinite (cannot be calculated). The example of a "Gravitational Capacitor" is contrived and can only be calculated in the case of electrostatics, where the field goes to zero on the conducting plates and is considered constant between the plates. The energy in the electrostatic capacitor is not being mapped to the self-energy of the EXTRA electrons in the plate, but they should. What you calculate there is the energy of the setup. That always fails when you consider a Coulomb potential.
Even there, if you allow for the existence of charges, the self-energy would become infinite.
So, by accident and carelessness, the difference in "Gravitational Field Energy" becomes "similar" to Potential Energy.
Of course, in the case of potential energy, the value of r is defined by the distance between the centers of mass of the two bodies.
In your case, r_0 has no meaning since, in your case, you are changing the mass of one of the bodies to become M+m. There is no physical process of moving masses or anything defining a geometry.
That is when the Fallacy was born.
From that, you started believing in the existence of a Gravitational Field Energy that is pervasive and not connected to the capacity of producing work (as it is in the definition of Potential Energy).
Since you started believing in your mistake, you conjured up a POSITIVE COSMIC GRAVITATIONAL FIELD ENERGY...
Since the positive energy nature of our Universe is already an unsurmountable problem in Physics (for the garden-variety scientists), adding more positive energy makes NO SENSE.
If Gravitational Field Energy is a real Physics construct, it should be well-defined irrespective of the interacting masses. For example, Newton's Gravitational force is GMm/r^2 irrespective of the interacting masses.
This means that one should be able to define it for each mass (M, m, and M+m). So, one should be able to calculate them independently and then calculate the difference.
Separate the integral into three integrals. One for M, another for m, and another for M+m. This should be done because U_g(M) is a concept that is defined for a mass M, so it shouldn't be dependent upon the other masses.
Call the smallest radii r_0, r_1, and r_2.
Now explain to us why r_0 should be equal to r_1, and r_2?
This is important because only when they are identical, is that one recover Newtonian Dynamics. Would would the radius used to calculate the Field Energy of a hydrogen atom be the same as the radius used to calculate the Gravitational Field Energy of TON618?
The point here is that you forced them to be equal to recover the value of Potential Energy, and that is where your sleight of hand took place.
There is no justification for it.
This is my full argument. Below is a rebuttal to a counterargument.
"Gedanken Experiment" is not an explanation since another "Gedanken Experiment" where the two masses join without just touching each other would give a distinct result. In other words, the Gravitational Energy Field definition should not depend on how the masses joined.
Marco Pereira
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