THE FRACEP MODEL, Part 7:
A Universal Potential for All Scales from Quantum to Cosmic
May 17, 2007 (Revised: 5/27/15)

Physics has identified four fundamental forces that drive matter interactions: the electromagnetic, the nuclear strong, the nuclear weak, and gravity. The potential functions that describe the particle and object interactions for each of the four forces are different but have some characteristic similarities. It is assumed here that the interaction of matter, regardless of scale, follows the same potential rule – that is, it can be modeled with the same mathematical function. The difference in the appearance (apparent functional form) of the potential at the different (nuclear, macro, or cosmic) scales is the result of the limiting behavior of the function at different scales. The FRACEP potential presented here represents the initial effort to develop a universal, multi-term function that agrees within reasonable limits at the typical quantum scales with the modified Yukawa (nuclear) potential. The FRACEP potential also agrees within the measurement uncertainty with the Newtonian potential at the typical macro and cosmic scales. In addition, this new potential also demonstrates a smooth transition between the quantum scales and the smallest macro scales of the Newtonian potential – a transition that resembles the attractive and repulsive appearance of the Casimir effect.

CONTENTS

1.0 INTRODUCTION

1.1 Two Well-known Interaction Potentials

1.1.1 The Interaction of Gravity (Mass Effect)

1.1.2 The Nuclear Interaction (Quantum Effect)

2.0 THE FRACEP POTENTIAL

2.1 The Mathematical Form of the FRACEP Potential (V_FRACEP)

2.2 The Quantum and the Fundamental Scales Behavior

2.3 The Macro Scale Behavior

2.3.1 The Ordinary Macro Scales of Every-Day Life

2.3.2 The Cosmic Macro Scales

2.4 A Further Analysis of the V_FRACEP Function

2.4.1 A Look at the Leading Coefficient

2.4.2 A Look at the exponential Factor

2.4.3 A Look at the Sine Factor

3.0 CONCLUSIONS

4.0 REFERENCES

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1.0 INTRODUCTION

Physics today has identified four fundamental forces that drive matter interactions: the electromagnetic, the nuclear strong, the nuclear weak, and gravity. The two nuclear forces operate by color charge or weak charge interactions at quantum scales between atomic and sub-atomic sized particles. The electromagnetic force operates by electric charge interactions that can have both quantum scale effects and macro scale effects. Gravity is a macro-scale effect operating by mass interactions between small every day-sized objects to cosmic-sized bodies.

Theoretically, the electromagnetic and the nuclear weak forces have been unified through a relation between the electric charge and the weak charge. This relation allows the mathematical description of the weak interaction to be expressed in terms of the mathematical description of the electromagnetic interaction. The theory is known as Quantum Electrodynamics (QED).

Efforts to unite QED with the strong force have provided numerous models – none of which is accepted as the “unique” unification being sought. These models (with parameters that vary from model to model) provide the basis of quark interactions through the color charge. The models are known as Quantum Chromodynamics (QCD). The combination of QED and QCD is the basis of the Standard Model (SM) that defines the interactions of the sub-atomic fundamental particles like the electrons, neutrinos and quarks.

To date, no completely successful model has been developed that unifies QED or QCD with gravity. A variety of possibilities (including string theory, super symmetry, loop quantum gravity, and higher unseen dimensions models) are being explored in hopes of providing the unification being sought.

The potential functions that describe the particle and object interactions for each of the four forces are different but have some characteristic similarities. This has led us to conclude that there might be a common rule applying to all four forces. We hypothesized that it might be possible to find a single (multi-term) potential function to characterize that rule. All terms in that function need not necessarily apply to all scales (forces). The goal of this work was to construct that universal potential. The FRACEP potential presented here is the initial result of that effort.

The FRACEP Model was developed to address the feasibility of describing the indivisible SM fundamental particles as composite particles. It provides the structure (Ref. 1) and sizes (Ref. 2) of the composite particles based on a minimum set of two fundamental particles (G0p and G0m) where G0m = -G0p.

The negative mass concept is not consistent with the SM concept; but, it is not unique to FRACEP. Hoyle, Burbidge and Narlikai show theoretically that the possibility of negative mass particles is consistent with an alternate cosmological picture that is consistent with observed data (Ref. 3).

The initial results presented here directly address the positive-mass particle interaction to the smallest (sub-SM fundamental) particle sizes. Elaboration on the negative-positive mass interaction is in progress.

We begin with a discussion of two well-known interaction potentials. The first is for the quantum or nuclear world (the Yukawa potential) and the other for the macro world (Newton’s gravity potential). We then present the FRACEP (universal) potential and discuss its behavior. We compare it with the Yukawa potential in the quantum regime. Then, we compare its behavior in the macro and cosmic regimes with Newton’s potential. Finally, we discuss the expected behavior of the universal FRACEP potential.

1.1 Two Well-known Interaction Potentials

1.1.1 The Interaction of Gravity (Mass Effect)

In Newtonian physics (Ref. 4), gravity is an action-at-a-distance force. That means that the interaction of two bodies does not result from contact between them. Gravity is based on an attractive potential (V_N) between two masses (m₁ and m₂). The potential is defined as:

(1)                                        V_N = -Gm₁ m₂ / r

The “G” is the gravitational constant. The m₁ and m₂ are the masses of the two interacting bodies. The "r" is the distance between the two bodies. Equation (1) assumes spherical symmetry in the potential around each body. For this model, the sizes of the masses are small relative to their separation, making the masses appear as point sources to each other.

Figure 1 shows the behavior of V_N as a function of separation distance between m₁ and m₂. In this figure, V_N is normalized by Gm₂ and the measured value in of V_N in Joules would be the plotted value multiplied by Gm₂. For example, for the case where m₁ = 1 kg at r = 1 m, the plotted normalized V_N = 1 J. Using the conventional value of m₂ = 1 kg and A = Gm₂ = -6.67432x10^-11, the properly scaled V_N = -6.67432x10^-11 J.

We see that V_N falls of monotonically with “r”. (That is as r is increased, each value of VN is smaller than the one before.) Experiments (Ref. 5) show that V_N reflects nature to sub-millimeter separation distances. For separations much smaller than that, it becomes difficult to distinguish between measurement uncertainty (e.g., instrument and experimental technique limitations, and possible variations in “G”). At the smallest distances, V_N grows infinitely large, but it is uncertain if nature necessarily follows this rule.

In his work (Ref. 6), Newton debated whether the 1/r was the best form for the potential. He considered higher orders of 1/r and exponential functions of r. However, his formulation of the orbit computations found that only 1/r gave an exact, closed-form solution that agreed with the data. Lacking any data interpreted as indicating another possible form, Newton’s 1/r potential comes to us as the standard.

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Figure 1. This shows Newton’s gravity potential as a function of distance. The value of G =
6.67432x10^-11Jm/kg². Convention would have V_N as negative but for the plot the negative sign is
suppressed. Here m₁ is the source body and m₂ is the test body that, by convention, is a 1 kg mass.

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The Newtonian formulation works well for macroscopic bodies (i.e., everything from apples to orbiting bodies - though not quite as well for the largest masses). The attractive potential causes all masses to pull toward each other. This attraction is believed responsible for the accretion of mass and contraction that leads to the formation of stars and planets.

