Reality

 

I Can See for Miles

Well, here's a poke at you
You're gonna choke on it too
You're gonna lose that smile
Because all the while

I can see for miles and miles
I can see for miles and miles
I can see for miles and miles
And miles and miles and miles

But is there a limit to what you can see?

A moving object in the reality of space-time can be described by its vector. A vector has a magnitude and a direction. That magnitude can be expressed as a scalar, but by definition that magnitude can NEVER be negative. Thus a vector can be opposed by the same magnitude in the opposite direction, but it can NOT be opposed by the opposite magnitude in the same direction.

This has a bearing on space as a dimension. A particle can be described as a vector. But its position in space can NEVER be negative. An exponential distribution, which also does not allow negative values, can  be viewed as a magnitude. The three dimensions of space: length, width, and height; should NOT be expressed as negative numbers. They can be expressed as relative, positional, negatives with respect to a fixed location, but this is merely a translation, not an allowance of absolute negative numbers. Time as a dimension by contrast can be expressed as negative numbers with now defined as zero, the past as a  negative number, and the future as a positive number. If the three dimensions of space are collapsed into a single dimension such as in Minkowski space, where time is retained as separate dimension, the limitation with space, x, no negative numbers,  but with time as both positive and negative values still applies. If this surface, constrained in space and unconstrained in time, is rotated about a third axis, say an axis of imaginary numbers,  that surface becomes a volume. Rotation is infinite but repeating, and rotation can be either positive or negative depending on the direction of rotation. Reality can be considered to be an imaginary number of zero. A clockwise rotation might be considered to be positive, a daydream,  while a negative, counterclockwise,  rotation might be considered to be a nightmare. The rotation of the surface of space-time about an imaginary axis defines a volume that is a cylinder. A vector in cylindrical coordinates can be expressed as re^iθ, where r is the real radius in space-time and θ is the angle with respect to the imaginary axis. This is a special case of Euler’s Formula,  e^ix=cos(x)+i*sin(x), where r =1 and x=θ, and thus re^iθ=r*cos(θ) + r*sin(θ)*i. It is true that θ=arctan( (r*sin(θ))/( r*cos(θ))) regardless of the surface being rotated. It is also true that r²=( r*cos(θ))²+(r*sin(θ))². But this has a different solution for r depending on the type of surface which is being rotated.

If the surface being rotated is flat, Euclidean, it has the conventional solution of

r=√(r*cos(θ))²+(r*sin(θ))²),

but this is only because on such a surface

cos(r)=cos(r*cos(θ))*cos(r*sin(θ))

and the elliptical/circular identity that 1=cos²+sin².  If the surface being rotated was spherical, then the solution would be

cos(r/R)=cos(r*cos(θ)/R))*cos(r*sin(θ/R))

where R is the radius of the spherical surface. As R approaches infinity this also becomes, r=√(r*cos(θ))²+r*sin(θ))²). However if the surface being rotated is hyperbolic, the solutions must satisfy,

cosh(r)=cosh(r*cos(θ))*cosh(r*sin(θ)).

If the surface being rotated must also pass through the origin, then this has two solutions,

  r=ln(0 ± 2*cosh(r*cos(θ))). 

This is because while sin(0) and sin(π) both pass though the origin, are zero, but  cos(0)=1 and cos(π)=-1; the hyperbolic identity that 1=cosh²-sinh²; and cosh(x)=cosh(-x), i.e. cosh is an odd function.

What is conventionally described as two intersecting inverted light cones in Minkowski space assumes that light travels on a flat, Euclidean, surface. If it instead light travels on a hyperbolic surface, the two inverted cones, become a two-sheeted hyperboloid where the sheets intersect at the origin. There is one solution in each sheet of the hyperboloid. However within each sheet of the hyperboloid, the solutions may appear, as 0 and ln(2*cosh(r*cos(θ))) because a particle in one sheet can not pass though the origin unless it is rotated by π, i.e. changes signs, and the other solution is in the other sheet. The average of 0 and ln(2*cosh(r*cos(θ))) is locally r=√(r*cos(θ))²+(r*sin(θ))²).  This approximation holds true until r is more than 5/6 of the absolute, is hyperbolic globally. Thus Pythagoras’ Theorem is true locally, but is not true globally, if the surface of the universe is hyperbolic. And you can see for miles, but if you are in one sheet of the hyperboloid, then you can not see past the origin into the other sheet and you might thus think the solution in that sheet is zero.