An “ab initio” Model for Quantum Theory and Relativity ()

Sebastiano Tosto^{}

ENEA Casaccia, via Anguillarese 127, Roma, Italy.

**DOI: **10.4236/jamp.2021.97114
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ENEA Casaccia, via Anguillarese 127, Roma, Italy.

The paper introduces a theoretical model aimed to show how the relativity can be made consistent with the non reality and non locality of the quantum physics. The concepts of quantization and superposition of states, usually regarded as distinctive properties of the quantum world, can be extended also to the relativity.

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Tosto, S. (2021) An “ab initio” Model for Quantum Theory and Relativity. *Journal of Applied Mathematics and Physics*, **9**, 1718-1807. doi: 10.4236/jamp.2021.97114.

1. Introduction

The quantum theory and the relativity have stimulated influential ideas and experimental efforts to investigate and understand a huge number of natural phenomena from atomic to cosmic scale [1] [2]. However, with space ranges spreading from ~10^{−}^{18} m to ~10^{26} m by about 44 orders of magnitude, is comprehensibly problematic the attempt to unify in the frame of a unique theory the whole variety of related natural phenomena. Yet, is symptomatic the fact that similar difficulties often arise even in formulating a more selective class of specific physical problems. In the case of the relativity, for example, something relevant should be still missing even at the mere cosmic scale; despite the great amount of its previsions and discoveries, remain problematic crucial topics like the progressive acceleration of universe expansion, the MOND (modified Newton), the dark matter and dark energy. A possible hint to overcome these difficulties is to identify an appropriate background of ideas that integrate or modify the preexisting ones; for example, at the quantum level, the major problem of the relativity is its link to the non-reality and non-locality of the quantum theory [3]. Also, the “handwritten” cosmological constant reluctantly introduced by Einstein after the Hubble experimental hint, is a further example of necessary revision of the general relativity even in its most representative cultural frame *i.e.* the cosmology. Is reasonable the suspect that focusing greater attention on the existing conceptual background is not an additional difficulty but a possible solution? To provide a contribution to this problem, the paper [4] has introduced an operative definition of space time

$\frac{\hslash G}{{c}^{2}}$ (1.1)

that implies as a corollary the statistical formulation of quantum uncertainty

$\delta x\delta {p}_{x}=n\hslash =\delta \epsilon \delta t,$ (1.2)

being *n* an arbitrary integer. The purpose of this paper is to examine the physical information deductible by the definition (1.1): as it merges
$\hslash $ and
$G/{c}^{2}$, in principle, it seems reasonably valuable for quantum and relativistic implications. The most intuitive way to convert (1.1) into an effective equation allowing successive calculations is to introduce explicitly its physical dimensions

$\frac{\hslash G}{{c}^{2}}=\frac{lengt{h}^{3}}{time},$ (1.3)

which also calculates

$\frac{{\hslash}^{2}G}{{c}^{2}}=8.2\times {10}^{-96}\text{\hspace{0.17em}}\text{J}\cdot {\text{m}}^{3}\text{\hspace{1em}}\frac{\hslash}{time}=\u03f5.$ (1.4)

The second (1.4) linked to (1.2) defines
$\delta \epsilon ={\u03f5}^{\prime}-{\u03f5}^{\u2033}$, where
${\u03f5}^{\prime}$ and
${\u03f5}^{\u2033}$ are two arbitrary boundary energies, whereas the right hand side implies
$\delta \epsilon =n\hslash \omega $ with
$\omega =\delta {t}^{-1}$ and
${\epsilon}^{\prime}\le n\hslash \omega \le {\epsilon}^{\u2033}$ ; this means that
$\hslash \omega $ is an energy in the range of allowed values
$n\hslash \omega $ falling within
$\delta \epsilon $ with
$n=1,2,\cdots $. The first equality (1.2) reads
$2\pi \delta x=nh/\delta p=n\delta \left({\lambda}^{-1}\right)$, having put
$\delta x={x}^{\prime}-{x}^{\u2033}$ and
$h\delta \left({p}^{-1}\right)=h/{p}^{\prime}-h/{p}^{\u2033}={{\lambda}^{\prime}}^{-1}-{{\lambda}^{\u2033}}^{-1}$ coherently with
$\delta \epsilon $ ; then to any
${x}^{\prime}\le x\le {x}^{\u2033}$ corresponds
$h/{\lambda}^{\prime}\le p\le h/{\lambda}^{\u2033}$ with
$p=h/\lambda $ and that
$2\pi x=n\lambda $ as well. Hence (1.2) summarizes contextually three fundamental statements of quantum physics. Now the crucial task of the present physical model is how to define specifically *length* and *time* of (1.3) to infer physical information. To this aim, (1.4) will be implemented via some fundamental parameters of the universe [5]

$\begin{array}{l}{t}_{u}^{estim}=4.35\times {10}^{17}\text{\hspace{0.17em}}\text{s},\text{\hspace{1em}}{r}_{u}^{estim}=4.35\times {10}^{26}\text{\hspace{0.17em}}\text{m},\\ \Lambda =1.9\times {10}^{-35}\text{\hspace{0.17em}}{\text{s}}^{-2},\text{\hspace{1em}}{H}_{u}=2.2\times {10}^{-18}\text{\hspace{0.17em}}{\text{s}}^{-1}\mathrm{,}\end{array}$ (1.5)

*i.e.* the estimated age and radius of the Universe, from now on quoted with shortened notation
${t}_{u}$ and
${r}_{u}$, the Einstein cosmological factor
$\Lambda $ and the Hubble constant
${H}_{u}$. Two quantities of interest are the mass
${m}_{ob}$ detectable in the universe counting the stars only [5] and the critical density
${\rho}_{cr}$ of Friedman equations [6]

${m}_{ob}=3\times {10}^{52}\text{\hspace{0.17em}}\text{kg},\text{\hspace{1em}}{\rho}_{cr}=\frac{3{H}_{u}^{2}}{8\pi G}=8.6\times {10}^{-27}\text{kg}/{\text{m}}^{\text{3}}\mathrm{.}$ (1.6)

Of course these numerical data must be intended as today’s values. The mass ${m}_{ob}$ is interesting, although approximated by defect for two reasons: because it refers to stars only and because it concerns by definition stars whose light has in fact reached us during the life time of the Universe.

The present theoretical model implements systematically uncertainty ranges to calculate quantum and relativistic quantities according to the logical step “local values $\to $ uncertainty ranges”. From a formal point of view this statement can be acknowledged reminding the standard concept of measure errors: just as no one trusts the reliability of a single value measured in its experimental error bar, likewise (1.2) waive the signification of a local dynamical variable in its uncertainty range. Yet the true physical meaning of this replacement is one among the crucial points of the model, as it will be more thoroughly shown below; some examples of calculated results are also reported in the Section 6 to confirm the concepts exposed in the Sections 1 to 5. The uncertainty ranges are defined via the standard notation

$\delta \left(any\text{\hspace{0.17em}}function\text{\hspace{0.17em}}f\right)={f}^{\u2033}-{f}^{\prime},$ (1.7)

being
${f}^{\u2033}$ and
${f}^{\prime}$ the range boundaries defined by two arbitrary values allowed to the concerned function; these values are arbitrary, unknown and unknowable by definition of quantum uncertainty. In general, both of them can be variables or constants. As
$\delta x$ implies *x* variable in an appropriate interval of values
${x}^{\prime}\le x\le {x}^{\u2033}$, the dynamical variable *x* is assumed to take a range of random values between arbitrary boundaries
${x}^{\prime}$ and
${x}^{\u2033}$. Also, since the symbol
$\delta $ introduces the meaning of change by definition, it also indicates the differential of *f* through the formal identity
${f}^{\u2033}={f}^{\prime}+\delta f$ ; then the further identity

${f}^{\u2033}={f}^{\prime}+\frac{\delta f}{\delta x}\delta x,\text{\hspace{1em}}\delta x={x}^{\u2033}-{x}^{\prime},\text{\hspace{1em}}{x}^{\prime}\le x\le {x}^{\u2033}$ (1.8)

introduces the ratio
$\delta f/\delta x$ that in turn takes physical meaning under appropriate conditions. A further way to implement the ranges is that already highlighted about
$\delta \epsilon $, *i.e.*
${n}^{\prime}\hslash \omega \le n\hslash \omega \le {n}^{\u2033}\hslash \omega $, which means

${n}^{\prime}\le n\le {n}^{\u2033}\mathrm{,}$ (1.9)

being of course
${n}^{\prime}$ and
${n}^{\u2033}$ arbitrary and unknown integers. In the following the shortened notations
$\delta {x}^{2}$ and
$\delta {\left(x\right)}^{2}$ mean respectively
${\left(\delta x\right)}^{2}={\left({x}^{\u2033}-{x}^{\prime}\right)}^{2}$ and
${\left({x}^{\u2033}\right)}^{2}-{\left({x}^{\prime}\right)}^{2}$. Eventually note that in principle both signs are allowed for any range; for example nothing hinders that
$\delta x=x-{x}_{0}$ is defined by
${x}_{0}\lessgtr 0$, so that
$\delta x=x\pm {x}_{0}=\pm \left|x\pm {x}_{0}\right|$ being both *x* and
${x}_{0}$ arbitrary. Sometimes in the following text a given result is obtained more than once in different contexts: this is not a redundant repetition, rather it must be intended as a check confirming that all conceptual steps progressively exposed are consistently linked each other. Despite the agnostic way to introduce (1.2), the remainder of this paper is able to formulate a self consistent theoretical physical model. The text is exposed in order to be as self contained as possible.

2. Preliminary Considerations

This section introduces some considerations having general character of straightforward corollaries of (1.1) to demonstrate that this definition of space time is physically sensible. All concepts introduced below are listed sequentially without calculations, while emphasizing their physical meaning; the validity of the various formulas inferred through the model will be concerned in the next section 6.

2.1. Energy and Energy Density in the Space Time

Implement first the dimensional analysis of (1.1), (1.3) and (1.2) defining ${\hslash}^{2}G/{c}^{2}=\u03f5{\mathcal{l}}^{3}$ and $\eta =\u03f5/{\mathcal{l}}^{3}$, where $\u03f5$ stands for energy and $\mathcal{l}$ for length to introduce the concept of energy density $\eta $. Then multiplying and dividing side by side these two equations one finds

$\frac{{\hslash}^{2}G/{c}^{2}}{\eta}=\frac{\u03f5{\mathcal{l}}^{3}}{\u03f5/{\mathcal{l}}^{3}}={\mathcal{l}}^{6},\text{\hspace{1em}}\frac{{\hslash}^{2}G}{{c}^{2}}\eta ={\u03f5}^{2},$ (2.1)

having identified $\u03f5$ pertinent to $\eta $ with that defined by the second equation. Equation (2.1) yields two equations. Owing to (1.4) the first one is

$\eta ={\left(\frac{\u03f5c}{\hslash}\right)}^{2}{G}^{-1}={\left(\frac{c}{\tau}\right)}^{2}{G}^{-1},\text{\hspace{1em}}\tau =\frac{\hslash}{\u03f5},$ (2.2)

according which the value of
$\eta $ depends on that of
$\u03f5$ via fundamental constants only: *i.e.*
$\eta =\eta \left(t\right)$ if
$\u03f5=\u03f5\left(t\right)$, whereas instead
$\eta =const$ if
$\epsilon =cons{t}^{\prime}$. It is necessary to make (2.2) consistent with (1.2), *i.e.* to regard
$\u03f5$ as an uncertainty range
$\u03f5-0$ ; as the boundaries of any range are arbitrary, in this particular case it means considering the energies
$0\le {\u03f5}^{\prime}\le \u03f5$ enclosed in the given boundaries. So
$\eta =0$ would be the deterministic value of energy density related to
$\eta =0$, whereas instead the actual value of
$\eta =\eta -0$ corresponds to the range size
$\u03f5-0$. Also, the dimensional relationships (2.1) yield the following equations

$\delta \mathcal{l}={\left(\frac{{\hslash}^{2}G/{c}^{2}}{\eta}\right)}^{1/6}={\left(\frac{\hslash G}{{c}^{2}}\delta t\right)}^{1/3},\text{\hspace{1em}}\eta =\eta \left(\delta t\right),\text{\hspace{1em}}\delta t\ne 0,\text{\hspace{0.05em}}$ (2.3)

being
$\delta t$ an appropriate time range corresponding to the space range
$\delta \mathcal{l}$ ; the second equality is nothing else but the definition (1.3) of space time multiplied by
$\delta t$. Let
$\delta t$ be equal in general to
$\tau +t-{t}_{0}$, being the reference time
${t}_{0}$ an arbitrary constant and
$\tau $ a further time constant. From a formal point of view, this is by definition the uncertainty time lapse
$t-{t}_{1}$ with *t* variable and
${t}_{1}={t}_{0}-\tau =cons{t}_{t}$. The corresponding space range
$\delta \mathcal{l}=\mathcal{l}-{\mathcal{l}}_{1}$, with
${\mathcal{l}}_{1}=cons{t}_{\mathcal{l}}$, is such that dividing both sides by
$\delta t$ one finds

