I. INTRODUCTION
Preliminaries: The SSCP (also referred to as the Discrete Fractal Paradigm or Discrete Scale Relativity) has led inevitably to a modified scaling for gravitation (see Papers #11 and #12 of the Selected Papers section). Rather than having the standard Newtonian value of G_{0} (= 6.67 x 10^{-8} cm^{3}/g sec^{2}) apply universally at all scales of nature’s hierarchy, the SSCP requires that each cosmological Scale has a different value G_{Ψ}, such that
G_{Ψ}= [Λ^{1-D}] G_{Ψ-1} , (1)
where Λ = 5.2 x 10^{17} and D = 3.174. Given G_{0} and Eq. (1), we can calculate that the Atomic Scale value G_{-1} for the gravitational coupling factor is about 2.18 x 10^{31} cm^{3}/g sec^{2}, and applies within gravitationally bound Atomic Scale systems. Rather than being virtually negligible in the microcosm of atoms and subatomic particles, General Relativity actually dominates the interior physics of these systems (see the 2008 New Developments section: “The Meaning of Planck’s Constant”). The new scaling for gravitation has also led to a major revision of the Planck scale, as discussed in the 2006 Technical Note on that subject.
Purpose of this Technical Note: Given the revised scaling for gravitational interactions, it seems very likely that the best currently available models for describing subatomic particles and nuclei are the Kerr-Newman solutions of General Relativity coupled with Maxwell electrodynamics. These solutions describe charged, rotating black holes and “naked” singularities, hereafter referred to collectively as ultracompact objects. In this technical note we will explore the degree to which Kerr-Newman ultracompacts provide accurate models for subatomic particles and nuclei. In cases where the total angular momentum and/or the charge of the ultracompact is zero, the Kerr-Newman solutions reduce to Kerr, Reissner-Nordstrom or Schwarzschild solutions. We will find that the mass, charge, angular momentum and radius relationships for subatomic particles and nuclei, as well as some of their more subtle phenomena, are highly consistent with the counterpart relationships for Kerr-Newman utracompacts, when G_{-1} is substituted for G_{0} in the calculations.
Theoretical Source Material: The reference source for the present discussion is the readily available book Gravitation by Misner, Wheeler and Thorne (W. H. Freeman & Co., San Francisco, 1973). Most of the relevant theoretical material is found in Chapter 33 of that book, with emphasis on Box 33.2 to Box 33.4. For the most part we will follow their use of cgs units and their notation, except that we will use J as the symbol for total angular momentum and q as the symbol for charge, so as to maintain consistency with other research presented at this website.
Geometrized Units: In working with the extremely complex partial differential equations of the Einstein-Maxwell equations it is highly useful to simplify matters by adopting geometrized units, wherein the masses (M), charges (q), angular momenta (J) and angular momenta per unit mass (a = J/M) are converted into lengths via the following method.
QUANTITY |
MULTIPLY BY |
RESULT |
Mass |
G/c^{2} |
Length |
Charge |
G^{1/2}/c^{2} |
Length |
Angular Momentum |
G/c^{3} |
Length |
a = J/M |
1/c |
Length |
At any point it is possible to revert back to the regular units (multiplying by the reciprocal of the appropriate geometrizing factor) in order to obtain the values of measurable physical quantitites. The use of geometrized units reduces the mathematical complexity without any loss of accuracy, and is highly consistent with the essence of General Relativity, i.e., emphasizing the primary role of spacetime geometry in the structure and dynamics of nature.
In this technical note I will give the relevant equations and present the results of calculations, which have been done at least twice. Feel free to check these calculations.
II. BASIC PROPERTIES OF KERR-NEWMAN ULTRACOMPACTS
1. The metrics for Kerr-Newman ultracompacts are stationary, axisymmetric around their rotational axis, and asymptotically flat, such that they go over to the Minkowski metric as r → ∞.
2. Relativistic ultracompacts have undergone full gravitational collapse and are dynamically dominated by a central point or ring singularity.
3. Ultracompacts are uniquely and almost completely characterized by their M, J and q values. This remarkable simplicity and “elementarity” is shared by subatomic particles.
4. In the dynamical physics of Kerr-Newman ultracompacts total M, J and q are highly conserved, as is the case for subatomic particles.
5. Based on their M, J and q values, both Kerr-Newman ultracompacts and subatomic particles form disjoint families. The basic Kerr-Newman families are shown below.
