The Upgraded Planck System of Units that Reaches from the Known Planck Scale All the Way Down to Subatomic Scales

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The Upgraded Planck System of Units that Reaches from the Known Planck Scale All the Way Down to Subatomic Scales

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A peer-reviewed article of this preprint also exists.

Dimitris M. Christodoulou^*

Demosthenes Kazanas

Dimitris M. Christodoulou^*

Demosthenes Kazanas

This version is not peer-reviewed

Submitted:

10 January 2023

Posted:

12 January 2023

Read the latest preprint version here

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Abstract

Natural systems of units {Ui} need to be overhauled to include the dimensionless coupling constants {αUi} of the natural forces. Otherwise, they cannot quantify all forces of nature in a unified manner. Thus, each force must furnish a system of units with at least one dimensional and one dimensionless constant. We revisit three natural systems of units (atomic, cosmological, and Planck). The Planck system is easier to rectify, and we do so in this work. The atomic system discounts {G,αG}, thus it cannot account for gravitation. The cosmological system discounts {h,α**h**}, thus it cannot account for quantum physics. Here, the symbols have their usual meanings; in particular, αG is the gravitational coupling constant and α**h** is Dirac’s fine-structure constant. The speed of light c and the impedance of free space Z0 are resistive properties imposed by the vacuum itself, thus they must be present in all systems of units. The upgraded Planck system with fundamental units

Keywords:

Subject: Physical Sciences - Particle and Field Physics

1. Introduction and Motivation

1.1. Three Fundamental Systems of Units Under Consideration

In a recent paper [1], we used a cosmological system of units based on the speed of light c, Newton’s gravitational constant G, and MOND’s characteristic acceleration

a_{0}

[2,3,4]. Since G is a building block of this system, then

a_{0}

can substitute for the universal MOND unit, the mysterious

A_{0} \equiv a_{0} G

. Besides

A_{0}

, pairs of the fundamental units produced two more de facto important dynamical units: the surface density

σ_{0} = a_{0} / G

, and the force

F_{0} = c^{4} / G

. Notice the breaking of the symmetry implicit in

F_{0}

: the inversion of G produces the term

c^{4} G

which is only a composite unit of no particular interest with dimensions of [M]⁻¹[L]⁷[T]⁻⁶

= c^{8} / F_{0}

. This deviation singles out the unit of force

F_{0} = c^{4} / G

as an important component of the cosmological system, but there is not much more one can do with it at this point, besides noting that the same unit of force appears in the famous Planck system of units as well, and that

F_{0}

is a geometry-independent quantity, since both c and G do not carry an imprint of the geometry of our four-dimensional spacetime.

The cosmological system of units does not use Planck’s constant h [5,6], which turns out to be a derived unit of no particular interest, but current thinking forgives the oversight—after all, this is a cosmological system designed for measurements in universal scales. Nevertheless, we were drawn into comparisons with the Planck system which now uses Dirac’s

h = h / (2 π)

as a fundamental unit [7,8,9]; and, soon enough, we also added Hartree’s atomic system of units [10], which paradoxically does not use c (or G, for that matter) as a fundamental unit. The immediate implication is that the speed of light is not an upper limit in the atomic world, where the unit of speed is

α_{h} c ≪ c

, where

α_{h} = 1 / 137.036

is the fine-structure constant. Under the circumstances, we ended up juggling three different fundamental systems of units, comparing and contrasting their building blocks and the assumptions that went into their architectures.

1.2. Dirac’s Problematic Constant $h$ and the Three Widely-Used Atomic Radii

The modern atomic and Planck systems use Dirac’s

h = h / (2 π)

[7] instead of Planck’s original and purely physical constant h [5]. This modification is not trivial because the

2 π

carries the unit of radians. This unit has been dropped by many authors and also by the SI system of units, leading to a widespread misunderstanding that

h

simply absorbs a numerical factor of

2 π

with no further ramifications. The inconsistency was noted by Bunker et al. [11] who inserted the unit of radian in the definition of

h

and the unit of cycle in the definition of h. The SI system must reinstate at least the radian unit, because it alerts us to the presence of geometry (see below).

Dirac believed that

h

is the true universal constant, and we can only guess the reason why: the

2 π

h

has introduced 2-D geometry into the constant, so, unlike Planck’s h, the constant

h

is not purely physical, it is a composite constant. This fact was effectively proven by Leblanc et al. [12] who showed that the Compton radius

r_{c}

(where

r_{c} \propto h

) also includes a geometric component. This result has important consequences in physics that become visible when we write side-by-side the three electronic radii of the atomic world (where

m_{e}

is the electron mass and e is the fundamental charge):

\begin{matrix} Classical radius : & r_{e} = e^{2} / (ϵ_{0} m_{e} c^{2}) = r_{c} α_{h} \\ Compton radius : & r_{c} = h / (m_{e} c) \\ Bohr radius : & r_{b} = ϵ_{0} h^{2} / (m_{e} e^{2}) = r_{c} / α_{h} \end{matrix} .

(1)

Here,

α_{h}

is the fine-structure constant and

ϵ_{0}

is the reduced vacuum permittivity defined by

ϵ_{0} \equiv 4 π ϵ_{0}

, an equation that shows how the stereometry of space modifies the physical unit

ϵ_{0}

of the vacuum. Therefore, we have an SI unit problem here too, just as Bunker et al. [11] discovered for

h

. The vacuum is a three-dimensional space, hence the stereometric term of

4 π

; thus, the units of

ϵ_{0}

must also include steradians.1 Now, these geometric considerations show why three different radii exist in atomic physics: although they have the same physical dimension, they capture entirely different geometries; the electrons in the atoms venture in 3-D space (hence, the

1 / (4 π)

r_{e}

); the emitted photons only “see” two dimensions (see footnote 1); and the electrons in the Bohr model of the atom are quantized and they see only discrete sectors embedded in 3-D space (hence, the

1 / π

factor in

r_{b}

). The factor of 1/4 “missing” from the

1 / π

is however applied to the energy levels, because this factor is included in the Rydberg energy

E_{R}

(see below).

Deriving the geometric pattern of the quantized radii

r_{n}

of the Bohr model is a little harder, yet within our grasp.2 In any case, the factor of 1/4 is necessarily missing from

r_{b}

, so that the quantized angular momentum

L_{n}

is truly a 2-D quantity (

L_{n} \propto h \propto 1 / (2 π)

), and the associated Rydberg energy

E_{R}

is independent of geometry (although the abolished geometry contributes a unitless constant of 1/4, i.e.,

E_{R} \propto 1 / {(ϵ_{0} h)}^{2} \propto 1 / {[4 π / (2 π)]}^{2} = 1 / 4

, the same factor as that “missing” from

r_{b}

Lastly, the Bohr radius is the fundamental unit of length in the atomic system [10], but we argue that the Compton radius is actually the most important unit because its definition in equation (1) does not contain the fine-structure constant

α_{h}

. Furthermore, there is more circumstantial evidence that

r_{c}

is important among the three radii shown in equation (1):

r_{c}

is the geometric mean of

r_{e}

and

r_{b}

(i.e.,

r_{c} = \sqrt{r_{e} r_{b}}

), and this implies that the Compton radius

r_{c}

is also the geometric mean of all three length scales combined, viz.

r_{c} = \sqrt[3]{r_{e} r_{c} r_{b}} .

(2)

Thus,

r_{c}

is singled out among the three electronic radii for further duty in all systems of units (but probably without the

2 π

term, in order to remove the artificially inserted 2-D geometry). Furthermore, it may not be as obvious yet, but geometric averaging plays a huge role in nature. The above geometric means (hereafter, G-Ms, to avoid confusion with “

G M

”) are only a prelude to their ubiquitous appearances in many G-M combinations of natural constants and physical quantities as well.

1.3. Dimensionless Constants

The general notion about constructing a system of units is that one is free to choose any units to be the building blocks. Dimensionless constants do not have units to offer, thus they are not chosen as building blocks. They remain as passive invariants in any adopted system of units, and they serve mostly as cross-checks of the various calculations performed between dimensional quantities. The current thinking is summarized in the following excerpt from Zeidler [14]:

“A special role is played by those physical quantities which are dimensionless in the SI system. We expect that such quantities are related to important physical effects. The experience of physicists confirms this.”

So, we suspect that such constants are important in physics, but we do not really know what to do with them beyond their ascertained invariance, simply because they lack units.

The general notion about constructing a physical system of units is wrong on two counts: (1) Although unitless constants do not have units to offer, they must be actively included in systems of units, because they introduce the natural forces that cause the important physical effects mentioned by Zeidler [14]. (2) We are not free to choose any dimensional units as building blocks; we must choose wisely the units that measure the fundamental forces and, in addition, those units dictated by the vacuum itself. In particular, choosing a favorite particle to supply its properties for building blocks (as in the three atomic radii) is a bad idea and the reason is that such favoritism would violate a principle of fairness in this world. As will be seen below, nature does not at all favor or neglect any particle or force field, not even the “very small” ones against the “very large” ones, and vice versa.

One or both of the above defects have crept into our systems of units, where they selectively crippled or eliminated entirely some fundamental forces of nature. A natural force is crippled when its dimensional or its dimensionless constant is not included as a building block of a system of units; and a force is eliminated entirely when both of its defining constants are not included in a fundamental system of units.

1.4. Outline

In this work, we construct a self-consistent system of units that does not suffer from the above defects and that includes gravity, electromagnetism, and the weak interaction. Since all forces besides gravity have been unified, leaving out the coupling constant of the strong interaction may not hurt the effort too much—and we are going to test our system’s performance in atomic and subatomic scales, and in the Planck and the macroscopic scales as well.

It turns out that the Planck system is easier to upgrade, because it already includes the appropriate dimensional constants

{c, Z_{0}, G, h}

, although the impedance of free space

Z_{0}

has so far been sidelined. So, what we need to do for the upgrade is to activate the unitless coupling constants

α_{G}

(gravitational coupling constant) and

α_{h}

(fine-structure constant); and to repair the damage that

h

has caused by inadvertently introducing geometry in them (see footnote 6 for details), besides the well-intended quantum forces.

In Section 2, we describe the building blocks of the upgraded Planck system of units. In Section 3, we collect the new results concerning masses, charges, and lengths in the new system. In Section 4, we discuss the results, and, in Section 5, we summarize potential issues still lingering in this system of units, as well as some future research prospects.

Finally, in Section 6, we list the most important highlights of our investigation, including the results obtained in the Appendices. In Appendix A, we derive the long-sought physical significance of Koide’s lepton constant [15] of atomic physics, and in Appendix B, we discuss the universality of the Tully-Fisher/Faber-Jackson relation [16,17] discovered in spiral and elliptical galaxies, respectively. This fundamental relation with a dynamical quantity to the fourth power signifies a new universal law of nature that has manifestations in several other parts of physical science (see Appendix B).

2. The Building Blocks of the Upgraded Planck System

The upgraded Planck system (UPS) includes the following building blocks at the very least:

UPS : = {c, Z_{0}, G, α_{G}, h, α_{h}},

(3)

where

{c, Z_{0}, G, h}

are dimensional units and

{α_{G}, α_{h}}

are dimensionless units. We use a slash to indicate the presence of geometric units (which have been inadvertently dropped from the systems of units), but the opposite applies to the coupling constants, in which the role of geometry has been reversed due to the erroneous use of

h

in their definitions (see below). This is a problem with which we have to contend with throughout this work.

2.1. Dimensional Units

For future reference, we need to recall and emphasize a gem of natural units, the fundamental dimensional relation between gravity (supplying G) and electromagnetic (EM) forces (supplying also the vacuum’s constant

{(4 π ϵ_{0})}^{- 1}

). This is obtained by equating the dimensions of the forces in Newton’s gravitational law and Coulomb’s law for two electrons. We find, in dimensional form, that

G m_{e}^{2} \sim {(4 π ϵ_{0})}^{- 1} e^{2},

(4)

where

m_{e}

is the mass of the electron and e is the elementary positive charge. Here, we write down explicitly the vacuum permittivity

ϵ_{0}

4 π ϵ_{0}

to ensure that its geometric content (the

4 π

factor in the EM term) is clearly noticeable.

Each side of equation (4) becomes unitless when divided by

h c

, as is done separately in the definitions of the two fundamental coupling constants. Unfortunately, the

h

introduces then additional geometry into the gravitational part and eliminates geometry from the EM part, clearly altering the original geometrical characteristics of the two coupling constants (see Section 2.1.2 below). The unit of [rad]⁻¹ has been dropped from

h

by international agreement, so this intrusion of geometry is no longer visible [11]; going as far back as Schrödinger [18], our community is under the mistaken impression that it is only a pure numerical factor of

2 π

which has been absorbed in the definition of

h

. This is a terrible mistake that has held us back for the past 100 years or so.

2.1.1. Constants Imposed by the Vacuum

The speed of light is not related to a natural force. It is passively produced by the vacuum in order to dictate the speed of EM waves, and it also sets an upper limit to the motion of material objects possessing mass. The vacuum does that incidentally, by providing the smallest possible natural resistance to any kind of motion. The magnitude of c is set by the G-M of two inverse properties of the vacuum, viz.

c = \sqrt{ϵ_{0}^{- 1} μ_{0}^{- 1}} .

