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Chapter II: The UG Characteristics at Non-Relativistic Velocities

Before proceeding further, it is important to address the oscillating behavior of the UG potential energy demonstrated in figures 2-3 to 2-6.

Equation 2-1-1


Whereas the gravitational force described by Newton’s equation increases monotonically with a reduction in the distance between a pair of particles and is purely attractive, the UG potential equation incorporates a cosine term. The cosine enables a cyclical function to alternate between - and , and its inclusion in equation 2-1-1 implies that at short distances the UG equation alternates between zones of attraction and zones of repulsion. The cosine term further implies that there are an infinite number of distances at which the UG force or potential energy cross zero, as well as an infinite number of distances at which they assume local maxima or minima.

Figure 2-1 illustrates that the cyclical behavior of the UG equation can be restricted to very short distances, well below the range of distances where gravitation has been tested experimentally. The graph uses an example of the general function compared to unity. When, . As the value of decreases, reduces monotonically, yet remains almost indistinguishable from 1 until a distance of about , which is slightly larger than the size of a typical atom or molecule. As is further reduced, continues to drop monotonically until it becomes equal to zero at. At , reaches its first minimum at . As continues to decrease, begins oscillating between and with growing frequency. Similarly, figure 2.2 compares the behavior of with the behavior of. The two functions are indistinguishable from each other at distances of Therefore, using the set of numbers given in the previous example, both terms, and are, for all practical purposes, indistinguishable from and respectively at distances of .1 In figures 2-3 to 2-6, the exponent and the cosine terms are combined into the UG potential energy equation 2-1-1. Again, using the same selected values and the given display scales, the UG potential energy can hardly be distinguished from the Newtonian potential at distances of Under the assumption that the UG force is a conserving force, the UG gravitational force between two particles of respective mass and can be calculated by computing the gradient of , given by equation 2-1-1. As the force is further assumed to be a central force, equation 2-1-1 is dependent only on the distance , and independent of any orientation angle.2 Therefore, only the radial derivative must be taken into account, leading to

Equation 2-1-2


Figure 2-1: Comparison between and using , The x axis provides the distance in . The two functions converge at about .



Figure 2-2: Compares , denoted as , to , denoted as Y, where .


In the general case of two objects consisting of a variety of particles, where the first object is composed of an ensemble of particles of particle mass   and the second object is composed of an ensemble of particles of particle mass , the UG potential energy between the two objects is given by


Equation 2-1-3




Figure 2-3: The UG and Newtonian potential equations are compared in a display range of , using values of , , , and The two functions converge at about .

Figure 2-4: Displays the same comparison as figure 2.3, viewed in a display range of . In this range, as long as , the UG potential energy oscillates with a large, almost constant amplitude (relative to the Newtonian potential), and with increasing frequency as approaches zero. Note that is essentially positive, and only briefly negative near its minima that reside on the Newtonian curve.




 

Figures 2-5 and 2-6: Comparison between the UG and Newtonian potential equations, viewed in display ranges of and respectively. Note that at low values (below about) there is an explosive increase in amplitude, saturating the display scale almost instantly.


The summations over and include all of the particles in objects 1 and 2 respectively, and is the distance between the particle of mass in the first object and the particle of mass in the second object.

For simplicity, let it be assumed that the first object is composed of identical particles of particle mass and the second object is composed of identical particles of particle mass . Furthermore, the distance between the two objects is assumed to be substantially larger than the size (or diameter) of either object. Consequently, the potential energy equation is given by

Equation 2-1-4


Similarly, the unified gravitational force is provided by

Equation 2-1-5

Therefore, at distances, and, where and approach 1 and approaches zero, the UG force equation converges to

Equation 2-1-6


where denotes the Newtonian force, therebydemonstrating that the UG and Newtonian forces converge at far distances.3 In general, at the limit . Thus, both theories provide the same results for any gravitational system that is composed of ordinary particles, as long as the constants and are sufficiently small, and the distance is not sub-microscopic.