$\frac{\delta \mathcal{l}}{\delta t}=\delta \stackrel{\dot{}}{\mathcal{l}}={\left(\frac{\hslash G}{{c}^{2}}\right)}^{1/3}\delta {t}^{-2/3}$ (2.4)

and thus also

$\delta \mathcal{l}=\pm \sqrt{\frac{\hslash G}{\delta \stackrel{\dot{}}{\mathcal{l}}{c}^{2}}};$ (2.5)

clearly this result merely rewrites (1.3) as
$\delta {\mathcal{l}}^{3}/\delta t$. Note that
$\delta \stackrel{\dot{}}{\mathcal{l}}$ has physical dimensions of velocity; so, regarding
$\delta \stackrel{\dot{}}{\mathcal{l}}\le c$, at any *t* (2.5) yields in particular

$\delta {\mathcal{l}}_{min}=\pm \sqrt{\frac{\hslash G}{{c}^{3}}}=\pm {\mathcal{l}}_{P}\mathrm{.}$ (2.6)

The Planck length appears to be the smallest space range physically inferable through the definition (1.1) of space time, whereas the Planck time and energy can be nothing else but ${t}_{P}={\mathcal{l}}_{P}/c$ and ${\u03f5}_{P}=\hslash /{t}_{P}$ according to (1.4), whence the Planck mass ${m}_{P}={\u03f5}_{P}/{c}^{2}$ too by dimensional reasons. Instead at $t={t}_{0}$ by definition $\delta t\equiv \tau =cons{t}_{t}$ and thus $\delta \mathcal{l}=cons{t}_{\mathcal{l}}$ ; so (2.4) turns into

${\frac{\delta \mathcal{l}}{\delta t}|}_{t=\tau}=\frac{cons{t}_{\mathcal{l}}}{cons{t}_{\tau}}={\left(\frac{\hslash G}{{c}^{2}}\right)}^{1/3}{\tau}^{-2/3}={\left(\frac{\hslash G}{{c}^{2}{\tau}^{2}}\right)}^{1/3}.$ (2.7)

The left hand side introduces a velocity by definition constant. In particular, $\tau ={t}_{P}$ is consistent with the ratio of constants $cons{t}_{\mathcal{l}}/cons{t}_{t}=c$, in analogy with ${\left({\u03f5}_{0}{\mu}_{0}\right)}^{-1/2}$ of the classical electrodynamics.

2.2. Classical Newton law

Since $\u03f5c/\hslash $ in (2.2) has physical dimensions of acceleration, write thus

$acceleration=\frac{\u03f5c}{\hslash}=2.9\times {10}^{42}\u03f5\text{\hspace{0.17em}}\text{m}/{\text{s}}^{\text{2}}\mathrm{;}$ (2.8)

then it must be possible to define via dimensional reasons also the related velocity, force and energy

$\delta t\frac{\u03f5c}{\hslash}=velocity,\text{\hspace{1em}}m\frac{\u03f5c}{\hslash}=force,\text{\hspace{1em}}m\delta \mathcal{l}\frac{\u03f5c}{\hslash}=energy;$ (2.9)

both $\delta t$ and $\delta \mathcal{l}$ are arbitrary because they are introduced without hypothesis or conceptual constrain. Also,

$\frac{\u03f5c}{\hslash}=\frac{\delta v}{\delta t}=\stackrel{\dot{}}{v},\text{\hspace{1em}}v=\frac{\delta \mathcal{l}}{\delta t},$ (2.10)

whereas holds the dimensional relationship
$m\delta \mathcal{l}\u03f5=mas{s}^{2}lengt{h}^{3}/tim{e}^{2}$. As this latter defines in turn at the right hand side the physical dimensions of
$G\times mas{s}^{3}$, it is possible to write
$mass\delta \mathcal{l}\u03f5=Gmas{s}^{3}$ *i.e.*

$\u03f5=\pm G\frac{mas{s}^{3}}{mass\times length}=\pm G\frac{{m}^{2}}{\delta \mathcal{l}}=\pm G\frac{{m}_{1}{m}_{2}}{\delta \mathcal{l}};$ (2.11)

the double sign being in agreement with (2.5). The energy
$\u03f5$ is here identifiable with the analytical form of the classical Newton law, if
$\delta \mathcal{l}$ is regarded as the uncertainty range corresponding to the random distance between
${m}_{1}$ and
${m}_{1}$. First of all, with this available dimensional information only, it is impossible to identify the respective role of either mass; otherwise stated, as
${m}_{1}$ and
${m}_{2}$ were both formally inferred from a unique
${m}^{2}$, the quantum uncertainty requires that inertial and gravitational mass are physically equivalent. Moreover nothing is known in fact about
$\delta \mathcal{l}$, introduced in (2.1) simply as an arbitrary dimensional length. This suggests that actually (2.11), quoted here only as a preliminary check of (2.9), is the classical formula of a more general force, which in fact will be considered again in the next section 4. Yet it worth emphasizing since now that (2.11) is conceptually different from
$G{m}_{1}{m}_{2}/r$, seemingly analogous but actually wrong: indeed this latter implies an instantaneous action at distance *r*, whereas (2.11) implies the finite propagation time of the gravitational interaction through the range
$\delta \mathcal{l}$. This point will be concerned later, it is enough to anticipate here that the space range
$\delta \mathcal{l}$ contains inherently time information according to (1.2). The chance of defining
$\stackrel{\dot{}}{v}$ in (2.10) is justified in principle via (1.2) introducing first
$v=\left|v\right|$ as

$\frac{\delta x}{\delta t}=v=\frac{\delta \epsilon}{\delta p};$ (2.12)

this link implies that in general the size of all uncertainty ranges defining *v* are time dependent themselves. Thus is sensible in principle
$\stackrel{\dot{}}{v}$ anticipated in (2.10). The fact that the boundary coordinates of
$\delta x$ are arbitrary does not exclude for example
$\delta x={x}_{2}\left(t\right)-{x}_{1}\left(t\right)$ ; so the uncertainty allows defining the possible time dependence of velocity modulus as change rate of
$\delta x=\delta x\left(\delta t\right)$ as a function of time lapse
$\delta t$. The way (2.12) of expressing (1.2) is significant for at least four reasons: (i)
$v$ is in fact four-vector because, being defined in (2.10) via uncertainty ranges, it must be regarded according to (1.2). (ii) It agrees in principle with the idea of replacing the concept of derivative with that of ratio of uncertainty ranges, as it will be more thoroughly confirmed in the next sections 2.6 and 4. (iii) Despite the quantum nature of (1.2), the gravity fits in a natural way the present model based on (1.1). (iv) Equation (1.2) requires that *v* must be upper bound. Indeed, consider any local momentum *p* included in a range of allowed values
${p}_{0}\le p\le {p}_{1}$ ; if *v* would tend to infinity then
$\delta \epsilon $ related to
$v\delta p$ should be compatible with an infinite local energy, whereas instead
${\u03f5}_{0}\le \u03f5\le {\u03f5}_{1}$ correspond to a finite range of values allowed to the momentum. This absurd conclusion, *i.e.* finite
$\delta \epsilon $ and thus
$\delta p$ for infinite
$v\delta p$ because of *v*, requires an upper finite value *c* of *v* in agreement with (2.7). This property of *c* is a corollary of (1.2), not a postulate; as such it must hold in any reference system.

2.3. Quantum Uncertainty and Space Time

The energy density (2.2) is inherent the concept of space time according to its own origin (1.3) and has several implications, first of all the existence of a pressure
${P}_{st}$ internal to space time volume
$\delta {x}^{3}$ previously symbolized as
$lengt{h}^{3}$ ; the subscript means *space* *time* to emphasize that it is inherently based on the definition (1.1) only. As sketched in the Appendix A [4], one finds

${P}_{st}=\xi {\eta}_{st},\text{\hspace{1em}}\xi =\frac{1}{3},\text{\hspace{0.17em}}\frac{2}{3},\text{\hspace{0.17em}}1.$ (2.13)

For example, assuming a spherical volume of space time of radius ${r}_{st}$ crossed by a diametric light beam completely absorbed at its internal boundary, one calculates an internal outwards force

${F}_{st}=4\pi {r}_{st}^{2}\xi {\eta}_{st},\text{\hspace{1em}}\xi =\frac{1}{3}.$ (2.14)

Moreover (2.2) reads according to the first (2.9)

$\eta =\frac{velocit{y}^{2}}{\delta {t}^{2}G}=\frac{{a}^{2}}{G},\text{\hspace{1em}}a=\frac{velocity}{\delta t},$ (2.15)

so that $velocity\le c$ implies

$\eta \le \frac{{c}^{2}}{\delta {t}^{2}G}\mathrm{;}$ (2.16)

hence the first (2.9) reads

$\u03f5=\frac{1}{\text{n}}\frac{\hslash}{\delta t},\text{\hspace{1em}}\text{n}=\frac{c}{velocity}\ge \mathrm{1,}$ (2.17)

which requires in turn

${\u03f5}^{\mathrm{*}}>\frac{\hslash}{\delta t},\text{\hspace{1em}}{\u03f5}^{\mathrm{*}}=\text{n}\u03f5\mathrm{.}$ (2.18)

This result compares the time range $\delta t$ and the energy ${\u03f5}^{\mathrm{*}}$ to it related via $\hslash $ only for any physical reason. Is remarkable the fact that this familiar inequality of quantum mechanics is here consequence of the other relativistic one $v\le c$, which actually reads $v<c$ for matter particles; so (2.18) supports the validity of the first (2.9). It is worth considering also the second and third dimensional equations (2.9) that read

$force=\frac{energy}{{\u019b}_{m}},\text{\hspace{1em}}energy=\epsilon \frac{\delta \mathcal{l}}{{\u019b}_{m}},\text{\hspace{1em}}{\u019b}_{m}=\frac{\hslash}{mc},$ (2.19)

where
${\u019b}_{m}$ is the reduced Compton length of *m*. It appears reasonable to assume that the range size
$\delta \mathcal{l}$ is an integer number *n* of reduced Compton wavelengths, here regarded as the shortest wavelength relatable to one particle. If so, then putting

$\delta \mathcal{l}=n{\u019b}_{m}$ (2.20)

the second (2.19) reads

$energy=n\u03f5\mathrm{.}$ (2.21)

whereas the third (2.19) reads

$2\pi \delta \mathcal{l}=n{\lambda}_{m},\text{\hspace{1em}}\frac{h}{{\lambda}_{m}}=mc.$ (2.22)

Appears here as a corollary an early postulate of the old quantum mechanics, *i.e.* the energy quantization due to the integer number *n* of steady waves allowed for a particle traveling in a closed path. This result is generalized via an arbitrary factor
$\xi <1$ in order that
$\hslash /{\u019b}_{m}$ having physical dimensions of momentum reads

$2\pi \delta {\mathcal{l}}_{\xi}=n{\lambda}_{\xi},\text{\hspace{1em}}momentum=\frac{h}{{\lambda}_{\xi}}=m{v}_{\xi},\text{\hspace{1em}}{v}_{\xi}=\xi c,\text{\hspace{1em}}{\lambda}_{\xi}=\xi {\lambda}_{m}.$ (2.23)

All of this agrees with the results preliminarily obtained in section 1. A full paper [7] has been devoted to highlight the implications of (2.20), which in fact transfers *n* from its basic definition in (1.2) into the specific physical problem of a bound particle moving circularly around a central force field.

On the one hand (2.20) is justified by its direct corollaries (2.22) and (2.23), which however represent a particular case of boundary condition allowing steady wavelengths. While
$\delta {\mathcal{l}}_{\xi}$ is uniquely definable as the radius of a circumference, it must be replaced by a combination of minor and major semi-axes of an ellipse in order that (2.23) describes still the integer number of steady wavelengths along an elliptic perimeter. Are known in this respect various formulas, e.g. [8], that calculate this perimeter; the next subsection 5.7 will show how to infer through this reasoning the perihelion precession of orbiting planets. On the other hand the definition (2.23) of momentum is merely formal, being based on a dimensional assessment compliant with the condition
$v<c$ via the arbitrary factor
$\xi $. Nevertheless also in this reasoning energy quantization and De Broglie momentum are contextual. It is shown soon below that n of (2.17) is the refractive index of a dispersive medium. These considerations, crucial for the birth of the old quantum mechanics and here inferred as corollaries, suggest the necessity of defining more in detail the actual physical meaning of *v*.

2.4. Quantum Velocity and Space Time

The steps introduced by (2.9) are significant: whereas *c* in (1.1) is self-evident, it is a constant of nature, now rises the problem of clarifying further the physical meaning of velocity modulus *v* as a property of massive particles moving through the space time. To introduce *v* from a first principle, note that (2.12) yield

$\delta \epsilon =v\delta p=\frac{v}{c}\delta \left(pc\right),\text{\hspace{1em}}v=\frac{\delta x}{\delta t};$

since
$mc\ge mv$, rewrite (2.23) according to (2.17); *i.e.*
$mc=m\text{n}v$ yields

$p=mv=\frac{h}{\lambda},\text{\hspace{1em}}\text{n}=\frac{c}{v}\ge 1,\text{\hspace{1em}}\lambda =\text{n}{\lambda}_{m}\mathrm{.}$ (2.24)

Introduce an arbitrary frequency $\nu $ of matter wave inherent the De Broglie momentum and write now

$\delta \left(\frac{1}{\lambda}\right)=\delta \left(\frac{\nu}{v}\right)=\frac{1}{c}\delta \left(\text{n}\nu \right)=\frac{\delta \nu}{c}\frac{\delta \left(\text{n}\nu \right)}{\delta \nu},\text{\hspace{1em}}\nu =\frac{v}{\lambda}\mathrm{,}$

so that

$\frac{1}{\delta \nu}\delta \left(\frac{1}{\lambda}\right)=\frac{1}{c}\frac{\delta \left(\text{n}\nu \right)}{\delta \nu}=\frac{1}{{v}_{g}}\mathrm{:}$

the second equality defining a new reciprocal velocity ${v}_{g}^{-1}$ is justified by dimensional reasons according to the ratio at the left hand side. This yields the modulus of this velocity

${v}_{g}=\frac{c}{\delta \left(\text{n}\nu \right)/\delta \nu}\mathrm{;}$ (2.25)

clearly
${v}_{g}$ is the group velocity of a wave packet in a dispersive medium if
$v=v\left(\nu \right)$ as a function of De Broglie wave frequency
$\nu $. Actually this holds for both
$\delta \to 0$, *i.e.* the ratio of range sizes tends to the usual way to define the classical derivative for very small range sizes
$\delta \left(\text{n}\nu \right)$ and
$\delta \nu $. Hence (2.12) is sensible.