(5)

The SI value of c is

c = 2.9979 \times 10^{8} m s^{- 1}

, and its dimensions are [length][time]⁻¹ [19].

In equation (5), the reduced values of vacuum permittivity and vacuum permeability combine in a way that removes geometric constraints from this speed (see also footnote 1); the maximum permitted velocity of a combined EM wave or a massive object must be the same in any direction. In contrast, it is understood that a static electric field in vacuum must adjust to the geometric constraint imposed by

ϵ_{0}^{- 1}

, and this is why the vacuum’s inverse permittivity appears in Coulomb’s law. In fact,

ϵ_{0}^{- 1}

is the slope between the electric field

E

and the surface charge density

e / r^{2}

[1], where r represents distance (see also Appendix B for the role that various surface densities play in disjoint parts of physics).

Since equation (5) can be written in the equivalent form

c = \sqrt{ϵ_{0}^{- 1} μ_{0}^{- 1}},

(6)

we can surmise that the physical quantities

ϵ_{0}

and

μ_{0}

are geometry-free. Indeed, after some manipulations involving also the geometry-free fine-structure constant

α_{h}

(as this was inadvertently defined long ago using Dirac’s

h

), we find that

ϵ_{0}^{- 1} = c (\frac{2 h α_{h}}{e^{2}}),

(7)

and

μ_{0}^{- 1} = c (\frac{e^{2}}{2 h α_{h}}) .

(8)

There is no geometric influence on the right-hand sides of these equations. The quantity that is inverted from one equation to the other,

h / e^{2} = μ_{0} c / (2 α_{h})

, is proportional to the impedance of free space

Z_{0} = \sqrt{μ_{0} / ϵ_{0}} = 376.730 Ω

, which is the G-M of

μ_{0}

and

ϵ_{0}^{- 1}

; thus,

h / e^{2}

has dimension of [ohmic resistance] (see [20] and footnote 20 below). Thus,

ϵ_{0}^{- 1}

and

μ_{0}^{- 1}

can effectively be expressed as G-Ms involving the squares of c and

Z_{0}

(i.e.,

\sqrt{c^{2} Z_{0}^{2}}

and

\sqrt{c^{2} (1 / Z_{0}^{2})}

, respectively); the first G-M involves a direct multiplication of the two constants involved, whereas the second G-M uses the Lie-type inversion of one of the two constants [13]. We will pick up again this important inference in Section 4.2 below.

2.1.2. Dirac’s Constant $h$

Dirac’s constant is the slope between the energy E carried by a single photon and its angular frequency

ω

, viz.

E = h ω .

(9)

Its SI value is

h = 1.0546 \times 10^{- 34} J s {rad}^{- 1}

[11,19], and its dimensions are

{[action] [rad]}^{- 1}

or, equivalently, [moment of inertia] [second]⁻¹ [rad]⁻¹.

This new awareness, that inertia is built into

h

(and Planck’s h), may be the spark we need to theorize that the weak equivalence principle [21] is embedded into the microcosm as well, where gravity is not important. Action integrals [22], in particular, may be viewed as carrying the units of [moment of inertia] [second]⁻¹, thus each action is a measure of the rate of change of moment of inertia at all scales of the universe, large and small (see also Section 3.5 below).

In the spirit of equations (7) and (8), Planck’s reduced constant may also be split into a product of two G-Ms, viz.

h = \sqrt{h (ϵ_{0} c)} \sqrt{h (\frac{1}{ϵ_{0} c})} = e (\frac{h}{e});

(10)

the first G-M on the right-hand side is geometry-independent; the next G-M is influenced by 2-D geometry since it is directly proportional to

\sqrt{h (ϵ_{0}^{- 1})} \propto \sqrt{{(4 π^{2})}^{- 1} {(2 ϵ_{0})}^{- 1}} \propto {(2 π)}^{- 1} .

This G-M that reduces to

(h / e)

has dimensions of [magnetic flux] = [magnetic field][area].3 It is understood from the G-M decomposition (10) that the vacuum quantity

ϵ_{0} c = 4 π / Z_{0}

can couple to

h

, and thus influence quantum phenomena; and it does so in the definition of the fine-structure constant (Section 2.2.1).

2.1.3. Newton’s Gravitational Constant G

Newton’s gravitational constant G is the slope between the gravitational field

a (r)

(i.e., acceleration) and the surface mass density

σ (r) \equiv M (r) / r^{2}

[1] on the surface of a sphere of radius r enclosing a total mass of

M (r)

, viz.

a (r) = G σ (r) .

(11)

Its SI value is

G = 6.67430 \times 10^{- 11} m^{3} {kg}^{- 1} s^{- 2}

[19], with dimensions of [acceleration] [surface density]⁻¹.

In the spirit of equation (10), Newton’s gravitational constant can also be split into a product of two G-Ms, viz.

G = \sqrt{G M^{2}} \sqrt{G (\frac{1}{M^{2}})},

(12)

which shows the potential of mass M to couple to G, and thus influence gravitation. The

M^{2}

term does precisely that in the definition of the gravitational coupling constant (Section 2.2.2). Here, we think that the

M^{2}

is meant to include both the inertial and the gravitational mass on an equal footing, a manifestation of Mach’s famous principle [23].

2.2. Dimensionless Units

We now come to the operations and the properties of the unitless coupling constants

{α_{G}, α_{h}}

. We show how these units rectify the Planck system of units and make it functional over all non-nuclear scales of the universe, including atomic and subatomic scales as well.

2.2.1. Fine-Structure Constant $α_{h}$

The fine-structure constant has been defined as

α_{h} = \frac{e^{2}}{ϵ_{0} h c} .

(13)

Its value has been measured [24] to be very close to

{(137.036)}^{- 1}

(or

α_{h} = {(861.022576)}^{- 1}

for the wiser choice

h \to h

in the definition). Other than that,

α_{h}

brings no geometry and no units into the system of units. In particular, the geometry embedded in the electric field (and carried on by

ϵ_{0}

) has been inadvertently eliminated by the insertion of

h

in the modern definition (13).

Nevertheless, definition (13) provides a powerful tool (Section 2.2.3), which we have not taken advantage of in the past: being a measurable constant,

α_{h}

may serve as the reference UPS unit against which we can quantify all the other unitless coupling constants. For instance, the gravitational coupling constant

α_{G}

, which we describe next, acquires a quantitative meaning only by comparison to

α_{h}

in the ratio (

α_{G} / α_{h}

) (although, unfortunately, the reference value

α_{h}

introduces a unit of [rad] in this ratio and an artificial dependence on geometry) [25] .

2.2.2. Gravitational Coupling Constant $α_{G}$

Using G and the electron mass

m_{e}

, the gravitational coupling constant has been defined as

α_{G} = \frac{G m_{e}^{2}}{h c} .

(14)

Its value is

1.7518 \times 10^{- 45}

(or

α_{G} = 2.7881 \times 10^{- 46}

, the geometry-free value obtained for

h \to h

in the definition), as determined by calculation.

Comparing the definitions (14) and (13), we see that

α_{G}

is, unfortunately, geometry-dependent. This problem did not exist during Max Planck’s heydays, when h was in use and

h

did not exist. In general, the problem with the modern definitions of constants and variables is that

h

necessarily introduces 2-D geometry and a [rad] measure, in addition to the intended physical constant h. We must pronounce this Dirac’s error [7,8].

The geometry dependence so artificially inserted in

α_{G}

will be taken out entirely in the calculations that follow. We must emphasize up front that reinstating the true nature of

α_{G}

(and

α_{h}

) is necessary for the successful repair of the modern Planck system, and it leads to the determination of natural scales of mass, length, and charge for the chosen mass-to-charge ratio

(m_{e} / e)

of the electron (Section 2.2.3), or any other chosen particle for that matter (see footnote 7 for details).

2.2.3. Relative Strength of Gravitational Coupling $β_{G}$

Leaving aside the unfortunate introduction of [rad] in the above coupling constants, we come now to the only known method of actively using such dimensionless (pure) numbers. Being pure numbers, these constants have absolutely no meaning or practical use, but they become useful in ratios, in which their strengths are compared against other dimensionless constants; only in this comparative process, do these ratios acquire meaning via the ensuing comparisons, and then their relative strengths are, for all practical purposes, measurements of the same stature and importance as dimensional quantities (which, incidentally, are also measured by comparisons to international standards). One unitless coupling constant should however be included in the system in absolute terms in order to provide the reference value for comparisons.4

For the UPS, we choose

α_{h}

for this duty because it has been measured by experiment [24], and its physical meaning has now become clear (see Section 3.1 and Table A1 below): the factor

\sqrt{α_{h}} ≃ 1 / 30

(with

h \to h

in equation (13)) is a fundamental scale factor used by the Higgs field to assign mass to the bottom quark, which then becomes the gateway for the assignments of all lower particle masses.

In this study, we assume that the above two coupling constants do not vary in space. Then, their invariant nature tells us that, if we determine their relative strength for a particular particle or object (an electron in the UPS), then this ratio will be the same at all scales within the utilized system of units. Thus, we calculate the universal UPS ratio

β_{G} \equiv \frac{α_{G}}{α_{h}} = \frac{ϵ_{0} G m_{e}^{2}}{e^{2}} = 2.4006 \times 10^{- 43},

(15)

a pure comparative number which is independent of

h

and h (and c, for that matter—as would be expected, the vacuum does not at all contribute to such a ratio of forces). This “measurement” of

β_{G}

represents the strength of gravitational coupling relative to that of the EM coupling obtained for electrons (see footnote 7 for considering protons instead, and constructing another UPS with different scales, but with the same elementary particles).

Compared to the relation between units shown in equation (4),

β_{G}

carries a lot more weight because it can be used in quantitative calculations (although it was equation (4) that gave us a reason to define

β_{G}

). The numerical value obtained in equation (15) does not tell us that gravity is weak and the EM force is strong;5 it only tells us about the relative couplings of these forces in the particular system of units that includes

β_{G}

as a building block. Gravity is attractive and has always had a chance to grow past the other forces in extraordinarily massive settings (

M ≫ m_{e}

)—something that is actively occurring in many places in the present universe. In contrast, the Coulomb force cannot do the same, because its attraction brings together charges of opposite signs that cancel each other out [26].

Also, equation (15) does not tell us that, in the distant past, gravity could have been much stronger in the atomic world, and it only got weaker going forward in time [8,9] because of the expansion of the universe. We think this is utter nonsense. The gravitational force has always been weak in the atomic world because the characteristic atomic masses are too small (much smaller than the Planck mass). So, if it should come to a bout of conjectures, probably the safest choice is the assumption that these constants do not vary in time either, just as they do not vary in space.

2.3. Determining a New Atomic Mass Scale

Definitions (13) and (14) have both incorporated

h

, thus the coupling constants have been defined in the microcosm. Here, we use the above results to establish a new atomic mass scale, after correcting for the unintended insertion of planar geometry into the coupling constants.

Looking at equation (14), we see two problems that need to be addressed: (a) Despite the apparent lack of units (not entirely true, since

h

also carries radians),

α_{G}

is not influenced by EM coupling (there is no e in the definition, only mass

m_{e}

, and the two forces are not linked to one another, although they do so in the real world). (b) The unfortunate use of

h

has had the unforeseen consequence of arbitrarily adding more geometry (and a [rad] unit) into the dimensionless mix.6

We can solve both problems by adopting equation (15) to help us define a new atomic mass scale

M_{A}

in the UPS. The relative ratio

β_{G}

carries both forces, and the composite unit

h

, which was not appropriate in the first place, has been eliminated (correcting thus a century-old oversight). One unavoidable conclusion is that the geometry of the vacuum (the

ϵ_{0}

) is still present in

β_{G}

. This comes from the geometric dependence of the electric field, which will now overtly influence the new mass scale

M_{A}

(see also Section 5 below). This appears to be a realistic relationship since mass also carries charge and the carried electric field is influenced by the vacuum’s

ϵ_{0}

Based on these considerations, we return to equation (14), and we rewrite this definition by making the following substitutions:

β_{G} \to α_{G}

h \to h

, and

M_{A} \to m_{e}

. We find a new equation, viz.

β_{G} = G M_{A}^{2} / (h c),

(16)

in which both sides are dimensionless; and substituting for

β_{G}

from equation (15), we obtain the new atomic mass scale

M_{A} = \sqrt{ϵ_{0} h c} (\frac{m_{e}}{e}) = 2.6730 \times 10^{- 29} kg .

(17)

We repeat here the two ingredients that form the physical basis for this mass: (i) the unitless ratio

β_{G}

in equation (15) has no dependence on

h

, or h, or c; and (ii) the substitution

h \to h

produces a truly unitless equation (16); there are no loose [rad] units in this equation, covertly suppressed by SI conventional practices (although here the “cycle” unit has been suppressed in h, since it does not signify insertion of geometry; see also [11]).