It is well known that, in General Relativity (GR), gravity is treated as a field, and, that the mass of a particle represents a deformation of the fabric of space-time (Ref. 7, Ref. 7a). In its simplest form, with the appropriate assumptions and using the Newtonian gravitational potential, GR yields the same results as Newton's action-at-a-distance force. However, GR's effects also extend to the realm of the super massive, the most notable being the infamous black holes, which Newton’s gravitational expression does not adequately model.

Unlike the Newtonian formulation, GR has a “cosmological constant” that allows for negative, i.e., repulsive energy. This negative energy is considered responsible for the apparent acceleration of the expansion of the universe – an effect Newton cannot address.

Neither Newton nor GR extend adequately to the sub-atomic levels. For that, the probabilistic formulation of quantum mechanics (the foundation of the Standard Model (SM)) is required.

1.1.2 The Nuclear Interaction (Quantum Effect

Within the framework of the SM, the nuclear force describes interactions between particles at quantum levels (Ref. 8), that is, atom-atom interactions electron-atomic nucleus interactions, and quark-quark interactions within protons and neutrons.

The model of this force is based, to first order, on the well-known Yukawa potential that takes the form:

(2)                                        V_Y = -g² x e^-br / r

Here, b = mc/h-bar where h-bar is Planck’s constant divided by 2p, and "g" is analogous to the Newtonian gravitational constant “G”. To completely describe the nuclear force, V_Y is modified by several terms that are used to model specific physical effects (Ref. 9). These terms vary as higher order powers of 1/r with measured scaling parameters. The modified potential takes a variety of forms depending on the type of interaction being modeled. For example, one expression of the strong nuclear force describing neutron-neutron scattering takes the form:

(3)                                        V_Y-mod ~ A V_Y [1 + S(r){1 + 3/m x r + 3/m₂ x r²}] - d(m_p,r)

Here, d (m_p,r) is a function of r and the mass of the pi-meson (also known as the pion, indentified as p). S(r), d (m_p,r) and A are empirically determined.

The SM hypothesizes that the strong nuclear force between two sub-atomic particles results from the exchange of SM fundamental particles called gluons, but, the theory is not yet developed to model this exchange process. Current modeling relies on the exchange of larger non-fundamental particles like the pion (p), rho (r) or omega (w) particles.

The usual range for quantum level fundamental particle interactions is ~0.3 fm to ~2.5 fm (1 fm = 10^-15 m). Over this range, the constant b = mc/h-bar varies depending on the mass of the exchange particle. It appears that for r > 0.3 fm, the nuclear force is dominated by pion exchange giving b ~ 0.7082/fm. Below ~0.3 fm, the particles r and w are considered the dominant exchange particles giving b ~ 4.054/fm. Equation 3 represents the range where ther is pion exchange. The behavior of the nuclear potential is shown in Figure 2 as represented by V_Y-mod. The data for V_Y-mod shown in the figure are taken graphically from Bertulani (Ref. 9).

The potential is repulsive at very small distances (<0.8 fm) indicated by the negative slope of the curve. It becomes attractive at larger distances (~1 fm) as the slope changes to positive. This provides a mechanism for binding particles within a nucleus, but not allowing them to get too close. The adjacent repulsive and attractive behavior in the central region of V_Y-mod (0.5 < r < 2 fm) keeps the quarks inside the proton - not-too-far but not-too-close to each other, maintaining a quasi-stable particle configuration.

At larger distances as the radius approaches ~10 fm, the potential exponentially converges to zero making it inconsequential at macro distances. For the smallest distances (r < 0.5 fm), V_Y-mod exponentially grows infinitely large; but like Newton’s potential (V_N), it is not obvious that nature behaves this way becoming so large.

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Figure 2. This shows the modified nuclear potential as a function of distance.
The points on the curve were taken graphically from figure 3.3 in Bertulani (Ref. 9).

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Having discussed the two familiar interaction potential, we can now proceed to the FRACEP potential, V_FRACEP. This new potential addresses the interaction strength between any two of the smallest FRACEP particles (the G0p, the G0m) or of the components of the Standard Model fundamental particles taken as composite particles. Measurements at the smallest scales of interest to V_FRACEP (around G0p) are not possible with the current technology so it does not extrapolate down to those smallest scales using the standard nuclear potential because the standard potential grows without bound. Instead, it uses current data at sub-atomic scales as an intermediate step between the smallest scales of interest and the macro scale data of Newton. These two data sets are used to develop a universal potential model. It is this universal potential that we then extrapolate down to the smallest scales.

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2.0 THE FRACEP POTENTIAL

The new potential presented here is intended to describe the spatial interaction of the FRACEP Intermediate Building Block components of the composite fermions. Extending to the next smaller scale, it also must describe the interaction of the internal components of the Intermediate Building Blocks. And, finally to extent to the smallest scale it must describe the interaction of any two fundamental particles (the G0p’s and G0m’s). The separation distances for these interactions are smaller by many orders of magnitude than the interaction distances we see in the nuclear-scale and macro-scale processes.

In no case does either the Newtonian (V_N) or the modified Yukawa (V_Y-mod) potential address the sub-particle interactions or distances represented by the FRACEP particles. To address the FRACEP particle interactions, which lead to the binding of the stable fermions or repulsion of the unstable ones, a new interaction potential must be considered. Since at this time there is no way to measure this interaction potential, it is necessary to take a broader approach in viewing the new potential.

It is assumed here that the interaction of matter, regardless of scale, follows the same potential rule – that is, it can be modeled with the same mathematical function. The difference in the appearance (apparent functional form) of the potential at the different (nuclear, macro or cosmic) scales is the result of the limiting behavior of the function at different scales. The following sections of this paper present the functional forms and show a comparison of the V_FRACEP with the V_Y-mod and V_N potentials. By showing a universal potential that agrees with behavior at the larger scales (from macro down to nuclear), further extension down to the fundamental scales is cautiously assumed to be reasonable in the absence of contrary evidence.

2.1 The Mathematical Form of the FRACEP Potential (V_FRACEP)

The FRACEP potential, in its most general form, is a function of both time and space; that is, V_FRACEP = F(t) x V(r). The F(t) describes the temporal behavior that is important to the decay times of the composite fermions, as well as, the time oscillations associated with the charge and spin characteristics of the charge and spin carriers. The V(r) describes the special behavior only. For the purpose of this paper, V_FRACEP = V(r) only. The time dependent part, F(t), will be considered in a later work.

As previously stated, the intent for V_FRACEP is an expression that reproduces the V_N behavior at macro scales, and that agrees with the shape and magnitude of V_Y-mod at quantum scales. Another consideration is that V_FRACEP be extendable down to the smallest fundamental scales without producing a singularity.

Several observations are made which contribute to the construction of V_FRACEP. First, both V_N and V_Y-mod to first order have a “1/r” component to their behavior. So the V_FRACEP needs to reflect this.

Second, the behavior of V_Y-mod in Figure 2 is suggestive of a short segment of a damped oscillation – reminiscent of a stretched spring that is increasingly compressed in amplitude along its length. The damping (amplitude compression) is explicitly represented in the nuclear potential by an exponential decay function (in equation 2). The oscillation is less obvious.

One obvious way to express an oscillation is with a sine function. The sine function is a transcendental function in algebra that is computed as a series expansion. For an argument of 1/r, it has the form:

sin(1/r) = 1/r – (1/r)3(1/3!) + (1/r)5(1/5!) – ….