Recall now the eq $2\pi \delta r=n\lambda $ found in (2.23), whose physical meaning introduces a crucial condition on any mass ${m}_{2}$ orbiting according to (2.11) around ${m}_{1}$ at constant distance having modulus $\delta r$ : owing to the dual nature wave/corpuscle of matter, a steady ${m}_{2}$ wave is required to describe a stable orbiting system around ${m}_{1}$ at radial distance $\delta r$. Consider thus in this respect the De Broglie wave inferred in (2.24) and write

$2\pi =\frac{n\lambda}{\delta r}=\frac{h}{\hslash},$ (2.26)

which brings, in agreement with (2.23), directly to

$\frac{n\hslash}{\delta r}=\delta {p}_{r}=\frac{h}{\lambda}={p}_{\lambda}.$ (2.27)

At the right hand side appears the momentum
${p}_{\lambda}$ of a De Broglie wave delocalized within
$2\pi \delta r$. At the left hand side appears the radial momentum range
$\delta {p}_{r}$ of a corpuscle delocalized in
$\delta r$ : for example in the case of (2.11) it means that the space gap
$\delta r$ between the masses
${m}_{1}$ and
${m}_{2}$ implies a steady wavelength
$\lambda $ along the orbital path of the running mass around the rest mass, which confirms the indistinguishability of gravitational and inertial mass introduced in section 2.2. Rewrite now
$\delta {p}_{r}={p}_{r}-{p}_{r0}={p}_{\lambda}-0$ of (2.27) with vector notation, noting that
$\delta {p}_{r}$ is radial momentum range around the rest mass, whereas
$\delta {p}_{\lambda}={p}_{\lambda}-0$ is momentum range size along the path of the orbiting mass in agreement with (1.9); then
$\delta {p}_{r}={p}_{r}-{p}_{r0}={p}_{\lambda}$ implies
$\delta p={p}_{r}-{p}_{\lambda}={p}_{r0}$. Regard first
${p}_{r0}=0$, as it is possible because the range sizes are arbitrary, in which case
${p}_{r}={p}_{\lambda}$ reads more expressively
${p}_{\parallel}={p}_{\perp}$, *i.e.* radial and tangential vectors; this result implies a circular orbit where a unique
$\left|\delta r\right|$ implies a unique
$\left|\delta {p}_{r}\right|$ and thus constant orbital
$\left|{p}_{\lambda}\right|$. Of course in general
${p}_{r0}\ne 0$, which suggests that not necessarily the orbit must be circular; even admitting
$\delta {p}_{r}$ constant,
${p}_{r0}={p}_{r0}\left(t\right)$ implies
${p}_{\lambda}$ variable. Even so, however, it is possible that
$\delta {p}_{r}=\delta {p}_{\lambda}$ simply thinking
$\delta p=\left({p}_{r}-{{p}^{\prime}}_{r0}\right)-\left({p}_{\lambda}-{{p}^{\u2033}}_{r0}\right)=0$ ; *i.e.*
${p}_{r0}$ has been split itself into its radial and tangential components, so that
${p}_{r}$ and
${p}_{\lambda}$ change in the respective ranges

$\left|\delta {p}_{r}\right|=\left|\delta {p}_{\lambda}\right|\mathrm{:}$ (2.28)

this implies the momentum range conservation of the running mass regarded first via its radial delocalization momentum range
$\delta {p}_{r}$ and then also via its orbital delocalization momentum range
$\delta {p}_{\lambda}$, normal to the radial distance in a circular path. Clearly this holds for the ranges
$\delta {p}_{\parallel}$ and
$\delta {p}_{\perp}$, whatever the local
${p}_{\parallel}$ and
${p}_{\perp}$ might be. Nonetheless two considerations support significantly the present way of reasoning. (i) From a relativistic point of view
${p}_{r0}={p}_{r0}\left(\delta t\right)$ justifies the definition of the momentum vectors as 4-vectors owing to (1.2) and makes compatible the present conclusions even with another major effect: e.g. the orbit instability, sketched later in the subsection 5.3.4 as due to emission of gravitational waves. (ii) Examine a further consequence of (2.28) from a quantum point of view; as such this result should have a general validity, being it direct corollary of first principles. Consider a point source of particles, be they photons or matter corpuscles, and assume the tangential advancement of their wave front described by
${p}_{\lambda}$ correspondingly to the radial advancement of the corpuscle beam
${p}_{r}$. As the uncertainty requires considering the respective momenta consistently with the lack of deterministic trajectory, (2.28) suggests that to a wavelike radial propagation of the particle beam corresponds a wavelike propagation along a normal direction too. If a beam of corpuscles illuminates a solid plane with two slits, it is natural that the momentum
$\delta {p}_{\lambda}$ of running waves yields an interference pattern on a screen placed at distance
$\delta r=n\hslash /\delta {p}_{r}$ behind the slit plane. More specifically, *n* controls the various distances on the interference plane with respect to the slits. In other words (2.28) anyway hold and account for the duality wave/corpuscle of matter revealed by *n* tangential waves that interfere on the screen simply because two slits generate two beams according to the Huygens principle. All of this is self consistent regardless of how are defined the relativistic
${p}_{\parallel}$ and
${p}_{\perp}$ of particles of the beam.

2.5. Relativistic Outcomes

The relativistic worth of these results follows straightforwardly, first of all because even the time is inherently involved by (1.2); multiplying side by side the first and third (2.9) one finds

$\frac{\delta t\delta \mathcal{l}{\u03f5}^{2}}{{\hslash}^{2}}m=\frac{energy\times velocity}{{c}^{2}}$

and thus, owing to the first (2.17),

$\frac{m}{{\text{n}}^{2}}\frac{\delta \mathcal{l}}{\delta t}={p}_{\mathcal{l}}=\frac{energy\times velocity}{{c}^{2}},\text{\hspace{1em}}{v}_{\mathcal{l}}=\frac{1}{{\text{n}}^{2}}\frac{\delta \mathcal{l}}{\delta t},\text{\hspace{1em}}m{v}_{\mathcal{l}}={p}_{\mathcal{l}}=\frac{{v}_{\mathcal{l}}{\u03f5}_{\mathcal{l}}}{{c}^{2}}\mathrm{.}$ (2.29)

Note that ${\text{n}}^{2}$ is here mere dimensionless multiplicative factor of ${v}_{\mathcal{l}}$, which however cannot be specifically calculated because both $\delta \mathcal{l}$ and $\delta t$ are uncertainty ranges unknown and conceptually unknowable; so ${\text{n}}^{2}$ contributes in determining the resulting ${v}_{\mathcal{l}}$, whatever its value might be. These dimensional equations introduce the velocity and relativistic momentum components ${v}_{\mathcal{l}}$ and ${p}_{\mathcal{l}}$. Moreover write according to (2.12)

${v}_{\mathcal{l}}=\frac{\delta {\u03f5}_{\mathcal{l}}}{\delta {p}_{\mathcal{l}}}\mathrm{,}$ (2.30)

Multiplying side by side this velocity and (2.29) one finds

$\frac{\delta {\u03f5}_{\mathcal{l}}}{\delta {p}_{\mathcal{l}}}{\u03f5}_{\mathcal{l}}={p}_{\mathcal{l}}{c}^{2}\mathrm{.}$

Then it is possible to write

$\delta {\left({\u03f5}_{\mathcal{l}}\right)}^{2}=\delta {\left({p}_{\mathcal{l}}c\right)}^{2}$ (2.31)

*i.e.* identically
$\delta {\left({\u03f5}_{\mathcal{l}}\right)}^{2}=\delta \left({\left({p}_{\mathcal{l}}c\right)}^{2}+const\right)$, because of course
$\delta \left(const\right)=0$. Hence (2.31) is compatible with

${\u03f5}_{\mathcal{l}}^{2}={\left({p}_{\mathcal{l}}c\right)}^{2}+const\mathrm{.}$ (2.32)

Note that

$\underset{v\to 0}{lim}\frac{pc}{v}={m}_{0}{c}^{2}$ (2.33)

and then, being ${m}_{0}$ the rest mass of the particle, one finds

${\u03f5}_{\mathcal{l}}^{2}={\left({p}_{\mathcal{l}}c\right)}^{2}+{\left({m}_{0}{c}^{2}\right)}^{2}\mathrm{.}$ (2.34)

A simple reasoning shows therefore that the statistical formulation (1.2) of quantum uncertainty is immediate consequence of the space time definition (1.1) and that the energy and momentum equations of special relativity are actually quantum equations, likewise the Lorentz transformations; indeed merging (2.34), (2.27) and (2.29) one finds

${\u03f5}_{\mathcal{l}}=\frac{{m}_{0}{c}^{2}}{\sqrt{1-{v}_{\mathcal{l}}^{2}/{c}^{2}}},\text{\hspace{1em}}{p}_{\mathcal{l}}=\frac{{m}_{0}{v}_{\mathcal{l}}}{\sqrt{1-{v}_{\mathcal{l}}^{2}/{c}^{2}}}=\frac{{\u03f5}_{\mathcal{l}}{v}_{\mathcal{l}}}{{c}^{2}}=\frac{h}{\lambda}.$ (2.35)

Note that this definition of energy and momentum agrees with that of (2.23) simply putting

${\u03f5}_{\mathcal{l}}={m}_{\mathcal{l}}{c}^{2},\text{\hspace{1em}}{p}_{\mathcal{l}}={m}_{\mathcal{l}}{v}_{\mathcal{l}},\text{\hspace{1em}}{m}_{\mathcal{l}}=\frac{{m}_{0}}{\sqrt{1-{v}_{\mathcal{l}}^{2}/{c}^{2}}}.$ (2.36)

The definition of ${m}_{\mathcal{l}}$ is essential to overcome an evident difficulty about ${p}_{\mathcal{l}}$, which diverges for ${v}_{\mathcal{l}}\to c$ when expressed via the Lorentz factor but remains finite when expressed via ${\u03f5}_{\mathcal{l}}$. Clearly this is due to the fact that the former definition introduces explicitly the mass, the latter does not; so if is the mass that diverges because of the Lorentz factor, then both ${\u03f5}_{\mathcal{l}}$ and ${p}_{\mathcal{l}}$ coherently increase till then their asymptotic ratio becomes ${\u03f5}_{\mathcal{l}}/{p}_{\mathcal{l}}=c$. However there is a further reason, more subtle, to explain this point even better; this reason will be concerned in the section 5.5.8.

2.6. The Uncertainty Equations

Merging the third (2.9) and (2.22) one finds

$mc\delta \mathcal{l}=n\hslash $ (2.37)

and then also

$m{c}^{2}\frac{\delta \mathcal{l}}{c}=m{c}^{2}\delta t=n\hslash ,\text{\hspace{1em}}\delta t=\frac{\delta \mathcal{l}}{c}.$ (2.38)

Since *m* is arbitrary, consider now two masses
${m}^{\prime}$ and
${m}^{\u2033}$ such that
${m}^{\prime}{c}^{2}\left(\delta \mathcal{l}/c\right)={n}^{\prime}\hslash $ and
${m}^{\u2033}{c}^{2}\left(\delta \mathcal{l}/c\right)={n}^{\u2033}\hslash $. Subtracting side by side these equations the first (2.38) yields

$\left({m}^{\prime}-{m}^{\u2033}\right){c}^{2}\delta t=\left({n}^{\prime}-{n}^{\u2033}\right)\hslash ,\text{\hspace{1em}}\delta t=\delta \mathcal{l}/c;$

as of course ${n}^{\prime}-{n}^{\u2033}$ is still an arbitrary integer, this result reads

$\delta \u03f5\delta t={n}^{\u2034}\hslash ,\text{\hspace{1em}}\delta \u03f5=\left({m}^{\prime}-{m}^{\u2033}\right){c}^{2},\text{\hspace{1em}}{n}^{\prime}\le {n}^{\u2034}\le {n}^{\u2033}.$ (2.39)

So even *n* is defined in its own range of integer values, as stated in (1.9). Multiplying and dividing the left hand side of the first equation by an arbitrary velocity modulus *v*, by dimensional reasons one finds
$\delta p=\delta \u03f5/v$ and
$\delta x=v\delta t$ so that
$\delta p\delta x=n\hslash $. Thus (2.37) implies via (2.22) and (2.19) the uncertainty equations

$\delta {\epsilon}_{\mathcal{l}}\delta {t}_{\mathcal{l}}=n\hslash ,\text{\hspace{1em}}\delta \mathcal{l}\delta {p}_{\mathcal{l}}=n\hslash ,$ (2.40)