It is quite interesting that only the ratio

(m_{e} / e)

of the characteristic parameters of the electron ends up being a building block of the new mass scale

M_{A}

. The reciprocal ratio, i.e.,

e / m_{e} = 1.7588 \times 10^{11}

C kg⁻¹, was first measured by J. J. Thomson [27], years before the electronic charge itself was finally measured by experiment (see also Section 4.2).

The presence of

ϵ_{0}

(coupled to h, as shown in the G-Ms given in Section 2.1.2) in the new mass scale

M_{A}

is necessary (the vacuum’s

ϵ_{0}

is a building block of the electrostatic field): after some algebraic manipulations, we recast equation (17) (or equation (16)) to the equivalent form

M_{A} = \sqrt{(\frac{h c}{G}) (\frac{E_{grav}}{E_{elec}})},

(18)

where the ratio of energies,

\frac{E_{grav}}{E_{elec}} = β_{G},

was determined from the corresponding forces acting between two interacting electrons.7 In dimensional analysis, this ratio is 1 (see equation (4) above), but here,

β_{G}

plays an important quantitative role: the unitless factor

\sqrt{β_{G}} = 4.900 \times 10^{- 22}

(19)

scales the original Planck mass [5]

(M_{p} = \sqrt{h c / G} = 5.4555 \times 10^{- 8} kg)

down to the atomic world. This scaling is a significant result of our work, as it connects the original Planck mass scale with the

M_{A}

scale of the atomic world,8 viz.

M_{A} = M_{p} \sqrt{β_{G}} .

(20)

We note that

M_{A}

and

M_{p}

are mass scales related by this equation, and they do not (cannot) correspond to any real particle or object in nature (see also Section 4.3).

Notice the complete absence of

h

from equations (15)-(18). The only geometric dependence entering these equations is that which is imposed by the vacuum on to the electrostatic field (hence,

M_{A} \propto ϵ_{0}^{1 / 2} \propto 2 \sqrt{π}

). The

\sqrt{π}

does not carry any angular units since

E_{elec} \sim e^{2} / ϵ_{0} \sim [Joule]

in equation (18)—just like the

1 / π

in the Bohr radius and the factor of 1/4 in the Rydberg energy (see the analysis following equation (1) in Section 1.2). Therefore, besides introducing the speed of light, the vacuum manages to imprint

M_{A}

only with a unitless, purely numerical constant⁹ (see also footnote 3).

3. Results within the UPS Realm

3.1. Subatomic Masses

This new UPS mass scale (17) corresponds to

M_{A} = 15.0 MeV / c^{2},

(21)

thus it lands near the subatomic world of the low-mass up and down quarks, with corresponding masses

m_{u} = 2.16

MeV/

c^{2}

and

m_{d} = 4.67

MeV/

c^{2}

[29]; and it is smaller than the G-M defined for the hydrogen atom

\sqrt{m_{e} m_{p}} = 21.9 MeV / c^{2},

(22)

where

m_{p}

is the proton mass.

The new mass scale

M_{A}

appears to be important for the standard model of particle physics, and it should be investigated further theoretically (there is no elementary particle corresponding to this energy). So far, we have derived the following useful relations (sufficient to lead us to a clear physical interpretation of Koide’s enigmatic constant and other constants of the same type; for details, see Appendix A):

(1): The mismatch between $M_{A}$ and $\sqrt{m_{e} m_{p}}$ may be related to Koide’s K-constant, $K = 2 / 3$ [15], viz.

$M_{A} / \sqrt{m_{e} m_{p}} = 0.685,$

(23)

connecting thus the masses of leptons to the atomic constants $M_{A}$ and $m_{p}$ .
(2): Using the above values of first-generation quark masses and the mass of the strange quark, $m_{s} = 93.4$ MeV/ $c^{2}$ [29], we find that

$\sqrt{m_{u} m_{s}} / M_{A} ≃ 0.95,$

(24)

and

$\sqrt{m_{d} m_{s}} / \sqrt{m_{e} m_{p}} ≃ 0.95,$

(25)

showing only a 5% deviation of both quark G-Ms from the two atomic mass constants. The results indicate that the mass of the second-generation strange quark is connected to both $M_{A}$ and the masses of the first-generation quarks. Thus, a connection should exist for the charm quark too,10 and so on for the third generation of quarks as well.
(3): It certainly appears that there exists a ladder-type mechanism that uses G-Ms (and some scaling coefficients) to relate various particle masses (see also Table A1 in Appendix A below). Some examples (and their corresponding deviations from experiment) are:

$m_{s} = \sqrt{m_{d} m_{τ}} (2.5 %),$

(26)

where $m_{τ} = 1.777$ GeV/ $c^{2}$ is the tauon mass;

$m_{s} = \sqrt{m_{u} m_{b}} (1.7 %);$

(27)

$m_{c} = \sqrt{m_{p} m_{τ}} (1.7 %);$

(28)

$m_{c} = \sqrt{2 m_{d} m_{t}} (0.054 %),$

(29)

where $m_{t} = 172.5$ GeV/ $c^{2}$ is the top quark mass;

$m_{u} = \sqrt{2 m_{d} m_{e}} (1.1 %);$

(30)

$m_{p} = \sqrt{2 m_{μ} m_{b}} (0.17 %),$

(31)

where $m_{μ} = 105.66$ MeV/ $c^{2}$ is the muon mass;

$m_{b} = \sqrt{m_{μ} m_{t}} (2.1 %);$

(32)

and

$M_{A} = \sqrt{m_{μ} m_{u}} (0.71 %),$

(33)

$M_{A} = \sqrt{\sqrt{m_{e} m_{μ}} \sqrt{m_{e} m_{τ}}} (0.80 %) .$

(34)
(4): The Higgs boson ( $m_{H} = 125.25$ GeV/ $c^{2}$ ) is certainly special, although unavoidably a part of the mass ladder. This is the only particle that is not involved in simple G-Ms with the low-mass particles. Two of its complex relations are the following:

$m_{b} = \sqrt{m_{s} (m_{H} / K)} (0.21 %),$

(35)

where $K = 2 / 3$ [15]; and

$\frac{m_{H}}{m_{b}} = 30.0 ≃ \frac{M_{A}}{m_{e}} (2.0 %) .$

(36)

This relation shows how the Higgs boson manages to assign mass to the much lower-mass bottom quark by using a novel mechanism, not related to a G-M or Koide’s scale factor (see below).
(5): The vacuum expectation value (VEV) of the Higgs field is $v = 246.22$ GeV/ $c^{2}$ [30]. To within a deviation of 1.8%, we find for the compact11 triplet H-t-v that

$m_{t} = \sqrt{m_{H} v},$

(37)

which shows exactly where the most massive quark is located at the top of the mass ladder. Furthermore, the Higgs mass is the G-M of the top quark mass and the mass of the Z⁰ boson $m_{Z^{0}} = 91.1876$ GeV/ $c^{2}$ (a deviation of only 0.13%), viz.

$m_{H} = \sqrt{m_{t} m_{Z^{0}}} .$

(38)

Obviously, the top quark receives its mass from the Higgs field, and then it participates in the G-Ms that define the masses of the other particles (see Table A1 in Appendix A). The high-mass geometric sequence Z⁰-H-t-v appears to be very compact indeed (footnote 11), and its common ratio is about 1.38.¹² We note that W^± (mass $m_{W^{\pm}} = 80.377$ GeV/ $c^{2}$ ) is not a member of this sequence since $m_{Z^{0}} / m_{W^{\pm}} = K^{- 1 / 4} ≃ 1.11$ .13 This relation provides another definition of Koide’s K in terms of the decay products of the Higgs boson (deviation 2.5%).
(6): On the other hand, the G-M of $m_{H}$ and $m_{W^{\pm}}$ is 10% larger than $m_{Z^{0}}$ ; but using empirically Koide’s constant, we find that

$m_{Z^{0}} = \sqrt{(K^{1 / 2} m_{H}) m_{W^{\pm}}},$

(39)

an important relation with a deviation of the G-M from the measured $m_{Z^{0}}$ value of only 0.57%. Furthermore, the relation $m_{W^{\pm}} = \sqrt{K m_{H} m_{W^{\pm}}}$ also appears to hold (1.9% deviation), which then implies that

$m_{W^{\pm}} = K m_{H} .$

(40)

This relation helps us understand the important role of the exact constant $K = 2 / 3$ [15]: K is a numerical scale factor that relates some close pairs of particle masses. Here, the Higgs field connects to Z⁰ by an inverse-mapping G-M,14 viz.

$m_{Z^{0}} = \sqrt{m_{H}^{3} (1 / v)},$

and to W^± by the simple scale factor K, as seen in Equation (40). In hindsight, the Higgs field could not assign two different (but comparable) masses to Z⁰ and W^±, both by using G-M averages, so it used two different methods.
(7): Returning now to Equation (35), we see the Higgs mass is scaled inversely (by $1 / K$ ) to participate in a G-M with $m_{s}$ and $m_{b}$ . Although we have only a partial view of the dynamics of the Higgs field in the above equations, it is obvious that it follows a set of scaling rules in addition to participating in G-Ms. The origin of these scaling rules is unknown to us at this moment, but we feel confident that we have made a step in the right direction with this analysis. The relevant scaling factors may not be visible in the action integrals, but they should then appear in the solutions (a manifestation of symmetry breaking).
(8): The next and considerably more difficult step concerns the assignment of mass to the bottom quark, whose mass is much lower than the Higgs mass and the masses of its decay products. We were surprised to find that yet another method is used by the Higgs boson for this assignment.15

Notice the unitless factor of 30.0 in Equation (36). This equation suggests that the mass scale $M_{A}$ and the electron mass $m_{e}$ are related to the mass ratio $m_{H} / m_{b}$ . But $m_{e}$ is not a mass scale, so the proportion $M_{A} / m_{e}$ in Equation (36) is at least obscure, if not superficial (it does not represent a ratio of mass scales or a ratio of particle masses). Using Equation (58) derived below and the equations of Section 2.3, we rewrite this proportion (Equation (36)) in a palatable (physical) form, viz.

$\frac{m_{b}}{m_{H}} = \frac{1}{30} ≃ \sqrt{α_{h}} (2.1 %),$

(41)

where $α_{h} = {(861.022576)}^{- 1}$ is given by Equation (13) with the corrective substitution $h \to h$ [5,6].

Thus, the mass of the bottom quark $m_{b}$ , which is 30 times lower than $m_{H}$ , is determined self-consistently from this scaling equation by effectively using the ratio of scales $M_{p} / M_{A}$ in intermediate steps and the Planckian fine-structure constant16 $α_{h} = e^{2} / (ϵ_{0} h c)$ in the final step. This is the third method employed by the Higgs boson to assign mass. In particular, it uses this 1/30 scaling to get down to the bottom quark and, then, into the regime of the lower particle masses (see also Table A1 below). If the $m_{b}$ assignment also involves the W⁻ boson (which carries Koide’s scale factor K) to deliver charge to the bottom quark, then equations (40) and (41) combine to show that $m_{b} = (m_{W^{-}}) (\sqrt{α_{h}} / K) ≃ 0.05 (m_{W^{-}})$ . The physical significance of Koide’s factor for the high-mass quarks is discussed in detail in Appendix A.2.

3.2. The Planck Charge

The Planck charge

q_{p}

is a prime example of the state of confusion in the field: not understanding the significance of the meddling of geometry in the modern Planck units, people adopted different definitions of

q_{p}

by arbitrarily choosing between

ϵ_{0}

and

ϵ_{0}

and between

h

and h. In the end, this unit, along with the Planck units of magnetic flux

{[h / (ϵ_{0} c)]}^{1 / 2}

and ohmic resistance

{(ϵ_{0} c)}^{- 1}

, fell out of favor.17

Now, we know better. The definition of the Planck charge

q_{p}

must be geometry-free, viz.

q_{p} \equiv \sqrt{ϵ_{0} h c} = \sqrt{2 ϵ_{0} h c} .

(42)

Absence of geometry is required, first because this is a unit of charge, and second because

q_{p}

provides an alternative definition of the fine-structure constant (which is geometry-independent in its current definition (13)), viz.

α_{h} = {(\frac{e}{q_{p}})}^{2} .

(43)

We find that

q_{p} = 1.8755 \times 10^{- 18} C = 11.7062 e

(where

11.7062 = \sqrt{137.036}

). Once again, nature shows us here her principle of fairness (or impartiality). As in the case of the electron mass

m_{e}

, the elementary charge e here is not related to the fundamental unit of charge

q_{p}

by a rational numerical factor; instead,

q_{p}

is chosen as the UPS scale of charge that does not correspond to the charge multiple of any specific particle or field.

3.3. A New Atomic Length Scale

Equation (1) can help us determine a new length scale for the UPS, a scale that certainly does not correspond to any of the three atomic radii in Equation (1): based on nature’s apparent principle of fairness, we understand that none of the known electronic radii can be the fundamental unit of length. But we know that scale values generally fall between particle values and vice versa. To proceed then, we use the G-M of

r_{e}

and

r_{c}

to determine a new atomic length scale

L_{A}

.18

The G-M of

r_{e}

and

r_{c}

gives

L_{A} = r_{c} \sqrt{α_{h}} = \sqrt{\frac{h}{ϵ_{0} c^{3}}} (\frac{e}{m_{e}}),

(44)

and

L_{A} = 3.2987 \times 10^{- 14} m = r_{c} / 11.7062

. The numerical value 11.7062 is the same with that found for the ratio

q_{p} / e

(Equation (43)) because

α_{h} = {(L_{A} / r_{c})}^{2},

(45)

and then, the following proportion (cross-multiplied) holds exactly:

L_{A} q_{p} = r_{c} e .

