#79. A Cosmological Setting for a General Relativity-Quantum Mechanics Unification

PH

Red, theory; black, fact

Figure 1. The expanding 5-ball

Figure 2. A wave packet

The spacetime of general relativity (GM) is here considered to be an expanding 4D hyperball (4-ball) on the surface of an expanding 5D hyperball (5-ball). The latter is surrounded by subatomic-sized 5-balls ("paramorfs") that can fuse with the big, nearby 5-ball, which is the mechanism by which the latter enlarges. (Technically, a “sphere” is just a surface, with a dimensionality one less than the embedding space. I use “ball” here to refer to the embedding space dimensionality.)

Each fusion event sends out a ripple on the surface of the big 5-ball that travels at the speed of light in vacuum. A sequence of fusions happening in the correct order causes the ripples to add up to a shock wave at some point. At the maximum of the shock wave, the surface of the 5-ball is thrown out especially far into the surrounding emulsion of paramorfs, where it makes contact with yet another paramorf, resulting in yet another fusion event and another ripple, which has the correct phase to add to the shock wave. The result is a self-sustaining cycle that leads to persistence and thus observable particle-like phenomena. The crest of the shock wave may have to pass through a depletion zone to reach the nearest paramorf. This mechanism is based on Born's rule of quantum mechanics. <03-30-24: If wave curvature rather than displacement amplitude determines paramorf fusion probability, then we get something even closer to Born’s rule, which states that the square of the wave function is proportional to the probability of observing a particle. The curvature of a sine wave is not its square, but the resemblance is striking. Perhaps an experimental verification of Born’s rule with unprecedented accuracy is warranted to distinguish the two theories.>

Particle formation is entropically disfavored (requires a precise configuration unlikely to arise by chance) and thus only happens when paramorf fusions are frequent due causes other than the presence of particles. Postulating that spontaneous fusions are more frequent when the curvature of the 5-ball is greater, spontaneous fusions will be abundant when the growing 5-ball is still tiny and thus intensely curved. This would be seen in our 4-ball as the inflationary era of the Big Bang. The surface of our 4-ball would be formed by the stable particles radiating out of our local inflationary zone on the 5-ball into newly-created, blank 4-surface (see Figure 1). This radiation would define the post-inflationary era. Our time dimension would be one of the radii.

The illustrated mechanism of particle creation (see Figure 2) is periodic-deterministic and may account for photons and leptons. The corresponding chaotic mechanism may account for baryons, and the corresponding probabilistic mechanism may account for dark matter. The close relationship we see today between protons and electrons could have been due to their relationship during the inflationary era; the vicinity of one could have served as an incubator for the other. The multitude of expanding spacetime ripples predicted to be around any massive object would comprise the spacetime curvature referred to by the Einstein tensor of the relativistic field equations. 

The asymmetry of the wave packet that leads to the shock wave accounts for momentum. According to special relativity, mass-equivalent energy is just the spacetime component of the momentum along the time axis.

Fixing radius = 1, the 5-ball has the greatest volume of any ball dimensionality. (See the Wiki on “n-sphere”) Thus, this dimensionality could have been forced by some principle of minimizing the radius-to-volume ratio, call it a compaction principle (in a physical, not topological sense), the existence of which is already implied by the assumed ball shape. We cannot invoke gravity here to produce compaction because gravity emerges at a higher level of description than this. A surface tension-like effect related to the permittivity of free space may serve, which is already implied by invoking ripples on the surface. However, mention of ripples implies that the governing differential equation has oscillatory solutions, which seems to also require a medium with inertia, which may be related to the permeability of free space.

A limitation of this theory is that it does not explain the assumed presence of discrete, ancient inflationary zones on the surface of the 5-ball.

03-20-24: Close inspection of the volume versus dimensionality curve for n-balls of radius 1 suggests that maximum volume occurs at a fractional dimensionality somewhat above 5, which looks to be about five and a quarter. Under the compaction principle, this circumstance would lead to a squashed (oblate) 6-ball about one-quarter as thick as it is wide, with greatest curvature at the equator. (Here I am making an analogy with the Earth’s surface, which is an oblate spheroid.) This uneven distribution of curvature would result in the equatorial region losing its inflationary status later than at the poles, suggesting that the universal equatorial region spawned all the particles we can now see during the late inflationary era and that our familiar 3-space corresponds to a line of latitude on the oblate 6-ball travelling steadily toward a pole. This scenario allows the existence of ancient, dilute matter of non-equatorial origin coexisting with our 3-space. <04-11-24: This ancient dilute matter could account for cosmic rays and some of the diffuse cosmic gamma glow. Some of these ancient particles would by chance approach us in our future light cones and would therefore interact with our 3-space as antimatter. The resulting annihilation events would produce gamma rays and neutrinos. Those particles that escape annihilation could potentially re-emerge from our spacetime in our past light cones and at a different point, becoming matter cosmic rays. Cosmic particles following spacelike trajectories may not interact strongly with us, like two waves crossing at right angles, but Born's rule predicts some interaction.>

03-25-24: A second limitation of this theory is that relativity theory denies the existence of an absolute frame of reference, which I have just re-introduced in the form of the surface of a large ball. <04-27-25: Perhaps this limitation can be addressed by showing that the concept of no absolute frame of reference can be replaced with the concept of Doppler-shifted matter.>

Etymology: "warped spacetime," Greek: paramorfoménos chorochrónos, thus: "paramorf."