*i.e.* just (1.2) merely with a different notation of the conjugate dynamical variables. These equations imply the indistinguishability of identical particles, because actually they concern the phase space rather than the particles themselves; in other words it is impossible to distinguish electron 1 from electron 2 delocalized in a region of space time if nothing in known about them. Indeed it has been shown in (2.29) to (2.35) that, for example, momentum and energy are directly related to the range sizes (2.40) regardless of any hypothesis about the particles themselves. On the one hand is remarkable the fact that the Newtonian definitions (2.9) imply the concept of uncertainty, thus confirming that actually even the classical gravity is rooted in the quantum equations (1.2). On the other hand the agnostic meaning of uncertainty, which implies lack of information about the boundaries of the ranges and about the local values of the dynamical variables allowed in their ranges, is not a postulate but a corollary of the way to introduce (2.40). The agnostic meaning of (1.2) follows from (2.9), in turn deductible themselves without need of further considerations besides the dimensional analysis of the physical definitions of time and energy. In fact is enough a general idea only:
${\u03f5}^{\prime}\le \u03f5\le {\u03f5}^{\u2033}$ implied by
${m}^{\prime}\le m\le {m}^{\u2033}$ follows from and corresponds to the number of quantum states
${n}^{\prime}\le n\le {n}^{\u2033}$ allowed at any
${t}^{\prime}\le t\le {t}^{\u2033}$. By consequence space and time coordinates lose their deterministic local meaning of classical physics, while however their uncertainty ranges fulfill the respective Lorentz transformations (2.35). Of course owing to (2.24) it also holds for the wave definitions of dynamical variables
${\lambda}^{\prime}\le \lambda \le {\lambda}^{\u2033}$ of *p* and
${v}^{\prime}\le v\le {v}^{\u2033}$ itself. In effect it follows because the velocity is defined as ratio of uncertainty ranges as shown in (2.29), (2.40) and (2.12); a deterministic value of *v* is pertinent only in classical physics and in Einstein relativity. Although the reasoning to infer (2.39) has quantum character, it involves the relativistic
$\u03f5=m{c}^{2}$ and quantum
$p=h/\lambda $ local values of dynamical variables randomly falling within the respective uncertainty ranges. It has been shown in [9] that *n* must be arbitrary integer in order that (1.2) are independent of any particular reference system. Summarizing shortly the reasoning therein carried out, let
$\delta x\delta p$ be defined by range sizes in a given reference system *R* and
$\delta {x}^{\prime}\delta {p}^{\prime}$ in another
${R}^{\prime}$. Since the respective products are equal to *n* and
${n}^{\prime}$ times
$\hslash $, it is clear that (1.2) are actually indistinguishable in *R* and
${R}^{\prime}$, because *n* and
${n}^{\prime}$ are indistinguishable themselves; indeed *n* and
${n}^{\prime}$ are not specific values, rather they symbolize sets of arbitrary integers, so that any allowed *n* cannot be distinguished from any allowed
${n}^{\prime}$. In other words the quantization makes indistinguishable the reference systems because the unique sequence
$\hslash \mathrm{,2}\hslash \mathrm{,3}\hslash \mathrm{,}\cdots $ is identically compatible with
$\delta p\delta x$ in *R* and
$\delta {p}^{\prime}\delta {x}^{\prime}$ in
${R}^{\prime}$. Moreover all uncertainty ranges, e.g.
$\delta x={x}^{\u2033}-{x}^{\prime}$, contain themselves one boundary value, say
${x}^{\prime}$, that in principle could be referred to the origin of its own *R*, whereas the other boundary, say
${x}^{\u2033}$, determines the range size. However, being both boundary coordinates by definition unknown and conceptually unknowable, any link of
$\delta x$ to a specific *R* is conceptually missing as well. This conclusion is further confirmed in a more substantial way as shown in [9] and again sketched also here. Expressing the range sizes in the Planck units previously found, see (2.6) and following, (1.2) read in fact

${n}_{x}^{*}{n}_{p}^{*}=n={n}_{t}^{*}{n}_{\epsilon}^{*}$ (2.41)

where *n* is the arbitrary integer of quantization whereas
${n}_{j}^{\mathrm{*}}$ are arbitrary real numbers defining the*j*-th range sizes as multiple of the respective Planck units, e.g.
$\delta t={n}_{t}^{*}{t}_{P}$ with
${{n}^{\prime}}_{t}^{\mathrm{*}}\le {n}_{t}^{\mathrm{*}}\le {{n}^{\u2033}}_{t}^{\mathrm{*}}$ as in (1.9); the primed and double primed notation of
${n}_{t}^{\mathrm{*}}$ indicates instead specific time values
${n}_{t}^{\mathrm{*}}$, of course arbitrary. According to the link (2.41) between mere numbers it is in fact impossible even to introduce *R* itself, in agreement with the conclusion that by definition (1.2) hold in any reference system *R* inertial or not. Obviously an analogous conclusion holds for the ratios of uncertainty range sizes too, e.g. for
$v=\delta x/\delta t$ of (2.12)

$v=\frac{{n}_{x}^{*}}{{n}_{t}^{*}}\frac{{\mathcal{l}}_{P}}{{t}_{P}}=\frac{{n}_{x}^{*}}{{n}_{t}^{*}}c:$ (2.42)

owing to the arbitrariness of all range sizes, for example, this velocity can be intended in particular as an average value
$\stackrel{\xaf}{v}$ of all ratios admissible for
${x}^{\prime}\le x\le {x}^{\u2033}$ and
${t}^{\prime}\le t\le {t}^{\u2033}$ or as the limit of both
$\delta \to 0$, *i.e.* as
$v\equiv {v}_{lim}=\partial x/\partial t$. Despite the notation, even
$\partial x$ and
$\partial t$ still have in principle the meaning of very small uncertainty range sizes whose ratio defines the modulus *v* as a new dynamical variable; yet the local value of
${v}_{lim}$ is still numerically undefinable in the present quantum model because are not known the time and space coordinates
${n}_{t}^{\mathrm{*}}$ and
${n}_{x}^{\mathrm{*}}$ falling within
$\partial t$ and
$\partial x$. However, this is true if the infinitesimal
$\partial x$ and
$\partial t$ are regarded in fact as independent ranges, as so far implicitly assumed for
$\delta x$ and
$\delta t$. If both ranges are concurrently vanishing, as it happens in the usual concept of derivative, then their shared property of contextual vanishing adds supplementary information on the definition of *v*, which reads now
${v}_{cl}$ : indeed it modifies the concept of quantum total uncertainty hitherto intuitively acknowledged for
$\delta x/\delta t$. This is why
${v}_{cl}$ is conceptually knowable in classical physics, but not in general in the present agnostic model based on (1.2) only where all
${n}_{j}^{\mathrm{*}}$ of (2.41) are random in their own
${{n}^{\prime}}_{j}^{\mathrm{*}}\le {n}_{j}^{\mathrm{*}}\le {{n}^{\u2033}}_{j}^{\mathrm{*}}$ and thus independent. This point deserves attention and is further explained in the next subsection. Here it is enough to remark the conceptual difference between the modulus of the four-vector
${v}_{rel}=\left|{v}_{rel}\right|$ of special relativity and the quantum *v*, ratio of two independent and arbitrary uncertainty ranges defined in (2.29), (2.40) and (2.30). On the one hand the local moduli
${v}_{cl}$ classical or
${v}_{rel}$ relativistic are calculable, whereas the local *v* of (2.12) does not likewise any function whose local time and space coordinates are conceptually missing. On the other hand
${v}_{rel}$ must be related to the reference system where is available deterministic information on local coordinates and time, whereas *v* waives “a priori” its own *R* because of (2.41). The fact that in principle *v* can be introduced without explicit link to a specific reference system has a further implication. Let *v* enter in the formula of a physical amount *f* written in terms of uncertainty ranges of dynamical variables only: if
$f=f\left(\delta x\mathrm{,}\delta p\mathrm{,}\delta \epsilon \mathrm{,}\delta t\right)$ turns into
$f=f\left({n}_{x}^{*},{n}_{p}^{*},{n}_{\epsilon}^{*},{n}_{t}^{*}\right)$ while
$v=v\left({n}_{x}^{\mathrm{*}}/{n}_{t}^{\mathrm{*}}\right)$, then it is possible to introduce *f* itself without defining explicitly its *R*, whatever the specific physical meaning of *v* and *f* might be. Nevertheless the lack of a specific reference system does not imply in fact any ambiguity, as it will appear in the following section 6 where are calculated some numerical outcomes of the present model; rather the physical meaning of *v* results from that of
$\delta x$ and
$\delta t$ or
$\delta \epsilon $ and
$\delta p$ themselves according to (2.12). Although this physical model seems too agnostic to infer valuable information, note that these conceptual premises have been enough to infer the fundamental (1.2) from (1.1) and even preliminary relativistic results. The remainder of the paper aims to show that just this conceptual agnosticism allows to overcome the determinism of Einstein general relativity and plugs it into the elusive quantum world; the calculations will be carried only after having completed adequately the theoretical frame so far introduced.

