Particles from Gravity

Updated at (6) (PDF) Particles from Gravity (Jul 24, 2021) (researchgate.net)

3 May 2021

Article type: Original paper

Title:?Particles from Gravity

Author: Warren Frisina?

Keywords:?gravitation, fundamental constants, foundations, special relativity, quantum to classical crossover, cosmology, astronomy, astrophysics?

Abstract

Rather than attempting to fit gravitation into the quantum format, quanta are derived from gravitation, i.e.?

me3 = k2 (H/G) (e/c)4, electron mass;?

?m03 = (H/G) (h/c)2, sub-proton mass;?

?γ2 = (n/2) αg-1, relativity/quantum bridge;?

Fg,1 = 0.021(c17hH2/G5)1/6, Strong force;?

where Hubble parameter, H, is in proposed acceleration units.??Particles are seen as gravitational sinks instead of sources, possibly avoiding renormalization and the overwhelming calculated mass of the quantum vacuum.?General mass density fall-off from visible astronomical objects and systems in non-expanding space, ρ = (3/4Π)(H/G)r -1 kg/m3, was derived from the unmodified Newtonian gravitation form (1/r2) applied to accelerated expanding space; the inverse linear form has been shown to describe, for instance, flat galaxy rotation curves.?

[EQUATION KEY: Equations are complete but didn't translate exactly. When a number follows a letter, the number is usually an exponent. When a letter follows a letter, the second letter is usually a subscript. When a number follows two letters, the number is an exponent of the combination of the two letters (which is a letter and a subscript). The letter "c" is always light speed. The equations are in proper form in a draft of this manuscript at,?https://www.researchgate.net/profile/Warren-Frisina-3/publications; this draft is titled "Another view of Newton's law of gravity."]

Declarations

? - No funding was received for conducting this study.

? -The author approves of the ethical standard.?

? -The author has no conflicts of interest to declare.?

1.?Introduction

Given the multifaceted nature of linking quanta and gravitation, it is possibly best to?begin with an unexplained observation to bring some focus.?The axes of quasars are aligned with the fine webwork of the large-scale structure of the Universe [1] [Figure 1].?

[See image over title, above]

Figure 1.?Large-scale structure of the?Universe, 2-D schematic emphasizing?a primary repeatable structural "cell," and quasar axes alignments with cell wall consisting of galactic?superclusters; vectors normal to individual quasar axes represent?the accelerated dark energy field.??

To reach the scale of this figure requires stepping from the galaxy, to clusters of galaxies, to galactic superclusters (gray ring, millions of light years thick), then finally to a basic repeatable "cell," the center of which is the large-scale void of dark energy, billions of light years across,?driver of the accelerated Hubble expansion.?These cells were better described as clashings of "soap bubbles" competing for dominance, in that the basically spherical voids can be thousands of times the thickness of the walls, and this analog provides a sense of proportion, variety and dynamism. The expansion begins at the supercluster level, so that as the great void is expanding?the gray wall is being stretched or constituents are undergoing mutual repulsion, as the wall is simultaneously increasing in length and thickening, as permitted by surrounding cells in three dimensions.??

????At this scale there is no viable means of aligning individual quasars with one another, nor?with the overall large-scale pattern.?Conventional gravitation is not helpful here, rather the cosmological constant predominates, which is not native to the formal mathematics of general relativity; classical Newtonian gravity, nor variants that differ from 1/r2 are also not appropriate.?Some other effect dominates, conventionally referred to as the unknown "dark energy," in that repulsion is the principal effect both along and across the perimeter, and radially from the void center.??

?????However, there are common elements in these?alignments -- the circumferences of adjacent large-scale voids, the tangents of which are parallel to the quasar axes.?Alternatively, the repulsive radial dark energy fields of these voids are common as well with these alignments; the quasar axes are normal to the radial accelerated expanding field of the void, indicated by typical directed field lines at specific quasars in this figure.

????Negative mass has been discussed to account for such repulsion [2].?But this is problematical in that two types of mass must be thoroughly accounted for; there is no experience at hand where such a negative mass repels a common particle of positive mass.?However, if it can be shown that all mass/energy is?fundamentally repulsive and only conditionally attractive, such that current evidence is not contradicted, then such mass is mathematically equivalent in classical Newtonian gravity (two negative masses) to two positive masses and should be mutually attracted when sufficiently close initially; only masses with opposite signs here should be mutually repulsive.??

