In this chapter, we discuss fundamental forces and explain how and why they emerge. However, this chapter assumes that the reader has a basic knowledge on both fundamental forces and previous chapters of Geometric Generalization.

Once again, we should utilize our balloon example (Section 7.1) in order to emphasize a few points.

In previous chapters, we have deeply discussed that Geometric Generalization considers elementary “particles” as strain formations like knots on the surface of a balloon. Knots on the balloon affect the whole surface of the balloon by both wrinkling the balloons surface and decreasing the balloon’s overall circumference. Hence, according to our balloon example, we may assume that if an extra stress is formed in knots on the balloon’s surface, this extra stress should affect the rest of balloon’s geometry.

The mechanism that causes confinement volumes to contract more when they are gathered in a lump was discussed in the previous section. Effects of this extra tightening of the confinement volume on the overall space-time geometry can practically be visualized by analyzing the effects of a knot on the surface of a balloon:

 - Overall circumference of the balloon is tightened by a knot, since some portion of the balloon’s surface is tied into that knot. Consequently, the knot increases the inner pressure of that balloon, which is constant everywhere in the balloon according to Pascal’s principle.

 - The knot deforms the balloon’s surface around itself as if the balloon’s surface is depressed locally. Consequently, the knot changes the wall tension around itself (LaPlace’s Law).

Therefore, any variation in the stress of the balloon knots should change the knot’s effects on the overall geometry of the balloon surface.

It is not very complicated to calculate how a knot on the surface of a balloon affects the overall shape of the balloon (ignoring the elasticity of a balloon’s material). Some area of surface of our balloon is tied into a knot. The basic parameter that describes the variation on the balloon’s surface is size of the area that is tied into a knot. Hence, if the size of the area that is tied into a knot varies somehow, then the effect of the knot on the overall size of the balloon will vary consequently.

However, it sure that our expanding space is not an elastic material like the balloon’s static surface; distinctions of the balloon example have already been discussed in previous chapters. However, we may assume that the effect of the knot (the confinement volume) on the overall space-time geometry is directly dependent on the tightness of that the confinement volume alone, which also determines the degree of mass of that knot. Therefore, we can now state that any additional contraction in the confinement volumes of strain packages (knots or vortexes) changes the effects of those strain packages to the overall expanding space.

From the beginning of this paper, we described how the expansion is constant in compressed or confined areas, although it collapses onto itself against the universal strain on the expansion within the wrinkling epoch. In fact, our hypothesis is based on this idea, and in initial chapters, mathematical definitions were developed based on this constancy.

The graph below demonstrates the universal strain on the expansion in our space-time geometry. Please note that the length of the circulating (helical) radial distance (A₁ - A₄) in the wrinkling epoch (Hubble’s expansion) is equal to the original unwrinkled one (the inflationary epoch), (Section 5.4).

Figure 7.13 The inflationary epoch to wrinkling epoch

In the previous section, we discussed the mechanism of gravity at the smallest scale, which suggests that when quanta of matter are lumped together they compress each other’s confinement volumes mutually. It seems that we are faced with a new formation this time. As a result, the extra compression of the confinement volumes (like compression on the proton that is in the center of a uniformly (+) charged spherical shell) changes (restrains) the constant expansion in the confinement volume.

Figure 7.14 Gravitational restraint on the expansion

Practically, with extra compression of the confinement volumes, Estring length decreases and obliquity angle increases regionally.

Such a compression mechanism implies that constant and continuous fundamental expansion, which is preserved even in the state of the universal strain on the expansion, is now restrained, and it decreases regionally.

According to Geometric Generalization, gravity is the regional restraint and compression on the expansion.

The regional restraint on the expansion eventually decreases the rate of the expansion (practically Estring length), and deforms (curves-buckles) space. Hence, it should be considered as a secondary effect on the expansion. Additionally, such a regional restraint on the expansion breaks down the universal homogeneity in the expansion as if there is a stress towards spatial directions.

Essentially, there is a surprising similarity between the strong force and gravity. Both of them are based on similar mechanisms in which the intrinsic tendency to expand creates a compression on itself. However, while strong force locally confines the expansion in local strain formations, gravity causes regional restraints in the expansion.

The mechanism that compresses the confinement volume increases the tightness and decreases the Estring length in that confinement volume. This variation has significant consequences.

First, according to our metric definitions (of distancespatial, timeas clock-ticks, and mass), it can be said that towards a gravitational well, spatial distance metric contracts and rate of clock-ticks decreases relatively in that confinement volume’s inertial reference frame. Additionally, when such an extra contraction occurs, mass of that confinement volume increases relatively.

Practically, this effect, which tightens the confinement volumes, causes metrics of different observers to show variation as if metric of space-time changes (contracts). This is the case, even though Nature is not formed of a medium or fabric. As a result, formulation of the variation of metric at each location enables the formulation of the effect of gravity.

Figure 7.15 Metric contraction towards a gravitational well due to Estring length contraction.
Blue lines (space) illustrate the Einsteinian curvature, where space is deformed (buckled)
because of the regional restraint on the expansion.  

At this point, let us examine general relativity before going any further.

In certain aspects, The Theory of General Relativity mathematically describes this variation in space-time geometry. According to general relativity, matter curves space-time, and consequently, curvature influences matters’ motion. In a limited sense, the space-time deformation suggested by general relativity can be visualized by examining the deformation on the surface curvature of the (inelastic) balloon by the knot on its surface. In the beginning, general relativity was formulated assuming that the universe was a static entity like our balloon example, and contents of this static universe were in motion relatively to each other.

This paper assumes that geometry of space-time has a fundamental tendency to expand, which is a result of logical principles (Section 4.3 “The Cause”), and quanta of matter are strain formations like knots or vortexes on this expanding geometry (Section 5.3).