2.7. Uncertainty, Covariance, Simultaneity

In general the choice of the reference system *R* is crucial in any classical physical model that implements deterministic local coordinates; in the Einstein relativity, the equations are required to be invariant with respect to the reference systems, including the non-inertial ones. Consider however a quantum problem formulated *only* via uncertainty ranges; in fact (2.41) shows that if all
${n}_{j}^{\mathrm{*}}$ and *n* are arbitrary, then there is no direct correlation between range sizes of dynamical variables and reference systems just because the former do not contain any information someway related to the latter. As it has been remarked for *n*, the only available information is that the product of two range sizes of conjugate dynamical variables must be quantized; *i.e.* both products
${n}_{j}^{\mathrm{*}}{n}_{{j}^{\prime}}^{\mathrm{*}}$ of (2.41) must yield an arbitrary integer whatever the local values of the respective dynamical variables *j* and *j’ *might be. So, if the local coordinates are replaced by ranges that fulfill (1.2) and (2.41), then is missing “a priori” the existence of privileged reference systems; moreover it is easy to show that, by consequence, the requirement of the different form of equations in *R* and *R’* becomes inessential. Is instructive in this respect the classical example reported in various textbooks, e.g. [10], of a point mass *m* tethered by a massless and inextensible wire, so that the mass moves circularly around a fixed coordinate. This example becomes significant noting that if the wire is broken, e.g. by the centrifugal force itself, thereafter the motion of the mass is rectilinear uniform along the tangent to the circumference in the breakdown point. This is true in *R* with origin fixed on the rotation center of the mass. In *R’* fixed on the moving mass, instead, the mass is at rest; when the wire is broken the mass deviates from its initial path, it follows a curved trajectory. The classical physics implements
$F=ma$ and
${F}^{\prime}=m{a}^{\prime}$ respectively: elementary considerations show that
${a}^{\prime}={a}_{r}+{a}_{ce}+{a}_{co}$, where the subscripts stand for real, centrifugal and Coriolis terms. This is a typical example where the motion of the mass described in *R* and *R’* implies acceleration terms appearing in the non-inertial *R’* only. Einstein felt then the necessity of a covariant theory including the gravity. Classically it is possible to introduce *R* and *R’* along with any other *R’’* arbitrarily chosen. Just for this reason the Einstein relativity aims to describe in general any physical system independently of a specific *R*, thus excluding the existence of such a “privileged” *R* but admitting however that in effect all these various *R* are actually definable. In any approach formulated via (1.2), instead, neither any deterministic coordinate of the tethered system nor its own *R* are actually definable owing to (2.41); in general it is possible only to say that exist arbitrary reference systems and that if two of them are inertial then hold (2.35). Consider indeed the dynamics of the rotating mass described by the Newtonian (2.9): is crucial the fact that the acceleration is defined as
$a=\left(c/\hslash \right)\u03f5$, where the unique variable is
$\u03f5$. This latter however is a local energy that must be implemented solely via its uncertainty range
$\delta \epsilon ={\u03f5}_{2}-{\u03f5}_{1}$ : *i.e.*
$\u03f5$ is to be regarded as
${\u03f5}_{1}\le \u03f5\le {\u03f5}_{2}$, whereas
$a$ is a random value included within a range
${a}_{1}\le a\le {a}_{2}$ of values. Also, in the present quantum model we should compare the ranges
$\delta \epsilon $ and
$\delta {\epsilon}^{\prime}={{\u03f5}^{\prime}}_{2}-{{\u03f5}^{\prime}}_{1}$ before and after the breakdown of the tethered system; in other words *a’* is now a different local value included within a different range
${{a}^{\prime}}_{1}\le {a}^{\prime}\le {{a}^{\prime}}_{2}$ of values. But actually this comparison between inertial and non-inertial reference systems is conceptually meaningless in the present model: the range boundaries and the local values of dynamical variables are unknowable and irrelevant as concerns the physical description of any quantum problem, so it is irrelevant the local disagreement
$a\ne {a}^{\prime}$ and the fact that *a’* includes various additional terms with respect to
$a$. The agnosticism implied by (1.2) compels considering these ranges before and after the wire breakdown regardless of how the corresponding local accelerations are made of; in other words the chance
$\delta \epsilon =\delta {\epsilon}^{\prime}$ of including the respective
$a$ and *a’*, in principle possible because
$a$ and
$\u03f5$ differ by a proportionality constant factor, bypasses the necessity of discriminating *R* and *R*’ to describe the tethered system with or without its breakdown. Since this holds for any uncertainty range by definition, in fact (2.41) waives the necessity of specifying either reference system to concern the dynamical variables. Accordingly it is possible to regard all ranges of (1.2) independently of their definition in a specific *R*: rather it is possible to introduce
$\delta x$ and
$\delta t$ independently of the local space and time coordinates
${{n}^{\prime}}_{x}^{\mathrm{*}}$ and
${{n}^{\prime}}_{t}^{\mathrm{*}}$ in Plank units, equivalent to
$x=x\left(t\right)$ and
${x}^{\prime}={x}^{\prime}\left({t}^{\prime}\right)$, and regardless of their own
${R}_{x}$ and
${R}_{t}$. It is clear now the last statement of the previous subsection: distinguishing
${n}_{x}^{*}={n}_{x}^{*}\left({n}_{t}^{*}\right)$ and
${{n}^{\prime}}_{x}^{*}={{n}^{\prime}}_{x}^{*}\left({{n}^{\prime}}_{t}^{*}\right)$ to obtain the local velocity through the usual concept of derivative violates the concept of total uncertainty and implies turning the formulation of the physical problem into the deterministic definition of classical dynamical variables. This conclusion is reasonably extrapolable also to different times
${{n}^{\prime}}_{t}^{\mathrm{*}}$ and
${{n}^{\u2033}}_{t}^{\mathrm{*}}$ in two different *R*: *i.e.*
${{n}^{\prime}}_{t}^{\mathrm{*}}={{n}^{\u2033}}_{t}^{\mathrm{*}}$, for example, would violate the principle of total lack of information about the local time variable, whereas the uncertainty leaves out any form of local determinism in any *R*. So
${{n}^{\prime}}_{t}^{\mathrm{*}}$ and
${{n}^{\u2033}}_{t}^{\mathrm{*}}$ must fulfill the condition
${{n}^{\prime}}_{t}^{\mathrm{*}}\perp \perp {{n}^{\u2033}}_{t}^{\mathrm{*}}$, where the symbol means “independent of”: *i.e.* it states that
${{n}^{\prime}}_{t}^{\mathrm{*}}$ and
${{n}^{\u2033}}_{t}^{\mathrm{*}}$ are independent local times in
${t}_{P}$ units and introduces two crucial corollaries. On the one hand if
${{n}^{\prime}}_{t}^{\mathrm{*}}$ and
${{n}^{\u2033}}_{t}^{\mathrm{*}}$ are independent local times in different reference systems, then
${{n}^{\prime}}_{t}^{\mathrm{*}}\perp \perp {{n}^{\u2033}}_{t}^{\mathrm{*}}$ excludes the concept of simultaneity in *R* and *R’*; as any *t* in *R* is not numerically correlatable to *t’* in *R’* in a deterministic way, two events simultaneous in *R* are not automatically simultaneous in *R'*. On the other hand the local time
${{n}^{\prime}}_{t}^{\mathrm{*}}$ is illusory itself likewise the local coordinate
${{n}^{\prime}}_{x}^{\mathrm{*}}$ ; indeed during a given time lapse of length
${n}_{t}^{\mathrm{*}}{t}_{P}$ it is meaningless to distinguish
${{n}^{\prime}}_{t}^{*}>{{n}^{\u2033}}_{t}^{*}$ or
${{n}^{\prime}}_{t}^{*}<{{n}^{\u2033}}_{t}^{*}$. The lack of the concept of simultaneity in the special relativity is obvious: if *c* is finite and invariant in all reference systems whereas instead space lengths and time lapses are subjected to (2.35) via
$\delta p$ and
$\delta \epsilon $ of (1.2), it is trivial to conclude that the time lapses
$\delta t$ and
$\delta {t}^{\prime}$ in different reference systems cannot imply the simultaneity of a given event for two observers in reciprocal motion. From a quantum point of view the same conclusion is due to the lack of single time coordinates to be compared in a deterministic way, being significant instead time ranges to be compared e.g. in two different reference systems. These statements, well known since the birth of the special relativity by consequence of the finite value of invariant *c*, appear here as straightforward corollaries of the quantum uncertainty. Nevertheless the present way to regard the quantum physics based on (1.2) has not only relativistic implications, partially already inferred in the subsection 2.5 and further considered in the next section 3, but also quantum implications. Here are two short examples of corollaries of (1.2) pertinent to the present reasoning.

(i) Quantum implication: the hydrogenlike atoms. Write

$\delta {p}_{r}={{p}^{\prime}}_{r}-0,\text{\hspace{1em}}\delta \epsilon ={\epsilon}^{\prime}-0:$ (2.43)

since the unprimed range sizes at the left hand sizes are arbitrary, the same must hold for the primed range sizes at the right hand side. Then it must be possible to implement identically both of them, regarding thus the lower boundary value 0 as a particular but not deterministic case. To infer physical information from these statements and check their validity, find the classical energy ${\u03f5}^{\prime}$ of hydrogenlike atoms. Implementing (1.2) for ${{p}^{\prime}}_{r}$, which actually owing to (2.43) is by definition radial range despite its notation, one finds

${\u03f5}^{\prime}=\frac{{{p}^{\prime}}_{r}^{2}}{2m}=\frac{{\left(n\hslash \right)}^{2}}{2m{r}^{\prime}}=\frac{2{\left(n\hslash \right)}^{2}}{m{\left(2{r}^{\prime}\right)}^{2}}=\frac{2{\left(n\hslash \right)}^{2}}{m\delta {r}^{2}},\text{\hspace{1em}}\delta r=2{r}^{\prime};\text{\hspace{0.05em}}$ (2.44)

clearly ${r}^{\prime}$ has been introduced as a length by dimensional reasons and symbolizes the range ${r}^{\prime}-0$. Thus

$\delta r=\pm \frac{2{\left(n\hslash \right)}^{2}}{m{\u03f5}^{\prime}\delta r}=\pm \frac{2{\left(n\hslash \right)}^{2}}{m\left(-Z{e}^{2}/\delta r\right)\delta r}=\mp \frac{2{\left(n\hslash \right)}^{2}}{mZ{e}^{2}},\text{\hspace{1em}}{\u03f5}^{\prime}=-\frac{Z{e}^{2}}{\delta r},$ (2.45)

having introduced appropriate information, *i.e.* the specific electromagnetic interaction between nucleus and electron charges. One finds, with the second equation reasonably suggested by (2.44) and first (2.45) itself

$\delta r=2{r}_{B},\text{\hspace{1em}}{r}_{B}={r}^{\prime}=\frac{{\left(n\hslash \right)}^{2}}{mZ{e}^{2}}$

that yield Bohr radius and energy

${\u03f5}^{\prime}={\epsilon}_{B}=-\frac{Z{e}^{2}}{\delta r}=-\frac{Z{e}^{2}}{2{r}^{\prime}}=-\frac{{\left(Z{e}^{2}\right)}^{2}m}{2{\left(n\hslash \right)}^{2}}.$ (2.46)

Now it should be clear why range sizes and boundary coordinates are irrelevant as concerns the quantum problems, as in effect it has been demonstrated for various systems [11] [12]. In particular it is not necessary to specify *R* centered on the nucleus, it is enough to state that nucleus and electron are
$\delta r$ apart; the radial range size is then defined by the non deterministic Bohr radius via the integer
$1\le n\le \infty $. Is clear thus the meaning of the coefficient 2 in the second (2.44): an electron
${r}^{\prime}$ apart from the nucleus has total radial delocalization range
$2{r}^{\prime}$. In this model the quantum numbers are in fact numbers of quantum states.

(ii) Relativistic implication: the invariant equations. Write (1.2) as follows

$\delta x\delta t=\delta x\frac{n\hslash}{\delta \epsilon}=\delta {x}^{2}\frac{\delta p}{\delta \epsilon},\text{\hspace{1em}}\delta x\delta t=\delta t\frac{n\hslash}{\delta p}=\delta {t}^{2}\frac{\delta \epsilon}{\delta p}$

so that, subtracting side by side,

${c}^{2}\delta {t}^{2}\frac{\delta \epsilon /c}{\delta \left(pc\right)}-\delta {x}^{2}\frac{\delta p}{\delta \epsilon}=0.$

Let be now, without loss of generality,

$\frac{\delta \epsilon /c}{\delta \left(pc\right)}=\frac{\delta p}{\delta \epsilon}-Y,$

being *Y* a function with physical dimensions of reciprocal velocity to be defined. Hence

$\left({c}^{2}\delta {t}^{2}-\delta {x}^{2}\right)\frac{\delta p}{\delta \epsilon}-Y{c}^{2}\delta {t}^{2}=0,$

*i.e. *

${c}^{2}\delta {t}^{2}-\delta {x}^{2}=Y{c}^{2}\delta {t}^{2}\frac{\delta \epsilon}{\delta p}=Y{c}^{2}\delta t\frac{n\hslash}{\delta p}=Y{c}^{2}\delta t\delta x.$ (2.47)

The left hand side reads, again because of (1.2) and (2.34),

$\begin{array}{c}{c}^{2}\frac{{n}^{2}{\hslash}^{2}}{\delta {\epsilon}^{2}}-\frac{{n}^{2}{\hslash}^{2}}{\delta {p}^{2}}={\left(n\hslash c\right)}^{2}\left(\frac{1}{\delta {\epsilon}^{2}}-\frac{1}{\delta \left({p}^{2}{c}^{2}\right)}\right)\\ =\frac{{\left(n\hslash c\right)}^{2}}{\delta {\epsilon}^{2}\delta \left({p}^{2}{c}^{2}\right)}\left(\delta \left({p}^{2}{c}^{2}\right)-\delta {\epsilon}^{2}\right)\\ =-\frac{{\left(n\hslash c\right)}^{2}{\left(m{c}^{2}\right)}^{2}}{\delta {\epsilon}^{2}\delta \left({p}^{2}{c}^{2}\right)}.\end{array}$ (2.48)

If *Y* is a constant, *i.e.*
$Y={c}^{-1}$, (2.47) shows two invariant quantities of the special relativity correlated each other;
${c}^{2}\delta {t}^{2}-\delta {x}^{2}$ is particularly important as it has been demonstrated in [13] to be conceptual foundation of the special relativity. Special attention deserves in this respect the operator formalism of quantum mechanics, which regards since the beginning the particles as waves; instead the last equations have concerned the corpuscular properties of matter. The next section shows how to introduce in this conceptual frame also the wave formalism, in agreement with the corpuscular/wave nature of the particles.

2.8. The Wave Formalism

Rewrite identically (2.32) as

$\left({\u03f5}_{\mathcal{l}}+{p}_{\mathcal{l}}c\right)\left({\u03f5}_{\mathcal{l}}-{p}_{\mathcal{l}}c\right)=const,$ (2.49)

which is trivially consistent with

$\begin{array}{l}{{\u03f5}^{\prime}}_{\mathcal{l}}+{p}_{\mathcal{l}}c=cons{t}^{\prime},\text{\hspace{1em}}{{\u03f5}^{\u2033}}_{\mathcal{l}}-{p}_{\mathcal{l}}c=cons{t}^{\u2033},\text{\hspace{1em}}{{\u03f5}^{\prime}}_{\mathcal{l}}={{\u03f5}^{\u2033}}_{\mathcal{l}}=\pm {\u03f5}_{\mathcal{l}},\\ cons{t}^{\prime}=cons{t}^{\u2033}=\pm \sqrt{const};\end{array}$ (2.50)

it is trivially evident that multiplying side by side with the help of the third and fourth conditions (2.50) one obtains (2.49) and thus (2.32). Yet two more conditions make the first two (2.50) compatible with (2.32). Multiplying side by side ${{\u03f5}^{\prime}}_{\mathcal{l}}=cons{t}^{\prime}-{p}_{\mathcal{l}}c$ and ${{\u03f5}^{\u2033}}_{\mathcal{l}}=cons{t}^{\u2033}+{p}_{\mathcal{l}}c$ one finds

${{\u03f5}^{\prime}}_{\mathcal{l}}{{\u03f5}^{\u2033}}_{\mathcal{l}}=-{p}_{\mathcal{l}}^{2}{c}^{2}+\left(cons{t}^{\prime}-cons{t}^{\u2033}\right){p}_{\mathcal{l}}c+cons{t}^{\prime}cons{t}^{\u2033},$ (2.51)

whence the two chances allowed by $cons{t}^{\prime}=cons{t}^{\u2033}$ :

(1) ${{\u03f5}^{\prime}}_{\mathcal{l}}{{\u03f5}^{\u2033}}_{\mathcal{l}}={\epsilon}^{*2},\text{\hspace{1em}}{p}_{\mathcal{l}}=\pm i{p}^{*},\text{\hspace{1em}}cons{t}^{\prime}=cons{t}^{\u2033}=\pm \sqrt{const},$ (2.52)

which yields

${\u03f5}^{\mathrm{*2}}={\left({p}^{\mathrm{*}}c\right)}^{2}+cons{t}^{2}\mathrm{,}$

and

(2) ${{\u03f5}^{\prime}}_{\mathcal{l}}{{\u03f5}^{\u2033}}_{\mathcal{l}}=-{\u03f5}^{*2}={\left(i{\u03f5}^{*}\right)}^{2},\text{\hspace{1em}}{p}_{\mathcal{l}}={p}^{*},\text{\hspace{1em}}cons{t}^{\prime}=cons{t}^{\u2033}=\pm i\sqrt{const},$ (2.53)

which yields

$-{\u03f5}_{\mathcal{l}}^{\mathrm{*2}}=-{\left({p}^{\mathrm{*}}c\right)}^{2}-cons{t}^{2}\mathrm{.}$