This indicates that the observed “1/r” behavior in V_Y-mod could be the first order approximation to a sine function. Thinking ahead, it could also explain the “1/r” behavior at the macro scales under the right conditions.

The functional representation of the stretching is less obvious. It appears to be the result of the higher order terms in V_Y-mod. This stretching effect in V_FRACEP is determined empirically by fitting to V_Y-mod.

Equation 4 shows the functional form determined for V_FRACEP based on the above considerations.

(4a)                                        V_FRACEP = (A₀(M) + B₀(M)) x sin(arg(r,M)) x exp(K₄ x r)

(4b)                                        arg(r,M) = (150p/180) [K₁r² + K₂/r + f/r³+ K₃]

(4c)                                        A₀ = 1/(0.18 x M) ; B₀ = 9.2095x10^-8 x M^1/2 ;
                                               K₁ = -0.09/M² ; K₂ = M^1/2[1/M - (0.00006/m_p)];
                                                f = -6.0x10^-7 x [(10¹⁵)³ x m_p_Au³ ] x [M/8x10⁶⁰]²;
                                                K₃ = 1/M^1/2 ; K₄ = -2.4/M

In (4c), M = m_s/ m _p where m_s is the mass of the source particle around which the potential is computed. The m_p = 139.57 MeV/c² (the mass of the pi meson), and p in (4b) is the constant 3.14159. The m_s and m_p are in MeV/c² and r is in fermis where fm_p_m = 10¹⁵. (That is, there are 10¹⁵ fermis per meter.) The conversion factor from meters to astronomical units is m_p_Au = 1.496x10¹¹. The conversion factor for macro mass units of kilograms to nuclear mass units MeV/c² is kg_p_MeV = 1 / 5.60949552x10³².

In the next two sections, V_FRACEP is compared at the quantum scales with V_Y-mod, and at the macro scales with V_N.

2.2 The Quantum and the Fundamental Scales Behavior

Figure 3 shows a comparison of V_FRACEP with V_Y-mod at the quantum scales down to the fundamental scales. V_FRACEP agrees with V_Y-mod in shape and amplitude generally well within the usual range for quantum level elementary particle interactions (~0.3 fm to ~2.5 fm). Note that although this central region agrees, the two tail regions (approximately r < 2.2 fm and r > 2.8 fm) have different behavior.

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FIGURE 3. This shows a comparison of the V_FRACEP and V_Y-mod potentials. The potentials are
given in GeV and the radius is in fermis (10^-15 meters). The dashed line is V_Y-mod and the solid
line is V_FRACEP. The larger curves represent the potentials in the usual nuclear range. The values
for the potential V_Y-mod were taken graphically from figure 3.3 Ref.9. As noted, V_Y-mod increases
monotonically to the left in the region for r less than about 0.75 fm. Insert (a) shows that in this same
region V_FRACEP oscillates while increasing to a maximum of ~+1.234 in amplitude as the radius
approaches zero. At the large radius end, V_Y-mod monotonically decreases to zero; while insert (b)
shows the damped oscillation in V_FRACEP which decreases in amplitude to zero as the radius increases.

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Above r ~ 2 fm, both V_Y-mod and V_FRACEP damp exponentially approaching zero as the radius approaches ~10 fm. However, V_FRACEP begins ringing (oscillating) for r > 2.8 fm where V_Y-mod monotonically decreases. At the limits of measurement ability in this region, both V_FRACEP and V_Y-mod would appear approximately the same.

Below r ~ 0.75 fm, the behavior of V_FRACEP again is different from V_Y-mod. V_FRACEP rings (oscillates) for r < 0.5 fm. The maximum amplitude approaches V_FRACEP ~ 1.234 GeV as the radius approaches zero. On the other hand, V_Y-mod monotonically grows for r < 0.5 approaching infinity as the radius approaches zero.

Qualitatively, the ringing at both ends of V_FRACEP serves a purpose similar to that of the oscillating characteristic behavior in the central region of V_Y-mod (0.5 < r < 2 fm). In V_Y-mod, this region of changing slope is seen as adjacent areas of repulsion and attraction that keep the quarks inside a proton - not-too-far but not-too-close to each other and thus maintaining a stable particle configuration.

It is the ever increasing frequency of oscillation of V_FRACEP that remains finite in amplitude at the smallest scales that is particularly relevant to the V_FRACEP fundamental particle interactions. In this region, the ringing provides a mechanism to shield the fundamental particles from collapsing into each other. At the same time, it keeps the potential at the origin from growing without bound which protects the integrity of the space-time fabric.

V_FRACEP also exhibits the same adjacent repulsive and attractive behavior in the larger “r” regime. The damped oscillation of V_FRACEP at the larger distances in the nuclear range provides the basis of the transition to the 1/r form of V_NEWTON at the macro scale distances.

2.3 The Macro Scale Behavior

The discussion of the macro scale behavior is more challenging than the discussion of the quantum scale behavior. This is because the range of the macro effect is much broader than for the quantum effect.

The range of the quantum effect is very short, keeping the interacting particles very close to one another but not too close. This short range gives stability to particles like the proton. The macro behavior of gravity, on the other hand, is very long-ranged. It affects everything from tiny masses at micrometer separations to large astronomical bodies at cosmic scales. For this reason, the following discussion is separated into two parts: 1) the ordinary scales of every-day life, and 2) the cosmic scales.

2.3.1 The Ordinary Macro Scales of Every-Day Life

For the purpose here, the discussion of ordinary macro scales encompasses sizes, masses and separation distances of objects seen on earth. Although technically the separation distance of any two such objects is from center to center, for practical purposes it is seen as the measurable distance between the two surfaces. The lower limit of the size, mass and separation distance is such that quantum effects are not seen in ordinary macro scale applications; nor, are they reflected in Newton’s laws. However, because FRACEP is designed to span scales from the smallest to the largest, this section addresses the question of the transition from ordinary macro scales down to quantum scales.

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FIGURE 4. This shows a comparison of the V_FRACEP and V_Newton potentials. The potentials
are given in Joules and the radius is in meters. The dashed line is V_Newton and the solid line is
V_FRACEP. The dash-dot line shows the point where V_FRACEP diverges from V_Newton.

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Figure 4 shows a comparison of V_FRACEP with V_NEWTON at macro separation scales on the order of thousands of meters down to the transition to quantum scales (around micrometers at the smallest mass considered on the plot). The three plotted masses (5500 kg, 55 kg and 0.55 kg) show the general behavior of V_FRACEP.

The plot shows that V_FRACEP agrees with V_NEWTON in shape and amplitude at the larger separation distances, “r”. The horizontal axis plots the natural log of the separation because of the wide range in “r” (3x10^-7 m to 9x10³ m). Similarly, the vertical axis plots the natural log of the potential divided by the Newtonian gravitational constant “G”. This quantity ranges from -2x10^-9 J to +2x10⁴ J.

V_NEWTON is an attractive potential that monotonically decreases with increasing r. By definition it is always represented as a negative number. Since the natural logarithm function (ln) is not valid for negative arguments, the log was taken using the absolute value of the scaled potential (|V|/G) and plotted with the sign of the potential. In the log-log format the 1/r behavior becomes a straight line. In the figure, ln(|V_NEWTON|/G) takes on positive values for V_NEWTON less than the value of G but a negative value for V_NEWTON greater than G.