(46)

This relation implies that the G-M of the new scales

L_{A}

and

q_{p}

is equal to the G-M of the traditional and widely-used electronic constants

r_{c}

and e; and it brings to light a previously unused combination of units with dimensions of [length][charge]. These dimensions are equivalent to

\frac{[momentum flux]}{[magnetic flux]} = \frac{[momentum]}{[magnetic field]} = \frac{[energy]}{[electric field]};

these interesting units compare mass flows (“matter waves”) to EM waves (“energy flows”), and energy/momentum to EM field components. These quotients also indicate a close correspondence between the relativistic energy-momentum (E-p) equation

E = c p,

(47)

and Maxwell’s EM amplitudes (

E_{0}

B_{0}

; [34]) relation

E_{0} = c B_{0} .

(48)

The above dimensional ratios of units are obtained easily by dividing these two equations. We see then that

B_{0}

(current flow) is to EM waves what momentum p (mass flow) is to dynamics, and similarly for amplitude

E_{0}

and energy E.

Length

L_{A}

is much larger than the modern Planck length

L_{p} = \sqrt{h G / c^{3}} = 1.6163 \times 10^{- 35}

m. (The modern definition of

L_{p}

must be used here, because

r_{c}

in Equation (44) brought its 2-D geometry into

L_{A}

, and

α_{h}

is accidentally geometry-free.) In this case,

L_{A}

must be scaled down to produce

L_{p}

; thus,

L_{p} = L_{A} \sqrt{β_{G}}

. This scaling-down of

L_{A}

should be contrasted to the scaling-up of

M_{A}

to produce the original Planck mass

M_{p}

(i.e.,

M_{p} = M_{A} / \sqrt{β_{G}}

; see Section 2.3).

3.4. Cosmological Scales and Some Ambivalent Superatomic Particles

In Section 2.3 and Section 3.3 above, we rescaled the fundamental scales of the UPS to obtain the corresponding Planck scales. These “A” and “p” values do not describe any specific particle or object in the universe; that would not be a fair choice by nature. But we can extend both scales into the macrocosm by running the G-Ms toward larger masses and lengths.

(a) Cosmological Mass Scales.—We evaluate a geometric progression that starts with scales

M_{A}

and

M_{p}

, and moves on to larger mass scales:

{M_{B}, M_{C}, M_{D}} = {1.113 \times 10^{14}, 2.271 \times 10^{35}, 4.633 \times 10^{56}} kg .

(49)

Mass scale

M_{D}

is 2-3 orders of magnitude larger than the current estimates of the mass of the universe [1], so we can halt the sequence at

M_{D}

. The common ratio of the geometric progression is

M_{p} / M_{A} = 1 / \sqrt{β_{G}} = 2.041 \times 10^{21}

. The G-M of

M_{B}

and

M_{C}

is equal to 0.84 earth masses; and the G-M of

M_{C}

and

M_{D}

5 \times 10^{15}

solar masses and identifies universal structures much larger than individual galaxies (e.g., galaxy clusters).

(b) Cosmological Length Scales.—We evaluate a geometric progression that starts with scales

L_{p}

and

L_{A}

, and moves on to longer length scales:

{L_{B}, L_{C}} = {6.730 \times 10^{7}, 1.373 \times 10^{29}} m .

(50)

Length scale

L_{C}

is 2-3 orders of magnitude larger than the current estimates of the size of the universe [1], so we can halt the sequence at

L_{C}

. The common ratio of this geometric progression is

L_{A} / L_{p} = 1 / \sqrt{β_{G}} = 2.041 \times 10^{21}

, the same as the common mass ratio given in item (a) above. The G-M of

L_{B}

and

L_{C}

is equal to 98.5 parsecs, a value typical of giant molecular cloud complexes in spiral galaxies.

T_{CMB} = 2.7255

K, to an equivalent mass

m_{CMB} = 3.52 \times 10^{- 10}

MeV/

c^{2}

. Since

m_{CMB} ≪ M_{A}

, we need to extend the geometric progression of mass scales to much lower masses as well. At the low-mass end of the geometric sequence

{M_{0}, M_{A}, M_{p}}

, the tiny mass scale

M_{0}

is found to be

M_{0} = 7.35 \times 10^{- 21}

MeV/

c^{2}

. Then, the G-M relation

m_{CMB} = \sqrt{M_{0} M_{A}},

(51)

holds to within a 5.7% deviation between the two sides. This deviation is relatively small, given the enormous difference in scales (by 21.31 orders of magnitude) involved on the right-hand side of Equation (51).

(d) A Superatomic Particle Near the Planck Mass?—The equivalent mass of the CMB photons is so low, that extending the geometric sequence of

{m_{CMB}, m_{H}}

to higher masses, we obtain a potential particle mass of

M_{S} = m_{H}^{2} / m_{CMB} = 4.453 \times 10^{16}

GeV/

c^{2} ≃ 1.455 \times 10^{- 3} M_{p}

, which is at the scales where the strong force supposedly joins in with the other forces [35]. Since the Higgs mass is

m_{H} = 125.25

GeV/

c^{2}

, then the energy ratio

\sqrt{β_{W}}

(analogous to

\sqrt{β_{G}}

in Section 2.3) that scales the strong interaction down to the weak interaction is

\sqrt{β_{W}} = \sqrt{\frac{E_{W}}{E_{S}}} = \sqrt{\frac{m_{H}}{M_{S}}} = 5.30 \times 10^{- 8} .

(52)

This value is smaller by a factor of 20 compared to the usually quoted coupling constant ratio of the weak to the strong interaction. One reason is that the quoted estimates of this ratio in particle physics depend on microphysics [35]; these values are not really constants, since they show some secular dependence on particle energy [36,37]. In any case, it is doubtful that the Higgs field can assign masses above its VEV of 246.22 GeV/

c^{2}

[38].

(e) Sub-TeV Particles?—In the atomic world, the Higgs VEV appears to be a barrier against growing more massive nuclei and particles. Nevertheless, researchers are searching the TeV scales in hopes of discovering such particles [39]. If there is a way to jump across the Higgs VEV (which we do not currently see; see also [38]), then the next few particle slots generated by the high-mass geometric progression Z⁰-H-t-

v \dots

will have energies of 0.351, 0.502, 0.716, and 1.022 TeV.

3.5. Units Jumping to the Forefront

Consider the units of mass M and area

R^{2}

; think of them, in abstract terms, as gravity and 2-D geometry. It has long been known that one can form two new units from them, the surface density

σ = M / R^{2}

and the moment of inertia

I = M R^{2}

. These have been used in physics applications, but they are not very popular, certainly not as popular as the unit of volume density we get from M and

R^{3}

. It has become clear to us that

σ

and I have not been given due attention in the past, but now they seem to come to the forefront in our analyses of various astrophysical problems.

Surface density

σ

plays an important role in our recent work on varying-G gravity [1], and provided the cosmological system of units with an important relation used in unit reductions and in the sourcing of the gravitational field:

σ \sim a / G

, where a is acceleration (or, equivalently, the gravitational field), so that

[surface density] = \frac{[gravitational field]}{[gravitational constant]},

indicating that the surface density

σ (r) = M (r) / r^{2}

is a dynamical quantity of considerable influence. Moment of inertia I has already played a role in our calculations of units, as we hinted in Section 2.1.2. Here, we discuss its potential significance for inertial masses and the weak equivalence principle [21]. We use dimensional analysis throughout this subsection, so all symbols (having their usual meanings) represent units.

Bunker et al. [11] righted a wrong in physics when they pointed out that, in units,

h = E / Ω

. (In this notation, [rad] is a unit carried by

Ω \sim

[rad][second]⁻¹.) The current SI system suffers from this inconsistency, and it should be repaired as soon as possible. The problem that concerns us here is the following: in the original Planck system, one of the fundamental units was h, and action integrals had dimensions of

h = E T = M R^{2} T^{- 1} = L

, where

L

denotes angular momentum. This description has been in the books since Dirac and Landau’s times circa 1930 [8,9,22].

In their dimensional analysis of Newton’s G, Landau & Lifshitz [22] used only basic cgs-system units, and they did not mention any interpretation of the units of action. Other authors have mentioned that the action integrals

S

appear to have the same unit as angular momentum

L

, so

S = L = E T

. This is an interesting conclusion, but not a fundamental one, since there exist objects and fields in nature that neither rotate nor revolve (e.g., free spin-0 particles in vacuum and tenuous gases in galaxy clusters).

We have a different interpretation of the units of action

S

, which appears to be fundamental, viz.

S = M R^{2} / T = I / T;

that is,

S

describes the rate of change of the moment of inertia

I / T

, where

I = M R^{2}

, and M is bona fide inertial mass. We believe that this interpretation has important ramifications going forward, because it introduces the concept of inertia into the units of all action integrals, including those of non-gravitating (atomic and nuclear) fields. But if the massless fields know about inertia in their variations, then the weak equivalence principle (and its inertial mass) may have already permeated all other physical processes besides gravity. In other words, macroscopic mass is, in a major way, gravity; but its inertia appears to be a universal property,19 communicated by the vacuum itself (i.e., by the “resisting” properties c and

Z_{0}

) in the same way that geometry-dependent vacuum properties (

ϵ_{0}

and

μ_{0}

) are also installed in the universe.

To conclude, we believe that force actions cannot be unified before the influence of the vacuum on every single field is delineated and becomes fully understood. This is not going to be an easy task; as we mentioned above, the vacuum interferes in the universe by three distinct ways: dimensional geometry-dependent constants (

ϵ_{0}

μ_{0}

), dimensional geometry-free constants (c,

Z_{0}

), and purely numerical imprinted values. (We note that it is this vacuum “activities list” that prompted us to include

Z_{0}

as another building block of the UPS, as shown in Equation (3).)

4. Discussion

4.1. Pairs of Fundamental Dimensional Units

Equation (45) shows that two lengths are needed to produce the fine-structure constant

α_{h}

in any system of units: the fundamental scale

L_{A}

and a Compton-type scale such as

r_{c}

. This subsidiary scale cannot be defined by using the fundamental mass scale (then, one gets

α_{h} = 1

). Therefore, Equation (45) defines

r_{c}

independently of mass

M_{A}

. In our case, this definition is obtained easier from Equation (46):

r_{c} = L_{A} (q_{p} / e)

. Using the definition of the fine-structure constant is an integral part of the above derivation of

r_{c}

; and this example justifies our statement that all systems besides the UPS are incomplete, missing at least the unitless coupling constants, and thus, they are incapable of describing all scales and forces in the universe.

Next, we consider Planck’s original set of dimensional units

{c, G, h}

, with h in place of

h

to avoid misunderstandings from the introduction of geometry into the units. The speed of light barrier is applicable to all systems of units, but h is not fundamental in the cosmological system and G is not fundamental in the atomic system for “obvious” (now known to be obviously wrong) reasons: negligibly weak influences should not be building blocks at the core of a system. We realize now that all three constants are necessary building blocks, and that the vacuum-force pairs

{c, G}

and

{c, h}

serve two different (but complementary) functions within the UPS:

(a) The pair of constants

{c, G}

with its universal unit of force20

F_{0} = c^{4} / G

and the corresponding unit imprint of the famous Tully-Fisher/Faber-Jackson relation [16,17]

c^{4} = G M a_{0}

(where

F_{0} = M a_{0}

; [2,3,4]) was analyzed in our companion paper [1] in the cosmological system of units. (We discuss the universality of this relation in Appendix B.) Combined with Newton’s G, powers of c define units whose purpose is to monitor the effectiveness of forces F in producing motion (speed v). Some of these units are very well-known:

c^{2} / G \sim F / v^{2} = M / R_{S}

c^{3} / G \sim F / v = Z_{m}

c^{4} / G \sim F

, and

c^{5} / G \sim F v = P

. Here, M is mass,

R_{S}

is (Schwarzschild) radius,

Z_{m}

is mechanical impedance, and P is power.

(b) With the notable exceptions of

\sqrt{ϵ_{0} h c} \sim q

(charge) and

\sqrt{h / (ϵ_{0} c)} \sim Φ_{B}

(magnetic flux) (Section 2.1.2), the pair of constants

{c, h}

can only generate composite units, which cannot be viewed as fundamental units in the physical world; although these units do afford some interesting symmetries. For instance, examine the sequence of units

h c \to [E] [L]

h \to [E] [T]

h / c \to [M] [L]

, and

h / c^{2} \to [M] [T]

, before the next powers of c generate some lower-level subsidiary units, e.g.,

h / c^{3} \to [M] {[a]}^{- 1}

. Combining powers of c with Planck’s h, these units are designed to monitor the action integral

S

(i.e., integrated energy in time) during motion, although they are not as well-known:

h / c^{3} \sim S / v^{3}

h / c^{2} \sim S / v^{2}

h / c \sim S / v

h \sim S

, and

h c \sim S v

. Since action

S

determines both speed v and acceleration a, this sequence of units can also be interpreted as:

h / c^{3} \sim (E / v^{2}) / a = M / a

h / c^{2} \sim (E / v) / a = p / a

h / c \sim E / a

h \sim (E v) / a

, and

h c \sim (E v^{2}) / a

, where E represents energy and p represents momentum.

