Searching for Quantum gravity. what is distance? (3)
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 post, I argued a radical, sounding absurd idea that applying the mathematical abstraction directly to getting the measurable or observable property in the physical world such as finding distance between two separate points in space could be dangerous in that it can distort the true reality so may yield a wrong prediction.
To support this view, I adopted the notion of cost which is time for the case of determining distance in space and it can help us to recognize the pitfall. In mathematics, subtracting two numbers what is normally used to get the distance between two points over number line is just an abstracted description human can think of in mind and such abstracted description can be distinguishable from the specific physical implementation of it. Why? Because the mathematical abstracted operation is time free in that it does not care about the time cost needed for the determination of it. Meanwhile, the notion of time cost arises in the actual implementation in physical world. For example, computer can calculate the subtraction of two large numbers much faster than human with only bare brain. We cannot tell which one is better than the other and both are just valid but “approximated” implementations by either human or computer showing different time cost. For C++ programmers, the point that this distinction between abstraction and physical implementation must be treated cautiously can be easily understood. The distance between two points is just a final result or objective we want to see and mathematics do not care about what a specific route to final goal will be chosen to get the distance. We human need to make a choice on the various routes which will take much or less time to get there. Knowing that the time cost does not matter in mathematical abstraction which is ideal case and time cost actually matters in the physical implementation is crucial to keep going to next discussion.
For those who are still not sure about my reasoning so far, let me take another thought experiment. Here let assume that there are two observers living their own planets A and B separated at a distance of 1 billion light years. Also there is a very large rigid ruler connecting two planet so both observers know their location relative to the ruler by reading it. Here the ruler plays the same role described in the Einstein’s special relativity paper. By reading the ruler, observer A’ at A can read a scalar value X for the exact location in space and other observer B’ at B can read a scalar value Y. Also, at initial time t0, assume that they know others location that the distance between two is |X-Y|. For now let’s suppose that a new planet C suddenly appeared near B at t0 and observer B’ soon know C is created nearby. It is fine because it is thought experiment. Now the question is how to determine the distance between A and C. For observer B’ located at B near C, determining the distance between A and C can be done almost instantaneously by doing mathematical operation because he is very close to C. First he can read the approximated value for the location of C by reading the ruler, let’s say it is Z, and determine the distance by mathematical operation by subtracting two scalar values X and Z in almost no time.
Meanwhile, the observer A’ at the planet A at t0 when C just appeared at a distance of 1 billion light years away has a no way to know that event happened at B. If he tries to observe and take a photo to the direction of planet B soon after t0, what he can see in the photo at t0 is just only B at r0 even though C actually exists near B until 1 billion light years flow since t0. According to Einstein’s special relativity, nothing can move faster than light so if A’ want to measure the distance between A and C, he need to wait 1 billion light years and it’s the fastest way to measure although B’ can determine the distance between A and C without waiting 1 billion light years with mathematical abstracted operation.
From the exampled thought experiment, we can see there are two ways in determining the distance in space. The observer A’ can use only the physically feasible method of which time cost needed to complete the determination is proportional to the magnitude of distance but B’ at B near C can use the mathematical abstracted operation so that he can completed the determination much faster than any physically feasible way. I argue this shows clearly that there is some sort of inequality between the mathematical abstracted operation and the physically possible operation yielding almost same result and the gab is impossible to be filled by anything relevant to physical world.
The main goal in physics I think is to make a perfect prediction repeatedly. That being said, the main question I want to raise here is what applying mathematical abstraction to physical world directly will affect the prediction we want to see. Will it show us what really exists as it is without any distortion so we can observe the exact thing as predicted? Someone who knows well about quantum weirdness such as measurement problem, wave function collapse which is detectable at physically measuring state of the system may guess the answer would be negative. The double slit experiment shows that calculating the probability to find the location of elementary particle, the only one available option we can take for now, does not provide the precise prediction on where the particle to be found in near future. Simply putting, people do not have a clear understanding on what applying mathematical abstraction to physical world really means and what side effect we could not expect will arise when it is done in physical world.
What makes discrepancy between continuum and discontinuum?
In previous post, I argued that there is no physically feasible way to define the distance in continuum. Instead, we usually use a mathematical abstracted operation, subtracting two scalars which is the only one available option we know so far. And such mathematical operation does not limited by Einstein’s famous second postulate in his special relativity saying nothing moves faster than light as we can see previous thought experiment. Here what I want to do is to find the physical intuitive meaning of that mathematical operation can be completed much faster than speed of light which is the absolute limiting factor residing in every event happening in our universe.
To be continued …