The underlying postulate that the unified gravitational force equation can also be applied to the strong interaction suggests that in the nucleus of an atom, where the distance between the nucleons is about , the UG potential energy between two protons will be virtually equal in amplitude to the Coulomb potential. Therefore,4 using equation 2-1-1 and Coulomb’s law,

Equation 2-1-7


Multiplying by the constant and taking the log of both sides provides

Equation 2-1-8

Substituting for the proton charge, for the proton mass, , , and , and solving equation 2-1-8 via iterations provides the approximate value of the constant .

Evaluating the exponential values at distances of , , and , with the assumption that will increase the UG force by a factor of about between 1 and , and by a factor of about between and . Therefore, equations 2-1-1 and 2-1-2 provide the necessary explosive growth at approximately and below, while asymptotically converging to the Newtonian force for ordinary particles at distances significantly larger than about .

Finding the value of the constant is more complicated, as the oscillations could have started at distance ranges where the amplitude of the UG potential energy is negligible compared with the electromagnetic potential energy, and are thus difficult to detect.

A lower bound for the constant can be attained by taking into account the stability of the nucleus, where the distances between the nucleons average about , requiring that a minimum occur at about this distance. For a minimum to occur, the interaction between the nucleons at must be within the oscillation range of the cosine term, or , leading to .5



Figure 2-7: The graph demonstrates the explosive growth of the UG potential energy at the nuclear boundaries. As calculated in the case of two proton interactions, the amplitude of the UG potential energy is equal in magnitude to the Coulomb potential energy at about . At distances below , the UG potential energy completely dominates over the electromagnetic potential energy (with the exception of the UG zero crossings). At just above , however, the UG potential energy becomes negligible.



The requirement that the UG force equation be consistent with the Newtonian force equation within the level of measurement reliability can be used for estimating an upper bound for the value of the constant  . To date, all measurements of have been conducted at distances , where . Therefore, the exponential term can be replaced by a value of , leading to , assuring that the two forces are indistinguishable at the distance range of interest when the value of is sufficiently small. The variation in the measurement of the gravitational constant (one part in for (Gillies, 1997)) can provide the upper bound of the constant by calculating the range of the acceptable deviation between the forces calculated by the UG and the Newtonian equations. This level of variation in may be caused by the following:

1. Measurement error- Since the gravitational force is significantly weaker than the electromagnetic force, the signal-to-noise ratio is low. In addition, as there is no negative mass, the external gravitational fields cannot be masked out.

2. The presence of additional (relatively light) superheavy particles that interact according to equations 2-1-1 and 2-1-2, but do not dominate the interaction.

3. Variation between the gravitational force and Newton’s equations when applied to ordinary matter.

The failure to find an exact value of despite substantial improvements in measurement technology suggests that measurement error alone cannot entirely account for the deviation. As no stable SHP has ever been detected on the surface of Earth, the second option seems unlikely. Therefore, the variability of is presumably a consequence of the deviation of gravity from the Newtonian equation. Finding the accurate value of requires a detailed analysis that takes into account the geometry of the specific experiment, as well as the fact that ordinary matter molecules contain electrons, protons and neutrons of different masses.

As an example, assume that the experiment for measuring the gravitational constant is conducted by measuring the force between two identical homogeneous spheres of density (in units of number of molecules per cubic meter) with radius . When measurements take place, the two centers of the spheres are meters apart along the axis. Each molecule of the matter enclosed in the two spheres contains protons of mass , an average of neutrons of mass (the number of neutrons must be averaged from all stable isotopes), and electrons of mass (due to molecular neutrality ).

According to Newton’s law,

Equation 2-1-9a


while according to the UG theory,


Equation 2-1-9b 


where , , and

.