The above symmetries are naturally propagated also to derivative units. As a typical case, we discuss the sequence of composite units

M / T^{n}

(for integer n) generated by the widely-used pair of units of mass and time

{M, T}

, because this sequence holds some surprises. These units apparently measure resistive properties in the material world:

\begin{matrix} M / T & \sim F / (R / T) & = Z_{m} & [mechanical impedance] \\ M / T^{2} & \sim F / R & = S_{m} & [mechanical stiffness] \\ M / T^{3} & \sim F / (R T) & = σ_{P} & [power] {[area]}^{- 1} \end{matrix},

(53)

where R represents length and index P represents power. It is surprising that the unit

M / T

(of the ubiquitous

\dot{M}

of accretion physics) turns out to be a resistive property of inflowing matter. It is also quite surprising that the “power surface density”

σ_{P}

is a member of this sequence of units that describe the various types of mechanical resistance. In Appendix B, we find that power surface density is a universal dynamical quantity, although it appears prominently only in the Stefan-Boltzmann law [40,41]. Its resistive character becomes apparent when we rewrite it in terms of force F and moment of inertia I, viz.

σ_{P} = F^{2} / (I / T),

(54)

where

(I / T)

represents resistance due to the rate of change of the moment of inertia. In this equation, we recognize the importance of the force squared

F^{2}

σ_{P} \sim M / T^{3}

. Coming full circle to expressing the resistances in terms of

F^{2}

, we find for the impedance and the stiffness that

Z_{m} = F^{2} / P

and

S_{m} = F^{2} / E

, respectively, where E represents energy. Therefore, the magnitude of

F^{2}

appears to be regulated by power in impedance, by energy in stiffness, and by inertial changes in power surface density.

Furthermore, the inertial magnitude itself appears in the next term of the sequence (53), i.e.,

M / T^{4} \sim F^{2} / I

, and the integrated quantity

(I T)

appears next in

M / T^{5} \sim F^{2} / (I T)

. Obviously then, the units of the sequence

M / T^{n}

describe resistive properties in which

F^{2}

is regulated by the temporal variations of inertia.

4.2. The Varied Contributions of the Vacuum

The free space known as the vacuum is described by four interdependent constants (

ϵ_{0}

μ_{0}

c = 1 / \sqrt{ϵ_{0} μ_{0}}

Z_{0} = \sqrt{μ_{0} / ϵ_{0}}

). When the vacuum wishes to also promote geometry in some parts of the natural world, then it introduces either

ϵ_{0} \equiv 4 π ϵ_{0}

μ_{0} \equiv μ_{0} / (4 π)

or both, provided they are not introduced in a product (there is no geometry in

ϵ_{0} μ_{0} = 1 / c^{2}

From the nongeometric vacuum quantities

ϵ_{0}

and

μ_{0}

, only two additional purely physical quantities can be constructed by G-Ms, the speed of light c and the impedance of free space

Z_{0}

(Section 2.1.1). They both represent upper limits in nature, the only known upper limits communicated by the vacuum to all scales and in all directions within the universe. Their origin is the least (but nonzero) resistance that the vacuum mounts passively against all motions in the material world (see also Section 5 below).

Next, we wish to track down the geometry that is imposed selectively by the vacuum, so we rewrite the fundamental G-Ms discussed in Section 2.1.1 as follows:

\sqrt{ϵ_{0}^{- 1} μ_{0}^{- 1}} = c,

(55)

and

\sqrt{ϵ_{0}^{- 1} (\frac{1}{μ_{0}^{- 1}})} = \frac{Z_{0}}{4 π} .

(56)

The G-M (55) is clearly geometry-free, whereas the G-M (56) attaches the

4 π

of 3-D space to the geometry-free impedance of free space

Z_{0}

.21 This is an important conclusion: when

ϵ_{0}^{- 1}

μ_{0}^{- 1}

appear on their own in equations, or in any combination other than their product (55), they do carry geometry with them. These composite vacuum constants show us how free space manages to interfere in the construction and evolution of additional (ready-to-interact with one other) physical entities, such as mass and electric charge, that characterize the underlying force fields.

We emphasize here that mass and charge are not actually fundamental quantities, as is widely believed; they can only be derived and clearly understood, if the contributions of the vacuum and the unitless coupling constants are also taken into account. We demonstrate this point here, with exact calculations:

(a) Consider first Equation (13). Solving for the charge e, we obtain a scaled-down G-M relation of the form

e = α_{h}^{1 / 2} \sqrt{h (2 ϵ_{0} c)} = α_{h}^{1 / 2} q_{p} .

(57)

So, Planck’s physical constant h and the vacuum’s combination of

(2 ϵ_{0} c)

determine e as a geometry-free, G-M quantity. From this point of view, we can also see how dimensionless constants resize properties of the material world: the G-M is scaled down by the geometry-free factor of

α_{h}^{1 / 2} ≃ 1 / \sqrt{137}

(see also Section 3.2).

(b) Consider next Equation (14). Solving for the mass

m_{e}

, we obtain a G-M relation of the form

m_{e} = {(\frac{α_{G}}{2 π})}^{1 / 2} \sqrt{(\frac{h}{G}) c} = {(\frac{α_{G}}{2 π})}^{1 / 2} M_{p} .

(58)

In this case,

m_{e}

is determined by the G-M of the composite physical constant

h / G

and the vacuum’s c. (G participates because this is a mass assignment.) The G-M is scaled down by a factor of

{[α_{G} / (2 π)]}^{1 / 2} = 1.670 \times 10^{- 23}

relative to

M_{p}

. Because of the inclusion of

2 π

, this factor is geometry-free, and so is

m_{e}

By dividing equations (57) and (58), and neglecting for the moment the dimensionless, geometry-free factor

{(4 π / β_{G})}^{1 / 2} = 7.235 \times 10^{21}

, we obtain a geometry-independent G-M for the electron’s charge-to-mass ratio, viz.

\frac{e}{m_{e}} \propto \sqrt{ϵ_{0} G} .

(59)

Thus, the ratio

e / m_{e}

is determined mainly by the G-M of the nongeometric constants

ϵ_{0}

and G (vacuum and gravity, respectively); and the neglected scale factor carries the relative strength of the two unitless coupling constants (

\sqrt{4 π / β_{G}} \propto \sqrt{α_{h} / α_{G}}

) with the geometry taken out of the ratio

β_{G}

4.3. Sparse Geometric Averaging in Nature

We think we understand why virtually all pairs of constants and units (

U_{1}, U_{2}

) combine in G-Ms22 involving the direct form

U_{1} U_{2}

or the inversion form

U_{1} U_{2}^{- 1}

(or

U_{1}^{- 1} U_{2}

). Physically, two basic G-M quantities can be derived from each pair of units. Mathematically, these two operations result in mappings that are aways “smooth” since they involve constants; thus, the units of a system of units form a Lie group [13], and the associated Lie algebra can be carried out with ease.

One remaining question is why there are also square roots on top of these unit combinations, establishing thus G-Ms. We fall back to what is already known about G-Ms: compared to the commonly used arithmetic means, G-Ms place significant more weight to the smaller of the two values. Thus, the most obvious property of the geometric averages

\sqrt{U_{1} U_{2}^{\pm 1}}

is that they assist the smaller physical constants to leave their indelible marks in their combinations with larger constants. In a sense, by not letting small constants become negligible (or dominant) when they are combined with large constants,23 nature seems to follow a principle of fairness or impartiality at all scales of the universe. The degree of support for the small constants can be quite dramatic for much differing constants, as equations (15) and (19) vividly demonstrate: the G-M

\sqrt{β_{G}}

gains 21.31 orders of magnitude relative to the ratio

β_{G}

, and it connects the Planck scale with the atomic world.

Consider now the subatomic particles discussed in Section 3.1. Nature did not make a particle in each individual G-M slot. The mass ladder is mostly empty, and just a few actual particles have materialized in the subatomic scales of the universe [35]. So, there are additional selection criteria (scaling laws) on top of the G-Ms that regulate the creation of particles. Besides the factors of 2 and

\sqrt{2}

in the equations of Section 3.1, we have seen that the Higgs boson does not rely on pure G-Ms to reach down to lower masses; it uses, in addition, two different scale factors, Koide’s

K = 2 / 3

and

\sqrt{α_{h}} ≃ 1 / 30

(Equation (41)), to bypass many available particle slots (see also Table A1 below). In particular, the dramatic drop from the Higgs mass to the mass of the bottom quark can only be described as a deflation of particle mass that bypasses 10 G-M particle slots intervening between

m_{Z^{0}}

and

m_{b}

5. Lingering Issues, Future Prospects, and a Brief Summary

The UPS was summarized in Equation (3). The system is not flawless yet, and several issues must be investigated and resolved in the future (see, e.g., footnote 24 below). These issues can be traced to Dirac’s introduction of

h = h / (2 π)

in place of Planck’s h.

It is certainly true that in quantum mechanics, Dirac’s composite constant

h / (2 π)

always appears in form, and this also prompted Schrödinger [11,18] to absorb the

2 π

into a convenient new constant K. This tactic tells us that Schrödinger was not aware that he was including geometry into his constant K. Dirac [7,8,9], on the other hand, believed that

h = K

is the true constant (not h), so we can guess that he sensed that the two constants are fundamentally different in their makeup (see Section 1.2 for more details).

Dirac’s reform has modified quite substantially the systems of units that have adopted

h

, but this modification came with a heavy price. Planck’s purely physical constant h cannot be dropped so nimbly, because then, we introduce errors in the definitions of the coupling constants. Dimensionless coupling constants should not include geometric dependencies other than

ϵ_{0}

μ_{0}

(and these enter only via EM terms); geometry would give the constants an additional unit of [rad] and it would alter their nature. On the dimensional side of vacuum units, c and

Z_{0}

(Section 4.2) are also geometry-free constants for a good reason: they represent upper limits set by the vacuum to be applicable in any direction of space irrespective of the dimensionality of space.

We note another issue concerning

h

: In the dimensional part of the UPS, the constant

h

is the only fundamental dimensional unit that introduces geometry in the physical units. This is unusual and a singular property. Although we were inclined to adopt Planck’s h in place of

h

, we did not do so, because we do not know how to choose between the two constants. It seems from the calculations above that the use of h in the definitions of scales (Planck units, coupling constants) is mandatory, but then

h

may be more appropriate to be retained for particles and fields, as Dirac [7,8,9] also thought. Perhaps, both constants should be retained in a modified UPS, along with

α_{h}

and

β_{G}

(see the UPS as described in footnote 24).

Examining now the definitions of the dimensionless units that we summarized in Section 2.2 (Equations (13) and (14)), we see that

α_{h}

is indeed geometry-free (

ϵ_{0} h = 2 ϵ_{0} h

), but

α_{G}

is not (

α_{G} \propto 1 / h

). We think this is an enormous oversight flying undercover at least since Dirac [8] introduced his “large numbers hypothesis;” and it has prevented physicists from defining an atomic mass scale in the modern Planck system, creating thus an insurmountable obstacle to force unification. The state of confusion can best be seen in the widespread misconception “that G carries units into the action of general relativity, thus gravity is not like the other forces of nature,” taught to thousands upon thousands of physics students for nearly a century. We now understand that gravity is just like the other forces, and it enters the “ring” with one dimensionless (

α_{G}

) and one dimensional (G) constant, just as the EM forces do too.

Owing to the omnidirectional nature of the gravitational force, both of its constants should be geometry-free. For this reason, we tried to bypass the problem with the definition (14) of

α_{G}

(it effectively carries a unit of [rad], thus it cannot be utilized) and to define new consistent atomic units within the UPS. First, we created a dimensionless ratio

β_{G} = α_{G} / α_{h}

of the coupling constants that describes their relative strength; the

h

does not partake in this ratio, and the only geometric influence left comes from the EM field. But this does not affect the makeup of the relative strength

β_{G}

, since

β_{G}

is expressed as a ratio of energies.24

Next, we created a dimensionless geometry-free combination of fundamental units to attach to

β_{G}

, viz.

β_{G} = G M_{A}^{2} / (h c) = {(M_{A} / M_{p})}^{2},

where

M_{p}

is the mass scale of the original [5] system of units. Finally, the new atomic mass scale

M_{A}

was derived from the known values of

β_{G}

and

M_{p}

, viz.

M_{A} = M_{p} \sqrt{β_{G}} .