M |
J |
Q |
SOLUTION |
SINGULARITY TYPE |
SUBATOMIC ANALOGUE |
yes |
0 |
0 |
Schwarzschild |
point |
? |
yes |
yes |
0 |
Kerr |
ring |
neutron |
yes |
0 |
yes |
Reissner-Nordstrom |
point |
^{4}He^{++} (e^{-}?) |
yes |
yes |
yes |
Kerr-Newman |
ring |
proton (e^{-}?) |
6. Kerr-Newman ultracompacts have magnetic dipole moments (μ), but they do not have electric dipole moments. In terms of their electromagnetic properties, there is a high degree of correspondence between Kerr-Newman ultracompacts and subatomic particles/nuclei.
7. Black hole solutions (M^{2} ³ a^{2} + q^{2}) have event horizons, whereas naked singularity solutions (a^{2} + q^{2} > M^{2}) do not have event horizons.
8. Kerr-Newman ultracompacts are “time independent” stable systems, as are the stable subatomic particles and nuclei.
9. Kerr-Newman ultracompacts are associated with J µ kM^{2} relationships, as are subatomic particles (e.g., the Regge trajectories of resonance families).
10. Two general laws of black hole dynamics are as follows.
First Law:
In black hole dynamics total energy is conserved, as are total momentum, angular momentun and charge.
Second Law:
For interacting black holes, neither the sum of the surface areas, nor the sum of the squares of the irreducible masses, can decrease.
The surface area of a Kerr-Newman black hole is: A = 4π ([M + {M^{2} – q^{2} – a^{2}}^{1/2 }]^{ 2} + a^{2}), and while M, q and a for individual ultracompacts can change due to interactions, A is never reduced.
The potential for a strong and direct self-similarity relationship between Kerr-Newman utracompacts and subatomic particles/nuclei seems quite compelling. Below we will see that this discrete self-similarity goes well beyond qualitative phenomena to the quantitative determination of physical relationships and fundamental properties.
III. THE PHYSICAL DISTINCTION BETWEEN LEPTONS AND HADRONS
An important relation for Kerr-Newman ultracompacts is:
M^{2} ³ a^{2} + q^{2}, (2)
where a = J/M. When this relation is obeyed, the solution is a black hole and the central singularity is enclosed within an event horizon. When the relation is not obeyed, i.e., when:
M^{2} < a^{2} + q^{2}, (3)
then the solution has a naked singularity at its center and there is no event horizon enclosing the singularity.
It is extremely interesting to test this relation in the case of subatomic particles/nuclei now that we have the correct value for G_{Ψ}, and the table below summarizes the results of the calculations.
PARTICLE/NUCLEI |
M^{2}/a^{2} + q^{2} |
SOLUTION TYPE |
e^{-} |
4.36 x 10^{-13} |
Naked singularity |
Planck Mass (M) |
1.31 |
Black hole |
p^{+} |
4.86 |
Black hole |
^{4}He^{++} |
1059.76 |
Black hole |
^{56}Fe^{+26} |
1056.91 |
Black hole |
From this exercise we have learned two important facts.
(1). The distinction between leptons and hadrons appears to be that hadrons are Kerr-Newman ultracompacts that obey M^{2} ³ a^{2} + q^{2} and have event horizons (and ergospheres, as we will see later). Leptons, on the other hand, are Kerr-Newman ultracompacts that violate Eq. (2) such that a^{2} + q^{2} > M^{2}. They are horizonless singularities. This is consistent with the observational result that the electron appears to be structureless down to scales of at least 10^{-16} cm.
(2) We have also answered the key question raised in section 7a of the March 2008 New Development on the meaning of Planck’s constant: What is the significance of the fact that the Planck mass (M) appears to be slightly smaller than the mass of the proton. Our calculations show that M is a good approximation to the “tipping point” mass (M^{2} = a^{2} + q^{2}) that distinguishes horizonless and horizon-possessing ultracompacts.
If one plots log M versus log [M^{2}/a^{2} + q^{2}], it appears that one gets a sigmoidal curve that is very sensitively dependent upon M. The point of highest inflection (largest slope), where the curve goes through log [M^{2}/a^{2} + q^{2}] = 0, occurs at M » M. It is quite likely that M does not refer to the mass of an actual particle, but rather that it defines the lepton/hadron, or horizonless/horizon-possessing, boundary for Atomic Scale ultracompacts. It is anticipated that, with a mass of 105.66 Mev, the muon will undoubtedly violate the M^{2} ³ a^{2} + q^{2} relation and fit the pattern as a horizonless Kerr-Newman ultracompact. However, it is not immediately clear whether the tau particle with the relatively large mass of 1784 Mev also fits the pattern described above.