The terms and respectively provide the distance from the center, the azimuth and the elevation angles of the volume point within the first sphere, and , and are their counterparts in the second sphere. As the two spheres are not in contact, must be larger than and all possible pairs of particles split between the two spheres must be separated by a distance of at least . Therefore, the term can be replaced by .

Equation 2-1-9c


and are given by equations 2-1-9a and 2-1-9b, and the calculated is consequently dependent on the value of the constant . Therefore, the optimal value of can then be estimated by finding the range of values that provide the best fit between the calculated and measured at different values of and . It is clear from the discussion above that should approach zero as approaches zero, as long as and .

In the absence of such an experiment, the value of could be estimated from the reported range of variation of the value of . The gravitational contribution of an electron and the effect of the small difference between the proton and neutron masses are relatively small in comparison to the contribution of either a proton or a neutron. Therefore, the contribution of the electrons to the gravitational force is assumed to be negligible, while protons and neutrons are assumed for simplicity to have an equal mass of . In addition, the distance must be between . Using equations 2-1-9a, 2-1-9b and 2-1-9c for the case where and are significantly smaller, but not negligible compared with the distance , provides

Equation 2-1-9d


Given that or,


.


Since the deviation in can reach the value of from time to time, should be of the order of . As typical high precision laboratory-based experiments were conducted at distances of the order of , the value of the constant is likely to be in the general range of . This estimated value is only an approximation, however, and may be off by as much as one to two orders of magnitude. For the purpose of discussion, as the exact values of the constants and are not known, values of and will be assumed for the remainder of this book. Furthermore, the gravitational contribution of the electrons and the effect of the small difference between proton and neutron masses will also be assumed to be insignificant.6 Instead, a mass of will be used for either a nucleon or the hydrogen atom . The term “ordinary particle” (or “ordinary matter”) will be used throughout this book to refer to either a proton, a neutron or an electron (or to matter composed of these particles), while an ordinary particle of mass will refer specifically to either the hydrogen atom , a proton or a neutron. An object composed of ordinary matter that contains one or more atoms or molecules with one or more nucleons will be regarded as having particles of mass , where is the overall number of nucleons in the object. Finally, for the remainder of this chapter, particle velocities will be assumed to be non-relativistic.


Section II-1: The UG Gravitational Zones at Non-Relativistic Velocities


According to equations 2-1-1 and 2-1-2, the UG interaction between a particle of mass at a given location and a second particle of mass divides the space surrounding the first particle into zones that define the force and potential energy acting on the second particle by the first at any location within this space. The equations below have been developed for the non-relativistic case to quantify the widths and boundaries of the zones, and to find where they reach a local maximum or minimum.

The local potential energy maxima and minima (where the force is equal to zero) are given by

Equation 2-1-10


or

Equation 2-1-11


or

Equation 2-1-12

for


with minima at even values and maxima at odd values. Similarly, applying equation 2-1-1 and the relation may provide the zero intersections of the potential energy function,

Equation 2-1-13


or , thus

Equation 2-1-14

For distances , , and therefore7

Equation 2-1-15

, for

A zone is defined as the area of space enclosed between a minimum curve and an immediate neighboring maximum curve. Any zone , refers to the zone situated between the minimum indexed and the maximum indexed , while any zone refers to the zone situated between the same minimum and the maximum curve indexed . The distance between two successive maxima or two successive minima can be found by using equation 2-1-12 for the usual case of , or for the case of . Since ,



Equation 2-1-16

as


Similarly, it can be shown that as .

