Chapter 18: The General Theory of Relativity (the highly abbreviated version)

EinsteinThe General Theory of Relativity by Dr. Albert Einstein was, in many respects, the high point of his career. First published in 1915, and tinkered with for years thereafter, it provided a revolution of the way in which science perceives space in general, and specifically, gravity.

Building on the success of the Special Theory of Relativity, and realizing that it was not applicable to accelerated reference frames (that is, places where an acceleration force is present, like in the vicinity of a planet), Dr. Einstein set about to create a theoretical framework to provide a better explanation for the effects of gravity than the Newtonian formation that had dominated physics since its inception.

Newton’s equation for gravity is a force equation which states:

Fg = G (M1m2)

    r2

Where: Fg is the force due to gravity

M1 is the larger mass

m2 is the subject mass

G is the gravitational constant (6.674 X 10-11 Nm2/kg2)

And r is the distance between them

This is all very well and good and allowed Newton, and the physicists that followed him to calculate a multitude of observed phenomena, and make many, many validated predictions. It has withstood the test of the centuries, much like Euclidean geometry.

There are, however, a few theoretical shortcomings with this formulation.

First, and this limitation was also realized by Newton himself, there is no causative source for this force. In this interpretation, gravity is seen as a pure ‘action at a distance’, that is, being that it is only seen as a disconnected force, it will act on faraway objects without there being any other seeming connection to them, other than the force field that has been characterized, but not explained.

Secondly, Newton was unable to link gravitational mass to inertial mass. Although he understood that there must be some underlying connection between the two, he understood well that there was no structural link between the two quantities, since one had to do with impeding or continuing a given motion (inertia), and the other dealt with creating force and initiating motion (gravitational mass).

Because of this, free fall, the act of allowing something or oneself to accelerate with the force of gravity, rather than resisting it like we do by resting on a surface, is seen as an accelerated framework, rather than as a frame at rest. This creates problems in explaining the way that objects interact in free fall (or microgravity) situations, such as those experienced by astronauts aboard the International Space Station.

Thirdly, it creates a strictly linear equation to describe a situation that, by all accounts, is a volumetric phenomenon. It may appear like the Earth and the Moon interact only on a straight line strung between their centers, but it is very clear that each has a complete field that surrounds them, and that a full description of their interaction should encompass the larger effects created by the interplay of the entire field.

Finally, Newton’s formulation is a time independent equation, indeed, the only place in the equation in which time appears is in the constant, G (it’s buried within N = kg * m/s2).

And so it was that Dr. Einstein turned to the mathematical structure of curved space, created by Karl Gauss, to formulate a more appropriate description.

Dr. Einstein’s fundamental assumption upon which the General Theory of Relativity is based is this:

“we […] assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system.” (Einstein 1907)

Which pretty much gets us out of the whole gravitational mass versus inertial mass problem, except that this statement must be accepted based on anecdotal evidence rather than from some rigorous proof or deductive logic. We’ll talk more about this assumption and the thought experiment that Dr. Einstein used as evidence in later chapters, but for the time being, we’ll accept this and move on.

So Dr. Einstein built a set of field equations to describe a new perception of gravity, and in their most simple form, those equations can be reduced to the following equality:

Screen Shot 2014-01-26 at 7.20.17 PM

Which can be roughly stated to mean the following: The geometry of space (the ‘Gμν’ Matrix, on the left) is equal to and described by a constant (8πG/c4) multiplied by the Tensor matrix, Tμν, which contains a summary of all of the forces present in the locality under consideration.

This is a very powerful theory and description, and has been used to explain many old mysteries and to predict many seemingly impossible things. Black holes, a state of matter in which the mass becomes so compressed that the strength of gravity is so strong and concentrated that it overwhelms the speed (and acceleration!) of light, are a concept that was developed from the Einstein Field Equations. The inclusion of this phenomenon in the category of galactic specimens has greatly increased our ability to describe many of the strange and highly energetic sources that our constantly improving telescopes have shown us.

General Relativity also has more practical applications, like correctly describing the differences in the time-space environment between the Earth and an orbiting satellite; which allows GPS systems to work to the accuracy that they must.

General Relativity establishes that the mass and the energy of matter actually bends the unseen fabric of space, curves it if you will, so that when a beam of light is diverted around a star or another massive object, that it is actually following a ‘world line’ that represents the straightest possible course, like a boat going around a whirlpool. If the boat went straight into the whirlpool, its course would be much harder and longer (and much more dangerous), but by following a path near the edge of the edge of the vortex flow, it could safely traverse the water field created by the flow along a path of least resistance that represents a balance between the boat inertia and the water forces. That path would be curved and represents the ‘world line’ for the boat around the whirlpool.

