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.

This section is easy for readers who carefully examine the universal strain on the expansion in previous chapters. However, let us remind a few points in order to start discussing gravity.

Our space-time geometry has a fundamental tendency to expand. This tendency is permanent, even if the expansion has collapsed onto itself, and it is locally confined into knot-like formations (e.g. protons). However, because of the tying formation in knots, tendency to expand compresses itself (strong force). Additionally, there appears energy in the confinement volumes (equals to their rest mass) against the internal stresses that tie the expansion in knots. Finally, we asserted that electromagnetic field is the flow of strains like extending wrinkles in our expanding geometry.

Consequently, the confinement volume is formed of a few opposite effects, which are in equilibrium:

A) The confinement volume tends to expand inversely proportional to its tightness; this tendency is a result of logical geometric principles (Section 4.3 “The Cause”).

B) There appears a local stress in the locking mechanism of the confinement volumes with knot formation, which is against the expansion (the flux).

C) Additionally, electromagnetic interaction (spatial wrinkles) affects the tying mechanism of knots.

As we discussed in Chapter 5, the tightness of the confinement volume determines the energy content of that elementary “particle”, since our definition of energy is an interpretation of Hooke’s law, which relates strain and stress. Furthermore, tightness of the confinement volume determines relative clock-ticks rate and the spatial distance metric for that “particles’ ” inertial frame of reference.

Figure 7.9 Equilibrium in the confinement volume

The equilibrium in the confinement volume can vary in some special conditions. Let us imagine two protons, which are close to each other. These protons repel each other because of the electromagnetic interaction. Their confinement volumes become slightly tightened as the electric potential between them transforms into kinetic energy. This well-known simple scenario is agreeable with both empirical data and our previous discussions.

Figure 7.10 Inertial and accelerated helix

Now let us discuss a thought experiment in order to go beyond the above scenario. Let us assume there is a proton in the center of a uniformly (+) charged spherical shell. This time proton in the center stays stationary, and it does not gain any kinetic energy, since the net electrostatic force is zero in the center. On the other hand, we may assume that the proton in the center is under an isotropic force; therefore, it does not gain kinetic energy. This isotropic force affects the tying mechanism in the confinement volume, and this isotropic force increases the tightness of that confinement volume slightly. (In fact, this additional contraction is equivalent to the contraction that defines kinetic energy.)

Consequently, this contraction varies the equilibrium on the confinement volume as if there is an additional strain on the expansion at that region.

Figure 7.11 Squeezed helical path

Such an extra compression in the confinement volume has an important meaning. According to our discussions on relativity in Section 6.1 and 6.3, definitions of distancespatial, timeas clock-ticks, and mass are the functions of the tightness of the confinement volume in its inertial reference frame. Therefore, our stationary proton in the center of uniformly (+) charged spherical shell should be in an unusual state. It is as if it has gained a kinetic energy, because its confinement volume is relatively contracted.

Exact degree of any extra contraction can be calculated as if the confinement volume has gained kinetic energy because of the electrostatic force. This degree is the basis on which to calculate the effect caused by matter on space-time. In other words, this degree describes the amount that matter (local stress) affects the geometry of space-time.

Now, it is time to transform our abstract experiment to a more realistic scenario. This time, let us fill a box with many protons in such a way that protons cannot escape from that box, and compress the box to make it smaller against the protons’ electrostatic repulsion.

Practically, protons in that compressed box gain kinetic energy because of their electrostatic repulsion; hence, their confinement volumes contract as we formulated on Section 6.6. This situation can be observed as the temperature increases in the box.

Furthermore, we may assume that the confinement volumes of protons in that box also contracts slightly as if they gain kinetic energy, but without gaining kinetic energy. In this compressed state, confinement volumes behave like additional supports to each other and help each other become tighter.

This situation is similar with the previous example of proton, which is in the center of a uniformly (+) charged spherical shell. Hence, knot formation of the confinement volume is able to compress itself slightly more than the extra contraction that determines its kinetic energy. Here, the equilibrium in the confinement volumes of protons varies unusually, after the box is compressed. Same tying mechanism of knots causes a higher contraction of the size of the confinement volume due to the compression on the box.