V_FRACEP, on the other hand, has both negative and positive values depending on the combination of mass and r. It is the changing slope of the potential curve that gives V_FRACEP both attractive and repulsive behavior. (This behavior is shown more explicitly later in Figure 5).

Figure 4 shows that V_FRACEP agrees with V_NEWTON to within the measurement uncertainty for the three masses shown at reasonable macro scale ranges about r >0.001 m (ln(r) ~ 7). In this region, V_FRACEP is an attractive potential like V_NEWTON. The agreement with V_NEWTON extends to shorter ranges as the mass decreases.

The uncertainty in the potential measurement, and therefore a possible uncertainty in the shape of the potential, has been the subject of continuing study. The original value of the gravitational constant, G, was determined by Newton in the seventeenth century based on orbit calculations for the known planets and the moon. The tacit assumption after considering alternatives was that his law of gravitational attraction (Gm₁m₂/r) was the correct one (given the measurement capability of the day). In Newton’s function, only the constant G needed to be determined experimentally.

Historically, the astronomical determination of the gravitational constant was far more accurate than laboratory determinations as recently as the mid twentieth century (Ref. 10). Because the gravitational field is so weak, measurements in the laboratory are extremely sensitive to any outside interference (such as electromagnetic effects, vibrations or other large bodies nearby).

Earliest laboratory measurements date back to Henry Cavendish in the eighteenth century. His experiments placed heavy masses near a dumbbell-shaped pendulum hanging by a thread. By 1986, a value for G with a fractional error of 1.3x10^-4 was published by the National Institute of Standards and Technology.

A more recent laboratory experiment (Ref. 11) used a new type of torsion turntable that avoided the typical problems of earlier experiments such as the friction effects caused by the twisting thread. The experiment used relatively small masses (8 kg steel balls) and with separation distances in the sub-millimeter range. It achieved a fractional error of 1.4x10^-5for a value of G = 6.67432 +/- 0.00009 x10^-11 m3/kg-s² – the best measurement so far.

The sub-millimeter range on Figure 4 corresponds to ln(r) < -7. The figure shows that agreement of V_FRACEP with V_NEWTON is expected to continue to tenths of millimeters for the largest mass shown and to the micrometer range for the smallest mass shown.

Figure 4 also shows that as r decreases, V_FRACEP begins to diverge from V_NEWTON. The break-point shifts to shorter ranges as the mass decreases (as Dln(m)/Dln(r) ~ 2). It is at this point that the transition from ordinary macro scale behavior (1/r) begins to take place and V_FRACEP begins to oscillate. As shown, for m = 5500 kg, the transition begins at ln(r) ~ 8. For 55 kg, it is at ln(r) ~ 10. And for 0.55 kg, it is at ln(r) ~ 13. The turnover point for the macro scales occurs at approximately 7.4x10^-7 m_s^1/2 meters. This approximation appears to hold for masses as large as the sun (~10³⁰ kg). This approximation changes for the larger cosmic scales.

Consider for a moment the expected agreement of V_FRACEP with V_NEWTON for some common materials. We can determine the radius of a sphere of any material given its mass (m) and density ( r) using two relations: volume = 4/3 pr³ and volume = m/r. Eliminating volume from the two relations gives the radius of a sphere as:

(5)                                        r³ = 3m / (4pr)

Table 1 shows the size of spheres of several common materials for the three masses shown on Figure 4. The table includes the density and the ln(r) for easy comparison with the figure.

A comparison of the table with Figure 4 shows that for all of the common materials, except neutronium, V_FRACEP agrees with V_NEWTON for all separations outside the sphere surface. And further, agreement continues to well within the sphere’s surface. For example, in the 55 kg case, the break-point is 4.5x10^-5 m but the radii for all of the common materials are greater than 0.1 meter. This is four orders of magnitude larger than the break-point. This behavior is similar to what is seen for the other masses indicated.

Neutronium is not a common material on earth. It is the degenerate matter inside a neutron star. That means that the material within the star is made of only neutrons under sufficient pressure to keep them stable and closely packed – but not so much pressure that the neutrons tunnel into one another as would occur in a black hole. It is presented here as an extreme counter example to the other common materials shown.

As degenerate matter, neutronium is not expected to show good agreement even close to the sphere surface on the outside. In all cases, the gravitational field of V_FRACEP is in the transitional region and does not begin to agree with V_NEWTON until well outside the sphere for this material. The neutronium deviation for the 55 kg case begins about 5.5x10^-7m (ln(r) = -14.5) but the break-point is 4.5x10^-5 m – well outside the sphere surface. This same pattern is true for the other masses shown.

Table 1. This shows the radius of a sphere of some common materials given their
mass and density. The 8 kg steel ball is the size and material used in the experiment
described above to measure the latest, best value of G down to sub-millimeter ranges.
At this mass, the break-point of V_FRACEP from V_NEWTON is around 2x10^-5 m –
well into the sub-millimeter range of the experiment.

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Figure 5 shows the behavior of V_FRACEP in the transition region where it diverges from close agreement with VNewton. The plot is for the 55 kg case but it is characteristic of the behavior for the other masses considered. The only difference is the shift in the break-points.

The oscillatory behavior is similar to the behavior seen in the nuclear case (Figure 3b) in that the oscillation frequency increases as the range decreases. In both cases (nuclear and macro) a maximum, constant amplitude is reached though there is a difference. In the nuclear case (Figure 3b) the maximum is reached after an increasing trend; while, in the macro case (Figure 5), the maximum is reached almost immediately. As the macro mass and range combination approach the nuclear case the transition region evolves from the instantaneous and constant maximum oscillation limit (of the macro behavior) to the increasing to a maximum oscillation limit (of the quantum behavior).

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FIGURE 5. This shows a comparison of the V_FRACEP and V_Newton potentials in the transition
region between macro scales down to quantum (or nuclear) scales. The potentials are given
in Joules and the radius is in meters. The dashed line is V_Newton and the solid line is V_FRACEP.

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We can draw another analogy between the macro and nuclear cases. Recall in the nuclear case the repulsive part of the potential allowed the particles (like the quarks in a proton) to separate from one another far enough to maintain the integrity of the fabric of space. That is they did not tunnel into each other. At the same time, the attractive part of the potential kept the quarks sufficiently close to maintain the stability of the proton.

We hypothesize here that an analogous effect is occurring in the macro case. In the transition region, the repulsive part of the oscillation keeps objects from breaking the surface barrier of solid objects. This, in effect, prevents them from tunneling into one another while and the same time allows them to be in apparent contact.

The interest in the transition region is the subject of much study as of late. The reason for this interest comes from efforts to understand why gravity is so many orders of magnitude weaker (over 32) compared to the other forces of nature (Ref. 5).

String theorists argue that there are extra dimensions (above the three spatial ones with which we are familiar) that are curled up and not visible on the typical macro scales. The number and scale of extra dimensions would affect where the gravitational potential deviates from Newton’s 1/r potential.

Some generalized Newtonian-like potentials (Ref. 13) are theorized to have a form as:

(6)                                        V(r) = G* m₁m₂ / rⁿ⁺¹         for r << R.

