Unit Acceleration and Dimensionality

[bperet]

I was thinking about how "unit acceleration" (for lack of a better term) would interact with the progression to produce non-uniform motion, per Gopi's discussion on Photons and Electrons.

1. The speed of the progression is unity.
2. An acceleration would have to act upon BOTH aspects of motion, space and time. There could not be a preferred aspect, so the acceleration would have to be in the form of ds/dt.
3. When applied to unit speed, this would change the ratio from 1/1 to 2/2, but would not change the speed, since 2/2=1. This would have no observable effect, so probably isn't the case.
4. In order to increase a speed in space, the spatial aspect must increase but the temporal aspect must remain the same. In order to increase speed in time, the temporal aspect must increase but the spatial aspect must remain the same.
5. Any increase in one aspect is tantamount to a decrease in the other aspect. Therefore, a "unit acceleration" of ds/dt would cause a dichotomy to occur on unit speed:
   • the increase in space would cause a corresponding decrease in time, effectively neutralizing the unit outward motion of time due to the progression, resulting in a speed of 2/1.
   • the increase in time would cause a corresponding decrease in space, resulting in a speed of 1/2.
6. Where the magnitude of motion is insufficient in either case to effect a change of absolute location in either the material or cosmic sectors, we are now confronted with the difficulty of having two, different speeds in the SAME absolute locations.
7. I believe this gives rise to the concept of dimensionality -- 2/1 and 1/2 coexist in two, different dimensions at the same absolute location.

Looking at the dimensional relations, Gopi also states, "So multiplication comes prior to addition." It occurs to me that this is analogous to saying something like, "time comes prior to space," so that got me thinking...

• To increase speed, you add: s + s = 2s.
• To increase dimension, you multiply: s x s = s²

This looks like the same dichotomy we see in projective geometry, where:

• line + line = longer line (linear, 0-degree separation)
• line x line = plane (area, polar, 90-degree separation)

Whereas we are dealing with a scalar system at this point (no coordinates or direction), the "scalar" unit acceleration must also act in both aspects, having both additive and multiplicative effect on speed (analogous to Gopi's pulsation of the photon). The unit acceleration would double and square the scalar speed:



Which creates a non-unit speed in two dimensions, one speed 2/1 and the other 1/2, giving rise to Gopi's "scalar vibration."

[Gopi]

Firstly, the equation does not match up, on the left you have 2(s/t)² but on the right you have the product giving (s/t)².

If one extends that, then for 3/1 and 1/3 speeds to be possible, it must be necessary to cube the speed to maintain the relation, and in general a 1/n*n/1 would require (s/t)ⁿ. Seeing that speeds go through all numbers, but dimensions are observed to be fixed at 3 (or at least, finite for the physical Universe), there might be another connection here being missed.

[bperet]

Firstly, the equation does not match up, on the left you have 2(s/t)² but on the right you have the product giving (s/t)².

I redid my equation using the math 'tex' commands to make it more readable. Where is the mistake? Not seeing it.

If one extends that, then for 3/1 and 1/3 speeds to be possible, it must be necessary to cube the speed to maintain the relation, and in general a 1/n*n/1 would require (s/t)ⁿ. Seeing that speeds go through all numbers, but dimensions are observed to be fixed at 3 (or at least, finite for the physical Universe), there might be another connection here being missed.

That may BE the case, where the system n (s/t)ⁿ, wants to extend towards infinity. BUT, go back to Nehru's equation for dimensional stability: n(n-1)/2 = n, where the only solutions are: 0, 3 and infinity. This may explain why whe have a 3-dimensional universe with yang/point (0) in one aspect, and a yin/plane (infinity) as its dual, and three speed ranges in between. The building blocks then become speeds of 1/3 and 3/1, which have displacements of 2 and (2), respectively--the 2 natural units that define the two halves of the wave. In the yin/planar/polar aspect of the motion, these two halves of the wave would move from 0-pi, then from pi-0, creating a 1-unit electric rotation, rather than a 2-unit linear vibration.

(Also note that Nehru's equation is for INDEPENDENT dimensions. There can be any number of DEPENDENT dimensions.)

[Gopi]

The left hand side has 2(s/t)² while the right hand side has 2s²/2t² where the 2's in the numerator and the denominator cancel and give you back s²/t².

[bperet]

Ah, I was looking on the right side for the problem, and it was on the left. The doubling function would have to affect BOTH aspects equally, so mathematically it isn't 2, but 2/2. I corrected the original equation.

[bperet]

Going back to Nehru's equation for dimensional stability:



results in THREE possible solutions: zero, 3 and infinity. The "3" is just the classic 2-unit motion (0-1-infinity) in 3 dimensions.

It occurs to me that what this equation also suggests is that it is the nature of dimensional stability to create the duality of zero and infinity. Any fewer than 3 dimensions results in a rollback to zero dimensions (1=2, 0=1, 0=0-stable) and we have the "yang" aspect of motion. Any greater than 3 dimensions results in a roll forward to infinite dimensions (6=4, 10=5, 15=6,... infinity=infinity-stable) and we have the "yin" aspect of motion. The fulcrum of the system is unit motion in three dimensions.

This may also explain why we only have electric (1d) and magnetic (2d) motions, as fundamental motions. There is no 3-dimensional, fundamental motion in the RS, only compound motions generate the 3d, net effect. So what we observe in the electric and magnetic motions is sort of a "dimensional collapse," which makes sense in light of Nehru's birotation concept of dimensional reduction (towards zero dimensions), and Gopi's conjugate, dimensional expansion (towards infinite dimensions) and the Calculus concepts of differentiation and integration, respectively. So what we have is this relationship:

• 0d (absolute location, stable)
• 1d (electric, unstable, seeking 0d--birotation becoming SHM)
• 2d (magnetic, unstable, seeking 1d--solid birotation becoming RV)
• 3d (stable, mass, gravity)

This pretty much defines the three speed ranges of the material sector, 1-x, 2-x and 3-x, or in dimensional terms, 1-0, 2-1, 3-2. It also explains why we have chemical interaction--atoms trying to seek the 3-dimensional range of stability by "joining forces," so to speak.

That begs the question on the "infinity" side... the first thing we see there is that relations in the time region, the atomic configuration space, are 4-dimensional and want to head off to infinity, which may explain the recursive and series-expansion nature that is found in atomic energy levels and spectra. Larson explained this as "time replaces space" so s/t becomes 1/t², etc., to show the dimensional expansion. String theory, dealing with a configuration similar to the RS time region, has even more dimensions.

Wondering if the atomic energy levels are a dimensional increase towards infinity, rather than a geometric arrangement--the projection of such dimensional increase.

[bperet]

I was incorporating these concepts into the latest revision of the RS2 book (one more revision... then I'll update the main site), and it occurs to me that, since the dimensional datum is 3, that the unit speed datum is not 1, but 1³. Mathematically, 1 = 1² = 1³, but conceptually, not the same thing since there ARE units involved:



So anytime you are dealing with mathematical relationships that reference the progression of the natural reference system, there will be 3rd power exponents involved--as in the case of gravitational force.