Both conditions (2.52) and (2.53) agree with (2.32) exactly likewise (2.50). The former implies imaginary momentum
${p}_{\mathcal{l}}$ ; also, since there is no reason to exclude
${{\u03f5}^{\prime}}_{\mathcal{l}}>0$ and
${{\u03f5}^{\u2033}}_{\mathcal{l}}>0$, one must accept even imaginary energy
${\u03f5}_{\mathcal{l}}$, which opens a new conceptual frame along with the imaginary momentum too. The relativistic (2.32) needs therefore a new interpretation to be consistent with complex dynamical variables, which in turn must be acknowledged themselves: it is evident that all of this implies in fact the corpuscle/wave behavior of matter to fit both quantum and relativistic results. While the compatibility of (2.49) with the initial (2.32) is trivial, that of (2.52) and (2.53) with (2.32) is still possible even defining complex quantities. Consider first (2.52) to calculate via
${p}^{*}=\pm {p}_{\mathcal{l}}/i$ the corresponding complex range
$\delta {p}^{\mathrm{*}}=\pm \delta {p}_{\mathcal{l}}/i=\pm n\hslash /i\delta {x}_{\mathcal{l}}$ according to (1.2). Since any integer *n* can be expressed as a difference of two integers
${n}^{\prime}$ and
${n}^{\u2033}$ one finds

$\delta {p}^{*}={{p}^{\prime}}^{*}-{{p}^{\u2033}}^{*}=\pm \frac{\hslash}{i}\frac{n}{\delta {x}_{\mathcal{l}}}=\pm \frac{\hslash}{i}\frac{{n}^{\prime}}{\delta {x}_{\mathcal{l}}}\mp \frac{\hslash}{i}\frac{{n}^{\u2033}}{\delta {x}_{\mathcal{l}}},\text{\hspace{1em}}n={n}^{\prime}-{n}^{\u2033}$ (2.54)

and thus also

${{p}^{\prime}}^{*}\delta {{\psi}^{\prime}}_{\mathcal{l}}^{*}=\pm \frac{\hslash}{i}{n}^{\prime}\frac{\delta {{\psi}^{\prime}}_{\mathcal{l}}^{*}}{\delta {x}_{\mathcal{l}}},\text{\hspace{1em}}{{p}^{\u2033}}^{*}\delta {{\psi}^{\u2033}}_{\mathcal{l}}^{*}=\pm \frac{\hslash}{i}{n}^{\u2033}\frac{\delta {{\psi}^{\u2033}}_{\mathcal{l}}^{*}}{\delta {x}_{\mathcal{l}}},$ (2.55)

having multiplied both sides of these equations by $\delta {{\psi}^{\prime}}_{\mathcal{l}}^{\mathrm{*}}$ and $\delta {{\psi}^{\u2033}}_{\mathcal{l}}^{\mathrm{*}}$ with the purpose of obtaining again via (2.55) a real value of momentum consistent with the relativistic ${p}_{\mathcal{l}}$ of (2.49). Subtracting side by side (2.55) and writing explicitly $\delta {{\psi}^{\prime}}_{\mathcal{l}}^{*}={{\psi}^{\prime}}_{\mathcal{l}}^{*}-{{\psi}^{\prime}}_{\mathcal{l}0}$ and $\delta {{\psi}^{\u2033}}_{\mathcal{l}}^{*}={{\psi}^{\u2033}}_{\mathcal{l}}^{*}-{{\psi}^{\u2033}}_{\mathcal{l}0}^{*}$, (2.54) yields

$\begin{array}{l}{{p}^{\prime}}^{*}\left({{\psi}^{\prime}}_{\mathcal{l}}^{*}-{{\psi}^{\prime}}_{\mathcal{l}0}\right)-{{p}^{\u2033}}^{*}\left({{\psi}^{\u2033}}_{\mathcal{l}}^{*}-{{\psi}^{\u2033}}_{\mathcal{l}0}^{*}\right)\\ =\pm \frac{\hslash}{i}{n}^{\prime}\frac{\delta}{\delta {x}_{\mathcal{l}}}\left({{\psi}^{\prime}}_{\mathcal{l}}^{*}-{{\psi}^{\prime}}_{\mathcal{l}0}\right)\mp \frac{\hslash}{i}{n}^{\u2033}\frac{\delta}{\delta {x}_{\mathcal{l}}}\left({{\psi}^{\u2033}}_{\mathcal{l}}^{*}-{{\psi}^{\u2033}}_{\mathcal{l}0}\right).\end{array}$ (2.56)

If the range boundaries ${{\psi}^{\prime}}_{\mathcal{l}0}$ and ${{\psi}^{\u2033}}_{\mathcal{l}0}^{\mathrm{*}}$, arbitrary in principle, are defined such that

${{p}^{\prime}}^{*}{{\psi}^{\prime}}_{\mathcal{l}0}={{p}^{\u2033}}^{*}{{\psi}^{\u2033}}_{\mathcal{l}0}^{*},$

then (2.56) reads

${{p}^{\prime}}^{*}{{\psi}^{\prime}}_{\mathcal{l}}^{*}-{{p}^{\u2033}}^{*}{{\psi}^{\u2033}}_{\mathcal{l}}^{*}=\pm \frac{\hslash}{i}{n}^{\prime}\frac{\delta {{\psi}^{\prime}}_{\mathcal{l}}^{*}}{\delta {x}_{\mathcal{l}}}\mp \frac{\hslash}{i}{n}^{\u2033}\frac{\delta {{\psi}^{\u2033}}_{\mathcal{l}}^{*}}{\delta {x}_{\mathcal{l}}}$

and thus

${{p}^{\prime}}^{*}{{\psi}^{\prime}}_{\mathcal{l}}^{*}=\pm \frac{\hslash}{i}{n}^{\prime}\frac{\delta {{\psi}^{\prime}}_{\mathcal{l}}^{*}}{\delta {x}_{\mathcal{l}}},\text{\hspace{1em}}{{p}^{\u2033}}^{*}{{\psi}^{\u2033}}_{\mathcal{l}}^{*}=\pm \frac{\hslash}{i}{n}^{\u2033}\frac{\delta {{\psi}^{\u2033}}_{\mathcal{l}}^{*}}{\delta {x}_{\mathcal{l}}}.$ (2.57)

It is easy to acknowledge that for ${{\psi}^{\prime}}_{\mathcal{l}}^{\mathrm{*}}\to {{\psi}^{\prime}}_{\mathcal{l}0}$ and ${{\psi}^{\u2033}}_{\mathcal{l}}^{\mathrm{*}}\to {{\psi}^{\u2033}}_{\mathcal{l}0}$, along with $\delta {x}_{\mathcal{l}}\to 0$, all $\delta $ of (2.57) turn into the classical $\partial $ ; so both equations are nothing else but the classical momentum wave equations with respective real eigenvalues ${{p}^{\prime}}^{\mathrm{*}}$ and ${{p}^{\u2033}}^{\mathrm{*}}$ with an appropriate choice of ${{\psi}^{\prime}}_{\mathcal{l}0}$ and ${{\psi}^{\u2033}}_{\mathcal{l}0}$. This result, which introduces the quantization required by (1.2), is clearly the wave formulation of momentum equation via the complex wave function ${\psi}^{\mathrm{*}}$. Note that this result could have been obtained more shortly starting from (2.27) rewritten replacing ${p}_{\lambda}\equiv {p}_{\mathcal{l}}$ : regarding the De Broglie momentum as the imaginary momentum appearing in (2.52),

$\frac{n\hslash}{\delta r}={p}_{\lambda}$

of (2.27) turns into

$\frac{\hslash}{i}\frac{n}{\delta r}=\frac{{p}_{\mathcal{l}}}{i}=\pm {p}^{*},$

whence, multiplying by the function $\delta \psi $ both sides,

$\frac{\hslash}{i}\frac{\delta \psi}{\delta r}=\pm \frac{\delta \psi}{n}{p}^{*}.$

Thus one finds

$\frac{\hslash}{i}\frac{\delta \psi}{\delta r}=\pm \psi {p}^{*},\text{\hspace{1em}}\delta \psi =n\psi .$

The last position expresses the range
$\delta \psi $ as a function of *n*; *i.e.* it follows thinking the values of *n* in an arbitrary range
${n}^{\prime}\le n\le {n}^{\u2033}$ and thus regarding
$\delta \psi $ as a range of terms
${n}^{\prime}\psi \mathrm{,}\left({n}^{\prime}+1\right)\psi ,\cdots ,\left({n}^{\u2033}-1\right)\psi \mathrm{,}{n}^{\u2033}\psi $ through which one calculates the respective eigenvalues of momentum falling between
${n}^{\prime}{p}^{\mathrm{*}}$ and
${n}^{\u2033}{p}^{\mathrm{*}}$. Consider now (2.53) with imaginary energy
${\epsilon}_{\mathcal{l}}^{\mathrm{*}}$. An identical reasoning holds of course here; trivial algebraic steps analogous to that from (2.55) to (2.57) yield

${{\u03f5}^{\prime}}^{*}{{\psi}^{\prime}}_{\mathcal{l}}^{*}=\pm \frac{\hslash}{i}{n}^{\prime}\frac{\delta {{\psi}^{\prime}}_{\mathcal{l}}^{*}}{\delta {t}_{\mathcal{l}}},\text{\hspace{1em}}{{\u03f5}^{\u2033}}^{*}{{\psi}^{\u2033}}_{\mathcal{l}}^{*}=\pm \frac{\hslash}{i}{n}^{\u2033}\frac{\delta {{\psi}^{\u2033}}_{\mathcal{l}}^{*}}{\delta {t}_{\mathcal{l}}}$ (2.58)

compatibly with the existence of states of negative energy. Moreover
${{\psi}^{\prime}}_{\mathcal{l}}^{\mathrm{*}}$ and
${{\psi}^{\u2033}}_{\mathcal{l}}^{\mathrm{*}}$ resulting from the uncertainty range formalism of (1.2) represent a combination of the *n*-th quantum states allowed for momentum and energy of particles. Once more the reason is that the uncertainty range boundaries are unknown and arbitrary; hence one could rewrite validly (2.54) replacing the upper boundary value
${{p}^{\prime}}^{\mathrm{*}}$ with
$\xi {{p}^{\u2034}}^{\mathrm{*}}+{{p}^{\prime}}^{\mathrm{*}}$ so that
$\delta {p}^{*}=\xi {{p}^{\u2034}}^{*}+{{p}^{\prime}}^{*}-{{p}^{\u2033}}^{*}$, being
$\xi $ an arbitrary constant coefficient. If so, then (2.55) would result with
$\delta {{\psi}^{\prime}}_{\mathcal{l}}^{\mathrm{*}}$ and
$\delta {{\psi}^{\u2033}}_{\mathcal{l}}^{\mathrm{*}}$ replaced by
$\delta \left({{\psi}^{\prime}}_{\mathcal{l}}^{\mathrm{*}}+\xi {{\psi}^{\u2034}}_{\mathcal{l}}^{\mathrm{*}}\right)$ and
$\delta \left({{\psi}^{\u2033}}_{\mathcal{l}}^{\mathrm{*}}+\xi {{\psi}^{\u2034}}_{\mathcal{l}}^{\mathrm{*}}\right)$, whereas (2.57) would consist of two primed and double primed functions like this

${p}^{**}{\psi}_{\mathcal{l}}^{**}=\pm \frac{\hslash}{i}\frac{\delta {\psi}^{**}}{\delta {x}_{\mathcal{l}}},\text{\hspace{1em}}{\psi}_{\mathcal{l}}^{**}={{\psi}^{\prime}}_{\mathcal{l}}^{*}+\xi {{\psi}^{\u2034}}_{\mathcal{l}}^{*}.$ (2.59)

From these considerations inferred as corollaries of (1.2) and (2.32) without need of postulates, was born the early wave mechanics and the modern quantum mechanics.

3. Relativistic Corollaries

This subsection examines four main implications of (2.2) noting that if $\eta =const$ then $\u03f5=const$ as well, whereas $\eta =\eta \left(\delta t\right)$ implies $\u03f5=\u03f5\left(\delta t\right)$ too. Owing to (1.4) and (2.9), the definition (1.1) of space time appears compliant with the idea of a dynamic system characterized by matter, energy and forces; also, the equivalence of mass and energy of the special relativity inferred in (2.33) agrees with the feature of space time characterized by the energy density $\eta $ of (2.1) inherent to its definition (1.1). Without these results the space time would be an empty concept unavoidably abstract and unphysical. Instead, for reasons shown in the appendix B, does exist in principle an outwards pressure corresponding to the energy density $\eta $ in (2.2), which in turn can be partially or totally counterbalanced by the attractive gravitational effect of matter/energy possibly present in a given volume of space time according to (2.32) and (2.33). The space time is therefore a dynamical system, in principle in equilibrium or non-equilibrium conditions, which evolves as a function of time. This point in particular, which anyway governs its dynamics, is now concerned to justify the possible presence of mass in a volume $\delta {x}^{3}$ of space time. Implement (1.1) to find a further result based again on a dimensional reasoning. Note the possible correlation

$\frac{\hslash G}{{c}^{2}}\iff \frac{\hslash \delta \mathcal{l}}{m},\text{\hspace{1em}}m\ne \mathrm{0,}$ (3.1)

between quantities having the same physical dimensions; *m* is an arbitrary mass confined and delocalized within the arbitrary size
$\delta \mathcal{l}$ of an uncertainty space time range, thus without chance of information about its exact position. This section concerns just the physical conditions consistent with the delocalization of *m* in an uncertainty range, in agreement with (1.2).

