The Quantum-Geometer

Tuesday, June 18, 2013

Determining Positions and Trajectories of Gravitationally Interacting Objects

The laws of motion we have explained in earlier articles can be used to calculate and predict the position, speed and trajectory of an object from any initial state.

Let’s consider the simple case of an object $b$ interacting gravitationally with an object $a$ . We know from a previous article that for any object, changes in momentum must be multiple integers of its mass. That is, $\Delta {{P}_{b}}=x{{m}_{_{b}}}$ where $x\in {{N}^{+}}$ . This implies that changes in momentum of an object $b$ due to gravitational interactions occur at positions that are at distances from the center of gravity of $a$ such that $\left\| {{{\vec{P}}}_{b}}+\vec{G}\left( a;b \right) \right\|={{P}_{b}}+x{{m}_{b}}$ . We will call these positions transitional positions.

The spacing between the transitional positions along the trajectory of $b$ depends on its mass. The greater ${{m}_{b}}$ , the closer the transitory positions will be. Based on the QGD gravitational interaction equation, $G\left( a;b \right)={{m}_{a}}{{m}_{b}}\left( k-\frac{{{d}^{2}}+d}{2} \right)$ , we see that the spacing between the transitional positions is very nearly proportional to the square of the distance between $b$ and $a$ (see figure below).

Gravity induced changes in momentum occur at transitional positions are always equal to ${{m}_{b}}$ which corresponds to changes in speed that are equal to $\frac{{{P}_{b}}+{{m}_{b}}}{{{m}_{b}}}$ or $\frac{{{P}_{b}}}{{{m}_{b}}}+1$ . It follows that the speed of $b$ between two subsequent transitional positions ${{p}_{i}}$ and ${{p}_{i+1}}$ is constant and equal to $\frac{\left\| {{{\vec{P}}}_{\left( b/i \right)}} \right\|}{{{m}_{b}}}$ .

The momentum vector providing both direction and the speed until it reaches the next transitional position, the previous or subsequent locations of $b$ can be calculated from an arbitrarily chosen initial position ${{p}_{i}}$ . It is important to remember that classical time being a notion that has no physical meaning the distance $b$ travels is relative not to the mathematical dimension of time but relative to a chosen number of preon(+) leaps of any given preon(+) chosen as reference. The time reference is therefore replaced by a distance reference. For example, we do not talk about the distance an object $b$ travels over $n$ units of time, but rather the distance it travels will over $l$ preon(+) leaps. That is, $d=\frac{{{v}_{b}}}{c}l$ where $d$ is the distance travelled, ${{v}_{b}}$ the speed of $b$ , $l$ the number of leaps and $c$ the momentum of preon(+), which is also equal to its speed. Note that the although the reference we have used here plays a role similar to that of the classical notion of time has played in physics, it differs from it in that is based on a fundamental aspect of physical reality; the preon(+) leap which is the fundamental unit of distance. Thus, a fourth dimension, even a conceptual one, is unnecessary. Reality can be fully described without the concept of time or the purely mathematical dimension of time.

Application at the Newtonian Scale

As we have seen above, the QGD laws of motion allows us to know the momentum, speed, direction at any position along the trajectory of a body. To better illustrate the law of motion, we will consider the classic example of a body for which $\displaystyle {{\vec{P}}_{\left( b/0 \right)}}=0$ , that is, $b$ is a free falling body.