Unifying physics has meant bridging two great pillars, quantum and relativity theories, the small and large scale.?More recently attention is given to minute vibrating entities in string or loop form, quantum gravity.??

??Quantum and relativity theories work quite well in their own domains, but they seem too different to be compatible.??An approach is to begin with the quantum format, which has a passive background; then try to fit gravitation to it.?Great theoretical strides have been made, but experimental confirmation is difficult.?The present approach reverses this procedure, and begins with relativity and spacetime -- gravitation -- which has an active background; then derives quanta.?Then both small and large scales can be treated in the more general background, or, at least, where the backgrounds merge.??Hopefully, the present offering will provide clues to facilitate a fully satisfactory unified theory.?The present piece is little more than an outline, under restricted conditions, and concentrates on the fringes of quantum and relativity theories, v→0 and v→c, in an effort to help bridge the two theories, or provide clues for dedicated quantum gravity work.????

????In Part 1 Newtonian gravitation -- the v→0 aspect -- is more naturally and effectively seated in accelerated expanding space, while fundamentally maintaining the classic 1/r2 relation, indicating locality rather than unexplained "action at a distance," and where a principal cosmological parameter, H km/s per Mpc, is adapted to the subatomic scale as H m/s2, and in conjunction with the gravitational constant is prominent in subatomic relations in the ratio H/G kg/m2, having a pressure-like character.??

????In Part 2 at the subatomic scale -- the v→c aspect -- Newtonian gravitation is paired with special relativity, but under this restricted condition to avoid the fundamental incompatibility of the two formulations, so that quanta may be derived, since complete formal general relativity is not compatible with Planck's constant, and special relativity is commonly employed at this scale.???

???????????????????????Part 1 -- The large scale

2.?Reseating Newtonian gravitation

Consider mass n having non-relativistic motion with respect to another mass m, where |m|>>|n|.?If mass centers are sufficiently far apart initially, they accelerate away from one another according to the accelerated Hubble expansion.[3]?Also, since general relativity reverts to Newtonian gravitation under these conditions, and given the equivalence of gravitational and acceleration effects?

G(-m)n / d2 = nA????????????????????????????????????????????????????(1)

where (-m) is the mass/energy of an arbitrary spherical section of space (equivalent to the gravitational field according to general relativity), with or without visible matter, undergoing observed accelerated expansion, n is a common visible particle in the visible matter about the periphery of this section of space, and where A is the acceleration of n with respect to the center of mass of m due to this hypothetical Newtonian form of gravitational repulsion at the scale of galactic superclusters and above.?Negative mass has been discussed, but not accounted for with positive mass, as mentioned.[2]?Presently, all mass/energy will be shown to be negative, so that it may be defined positive.??

?????If r is the radius of mass m, and u is the radius of n, where r >> u, distance d between mass centers may be replaced by r,?thus

A= G(-mx) / rx2.???????????????????????????????????????????????????????????????????

Calibrating A:?given observed accelerated expansion at this scale, let mx ≈ 1015 Solar masses, the local supercluster;[4] this mass encompasses some 100 million light years (ly), so that let rx ≈ 50 x106 ly, then

-mx / rx2?=?A/ G?????????????????????????????????????????????????????(2)????????????????????????????????

where (-mx)?is the total mass/energy -- visible, dark energy and dark matter -- emcompassed by said volume of radius rx [Note 1], and where A is the acceleration of the local supercluster (equivalent to that of mass n at the periphery of the volume),?

A ≈ 6 × 10-13 m/s2;???????????(prediction)?????????????????????????????(3)???????????????????????????????

the value for different scales would likely differ somewhat (see Section 4).??

2.1?Electron mass??

Examining Equation 2 at the subatomic scale, let mx be that of the electron, me,?and rx be the classical electron radius, ke2/mec2 (derived by noting the coulomb energy necessary to hold two point charges, each equivalent to electron charge, e, a distance apart equivalent to the classical electron radius), so that

me3 = -k2(A/G)(e/c)4????????????????????????????????????????????????(4)???????????????????????????????????

the order of electron mass magnitude in terms of natural parameters, substantiating Equations 1, 2 and 3.??

