Before leaving the discussion about the current state of geometry, mathematics and physics (yes, it’s been a whirlwind tour), there are a few of points regarding the predictions of Special and General Relativity that bear mentioning. The first is the thought experiment that is generally referred to as the ‘Twin Paradox’.

‘Thought experiments’ are interesting logical arguments contrived to make some theoretical point. What one does for a thought experiment is to propose a set of assumptions that are generally fanciful in nature and apply the rules or precepts of a certain set of ideas to the situation to illustrate a predetermined outcome, at least in the eyes of the one proposing the ‘experiment’. Many times, it is an experiment that could not be performed. Nonetheless…

The Twin Paradox is a relativistic time travel experiment that goes like this: A pair of twins are recruited for an experiment. One remains on Earth, and the other is sent out travelling in a spaceship toward Alpha Centauri or some other ‘nearby’ star, in a vessel that can travel at speeds approaching that of light. The journey takes several years; in this case let’s assume that it takes 10 years, 5 years out and 5 years back. When the traveling twin returns, he finds that his sibling has aged 10 years, duh. But he, having been on the relativistic spaceship has aged only 6 years. How can this be?

The answer has to do with the effect that traveling at that velocity has on the way that the traveling twin experiences time. Because the speed of light is constant, then all of the other properties of matter must vary dependent on their speed, by factors quantified by the equations known as the Lorenz Transformations. As it turns out, length, mass and time are all altered by extreme velocity in a ratio described by the following equations:

As one approaches the speed of light, in the eyes of the stationary observer, the length of the traveler gets shorter, his mass increases and the ‘speed’ of his time slows down, so he ages less, by a factor determined by the square root of the difference between the speed of light squared minus the speed of the traveler squared. This seems very counter intuitive, but actually has a very real effect on products in general use in modern everyday life. You see, without taking this ‘time dilation’ effect into account, your GPS system could not attain the accuracy required to show you where you are on the map. The differences between clock time for you and clock time for the satellite that gives you the signals, which is in a geostationary orbit about 22,000 miles out, are slightly different due to the orbital velocity of the satellite. That difference is very small, but in the scheme of calculating your location from the signal of the satellite, it must be accounted for to determine your exact location (+/- about 10 feet) on Earth. Trippy.

The twin paradox has been discussed and modified many times, and at least according to some, the time dilation is caused by the acceleration and then the reverse acceleration required for the traveler to reach those speeds rather than by the velocity itself, and many now just accept the validity of this effect, especially since it has been so thoroughly validated by our GPS network (that is if you’re not using Apple Maps). And it does make some sense in the example where one of the observers is left on Earth, which has a constant gravitational field versus the traveler whose gravitational field varies with the acceleration (more on this in the next section), so it is no longer considered to be a true paradox.

The problem comes in if you reduce the experiment to its lowest form: that of two observers in free space. Because if you have only two observers, then the velocity and acceleration between them becomes completely relative and there can be no physical distinction to determine which is the observer and which is the observed, and therefore in this case, whose time slows down.

Now some will argue that the one whose spaceship is firing and therefore accelerating will be the one to experience the time shift, even though from his perspective, he would see his stationary twin doing the exact opposite motions that he himself is experiencing; and that makes a good point, since acceleration equals gravity as we’ll demonstrate in a thought experiment later (with a somewhat surprising conclusion), but the overarching takeaway from this example is this: For this to be true, there must be a ‘fabric’ even in otherwise empty space, because without a ‘fabric’ to space, the acceleration could not be felt. This ‘fabric’ is not exactly the same as the Luminiferous Aether introduced in an earlier chapter, but it still implies, or rather demands, that so-called empty space must have substance, a fabric, a structure if you will, at least in the presence of mass, that allows mass to react and experience acceleration.

The other points have to do with the space surrounding a black hole, specifically at the event horizon. This space has relativistic properties, too.

