Chapter I: The Theory of Unified Gravitation

The Unification of the Strong Interaction and the Gravitational Force

The theory of unified gravitation is based on the assumption that the nuclear strong interaction and gravitation are one and the same force, viewed at different distance scales, and that a single equation can describe both interactions on nuclear as well as on cosmological scales. In particular, it is assumed that similar to Newton’s theory of gravitation, the unified gravitational force is predominantly a central and conserving force between a pair of particles. At large distances, the UG force equation should asymptotically approach Newton’s classical force equation, when applied to ordinary matter composed of protons, neutrons and electrons, and should fully comply with all experimental measurements of gravitation that are currently available. In addition, the UG force must comply with the behavior of the nuclear forces between nucleons, which have been observed to be substantially stronger than the Coulomb interaction at below sub-Fermi distances. Accordingly, the UG force must produce an explosive growth at distances below ,¹ and is thus assumed to demonstrate exponential behavior. At a microscopic distance range of approximately , the UG force between two protons must be negligible in comparison to the Coulomb force. It is further assumed that the force is independent of velocity and spin, as their effects are either relatively small, or are part of the electromagnetic forces. However, it should be noted that as energy and force equations are not invariant under the Lorenz transformations, the force may not be central and may become dependent on particle velocity when the particles move at relativistic velocities. The UG theory must also be capable of explaining the observed nuclear resonance patterns, as well as planetary (and galactic) ring patterns, which consist of thousands of ringlets (oscillation patterns). The large area occupied by planetary and galactic rings rules out quantum mechanics as the cause of the oscillations, thus indicating that the UG potential must be driven by a periodic function. As most gravitational systems do not demonstrate cyclical behavior, the oscillating term must be suppressed under ordinary conditions.

Newton never published the reasoning that guided him to his gravitational force equation and never attempted to prove the equation on a theoretical basis. This is quite understandable, given that fundamental principles cannot be proven to be correct and may be regarded as valid only as long as they are not rendered false by experimental results. The basic UG equation 1-7 has the same limitation and cannot be proven correct. The equation will simply be required to survive a continuous inflow of experimental data that may either confirm or refute its accuracy. In the case that it is proven to be incorrect, the equation should either be adjusted or its theoretical foundation discarded. The logic underlying equation 1-7 is laid out in the following section of this chapter, where it will be shown that mathematically a family of very similar equations may be compatible with the aforementioned requirements. Equation 1-7 has been selected as virtually the simplest equation fulfilling the criteria.

Section I-1: The Logic leading to the Theory of Unified Gravitation

The UG theory for the strong interaction and gravitation is based on four postulates leading to a family of possible gravitational equations. While the different possible equations may provide different potential energy values, they share major properties that can be shown to provide a simpler and more accurate theory than the one currently used, with the ability to explain a large number of unexplained observed phenomena. The first UG postulate states,

UG Postulate I: The nuclear strong interaction and gravitation are actually one and the same interaction, viewed on different distance scales. Therefore, a single equation can describe both interactions from the sub-nuclear scale to the cosmological scale.

Newton’s gravitational equations 1-1 and 1-2 describe a central and conserving force between a pair of particles. The force depends on the distance between the two particles and on the product of their masses. Therefore, a natural starting point for the UG theory is to assume that the UG force is also a central and conserving force between a pair of particles, and that the force depends exclusively on the distance between the particles and on their respective masses. The short-range strong force was also shown to depend on inter particle distances and on particle masses, and to be dominated by central forces. However, the strong force has also been shown to contain a non-central component, and furthermore to depend relatively weakly on the charge and spin of the particles. Thus, the assumption of a central force that is entirely dependent on particle mass and distance can be challenged for distances of the order of . The weak dependency of the overall nuclear force on particle charge and spin can be attributed to electromagnetic interactions between the charged particles, or between their charges and the magnetic fields generated by their motion and spin. The existence of a non-central component can be attributed to a relativistic effect caused by high particle velocities. Whereas two charges at rest exert a central Coulomb force on each other, two relativistic charges are known to exert a non-central magnetic force on each other, in addition to the central Coulomb force. This phenomenon was explained by Einstein to be a simple relativistic effect. The underlying cause for this effect is that a force, under special relativity, is not an invariant entity. In other words, due to the relativistic phenomena of distance contraction and time dilation, force equations are not the same when viewed in different inertial frames that move at relativistic velocities relative to each other. Consequently, a central force between two particles at rest would not be viewed as a central force when both particles are moving at relativistic speeds relative to the observer. These relativistic effects provide a mechanism by which the central Coulomb force, as well as any other central force between two particles at rest, can create a substantial non-central force component (for further discussion see Chapter III). Therefore, the existence of a strong central UG force and an electromagnetic force between two massive charged particles can be consistent with observations of non-central components, with some weaker dependencies on the charge and spin of the interacting particles. This enables the UG assumption that when viewed from an inertial rest frame of one of the interacting particles, the unified gravitational force is predominantly a central and conserving force between a pair of particles, depending exclusively on the absolute value of the distance between the particles and on their respective masses, where the distance and masses are measured relative to the given inertial rest frame. In cases where the test particle has a relativistic velocity relative to the source particle, its perceived mass and distance are altered by the relativistic effect. For the time being, the discussion is limited to the non-relativistic case, where the particle velocities, relative to each other and relative to the observer, are significantly lower than the speed of light. The discussion will be broadened in the third chapter of this book to deal with relativistic particles as well.