Here R is the diameter of the curled up dimensions and G* is the appropriately adjusted gravitational constant. So far, measurements to as small as 0.2 mm have not found the transition region to support the extra dimension idea.

V_FRACEP is not attempting to model curled extra dimensions or other specific effects. Its purpose is to provide a generalized universal potential that spans from the largest to the smallest scales based on empirical observation. If correct, it would argue that the beginning of the transition, observed as deviations from V_NEWTON, would not be expected until you approach separations of about 0.002 mm depending on the masses being used.

Finally before proceeding to the cosmic scale discussions, there is one last effect at the smaller scales that should be mentioned. We noted previously that Newton’s potential is attractive and uniformly decreasing with distance. This guarantees that the force felt between two bodies is always attractive. This is the macro scale region we have been discussing.

At the nuclear scales, the modified Yukawa potential has a slowly oscillating behavior. That means the force also oscillates. That is, there is an attractive force region, followed by a repulsive force region, followed be an attractive force region, and so forth.

V_FRACEP agrees with both Yukawa at the quantum end and Newton at the macro (gravity) end. But V_FRACEP also shows an oscillating transition region that connects the two extremes.

It should be noted that even though V_NEWTON and V_Y-mod both have 1/r–like behavior at some level, the connection between gravity and quantum mechanics has not yet been definitively established.

The question to be considered is whether there is an observed effect that has an oscillating behavior in the same region as the V_FRACEP transition regardless of the expected source? The answer to this question is yes. The force in question is called the Casimir force.

The Casimir force was first proposed in the 1940sby Hendrik Casimir (Ref. 14, Ref. 15, Ref. 16). His work predicted a very weak force between any two objects a few micro-meters apart in a vacuum. He concentrated his work on perfect conductors for the materials. Later calculations generalized the work to non-perfect conductors and dielectric materials.

The force Casimir predicted is identified as a quantum mechanical effect that results from “zero-point electromagnetic fluctuations”. That means that, in the very tiny space between the two plates, virtual electrons pop into and out of existence at times that are too short to be measured. This constantly fluctuating charge and mass in the space gives rise to a weak force that is not related to any currents that might be induced in the two plates.

Usually, the weak force is attractive. But a weaker repulsive force has also been observed in the nano-meter range. It is this range where V_FRACEP sees the oscillating transition in its potential. So it is fair to ask if it is possible that the Casimir force is truly the transition force implied by the oscillating portion of V_FRACEP?

The main interest in the transition force is motivated by the desire to understand how the forces of nature are related. Efforts are being made to find a model that unifies all of the forces of nature (Ref. 17, Ref. 18). Total unification means that a set of mathematical expressions can be found to describe the generating field for one of the forces in terms of the generating field for one of the other forces.

A set of expressions currently shows the relation between electromagnetism and quantum mechanism. But a generally accepted standard relationship is lacking when it comes to unifying quantum mechanics and gravity. One interesting expression that hints at offering some help in resolving this unification issue comes from electro-magnetics which has an expression for charge density as a function of mass density. This, unfortunately, is not sufficient to unify gravity with electromagnetism or quantum mechanics.

What is really needed for unification is an expression for total mass as a function of total charge. This is needed to allow a reliably accurate and precise prediction of the masses of the sub-atomic particles. The standard model of fundamental particles has charge as an inherent characteristic. That is, the same electric charge characteristic is associated with sub-atomic particles of different masses. Further, there is no obvious characteristic in the particles that would cause them to have the same charge but different masses. So there is no obvious way to for that model to make an accurate and precise prediction of the mass of the particles with the same charge.

The FRACEP model on the other hand has the quantum level particles as composites with charge elements as components in the particles. The larger particles in this model have more charge carrying components. This may allow a unique charge verses mass relation to be developed. The existence of such a relation, combined with V_FRACEP’s oscillating transition region, may provide a possible basis for a totally unified theory.

We can now proceed to the cosmic scale discussion. This discussion shows a common behavior with the ordinary macro scales behavior we have been discussing – including the oscillating transition region at the lower range limits. This oscillating behavior is also seen at the cosmic scales. It hints at the quantum-like behavior predicted by the current models at the black hole level of mass.

2.3.2 The Cosmic Macro Scales

For the purpose here, the discussion of cosmic macro scales encompasses sizes, masses and separation distances of objects outside of earth. Because of the large distances involved, any two objects appear as point sources to each other. For this reason, the separation distances are treated as being from center-to- center without considering the possibility of extended source interactions. This is the usual use of Newton’s laws in this region. The discussion of the cosmic scales begins with the separation distance in the solar system.

Figure 6 shows a comparison of V_FRACEP with V_NEWTON for a source body with the mass of our sun (m_s = 1 SU = 1.99x10³⁰ kg.) The SU is an astronomical unit of mass termed Solar Units. In V_NEWTON (equation 6), m₁ is the source mass (m_s) and m₂ is the test mass (m_t). Convention for studies of this kind generally use m_t = 1 kg. In V_FRACEP (equation 4), the m_t = 1 kg is suppressed.

The plot shows close agreement between the two potentials over the range of the planetary orbits within the solar system. The departure of V_FRACEP from V_NEWTON occurs well within the orbit of Mercury. The insert in the figure shows the oscillation in V_FRACEP begins outside the sun’s surface. In other words, if you could get close enough to the sun you would feel the adjacent repulsive and attractive regions in the potential before actually entering the sun’s interior.

It should be noted for the sake of purity that the straight forward application of Newton’s law assumes a point source relative to the test mass at a distance. In reality, the sun is actually an extended source compared to the size and distance of Mercury. So, for the purpose of the comparison represented in the figure, the source m1 is assumes to be concentrated within a sufficiently small space relative to the orbit locations on the plot. In reality this is true for most of the planets.

The oscillation region for the cosmic scales is identical to the oscillations in the ordinary scale macro region previously seen in Figure 5. The turnover point for the cosmic scales occurs at around 4x10^-5 m_s(SU). This rough approximation appears to hold for the sun (1 SU), as well as, masses larger than the sun.

The oscillatory behavior of the potential is evident even for the largest masses with the first oscillation peak occurring outside the surface of the body. This means that starting at the center of the body of the source mass the potential begins to oscillate while inside the body and continues the oscillation for some distance even after passing through the surface. As one then moves farther out from the center, the oscillation transitions to the 1/r form of Newton at the point referred to here as the first peak. One exception to that rule is the Milky Way (our galaxy) which is an extended source relative to the sun’s location in it.

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FIGURE 6. This shows a comparison of the V_FRACEP and V_Newton potentials.
The potentials are given in Joules and the radius is in meters. The dashed line is V_Newton
and the solid line is V_FRACEP. The insert shows where V_FRACEP diverges from V_Newton.
The divergence begins outside the sun’s surface but well inside the orbit of Mercury.

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Table 2 shows the oscillation first peak for several cosmic bodies relative to the body’s surface position from its center. The neutron star and the black hole considered are examples of typical possible bodies. Not all neutron stars and black holes are of the masses and sizes reflected in the table. The nature of these bodies is a reflection of the stellar evolution process (Ref. 19).

In the sequence of the life events of a star, while the star is burning it producing energy by fusion. That means that two atoms, like hydrogen combine to produce one helium atom and in the process energy is released. This energy release creates a pressure that pushes its surface outward. At the same time, gravity tries to pull the surface inward. The result is that when the outward pressure balances the inward pull the star has a stable surface at a fixed position from its center.