Since $\left\| {{{\vec{P}}}_{\left( b/i \right)}} \right\|=\left\| {{{\vec{P}}}_{b/0}} \right\|+i{{m}_{b}}$ , the change in momentum between two transitional positions ${{p}_{q}}$ and ${{p}_{r}}$ is equal to $\left( r-q \right){{m}_{b}}$ , and total acceleration is equal to $r-q$ . But since the distance between two transitional positions is proportional to the difference between the squares of their distances from the center of gravity of $a$ , if we set ${{p}_{q}}$ at the center of gravity of $a$ then the change in momentum between ${{p}_{q}}$ and ${{p}_{r}}$ is given by

$\displaystyle r{{m}_{b}}={{m}_{a}}{{m}_{b}}\left( k-\frac{d_{q}^{2}+{{d}_{q}}}{2} \right)-{{m}_{a}}{{m}_{b}}\left( k-\frac{d_{r}^{2}+{{d}_{r}}}{2} \right)$ and since ${{d}_{q}}=0$

$r{{m}_{b}}={{m}_{a}}{{m}_{b}}k-{{m}_{a}}{{m}_{b}}\left( k-\frac{d_{r}^{2}+{{d}_{r}}}{2} \right)={{m}_{a}}{{m}_{b}}\left( \frac{d_{r}^{2}+{{d}_{r}}}{2} \right)$ , the speed at ${{p}_{q}}$ is equal to ${{v}_{r}}=r={{m}_{a}}\left( \frac{d_{r}^{2}+{{d}_{r}}}{2} \right)$ . Thus the rate acceleration of a body is proportional to the mass of $a$ and to the square of the distance of between its initial position and the center of gravity of $a$ , but is independent of the mass of $b$ , which explains why two bodies will have the same acceleration regardless of their mass. This is consistent with Newton’s law of gravity, which at the scale it is applied to, distances in leaps are large enough so that $\frac{d_{r}^{2}+{{d}_{r}}}{2}\approx \frac{d_{r}^{2}}{2}$ . Newton’s law of gravity emerges from the principles of QGD and corresponds to an approximation of its gravitational interaction equation.

The distance between two successive transitional positions is proportional to the difference between the square of their distances from the center of gravity. This is also consistent with the inverse square of the Newtonian equation (see figure below).

Conclusion and Experimental Prediction

As we have seen, according to QGD, an object interacting gravitationally doesn’t go through the infinite number of infinitesimal speed increments implied by Newton’s gravity equation. An object accelerates only at transitional positions. At non-fundamental scales, the relative distance between transitional positions is small and the acceleration of an object appears continuous. But the closer we get to the fundamental scale, the more evident it becomes that the acceleration is discrete rather than continuous. This is consistent with observation (see quantum leap).

Based on the notions we have introduced in this section, the discrete acceleration of objects at transitional positions should be observable. Data from a free fall experiment using an object coupled with a precise enough accelerometer should show that the speed of an object changes at transitional positions and that between them, its speed remains constant.

Sunday, May 26, 2013

The Dark Matter Effect

The subject of dark matter is probably one of the most intriguing in physics today. Hardly a day goes by that doesn’t have someone claiming to possess the theory that explains dark matter. Dark matter, or should I say the dark matter effect, is the subject of so much speculation and theories (most of which are mutually exclusive) that the last thing I wanted to do was to add to the noise which is why I have referred to it only within the larger context of gravitational interactions.

Another problem, if you can call It that, is that QGD ‘s explanation of the dark matter effect is too simple. The effect emerges naturally from QGD’s postulates. In fact, dark matter is at the very core of quantum-geometry dynamics. You see, if quantum-geometry dynamics is correct, the dark matter effect is simply the macroscopic effect of free preons(+). In other words, dark matter is made of free preons(+).

We have described preons(+) has being the fundamental particle of matter in detail. Preons(+) form all other particles, including photons. Individually, they interact orders of magnitude more weakly than the even the least massive photons, which is why no instruments can detect them directly, but over sufficiently large regions of space, their collective mass is sufficient to gravitationally interact with and affect the behavior of light and massive structures.

Dark matter, contrary to beliefs, is not dark. Dark, by definition, is said of something that does not emit light. QGD contends that dark matter has been observed and studied for nearly five decades. You see, according to QGD, the only matter that existed in the primordial universe was in the form of preons(+) which were uniformly distributed throughout quantum-geometrical space. We’ll call this state, the isotropic state, one in which nothing existed but dark matter.

