Thursday, August 15, 2013

A Remarkably Simple Proof of the Discreteness of Space

A postulate of quantum-geometry dynamics is that space is fundamentally discrete (quantum-geometrical, to be precise). Of course, proving this using our present technology may appear to be beyond difficult especially if, as QGD suggests, the discreteness of space exists at a scale that is orders of magnitude smaller than the Planck scale. The task of proving that space is made of preons(-) may even be impossible because, if as discussed in On Measuring the Immeasurable, fundamental reality lies beyond the limit of the observable. That said, in the same article I explain that though preons(-), which according to QGD is the discrete and fundamental unit of space, and preons(+), its predicted fundamental unit of matter, must be unobservable, their existence implies consequences and effects that must be observable at larger scales.

According to QGD, the momentum of a particle or structure is given by \left\| {{{\vec{P}}}_{a}} \right\|=\left\| \sum\limits_{i=1}^{{{m}_{a}}}{{{{\vec{c}}}_{i}}} \right\| where \left\| {{{\vec{P}}}_{a}} \right\| is the magnitude of the momentum vector of a particle or a structurea , {{\vec{c}}_{i}} the momentum vectors of the component preons(+) of a and {{m}_{a}} its mass measured in preons(+). The speed of particle is defined as {{v}_{a}}=\frac{\left\| {{{\vec{P}}}_{a}} \right\|}{{{m}_{a}}} . We saw that when a structure a absorbs a photon b of mass{{m}_{b}} , then its new momentum \displaystyle \left\| {{{{\vec{P}}'}}_{a}} \right\| is given by \displaystyle \left\| {{{{\vec{P}}'}}_{a}} \right\|=\left\| {{{\vec{P}}}_{a}}+{{{\vec{P}}}_{b}} \right\| . We also saw that when a is subjected to gravitational interaction,\displaystyle \vec{G}\left( a;b \right) , the change in momentum \Delta \left\| {{{\vec{P}}}_{a}} \right\| is equal to \displaystyle \left\| \vec{G}\left( a;b \right) \right\| so that\displaystyle \left\| {{{{\vec{P}}'}}_{a}} \right\|=\left\| {{{\vec{P}}}_{a}}+\vec{G}\left( a;b \right) \right\| . This is explained in more details in earlier articles. Now, let us see how QGD’s equations can be applied to explain and predict reality at our scale. To illustrate this, we will apply the QGD’s equations to baseball.

Leta be a baseball andb a baseball bat and let’s look at what happens when the ball, traveling towards the bat at speed {{v}_{a}} is hit by a baseball ball, itself going at speed{{v}_{b}} . Using the definitions above, we know that the momentums of a and b are respectively given by \left\| {{{\vec{P}}}_{a}} \right\|=\left\| \sum\limits_{i=1}^{{{m}_{a}}}{{{{\vec{c}}}_{i}}} \right\| and \left\| {{{\vec{P}}}_{b}} \right\|=\left\| \sum\limits_{i=1}^{{{m}_{b}}}{{{{\vec{c}}}_{i}}} \right\| and their speed by {{v}_{a}}=\frac{\left\| {{{\vec{P}}}_{a}} \right\|}{{{m}_{a}}} and{{v}_{b}}=\frac{\left\| {{{\vec{P}}}_{b}} \right\|}{{{m}_{b}}} . We also know that saw that, if space is quantum-geometrical, any change in momentum of an object must an exact multiple of it mass. That is : \Delta \left\| {{{{P}'}}_{a}} \right\|=x{{m}_{a}} . As a consequence, unless the mass of the bat is an exact multiple of the mass of the ball, it cannot transfer all of its momentum to it. Then x=\left\lfloor \frac{\left\| {{{\vec{P}}}_{b}} \right\|}{{{m}_{a}}} \right\rfloor and \Delta \left\| {{{\vec{P}}}_{a}} \right\|=\left\lfloor \frac{\left\| {{{\vec{P}}}_{b}} \right\|}{{{m}_{a}}} \right\rfloor {{m}_{a}} , where the brackets represent the floor function. Now what happens to the momentum that is not transferred to the ball? It is conserved the bat in the form of momentum, with part of dissipated by friction and distortion.

