FOURTH BRIEF: Dismissal of the Third Postulate of Cosmology

The Third Postulate of Cosmology says, “The Universe is flat (linear).” So specifically, just what is a flat (or linear) coordinate system?

A good 2-dimensional illustration of a linear coordinate system is represented by the Pythagorean Theorem. The basis of the therom states that the length of a side of a right triangle cannot exceed the length of the hypotenuse. This is shown mathematically as follows:

*a = *(*c *^{2 }–
*b*^{ 2 })^{ ½}

*b = *(*c *^{2 }– *a*^{ 2 })^{ ½}

The equations above express a right triangle with sides *a *and* b*, and hypotenuse *c*. As seen, if *a* or *b* is greater than *c*, the right side of the respective equation becomes the square root of a negative number, and there’s no such thing as the square root of a negative number. However, there’s a critical premise to this called the
“First Condition of Linearity” that says the continuum must be linear (flat) or the theorem doesn’t work. For this reason, the equations are known as “linear equations,” because they only work in a coordinate system that’s
linear.

This means **the right triangle must be drawn on a flat surface for the Pythagorean Theorem to work properly**. If the surface is not flat, all bets are off. For example,
if drawn on a curved surface such as a sphere, one or both sides of a right triangle can be greater than the hypotenuse. [Try drawing a right triangle on a globe, with the right angle at the North Pole. When the sides extend past the equater they are longer
than the hypotenuse.]

Now, the same phenomenon is true in a 3-dimentional coordinate system, as verified by the Lorentz Transformation shown following:

*m _{o }C/m = *(

*C*

^{2 }–

*v*

^{ 2 })

^{ ½}

Here we see, on the right side of the equation,
the velocity *v* of a mass *m* cannot be greater than the velocity of light *C*, because the result is the square root of a negative number. Again we see the First Condition of Linearity. Like the Pythagorean Theorem, the transformation
is a linear equation. So the velocity of light is an absolute constant in a linear coordinate system. And in accordance with the Third Postulate of Cosmology, the Universe must be flat (linear), because we don’t know of anything that exceeds the velocity
of light.

Or do we? Let’s take a look at a brief mind game that might surprise you.

First we must bear in mind that
the Lorentz Transformation does not include “force.” A force (gravity for example) would cause the mass to accelerate (Newton’s 2^{nd} Law). Thus, the transformation just pertains to a mass at some velocity, which excludes any applied
force. Second, when referring to relative velociy, there is no law of science that says we cannot place the reference point at any location we choose. So we’ll designate you as our reference
point.

In essence, we’re looking for something that exceeds the velocity of light with respect to you. Now let’s say, on a night, you are outdoors lying on a recliner, looking straight up into the sky, and you see a star. As you watch that star for a while, you’ll see that, relative to you, it’s slowly moving across the sky.

Well, not really slowly. Excluding the sun, Alfa Centauri is our closest star. Its distance is 4.3 ly. So the star you are looking at is traveling around you in a circle with minimum radius of 4.3 ly and circumference of 27 ly. So relative to you, that star is traveling a minimum distance of 27 ly in 24 hours.

Relative to you, that star is traveling farther in one hour than light travels in one year. In fact, relative to you almost everything in the Universe is exceeding the velocity of light.

How can that be? After all, the Lorentz Transformation proves that the velocity of light is an absolute constant. Doesn’t it?

No it doesn’t. It proves the velocity of light is an absolute constant in a linear coordinate system.

Apparently the Universe is not flat (linear). In opposition to The Third Postulate of Cosmology, the Universe is actually nonlinear in compliance with Einstein’s Non-Symmetric Field Theory.

Now, I must admit that this presents a rather perplexing problem, because for most people it’s very difficult to comprehend a nonlinear coordinate system. In essence, it’s not so hard to visualize a 1-dimensional curved line or a 2-dimensional curved surface. But how do we visualize a 3-dimensional coordinate system that’s curved?

Have you ever seen an exponential graph or a logarithm graph?