The interpretation of this relation is straightforward: the ratio of the two widely different mass scales

M_{A} / M_{p} = 4.9 \times 10^{- 22}

is precisely equal to the square root of the relative ratio of the two coupling constants

β_{G} = α_{G} / α_{h} = 2.4 \times 10^{- 43}

In Section 3, we tested the influence of this mass scale in the atomic and subatomic world, and the results appear to be strong. The mass constant

M_{A}

has no trouble meddling in G-Ms (Section 3.1) along with particle (sub)atomic masses that have been measured by experiment [19,29] (but see also footnote 7 for UPS^′, an alternative system of units based on proton’s parameters). In the process, we also clarified the confusion surrounding the so-called Planck charge (Section 3.2), and we also derived a new length scale that had no trouble meshing in G-M calculations with the already-known atomic radii (Section 3.3 and Section 1.2).

In Section 3.1 and Section 3.4, we calculated both mass scales and actual particle masses at practically all scales of the universe. The Higgs field uses a multitude of methods to distribute masses to selected atomic particles. This diversity of methods is, in part, responsible for hindering progress in the effort to unify the four fundamental forces of nature. The other part concerns the role of the vacuum (Section 2.1.1 and Section 4.2). The behavior of the vacuum is not at all what our books describe (e.g., [28,31,35]). As far as we can see, the vacuum is not subject to forcing of any kind, and it seems to be unresponsive to quantum fluctuations, which occur exclusively in fields. By and large, the vacuum appears to be a passive independent entity, with no physical properties of its own, that imposes implicitly certain rules (by resisting the least) to the material world, that all inhabitants must necessarily observe and obey (to within the bounds of the uncertainty principle of course; see also Appendix B.2). Under these circumstances, there is no back reaction from the material world on to the vacuum itself. In hindsight, this conclusion makes sense—how can anything tangible manage to tangle up that which is the epitome of nothingness?

6. Highlights

6.1. Summary of Results

(1): Current systems of units are incomplete and incapable of describing all aspects of this universe. They do not include some of the fundamental dimensional constants, the dimensionless coupling constants, and all the restrictions installed by the vacuum itself to the material world.
(2): Each force of nature must be represented in a system of units with a dimensional and a dimensionless coupling constant. If Planck’s h is dropped, then the system cannot measure quantities related to quantum phenomena. If Newton’s G is dropped, then the system does not include gravity.
(3a): The fine-structure constant $α_{h} = {(861.022576)}^{- 1}$ , not multiplied by $2 π$ , is the only coupling constant that must be included in absolute terms. It has been measured by experiment, and it provides the scale factor $\sqrt{α_{h}} ≃ 1 / 30$ used by the Higgs boson to deflate and assign mass to the bottom quark; and then, to reach down to all the other lower-mass particles (Table A1 below). On the other hand, the Higgs boson creates the masses of the Z⁰ and W^± bosons and the other quarks by G-M averaging and by using Koide’s scale of 2/3 in various incarnations.
(3b): All other unitless constants must be included in relative terms because only ratios of these coupling constants have physical meaning—they provide relative strengths, just as the dimensional quantities and the units also do.
(3c): The modern definitions of the unitless coupling constants are incorrect because $h$ was used instead of Planck’s physical constant h. Dirac’s $h$ is a composite constant that also carries planar 2-D geometry and a unit of [rad]⁻¹; the $2 π$ radians in $h$ have inadvertently reversed the influence of geometry on to the coupling constants.
(4): The vacuum is a passive entity and not subject to forcing of any kind by the material world. By providing the least (but nonzero) resistance to all motions that occur in its domain, the vacuum installs upper limits to the material world (c and $Z_{0}$ in nearly perfect dielectrics), which must then be included in systems of units as well. These two geometry-free constants also bring $4 π ϵ_{0}$ and $μ_{0} / (4 π)$ with them, in which the influence of 3-D geometry is apparent. (Here, the vacuum’s $ϵ_{0}$ and $μ_{0}$ are both lower limits.)
(5): The “dark energy” that permeates the universe and that drives its accelerated expansion is not produced by quantum gravity. The vacuum that we described possesses no physical property that could possibly oscillate, and it cannot produce a discrepancy of 120 orders of magnitude in energy density. Furthermore, in recent work, we showed that the spatial variation of Newton’s G, coupled to mass surface density, can apparently create positive dark pressure whose magnitude is comparable to the observed value and whose gradient accelerates the expansion of the universe [1].
(6): The rate of change of the moment of inertia is the universal unit of all action integrals, so it appears that inertia and the weak equivalence principle have been built into all scales of the universe, large and small, gravitating or not.
(7a): There exists a new atomic mass scale $M_{A} = 15.0$ MeV/ $c^{2}$ that we found by deflating the original Planck mass $M_{p}$ by $\sqrt{β_{G}} = 4.900 \times 10^{- 22}$ , where $β_{G}$ is the relative ratio of the coupling constants of gravity and fine structure. Of course, in our expanding universe, the event took place in reverse ( $M_{A} / \sqrt{β_{G}} \to M_{p}$ ). This inflation of scale accounted for 21.31 orders of magnitude in mass, and explains how the Planck scale is connected to the atomic world. (At the same time, the atomic scale of length was deflated by precisely the same amount to produce the tiny Planck length.)
(7b): No (sub)atomic particle is found to occupy a scale value, and the measured masses in the atomic world are connected mostly by G-M averaging. By using G-M averaging, nature (a) remains impartial to designating any one of the particles as being more significant than the other particles; and (b) assigns more weight to the smaller participant in the G-M, thereby assisting smaller forces to leave their marks on the universe.
(7c): We can relate characteristic atomic constants (charge e, mass $m_{e}$ , G-M $\sqrt{m_{e} m_{p}}$ , Compton radius $r_{c}$ ) to scale values ( $q_{p}, M_{p}, M_{A}, L_{A}$ , respectively), but this is not how these physical entities were created; they were created by the Higgs scalings (1/30 and 2/3) and by G-M averaging of other nearby physical entities.
(8): Leptons, quarks, and bosons get their masses from the Higgs field. The boson-quark mass ladder is shown in Table A1 below. How the Higgs field acquires its mass $m_{H}$ and its vacuum expectation value v remains a mystery; the only hint in the known masses [29] is that $m_{H} ≃ v / 2$ (to within a deviation of 1.7%).
(9): Koide’s lepton constant $K = 2 / 3$ is one of the scaling constants used by the Higgs field. We derived it from first principles in Appendix A, and it is also applicable to the high-mass quark triplet c-b-t. We also derived two additional Koide-type constants, $J = 4 / 7$ (from the low-mass quark triplet u-d-s) and $B = 0.336$ (from the Higgs bosons W^±-Z⁰-H). Constant B is barely 0.8% larger than the absolute minimum value of 1/3 that occurs for three equal masses.
(10): In Appendix B, we pointed out four instances of a universal law that has the general form

$(a surface density) \propto {(a kinetic scalar quantity)}^{4},$

in which the power of 4 is the sum of the 3 spatial degrees of freedom and 1 additional degree of freedom for the scale of the underlying scalar quantity. The three types of surface density involved describe force F, power P, and moment of inertia I, all divided by surface area A. Pressure $F / A$ appears in the Higgs field and the Casimir effect; intensity $P / A$ appears in the Stefan-Boltzmann law; and mass $I / A$ appears in the Tully-Fisher/Faber-Jackson relation in spiral/elliptical galaxies. It certainly appears that the dynamics of the present universe is driven by the surface densities of various fundamental quantities (see also Appendix B.2).

6.2. Critical Questions and Answers

(Q1): How does Planck mass relate to the atomic world?

—The atomic mass scale $M_{A} = 15$ MeV/ $c^{2}$ inflates to the Planck mass

$M_{A} / \sqrt{β_{G}} \to M_{p}$ , where $\sqrt{β_{G}} = 4.900 \times 10^{- 22}$ .
(Q2): What is the physical meaning of number 137?

—Number $137 = 861 / (2 π)$ , where the $2 π$ is a geometric term carrying the unit of [rad]; so, 137 is not unitless. The actual unitless constant is 861, and the scale factor $\sqrt{861} ≃ 30$ is used by the Higgs boson to assign masses to much lighter particles, starting with the bottom quark and moving on down the mass ladder (Table A1).
(Q3): What is the physical meaning of Koide’s constant?

—Koide’s $K = 2 / 3$ is another scale factor used by the Higgs boson to assign masses to lighter vector bosons (the particles W^± and Z⁰). Koide’s formula holds exactly for the leptons e- $μ$ - $τ$ and for the high-mass quarks c-b-t.
(Q4): How does the top quark get its mass?

—The top quark mass is the geometric mean of the Higgs mass and the Higgs field vacuum expectation value $v = 246.22$ GeV/ $c^{2}$ , so that $m_{t} = \sqrt{m_{H} v}$ .
(Q5): How do Higgs vector bosons get their masses?

—By two different mechanisms: In the ordered high-mass triplet, Z⁰-H-t, the Higgs mass is the geometric mean of the Z⁰ mass and the top quark mass, so $m_{Z^{0}} = m_{H}^{2} / m_{t}$ . In contrast, W^± gets its mass by Koide scaling: $W^{\pm} = K m_{H}$ , where $K = 2 / 3$ .
(Q6): How does the bottom quark get its mass?

—By a third mechanism: The Higgs mass is scaled down by the factor of 30 derived in (Q2) above, so that $m_{b} = m_{H} / 30$ . We have tried several other mechanisms and scalings, none of them worked. The assignment of mass to the bottom quark is becoming a major issue to resolve in the future by any theory that purports to explain mass assignments to low-mass quarks.
(Q7): Is $h$ , rather than h, the true universal constant?

—They both are, but Planck’s h is a pure physical constant, whereas Dirac’s $h = h / (2 π)$ is composite and includes also the geometric term $2 π$ that carries the unit of [rad]. Because of this geometric content, an error was committed in the post-Planckian era when $h$ was adopted for the modern definitions of the fine-structure constant and the gravitational coupling constant.
(Q8): Is dark energy a quantum phenomenon?

—No, it is not. The vacuum possesses no physical or tangible property, it only provides the least (albeit nonzero) resistance to all motions of its inhabitants, thereby indirectly installing upper limits to the flow speeds of matter/energy and electric currents (speed c and impedance $Z_{0}$ , respectively). According to this interpretation, and in agreement with recent observations and current cosmological models of accelerated universal expansion, we have obtained a reasonable estimate of the dark energy density ( $U_{0} = 1.35$ GeV m⁻³ corresponding to $ρ_{0} = 2.40 \times 10^{- 27}$ kg m⁻³) created in empty space by the spatial variation of Newton’s G [1].

Author Contributions

Both authors have worked on all aspects of the problem, and both read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

Not applicable.

Acknowledgments

NASA support over the years is gratefully acknowledged.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:

G-M	Geometric-Mean
MOND	Modified Newtonian Dynamics
UPS	An Upgraded Planck System Based on Electron Mass
UPS^′	An Upgraded Planck System Based on Proton Mass

Appendix A. The Physical Meaning of Koide’s Lepton Constant

Koide’s constant of 2/3 involves the three lepton masses and it is a puzzle in particle physics [15]. This constant is not a mere numerical coincidence, it is a fundamental scale that the Higgs boson uses to create the W^± bosons, and then it is propagated to lower masses according to the relations given in Section 3.1.

Koide’s constant is defined for the three leptons e-

μ

τ

(m_{e} + m_{μ} + m_{τ}) / {(\sqrt{m_{e}} + \sqrt{m_{μ}} + \sqrt{m_{τ}})}^{2} \equiv 2 / 3 .

(A1)

Mathematically, this relation is equivalent to an equation involving arithmetic means and G-Ms of paired quantities, viz.

f \equiv (m_{e} + m_{μ} + m_{τ}) / (\sqrt{m_{e} m_{μ}} + \sqrt{m_{e} m_{τ}} + \sqrt{m_{μ} m_{τ}}) = 4,

(A2)

where the numerator should be viewed as the triple sum of arithmetic means of paired quantities, viz.

m_{e} + m_{μ} + m_{τ} = \frac{1}{2} (m_{e} + m_{μ}) + \frac{1}{2} (m_{e} + m_{τ}) + \frac{1}{2} (m_{μ} + m_{τ}) .

Equation (A2) reveals a special relationship between the three arithmetic means in the numerator and the three G-Ms in the denominator: the ratio of their averages must be equal to 4; then, Equation (A1) is an identity, as Koide [15] discovered. This special relationship is recovered from the equations of Section 3.1. We carried out several reductions25 by evaluating lower masses starting from the Higgs mass and the masses of

m_{W^{\pm}}

m_{Z^{0}}

, and

m_{t}

. The endpoint is a pair of inversion G-M relations between lepton masses, viz.

\sqrt{m_{μ} / m_{e}} = 22 K,

(A3)

and

\sqrt{m_{τ} / m_{e}} = 40 K^{- 1} .

(A4)

Their product is independent of K, and its value (880) carries an error of 4% as compared to the experimental value of 848. The coefficients in these ratios were rounded off to obtain a 2% accuracy in each individual ratio (not rounded, the coefficient in Equation (A4) has a deviation of 5%, sufficiently large to produce a higher K-value of 0.70).

