As I was attending the 5:00 PM Sunday Mass today at Saint Leos in Fairfax VA, my mind kept wandering on the topic of what it would mean to travel so close to the speed of light that the degree of time dilation would be the least transfinite ordinal, often referred to as Omega or Aleph 0, the infinite number of integers. Ill often add my distinction to differentiate between Aleph 0 and Omega where I refer to Omega as the number of integers in a fractionally finite subset of the integers. I see a need to make such a distinction when I am contemplating the obtuse notion of infinite gamma factors. We thus are in a bit of a pickle having dared to rationalize infinite time dilation. For example, according to a strict tenant of Special Relativity, the speed of light is the same in all reference frames. If one wants to make this assertion an ontologically inviolable tenant, then we must also assume that even for travelers moving with a gamma factor of Omega or Aleph 0, they would also measure the speed of light to be equal to C. Crazy things begin to happen as one approaches the speed of light in inertial travel ever more closely. More and more of the universe becomes translated to appear in front of the spacecraft and more and more scrunched up to a location directly in front of the spacecraft. Carrying this process to the extremes at a gamma factor of Omega, the entire universe would be scrunched up to a point directly in front of the spacecraft while at the same time, the space in front of the spacecraft would be infinitely contracted. The crazy thing is that another spacecraft traveling along side a first spacecraft having an gamma factor of Omega but for which the relative velocities between these two spacecraft in their own reference frames is say, a mere millimeter per second would presumably be able to shine a flash-light beam back and forth while everything about one spacecraft with respect to the other would seem as if they were essentially not moving at all and docked together. However, Aleph 0 is not the end of the story in terms of transfinite or infinite ordinals. The next highest infinite ordinal is Aleph 1, which according to the perhaps improvable Continuum Hypothesis is the number of real numbers, a set that either way, is larger in number of elements than the number of integers by a multiplicative factor of infinity. So, here we have a scenario for which a spacecraft traveling at Omega or Aleph 0 experiences the entire background of the universe all compressed to a location directly in front of the spacecraft for which the relative degree of time dilation is Omega or Aleph 0. We are now left contemplating the notion that although one year of time passage on the spacecraft traveling with a gamma factor of Aleph 0 would correspond to Aleph 0 years of time passage in the background, a mere year of time passage on the spacecraft traveling with a gamma factor of Aleph 1 would correspond to an infinite number of years passing by on the Aleph 0 gamma factor spacecraft. We can generalize to additional spacecraft traveling with a gamma factor of Aleph 2, Aleph 3, Aleph 4, ..., Aleph (Omega), ... Aleph (Aleph 1), ... and so on with the huge caveat that the universe is commensurately big to support such transfinite gamma factors, something that perhaps no one knows except GOD HIMSELF. I enjoy speculation on the notion that there exist an effective universe or realm relative to an observer moving at any infinite gamma factor that is essentially distinct from any other observer moving at a gamma factor that is less that that of the first observer by a summand of a sub-canonical ensemble or a very large sub-ensemble number that is none-the-less finite. Thus, between two observers traveling at a gamma factor of Omega and (2)(Omega) respectfully, there would exist an [(Omega)/(sub-canonical ensemble)] number of relatively distinct realms in the universe of travel. Since an infinite quantity divided by a finite quantity no matter how large yields an infinite quantity, we are thus left with the possibility of the resulting infinite number of realms. Now, my use of transfinite Aleph numbers to append gamma factor values to spacecraft performance would no doubt be considered an anathema to conventional real analysis experts and will no doubt irk Set Theorists as being used outside of the conventional context for which such notions are intended, however, the strict Special Relativistic axiom of the speed of light being the same in all reference frames if taken to the logical extreme, will require that my musings be correct, if not in logic then in spirit. I will have more to say on this topic in the days to follow.
Posted on: Mon, 10 Mar 2014 00:59:11 +0000
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