IV. DETAILED RESULTS FOR THE PROTON
In this section we will present the algebraic equations for various parameters of a Kerr-Newman ultracompact, which are derived from the full Kerr-Newman family of solutions. We will find that these algebraic relationships are very well-suited for modeling the proton.
The data used in the proton calculations are as follows.
PARAMETER |
MEASURED VALUE |
GEOMETRIZED VALUE |
Mass (M) |
1.67 x 10^{-24} g |
4.07 x 10^{-14} cm |
Angular Momentum Per Unit Mass (a = J/M) |
5.49 x 10^{-4} cm^{2}/sec |
1.83 x 10^{-14} cm |
Charge (q) |
4.80 x 10^{-10} [g cm^{3}/sec^{2}]^{1/2} |
2.51 x 10^{-15} cm |
Angular Momentum (J) |
9.13 x 10^{-28} erg sec |
7.44 x 10^{-28} cm |
(1) General Event Horizon Relation
As noted above M^{2}/a^{2} + q^{2} = 4.86 for the proton.
(2) Radius of the Proton Event Horizon
The algebraic expression for the radius (r_{+}) of the proton’s event horizon is:
r_{+} = M + [M^{2} – a^{2} – q^{2}]^{1/2} . (4)
When the expression is evaluated, we get:
r_{+} = 7.7 x 10^{-14} cm.
Since the mass and charge radii for the proton are roughly 8 x 10^{-14} cm, our calculated value for r_{+} is encouraging.
(3) The Static Limit Radius
The static limit radius (r) is defined as the radius at which all test particles approaching a Kerr-Newman ultracompact orbit in the same direction as the rotation of the ultracompact.
Between r and r_{+}, referred to as the ergosphere, there is strong frame-dragging that increases as the distance to r_{+} decreases.
r º M + [M^{2} – q^{2} –a^{2} cos^{2} θ]^{1/2}. (5)
For the proton, Eq. (5) yields a static limit radius of 8.13 x 10^{-14} cm, which is quite close to the observed value for the radius of the proton.
(4) Surface Area of the Proton
The surface area (A) for a Kerr-Newman ultracompact is:
A = 4π [ r_{+}^{2} + a^{2}] = 4π ([M + {M^{2} – q^{2} – a^{2}}^{1/2 }]^{ 2} + a^{2}). (6)
In the case of the proton, Eq. (6) yields A = 7.87 x 10^{-26} cm^{2}.
(5) The Irreducible Mass of the Proton
Regarding interactions of Kerr-Newman ultracompacts, each object has a quantity referred to as its irreducible mass (M_{ir}), which is defined as follows.
M_{ir} º ½ [r_{+}^{2} + a^{2}]^{1/2} = [A/16 π]^{1/2}. (7)
In the case of the proton, Eq. (7) yields
M_{ir} = 1.62 x 10^{-24} g,
which is just below the proton’s observed mass of 1.67 x 10^{-24} g.
(6) The “Initial” Mass of the Proton
The initial mass (M) for a Kerr-Newman ulracompact, which is synonymous with its rest mass, is calculated by the following algebraic expression.
M^{2} = [M_{ir} + q^{2}/4M_{ir}]^{2} + J^{2}/4(M_{ir})^{2}. (8)
Note that the total mass/energy for the ultracompact is comprised of 3 parts: the irreducible mass, the electromagnetic mass/energy, and the rotational energy. Putting the proton’s values for Mir, q and J into Eq. (8) yields:
M = 1.67 x 10^{-24} g,
which is the observed rest mass of the proton. Since the SSCP’s empirically-derived scaling constants Λ and D have uncertainties in the range of 1-3%, the value of G_{-1} has at least a 1% level of uncertainty. This uncertainty puts limits on the accuracy to which we can test SSCP predictions at present, but refinement of the values for Λ and D is certainly possible and research devoted to that end would be a logical step in the development in the SSCP.