Substituting the above ratios into Equation (A2), we find a fourth-order polynomial equation of the form

484 K^{4} - 88 K^{3} - 3519 K^{2} - 160 K + 1600 = 0,

(A5)

The solutions of the fourth-order equation can be obtained analytically, and they are all real. The two positive roots are

K_{1} = 0.66641 and K_{2} = 2.7283 .

(A6)

Root

K_{1}

is Koide’s constant; it determines the lepton mass ratios, and they, in turn, satisfy Equation (A1) to within an error of 0.04% (Koide’s

K = 2 / 3

is an extremely robust physical constant, as experimenters have discovered).

Root

K_{2}

is rejected based on the experimental results. Substituting

K_{2}

into the equations of the lepton mass ratios reverses the two values, and results in

m_{μ} > m_{τ}

. This solution is obviously incompatible with the measured masses of these two particles [29].

Despite the rejection,

K_{2}

is of some theoretical interest: the ratio

K_{2} / K_{1} = 4

, the same 4 that appears in Equation (A2). Then, we find that the

f = 4

factor in Equation (A2) is the raw physical constant, and Koide’s K is derived from it:

K = \frac{f}{f + 2} = \frac{2}{3} .

(A7)

This relation is derived by substituting Equation (A2) into Equation (A1).

Appendix A.1. Physical Interpretation

The physical interpretation of nature’s choice of

f = 4

is deduced from Equation (A2), rewritten in the accessible form

\frac{1}{3} (m_{e} + m_{μ} + m_{τ}) = \frac{4}{3} (\sqrt{m_{e} m_{μ}} + \sqrt{m_{e} m_{τ}} + \sqrt{m_{μ} m_{τ}}) .

(A8)

The factor of 1/3 indicates that the left-hand side is the arithmetic mean of the lepton masses. The factor of 4/3 on the right-hand side (also the G-M

\sqrt{K_{1} K_{2}}

to within 1%) is

C_{F}

, the quadratic Casimir charge of the SU(3) fundamental representation [31]. It seems then that the assignment of masses to the leptons is constrained also by the delivery of charge, and this is why

f = 3 C_{F},

(A9)

appears in the right-hand side of Equation (A2). This last equation is a special case of the general formula ([31], section 4.5) of quantum chromodynamics, viz.

\frac{N_{A}}{2} = N C_{F},

(A10)

as applied to SU(3), where

N = 3

dimensions, the indices

1, 2, \dots, N_{A}

label the

N_{A} \times N_{A}

color generators in the “octet” quark-antiquark state,

N_{A} = N^{2} - 1 = 8

, and

f = N_{A} / 2

; thus, we find that

f = 4

and

C_{F} = 4 / 3

in SU(3). Finally, Koide’s constant turns out to depend only on the SU(3) octet number

N_{A}

, viz.

K = \frac{N_{A}}{N_{A} + 4} .

(A11)

Appendix A.2. Additional Koide-type Constants

The G-M relations of Section 3.1 can help us make physical sense of various other triple combinations of masses involving quarks or bosons that have been previously derived by numerology. Here, we summarize the calculations for three triplets that naturally come to mind, the quarks c-b-t and u-d-s, and the bosons W^±-Z⁰-H:

Table A1. Boson-Quark mass ladder in terms of the Higgs mass

m_{H}

. Two scales are used, Koide’s

K = 2 / 3

and the 1/30 deflation of

m_{H}

down to the bottom quark mass

m_{b}

Table A1. Boson-Quark mass ladder in terms of the Higgs mass

m_{H}

. Two scales are used, Koide’s

K = 2 / 3

and the 1/30 deflation of

m_{H}

down to the bottom quark mass

m_{b}

Particle	Mass Relation	Deviation* (%)
Bosons
Z⁰	$m_{Z^{0}} = K^{3 / 4} m_{H}$	+1.3
W^±	$m_{W^{\pm}} = K m_{H}$	+3.9
Quarks
top	$m_{t} = K^{- 3 / 4} m_{H}$	$- 1.6$
bottom	$m_{b} = m_{H} / 30$	$- 0.12$
charm	$m_{c} = 2 \sqrt{K / 30^{3}} m_{H}$	$- 2.0$
strange	$m_{s} = (K / 30^{2}) m_{H}$	$- 0.67$
down	$m_{d} = 2 (K^{7 / 4} / 30^{3}) m_{H}$	$- 2.3$
up	$m_{u} = (K^{2} / 30^{3}) m_{H}$	$- 4.5$

* Deviation = [(right-to-left side)

- 1

]

\times 100 %

High-mass quarks c-b-t.—Based on experimental masses, Equation (A1) with c-b-t values in place of e-

μ

τ

values produces a constant of 0.669 on the right-hand side, only 0.35% higher than

K = 2 / 3

. This is a solid physical result. From the equations of Section 3.1, we find that

m_{t} = 30 m_{b} K^{- 3 / 4}

m_{c} = 2 m_{b} \sqrt{K / 30}

, and a corresponding constant of 0.668 with a deviation of 0.20% from

K = 2 / 3

Low-mass quarks u-d-s.—Koide’s K is not produced by the masses of the u-d-s triplet. Based on their experimental masses, Equation (A1) with u-d-s in place of e-

μ

τ

produces the constant

J = 0.567

on the right-hand side, probably a value of no interest to numerology. We, on the other hand, have derived this constant analytically by utilizing the G-M relations of Section 3.1 and by expressing the u-d-s quark masses in terms of the Higgs scales

m_{b} / m_{H} = 1 / 30

(deflation) and

K = 2 / 3

(Koide), used in assignments of masses lower than

m_{H}

. It turns out that the entire boson-quark mass ladder has to be calculated in the process. The results of our calculations are listed in Table A1. Using the values obtained for the u-d-s masses at the bottom of the mass ladder, we find that

(m_{u} + m_{d} + m_{s}) / {(\sqrt{m_{u}} + \sqrt{m_{d}} + \sqrt{m_{s}})}^{2} = 0.570,

(A12)

a constant that deviates only by 0.53% from the experimental value of

J = 0.567

. In this case, we find that

0.570 \to 4 / 7

f = 2 C_{F} = N_{A} / N = 8 / 3

, and

J = N_{A} / (N_{A} + 2 N) = 4 / 7

in SU(3) (in place of Equations (A9)-(A11)).

Higgs bosons W^±-Z⁰-H.—Based on experimental masses, Equation (A1) with W^±-Z⁰-H masses in place of e-

μ

τ

masses produces a constant of

B = 0.336

on the right-hand side, only 0.80% higher than the lowest attainable value of

1 / 3

obtained for three equal masses. From equations (39) and (40), we find that

m_{Z^{0}} = K^{3 / 4} m_{H}

and

m_{W^{\pm}} = K m_{H}

(see also Table A1), and then, Equation (A1) for the W^±-Z⁰-H triplet is transformed to

1 + K^{3 / 4} + K = 0.336 {(1 + K^{3 / 8} + K^{1 / 2})}^{2},

(A13)

with an accepted root at

K_{1} = 0.662

(deviation 0.70% from

K = 2 / 3

) and a rejected root at

K_{2} = 1.541

. We conclude that

K_{1}

is Koide’s constant, in which case,

f = 1.012

on the right-hand side of Equation (A2) and

B = f / (f + 2)

(let

B \to K

in Equation (A7)). The quark-antiquark color octet number

N_{A}

and the quadratic Casimir charge

C_{F}

are not involved in these calculations (Equations (A9)-(A11) are not applicable to bosons).

Appendix B. A Universal Natural Law Discovered in Widely Distant Scales

The work that we presented in this paper was triggered by the realization that the unit of force

F_{0} = c^{4} / G

is precisely the same in the cosmological and the Planck systems of units. Furthermore, this unit takes the form of [speed]⁴ in the Tully-Fisher/Faber-Jackson relation [16,17] discovered in galaxies (Section 4.1). This is astonishing, given the tremendous difference in scales between the two systems of units.

The implication is that such a fundamental relation ought to exist also in the Planck scale, and presumably in the microcosmic scales as well. Indeed, they do exist, and the first such relation dates back to Stefan [40], who discovered the famous [temperature]⁴ dependence of an emitting blackbody’s intensity or power surface density with units [power][area]⁻¹. We searched and found that such a universal law has been discovered in all of the above scales, but a connection has not been previously made because the corresponding subfields of physics are presently disjoint.

The universal law involves the surface densities of various fundamental quantities. (Such surface-density dynamical variables have become of primary importance in the work presented in our recent paper [1].) These surface densities are all related to the fourth power of kinetic terms which are limited by the various small resistances imposed by the vacuum. Specifically:

(1): In quantum gravity, the energy-density shift of the Higgs field $U_{H}$ resulting from spontaneous symmetry breaking (that prevents ultraviolet divergence) is $U_{H} \propto v^{4}$ , where v is the Higgs vacuum expectation value [28,46]. This relation is equivalent to

$σ_{F} \propto v^{4},$

(A14)

where the “force surface density” $σ_{F} \equiv F / A$ (where F is force, A is area, and $U_{H} = F / A$ has dimensions of [pressure]).
(2): In the macroscopic realization of the Casimir effect, the same force per unit area is proportional to the fourth power of the reciprocal of distance D between parallel plates [47,48], viz.

$σ_{F} \propto {(1 / D)}^{4} .$

(A15)

The units agree in the last two relations, since v above has dimensions of [distance]⁻¹ [28].
(3): In atomic physics, the celebrated Stefan-Boltzmann law [40,41] takes the equivalent form

$σ_{P} \propto Θ^{4},$

(A16)

where the “power surface density” (intensity) $σ_{P} \equiv P / A$ , P is power, and $Θ$ is mean temperature.
(4): In astrophysics, galaxies obey the relation $M \propto V^{4}$ , where M is mass and V is rotational speed or stellar velocity dispersion in spiral [16,49,50,51] and elliptical [17,52,53] galaxies, respectively. This relation is equivalent to

$σ_{I} \propto V^{4},$

(A17)

where the “moment-of-inertia surface density” $σ_{I} \equiv I / A$ (where I is moment of inertia and $I / A$ has dimensions of [mass]).

Appendix B.1. Dimensional Analysis of Surface Densities

Dimensional analysis can help us understand the meaning of these surface densities, but not by reducing their definitions to the fundamental units of the UPS. We have to search a little harder to find any common properties between these quantities. We begin with the power surface density (wave intensity)

σ_{P}

that assumes the simplest form among the surface densities:

σ_{P} = \frac{F^{2}}{(I / T)} .

(A18)

In EM interactions, the rate of change of moment of inertia can be replaced by

I / T = q^{2} Z_{0},

(A19)

where q is charge and

Z_{0} = \sqrt{μ_{0} / ϵ_{0}}

is vacuum impedance; we find that

σ_{P} = Z_{0}^{- 1} E^{2},

(A20)

where the electric field is given by

E = F / q

For gravitational power, the force in Equation (A18) is also modified by the vacuum, but only by coupling to Newton’s G. Rewriting Equation (A18) in terms of the gravitational field (acceleration) a, we find that

σ_{P} = (\frac{c}{G}) a^{2} .

(A21)

Force is

F = P / c

in terms of power P, and the force surface density

σ_{F}

takes the corresponding forms

σ_{F} = \frac{σ_{P}}{c} = ϵ_{0} E^{2} = G^{- 1} a^{2},

(A22)

where the vacuum’s c drops out from gravity’s

σ_{F}

. This is a fundamental difference as compared to the EM field’s

σ_{F}

, in which the vacuum (the

ϵ_{0}

here) is permanently attached.

Finally, as was probably expected, the moment-of-inertia surface density

σ_{I} = M

does not quite conform to the above picture. We find that

σ_{I} = \frac{F^{2}}{(I / T^{4})} = \frac{J^{2}}{(I / T^{2})},

(A23)

where

J

is impulse and

I / T^{2}

is energy. Mass is already built with inertia, and it is not surprising that it does not scale as

{(I / T)}^{- 1}

, as the other densities do. To find out how this force squared and impulse squared are regulated, we rewrite the terms in the denominators. It turns out that

I / T^{3} = P,

(A24)

so that Equation (A23) then takes the form

σ_{I} = \frac{F^{2}}{(P / T)} = \frac{J^{2}}{P T},

(A25)

where the integrated quantity

P T

represents energy E. Thus,

F^{2}

is regulated by the rate of change of power

P / T

, and

J^{2}

is regulated by energy

P T

, both of which are restricted by the speed of light.

