THE SYMBOLs METHOD

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THE SYMBOL'S METHOD - 10

3.5.- THE CONTRACTION FACTOR AND ITS ANGULAR EQUIVALENCE

In 1905, Albert Einstein published the first Relativity Theory (Restricted) and in 1916 the second or Generalized. In the first he openly treats the spatial contractions, keeping the concept of force, and in the second he openly introduces the concept of space curvature and completely avoids the force concept.

Yet, the concept of spatial contraction is older, as it appears published for the first time in 1892 by Hendrik A. Lorentz, and also independently by George Francis Fitzgerald, both of them induced by the negative attempt to measure the absolute speed of planet Earth, respect to an hypothetical still ether (Michelson-Morley experiment).

From that, the equation which gives the contraction factor is precisely called "contraction of Lorentz-Fitzgerald".

Einstein uses the spatial contraction, calling it "contraction of Lorentz", because he had directly learnt it from Lorentz. Even written notes by Einstein are conserved, where he ardently praises the clarity of Lorentz seminars where he assisted when he was a youngster.

The contraction factor was deducted to explain in full the negativity of Michelson-Morley experiment, but Heaviside and Lorentz also deducted that the force field created by a charge in movement was as the still field but smashed by a quantity resulting to multiply the charge by a factor which turned out to be the same as the experiment previously told, which is the following:

Contraction factor =

This means that measures of a system in motion, in the direction and sense of motion, should be multiplied by the square root of: 1 minus the square of the speed of motion divided by the speed of light.

The necessary condition to make the contraction factor to work, and it does so in an incredibly exact way even in the most refined calculations, is that the speed of light would be a universal constant independent from the emitting focus and the observer, whatever the speeds of both are.

Under these conditions, the Lorentz-Fitzgerald contractions holds all the elements to be resembled an angular function, and concretely to the sinus function. Indeed, if we name V the speed of the emitting focus and the observer and C the invariant light speed we can draw an sketch as follows:

Figure 22

With which we can say that:

D
Sin a = ------ y que D ² = C ² - V ²
C

If we now divide the second expression by C² we obtain that the contraction factor equals the sinus of the angle formed by the speed of light and the speed of the mobile. We obtain the following:

The contraction factor of a still body (factor = 1) is equivalent to forming 90º in respect to the speed of light and the contraction factor of a body travelling at light speed (factor = 0 ) is equivalent to form 0º with this latter speed, as it could not be otherwise as both speeds are equal.

Between the two limit cases we will have all the other angular equivalences which we can express, including the former, such as :

Angle between V and C = Arch sinus

Because if vector C is constant it happens that the angle which forms with vector V gives as a result that:

Angle sinus = Spatial Contraction Factor

3.6.- STRUCTURAL CONTRACTION FACTORS.

We know by section 3 that the exponent of i is expressed in degrees. Given that this exponent indicates the morphological structure of the space, it is possible to calculate the contraction factors affecting the structure of each of space dimensions, because the general expression of the angular spaces is:

i ^{( 2.n.G ) /
T}

Being:

i = Square root of –1
n = Number of spatial dimensions
G = Turning degrees ( independent variable)
T = Necessary Degrees to pass from a position to its contrary:

( Curvature = 0 ....... T = 180 ) - ( Curvature > 0 ....... T > 180 ) - ( Curvature < 0 ........T < 180 )

And from it, all the i expressions previously seen can be directly obtained:

Si T=180 y n=0 ...................... i ^{0 / 180} ............... Sin 180 = 0
Si T=180 y n=1 ...................... i ^{G / 90} ................ Sin 90 = 1
Si T=180 y n=2 ...................... i ^{G / 45} ................ Sin 45 = 0.7071067814
Si T=180 y n=3 ...................... i ^{G / 30} ................ Sin 30 = 0.5 y
Si T=180 y n=4 ...................... i ^{G / 22.5} .............. Sin 22.5 = 0.3826834325

We can notice that the structural contraction factors of the angles of different morphological structures vary according with the angular dimension variation.

Thus, for 0 dimension contraction is total (factor = 0), for dimension 1 contraction does not exist (factor = 1). From here, the contraction factors start to diminish, meaning that the spatial contraction start to increase again.