Figures 2-3 to 2-6 illustrate the behavior of the function (using equation 2-1-1)compared with the Newtonian gravitational potential at different distance ranges. As demonstrated in figure 2-3, the two functions are virtually identical at distances sufficiently greater than the first minimum (), which according to equation 2-1-12 occurs at

Equation 2-1-17a


Below this point the behaviors of the UG and the Newtonian functions diverge substantially, as the UG potential Vg changes direction and begins to exhibit an oscillation pattern with . Initially, when, the oscillation amplitude of the potential energy remains almost constant at a value close to. As approaches or becomes smaller than the constant , both the amplitude and frequency of the oscillations increase sharply with the reduction of distance . Since the oscillation frequency approaches infinity as , the potential energy continues to alternate between rapidly growing positive and negative potentials, which converge exponentially into positive and negative infinities. An infinite amount of energy is thus required to reduce the distance between the two particles to zero. Therefore, the particles can never “meet,” and singularity is avoided. If proven correct, this will have a profound effect on the concept of black holes (black holes will be discussed in section VII-2). , defined above as the distance of the first minimum, serves as an indicator for howclose the two particles must be from each another before their gravitational interaction begins to deviate strongly from the classical Newtonian behavior and proceeds into a pattern of sinusoidal zones. For its significance, the distance will be referred to as the zonal rangeof the particles. In the case of two objects consisting of many types of particles, the zonal rangebetweenthe heaviest particle in each object will be defined as the maximal zonal rangeof the objects. For example, in the comparison between and in figure 2-8, we can observe that sufficiently far beyond the zonal range distance, the UGand the Newtonian forces are indistinguishable. In other words, starting at somewhat beyond their zonal range , the force between a pair of superheavy particles of mass and is equivalent to the force between two objects that are positioned at the same distance from each other, where the first object is composed of ordinaryparticles and the second object consists of ordinary particles. This, however, is not the case for distances within the particles’ zonal range. For larger superheavy particles of mass , which will be shown to be capable of generating planetary rings, . The SHP zonal range with ordinary particles is given by equation 2-1-17a and is inversly proportional to . In cases where , is very small, and the SHP zonal range is located at an extremely large distance from the central core of the planet (see equation 2-1-17a), significantly beyond the distance where the UG potential energy oscillations cease. The amplitude at this zonal minimum is also negligible. For this reason, a second important distance, termed the zonal oscillation rangeisdefined as the radius of the first maximum () given by

Equation 2-1-17b


where the potential energy becomes virtually .


Figures 2-8 and 2-9: A comparison between the UG force and the Newtonian gravitational force, given the same parameters used in figures 2-3 to 2-6. The two functions converge at about ().



Section II-2: Superheavy Particles Embedded in Ordinary Matter

According to the UG equation, a single massive superheavy particle embedded in a large amount of ordinary matter can completely dominate a significant amount of the surrounding region, yet remain completely undetected at very long or short-range distances. In order to demonstrate this concept, consider the following example of a hypothetical object composed of   ordinary particles of particle mass arranged in spherically symmetric distribution within a distance of a single superheavy particle of mass . Figures 2-10 and 2-11 provide the comparison between the absolute value of the contribution of the ordinary particles and the contribution of the single SHP of mass to the potential energy of a given ordinary particle at a distance . At distances significantly greater than the SHP zonal range with ordinary matter, where the UG equation and the Newtonian equations converge, the overall larger mass of the ordinary matter dominates the interaction, and the SHP effect is completely negligible (smaller by a factor of ). As reduces to below about , the superheavy particle contribution begins to dominate. The SHP dominance peaks at the maximum, where its contribution becomes about 18.7 times larger than the contribution of ordinary particles. As reduces below , the ratio falls to a level at which the SHP effect is diminished by the excessive mass of ordinary particles, and the ordinary matter contribution once again becomes dominant. As continues to drop, the contribution of the superheavy particle becomes completely negligible, as long as remains greater than .



Figure 2-10: A comparison between the inverted contribution of ordinary particles and the contribution of a single SHP of mass to the potential energy of an ordinary particle at distance (axis). The ordinary particles clearly dominate at distances , while the SHPs begin to dominate at .



Figure 2-11: Using the same parameters as figure 2-10,the SHP dominance is terminated at , as ordinary matter begins to dominate the interaction at larger distances.