2.2?Negative mass??Gravitational sinks; dark energy; locality; confirmation of Bell's theorem

?As to the negative mass in Equation 4: considering the foregoing, ordinary particles might be seen as?gravitational sinks, rather than strictly local sources, with mass/energy seated in the surrounding voids as proposed primal negative (repulsive) gravitational energy -- conventionally referred to as dark energy -- commonly said to cause the observed accelerated universal expansion.?In that the particle as sink has the indicated negative sign of the adjacent expanding negative mass void, to which the particle as sink, n, is in direct physical contact, establishing locality, and that it cancels, all mass/energy in the Universe might be characterized as negative (see also Equation 13 and Section 11); only one type of mass/energy is retained, in conformance with observation.??Then all mass/energy may be defined as positive for convenience.?In that all mass/energy would be in effect merged, "non-locality" might be considered a misnomer.[5]?

?2.3?Mechanism of "attractive" gravitation??

When two ordinary particles as gravitational sinks of the same sign (both positive or both negative,dependent on said difinition, either set consistent with the classical Newtonian gravitational form) are sufficiently close (v < c domain), their incoming field lines from all directions are partially shadowed by one another, and the particles appear to be attracted to one another, while it is proposed that they are pushed together by their respective accelerated expanding voids;?gravity is seen here as fundamentally repulsive and apparently attractive, depending on scale of observation (or relative lack of expanding space among particles).?

2.4?Falsifiability?

Newly created particles as gravitational sinks should immediately connect to large-scale space, while newly created particles as sources should have gravitational fields expanding at finite speed, making the sequence falsifiable in principle.???

??

3.?Dark matter; or Mach's principle and Newtonian gravitation?

?Regarding alternate proposals, Primordial Black Holes and other Macroscopic Compact Halo Objects are less likely as dark matter candidates,[6] while Weakly Interacting Massive Particles in a negative experiment still remain viable.[7]??

?????In place of dark matter there might be a commensurate incoming stabilizing gravitational field from large-scale space (Mach's principle).?Then "dark matter" could be dark energy at a smaller scale, in that field lines from large-scale voids (billions of light years scale) point from void centers to galactic superclusters at the periphery (millions of light years scale, gray ring) to ever smaller scales. [Figure 1]????

?This might be verified with possible different patterns of light deflection for gravitational lensing due to hypothesized dark particulate matter possibly only a few times galaxy visible diameter, and proposed field differences, the latter being considerably more extensive and uniform suggested by Equation 2 or 5a -- an order of magnitude larger than the visible disc regarding Andromeda.[8]?From Equation 2 gravitational mass/energy should increase (from a galaxy toward sources in large-scale space) with the square of the radius of an astronomical body beyond the visible portion, while the volume of space increases by the cube of the radius.?This indicates that the apparent mass density of the body (including the stabilizing incoming field) should slowly decrease linearly with increasing radius,?

ρ = (3/4Π)(A/G)?r -1???????????????????????????????????????????????(5)????????????????????????????????

ρ ≈?0.002 r -1?kg/m3?????????????????????????????????????????????(5a)???????????????????????

in spherical symmetry about an astronomical body, i.e. galaxy, in the vicinity the local galactic supercluster; acceleration, A, should be recalibrated for constituants of other supercluster systems.?Similarly, Equation 2 is only applicable where this parameter is calculable, i.e. not applicable to the observable Universe itself, where this parameter might vary significantly from region to region, and/or scale variations, as about the "great attractor" and commensurate surrounding void. (See Section 4.)?It has been shown that the inverse linear equation form yields flat galaxy rotation curves.[9]

4. Is supercluster acceleration, A, related to the Hubble parameter?

Since an accelerating Universe implies a varying Hubble parameter, in addition to expansion effects, and observations are averaging to H0 = 70 km/s per Mpc [10] [Note 2], consider this rough estimate for uniformly accelerated motion?

H ≈ (v2-u2) / 2d??=?[(70 km/s)2- 02] / 2 Mpc?=?7.9 x 10-14 m/s2

where u=0 is with reference to an observer on Earth, and where H is about the order of acceleration A.?Then let H replace A,

?-mx?/ rx2 = H / G??????????????(prediction)????????????????(6)???????????????????

where H ≈ 6 x 10-13 m/s2 is a first approximation of the Hubble parameter in acceleration units.?

Alternatively, a similar calculation employing two means of measuring the Hubble parameter, referred to as the "Hubble tension" [11], with v ≈ 73 km/s, for the closer universe, and u ≈ 68 km/s for the distant universe, yields H0 = 8.5 x10^-14 m/s^2, the order of the above calculation. Considering the Hubble parameter as an acceleration could relieve this astronomical "tension," in that both measurement means are correct, and pointing to an accelerating Hubble parameter. This is logical in that the Universe is observed to be accelerating.