3.1. Real and Virtual Mass

Are reasonably conceivable two conditions on the correlation (3.1), here expressed as follows

$\frac{\hslash \delta \mathcal{l}}{m}=\xi \frac{G}{{c}^{2}}$ (3.2)

being $\xi $ an appropriate proportionality factor.

(i) One concerns the Lorentz invariance of both definitions (3.1): for the first one this condition is self-evident because it is a constant, for the second one the condition must be purposely required. Write owing to (2.35)

$\frac{\delta \mathcal{l}}{m}\equiv \frac{\beta \delta \mathcal{l}}{\beta m}=\frac{\delta {\mathcal{l}}^{\prime}}{{m}^{\prime}},\text{\hspace{1em}}\beta =\sqrt{1-{v}^{2}/{c}^{2}};$

since both
$\delta {\mathcal{l}}^{\prime}$ and
${m}^{\prime}$ are Lorentz transformations of
$\delta \mathcal{l}$ and *m*, it must be true that

$m=\frac{{m}^{\prime}}{\beta},\text{\hspace{1em}}\delta {\mathcal{l}}^{\prime}=\beta \delta \mathcal{l}.$ (3.3)

With the given definition of
$\beta $, for
$v=0$ clearly
$m\equiv {m}^{\prime}$ and
$\delta {\mathcal{l}}^{\prime}\equiv \delta \mathcal{l}$ ; *i.e.* *m* is the *v* dependent dynamic mass corresponding to the rest mass
${m}^{\prime}$ defined in (2.33), whereas
$\delta {\mathcal{l}}^{\prime}$ is the space contraction of the proper length
$\delta \mathcal{l}$. It is significant that (3.3) confirms the result (2.36) obtained via (2.35).

(ii) Consider now the limit of (3.2) for
$m\to 0$ ; it is reasonable to expect that this limit is nothing else but the definition (1.1) of empty space time, *i.e.*

$\hslash \underset{m\to 0}{\mathrm{lim}}\frac{\delta \mathcal{l}}{m}=\frac{G}{{c}^{2}}\underset{m\to 0}{\mathrm{lim}}\xi =\frac{G}{{c}^{2}}{\xi}_{0},\text{\hspace{1em}}{\xi}_{0}=\hslash .$ (3.4)

This limit ensures the consistency of the definitions (3.1) in agreement with the idea of *m* delocalized in
$\delta \mathcal{l}$ : if no particle is delocalized, the range size is null. This suggests putting by dimensional reasons

$\xi =\hslash +mv\delta \mathcal{l}=v\delta \mathcal{l}\left({m}_{o}+m\right),\text{\hspace{1em}}{m}_{o}=\frac{\hslash}{v\delta \mathcal{l}}=\frac{\delta {p}_{\mathcal{l}}}{nv},\text{\hspace{1em}}m\ne 0$ (3.5)

so that the second (3.5) reads ${m}_{o}=\u03f5\delta {p}_{\mathcal{l}}/\left(n{p}_{\mathcal{l}}{c}^{2}\right)$ thanks to (2.29) and thus

${m}_{o}{c}^{2}=\frac{\u03f5}{n}\frac{\delta {p}_{\mathcal{l}}}{{p}_{\mathcal{l}}},$ (3.6)

whereas (3.2) reads

$\hslash \frac{\delta \mathcal{l}}{m}=\left({m}_{o}+m\right)v\delta \mathcal{l}\frac{G}{{c}^{2}}\mathrm{.}$ (3.7)

Note that owing to (2.35), (2.26) and (2.27)

$\frac{\hslash}{mv}=\frac{\hslash}{p\beta}=\frac{\lambda}{2\pi \beta}=\frac{{\lambda}^{\prime}}{2\pi}=\frac{\delta {r}^{\prime}}{{n}^{\prime}},\text{\hspace{1em}}{\lambda}^{\prime}=\frac{\lambda}{\beta},\text{\hspace{1em}}2\pi \delta {r}^{\prime}={n}^{\prime}{\lambda}^{\prime};$ (3.8)

the last step of the chain means that whatever ${\lambda}^{\prime}$ might be, it is possible to define a corresponding $\delta {r}^{\prime}$ that must identically fulfill the condition (2.26) being inessential the primed notation. Hence (3.7) yields

$\frac{\delta {r}^{\prime}}{{n}^{\prime}}=\left({m}_{o}+m\right)\frac{G}{{c}^{2}}\mathrm{.}$ (3.9)

On the one hand multiplying and dividing the right hand side by ${m}_{o}m$ (3.9) yields

$\frac{\mu {c}^{2}}{{n}^{\prime}}=G\frac{{m}_{o}m}{\delta {r}^{\prime}},\text{\hspace{1em}}\mu =\frac{1}{{m}_{o}}+\frac{1}{m},$ (3.10)

*i.e.* still holds a Newton-like law but with positive sign, yet in principle still consistent with (2.11). On the other hand, examine (3.9) that holds in general for any
$m\ne {m}_{o}\ne 0$ and takes the meaning of quantum vacuum fluctuation consisting of the presence of two particles of masses *m* and
${m}_{o}$. Regard now (3.9) in the particular case especially important where these masses form a virtual pair particle/antiparticle. Specifying then with more expressive notation
${m}_{o}+m={m}_{\text{v}}+{m}_{\text{v}}^{\mathrm{*}}$, where the subscript v stands for *virtual*, write thus

$\delta {r}^{\prime}={n}^{\prime}\left({m}_{\text{v}}+{m}_{\text{v}}^{\mathrm{*}}\right)\frac{G}{{c}^{2}}={n}^{\prime}\frac{{\u03f5}_{\text{v}}}{{F}_{P}},\text{\hspace{1em}}{\u03f5}_{\text{v}}=\left({m}_{\text{v}}+{m}_{\text{v}}^{\mathrm{*}}\right){c}^{2},\text{\hspace{1em}}{F}_{P}=\frac{{c}^{4}}{G}$ (3.11)

via the Planck force ${F}_{P}$. Then

$\begin{array}{l}\delta {p}_{r}=\frac{n\hslash}{\delta {r}^{\prime}}=\frac{n}{{n}^{\prime}}\frac{h}{{\lambda}^{\prime}}=\frac{n}{{n}^{\prime}}{p}^{\prime}=\frac{n}{{n}^{\prime}}\frac{\hslash {F}_{P}}{{\u03f5}_{\text{v}}},\text{\hspace{1em}}\delta {t}_{\text{v}}=\frac{n\hslash}{\delta {\u03f5}_{\text{v}}}=n\frac{\hslash}{{F}_{P}\delta {r}^{\prime}},\\ \delta {\u03f5}_{\text{v}}={n}^{\prime}{\u03f5}_{\text{v}},\text{\hspace{1em}}{n}^{\prime}\ne n\mathrm{;}\end{array}$ (3.12)

the last position holds because
${n}^{\prime}$ is due to the integer number of wavelengths consistent with
$2\pi \delta {r}^{\prime}$ according to (2.26), whereas *n* is clearly due to (1.2). In turn,
$\delta {\u03f5}_{\text{v}}={n}^{\prime}{\u03f5}_{\text{v}}$ implies the rising of a random number
${n}^{\prime}$ of pairs of particles and antiparticles with total energy
${n}^{\prime}{\u03f5}_{\text{v}}$ allowed to exist during the time lapse
$\delta {t}_{\text{v}}$ ; it is significant that (3.1) implies
${n}^{\prime}$ fluctuation driven couples of particles/antiparticles. Examine now the chance of defining as a further particular case of (3.9) the lower limit value of
$\delta {r}^{\prime}$. Consider first to this purpose
${n}^{\prime}=n=1$, so that

$\delta r=\delta {{r}^{\prime}}_{n\mathrm{=1}}=\left({m}_{o}+m\right)\frac{G}{{c}^{2}}\mathrm{;}$ (3.13)

moreover it is possible to infer from the second (3.5)

$\delta {r}_{\mathrm{min}}=\frac{\left(m+{m}_{oc}\right)G}{{c}^{2}},\text{\hspace{1em}}{m}_{oc}=\frac{\hslash}{c\delta \mathcal{l}}=\frac{\hslash}{c{\u019b}_{mo}},\text{\hspace{1em}}n=1,\text{\hspace{1em}}v=c,\text{\hspace{0.05em}}$ (3.14)

where the last position implies consequently
$\delta \mathcal{l}={\u019b}_{mo}$ by definition whatever
${m}_{oc}$ might be. Eventually, since *m* and
${m}_{o}$ are arbitrary,
${m}_{o}$ in particular has been defined by dimensional reasons only, it is also possible to consider
${m}_{oc}=m\ne 0$ for the simple reason that nothing hinders this position; in this last particular case

$\delta {r}_{\mathrm{min}}\equiv \delta {r}_{bh}=2m\frac{G}{{c}^{2}},\text{\hspace{1em}}n=1,\text{\hspace{1em}}v=c,\text{\hspace{1em}}{m}_{oc}=m,$ (3.15)

whereas the last position requires

$m={m}_{oc}=\frac{\hslash}{c{\u019b}_{mo}}=\frac{h}{c{\lambda}_{mo}}=\frac{{p}_{mo}}{c}.$ (3.16)

Consider now the condition (3.5) to highlight when *m* can be assumed in fact confined in the range size
$\delta \mathcal{l}$ during an arbitrary time range
$\delta t$ : this requirement implies that
${v}_{\mathcal{l}}\delta t\le \delta \mathcal{l}$, which must hold for both components
$\pm {v}_{\mathcal{l}}$ of the displacement velocity vector v in principle necessary to introduce the delocalization of *m* along
$\delta \mathcal{l}$ ; if
$\delta t$ is defined in *R* fixed on *m*, then the confinement condition requires actually
$\left|{v}_{\mathcal{l}}\right|\delta t\le \delta \mathcal{l}/2$ to allow at least one chance for *m* to remain really confined when displacing at rate
$\left|{v}_{\mathcal{l}}\right|$ towards either possible direction from the middle coordinate of
$\delta \mathcal{l}$. Thus under the condition

$\frac{\hslash G}{{c}^{2}}=\frac{\hslash}{m}\frac{\delta \mathcal{l}}{2}$ (3.17)

the initial mass-free space time
$\hslash G/{c}^{2}$ includes now *m* during the time range
$\delta t$ whatever its
$\pm {v}_{\mathcal{l}}$ might be. This explains the link between
$\delta \mathcal{l}\ge 2\left|{v}_{\mathcal{l}}\right|\delta t$ defining
$\delta {r}_{\mathrm{min}}$ and the factor 2 defining
$\delta {r}_{bh}$. Hence the confinement condition of *m* in the given
$\delta {r}_{bh}$ implies (3.16), which in turn holds even for photons because
${m}_{oc}$ is defined via *c*. Note eventually that (3.16) takes an interesting form writing

$\frac{2mG}{{c}^{2}}=\frac{2m{c}^{2}G}{{c}^{4}}=\frac{2h{\nu}_{bh}}{{F}_{P}}=\delta {r}_{bh},\text{\hspace{1em}}m=\frac{h{\nu}_{bh}}{{c}^{2}}=\frac{h}{c{\lambda}_{mo}},\text{\hspace{1em}}h{\nu}_{bh}=\frac{1}{2}{F}_{P}\delta {r}_{bh};$ (3.18)

via the Planck force. So
$\delta {r}_{bh}$ takes the form of a zero point energy of a mass *m* oscillating with frequency
${\nu}_{bh}$ corresponding to (3.15) in its confinement range
$\delta \mathcal{l}$.