Between those extreme circumstances, however, there is a third condition of a “nearly Newtonian environment.” This environmental classification is comprised of ordinary matter and superheavy particles, where the distances between objects are adequately large that the SHP effect is detectable, but can be treated as a small perturbation to the Newtonian equations (an example of this is the formation of planetary rings, discussed in Chapter V). A relatively simple way to estimate whether a cosmological system is a “nearly Newtonian environment” is to verify that the orbital motions of its bodies are in accordance with Kepler’s laws of motion. The interactions between the Sun, planets, and other objects in our Solar System, for instance, are included in the “nearly Newtonian” class. However, in the case of galaxies, or deep inside stars, planets or large satellites, the environment can depart significantly from being “nearly Newtonian.”

The UG interaction between two objects can be broken down to the sum of two distinct groups according to the following equation:

Equation 2-1-18


where the first summation of index includes all of the particles in the first object, where the summation of index includes all of the particles in the second object, and where the summation on includes only the object 2 particles of mass that are within ten times of their oscillation zonal range with the particle (and therefore, does not include particles of the second object which comply with or ).8 Combined with the requirement that , the contributions to the UG potential energy of all pairs of particles of object 1 and particles of object 2 that are not in the sub-group are reduced to approximately their Newtonian interactions. As the overall mass of this group is equal to , where represents the total mass of the larger object 2, their overall contribution can be replaced by the first term of equation 2-1-18. While the first term describes a classical Newtonian gravitational potential, the second term includes the interactions of all pairs for which their calculated UG contribution to the potential energy of the first object departs significantly from their calculated Newtonian contribution. A similar approach can be taken to calculate the UG force, where

Equation 2-1-19


In the present case, both conditions of and (or and ) are required in order for the particle of object 2 to be in the first term of equation 2-1-19. The main benefit of distinguishing between the two terms is that the linear Newtonian term is in compliance with important Newtonian properties, specifically Newton’s shell theorem, which states the following:

1. A spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at its center point.

2. An object inside a spherically symmetric shell feels no gravitational force exerted by the shell, regardless of the object’s location within the shell.

3. Within a solid sphere of constant density, the gravitational force varies linearly with distance from the center, becoming zero at the center of the mass.

While the shell theorem holds true when applied to the first term (), the shell theorem does not hold true for the general UG force, and therefore does not hold true for the second terms of equations 2-1-18 and 2-1-19. In summary, when the second terms of equations 2-1-18 and 2-1-19 are small relative to the first Newtonian terms, but still detectable, the gravitational interaction between the two objects can be classified as “nearly Newtonian”9

Although equation 2- 1-12 provides the minima (for even ) of the potential energy, the smaller object can become trapped exactly at a minimum only if its velocity is equal to zero. In the more general case, where the small object velocity is not zero, the orbit will deviate slightly outward from the calculated circular radii of the potential minima. For example, consider the gravitational interaction between a large, spherically symmetric object of mass , and a significantly smaller spherically symmetric object of mass traveling in circular orbit around the large object at non-relativistic velocity. The larger object is assumed to consist of identical superheavy particles of mass residing in a relatively small core at the center of the object, and of ordinary particles of particle mass and an overall total ordinary particle mass of distributed throughout the object in a spherically symmetric form. Similarly, the small orbiting object is assumed to consist of identical particles of particle mass and a total mass of The circular orbital radius is assumed to be much greater than the sum of the radius of the small object, the radius of the large object, and the zonal range between the particles of masses and . Since the orbit is assumed to be circular,10

Equation 2-1-20



Therefore, the kinetic energy of the small object is given by

Equation 2-1-21


and the overall sum of the small object potential and kinetic energies is provided by

Equation 2-1-22


Substituting the number of particles in each object and their respective masses into equations 2-1-18 and 2-1-19 yields

Equation 2-1-23




Since is assumed to be much larger than both and the zonal range between particles of mass and (therefore and ), and since both objects are assumed to contain a spherically symmetric distribution of matter,