This would suggest two cosmological parameters appearing at the subatomic scale in the ratio H/G kg/m2 with pressure-like units -- Equation 4, 13 and 21c, where A = H .?This H would be analogous to the cosmological constant.?When refined, parameter H could be universal, in comparison with the relative locality of parameter A. Actually, the two calculations above might be considered the refinement, implying that the Hubble parameter is an acceleration that is everywhere and everywhen the same in the visible universe,

where H = 8.5 x 10^-14 m/s^2.

5.?Large-scale structure and dynamics

Equation 6 suggests a negatively curved Universe when scaled up, and A or H is only of the order of 10-14 m/s2 so that curvature is possibly imperceptible observationally regarding the large scale.?Since there is not 100% confidence that the Universe is perfectly flat, this possibility, and prediction, remains open.[12]

????When two large-scale voids are overlapped such that there is little or no visible matter between them,?

Fg?=?G(-m1)(-m2) /d2?=?H2r12r22 / G(r1+r2 -a)2??

where Fg is the force of apparent gravitational attraction, and where masses are given by Equation 6, d is the distance between mass centers, and a is overlap length; and where the relation?is mathematically equivalent to the classical Newtonian form for the Solar system scale, indicating apparent mutual attraction, subject to observational confirmation.?However, the mechanism is also physically repulsive in that the outward pointing fields of each mass cancel in the overlapped region, and expand otherwise, so that the result should be a single larger cell, subject to confirmation by observing different pairs in various stages of overlap.??

?????????

?????????????????????????????Part 2 -- The Small Scale??????

6. Relativistic kenetic energy [Note 3]

For constant velocity v→c, after acceleration,?so that special relativity is appropriate, let

E = xc2 + (y-x)c2???

?where?xc2, rest mass/energy,?is undefined and absent until derived??

?(y-x)c2 = particle precursor relativistic kenetic energy without a rest mass assumption, i.e. gross (non-virtual) gravitational field fluctuation (assuming space as equivalent to the gravitational field, from general relativity)??

y = precursor to particle total mass???????????????????????????????????????????????????????????????????????????????????????????????????????????

x = precursor to particle rest mass.[Note 4]

Defining,???????????????????????

Es ? (y-x)c2?????????????????????????????????????????????????????????????????????

y = x (1- v2/c2)-1/2??????????????????????????????????????????????????????????????

?y = -ix (v2/c2 -1)-1/2;?

combining,

Es = xc2 [(1- v2 / c2)-1/2?-1]????????????v→c?(all unspecified v)??????????????(7)??????????????

Es = -xc2 -ixc2(v2 / c2 -1)-1/2 ;??????????????????????????????????????????(7a)????????????????????

where constant velocity, v, in Equation 7 is that of a particle precursor relative to a similar particle precursor in the same mathematical domain; in Equation 7a the reference is c, the Cosmic Microwave Background (see below).?While the expressions are equivalent, they are not identical.?These two hypothetical particle precursors represent the energies of "spacetime waves," Es, just prior to particle rest mass establishment, to contrast with classical gravitational waves where v = c.??

When the ordinary form of the relativistic velocity term is apparent, v < c; when the complex form is seen, v > c, which does not refer to tachyons; "v > c" refers to "v different from c in expanding space" or (possibly imperceptable) negative curvature -- not v greater than c; similarly "v < c" refers to "v different from c in non-expanding or contracting space" or positive curvature -- not v less than c.?For all unspecified v, v→c from mathematically real or complex domains. [Note 5]

????Similarly for momentum,?

ps ? (y-x)v????????????????????????????????????????????????????????????????????????

ps = xv [(1- v2 / c2)-1/2?-1]????????????????????????????????????????????(8)

ps = -xv -ixv (v2 / c2 -1)-1/2.???????????????????????????????????????????(9)

7. Planck's constant, de Broglie wavelength, sub-proton mass

Consider????????????????????

?λs = λ (1- v2/c2)1/2????????????????????????????????????????????????(10)

?λs = i λ (v2/c2 -1)1/2???????????????????????????????????????????????(10a)

where λs is a particle precursor wavelength, λ to be determined (as with x).?Spacetime wave frequency would be?

Vs = v / λ s??????????????????????????????????????????????????????(11)

Substituting Equations 10 or 10a into Equation 11, Vs?increases with v→c, so that frequency appears directly related to energy,

K = Es / Vs??????????????(v→c)????????????????????????????????????(12)???????????????????????