As the UG force is assumed to be a central and conserving force, it can be written as a gradient of a potential energy scalar function

where the distance vector between the two particles is given by , where and are the locations of the interacting particles and where , and is defined as the unit vector in the direction of . The respective rest masses of the particles are denoted and , the potential energy function is given by , and for reasons of symmetry, the potential energy of each particle should be the same when the particle masses are exchanged. Therefore, .

Additionally, at sufficiently large distances, the UG potential energy equation converges toward Newton’s equation 1-2, which depends implicitly on the product of the masses of interacting particles, rather than on their individual masses. There is also no evidence of strong forces acting between particles where either one or both particles have zero mass. Therefore, for any particle mass M, . This strongly suggests that the potential energy function actually depends on only two variables: the distance , and the product of the two masses . Therefore, the second UG postulate can be stated as,

UG Postulate II: The unified gravitational force is a force between a pair of particles. When viewed at an inertial rest frame of one of the interacting particles (the source particle), the unified gravitational force applied on the second particle (the test particle) is predominantly a central and conserving force that depends exclusively on the absolute distance between the particles and on the product of their masses.²

This leads to

Equation 1-2-1

An important question that needs to be addressed is how the UG force and the UG potential energy depend on the distance (). In accordance with the first UG postulate, at large distances³ the UG force equation should asymptotically approach the classical Newtonian force equation and should fully comply with all experimental measurements of gravitation that are currently available, leading to

Equation 1-2-2

and at

where and are the masses of ordinary particles such as protons, neutrons and electrons.

In addition, according to the first postulate, the UG force must comply with the behavior of the nuclear forces between nucleons, which have been observed to be substantially stronger than the Coulomb interaction at below approximately . Observations further dictate that at distances , the nuclear force between two protons must be negligible in comparison to the Coulomb force acting between them. Accordingly, the UG potential energy must produce an explosive growth at about , thus leading to the third postulate.

UG Postulate III: The UG potential energy has an exponential dependency on the distance .

The simplest mathematical function that complies with the aforementioned requirements is the function , where is of the order of . This function demonstrates exponential growth, as well as substantial amplitude at , and relatively negligible amplitude at . In addition, this function and its derivative are practically indistinguishable from the Newtonian function and its derivative at , as long as the value of is sufficiently small. Expressed mathematically,

Equation 1-2-3

for

Therefore, if and then . Similarly, for the derivative of (or the force),

Equation 1-2-4

where . Note that while Newton’s force and potential energy equations depend linearly on the particle mass through the product of , the second UG postulate does not require that the UG force or potential energy be linear with . Such linearity is possible if the variable of equation 1-2-3 is proportional to , or if is independent of , and therefore a constant. However, in order to comply with the first UG postulate, the UG theory should be capable of explaining the patterns of nuclear resonances, as well as planetary and galactic rings patterns, which may consist of many thousands of ringlets, as observed in Saturn’s complex ring system. Both phenomena demonstrate strong oscillation patterns. The existence of nuclear resonances, which demonstrate a series of discrete nuclear energy levels that depend mainly on the masses of nucleons, strongly suggests the presence of a shell structure in the nucleus. The existence of such a shell structure, however, does not necessarily infer a cyclical UG potential energy. After all, the Coulomb potential energy of the proton-electron system given by the equation is non-cyclical, but still produces the atomic electron shell structure due to quantum effects. However, quantum effects occur on microscopic distance scales and thus cannot be responsible for planetary or galactic rings, as the large size of the area occupied by the planetary and galactic rings simply rules them out as a reasonable cause. Attempts to explain ring structures via electromagnetic forces have proven to be unsuccessful, and attempts to explain planetary rings and gaps as a result of orbital resonances between rings (or the gaps) and certain satellites have been only partially successful, as they do not explain the vastness of the ring systems (see Chapter V), nor the entirety of the observed rings and gaps. In order to provide a mechanism capable of producing the vast planetary ring systems, as well as the ring and spiral structures observed in galaxies (see Chapter IV), the UG potential energy equation must also contain a periodic function. As most of the gravitational systems familiar to us do not demonstrate cyclical behavior, the oscillating term must somehow be suppressed at distances larger than when applied to protons, neutrons and electrons. Whereas nuclear resonances occur at distances of approximately , planetary rings occur at distances below , and rings are observed in galaxies at distance ranges of the order of to .⁴ Therefore, the distance range of the oscillations must vary for different systems, yet must still depend exclusively on the mass of the interacting particles. Theoretically, ring and galaxy formations could also be explained by some odd distribution of dark matter, or by collisions between systems; however, to date these explanations have provided only limited success for a limited number of systems, while resulting in models with escalating complexity. Instead, a different approach is attempted by the UG theory.