On the other hand, general relativity is not based on fundamentally expanding geometry, and it assumes that matter curves (almost) static space-time. Consequently, general relativity is surely aware of the inseparable relationship between matter and space-time geometry, but it does not define why this relationship exists physically.

A pure concept of space involves only empty spatial dimensions, and it should not have a reason or capability of containing stress or strain. Any local curvature or deformation in space-time due to matter leads to a paradox that space is no more exactly space, but it is a kind of medium, which is able to react and resist matter. In fact, this physical and philosophical inconsistency appears in equations of general relativity, which equalizes geometric quantities (Einstein’s strain tensor) to force (stress-energy tensor).

E 7.8 Symbolic formulation of The Theory of General Relativity.
(G_ab is Einstein’s strain tensor, G is gravitational constant, c is speed of light, and T_ab is stress-energy tensor)

This paper also defines the concept of energy and mass as strain formations on expanding geometry like an interpretation of Hooke’s law, which relates stress to strain. However, this paper does not assume that energy emerges in physical reality, because space-time is deformed and space-time resists deformation. From the beginning of this paper, we clearly mentioned that energy-stress is on the tendency to expand, where the expansion is confined somehow, and the compression or confinement of the expansion creates the concept of energy and mass in physical reality. Strain (the tightness of the confinement volume) is the physical appearance of the energy, but not its cause, and magnitude of strain simply indicates the amount of compression or confinement on the expansion at that knot or vortex.

Finally, according to this paper, there is a mutual relation between matter and space-time; since, quantum of matter itself is just a strain formation in the expanding space. However, this paper does not accept a concept of space-time that has a capacity to resist being curved or deformed, but it assumes that the expansion resists compression. This is the point where this paper diverges from general relativity.

The formulation that Geometric Generalization suggests resembles the formulation of General Relativity, and it is based on the relation between matter and space-time. On the other hand, since this paper accepts quanta of matter as strain packages in expanding geometry, the relation is between local strains on the expansion (matter) and the global restraints on the expansion (global deformations on space-time).

One side of the equation describes the variation (the restraint) on the expansion. In other words, it describes the deformation in space-time geometry.

As we discussed just now, this deformation is about the variation (contraction) in our spatial distance metric. However, we need an additional variable, which we assumed, until this chapter, as a constant for everywhere in space, to allow us to formulate the variation in spatial distance metric. It is the degree of the universal strain on the expansion, and it appears as unit Estring length (the length of helical path for one rotation in space-time) in our equations. (Readers may examine the definition of Estring in Section 6.5.2)

In Section 5.2, we discussed that the expansion is permanent, even if it collapses onto itself during the wrinkling epoch. In fact, in Section 6.5.2, we noted that Estring length is constant for relative observers (accelerated). However, this time, under such extra restraint on the expansion, Estring length regionally varies under gravity. Practically, radius of helical path in the confinement volume is tightened by the restraint.

Consequently, unit Estring length (or its angle with spatial plane) defines the strain on the expansion at that location. As a result, it describes the gravitational effect (restraint-deformation in space-time) at that location.

Please note that Estring is not a spatial length, but it describes the obliquity of the original radial (time) dimension; hence, the restraint on Estring length describes the variation on the radius of curvature at a location. “Curvature of space-time”, which is defined as Einstein strain tensor (G_ab) in general relativity, is an interpretation of the restraint of Estring length. The variation in the rate of the expansion caused by matter can be visualized by the regional variation in the wall tension of our balloon example.

The other side of equation describes how matter (heaps of strain packages with mass) compresses and changes unit Estring length by heaping together. Variation of unit Estring length is a function of the restraint caused by matter, according to the mechanism that we discussed in previous chapter. The stress caused by the heap of matter appears as stress-energy tensor (T) in general relativity.

The universal strain on the expansion caused by a single quantum of matter can be visualized by stress caused by a single knot that is tied on the surface our example balloon. However, when these knots heap together their effect on the expansion varies, since they increase each other’s stress content by compressing each other’s confinement volumes.

In gravitational fields, the fundamental tendency to expand is self-compressed and restrained, and the expansion decreases relatively under the influence of gravitation, whether the expansion is confined in volumes or not.

Practically, this situation is observed when the path of light (electromagnetic radiation) is bent around a massive object. In fact, path of light is exactly the original direction of the radius (time dimension) in the inflationary epoch, which has had an oblique angle with spatial plane since it collapsed with the universal strain on the expansion. As a result, path of light in space-time represents the direction of the expansion (original radial direction). Path of light is directly affected by regional restraint on the expansion, as if light slows down in gravitational fields. More appropriately, wavelength of the electromagnetic radiation approaching the gravitational well contracts (blueshift) relatively, due to the regional restraint on the expansion.

Please note that “slowing down” does not mean that constant speed of light varies somehow. It simply means that, in gravitational fields, both spatial distance metric and time as quantity of clock-ticks contract mutually with the regional restraint on the expansion. On the other hand, speed of light is the constant relation between the spatial distance metric and time as quantity of clock-ticks, which is constant for any relative observer.

Additionally, we can also say that the circulating expansion in the confinement volume is also affected in the same manner. The expansion in the confinement volume circulates under gravitation, and the confinement volume contracts additionally in gravitational fields. Consequently, inert bodies accelerate and gain kinetic energy (according to distant observers) in gravitational fields, since the circulating action in their confinement volumes is affected exactly as the path of light is affected. Our definitions of distancespatial, timeas clock-ticks, and mass include mathematical formulation of this variation, since these definitions are based on the tightness of the confinement volumes (Section 5.3, 6.1, and 6.3).

Now, let us discuss the physical meaning of gravitational constant: