Saturday, September 24, 2011

Wormhole and the notion of Continuity as a mathematical ideal

It is natural to assume no limit for counting numbers. That is, accepting the existence of a largest number is contrary to intuition because there might be a case where we need a larger number, for instance the budget deficit of our country. We know any case of physical reality that we consider, like sands on earth, number of planets that we are aware of and even the budget deficit at any moment is finite. However, mathematics must be adequate for all imaginable cases. Whenever we set a limit for counting one can imagine something larger. Therefore, the limit for counting must not be imaginable which is why we assume no limit, or equivalently that the largest counting number is infinite.

A system chosen to cover all imaginable cases of reality must include all such cases as a proper subset possibly by including some ideal cases. Clearly, an ideal concept such as infinity does not need to correspond to our intuition. Furthermore, Cantor shows that once we allow a notion within mathematics that does not correspond to our intuition it is possible to derive unexpected results. Cantor demonstrates the existence of an infinite sequence of infinities each of a higher order from all previous ones.

Galileo observed that there are as many multiples of 5 as the counting numbers. The sameness of a proper subset to the entire set was so startling that Galileo never spoke of it and merely left it as a note. Vaguely speaking mathematics needs to be consistent so we can depend on results derived within it. When we include a notion that does not correspond to our intuition within this system, we should expect to see results that may not correspond to our intuition as well. A historical case is the interpretation of Euclid’s Fifth Postulate by Lobachevsky.

Superficially, there are two distinct cases of arriving at counterintuitive or paradoxical results. In order to solve problems of an aspect of reality we map a model to the reality under consideration. In doing so, we must approximate the reality by making certain assumptions, knowingly or not. Usually we refer to this process as an abstraction. For instance, Newtonian mechanics is a reasonable approximation for speeds negligible relative to the speed of light. The use of relativity in this context does not produce any measurable gain. On the other hand, relativity is a better model at speeds that are significant as compared to the speed of light. In this case getting unacceptable results through the use of Newtonian mechanics is a consequence of the inadequacy of the model mapped to the reality under consideration.

When an unacceptable result comes directly from inside of mathematics we refer to it as a paradox. Note that unacceptable here means that the result is mathematically sound but defies reality. In other words, we are not dealing with an unacceptable derivable result coming from our inaccurate approximation of reality with a mathematical model. What we are forgetting, though, is the ideal assumptions such as infinity that we have made part of our mathematics. For instance, the source of Zeno’s paradox is the ideal assumption that a line segment can be divided into two halves indefinitely. On the other hand, the assumption that any line segment has a midpoint is as necessary as the notion of infinity with regard to counting numbers.

Let us take a look at Zeno’s paradox by asking Arash the Persian Archer to hit a target at 100 meters, a child’s play for Arash. Zeno tells us that Arash's arrow will never reach its target, though he refuses to face Arash as a target. Zeno argues that before the arrow reaches its target it must pass through the midpoint at 50 meters from Arash. Next, the arrow must pass through another midpoint, which is at 75 meters from Arash. Continuing the argument he correctly tells us that the arrow must go through infinitely many points, which requires an infinite amount of time before reaching the 100-meter point. Thus the arrow will remain flying forever without ever reaching its target.

In a lot of cases, such as the imaginary numbers we have been able to map unintuitive notions to the reality around us and to find ways for using them to solve actual problems of physical reality. However, with regard to Zeno’s paradox the suggested mathematical answer is no more than simply accepting the paradox as the truth. The notion of infinite sum, or an infinite sequence of partial sums, appeared in the work of Archimedes when calculating the area under a curve. Pascal takes the other end and defines a tangent line as the limit of a sequence of secants, as opposed to a line touching a curve at a single point. Leibniz and Newton establish the connection between the two ends via the Fundamental Theorem of Calculus. There is no doubt in how important calculus has been. However, what matters for our discussion is the work of Archimedes.

Archimedes’ approach of an infinite sum converging to a finite number uses the ideal notions of convergence towards a limit point, which is equivalent to accepting Zeno’s paradox as a none-paradox.

A related theorem using the axiom of choice in addition to the notion of continuity is the Banach-Tarski paradox, which shows that a pea can be chopped to pieces and reassembled into a basketball. Here again we have a valid mathematical result that directly defies our intuition.

In general, intuition is not an accurate means for describing reality. It was not long ago when we thought earth is flat simply because we could not walk on ceilings. But it is hard to deny the fact that Arash’s arrow does reach its target. Perhaps in this case our intuition took a wrong turn at a different point. Our intuition leads us to believe that motion is continuous. Is this really true, or is it analogous to believing that earth is flat?

We all have seen the bottom of a swimming pool appearing shallower than the actual depth of the pool. This phenomenon is described as refraction of light. It is believed that when light passes from one medium to another its speed changes causing a change in direction. But does the actual speed of light change, or only its measurable speed changes?

Let us change our intuition to believing that a photon cannot move less than a small displacement, a quantum leap, the size of which depends on the medium. For instance the quantum leap (QL) for air is longer than QL for water. With this intuition we can say that the speed of light remains the same, only the length of leaps are different. Therefore, what we observe is that the measurable speed of light has changed. In other words, a photon is making the same number of leaps in both mediums but the length of leaps are different making it appear as if the photon is moving at a different speed.

The notion of QL resolves Zeno’s paradox by disallowing the bisecting of the motion of a particle, or any object for that matter, indefinitely. Thus, the arrow only passes through a finite number of points rather than the mathematical ideal of infinitely many midpoints.

Note that we are not postulating that there is a smallest QL. We are only saying that any motion comprises of displacements, or quantum leaps the lengths of which depend on the medium. Thus, the notion of continuity is still preserved. The QL near a black hole is very small and approaches zero when light can no longer escape.

The notion of QL provides a more reasonable view for a wormhole. It is argued that at Big Bang the universe expanded instantaneously to a certain point. The speed of light was established within the extent of the universe after its creation, or being. Imagine an area in the vastness of existence where there are no particles, a total void. There are no forces of any kind in a total void because any form of force field requires the presence of particles of some kind for its extent. What will happen when a photon enters one end of void? I believe the QL in a void is the length of the void. That is, the photon will reach the other end of the void in a single Quantum Leap.

At Big Bang our universe expanded instantaneously in all directions, not necessarily as a perfect sphere. Once particles reached a none-void medium their QL became finite and their motion slowed down. In turn, the boundary of our universe established the QL for particles within. Perhaps the center of the Big Bang contains less mass in an identifiable manner as compared to the density of the rest of the universe, even though indefinite number of explosions since then have sent back a lot of mass to this point.

It is hard to guess whether we need equipments at two ends of a wormhole, or one can create a wormhole from a point on earth to anywhere in the universe. In the latter case we can use the wormhole at least as a real-time telescope. I can only hope that no one will be tempted to observe the core of the sun in real time, well at least not from earth.

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