Figure 7.12 Confinement volumes compress each other

Eventually, this mechanism is the cause of gravity in small scale. Confinement volumes are tightened in this pressured box, and it results in relativistic consequences as we discussed and formulated in Chapter 6.

Let us review our definitions:

E 7.7 (2π r _electron is equal to the Compton wavelength of electron. Mass of an elementary “particle” is defined as a ratio of its Compton wavelength to electron’s Compton wavelength).

Simply, when the confinement volume of a proton has an extra contraction as explained above, according to our definitions, its mass increases, rate of its clock-ticks decreases and spatial metric in its inertial reference frame contracts relatively. Eventually, extra contraction of the confinement volume causes an effect as if our box that is full of protons compresses itself.

Moreover, when our box is large enough to contain many protons, consequences of this effect becomes so dense that it holds protons together and compresses them without any external box pressure. Our sun and other stars are such lumps of matter, and this effect is known as gravity.

This section (7.6) examines the electromagnetic origin of the mechanism of gravity at the smallest scale. Naturally, if there is such a connection, there should also be consequent effects that can be observed experimentally.

Geometric Generalization assumes quanta of matter as knot-like or vortex-like local deformations on space-time, and electromagnetic field as flow of wrinkle-like deformations, which appear around the knots (charges).

Interestingly, General Relativity assumes that matter and space-time mutually interacts, and it formulates the effect of gravity as regional curvatures-deformations on space-time.

In the following sections, we will discuss in depth how our approach develops the consequent effect of gravity. However, we should now note that both well-known formulation of gravity by General Relativity and our definition of matter and energy are based on the concept of deformations in space-time in a similar manner.

General Relativity was confirmed by observing the bending (curvature) of the path of light in gravitational fields. Therefore, the same effect (the bending of the path of light by the deformations on space-time) should similarly be expected at the small scale.

According to Geometric Generalization, electron is a vortex of flowing wrinkles-strains in space-time. This vortex itself is a vertical deformation, and it affects the diffuse of light, which is a well-known fact. When light encounters a transparent medium, it slows down and refracts, due to its interaction with electrons. As we can roughly visualize this phenomenon, light winds around the vortex (tightens and relaxes the vortex), and it delays. Similarly, the path of light is bent by the gravitational field. (Quantum Mechanics formulates the algorithm of this phenomenon as the phase change in the wave function.) (In fact, reflection of light also depends on the same mechanism.)

Eventually, electromagnetic field itself is also a deformation in space-time, since this paper treats electromagnetic field as flow of wrinkle-like deformations that emerge around the knots. If this proposition is true, then it should be possible to observe the consequent effects of the electromagnetic field in medium (lab) scales.

Actually, our assumption on the proton in the center of a uniformly (+) charged spherical shell is a result of this point of view. This prediction, which suggests a relative decrease in the clock-ticks in that protons frame, signifies the electromagnetic origin of the gravity at the particle scale.

Our proton example, which we discussed in this section, can be generalized, leading us to the conclusion that the electromagnetic field (wrinkle-like deformations on space) slows down and bends the path of light (quanta of electromagnetic radiation) in certain conditions in a similar way that gravitational field bends the path of light. Consequently, it should be expected to measure a slowing down of the speed of the light that crosses through the electromagnetic field, depending on various conditions. 

Unfortunately, there appears a problem here; it does not seem to be easy to design a practical experiment to observe this effect. We can visualize the analogy as follows: Slowing down of the light in the electromagnetic field looks like slowing down of the speed of small ripples in sea by larger waves. (Because the vertical displacement caused by large waves increases the distance between the two coordinates that the small ripple should travel on the sea surface.) In order to observe such an effect, appropriately arranged large waves are needed. In other words, such an effect could be observed if a sufficient electromagnetic field is created by vortexes and knots, which are specifically arranged and oriented in order to amplify crest and troughs.

Please note that the effect that is discussed here is very similar to the effect caused by the gravitational field. The expression “slowing down of the speed of light” should not be misinterpreted. To be more exact, light does not slow down, but it only appears to slow down, since the path that it should travel increases by the vertical deformations (strains) in space. (According our hypothesis all existence travels at the constant speed of light; however, final elements of mass travel the same distance by circulating.) As a result, the speed of light is kept constant in all frames without any exception.