At the end of a star’s life, energy production ceases and gravity forces the star to collapse. Neutron stars or black holes result depending on the starting mass of the star.

Table 2. This shows examples of where the potential’s oscillation begins for
several astronomical bodies. Column 2 shows the mass of the body in SU, the
radius to the body’s surface in AU and (in meters), and the location of the first
oscillation in AU and (in meters). Note that our Solar System is located
~2x10⁸ AU (~3x10¹⁹ m) from the center of our Milky Way galaxy.

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For bodies less than around 0.08 SU (~80 times the mass of Jupiter), fusion never begins. The only heat they possess is what is caused by gravitational contraction during formation. These bodies slowly cool over time, leaving a dead planet sized body of normal rocky-like matter.

For low mass stars (0.08 to 1.4 SU) the star begins to collapse when energy production ceases. This cause the temperature and pressure to rise resulting in a flash of helium burning that expands the star to a red giant. It loses a portion of its outer mass and then collapses to a white dwarf – a moon sized body that gradually cools.

For larger mass stars, things get more complicated. For masses >1.4 SU, the collapse leads to a neutron star. When energy production ceases, the star goes through the helium flash stage and then begins to collapse. Because of the greater mass of this star type, the density becomes sufficiently high that energy production can begin again causing fusion of even heavier atoms. Repeated cycles of collapse and re-ignition continue until the star reaches the iron atom. At this point all energy production ceases for good causing the star to become a super nova. At this time, the super nova explosion strips the star of most of its matter.

At the same time, there is an implosion of the remaining material. The force of the implosion accelerates the material to a sufficiently high speed by the end of the collapse that it breaks apart the atoms leaving only neutrons that are as closely packed together as they can get. This dense material is called neutronium and the resulting body is referred to as a neutron star.

From Table 2, we see that a typical neutron star (that began life with more mass than our sun) can lose so much of its material it ends its life at only around 3x10^-5 SU. Its size was originally much larger that of our sun, but it ends up much smaller (0.016x10⁶ m). According to the VFRACEP, its potential does not take on the 1/r form of Newton until well outside the collapsed star’s surface (~3x10⁶ m). The oscillation that begins within the neutron star and continues until well outside the surface indicates the possibility of quantum like behavior in the space surrounding the neutron star. It is not clear at this time if any quantum like behavior has been observed.

Consider now the black hole case. For stars > 40 SU, the collapse follows the same path as a neutron star. But, the mass is much greater yet when the final stage occurs. Here, the force of the implosion accelerates the material to a much high speed yet.

The collisions caused by these super high speeds not only break apart the atoms but also the more elementary matter (like electrons and quarks). This in turn results in a disruption of the fabric of space itself. The result is a black hole. Because of the overwhelming force of gravity associated with a black hole, when one gets close enough nothing can escape its pull – not even light.

Black holes cannot be seen. The only evidence of their existence is the effect of gravity on the space around them – like an anomalously strong pull on close objects in what appears to be empty space. There can also be a faint glow of light and x-ray emissions in the surrounding region. These are signs of quantum like processes in the space around the black hole. It is also predicted that time dilation (a special relativistic effect) would be experienced in the region around and into the black hole. This means that time passes slower and slower as you enter the influence of the black hole.

From Table 2, we see that a typical not-too-large black hole can begin life with more mass yet than the un-collapsed neutron star; but, it loses so much of its material it ends its life that it is only around 1.5x10^-4 SU. Its size was originally much larger than even the un-collapsed neutron star, but it ends up much smaller (0.005x10⁶ m). (This radius for the black hole is known as its event horizon.) According to the V_FRACEP, its potential does not take on the 1/r form of Newton until well outside the black hole’s event horizon (~15x10⁶ m). This oscillation of the potential, that begins within and continues until well outside the event horizon, could be interpreted as the source of the quantum-like behavior observed in the space surrounding the black hole (Ref. 20).

Among the black holes in space, are a collection of super massive ones that are believed to be within the heart of most galaxies. Within the heart of our Milky Way there is one of these super massive black holes (SgrA-* - Sagittarius A - star). As Table 2 indicates, it has the mass of about 2 million suns and is about 10¹⁰ times the size of our sun.

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FIGURE7. This shows the pinwheel structure of the Milky Way Galaxy. The shaded areas
contain the stars and dust of the dense regions of the galaxy. The area in-between is relatively
material free. The four spiral arms are roughly elliptical in shape as indicated by the black dashed
overlaid ellipses. The figure shows that the ellipses are rotated relative to one another reflecting
the different rotation speeds at the different distances from the galaxy center. The outer ring is a
roughly circular donut that surrounds the rest of the galaxy material. The lengths of the arrows
give one measure of the distances to the outer edges of the dense regions of each of the arms: (a)
8,500 lt-yrs, (b) 17,000 lt-yrs, (c) 22,000 lt-yrs, (d) 28,700 lt-yrs, (e) 35,400 lt-yrs,
(f) 50,000 lt-yrs. The sun is ~28,000 lt-yrs from the galaxy center.

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V_FRACEP indicates the oscillation of the potential of a super massive black hole with the mass of SgrA-* extends as far as 60 – 80 AU which is about twice as far as Pluto is from our sun. Our solar system is about 1.8 billion light-years (1.8x10⁹ AU) from SgrA-*. This is well outside the oscillatory region of its potential and so we see only 1/r pull from the Milky Way’s black hole. (A light-year is the distance light can travel in one year – 1 lt-yr = 63,240 AU = 9.46x10¹⁵ m.)

Our Milky Way Galaxy (Ref. 21) is what is referred to as a spiral galaxy. It has a roughly spherical core with a radius about 4.7x10⁸ AU (~7,500 lt-yrs) that contains the super massive black hole SgrA-* that was just discussed. Surrounding the core is a swirling flat disc of stars and dust (~2,000 lt-yrs thick) that extends in radius to about 3.2x10⁹ AU (~50,000 lt-yrs) from the center. Observing the disk from above, one sees the spiral arms of the disk – like a pinwheel around the core. The effect of the spiral arms gives the disk the appearance of bands of higher density material (the stars and dust) separated by very low density regions (Figure 7).

As Table 2 shows, the oscillation in the potential of the Milky Way Galaxy, according to FRACEP, begins between 4x10⁵ and 80x10⁵ AU and continues inward towards the center. This assumes the bulk of the mass is contained within a relatively small region of the core.

It has generally been believed in the past that this assumption was true. However, this is no longer the general consensus. It is now believed that a large portion of the Galaxy’s material also is distributed in the spiral arms. This means that the exact extent of the oscillation in the potential is likely more extensive than the simplified picture presented here. It also indicates that our Solar System is well within the extended source of the Galaxy. So, one must cautiously consider the Galaxy’s potential oscillations relative to the location of the Solar System.

The appearance of the Galaxy’s potential to objects a long distance from it is another matter. For distances larger than 80x10⁵ AU, V_FRACEP has the 1/r pull. The closest galaxy to the Milky Way is SgrDEG (Sagittarius Dwarf Elliptical Galaxy) at ~80,000 lt-yrs (~5x10⁹ AU) from the sun (Ref. 22). At this distance, the Milky Way’s potential is seen as 1/r – just as Newton would predict.

