The workbench of God
The mathematical characteristics of Schild’s model of reticular space-time tell us that it forms a cubic lattice. More precisely, we should consider it :
“hypercubic lattice”.
This means simply that the lattice represents more than just three-dimensional space. It must be seen as “space-time” and possess four dimensions. The mathematical model integrates time coherently with the three spatial dimensions to form a fourth dimension.
There is a way to glimpse the reality of this “fourth dimension” which will not overwhelm us with its mysteriousness. Think of our model as a three-dimensional cubic lattice in which the smallest unit of time of which we can conceive is the discrete time quantum T.
We can describe the quantum T as the time required for a perturbation to pass between two extremes (knots, event points) in a net of the lattice with the smallest possible length equal to the discrete length quantum L.
We can thus understand the elementary length L as the smallest possible measure of space between event points in the lattice (the only points that can be identified in the lattice at which anything occurs) that is not zero. As a result, any length can and must always be understood as a whole number of linear quanta L.
Similarly, we must understand elementary time T as the smallest possible measure of time that is not zero required to travel between two event points in the lattice. Every other measurement of time is made up of a whole number of time quanta T.
There can only be local variations in the lattice’s state of temporal uniformity in the discrete lattice of space-time. Such variations move about the lattice as perturbations in the “state” of the lattice itself at velocities that can be close (smaller) or at most equal to 300,000 kilometers per second.
Our goal in establishing the new physics will be to describe these variations in the lattice of Schild’s model of space-time. Once organized into certain precise geometric models, these variations can represent and describe all the characteristic properties we attribute to the basic categories of our physical experience: bodies, radiation and fields.
In this, there is nothing particularly new under the Sun.
As Jammer notes in his aforementioned book, William Kingdon Clifford published the work On the Space Theory of Matter as early as 1876. In it, Clifford carried out the daring project of demonstrating the ultimate identity of space and matter.
Clifford had translated into English the works of the German mathematician Herman Riemann, who in 1850 became the first person to study the possibility that the structure of space was non-Euclidean.
Space, in Clifford’s view, was not merely a stage on which physical events were performed. He understood it instead as the only material out of which physical reality was constructed.
Nothing else occurs in the physical world but this variation
of the curvature of space .
As Jammer put it:
Clifford was nonetheless unable to carry out his ambitious program and interpret the concept of mass in purely spatial or geometric terms.
We do not hide our ambition to do this, however, as we believe we are now able to interpret mass in purely geometrical terms. We therefore think we can now successfully carry out Clifford’s program, which had remained impossible in his own day.
To this end, we must seek to understand the principal characteristic that makes Schild’s model of discrete space-time the best candidate to play the role of aether in the new physics we derive from Wave Field Theory.
Time as the matrix of the variation.
Everything we know about time in one place is linked to the concept of duration. When we think about time in terms of different places, however, our consciousness of it is necessarily linked to the concept of velocity.
In the discrete lattice, we can speak of duration when we consider the passage of two subsequent perturbations through a single event point in the lattice’s “state of uniformity”.
This already satisfies Einstein’s first condition for the existence of a medium to transport light: what moves about in our space-time-aether is not the aether itself but the perturbations in its state.
Time varies locally within this discrete, geometrically three-dimensional aether and produces a variation in its spatial and temporal structure that oscillates along the fourth dimension just as the geometry of the surface of the sea oscillates along the third dimension.
This in fact creates a variation in the duration of the quanta of time in places that are discretely adjacent to one another.
What one sees, however, is a variation in the geometry of the structure of space-time in places that are discretely close to one another.
We can observe this variation by using the behavior of light as a perturbation in space-time to ascertain the kind of geometry that can be attributed to a particular zone in space.
It is the perturbation in Schild’s space-time that makes the variations of the uniformity of the structure’s state, and these vary in adjacent places.
The lattice itself does not move, only the variations of the lattice structure move in the lattice. Like waves on the surface of the sea, it is not the water that moves, but the variations in the geometry of its surface.
We can therefore say that the state of the net’s uniformity varies between the passage of two perturbations through a single “event point” for a temporal duration nT where “n” is an integer number and “T” is the temporal quantum.
Velocity is considered the ratio of a length to the time it takes to cross it. In the lattice, we can thus only talk of a thing as possessing velocity when we examine the movement of a perturbation from one event point to another in the lattice.
The ratio between the “discrete” linear quantum L and the temporal quantum T is equal to the “finite” velocity of the perturbation in the state of the lattice perceived in its passage from one event point to another, which is as close as possible to the first.
We must explain a peculiar characteristic of the discrete model of space-time to make Schild’s net comply with the mathematical properties required to make it consistent with itself.
To make it conform to the dictates of Relativity, the temporal quanta T must possess a certain kind of elasticity.
The mathematical structure Schild had to give the lattice in fact explains why the time quantum T can possess a specific degree of variability, a predefined property that determines its elasticity.
The time it takes a perturbation to move from one event point to the point that is as close to it as possible varies between a minimum and a maximum.
In practice, to make Schild’s space-time mathematically consistent, we must allow that the time quantum T is elastic. Although variable within certain precise limits, it must constitute an invariable quantity, a unit of time, an indivisible time quantum.
This makes Schild’s model of space-time the best candidate to describe the variability of the geometry of space-time. When we admit the possibility that the lattice’s state of uniformity can vary, we can conceive of the birth and movement of variations in its uniformity.