The simplest cyclical function that can concurrently provide diminished oscillations as and produce a derivative that, similar to the Newtonian force, is proportional to , is the function , which oscillates at and approaches the constant value as . If were independent of particle mass, rings should always occur at the exact same interparticle distances, regardless of the masses of the two interacting particles, in disagreement with observations that different planetary and galactic systems often demonstrate concentric sets of rings at varying radii. This leads to the conclusion that must be dependent on the masses of the particles, or specifically on their product , as required by the second postulate. Therefore, the simplest functions that comply with the above requirements are of the form

Equation 1-2-5

where must be dependent on , and and may either be constants or functions of . The term was added to assure that .⁵ The requirement that at large distances should approach its Newtonian counterpart leads to

Equation 1-2-6

or

Equation 1-2-7

Using the same process for its derivative (or force),

Equation 1-2-8

Again, in agreement with equation 1-2-7,

Equation 1-2-9

This leaves us with three degrees of freedom, where setting three out of the four functions ,, or will uniquely determine the fourth function.⁶ However, there are a few constraints. As discussed above, must be dependent on , otherwise all planets and galaxies would demonstrate rings at exactly the same sets of radii. According to equation 1-2-7 (or the identical equation 1-2-9), either , or alternatively , must be equal to or approach zero if either are equal to or approach zero. Therefore, either one or both or must depend implicitly on .

If function is negative, the UG potential energy would be monotonically attenuated, becoming zero as approaches zero. The function must therefore be positive in order to explain both gravitation and the strong nuclear interaction. The existence of rings suggests that the UG cosine term of equation 1-2-5 oscillates within the radius of the farthest observed ring, requiring that . The fact that gravitational fields do not demonstrate explosive exponential growth in the vicinity of the observed galactic or planetary rings further requires that within the distance range of the rings, . In the range of distances , equation 1-2-5 can be reduced to , and therefore oscillates with virtually constant amplitude . It is reasonable to assume that particles of greater mass should produce a larger UG potential energy amplitude within this oscillation range. This suggests that B should also be dependent on . While these constraints eliminate many possible combinations of functions , , and , a substantial number of possibilities remain, and a significant amount of experimental data is required before the number of possible equations can be reduced to a bare minimum. However, once the values of and are set, , and can be treated as a set of constants (denoted , and ) and equation 1-2-5 becomes

Equation 1-2-10

Among the mathematically possible sets of functions , , and , the simplest and most logical are those where the exponent operand is a positive constant and the phase is set as zero. This leads to . As must depend on the particle masses, is selected for simplicity, where is a constant.

Using this simplified set, the UG potential energy can be written as

Equation 1-2-11

Equation 1-2-11 was derived by searching for the simplest function that is compatible with the UG postulates. While in agreement with the spirit of Occam’s Razor, the selection of the simplest equation is somewhat subjective, and cannot necessarily be regarded as proven. After all, nature is not required to guarantee simplicity. However, the simplest model is usually a good starting point. If experimental data is found to conflict with the predictions of equation 1-2-11, new sets of functions should be evaluated.

The inclusion of a cosine term (or in general, a cyclical term) in the UG equation, and the dependency of its operand on the particle masses, is probably the most fundamental deviation of the UG theory from Newton’s theory and general relativity. According to Newton’s equation, the external gravitational force applied on an object is always an attractive force that is linearly proportional to its total mass . Newton’s gravitational theory holds that the external force is the same whether a point-like object consists of few heavy fundamental particles or of many light fundamental particles, as long as the total sum of all of their masses are the same. For example, according to the Newtonian equation, the gravitational force applied by an external point-like and electromagnetically neutral object of mass on a point-like object containing a single particle of mass is virtually equal to the gravitational force applied by the same external object from the same distance on a point-like object containing particles of mass . The UG theory makes the distinction that the UG force is linear with the number of fundamental particles if they are all of the same mass. However, the UG force and potential energy are not linear functions of the fundamental particle masses, and therefore are not necessarily linear with the total mass of the object. As an example, the Newtonian gravitational potential energy between two atoms, one containing protons, neutrons and electrons of respective mass , and and the other containing protons, neutrons and electrons, is given by , where the distance between the two atoms is assumed to be significantly larger than the diameter of either atom.⁷ The total mass of each atom is given by and , and is the Newtonian gravitational constant.