2.4 A Further Analysis of the V_FRACEP Function

Having shown the general behavior of V_FRACEP in the three regimes (nuclear/quantum, macro and cosmic), let us consider the functional form in more detail. Recall, the general form of the potential:

(5)                                        V_FRACEP = F(M) x sin[arg(r,M)] x exp(-2.4r/M).

Here, M = m_s/m_p, where m_p is the mass of the pion particle. The arg(r, M) is a four term function where each of the terms varies with a different power of r. (See equation 4b). Each of its terms becomes significant in one or more of the regimes considered. This behavior is examined in detail in a moment, but first we consider the leading coefficient and the exponential factor.

To help facilitate the discussion, it is useful to understand the values of M and r that are appropriate to each regime. In the nuclear range, M is proportional to unity. That is m_s ~ m_p within a few orders of magnitude. For the pion, the typical particle associated with nuclear force exchange, M = 1. The range of the interaction is between r ~ 0.3 x 10^-15m and r ~2.5 x 10^-15m. The mean error between V_FRACEP and V_Y-mod (the modified Yukawa potential discussed in Section 1.2.2) for this interaction range is ~6%, with an RMS error of ~23%. This error level is not unusually large for this type comparison.

For the macro scale, M is on the order of 10²⁰ or greater. For example, an m_s = 55 kg has M ~ 10³² and an m_s = 5.5 mg would have M ~ 10²². The interaction range can be less than a millimeter or tens of thousands of meters depending on the value of ms. Outside of the transition region, where the oscillatory behavior has ceased, the mean error and the RMS error between V_FRACEP and V_Newton is less than 0.001%. Recall that the uncertainty in V_NEWTON is 0.00135%.

For the cosmic scales, M is on the order of 10⁵⁰ or greater. An m_s equal to the sun has M ~ 10⁶⁰. One equal to the earth has M ~ 10⁵⁴. The interaction range can be less than 400,000 km (the distance from earth to the moon) to the distance between galaxies (billions of light-years where 1 lt-yr ~ 10¹⁶m). Outside of the transition region, where the oscillatory behavior has ceased, the mean error and the RMS error between V_FRACEP and V_Newton are less than 0.001% when all four terms in arg(r, M) are used. This point will be address further in a moment.

2.4.1 A Look at the Leading Coefficient F(M)

The leading coefficient, F(M) = A₀(M) + B₀(M), is a two term function of M only. (It is not a function of r.) The A₀ term is most significant for the nuclear case, while the B₀ term is most significant for the macro and cosmic regimes. The transition region between nuclear and macro can have an appreciable contribution from both A₀ and B₀.

Recall A₀ = 1 / (0.18 x M), and B₀ = 9.2095x10^-8 x M^1/2. This means that in the macro and cosmic regimes, A₀ is as large as 10^-20 or smaller, while B₀ is as small as 100 or greater as the mass grows. So, B₀ is the more significant factor by over 20 orders of magnitude. In the nuclear regime, A₀ is around 6 while B₀ is around 10^-8. So A₀ is the more significant value by around 8 orders of magnitude. For masses near 6x10^-28 kg (~3x10⁵ MeV/c²), both A₀ and B₀ contribute significantly. Elementary particles with masses in this range include the heaviest (top) quark, the heavier nuclear-weak field exchange particles (W⁺ and W^-), and the higgs boson. This mass-range marks one type of transition from the nuclear regime to the macro regime. (Recall the pion particle has a mass mp = 139.6 MeV/c² and M = 1.)

2.4.2 A Look at the Exponential Factor

Now consider the exponential factor, exp(-2.4 x r / M), which varies as r/M. Here, r is in units of fermis (1 fm = 10^-15 m). As shown in Table 3, in the nuclear case the exponential term at the smaller ranges varies from 0.5 to 0.9 (column 4) depending on the particle mass. For example, for the pion, exp(-2.4 x 0.3 / 1) ~ 0.49. At the upper range the exponential varies from ~0.002 to ~0.6 (columns 5 and 6). As the table shows, in the nuclear regime, the exponential reduces the value of the potential by a significant amount, especially at the smaller masses and larger ranges. It is this factor that drives the potential to zero as the interaction range (for nuclear-scale particles) approaches the transition region to the macro ranges.

.

Table 3. This shows examples of the value of the exponential factor of
V_FRACEP for the nuclear case. Masses are given in MeV/c² and range is
given in fermis. For emphasis, the last column provides the value of the
exponential for a range beyond the limits of the usual nuclear interaction range.

.

For the macro and cosmic cases, the exponential factor is unity within their respective interaction ranges as shown in Table 4. For example, for a 5.5 mg mass (5.5x10^-9 kg) the M = 10²². At r = 10^-3 m (10¹² fm), the exponential is exp(-2.4 x 10¹² / 10²²) ~ 1. However, by the time r reaches cosmic scales (10⁵ km), the exponential for such a small mass has effectively gone to zero. This means the potential due to such a small mass is negligible no matter what the value of the other factors.

Finally, considering the range of values for the leading coefficient and the exponential factor, the limiting form of V_FRACEP for the three regimes (nuclear, macro and cosmic) can be summarized as follows. For the nuclear case

V_FRACEP ~ A₀ x exp(-2.4 x r / M) x sin[arg(r, M)].

And in the macro and cosmic cases

V_FRACEP ~ B₀ x sin[arg(r, M)].

In the transition region between nuclear and macro, the full form is required. That is

V_FRACEP ~ (A₀ + B₀) x exp(-2.4. r / M) x sin[arg(r, M)].

At this point we now consider the effect of the sine factor.

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Table 4. This shows examples of the value of the exponential factor of
V_FRACEP for the macro and cosmic cases. Masses are given in kilograms
and range is given in meters. Ranges from the lower end of the macro scale
to the mid-cosmic scales are considered. Recall that the exponential
actor requires r to be input in fermis.

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2.4.3 A Look at the SineFactor

The sine factor, sin[arg(r, M)], is a bounded function that varies cyclically between +1 and -1. It has four terms each of which vary with different powers of M and r. For a sufficiently small value of the argument, sin[arg(r, M)] = arg(r, M).We now look at the four terms of the argument and consider their relative values in the primary (non-transition) regions of each of the three regimes.

Recall the form of the terms in the argument (arg) from equation 4:

                                        arg(r,M) = (150 p/180) x [K₁r² + K₃ + K_2a/r + K_2b/r + f/r³]

                                        K₁ = -0.09 / M²

                                        K₃ = 1/M^1/2

                                        K_2a = 1/M^1/2 and K_2b = -M^1/2 x 0.00006 / m_p

                                        f = -0.0000006 x (10¹⁵)³ x m_p_Au³ x [M/8x10^-60]² /150 ;

In these expressions r is in fermis and M is unit-less. The factor (p/180) converts the argument from degrees to radians. The factor [(10¹⁵)³ x m_p_Au³] converts fermis to astronomical units (AU).

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Table 5. This shows a comparison of the values of the terms in arg(r, M). The range
of r values considered falls into the non-transition region for each of the regimes.
The terms are given in degrees and the r is converted to appropriate units for the
regime. The full argument is arg(r, m) = (p/180) . [K₁r² + K₃ + K_2a/r + K_2b/r + f/r³].
The last two rows show the mean error and the RMS error between V_FRACEP and
V_NEWTON for the macro and cosmic cases, and between V_FRACEP and V_Y-mod for
the nuclear case over the ranges considered. The uncertainty in V_NEWTON is 0.0135%.
The nuclear case uncertainty limit is less well defined but not unreasonably large.