K = x λ v.???????????????????????????????????????????????????????(12a)????????

Since precursor velocity is near light speed here there could be sufficient relativistic kenetic energy to produce a non-zero rest mass particle.?It might be suspected that λ is a de Broglie wavelength, if K is Planck's constant, h, and x is at least a rest mass.?

????In Equation 2, let mx = x, rx = K/xv from Equation 12a.?Then?

?mo3 = -(H/G)(h/c)2??????????????????????????????????????????????(13)

suggesting a stable bound light quark.[13]?It might now be indicated that in Equation 12a precursor mass, x, is now at least a rest mass, mo; K = h; and λ is a de Broglie wavelength,

λ=?h/mv?????????????????????????????????????????????????????(12b)

and that?A = H of Part 1.??The negative particle mass is consistent with that of the electron, so that all mass of the principal matter particles (as well as dark mass/energy, i.e. Equation 2, and electromagnetic radiation, Section 10 and 11) is negative, and may be defined as positive, as mentioned.

?8.?Relativity/quantum bridge,?gravitational coupling constant

Provisionally, at the subatomic level only,????

?Fs = dps / dt?????????????????

?Fs = xA { [ (1- v2 / c2)-1/2 -1 ] + (v2 / c2)(1- v2 / c2)-3/2}?????????????????????(14)

where ps is Equation 8, and x is at least a rest mass, m.??Equating gravitational and inertial forces, Fg?= Fs, where Fg is Newtonian gravity and Fs is Equation 14.?To balance the relation Fg is multiplied by the velocity term of Equation 14, so that provisionally

Gs = G {[ (1- v2/c2)-1/2 -1] + (v2/c2)(1- v2/c2)-3/2 }????????(subatomic level only, v→c)??

?and that the vis viva equation becomes

?v2 = Gs (m1 + m2) / r??????????????????????????????????????????????(15)

where in a planetary model v is the tangential velocity of a quark or nucleon m2 relative to quarks or nucleon(s) m1, and r is the distance between mass centers.?

????As with electrons in classical (Bohr) atoms let

?n λs / 2 = 2 Π r???????????????????????????????????????????????????(16)

where λs is wavelength of a spacetime wave, accounting for the fundamental wavelength, n = 1,2,3 ...?

?It can be shown that?????????

Es /ps = c2/v.?????????????????????????????????????????????????????(17)

Combining Equation 11, 12 and 17, where v→c????

?λs = h / ps??????????????????????????????????????????????????????(18)

which is equivalent to a deBroglie wavelength at v→c?(Equation 12b).???Combining Equations 16 and 18,

ps r = (n/2)??????????????????????????????????????????????????????(19)

expressing angular momentum; when n = 1,3,5 ..., there is an indication of fermions, while n = 2,4,?bosons.??Eliminating r in Equation 15 and 19,?

Gs ps / v2 = (n/2)??/?(m1 + m2).

Let????????????

U1 = { [ (1 - v2/c2)-1/2 -1] + (v2/c2)(1 - v2/c2)-3/2 }

U2 = [ (1 - v2/c2)-1/2 -1 ]

so that

U1 U2/v = (n/2)?c / G(m1 + m2)m2

which reduces to?

?(1 - vn2/c2)-1 = (n/2)?c / G(m1 +m2)m2?????????(v→c)??????????????????(20)

?γ2 = (n/2) αg-1??????????????????????????????????????????????????(20a)

where the left side has a relativistic character, gamma squared, and the right a quantum -- the inverse of the gravitational coupling constant, αg, when the mass term is a proton squared.?Plots of the complete and reduced velocity functions, f(v) vs. v, remain close to zero until v→c, where they overlap.??Equation 20 can be termed the "relativity/quantum bridge."

9.?Strong force?

For an isolated particle precursor the force maintaining particle integrity might be??

Fg?=?Gs m1m2 /r2??????????(subatomic level only)?

Fg = [ G (m/2)(m/2) / (h/mc)2 ]?{ [ (1 - v2/c2)-1/2 -1] + (v2/c2)(1 - v2/c2)-3/2 }?

Fg = [ G (m/2)(m/2) / (h/mc)2 ]??[ 1.3 (1 - v2/c2)-3/2 ]???????????(v→c)?????

where the velocity term in the reduced version overlaps that in the original expression in a plot at v→c.?The distance term is from?Equation 12b.??Combining the immediately above and Equation 20,

Fg,n = [ (n/2)1/2 ?αg-1/2 + (n/2)3/2??αg-3/2 ]??[ Gc2/4h2 ] m4????????????????????(21)

Fg,n = (n/2)3/2 (0.058) (c7/Gh)1/2 m.???????????????????????????????????(21a)?

