Mechanics of space-time’s geometry
This subtitle must certainly appear provocative to anyone who consider the current knowledge of physics and geometry to be untouchable. On the other hand, it is our job to reshuffle the deck and deal a new hand.
“Les jeux ne sont pas encore faits!”
To this end, we must examine the mathematical properties of Schild’s space-time that have a geometrical form and can be illustrated.
Consider a single surface quantum propagating in Schild’s space-time originating in a body that is a source of elementary waves and examine the way of the quantized portion of the surface expands as it increases the radius of its spherical surface.
Suppose here that the space-time in which it is expanding is flat and thus very distant from any other source of spherical waves so that its portion of the discrete lattice comes very close to being perfectly cubic.
In a real physical situation, as opposed to the ideal conditions of our isolated example, although cubic in shape, the lattice does “not” establish specific preferential directions in that zone of space-time by its specific orientation. This occurs because all the flat lattices created by spherical waves originating in the very distant sources that populate the material horizon of every ideal experiment that is supposedly “free of significant fields” coexist in the same zone.
No defined or preordained geometry exists in that zone, therefore, as none of the geometries imposed by the distant sources predominates over the others.
Observe a small portion of the spherical perturbation surface in the immediate vicinity of the source at the quantum level.
The wave surface is made up of individual surface quanta L2 that stretch out and become deformed during propagation according to the established rules that underlie the mathematical consistency of Schild’s space-time.
Quantized spherical perturbation surface propagates in space growing in a series of leaps as a function of the square of its distance from the source from which it comes.
Picture 51. A cubical lattice in Schild’s space-time with a spherical modification.
The linear quanta that mark the edges of the surface “extend” in the space between two successive jumps without in the process ceasing to remain constantly the same units of length. Elementary time T rather than elementary length L is considered elastic in this space-time.
As Milne demonstrated in his famous challenge of the primitive assumptions of General Relativity, it is not indispensable to assume the existence of a rigid measuring rod with which to assess the linear dimensions of space. In fact, a clock and a system of light projections and reflections can be used for this purpose.
We can send a light signal toward the target, which for example could be a mirror, in order to measure its distance. We start the clock when we send out the light and stop it when its reflection returns. Knowing the speed of light, we can measure the interval in space in terms of the interval in time.
Picture 52. The quantized spherical surface increases by squares as it propagates.
Hypothetical measurements by a subquantum observer equipped with a mirror and a light source can teach us about the kind of geometry in which it exists.
There is much more to Milne’s idea, however, than the discovery of the correspondence between intervals of time and space.
Time assumes preeminence over space, as what counts in measuring intervals in space-time is no longer to know how much space light traversed going and coming (which presupposes the existence of an impossible “rigid measuring rod”), but how much time it took to do so. This simply requires the existence of local temporal correspondences, which any clock at rest with respect to the observer can easily provide.
As we have already observed, paths of light called “geodetic” that pass across material fields can be influenced by variations in these fields’ mass and radius. Thus, paths through space may differ significantly from paths in time. The observer’s clock provides the common denominator.
Regardless of whether space-time is continuous or discrete, an observer taking time measurements always has a result that is closer to the reality of the geometry present in that zone of space than one who uses a “rigid measuring rod”.
We can nonetheless already see at this stage of the theory that the influence of the geometry of the perturbation surfaces in the gravitational phenomena condition the effects of the Principle of Relative Symmetry by an inverse function of the square of their radius.
And it is clear that these effects regulate the reaction to this Principle. In the gravitational wave interaction, this is a function of the parallelism of the wave surfaces outside the system of the two bodies. In the next chapter, we will seek to draw what conclusions are possible respecting this argument. These conclusions will allow us to proceed in providing an explanation of gravity based on waves.
The wave nature of gravity
The radius of the curvature of the elementary waves that interact in the phenomenon of gravity is of determining importance. The waves that spread out from the two bodies act according to the Principle of Relative Symmetry outside the system of the two bodies and their ability to act must be assessed as a function of the respective radiuses of their curvatures.