Now, since \displaystyle \Delta \left\| {{{\vec{P}}}_{b}} \right\|=y{{m}_{b}} and since \displaystyle \left\lfloor \frac{\left\| {{{\vec{P}}}_{a}} \right\|}{{{m}_{b}}} \right\rfloor =0 , the ball cannot transfer momentum to the bat. The ball retains the momentum it had before the impact (with direction reversed) so that the total momentum of the ball immediately after impact is given by \displaystyle \left\| \vec{P}_{a}^{'} \right\|=\left\| {{{\vec{P}}}_{a}} \right\|+\left\lfloor \frac{\left\| {{{\vec{P}}}_{b}} \right\|}{{{m}_{a}}} \right\rfloor {{m}_{a}} and its speed \displaystyle {{{v}'}_{a}}=\frac{\left\| \vec{P}_{a}^{'} \right\|}{{{m}_{a}}} .

This example shows how the equations used to describe the effect of optical reflection introduced in an earlier article are directly applicable at our scale.

Conservation of momentum is described somewhat accurately by dominant physics theories, but different scales use different sets of equations and even at our scale; the explanations are based on principles which, themselves are not really explained (see the Wikipedia article on Momentum). The problem is that the explanation of conservation of momentum provided by accepted physics theories contains a huge fallacy; one that, though it doesn’t affect its practical usefulness, results in one important oversight. Accepted physics theories can’t describe the acceleration after impact of a baseball, or the acceleration after impact of a billiard ball or that of the steel balls in Newton’s Cradle.

The classical equation for acceleration is given by that equation F=m\frac{\Delta v}{\Delta t}, which relate force F to mass m and acceleration. Using this, how exactly does one determine the acceleration of ball after impact with a bat? To do so within the framework of classical physics, we would need to know {{v}_{a}},the speed of the ball before impact, and {{{v}'}_{a}}, the speed of the ball after impact, and \Delta t the time interval over acceleration occurred. From these we can calculate given by \frac{{{{{v}'}}_{a}}-{{v}_{a}}}{\Delta t}. Measuring {{v}_{a}} and {{{v}'}_{b}} is relatively easy. The difficulty is in measuring the time interval \Delta t? Is it a thousandth of a second, a millionth of a second?

No one really knows, but we can safely assume that the speed of the ball, as soon as it is no longer in physical contact with the bat, is equal to{{{v}'}_{a}}. So all we really need to know how much time elapses between the moment the ball is in contact with the ball and the moment when it is no longer in contact. Now, that can vary depending on whether the ball is hit head on or at an angle. For the purpose of simplification, let’s assume that the baseball is hit at an angle so that the time of contact is the shortest possible. So how much time elapsed between the “in contact” and “not in contact” states? No existing apparatus can measure this interval for rigid objects, but let’s put the question in equivalent terms of distance and ask: Over what distance does the acceleration occur? The shorter the distance, the shorter the time interval and, using the classical definition of acceleration, the greater the acceleration must be.

Now, if space is continuous, the distance that separate the “in contact” to the “not in contact” states is infinitely small. That is, as small as the distance may be, we can find an intermediate point between the ball and the bat where the bat is not in contact. And if the distance over which the acceleration occurs is infinitesimal, then the \Delta t is also infinitesimal, that is, \displaystyle \Delta t=\frac{d}{{{{{v}'}}_{a}}} so that \underset{d\to 0}{\mathop{\lim }}\,\Delta t=0 . And, using Newton’s second law of motion, F=m\frac{\Delta v}{\Delta t}, since \underset{d\to 0}{\mathop{\lim }}\,\Delta t=0, then \underset{d\to 0}{\mathop{\lim }}\,F=\infty . That is, if space is continuous, an infinite amount of force would be necessary to accelerate a baseball in the opposite direction from the point of impact.

Now, some will argue that the time interval of acceleration depends on the rigidity of the materials of the ball and the bat, and since no materials are absolutely rigid, the baseball and bat will remain in contact over a non-infinitesimal distance so that \Delta t is not infinitesimal and the required force, not infinite. Even if we allow a longer time of contact, the acceleration of the ball only continues for as long as it remains in contact with the bat. The ball stops accelerating as soon as contact is broken. Then, that last bit of acceleration in the transition between the “in contact” and “not in contact” states, must has a definite value, and however small that definite value is, if space is continuous, the distance between those states must infinitesimal and the force required for that last bit of acceleration must be infinite. We know of course that the force is not infinite and that the infinity here is a consequence of the assumption that space is continuous.

Since the distance between the states of “in contact” and “not in contact” cannot be infinitesimal, it follows that space cannot be continuous. And if space is not continuous, then it must be quantum-geometrical. Also, the discreteness of space implies the discreteness of matter. Hence the observation of the conservation of momentum in baseball supports the existence of both preons(-) and preons(+).








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