In a similar fashion, light is bent around a gravitational field, by following the path of least resistance, which in this case is curved.

The most stunning confirmation of this assumption has been the discovery of gravitational lenses within the cosmos. These features, predicted by the General Theory, have large gravitational fields that are so strong and bend the space around them so severely that they act like glass lenses and bend and refocus the light that passes by. Since the initial discovery of one in 1979, many more have been found, including one so complete (called an Einstein Ring) that it actually allows an improved magnification of the space on the other side, providing a closer, if somewhat distorted view into the far, far field. This technique was used to view through a cluster called Abell 2218 in 2004 to observe the oldest galaxy yet discovered, whose age is approximated at 13 billion years.

Oh, and by the way, it solves that little ‘free fall’ problem, too.

General Relativity is, at its most fundamental level, a series of linear (differential) equations regarding force (the Tensor matrix) that is utilized to create a curvilinear space, which is used to describe the motions of the things that go through it. It allows gravity to be spherical, and to create the forms that we know must exist because of the things that we observe in the heavens, unlike the strictly linear interpretation proscribed by Newton. It also gives gravity a speed, a velocity (c, the speed of light), which comes from the Tensor and better describes the fact that the gravity is connected to the object which is its source.

There are some problems with this formulation, however, some of which are acknowledged by those who really study this stuff and can actually do the math. The first is, that since it utilizes a series of co-dependant non-linear partial differential equations, it is extremely difficult to find an exact solution without making a lot of simplifying assumptions. This will not be demonstrated here. There are actually very few, but numerical simulations have provided great insight into how relativistic systems behave.

Another problem is that this solution has, so far, been impossible to integrate into Quantum Mechanics and the Standard Model. While the effects predicted by the Special Theory of Relativity have been included and are an integral part of the formulation, as alluded to earlier, all of the efforts to mesh General Relativity with quantum formulations to date have failed. Even though we have these two great theories, they each work only at their specific end of the size spectrum, and even General Relativity has not managed to bridge that gap.

The other problems that will be discussed here are more specific to this document and the issues that exist with conventional thinking and geometry that have been mentioned in previous chapters.

First off, General Relativity is still a point-wise solution. Although it does create curved space and fields, these are still described in a basic Euclidean-Newtonian framework wherein curves are described by a series of points. It has already been shown how this is, by definition, only an approximation.

Secondly, the General Theory relies on the Newtonian definition of gravity. It is the “G” in the constant term. This presents a problem at close distances, which we’ll talk about in a following chapter. But for now, we’ll just note that the finest description of gravity yet concocted relies on a 17th century definition of a force, which is linear by nature.

Finally, General Relativity still maintains that time is a separate, independent dimension and carries it as such, as though it is a totally separate, linear dimension like those of distance, that may have both positive and negative values.

Dr. Schrödinger was one of the most interesting people that I encountered during the process of writing this book. His affinity for young women, and his transformation from ‘genius’ to ‘goat’ is both fascinating and in my opinion, predictable, because, while the general physics community has appropriated his equations and ensconced his ideas in the Standard Model, I just refuse to accept any theory that has to use imaginary numbers to find an answer. I don’t dispute that it works, but just because it seems to give the correct answers that doesn’t mean that the methodology is essentially correct.

You can catch fish by electrocuting them in the water, but that doesn’t necessarily mean that it’s the best or only way to do it.

And sometimes, that can end badly. ( http://wilk4.com/humor/humorm12.htm; see nominee #13)

While the General Theory of Relativity was, and remains quite a triumph of thought, the fact that it relies on a point-wise formulation for the force limits its applicability to volumetric space. Although it works fine as a solution for any body-to-body problem, and to my mind, rightfully assumes that gravity warps space, because of that  formulation and its use of the Newtonian equation for gravity, it never fully captures the nature and form of gravity itself, as we’ll discuss in the next section.

We’ll also spend a little time discussing the nature of time itself, and the notion of time travel, which, as we’ll see, is a concept based mostly on the hubris of our species and not on established, observed phenomena.

It’s not that I don’t like the movies and stories about it, it’s just that when you really think about it, time travel is just, well, silly.

We’ll also talk briefly about String Theory.

So bear with me, dear reader, we’re almost there.

– O. Penurmind

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