Equation 2-1-24


or using ,

Equation 2-1-25


Similarly, the force can be derived from equation 2-1-19, leading to

Equation 2-1-26


Substituting equations 2-1-25 and 2-2-26 into equation 2-1-22,

Equation 2-1-27


leading to

Equation 2-1-28


Thus,

Equation 2-1-29



+



=


+



=


If the large object does not consist of any ordinary particles (), the orbit will comply with the following equation:


or

Equation 2-1-30


To calculate the closest maximum or minimum in the vicinity of ,

Equation 2-1-31


In cases where ,



Equation 2-1-32


When and ,

Equation 2-1-33


When and ,

Equation 2-1-34


In cases where ,

Equation 2-1-35


When and ,

Equation 2-1-36


When ,

Equation 2-1-37


However, in most cases the overall mass of ordinary matter is expected to exceed the total SHP mass, or . At distances far greater than the size of an atom nucleus, , and can be replaced by . Therefore, equation 2-1-29 will become

Equation 2-1-38


For the usual case of and (where the orbiting object is within the zonal oscillation range), this equation reduces to

Equation 2-1-39


The case of :


In the case of equation 2-1-39 leads to or to the same conclusion as in equation 2-1-36

Equation 2-1-40

where , ,


with minima at even values and maxima at odd values. Thus minima occur at

, where , ,


The case of :


In the case of , the term in equation 2-1-39 is negligible (as , thus


Equation 2-1-41


Consequently, minima and maxima occur at

Equation 2-1-42

where and or , and where minima occur at the

.


Equation 2-1-41 can be solved only if . Therefore, for minima (and maxima) to occur, the distance must comply with


Equation 2-1-43

Note that since , the arccosine term is limited to the range and . Additionally, equation 2-1-41 becomes accurate only where . As , Therefore, for all practical purposes, if the condition of equation 2-1-42 is fulfilled, the minima are expected to occur at

Equation 2-1-44


As a reminder, the given scenario assumes that , ,

, and that .

The case of

In cases of , equation 2-1-39 leads to . Therefore, , and consequently, for an integer,

, where minima occur at approximately .









1 The reasoning behind the specific selected values for the constants and will be explained later on in this chapter.

2 The assumption that the UG force is a central force is valid only as long as the particle velocities are non-relativistic. Relativistic velocities will be discussed in Chapter III.

3 See figure 2-8.

4 The Coulomb interaction is clearly dominant at and negligible compared with the strong interaction at approximately . Therefore, the UG potential energy must be approximately equal to the electromagnetic (Coulomb’s) potential energy somewhere in between those two distances. This provides a margin of error of about for the constant .

5 The term is used here for either the mass of a proton , the mass of a neutron , or the mass of Hydrogen . Therefore, an approximate average of   was used for the calculation.

6 This is justified as the number of electrons must equal the number of protons (for reasons of electrical neutrality), and the mass of a proton is about 1,837 times heavier than the mass of an electron. Similarly, the difference between the masses of protons and neutrons is only about of the proton mass.

7 Note, however, that at   each zero is split into two very close consecutive zeroes, since the relatively large positive potential energy very briefly becomes negative, with a minimum coinciding with the relatively negligible Newtonian potential energy (see figures 2-4 and 2-5).

8 The number   was arbitrarily selected. The value needs to be sufficiently large to allow the convergence of the UG and Newtonian potentials within the required level of accuracy.

9The process applied here, of separating the contribution of the interaction into a Newtonian term and a non-Newtonian term under the assumption that ordinary matter is positioned in spherically symmetric distribution within either or both objects, significantly reduced the complexity of the equations. Therefore, this approach will often be used in the following chapters.

10 The assumption of a circular orbit is very reasonable, given that at non-relativistic velocities the minima contours of the UG potential energy are spherically symmetric with steep slopes.



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