Reference: A Conceptual Model of Inertia, Mass & Location
The inconsistency I am faced with is “a continuous space being defined as a set of discrete points.” Mathematics fails to define the relationship between point and space properly. It seems that there is a better explanation underlying this inconsistency.
Science has progressed to a point where it is looking at the process of observation itself. It is difficult to come up with some absolute reference point, from which to look at the rest of existence. God seems to be a placeholder for what we do not understand. The only possible avenue to take is to take one inconsistency at a time and resolve it as best as one can, keeping in mind that one may have to come back to it later to resolve it further.
The universe is obviously finite in terms of what one is actually able to observe. We may postulate the universe to be infinite but that does not add anything more to our actual awareness or understanding. The idea of location is limited to the mechanical universe that we actually observe, independent of what one might postulate mathematically or otherwise.
We are now confronted with the idea of motion that goes beyond the classical concept of a physical body moving in space and time. A physical body consists of motion within itself at atomic levels. We cannot say if there is a point within the atom that is absolutely still. The stillness may be computed only in relative terms as difficulty involved in shifting a point. This is where the concept of inertia springs from. Thus, the idea of a location in space is intimately tied with the idea of inertia. This is what I meant by the concept of “centeredness” of CoM. I do not know if there is any limit to ideal centeredness. That is subject to observation.
An electromagnetic wave does not have its momentum concentrated at a point; rather it is evenly spread throughout. So, it has a physical presence that cannot be described as located at a point, as it can be done with mass. A photon may be described as having a physical presence as a wave packet, the size of which depends on its frequency. The higher is the frequency, the smaller is the region that describes its presence. One may say the location of a photon is spread-out in space, as opposed to being concentrated at a point as in case of mass. There is continuity that seems to exist within that “spread-out location”.
As frequency of the photon decreases, its location seems to “spread-out” further as proportional to its wavelength. This is looking at location in terms of inertia. The inertia lessens as frequency decreases. This spread may assume infinite proportions as frequency approaches zero. Here we achieve an approximation of space as a primitive concept that does not depend on the concept of point. In fact, space and point seem to appear at the opposite ends of a scale of inertia. The picture that I get is shown above.
So, an absence of “point location” does not mean “absence of location”. It means a location spread out in a manner of continuity that is very different from a “set of points.”
The statement, “A physical point location in space depends on the distribution of mass around it” means that there is the continuity of space corresponding to “zero inertia” that acts as the background. In this background exist point locations as “concentrated inertia.” I shall explain this as further questions arise.
Basically, I believe that physical space and location can be described more coherently in terms of inertia. This description bypasses the contradiction posed by a mathematical description of space as a set of points.
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Comments
Here is Einstein’s point of view with which the above paper agrees:
From Wikipedia:
“In addition, Einstein used the text to defend the utility of field theories amid the advances of quantum mechanics. The best way to do that was to view particles not as independent objects but as a special manifestation of the field itself: “Could we not reject the concept of matter and build a pure field physics? We could regard matter as the regions in space where the field is extremely strong. A thrown stone is, from this point of view, a changing field in which the states of the greatest field intensity travel through space with the velocity of the stone.”
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Here are some comments as a realist and not as a mathematician.
DISCRETE per dictionary means, “apart or detached from others; separate; distinct” as in “six discrete parts”; or “consisting of or characterized by distinct or individual parts; discontinuous.” I do not see discrete points becoming continuous, regardless of how you scale them, because these points are dimensionless. Points may be looked upon as fractals. The concept of infinity does not change their basic characteristic. They do not coalesce into each other.
The continuous space is a primitive notion on its own. Just becuase it may be filled with points, does not make its definition dependent on the primitive notion of point. “dimension of continuous space” and “dimensionlesss and discrete point” are two primitive notions that are very different from each other. A line is a “single-dimension space” that is continuous. It may be filled with infinity of discrete points, but there are still going to be gaps and breaks between those points. The very fact that a line has no gaps tells you that it is not made up of points. However, a single point may be stretched to form a continuous line. That would be consistent, but that is not allowed in classical mathematics.
There is nobody making the claim that physical spacetime behaves like the mathematical model. I simply see the mathematics failing to adequately model the motion in subatomic space with its “point-space” conceptualization. That is making Quantum Mechanics so math-heavy that it seems to have lost touch with reality, and it is simply floundering around. It is about time to move away from conceptualizing space as a set of points.
I have looked at Newton (& Leibniz) in this regard. Infinity of points may approximate space when one is dealing with inertia in that space either in the form of particle-mass or in the form of wave-frequency, but not as both. This condition is violated at subatomic regions.
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An electromagnetic wave is produced when there is an oscillating charge. The frequency of oscillation determines the frequency of the electromagnetic wave. The electromagnetic wave maintains that same frequency throughout its propagation. That is a good argument for inertia at wave level.
I see space as the resulting phenomenon as wave frequency approaches zero. The infinite wavelength represents the continuity of space. There is complete absence of inertia when there is no frequency. Space is pure extendedness. Space is not made up of discrete dimensionless points, because points have no extendedness. Space is extendedness that may be filled with dimensionless points.
Electromagnetic wave is made up of rotating electromagnetic field that moves forward in a corkscrew fashion. This is shown in a Wikipedia graphic below.
Electromagnetic Wave Animation
The frequency of electromagnetic rotation manifests as inertia. As this frequency increases the inertia solidifies.
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