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Table 5 shows that in the non-traditional regions K₁, K₃ and K_2a are significant for the nuclear case, but negligible for the macro and cosmic cases. The K_2b is significant for all regimes, and f is only significant for the cosmic case.

This means that, in the non-traditional regions for each of the regimes, V_FRACEP takes a different limiting form. In the nuclear case,

V_FRACEP(nuclear) ~ (1/0.18M) x exp(-2.4r/M) x
sin{(p/180) x [-13.5r²/M² + 150M^1/2 + 150/(r x M^1/2) – 0.009M^1/2/ (r x m_p)]}

In this case, the argument of the sine function is mostly large so sin(arg) is not equal to arg.

In the macro case,

V_FRACEP(macro) ~ 9.2095x10^-8 x M^1/2 x
sin{(p/180) x [-0.009 M^1/2 / (r x m_p)]}

In this case, in the non-traditional region, the argument is sufficiently small so as to make the sin(arg) = arg. This leads to the limiting form

V_FRACEP(macro) ~ 1.45x10^-11 x M^1/2 /(r x m_p)
= -g* M / r

This has the same form as V_NEWTON (-Gm/r) where G is analogous to g* x m_p^1/2 when the units of V_FRACEP and V_NEWTON are the same.

In the cosmic case,

V_FRACEP(cosmic) ~ 9.2095x10^-8 x M^1/2 x
sin{(p/180) {– 0.009M^1/2/(r x m_p) – 9.0 x10^-5 x (10¹⁵)³ x m_p_Au3 x [M/8x10^-60]²/r³}

Again in this case, in the non-traditional region, the argument is sufficiently small so as to make sin(arg) = arg. This leads to the limiting form

V_FRACEP(cosmic) ~ -g* M/r - h*M^2.5/r³

The second term (the f-term in arg(r, M)) contributes significantly to the potential for the sun as far out as the asteroid belt beyond the orbit of Mars. The correction to the potential brings the error to within the uncertainty of V_NEWTON. Table 6 shows the effect of the f-term on the errors in the potential.

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Table 6.This shows the effect of adding the f-term to arg(r, M) in computing V_FRACEP. The
errors are given in percent. The uncertainty in V_NEWTON is 0.00135%. This shows that with
the f-term the error in V_FRACEP is unaffected beyond Mars and grows for the inner planets.

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3.0 CONCLUSIONS

This initial effort produced a universal potential that agrees at both the quantum scales and at the macro (through the cosmic) scales. V_FRACEP is a multi-term function that agrees within reasonable limits at the typical quantum scales with the modified Yukawa (nuclear) potential. It also agrees with the Newtonian potential at the typical macro and cosmic scales. In addition, this new potential also demonstrates a smooth transition between the quantum scales and the smallest macro scales of the Newtonian potential.

The potential, as developed, was not intended to model specific mechanisms. The intention was to produce an operational characterization of the field behavior in general. With four terms in the argument of the sine function, V_FRACEP reproduces the oscillatory behavior characteristic of particle confinement at quantum scales. But, it also provides a transition to macro scales that resembles the attractive and repulsive appearance of the Casimir effect. (This could imply that the Casimis effect – generally agreed to be a quantum effect – is a true transition between the macro and quantum scales.)

This oscillatory behavior also shows up at the cosmic scales. The combination of small radius with large mass produces oscillations that could be interpreted as quantum effects at macro scales – like those expected around black holes. Such effects are not predicted by Newtonian gravity or Einsteinian relativity – they require quantum gravity models.

Further modification of V_FRACEP will need to be considered to identify the simplest formulation of the mathematical form of the function. Also, how the function handles negative masses must be addressed. The existence of negative mass is an integral part of FRACEP, and the possibility of negative masses could offer some insight into the behavior and quantification of dark matter and dark energy assumed to be present in the universe.

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4.0 REFERENCES

¹ J.A. Giannini, "The FRACEP Model, Part 1a: A Look Inside the Elementary Particles of the Standard Model", Chapter 1 in Fractal Rings and Composite Elementary Particles (FRACEP) Model (2012) (http://www.jagnetbooks.org)

² J.A. Giannini, "The FRACEP Model, Part 1b: What Is the Size of the Composite Particles ", Chapter 2 in Fractal Rings and Composite Elementary Particles (FRACEP) Model (2012) (http://www.jagnetbooks.org)

³ F. Hoyle, G. Burbidge, and J.V. Narlikai, A Different Approach to Cosmology, Cambridge U. Press, Cambridge, UK (2000)

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⁵ B. Schwarzschild, “Theorists and Experimenters Seek to Learn Why Gravity Is So Weak”, Physics Today, September (2000) 22-24

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⁷ G. Rindler, Essential Relativity, VanNostrand Reinhold Co., NY (1969)

^7a M. Jammer, Concepts Of Space, Dover Publications Inc., Mineola, NY (1993).

⁸ B. Povh, K. Rith, C. Scholz, and F. Zetsche, Particles and Nuclei, ed. 2, Springer, NY (1999) Chapter 16.

⁹ C.A. Bertulani, Nuclear Physics in a Nutshell, Princeton U. Press, Princeton, NJ (2007) Chapter 3.4

¹⁰ V.M. Blanco and S.W. McCuskey, Basic Physics of the Solar System, Addison-Wesley Publishing Company, Inc., Reading, MA (1961) p. 135

¹¹ C. Seife, “A Slow Carousel Ride Gauges Gravity’s Pull”, Science, vol. 288 (2000) 944

¹² C. Speake, T. Quinn, “The Search for Newton’s Constant”, Physics Today, July (2014) 27-33

¹³ A. Arkani-Hamed, S. Dimpoulos, G. Dvali, “The Universe’s Unseen Dimensions”, Scientific American, August (2000) 62-69

¹⁴ H.B.G. Casimir, D. Polder, Phys. Rev. 73 (1948) 360

¹⁵ J. Miller, “Casimir Forces between Solids Can Be Repulsive”, Physics Today, February (2009) 19-22

¹⁶ “This month in Physics History: May 20 1948 Results of first experiments on the Casimir effect”, APS News, 21 no. 5 (2012) 2

¹⁷ S. Weinberg, "A Unified Physics by 2050", Scientific American, December (1999) 68-75

¹⁸ E.G. Adelberger for the EOT-Wash Group, "Sub-millimeter Tests of the Gravitational Inverse Square Law", arXiv:hep-ex/0202008, v1, February (2002).

¹⁹ T. Padmanabhan, An Invitation to Astrophysics, World Scientific Publishing Co. Ltd, Singapore (2006) Chapter 4 in Stars and Stellar Evolution, p 135.

²⁰ G. Dvali, “Quantum Black Holes”, Physics Today, January (2015) 38-43

²¹ P. Moore, "The Galaxy", Chapter 25 in The Data Book of Astronomy, Institute Of Physics Publishing, Midsomer Norton,UK (2000) p 310.

²² P. Moore, "Galaxies", Chapter 26 in The Data Book of Astronomy, Institute Of Physics Publishing, Midsomer Norton,UK (2000) p 313.

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