V. THE ELECTRON
If one thinks that the Kerr-Newman black hole solutions would fit any stable subatomic particle, one need only try to apply them in the case of the electron! Since the electron appears to be approximated by a naked singularity solution, i.e., with no horizon, the algebraic expressions we used to successfully model the proton are clearly invalid. For example, if we use the electron’s M, a and q values in Eq. (4) to derive a horizon radius, we get a radius of 3.36 x 10^{-11} cm, which grossly violates observational results indicating that the electron is structureless down to at least 10^{-16} cm. Even worse, Eq. (8) yields a rest mass value of 9.76 x 10^{-22} g, which is more than 500 times the rest mass of the proton! This clearly demonstrates that the Misner, Wheeler and Thorne algebraic expressions for Kerr-Newman black holes should not be applied to ultracompacts that are Kerr-Newman naked singularities. If we wish to test the degree to which the electron is consistent with one of the Kerr-Newman family of solutions, then we must develop new and more appropriate algebraic expressions or calculate physical properties based on the full Kerr-Newman solutions. A charged rotating Kerr-Newman naked singularity solution is arguably the most promising model for the electron. However as we will see below, a charged non-rotating Reissner-Nordstrom naked singularity solution is also a very promising, albeit very radical, possibility that may be uniquely consistent with the SSCP. Quantitative modeling of the electron as a Kerr-Newman ultracompact is a work-in-progress and will be the subject of a future publication. For the present, we will confine ourselves to conceptual and observational arguments relating to the fundamental nature of the electrons and subatomic nuclei.
VI. THE GALACTIC SCALE CONNECTION WITH SUBATOMIC ULTRACOMPACTS
In the section of this website entitled “Galactic Scale Self-Similarity” we discovered that Galactic Scale systems (galaxies and globular clusters) are self-similar analogues of subatomic particles and nuclei under conditions of extremely high temperature and density. The SSCP asserts that the observable universe is an infinitesimal volume deep in the interior of a Metagalactic Scale system that is undergoing a supernova-like event, hence the high temperature, high density, high velocities, relatively massive nuclear analogues, and global expansion. If galaxies and globular clusters are close analogues of subatomic nuclei and particles, then we should be able to recognize some of their qualitative and quantitative properties in the the Kerr-Newman solutions for ultracompacts.
It was shown early in the development of the SSCP that the radius range for galaxies is correctly related to the range of radii for Atomic Scale nuclei by our discrete self-similar Scale transformation equations. Most intriguingly, we see that Galactic Scale systems divide naturally into two very general families of systems: Spheroidal/Elliptical systems and Rotating Disk systems. There is an obvious and compelling analogy between this fundamental distinction on the Galactic Scale and the fact that Atomic Scale utracompacts also have a fundamental distinction between Boson systems and Fermion systems. Consistent with the character of bosons, spheroidal/elliptical galactic systems have varying approximations of spherical symmetry and the orbits of their constituent stars have a symmetric and somewhat stochastic distribution, rather than having large-scale coordinated motions. Like bosons, spheroidal galactic systems welcome company and tend to cluster. In possible analogy to fermions, disk galaxies have axisymmetric but highly flattened disks in which stellar orbits are globally coordinated. Also reminiscent of fermions, disk galaxies prefer maintaining their distance from one another and tend to avoid clustering much more than spheroidal galaxies.
A natural hypothesis, given the information presented in this technical note, is that fermions and disk galaxies represent Kerr-Newman ultracompacts with substantial angular momenta and central ring singularities. Bosons and spheroidal galactic systems, on the other hand, are hypothesized to represent systems with zero or low angular momenta, and thus represent Reissner-Nordstrom ultracompacts with point singularities. Continuing our analogy, it is likely that the spiral arms of disk galaxies reflect the frame-dragging phenomena between the static limit radius and event horizon of an ultracompact. Perhaps the region from the outer bulge to the inner halo of a typical massive disk galaxy corresponds to the ergosphere of a fast-rotating massive Kerr-Newman ultracompact, or a massive and rotationally-excited subatomic nucleus. Stars act as “tracers” defining the morphology of the spacetime in galactic disks. SSCP research on the general correspondence between Boson/Fermion Atomic Scale systems and Spheroidal/Disk Galactic Scale systems is ongoing and still has admittedly tentative aspects.
