Crossing the brachistochrone- Principle of synchronicity

The idea of synchronicity is an old one, and as far as my own influences are concerned, they are largely based on Carl Jung’s ideas, although his approach to the subject was mostly psychological.

Here I will formulate the basic idea of synchronicity, in the mathematical context of this document, as a set of rules, or principles, and I will explain soon afterwards.

These are the ‘rules:’

There is a one-to-one correspondence between physical phenomena and observation (principle of analogy).
The speed of light is constant (principle of relativity).
All accelerated objects move on brachistochrones (principle of least action).
For a given brachistochrone, the time of the brachistochrone is constant (principle of synchronicity).
The distance which an object travels at the period of a photon emitted by the object, is proportional to the distance the photon travels at the time of the brachistochrone on which the object travels (condition of simultaneity).

Here is a description of the rules:

The first two rules are the same with those in the theory of relativity. The first rule (principle of analogy), in a more physical context, means that the laws of nature are the same everywhere in the universe. This is formally stated as follows:

The principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference.
[https://en.wikipedia.org/wiki/Principle_of_relativity]

But the principle of analogy is a more general rule, which states that that the laws of nature are the same with the laws according to which the human mind functions. A related notion is the anthropic principle- that the universe is as we know it because we are able to know. If we understand the deeper meaning of the previous statement, then we may also understand how consciousness arises in the universe.

With respect to the principle of least action, according to Wikipedia, this is a formal definition: The path taken by a system between times t₁ and t₂ and configurations q₁ and q₂ is the one for which the action is stationary (no change) to first order.

An easier way to understand the principle, according to the same article of Wikipedia, is the following one: Pierre de Fermat postulated that light travels between two given points along the path of shortest time, which is known as the principle of least time, or Fermat’s principle.
[https://en.wikipedia.org/wiki/Principle_of_least_action]

If this is true, and since the brachistochrone is also the tautochrone (path of least time), then all (accelerating) objects will prefer this path. Non- accelerating objects, such as photons, will keep on moving on straight lines.

With respect to the time of the brachistochrone, it is said to be constant in the sense that, given the same brachistochrone, all objects falling on that brachistochrone will reach the bottom of the brachistochrone simultaneously, even if they fall from different heights. A consequence of this is that an object falling from a sufficient height will finally exceed the speed of light. There is nothing to prevent us from supposing so, and the equations presented in this document further support such an assumption.

Comparatively, the principle of relativity and the principle of synchronicity can be given by the following two formulas respectively,

If L and T stand for the length and the time of the brachistochrone respectively, as long as the length L is increased, the time T will also have to increase, so that the speed of light c is kept constant. But if we treat the time T as constant then the speed v of an object will increase as long as the length L of the brachistochrone is increased.

A way to bring together the previous couple of phenomenally incompatible formulas is in relation to the fifth rule. Instead of treating the time T of the brachistochrone as constant, we may treat the length L of the brachistochrone as constant. This can be done supposing that the acceleration of gravity g across the brachistochrone increases, while the length L of the brachistochrone (the distance to be traveled) stays the same. The additional amount of acceleration g, which also expresses the acceleration of the object moving on the brachistochrone, can be provided by the excited states n of the brachistochrone. Thus for a given distance L to be traveled, we have to equivalent expressions,

The two times T and t are not necessarily the same, since the speed c of the photon and the speed v of the object can be different. But if we relate these two times in such a way that at the first state of the brachistochrone (n=1) the two times are the same, then we have

This can be set as an initial condition. More generally, at any state n, it will be

The equation

is a condition of simultaneity, and expresses mathematically the principle of synchronicity.

Another way to formulate the previous condition, with respect to the period τ of the photon, is the following one. If we identify the states n of the brachistochrone with the harmonics of the reference photon, we have that

where λ_n and τ_n stand respectively for the wavelength and the period of the photon, at some state n.

If, for a given state n, we divide the length L, or L₀, of the brachistochrone into a number n of wavelengths λ_n, then, together with the initial condition,

we have an additional final condition

so that we take,

This is a mathematical description of the fifth rule, which relates the speed v_n of the object moving on the brachistochrone at some state n, to the period τ_n of the emitted (the reference) photon at the same harmonic n.

More generally, and also more accurately, the relationship between the properties of the object (e.g. its mass m₀, and speed v_n), and the properties of the reference photon (e.g. its wavelength λ_n, and period τ_n), and also the properties of the brachistochrone (e.g. its length L₀, and mass M_n), are given with respect to the energy of the brachistochrone, as we have already described. For example, in brief, we have for the energy

where the numbers n and N_n express, respectively, the harmonic of the photons, and the number of photons at any state n (if n=1, then N_n=N₀).

Given the fact that the numbers n and N are not necessarily the same, for the time of the brachistochrone we have that

where

For the speed of the object we have

while for the speed of light, as well as its wavelength and period, we have that

Thus we take the following ratios for the two different speeds,

Using the constant length L₀ of the brachistochrone as a guide,

we take the following products, which define the relationship between the speeds and the times,

where the time Δt≡t₀, as we have already mentioned, refers to an external observer, not travelling on the brachistochrone.

If we now suppose that as the speed v_n of the object increases, the number N_n of photons which fit in the distance L₀ is sufficiently large so that it is comparable to the harmonic n of the photons, we have that

If, consequently, we identify the distance L₀ with the photon’s wavelength λ₀ at the first harmonic, n=1, then it will be

If, additionally, we suppose that, as the number N_n of photons which fit in the distance L₀ approaches the harmonic n of the photons, so the time T_n of the brachistochrone approaches the period τ_n of the photons, then we have that

In that form, the last equation compares the period τ_n of the reference photons to the speed v_n of the object, at some higher harmonic n.

Here is another way to derive the previous product in a more general form, using the energy equation of the brachistochrone,

so that comparing the pair,

we take

where

Thus the constant of analogy in the fifth rule is

The condition of simultaneity

also reveals the non- local nature of synchronicity. Although the object can travel faster than light, the photon cannot. In fact the photon doesn’t need to move at all, if it is an oscillation of spacetime (in the same sense that a point on a sea wave oscillates, but it is not transferred by the wave). Such ‘points’ is what the observer measures, wherever he/she goes. Therefore his/her causal relationship with the surrounding environment is never lost.

Whether the disturbances of spacetime which are observed as photons, could also be perceived in the form of some other ‘particle,’ is another question. The point is that the principle of synchronicity sets the limits for an overall description of the problem, and can be the basis for an even more general consideration of the relationship between the observer (and his/her mind) and the phenomenon he/she observes.

A possible mathematical description of Consciousness, and its introduction into the equations, will be attempted later on.