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Singularity as the Strange Attractor

Chaos theory describes how chaotic systems that are apparently disordered have underlying order. Strange attractors in chaos theory are points where nearby trajectories converge to and diverge from it rapidly. The strange attractors and the trajectories ultimately create a kind of order in a chaotic system [7]. The bouncing ball analogy positions the proposed singularity as chaos theory’s strange attractor where pilot waves converge to and diverge from it.

Zero-Point Energy

From here, stochastic electrodynamics will continue to explain the scenario. This theory is based on the assumption that zero-point energy (ZPE) is pervasive throughout space and an integral part of the universe. I have associated this energy to the singularity in previous section. The theory explains many physical paradoxes just by adding the concept of ZPE to classical physics such as inertia, gravity, and the radiation paradox of the Bohr’s atom (based on fluctuating electromagnetic field associated with zero-point energy). According to stochastic electrodynamics, the ZPE field creates a range of superimposed random waves of all frequencies and phases in all directions, with a power spectrum proportional to the cube of the frequency.

Haisch, Rueda, and Puthoff , in the paper “Inertia as a Zero-Point Field Lorentz Force,” showed how inertia (the resistance of an object to acceleration) can be explained by the interaction of particles with zero-point fields. They assumed the following:

A fundamental particle (such as an electron) could be treated as a two-dimensional Planck oscillator driven by electric components (Ezp) of the ZPF to oscillate in the xy-plane. They then examined the effects of the magnetic components (Bzp) of the ZPF on the Planck oscillator under the condition of constant acceleration in the z-direction. The result was that the Lorentz force due to Bzp fluctuations proved to be proportional to the acceleration of the Planck oscillator, thus suggesting its interpretation as the reaction force due to inertia.[8]

Higg’s Field

According to the standard model, subatomic particles are mass-less. The hypothetic all pervasive Higg’s field is supposed to provide mass for the particles. However, the mass is obtained when a particle accelerate or decelerate throughout the Higg’s Field. This definition of the Higg’s field does not define the so-called rest mass which is the mass of a particle at rest or constant velocity. Then a particle at rest or at uniform motion has to be mass-less. Yet the pilot wave that constantly accelerates or decelerates within its wavelength, as described in this model, defines the rest mass of the particle within the Higg’s field even in uniform motion.

Heisenberg Uncertainty

In classical physics we are able to exactly measure a pair of properties an object possesses, such as its position and momentum. On the contrary, in quantum mechanics, if we pin down the exact location of a particle, the measurement of its momentum is uncertain by infinity. The uncertainty principle describes the complementarity relation between two properties of an object where two of them cannot be precisely determined. The more accurately we know about one property the less we can be certain about the other property.

To express the uncertainty between momentum (p) and position (x) of a particle, the Heisenberg equation is written as:
Δp Δx ≥ ħ/2π

Where ħ is the Dirac constant. Δp is uncertainty about the momentum, and Δx is the uncertainty about its position.

In 1922, Louis de Broglie postulated that any particle or object has an associated wave. This postulate is one of the pillars of quantum mechanics.

This model supposes that along the pilot wave, objects decelerate as they leave the x-axis and reach their linear velocity at the peak. Then they accelerate on their way back to x-axis, where they reach the speed of light and disappear. In this model we are denying the presence of mass in a portion of particle’s wavelength. However, while the object is present, the speed variance dictates the magnitude of the probability wave at any particular location.



The above diagram shows the changes in location and momentum as postulated in this model.

Please note that momentum is not fixed. Heisenberg uncertainty about position and momentum necessitates a variable momentum. In this scheme, at the peak, displacement along the x-axis is maximal. When a particle moves down the slope from the peak during its wave function, the changes in position along x-axis (Δx) are diminished. However, the change in its speed, and therefore its momentum (Δp), increases. Around the x-axis, the particle’s displacement is close to zero. At this moment the changes of the momentum is maximal. There will be a time that spatial displacement of the particle is zero (pass the light speed c).

On the other hand, close to the peak of spatial displacement, the particle has the least momentum and changes of momentum per unit of time. At the peak, there is a moment when the deflecting force disappears and the attracting force is about to start. At this instant the particle does not possess movement and therefore no changes of momentum. 

The Hamiltonian Η which represents the energy of a system, (the sum of kinetic, K and potential energy, V) can be written as,
Η = K + V,      K = p2/2m,           V = V (q)

Here q is the spatial displacement along the wavelength and p is the momentum;
Note that in the second Hamilton equation K is a function of p. It means that the momentum of the particle equals the rate at which it loses or gains kinetic energy with respect to changes in its location along the wavelength. When the kinetic energy is zero (an instant at the peak), momentum disappears.

This is kind of realization of Heisenberg location/momentum uncertainty principle. In the section “Mass and Gravity,” I offer a physical description of the Planck constant (h). The Planck constant can be taken as the total energy that is delivered by the particle to space-time during each wavelength. When the particle appears in space-time, its kinetic energy equals (h). At the peak, potential energy equals (h). This description of Planck constant confirms the close match between uncertainty principle and particle-wave behavior in this model.

For more information about chaos theory, please see http://en.wikipedia.org/wiki/Chaos_theory#Strange_attractors 
Bernard Haisch, et al., “Inertia as a Zero-Point Lorentz Force,” Physical Review A 49, no. 2 (February 1994). Accessed at  http://www.calphysics.org/articles/PRA94.pdf 
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