If?mass m is that of Equation 13, ?

Fg,1 = [ (1/2)1/2 ?αg-1/2 + (1/2)3/2 ?αg-3/2 ]??(1/4) [ (G2Hc4) / h4 ]1/3?????????????(21b)

Fg,1 = 0.021 ( c17 h H2 / G5 )1/6???????(prediction)???????????????????????(21c)???????????????????

the force perhaps providing quark integrity, and the Strong force.??

?

10.?Electron charge?

Let???????Fg,1??≈>?Fcoulomb.???????(lemma)

In Equation 21a when m is the Planck mass and n = 1, then the?charge, q, of the above is about the unit charge,?

10-18 coul ≈> q ≈> e??????????????(proof)

also substantiating Section 9.??

11.?Suggestion of photons and Weak force

There appears to be a similarity between spacetime waves and photons; from Section 7,???????????????????????????????

Es = hVs?????????????????????(v→ c)?????????????????????????????????????????????????????????????????????????????????????????????????

and commonly E = hV, though not yet established in the present context.?Thus????????????????????

Es / Vs = E / V = h.????????????????????(provisional)?????????????????????(22)

Let????Es'?=?-i h V (v2/c2 -1)-1/2???????(preliminary)??????

be a photon spacetime wave of velocity v→c (i.e. relative to the Cosmic Microwave Background).?However, spacetime waves for non-zero rest mass particles were developed from pure relativistic kenetic energy in a curved or gross (non-virtual) fluctuating field, where it was not assumed the particle had any independent existence beforehand; therefore

Es' = -hV -i hV (v2/c2 -1)-1/2????????????????????????????????????????????(23)

would seem more proper, according to the relativistic kenetic energy of Section 6. It can be shown that

Es / Vs?=?Es' / Vs'?=?h λV / v,?????????(v→c?in complex domain, v = c+)

and????λV =?c+,?????????????????????????????????????????????????????????????????????????????????????????????????????????

so that a light wave in highly curved expanding space is suggested.??In Equation 23, at v = c+, Es'?might have sufficient energy to produce a W particle, conventional agent of the Weak force.??Also, in flat space, v→0 (again, relative to the Cosmic Microwave Background) in Equation 23, then?

Es' = -2hV = -2E??????????????????????????????????????????????????????????

substantiating Equations 22 and 23.??

12. Concluding remarks?

While certainly not having the scope of string or loop quantum gravity, applying the conditions indicated to unmodified Newtonian gravitation and special relativity, natural parameters are adapted, affirmed, and derived, masses in terms of natural parameters are derived for the principal particles, and the four forces are outlined.?Hopefully clues have been provided to facilitate the principal quantum gravity approaches, such as suggesting lower energy experiments for confirmation so that current accelerator facilities are adequate, since, for example, the proposed the Strong force is theoretically substantiated, and possibly suitable in this regard.??

Also, the Hubble tension might be relieved by considering the Hubble parameter as an acceleration that is the same everywhere and everywhen in the visible Universe.

???????

Received:

Revised:

Published online:

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Footnotes?

Note 1. Mass (-mx) is treated as a particle with all mass concentrated at a point at the mass center, a distance rx from mass n.

Note 2.?The two principal methods of experimentally determining the Hubble parameter have been diverging to a difference of about 6 km/s per Mpc.??

Note 3.?The common general formula, E2 = (pc)2 + (m0c2)2, is not helpful in the sequence in that zero rest mass particles are treated separately in Section 11; also, it assumes flat spacetime; and relativistic momentum is treated separately as well.??

Note 4.??Rest mass/energy not being defined does not necessarily imply x = 0.?Rest mass/energy is a fixed quantity; x is not yet fixed at any quantity.?

Note 5.?Generally, the complex energy form is more in keeping with expanding space in that it has the same sign as the electron of Equation 4 as developed in this space, and subsequently Equation 13 and Section 11; also there is no velocity restriction as with the real mathematical form in large-scale space; velocity, v, is with respect to c, the Cosmic Microwave Background, not with another particle in the same complex domain;?and the form does not break down regardless of the size of the Universe regarding the empirical Hubble law; the restriction is that the lower limit is c. This is academic in this context, however, in that primarily velocity is restricted to v→c, or v = c+ in this case.