Temporal Density & Relativity

Virtual Chaos

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This section is taken verbatim from appendix II of the book "Relativity" by Albert Einstein. The only thing I offer here is that his reference to imaginary time, is referred to in Virtual Chaos as temporal density (tp).

APPENDIX II

MINKOWSKI'S FOUR - DIMENSIONAL SPACE
("WORLD") [Supplementary to Section XVII]

We can characterize the Lorentz Transformation still more simply if we introduce the imaginary sqr(-1) times ct in place of t as time-variable. If, in accordance with this, we insert
and similar for the accorded system K', then the condition which is identically satisfied by the transformation can be expresses thus:

(I2)

X'12 + X'22 + X'32 + X'42 = X12 + X22 + X32 + X42

That is, by the afore-mentioned choice of " co-ordinates," (IIa) is transformed into this equation.

We see from (I2) that the imaginary time co-ordinate x4 enters into the condition of transformation in exactly the same way as the space co-ordinates x1, x2, x3.It is due to this fact that, according to the theory of relativity, the " time " x 4 enters into natural laws in the same form as space co-ordinates x1, x2, x3.

A four dimensional continuum described by the " co-ordinates " x1, x2, x3, x4, was called " world " by Minkowski, who also termed point-event a " world-point. " From a " happening " in three dimensional space, physics becomes, as it were, an " existence " in the four dimensional " world. "

This four dimensional " world " bear a close similarity to the three dimensional " space " of (Euclidean) analytical geometry. If we introduce into the latter a new Cartesian co-ordinate system (x'1, x'2, x'3) with the same origin, then system x'1, x'2, x'3 are linear homogeneous functions of x1, x2, x3, which identically satisfy the equation

X'12 + X'22 + X'32 = X12 + X22 + X32

The analogy with (I2) is a complete one. We can regard Minkowski's " world " in a formal manner as a four dimensional Euclidean space (with imaginary time co-ordinate) ; the Lorentz transformation corresponds to a " rotation " of the co-ordinate system in the four dimensional "world. "

[END OF APPENDIX II]

So many of you have been logging straight to this piece, that we decided to move it into the new stuff where it is more easily accessible, and add a bit more of the math, because that's obviously what you're after.

This added math will be used later [hopefully] to show:

We are including this stuff without the usual essay to explain it, because if you follow the preceeding appendix, this stuff should be no problem for you to understand. We have made several attempts at the essay so far, but unfortunately none of them read very well, and all of them would have left the general public with their head spinning and their ears bleeding.

So, we will keep trying to translate this into plain English that the general public can understand. Any suggestions are more than welcome, but in the meantime, heres a headstart on forthcoming virtual chaos...

I. Diamond Sphere

  1. Xd2 + Yd2 = Ha2
  2. Ha2 + Zd2 = Hb2
  3. Hb2 = Xd2 + Yd2 + Zd2

II. Absolute Spatial Sphere

  1. r2 = (X-A)2 + (Y-B)2 + (Z-C)2
  2. r2 = Xd2 + Yd2 + Zd2

III. True Spatial Sphere

  1. r2 = (XdTx)2 + (YdTy)2 + (ZdTz)2
  2. r2 = (Xd/Xd)2 + (Yd/Yd)2 + (Zd/Zd)2
  3. r2 = 12 + 12 + 12
  4. r2 = 3
  5. r = +/- sqr(3)
  6. Ra = r(theta)
  7. Ra = r(Pi)/2
  8. Ra = +/- (sqr(3))(Pi/2)

I. Diamond Sphere

  1. The basis for the total spatial sphere is a diamond shaped object. Each hemisphere of the diamond consists of four right angle pyramids(angle xy is Pi/2 as is yz and xz). Using the Pythagorean Theorem the first hypotenuse is calculated using the distance in the X plane and the distance in the Y plane.
  2. Next, the value for the first hypotenuse(I1) is used with the distance in the Z plane to calculate the greater hypotenuse.
  3. With the equations from (I1) and (I2) a composite equation is formed which calculates the greater hypotenuse length. This arm forms an edge of the diamond sphere.

II. Absolute Spatial Sphere

  1. In order to calculate the radius for the Absolute Spatial Sphere, the equation for a sphere is used.
  2. The values for (X-A) and its comparable Y and Z distances are reduced to the notation of Pd, the distance in the given plane P.

III. True Spatial Sphere

  1. Although the Absolute Spatial Sphere gives the absolute value of the volume of space, it is not the actual value. Each distance in plane P has a temporal density or Tp for short. Multiplying these two values give the true value in plane P. By substituting the true value in place of the absolute distance in section II the equation for the True Spatial Sphere is achieved.
  2. Mass and energy is conserved, for Tp is actually the inverse of Pd. Therefore as the volume of space increases the temporal density decreases to compensate.
  3. Xd divided by Xd is one, simplifying the equation immensely.
  4. Carrying out the algebra, the equation reduces to the square of the radius is equal to 3.
  5. Taking the square root of both sides the true value of the radius is determined to equal plus or minus the square root of 3.
  6. The equation for radial arm length is used to calculate the true value for a radial arm, or Ra, of a pie wedge of the total sphere.
  7. Since the total sphere is divided into the eight wedges, the angle is always Pi/2.
  8. Substituting in the value for the radius from III5 the value for Ra is determined to be plus or minus Pi/2 times the square root of 3.

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