The UG potential energy of the same interaction is provided by

and are not identical at sufficiently short distances where is distinguishable from , or if is still oscillating or has not yet converged to .⁸ At these distance ranges, the cosine terms within are not all equal, and the total sum within the brackets does not add up to the product of the overall mass of the atoms. At large distances, where and , does converge toward , and with , the two potential energies converge.

Therefore, with a constant that is significantly less than the diameter of a hydrogen atom, and with a constant b that complies with , the two potential energy functions are indistinguishable at distances greater than . However, for interactions involving fundamental particles of a mass significantly larger than the mass of a proton or neutron, the deviation from the Newtonian equation would become significant at greater distance ranges. Theoretically, these distances can extend to tens of thousands of kilometers if sufficiently large fundamental particles exist at the center of planets, or may further extend to the order of tens of if substantially heavier particles exist at the galactic centers, leading to the fourth UG postulate.

UG Postulate IV: The extreme temperature and pressure conditions that exist at the cores of entities such as planets, stars and galaxies produce relatively stable superheavy particles (SHPs). Substantially higher temperature and pressure produce substantially more massive fundamental particles.

The production of very massive fundamental particles, coupled with the UG equations (equations 1-2-5, 1-2-10 or 1-2-11), can provide a mechanism for generating planetary or galactic rings within the distance range where the cosine term demonstrates oscillations. High-energy collisions between electrons and positrons have been shown to create particles that are heavier than times their mass. Consistent with postulate IV, experimental data shows that higher energy particle collisions are capable of producing more massive particles. However, the masses of the heaviest particles produced via collision experiments are far smaller than the extreme masses required for the production of planetary or galactic rings via equation 1-2-11. Furthermore, the heaviest particles produced by high-energy collision experiments exist for only an instant before decaying into a barrage of lighter particles. Note, however, that the energy levels at the cores of planets, stars and galaxies are many orders of magnitude greater than those created by particle accelerators, and therefore the postulation that they can produce particles of substantially greater mass is logical. The relative stability of planetary ring systems suggests that if rings are in fact produced by superheavy particles in the vicinity of the central core of a planet, then the number of superheavy particles involved must be about constant. This can be explained via two different processes. First, the steady-state conditions within the core of a planet dictate that on average, the rate of production of a particular type of SHP should be equal to its rate of decay. Second, although free neutrons are known to be unstable and to decay, a neutron becomes stable when bonded with a proton in an atom nucleus. A similar mechanism may be at work for the large superheavy particles within galactic, stellar or planetary cores, where the superheavy particles may become bonded to each other or to the central core of ordinary matter, and thus become stable.⁹ A more restrictive assertion, known as quark confinement, is made by the standard model, in which quarks are assumed to be stable only when they are bonded together.

We must also question whether it is reasonable to assume the possibility of the existence of fundamental particles of masses of many orders of magnitude greater than the most massive particle ever observed ( times heavier). To answer this question we should bear in mind that according to the UG postulate IV, such large particles can be produced and maintained only within violent environments of extremely high energy, temperature and pressure, as exist at the cores of planets, stars and galaxies. These violent environments are simply not accessible for direct and close observation. However, according to the UG theory, their effects can be observed from far distances in the form of planetary rings, ring and spiral galaxies, and as rejection forces that drive galaxies away from each other. The idea that a gravitational collapse could generate massive superheavy particles of many orders of magnitude larger than the heaviest particles observed can understandably be viewed as almost inconceivable. It should be noted, however, that this postulate is very mild relative to the accepted notion of the existence of black holes. According to the no-hair theorem,¹⁰ a black hole is viewed by an external observer to behave like a particle with enormous mass. However, the masses of black holes, which are believed to be produced by gravitational collapse, may exceed the SHP masses contemplated here by factors of to . Therefore, to make it easier to comprehend the fourth postulate in terms of current belief, the postulate can be described as an assumption that the extreme conditions generated by the collapse of gas clouds produces small black holes (SHPs) near the center of the planet, star or galaxy. With this approach, the main and fundamental difference between the UG theory and the current paradigm (Newton’s gravitational theory and general relativity) is the use of equation 1-2-11, rather than equation 1-2, to describe the potential energy of the interaction.