Gravity, inertia and the Banach-Tarski Theorem
v. 6 n. 32
NOTICE
The possible mutual benefit to quantum theory and set theory mathematics was introduced in the previous Letter, [1] and now limits and possibilites for cosmology are being considered.
Essentially, this theorem of pure mathematics poses that, for instance, a solid ball can be separated into parts, with those parts reassembled into two solid balls of the same size without distorting the parts and without there being any space among the reassembled solid balls.
The weak point in this scene is that the distances of the mathematical points within each of the sub-parts are not clear, but possibly firmed up with the introduction of the uncertainty principle of quantum physics, as discussed.
With this physical modification the mathematics become more and more "constructive" or realizable in the physical world the closer the mathematical points in the sub-parts remain the same distance apart. If this restriction is in place the original ball may be disassembled into parts and reassembled into two balls, but the products would each be smaller than the original, because there must remain the same number of points in the original and the sum of the products; there is a "conservation of mathematical points" implied in this theorem. In this case there is less drama, but more utility. This is one limiting situation.
Another limiting situation occurs when the concept of energy or mass is introduced. Energy or mass enters because of the introduction of the energy-time expression of the uncertainty principle. An average energy of the virtual particles making up each sub-part as defined can be selected so that when the original "ball" undergoes an accelerated expansion, i.e., conventionally, inflation, the ball retains its configuration (instead of separating into two) but becomes larger (after disassembly and reassembly); this is within the purview of the theorem. The theorem overlaps with physics because the mathematical points of the theorem can be considered the centers of the virtual particles of quantum mechanics, for the time being at least.
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This "inflation" occurs because the mathematical points are permitted to be a greater distance apart in each sub-part. In terms of the physics, the virtual particles become larger or their wavelenghs increase so that their energy per particle decreases; potential is being converted to kinetic energy. This may serve as a physical rationale for the distance between mathematical points increasing in the pure mathematics aspect of the situation.
Now the question, what is the limit of the expansion? Exactly how large will the original ball get in the single-ball model, combining the mathematics and physics?
This depends partly on the average initial energy of a virtual particle in the original ball before expansion. If too small, the ball will inflate to some finite size; then undergo gravitational collapse if gravity is assumed fundamentally attractive in this isolated model. Gravity enters the situation because of the finite energy of each virtual particle within the ball, given the mass-energy equivalence. [Note 1] Even though conventionally the single virtual particle mass-energy cannot be measured, the average over a defined sub-part can be selected to have an average measurable mass, i.e., zero-point energy of space.
The initial ball is disassembled and reassembled into a larger ball, according to the theorem. How large in this isolated model? There is nothing outside this model to restrict expansion, and expansion will continue unabated indefinitely if gravity is considered fundamentally repulsive. [2] However, in the real world there is always something outside of an isolated model. Consider another similar ball adjacent to the initial ball, and another, etc. Now there will be resistance external to the initial ball, and expansion will be controlled. Cosmologically, "... In this scene, the direction of acceleration of the Universe and gravity -- whether gravity appears attractive or repulsive -- depends on scale. As originally conceived, neither Newtonian gravity nor general relativity accounted for scale and the inertia of space. [3]
[1] Calculation below the Planck scale from the uncertainty principle and virtual particles? | LinkedIn
[Note 1] In these Letters gravity is taken to be fundamentally repulsive from the outset, rather than accepting the conventional "inflation" hypothesis, where the inflation is said to be short lived, replaced by "coasting," which in turn is replaced by dark energy, with lack of rationale for each transition. [2]