As almost everyone knows, a black hole is thought to be a small section of space in which gravity has become so intense that even light becomes trapped and cannot escape. Usually, these spaces are believed to be created by the implosion effects of an exploding star, during which, the interior mass becomes so compacted that it goes beyond the density of a star remnant composed of only neutrons to a compact mass of only, well we don’t know because the equations break down at the boundary, the event horizon, and we can’t ever know what actually happens inside because no information can escape.

Except maybe it can. Hawking says that the black hole has as temperature and therefore it must emit radiation. Black hole radiation, actually, Hawking radiation; which is a form of electromagnetic radiation, but whatever.

Relativity claims that an object approaching a black hole never actually makes it past the event horizon because time slows to a complete stop at that boundary, and the object can never be seen to cross into the interior. Of course, the observer going into the black hole sees no interruption of time and proceeds across the boundary hardly noticing (well, except that the gravity differential would tear him to bits, he couldn’t lift his limbs and he’d probably have passed out quite a ways away) because all of the light drawn in would mimic the universe that he left.

Many think that the rules of General Relativity become approximate at the event horizon, and that the rules provided by Quantum Gravity will apply at that point, which is a great dodge since, as pointed out previously, there is no theory for gravity within Quantum Mechanics. Once we understand one though, we’ll be able to explain the other.

There is also the little problem of the cosmic jets that seem to be a regular feature of spinning black holes, but we’ll have to wait a while to discuss that.

So while the General Theory of Relativity is an amazing feat of logic and mathematics, it does have some proven limitations that prohibit it from providing a consistent description of even the macro portion of the universe, much less the subatomic, and somewhat surprisingly, Relativity relies on an assumption that there is a structure to ‘free’ space in that one can experience acceleration in an environment unencumbered by additional objects, energy or gravity. That claim is not inherently wrong, in fact it’s quite provable, but it is not entered as an explicit assumption in the development of the proof for General Relativity. It is not quite the same as saying that acceleration in space is equivalent to gravity, which is at the core of the theory.

Empty space must have inherent structure.

Aside:* I’ve been struggling with this conclusion for years. Empty space should be, you know, empty; having no fields, charges, motion or (I thought) structure. But it must have a structure to allow mass to feel acceleration in the absence of a gravitational field, otherwise the mass could not ‘know’ whether it was accelerating or at rest. *

Einstein was quite right about acceleration and gravity.

Well almost.

…

*Thus ends the history of physics as told by yours truly, O. Penurmind. I’m sure that there will be many that will dispute my interpretations, and there may be some sections that I’ve misunderstood partially or that have been superseded by ongoing discoveries. Attempting to summarize and intellectually criticize 2500 years of mathematics and physics has been quite a daunting task, especially attempting to do so in as few pages as I have. One could easily expand the prior portion of this book into volumes and still not have covered the topics completely.*

*But we don’t really have time for that.*

*The point is that even if there are some, shall we say, ‘irregularities’ in the text thus far, it is fairly obvious that there are many logical and empirical problems with the Geometry and Physics concepts that are accepted as axioms today and that those ‘problems’ are really hindering our ability to understand the true nature of the forces and energy forms that compose the universe as we know it. And while I’ve had some fun poking holes in the structure and logic of the systems in use today, we, as a civilization and species, have really come quite a long way from living in trees.*

*That is, if you believe in evolution. If you don’t, then you should have quit reading a long time ago, if you started at all.*

*So in the next installment, we’ll begin describing a new and better way of looking at things; a logical extension of Dr. Einstein’s assumption that ‘the speed of light is constant’.*

*We’ll be kicking poor Euclid completely out the door. Poor guy. He has held sway for all this time, and quite frankly, it’s about time that we abandon his way of thinking and the structure of the universe that he envisioned. Thanks, old buddy, but don’t let the door impact your posterior on the way out.*

*It would be best if you could enter: ‘**delete *.***’ inside the geometry and number theory folder in your brain before reading the next section. I’ll give you a little time to get that done. Until next time,*

* – O. Penurmind*

…

To get notified when new posts are up, find the author on Twitter (@O_penurmind) or Facebook.