During the isotropic state, preons(+), as a consequence of the attractive force acting between them, started to form the simplest of all structures; neutrinos and photons. And because preons(+) were distributed isotropically, so were these newly formed photons. These isotropically distributed photons have been discovered in 1964 by Arno Penzias and Robert Wilson and called the comic microwave background radiation.

A number of theories can satisfactorily describe physical phenomena and at the same time be coherent, consistent with reality while being mutually exclusive. Mutually exclusive theories can’t all be right so the ultimate test, the only valid test of a theory is the predictions that it makes that are original to it and can be verified experimentally or observationally. So what original predictions can be drawn from QGD that can be tested in the real world? And how do can we know that QGD is correct in its description of dark matter?

One of the most obvious implications of QGD is that sufficiently large regions of quantum-geometrical space (minimally the size of a small galaxy) should contain the same amount of preons(+), or, since the preons(+) is the fundamental unit of mass, have the same mass. That is, ${{m}_{{{R}_{1}}}}={{m}_{{{R}_{2}}}}$ where ${{R}_{1}}$ and ${{R}_{2}}$ are regions of the same volume (the volume being defined quantum-geometrically as the number of preons(-) it contains).

Also, the mass of any regions of space is the sum of its free preons(+), ${{p}^{\left( + \right)}}$ , and its bounded preons(+), ${{p}^{\left\langle + \right\rangle }}$ , that is: $\displaystyle {{m}_{{{R}_{i}}}}={{m}_{p_{i}^{\left( + \right)}}}+{{m}_{p_{i}^{\left\langle + \right\rangle }}}$ where $\displaystyle {{m}_{p_{i}^{\left( + \right)}}}$ and $\displaystyle {{m}_{p_{i}^{\left\langle + \right\rangle }}}$ are respectively the mass of free preons(+) which form dark matter, and bounded preons(-) which for visible matter. To give an example, a region which may appear to be empty must have the same mass as a region of comparable size that is occupied by a galaxy or galaxies. The difference being that in the latter a great number of preons(+) are bound, hence concentrated, in material structures.

QGD Prediction

From the above, since the intensity of the CMBR within a region of space must be proportional to the number of free preons(+) it contains and inversely proportional to the amount of visible matter, the more visible matter a region contains, the weaker the CMBR should be. QGD predicts an inverse correlation between the amount of visible matter and the intensity of the CMBR. Thus, a sufficiently detailed CMBR map is expected to provide a snap shot of the distribution of free preons(+) or what scientist call dark matter.

Dark Matter and the Pioneer and Mercury Anomalies

When taken into account, the dark matter in our own solar system provides a simple explanation of the Pioneer anomaly and the perihelion precession of Mercury.

Supporting Observations

Interested readers may find some the descriptions of supporting observations in the following articles.

http://en.wikipedia.org/wiki/Low_surface_brightness_galaxy

http://en.wikipedia.org/wiki/VIRGOHI21

http://arxiv.org/abs/1010.5783

For who is willing to do a little bit of research, there is an enormous amount of observational data that supports the QGD’s explanation and predictions about the dark matter effect. A more extensive list will be provide in the second edition of Introduction to Quantum-Geometry Dynamics.

Monday, January 28, 2013

Update of Introduction to Quantum-Geometry Dynamics

There has been much progress in the six years since a draft of Introduction to Quantum-Geometry Dynamics has been made available. The blog reflects some of the progress made and is generally more up to date.

One important, if not essential change, is the shift away from continuous geometry and mathematics, which for lack of better tools were used to approximate interactions in discrete space as defined by quantum-geometry dynamics. The upcoming update of Introduction to Quantum-Geometry Dynamics will make use combinatorial mathematics and finite math exclusively and by doing so will avoid much of the pitfalls associated with the assumptions that space is infinite and infinitesimal.

The updated version will be available in the next couple of months. In the meanwhile, I will post new articles that will further expose the ideas of QGD.

Thank you for your patience and understanding,

Daniel L. Burnstein