Variations in the “state” of reticular space-time are solely and uniquely temporal. This observation’s effects the value of the velocities at which perturbations move about the lattice in a precise way.
This represents the geometrical properties of the zones of the lattice crossed by the perturbations. To form a mental picture of this, picture a line of people arm in arm crossing a broken terrain that includes both muddy and paved areas.
Suppose further that everyone is trying equally hard to maintain the greatest possible speed. You will observe that the line changes direction as it marches in proportion to the difficulties each person in the line encounters in crossing the terrain.
As it has to cross a mixed terrain consisting of areas of both mud and pavement, the line will be forced to fold in the direction of those people who have the greatest difficulty moving forward.
Suppose you are too far away to perceive the actual condition of the terrain and believe it to be level and uniform. You might think the trajectory varies because the people who make up the line do not know how or wish to maintain the same velocity.
If I assure you that careful tests have shown that these people all consume the same energy as they move forward and that this confirms that they are making an identical physical effort, you would seek a different explanation of the deviations in their trajectory. You might suggest that these people had different metabolisms.
To which I would respond by pointing out that the same people were proceeding across the terrain at differing velocities as is clear from the resulting zigzag trajectory. This is evidence that their metabolisms are statistically uniform.
At this point, you might begin to doubt your premises. You would seek to analyze them carefully to find a physical explanation external to the physical state of the people making up the line that could explain your observations.
You would notice immediately that your underlying assumptions were: first, that the terrain was uniform and, second, that each person took the same time to travel the same distance.
You could apply variations to both of these presupposed conditions, but you would quickly realize that the simplest solution would be to accept that the condition of the terrain the people were crossing was not uniform.
By changing this one precondition, you could also explain the second: the time required to cross certain portions of the terrain was different for the different parts of the line.
By analogy, variations in the geometry of the lattice from zone to zone explain the existence of variations in the velocity with which perturbations propagate. Variations in the duration of temporal quanta in different zones can also account for this variation in velocity.
If we accept this elementary possibility and seek to simplify ideas as was so important to Occam, we can continue with our investigation of the behaviors of perturbations in discrete space-time generally.
An elastic temporal quantum can simultaneously present different values in different places, which even if contiguous remain distinct in the lattice
(and represent different event points). Different elements in the front of a single moving perturbation can thus simultaneously present different velocities.
As a result, the quantized portions of a single perturbation front can move at sufficiently different velocities to produce deviations in the trajectories of the different quantized perturbation surfaces.
Picture 4. An elementary front propagating in the lattice.
The result of applying these diverse velocities to the propagation of discrete portions of the same front is perceived as a deviation in the trajectory of the propagation of the front deviating from a straight line.
Think of these perturbations as a succession of periodic waves, one after another. Let us call these waves “elementary” to distinguish them from waves of every other type.
When elementary waves propagate in their own environment, their presence conditions the geometry of the space in which they propagate. In this same environment, they condition the elementary dimension of time. The properties of variations in the geometry of the lattice as a three-dimensional structure of space are thus linked to variations in the temporal quanta.
The geometric form of the resulting structure is linked to its temporal behavior. As it is conditioned by this elasticity, its geometry “measured in terms of the velocity and behavior of its perturbations as they propagate” can vary locally.
The property by which reticular space-time conditions the geometry of space, fits the ideal characteristics of space-time in Relativity perfectly.
In General Relativity, Einstein in fact tells us that space-time is characterized by its geometry and that this geometry is a function of the fields of the material masses contained in it.
The gravitational field of a mass modifies the geometry of space-time around itself so that light that is passing through it moves in space-time according to the geometries it encounters along its path. It is here that we find the field, the third of our fundamental categories, in our discrete aether.
We can now see how the field, when seen as a tool to describe variations in space-time, not only invades the area described by radiation and matter, but can absorb all the properties characteristic of these two categories into itself and render it unnecessary to continue to see them as separate entities.
The gravitational field of a mass is only one of the fields we know. Other fields include the electromagnetic field of an electric charge or magnet and the nuclear field of a proton.
Fields were initially conceived as possessing an uncertain nature that at different times was interpreted in two antithetical ways precociously proclaiming the type of duality we now wish to challenge.
The concept field has often been seen as a construct, a complex construction consisting of data describing the properties of the space surrounding a mass or charge. This construction was designed to allow us to assess the differing characteristics of what exists and occurs around a charge or mass as a single entity.
Over time, the concept of field developed in an extraordinary fashion and its ability to describe the actions masses and charges exchange characterized their properties so well that they took on an autonomous role that was no longer simply descriptive.
To understand the new role the concept of the field plays in the justification of mass and radiation within our new system-aether, we must now formulate some hypotheses concerning the wave nature of energy and mass.
We will then assess whether and how these hypotheses can be supported by the laws of physics we already know and examine how their behaviors can be confirmed experimentally.
We will then examine how these hypotheses can be unified consistently when they have been immersed in the context of the discrete and reticular space-time we have just described as aether.
We will assess primarily whether the introduction of this concept into physics would allow us to form a better understanding of the world.
Along the path that leads us toward the confirmation of these hypotheses, we will identify new laws that are more general and unified than those which have guided physics up to this point.
These new laws will allow us to predict and explain new phenomenons and link all levels of our investigation from the micro-physics to the macro-physics and the mega-physics in a single plan.