VII. AN ALTERNATIVE MODEL FOR THE UNBOUND ELECTRON
It is commonly believed that the unbound electron has an intrinsic j = ½ and J = 0.866ħ, like the proton. However, the SSCP suggests that there is considerable evidence for a unique connection between globular clusters and Galactic Scale electrons. The morphology of globular clusters, possibly the most spherically symmetric objects in nature, are consistent with a central point singularity, a spherically symmetric spacetime, and a very low rate of rotation. Could it be that the electron only attains j = ½ and J = 0.866ħ when the electron becomes bound to a nucleus and decomposes (fully or partially) into shells of charged Subquantum Scale particles? Perhaps the unbound or free electron, like its globular cluster counterpart, is a very slow rotator. Perhaps for the unbound electron j = J = 0. If this were the case, the unbound electron would have a M^{2}/a^{2} + q^{2} value of 7.38 x 10^{-5} and it would therefore still represent a naked singularity ultracompact, with no event horizon and complete spherical symmetry. However the central singularity in the J = 0 case would be a point singularity and the relevant solution from the Kerr-Newman family of solutions would be the charged/J = 0 Reissner-Nordstrom solution.
The idea that j = 0 for the unbound electron is serious anathema to theoretical physicists who assume that the intrinsic j for the isolated electron is ½, but if the analogy between electrons and globular clusters is correct, then they need to go back to the laboratory and carefully reassess the angular momentum of the isolated electron. As discussed in detail by B. M. Garraway and S. Stenholm in 2002 (Contemporary Physics, vol. 43, no. 3, pgs. 147-160, 2002), the spin of the isolated electron has never been observed/measured. Only the spin of the bound electron has been measured. These authors come to the conclusion, based on their research and that of others, that it is in principle possible to determine the spin of the free electron, but that a 10-fold improvement in technical capabilities would be required. Perhaps the conventional value of j = 1/2 for the unbound electron is in reality based on reasonable, but ultimately incorrect, extrapolations. I suspect that a Reissner-Nordstrom naked singularity solution will make an excellent approximate model for both globular clusters and free electrons If this is true then careful research will vindicate the j = 0 hypothesis for the free electron, and the physical characteristics and interaction dynamics of globular clusters will be found to be fully consistent with the proposed globular cluster/electron analogy. If the analogy is vindicated, then one can anticipate new developments in our theoretical understanding of phenomena like superconductivity and the “anomalous” behavior of electrons in graphene monolayers.
VIII. CONCLUSIONS
If G_{-1} is the correct gravitational coupling constant for the internal dynamics of Atomic Scale systems like atoms and subatomic particles/nuclei, then the following changes to theoretical atomic and subatomic physics are inevitable.
(1) There is a revised Planck scale, based on G_{-1} instead of G_{0}, that is closely related to the scale of the proton, and that defines the “bottom” of the hadronic subhierarchy for the Atomic Scale.
(2) The internal dynamics of atoms and subatomic ultracompacts is dominated by gravitation rather than electromagnetism by a factor of 1/α = 137.036.
(3) The external dynamics that applies to unbound atoms, ions and subatomic ultracompacts is dominated by electrodynamics by a factor of Λ^{D-1} = 3.27 x 10^{38}, since this is the factor by which gravitational interactions decrease in “strength” (which is more appropriately thought of as the degree of coupling between mass/energy and spacetime geometry) when we pass from Atomic Scale to Stellar Scale spacetime.
(4) The Kerr-Newman family of solutions to the Einstein-Maxwell equations provides the best currently available models for subatomic ultracompacts, as demonstrated by the unique fit between the physical properties of the proton and the mathematical relationships defined by the Kerr-Newman formalism. It can be predicted that other subatomic particles/nuclei will also turn out to be uniquely characterized in terms of the family of Kerr-Newman ultracompacts, including their M, q, J and r interrelationships, their qualitative characteristics and their dynamical interaction properties.
(5) Revising General Relativity and Maxwell’s Electrodynamics so that they correctly incorporate the discrete self-similarity of Discrete Scale Relativity should further improve the fit between the physical properties of actual ultracompacts (on any cosmological Scale) and the specific mathematical models we use to describe them.
(6) Research into how well the electrodynamic properties (magnetic dipole moments, electric fields, gyromagnetic ratios, etc.) of subatomic particles/nuclei are modeled by Kerr-Newman solutions using G_{-1} would seem to be the next logical step in the research program outlined in this technical note. Determining the actual physical value of the free electron spin, as opposed to the “theoretical” value, is also a high priority item.