Their energies combine along the straight line that passes through the centers of the two bodies “only” in the zones where the spherical surfaces of the waves originating in both masses are “parallel” to each other.
Picture 55. Wave surfaces with differing radiuses in propagation.
The figure depicting the wave interaction clearly shows that the parallelism between these waves is clearer the more similar the lengths of the radiuses of the curvature of the wave surfaces.
This parallelism can be assessed numerically by examining the number of quanta of the wave surfaces L ² that are actually parallel in the combination resulting in the sum of the spherical waves originating in the two masses.
The parallelism of the wave surfaces outside the system determines the value of the pressure due to the Principle of Relative Symmetry. It thus determines the value of the velocity of the two bodies when the strings that held them at a fixed distance are removed and they are left free to fall into the gap of negative energy that opens and closes cyclically between them at the center of the system.
Thus, given the quantization of the distance and consequent quantization of the surfaces, the combined parameter of the surface proportionate to the velocity of bodies must be:
K = N L ².
Where N is the number of surface quanta L ² of the wave surface “sufficient for the purposes of the sum” and is thus in a parallel state.
As a wave that is effectively parallel is nonetheless a portion of a spherical surface, its effectiveness will be inversely proportional to the square of its radius. Its effectiveness will thus be inversely proportional to the square of the distance between the two bodies.
Two variables figure in this formula:
- L, the discrete elementary length and,
- N, the number of discrete elementary surfaces in the effective wave surfaces that make up the wave trains working by the Principle of Relative Symmetry.
To form some kind of impression of these variables’ numerical value, let us formulate the following hypothesis that should be considered arbitrary at this point.
(Really this hypothesis depends in part on considerations
of the symmetry of the domains of the natural constants ” h ” and “c”)
L = ( h / c ) 2
λ min. = L = 4.8843 . 10 -84 meters.
We now have a hypothesis of the value of the discrete length equal to the dimensional quantum L (we will see how reasonable it is a little further on).
We can use this hypothesis to calculate the number N of the quantum of the wave surface ( L ² ) effective in the wave formula of gravitation.
We deduce N from the gravitational constant of Newton G .
When we remove the combined parameter N from the experimental results using the value of the elementary length L, we are confronted with the result that the value of N is:
N =1/L . 1.877887684715i . 10 14
(where 1/L is a pure number adjusted to the unit of measurement, that tells us the number of linear quanta L in one meter).
The number N may at first sight appear to lack physical significance, but it can become surprisingly relevant when broken down into the following factors:
N = 1/L . ( 137, 0360 . 102 ) 5
where the number 137,0360 looks too much like the inverse of the Fine Structure Constant:
1 / α = (2 π e2) / (4 π eo h c) = 137,03599976(50)
for us not to identify the two numbers.
The Fine Structure Constant “α” is the parameter that characterizes the electrical interaction between protons and electrons in the atom and it is considered the characteristic constant of electromagnetic interactions.
It is the only truly a-dimensional numerical constant known to physics up to now that has proved fundamental in the structure of matter and does not depend from the anthropomorphic parameters.
We must now introduce a third term into the wave interaction between the masses that will limit their gravitational interaction.
This third term is not new. What is new is the approach we use to arrive at it and the application we make of it.
Picture 56. We can derive the dependency on the inverse square of the distance and deduce a quantum gravitational force in which we find the quantization of space-time in addition to the quantization of energy.
If we can corroborate this identification with other elements and considerations, we will have found the first plausible relationship to link elementary waves and their gravitational effect on masses to the electromagnetic interactions.
We will be in a position to present more solid arguments to support this thesis below. But even here we can assess the model’s ability to describe gravitational interactions and observe that we are doing something that is more than a simple exercise in numerology.
In our discussion of waves’ ability to influence gravitation as a function of their degree of parallelism, we have seen how the gravitational wave interaction completes the two terms involved in classical gravitation.
It allows us to derive the product of the masses and the inverse of the square of the distance between the masses.
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