Appendix B.2. Physical Properties of Surface Densities

We conclude with a summary of properties of the above three surface densities:

(a): The densities $σ_{P}$ and $σ_{F}$ (wave intensity and pressure, respectively) are both modified by the rate of change of inertia $(I / T)$ at all scales (Equations (A18) and (A22)).
(b): Density $σ_{I}$ (i.e., mass) is not modified by inertia, it is inertia; instead, we can say that mass is force squared $F^{2}$ regulated by the rate of change of power $(P / T)$ , or impulse squared $J^{2}$ regulated by energy E (Equation (A25)), where E should be viewed here as the rate of change of the action integral, i.e., $(S / T)$ .
(c): Vacuum constants are explicitly present in $σ_{P}$ (equations (A20) and (A21), where $σ_{P}$ is written in terms of the force fields squared $E^{2}$ and $a^{2}$ , respectively).
(d): The vacuum remains present in the $σ_{F}$ of the EM field, but it drops out from the $σ_{F}$ of the gravitational field (both behaviors are shown in Equation (A22)).
(e): The force surface density $σ_{F}$ (Equation (A22)) represents the conventional energy density of the force fields, whereas $σ_{P}$ (Equation (A18)) shows that inertia is present during the action of all forces.
(f): Both sides of Equation (A19) have dimensions of Planck’s constant h, thus $I / T \sim q^{2} Z_{0} = [action]$ . Higher powers of T in $I / T^{n}$ ( $n = 2, 3$ ) are also physically quite important: $I / T^{2} = [energy]$ and $I / T^{3} = [power]$ . Equation (A24) for $I / T^{3}$ then implies that power stems from the third time derivative of the moment of inertia, a property that is fundamental for the emission of gravitational waves. The same relation, applied to EM waves, produces the ohmic power with dimensions of [electric current]² [ohmic resistance].
(g): Equations (A14)-(A17) all have the characteristic form

$[a surface density] \propto {[a kinetic scalar quantity]}^{4},$

in which the power of 4 represents $N + 1$ degrees of freedom, with $N = 3$ for the spatial dimensions, plus 1 degree of freedom for the scale of the underlying scalar quantity.
(h): The intensity $σ_{P}$ and the pressure $σ_{F}$ are proportional to kinetic terms that express ${[specific energy]}^{4}$ . The Tully-Fisher/Faber-Jackson relation $M \sim V^{4} \sim {[specific energy]}^{2}$ [16,17] then indicates that, in units where $c = G = 1$ ([28], p. xix), it is the square of the mass

$M^{2} = {(σ_{I})}^{2} \sim {[specific energy]}^{4},$

that falls in the same category. Thus, we believe that the $M^{2}$ term here is meant to signify that the inertial and the gravitational mass are included on an equal footing (just as in Equation (12) discussed in Section 2.1.3).

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1	Similarly, $μ_{0} \equiv μ_{0} / (4 π)$ is the reduced vacuum permeability, and then, $ϵ_{0} μ_{0} = 1 / c^{2}$ . The stereometric $4 π$ terms cancel out nicely to produce the “definition” of c, which is a purely physical quantity. Then, Dirac’s $2 π$ term in $h$ tells us that Planck’s free photons only “see” two dimensions, no matter how they move in stereometric (3-D) space (in lines, or circles, or ellipses, etc.). We also learn that the fundamental natural constant c is produced by the vacuum itself, and it is the geometric mean of two smooth inverse Lie mappings of $ϵ_{0}$ and $μ_{0}$ [13].
2	In Bohr’s model, the (nongeometric) number-parts of energies $E_{n}$ and radii $r_{n}$ are related by $E_{n} \propto 1 / r_{n} = 1 / n^{2}$ . So, for pure numbers, we see that $\sqrt{r_{n}} = n$ , and the coefficients of the quantized radii are essentially produced by geometric averaging, viz., $r_{n + 1} = \sqrt{r_{n} r_{n + 2}} + 1$ . The +1 extends the sequence back to $n = 0, r_{0} = 0$ .
3	The unit of [area] justifies the deduced geometric factor $1 / (2 π)$ in the second G-M of equation (10). The additional factor of 2 attached to $ϵ_{0}$ is a unitless imprint, but it has a geometric origin. This type of imprinting is difficult to track down in the various equations of physics when they are presented in reduced, simplified form (see also the discussion in Section 1.2 about the numerical factor of 1/4 imprinted by geometry to the Rydberg energy).
4	The reference unitless constant ( $α_{h}$ here) plays the exact same role that the standard 1-meter ruler and the standard 1-kilogram cylinder play in the SI system of units of length and mass, respectively.
5	Had we used the mass and the charge of a supermassive black hole (e.g., [26]) in equation (15), we would have obtained a relative strength of couplings $β_{G} ≫ 1$ and a different system of units, which would be hard to relate to the Planck scale and even harder to use in the atomic world.
6	We point out again that using $h$ in definitions (13) and (14) reverses what nature intended. It makes $α_{h}$ be a geometry-free value, although the geometry should have been that of $ϵ_{0}$ coming from the electric field), and $α_{G}$ ends up with $2 π$ radians, although it should have been geometry-free.
7	An alternative choice, such as of two interacting protons with masses $m_{p}$ , leads to another complete system of units, say UPS^′. In this case, we find that $M_{A}^{'} = 27.5$ GeV/ $c^{2}$ and ${(β_{G}^{'})}^{1 / 2} = 9.00 \times 10^{- 19}$ , but equation (18) is still valid, and connects $M_{A}^{'}$ with the original Planck mass $M_{p} = \sqrt{h c / G}$ . Also, the scaling $M_{A}^{'} / M_{A} = m_{p} / m_{e}$ holds precisely between the two systems of units; and the relation $M_{A} M_{A}^{'} = (m_{e} m_{p}) / α_{h}$ is exact as well. Finally, referring to the upcoming UPS results in Section 3.1 below, the relation $M_{A}^{'} = \sqrt{m_{t} m_{b}}$ holds to within 2.5% in the UPS^′, where $m_{t}$ and $m_{b}$ are the masses of the top and bottom quarks, respectively; thus, $M_{A}^{'}$ actively participates in the mass ladder of the UPS^′, just as $M_{A}$ does in the UPS mass ladder of Section 3.1 and Appendix A.
8	We knew that a rescaling of the Planck mass $M_{p} = \sqrt{h c / G}$ by some power of $β_{G}$ would produce an atomic mass. But we did not know which power is appropriate to use. Here, we have shown that the appropriate coefficient of $M_{p}$ is the G-M of $E_{grav}$ and ${(E_{elec})}^{- 1}$ , if we are scaling $M_{p}$ down to lower masses. If we are scaling up, then the $- 1$ exponent naturally moves on top of $E_{grav}$ in the G-M (see Section 3.4 below). In retrospect, these two G-Ms make perfect sense in a “fair” world dominated by the pervasive G-M averaging of pairs of fundamental physical quantities.
9	The realization that the vacuum also leaves unitless numerical imprints (in addition to its dimensional constants $ϵ_{0}$ , $μ_{0}$ , c, $Z_{0}$ ) is new, unexpected, and it may prove very important in future work. Sooner or later, we will have to investigate such imprints of the vacuum to the nuclear world, especially in the strong interactions and the so-called beta functions [28].
10	The charm and bottom quarks have masses of $m_{c} = 1270$ MeV/ $c^{2}$ and $m_{b} = 4180$ MeV/ $c^{2}$ , respectively [29]. At such high masses, something must be changing in the dynamics: for the ordered by mass triplet s-c-b, we find, to within a 1.6% accuracy, that $m_{c} = 2 \sqrt{m_{s} m_{b}}$ . We also find that the charm quark participates rather “reluctantly” in just one pure/unscaled G-M (Equation (28), referring to the compact triplet p-c- $τ$ ); and even that one is unusual, as it involves the proton mass $m_{p}$ .
11	No other available particle slots in the domain.
12	It will become apparent in Appendix A that the ratio 1.38 approximates $C_{F} = 4 / 3$ (to within a deviation of 3.5%), where $C_{F}$ is the quadratic Casimir charge of the SU(3) fundamental representation of the quark potential (equation (4.45) in Reference [31]).
13	Eliminate $m_{H}$ between equations (39) and (40).
14	Eliminate $m_{t}$ between equations (37) and (38).
15	There is no way for the Higgs boson or its decay products to reach down to the bottom quark mass by G-Ms because of the barrier set by v. When we run the v-W^± $. . .$ G-M toward lower masses, i.e., ${(m_{W^{\pm}})}^{2} / v$ , it reaches a lowest possible mass slot that is $≃ 2 π$ times higher than $m_{b}$ . There is no justification for adopting yet another scaling factor of $2 π$ , as it does not appear in any other G-M relation. Furthermore, introducing geometry in mass assignments does not appear to be an appropriate practice.
16	We cannot help but wonder—if A. Sommerfeld, W. Pauli, C. Jung, R. Feynman, et al. [32] became familiar with this result, would they show the same fascination for number 861?
17	See Ref. [33] and also article `https://en.wikipedia.org/wiki/Planck_units`.
18	The remaining choice, the G-M of $r_{b}$ and $r_{c}$ , would give an equivalent result, scaled by a different power of $α_{h}$ ( $L_{A}^{'} = L_{A} / α_{h}$ ), such that $r_{c} = \sqrt{L_{A} L_{A}^{'}}$ .
19	In the absence of microcosmic inertia, a delta-function impulse delivering specific energy $E / M > c^{2}$ to an elementary particle would easily achieve motion with speed $v > c$ .
20	Besides combining with G to produce the units of force and power in the cosmological and Planck systems, c does something else that is notable: it combines with $ϵ_{0}$ to produce a surprise unit for ohmic resistance: $1 / (ϵ_{0} c) = Z_{0} = h / e^{2}$ (see Section 2.1.1).
21	However, we are not aware of a system in which the geometry-dependent term $Z_{0} / (4 π)$ is introduced. The impedance of EM modes in waveguides and in ideal dielectrics is a multiple of $Z_{0}$ [34] that does not involve the factor of $4 π$ .
22	Note that even actual planetary orbits [42] and also theoretical orbits in the virtual Hooke potential [43] show G-M averaging in many of their properties [44,45]. The two types of elliptical orbits have fundamentally different centers, but this is not enough to suppress or modify the ubiquitous geometric averaging that is so obvious in the parameters of the two sets of ellipses.
23	Arithmetic averaging would favor the large constant, whereas harmonic averaging would turn the tables and clearly favor the small constant. Compared to G-Ms, either one of these extreme averages treats “unfairly” one or the other participant.
24	We also timidly attempted a preliminary calculation of the scaling between weak and strong interactions, as a ratio of energies $β_{W}$ (Equation (52)). It seems that such energy ratios/comparisons are the way to incorporate consistently the dimensionless constants into the UPS. We can then imagine a complete $UPS : = {c, Z_{0}, G, h, α_{h}, β_{G}, β_{W}, \dots; ϵ_{0}, μ_{0}, h}$ that includes geometry-free and a set of geometry-dependent units, along with relative $β$ -ratios $≪ 1$ of unitless constants.
25	From (35) and (36), we get $m_{b} = 30 m_{s} / K$ (#1). From (32), (38)-(40), and (36), we get $m_{b} = 30 K^{- 3 / 4} m_{μ}$ (#2). From (#1) and (#2), we get $m_{s} = K^{1 / 4} m_{μ}$ (#3). From (27), (#3), and (#2), we get $m_{μ} = 30 K^{- 5 / 4} m_{u}$ (#4). From (26), (#3), (#4), and (30), we get Equation (A4). Finally, from (34), (36), and (A4), we get Equation (A3). Equations (28), (31), (33), and (37) were not used.

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The Upgraded Planck System of Units that Reaches from the Known Planck Scale All the Way Down to Subatomic Scales

Abstract

1. Introduction and Motivation

1.1. Three Fundamental Systems of Units Under Consideration

1.2. Dirac’s Problematic Constant h and the Three Widely-Used Atomic Radii

1.3. Dimensionless Constants

1.4. Outline

2. The Building Blocks of the Upgraded Planck System

2.1. Dimensional Units

2.1.1. Constants Imposed by the Vacuum

2.1.2. Dirac’s Constant h

2.1.3. Newton’s Gravitational Constant G

2.2. Dimensionless Units

2.2.1. Fine-Structure Constant α h

2.2.2. Gravitational Coupling Constant α G

2.2.3. Relative Strength of Gravitational Coupling β G

2.3. Determining a New Atomic Mass Scale

3. Results within the UPS Realm

3.1. Subatomic Masses

3.2. The Planck Charge

3.3. A New Atomic Length Scale

3.4. Cosmological Scales and Some Ambivalent Superatomic Particles

3.5. Units Jumping to the Forefront

4. Discussion

4.1. Pairs of Fundamental Dimensional Units

4.2. The Varied Contributions of the Vacuum

4.3. Sparse Geometric Averaging in Nature

5. Lingering Issues, Future Prospects, and a Brief Summary

6. Highlights

6.1. Summary of Results

6.2. Critical Questions and Answers

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

Abbreviations

Appendix A. The Physical Meaning of Koide’s Lepton Constant

Appendix A.1. Physical Interpretation

Appendix A.2. Additional Koide-type Constants

Appendix B. A Universal Natural Law Discovered in Widely Distant Scales

Appendix B.1. Dimensional Analysis of Surface Densities

Appendix B.2. Physical Properties of Surface Densities

References

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1.2. Dirac’s Problematic Constant $h$ and the Three Widely-Used Atomic Radii

2.1.2. Dirac’s Constant $h$

2.2.1. Fine-Structure Constant $α_{h}$

2.2.2. Gravitational Coupling Constant $α_{G}$

2.2.3. Relative Strength of Gravitational Coupling $β_{G}$