Searching for Quantum gravity. what is distance? (2)
Oh Youngjong
(dmqcka @ gmail.com)
(This article contains my own personal view and not finished yet so be noted that it may contain wrong information. It can be modified at any time without notice.)
In the previous writing, there proposed a radical idea that the notion of distance in any continuous dimension can be defined only after there is a way to generate the indivisible finite length unit out of it. The problem in defining the notion of distance can be traced back to ancient Greek philosophers. Specifically Dichotomy paradox which is one of Zeno’s paradoxes shows the paradoxical conclusion that traveling over any finite distance can neither be completed nor begun in any continuous dimension or space and so leading to another conclusion that distinguishing distance is meaningless.
Many scalars interested in it tried to provide reasonable explanation to resolve it. From the wiki page https://en.wikipedia.org/wiki/Zeno%27s_paradoxes, two suggestions for possible resolution would be worth to be mentioned for further discussion. First resolution is Hermann Weyl’s one that the paradox can be resolved if it is not true that between any two different points in space including time, there is always another point. I strongly support his view with the confidence that his assumption is true, which will be explained later with a new working model. Secondly, Pat Corvini claims that the paradox can be resolved by distinguishing the physical world from the abstract mathematics used to describe it. According to the wiki, it seems that Corvini distinguishes the abstract mathematical notion that between any two different points in space, there is always another point from the physical world of a moving object to traverse an infinite number of divisions. So I totally agree with the quote “The physical world requires a resolution amount used to distinguish distance while mathematics can use any resolution”.
Regarding Corvini’s view, I found a new concept would be quite useful to understand the necessity for distinction between the mathematical abstracted notion and the physical world and it is the cost which is needed to complete the determination of distance. In mathematics, i.e. over real number line which is a manifestation for any continuous dimension, the needed process to measuring distance is just a subtraction of two numbers which requires a constant cost over any distance. It means that no matter how far two numbers are separated from each other, the cost for the subtraction is all the same. For example, in the computer, the required time to complete subtracting two numbers in CPU is constant so it is independent on distance. On the other hand, in the physical world, our daily experiences show that the cost for measuring distance must increase as distance is getting larger. So regarding in determining distance, from this observation of time independence in mathematics and time dependence in physics, it is obvious that it would be too dangerous to apply a mathematical abstraction directly to the physical world.
To get the better understanding on the notion of distance in any dimension, especially space and time, I focus three things in this article. First is the conflict of quantum mechanics and theory of general relativity at singularity due to mainly (I think) the different notion of space and time. Since Max Planck opened a door leading to quantum mechanics by postulating the existence of quantized energy which was considered as continuous measurable property in classical physics, QM assume that space and time also must be quantized while GR dealing with cosmos scale objects treats space and time as continuous. In order to predict the behaviors of heavy matter such as stars and galaxies being absorbed by black hole, I believe that we must get a clear understanding about how heavy matters ruled by general relativity at larger continuous space moves at the scale of quantum mechanics assuming discrete Planck length inside of Schwarzschild radius of black hole.
Similar to what Corvini suggested that mathematical abstraction cannot be applied directly to the physical world, I think that there is a crucial boundary between the mathematical abstracted concepts and the physical tools used to describe the physical world. The key element making the separation is infinity. Actually to spoil my conclusion in advance, it is the mass. We human are all massive and have limited time, meanwhile light is massless so it has unlimited time. The mass plays a core role in assigning time and space to every physical object residing in the universe. The reason for this will be explained in details later.
Putting it with two suggested solutions for Zeno’s paradoxes mentioned above together, I argue that in physical world, there should be a way or model to generate an indivisible finite unit length in space and time which can be observable or measureable out of the absolute continuous space and time.
In next time, the common underlying model
for every elementary particle will be introduced.
References