3.2. Invariant Equations of Special Relativity.

Rewrite Identically (1.3) as

$\frac{\hslash G}{{c}^{2}}=\frac{{v}^{2}lengt{h}^{3}}{{v}^{2}time}\mathrm{,}$ (3.19)

being
$v\le c$ the modulus of an arbitrary velocity allowed in the space time containing mass, concerned in the previous subsection. In principle *v* could be the group velocity (2.25) of a wave packet propagating through space time volume filled with dispersive medium, or it could be the expansion rate of the boundary of space time volume compatible with (2.14), or eventually it could even be simply the velocity of a body of matter moving through the space time; it depends on how is defined *v*. To examine this point regard *v* as a possible velocity allowed in the space time, whatever it might represent in any reference system, and consider that (3.19) identically rewritten as

$\frac{\hslash G}{{v}^{2}}=\frac{{c}^{2}}{{v}^{2}}\frac{lengt{h}^{3}}{time}=\frac{{c}^{2}}{{v}^{2}}\frac{\delta {x}^{3}}{\delta t}$ (3.20)

describes the swelling of the early space time volume introduced in (1.3), here indicated as
$\delta {x}^{3}$ along with the factor
${c}^{2}/{v}^{2}$. This equation is justified by (2.13), (2.14), (2.2) and (2.9) and will be further implemented also in the next subsection 3.5. Rewriting explicitly (1.1) as a function of *v* one finds therefore

$\frac{\hslash G}{{v}^{2}}=\frac{\Delta {x}^{3}}{\delta t},\text{\hspace{1em}}{v}^{2}\le {c}^{2},$ (3.21)

where

$\Delta {x}^{3}=\frac{{c}^{2}}{{v}^{2}}\delta {x}^{3}\mathrm{;}$ (3.22)

accordingly the identity (3.19) becomes compliant with the space swelling rate during the time lapse $\delta t$, whereas (2.1) yields

${\u03f5}^{2}=\hslash \eta \frac{{v}^{2}}{{c}^{2}}\frac{\hslash G}{{v}^{2}}=\hslash \eta \frac{{v}^{2}}{{c}^{2}}\frac{\Delta {x}^{3}}{\delta t}.$ (3.23)

In principle this result is compatible with (2.4) and (2.14). A corollary of (3.22) follows starting again from (3.21) to write

$\frac{\hslash G}{{v}^{2}}=\frac{\hslash G}{{v}_{s}^{2}+{v}_{\mathcal{l}}^{2}}=\frac{\hslash G\delta {t}^{2}}{\delta {s}^{2}+\delta {\mathcal{l}}^{2}},\text{\hspace{1em}}{v}_{s}=\frac{\delta s}{\delta t},\text{\hspace{1em}}{v}_{\mathcal{l}}=\frac{\delta \mathcal{l}}{\delta t},$ (3.24)

being ${v}_{s}$ and ${v}_{\mathcal{l}}$ arbitrary velocities. Let be now $\delta {s}^{2}+\delta {\mathcal{l}}^{2}$ such that by definition

$\delta {s}_{o}^{2}+\delta {\mathcal{l}}_{o}^{2}={c}^{2}\delta {t}^{2}\mathrm{,}$ (3.25)

being $\delta {s}_{o}$ and $\delta {\mathcal{l}}_{o}$ specific values that for $\delta s\to \pm \delta {s}_{o}$ and $\delta \mathcal{l}\to \pm \delta {\mathcal{l}}_{o}$ verify (3.25). Hence

$\frac{\hslash G}{{v}_{o}^{2}}=\frac{\hslash G\delta {t}^{2}}{{c}^{2}\delta {t}^{2}},\text{\hspace{1em}}{v}_{o}^{2}\equiv {c}^{2}:$ (3.26)

if (3.25) is true, then (3.26) is nothing else but the initial definition (1.1) of space time itself, already found in (2.47), whereas it appears that either $\delta {s}_{o}^{2}$ or $\delta {\mathcal{l}}_{o}^{2}$ of (3.25) tend to the invariant interval of the special relativity. In other words, the step from (3.19) to (3.20) introducing the space time swelling implies the interval invariant rule. Consider indeed (3.22); Appendix B shows how to obtain from this equation the invariants

${c}^{2}\delta {t}^{2}-\delta {\mathcal{l}}^{2}={c}^{2}\delta {{t}^{\prime}}^{2}-\delta {{\mathcal{l}}^{\prime}}^{2},\text{\hspace{1em}}\delta \mathcal{l}\delta t=\delta {\mathcal{l}}^{\prime}\delta {t}^{\prime}$ (3.27)

according which trivial manipulations yield, as shown in (3.3),

$\delta {\mathcal{l}}^{2}=\delta {{\mathcal{l}}^{\prime}}^{2}\left(1-\frac{{v}^{2}}{{c}^{2}}\right),\text{\hspace{1em}}\delta {t}^{2}=\frac{\delta {{t}^{\prime}}^{2}}{1-{v}^{2}/{c}^{2}}.$ (3.28)

The algebraic steps show that Lorentz transformations and invariant interval in inertial *R* and *R'*, here introduced for simplicity via a one dimensional approach but immediately referable to a 4D formulation, are intrinsically inherent the space time definition (1.1) yield again space contraction and time dilation of special relativity. Equation (3.27) is particularly important because it is shown in [13] that the invariant interval is the conceptual basis of the special relativity, whence the chance of obtaining in particular (3.28).

3.3. Relativistic velocity

The results hitherto achieved compel explaining the concept of velocity. Multiplying both sides of (2.10) by $v/{c}^{3}$ one finds by consequence of (2.9) and according to (2.29)

$\frac{v}{{c}^{3}}\frac{\delta v}{\delta t}=\frac{v}{{c}^{3}}\frac{2\pi \u03f5c}{h}=\frac{2\pi \u03f5v}{h{c}^{2}}=\frac{2\pi p}{h},\text{\hspace{1em}}p=\frac{\u03f5v}{{c}^{2}};$ (3.29)

it follows thus

$\frac{2\pi}{\lambda}=\frac{v}{{c}^{3}}\frac{\delta v}{\delta t},\text{\hspace{1em}}p=\frac{h}{\lambda},$ (3.30)

whereas it is possible to define

$\frac{n}{\delta r}=\frac{v}{{c}^{3}}\frac{\delta v}{\delta t},\text{\hspace{1em}}\frac{2\pi}{\lambda}=\frac{n}{\delta r},\text{\hspace{1em}}2\pi \delta r=n\lambda $ (3.31)

that allows the last equation. Once more it is worth emphasizing that the invariant (3.29) has been obtained along with the third crucial equation. Therefore, merely examining the definition (2.8) of acceleration one finds quickly results already obtained in the Equations (2.29) to (2.35) starting from the three (2.9). But now there is more. Rewrite the first (3.31) as

$\frac{nc\delta t}{\delta r}=\frac{nc}{{v}_{r}}=\frac{v\delta v}{{c}^{2}},\text{\hspace{1em}}{v}_{r}=\frac{\delta r}{\delta t},$ (3.32)

which yields according to (2.22)

$\frac{{v}_{r}v}{n{c}^{2}}=\frac{c}{\delta v}$ (3.33)

that reads identically

$1-\frac{{v}_{r}v}{n{c}^{2}}=1-\frac{c}{\delta v}=\frac{\delta v-c}{\delta v}$

and thus, taking the reciprocals of both sides,

$\frac{1}{1-{v}_{r}v/n{c}^{2}}=\frac{\delta v}{\delta v-c};$

hence

$\frac{\delta v-c}{1-{v}_{r}v/n{c}^{2}}=\delta v.$ (3.34)

First of all eliminate *n*; it could be put equal to 1 by definition, yet it is easy to follow a general procedure valid for any *n*. With the positions

${V}_{r}=\frac{{v}_{r}}{\sqrt{n}},\text{\hspace{1em}}V=\frac{v}{\sqrt{n}},\text{\hspace{1em}}C=\frac{c}{\sqrt{n}},\text{\hspace{1em}}\delta V=\frac{\delta v}{\sqrt{n}}$ (3.35)

(3.34) reads

$\frac{\delta V-C}{1-{V}_{r}V/{c}^{2}}=\delta V=\delta V-C+C.$

Trivial manipulations of this equation yield

$\left(\delta V-C\right)\left(\frac{1}{1-{V}_{r}V/{c}^{2}}-1\right)=C$

*i.e. *

$\frac{\left(\delta V-C\right){V}_{r}V/{c}^{2}}{1-{V}_{r}V/{c}^{2}}=C\mathrm{.}$ (3.36)

Let us elaborate further this result in order to obtain a significant equation; is useful in particular the position

$\left(\delta V-C\right)\frac{{V}_{r}V}{{c}^{2}}=\xi \left({V}_{r}-V\right),$ (3.37)

where $\xi $ is arbitrary proportionality factor. With $\delta V$ in principle arbitrary as well because of the uncertainty, as previously stated, this position is allowed. This step appears important rewriting (3.36) via (3.37) as

$\frac{\left({V}_{r}-V\right)\xi}{1-{V}_{r}V/{c}^{2}}=C\mathrm{,}$

which in turn yields

$\frac{{V}_{r}-V}{1-{V}_{r}V/{c}^{2}}={V}^{*},\text{\hspace{1em}}{V}^{*}=\frac{C}{\xi}$ (3.38)

where the resulting
${V}^{\mathrm{*}}$ is still an arbitrary velocity. This formula is actually well known, as it relates in special relativity
${V}_{r}-V$ to
${V}^{\mathrm{*}}$ ; it is evident that in the particular cases where
$V=c$ or
${V}_{r}=c$ then
${V}^{*}=c$, *i.e.* the sum of *c* plus any velocity returns always *c*.

This reasoning is not at all redundant repetition of a result already known: (3.30) and (3.31) are quantum properties obtained contextually to (3.29) that is the invariant definition of relativistic momentum. Hence the reasoning implies merging of quantum and relativistic results concurring to the definition of *c* as an invariant limit velocity: this crucial statement of relativity is here required by (1.2). Note that (3.38) has been obtained via *V _{r}* and

$\frac{{v}_{r}v}{n{c}^{2}}=\frac{{V}_{r}V}{{c}^{2}}=\frac{C}{\delta V}:$ (3.39)

the problem of (3.33) is that the left hand side vanishes for
$n\to \infty $ incompatibly with the right hand side that never vanishes because
$\delta v\le c$. Owing to (3.35), instead, at the right hand side of (3.33) appear just the velocities leading to the result (3.38) of actual interest without contradicting the arbitrariness of *n*.

So (3.38) completes the conclusion (2.18), where a well known quantum inequality was inferred just from a physical property of *c*; here also this property of *c* appears as a further corollary of (1.1) and (1.2). Once more, as already shown in further papers [4] [11] [12], relativistic and quantum principles appear in the present approach as harmonically coexisting concepts without “ad hoc” hypotheses.

4. Euler-Lagrange Equations and Gravitational Potential

Start from (1.2) that yields

$\delta \stackrel{\dot{}}{x}=-\frac{n\hslash}{\delta {p}^{2}}\delta \stackrel{\dot{}}{p}=-\frac{\delta x}{\delta p}\delta \stackrel{\dot{}}{p};$ (4.1)

also, recalling the considerations of subsection 2.6, let us define

$\delta \stackrel{\dot{}}{x}=-\frac{\delta}{\delta t}\delta x,\text{\hspace{1em}}\delta \stackrel{\dot{}}{p}=\frac{\delta}{\delta t}\delta p,\text{\hspace{1em}}\delta \stackrel{\dot{}}{\epsilon}=\frac{\delta}{\delta t}\delta \epsilon .$ (4.2)

Note now that (4.1) can be rewritten as

$\delta \stackrel{\dot{}}{x}=-\frac{\delta x}{\delta p}\delta \stackrel{\dot{}}{p}=-\frac{\delta \stackrel{\dot{}}{p}}{\delta p}\delta x.$ (4.3)

The step from (4.1) to (4.3) is not trivial. As anticipated in the subsections 2.6 and 2.7, the chance of exchanging the place of $\delta x$ and $\delta \stackrel{\dot{}}{p}$ fulfills the specific concept of derivative in the physical frame of the quantum uncertainty; in fact $\delta \stackrel{\dot{}}{x}$ is mere ratio of arbitrary ranges finite by definition, to be regarded as independent differentials possibly but not necessarily tending both to zero. This subsection aims just to show that this way of intending the quantum derivative is physically sensible. Multiplying now both sides of the second equality (4.3) by $\delta x$, one finds

$\delta \stackrel{\dot{}}{x}\delta x=-\frac{\delta \stackrel{\dot{}}{p}}{\delta p}\delta {x}^{2}.$ (4.4)

Define now a function *f* consistent with this result, *i.e.* such that
$\delta f$ fulfills

$\delta \stackrel{\dot{}}{x}\delta x=\delta f=-\frac{\delta \stackrel{\dot{}}{p}}{\delta p}\delta {x}^{2},\text{\hspace{1em}}f=f\left(x,\stackrel{\dot{}}{x},p,\stackrel{\dot{}}{p}\right);$ (4.5)

in turn (4.4) and (4.5) are consistent with the positions

$\delta x=\frac{\delta f}{\delta \stackrel{\dot{}}{x}},\text{\hspace{1em}}-\frac{\delta \stackrel{\dot{}}{p}}{\delta p}\delta x=\frac{\delta f}{\delta x}.$ (4.6)

As concerns the first equation, the first (4.2) yields

$\delta \stackrel{\dot{}}{x}=\frac{\delta}{\delta t}\frac{\delta f}{\delta \stackrel{\dot{}}{x}},$

whereas (4.4) reads with the help of the second (4.6)

$\delta \stackrel{\dot{}}{x}=-\frac{\delta \stackrel{\dot{}}{p}}{\delta p}\delta x=\frac{\delta f}{\delta x}.$

Hence, merging the last two results, one finds

$\frac{\delta}{\delta t}\frac{\delta f}{\delta \stackrel{\dot{}}{x}}=\frac{\delta f}{\delta x}.$ (4.7)

According to (4.4) the function *f* has physical dimensions *length*^{2}/*time* and fulfills the same kind of equation of the Lagrangian
$\mathcal{L}$ of a physical system; in fact *f* is proportional to
$\mathcal{L}$ a multiplicative constant
${c}^{3}/G$ apart. Since
${f}^{\prime}=f{c}^{3}/G$ is an energy, this is in principle just the sought Lagrangian. Yet the way to obtain this equation via the proportionality constant does not require the condition
${E}_{kin}={E}_{kin}\left(\stackrel{\dot{}}{x}\right)$ and
${E}_{pot}={E}_{pot}\left(x\right)$.

The Euler-Lagrange equations are well known; yet the non-trivial fact is that they have been inferred here as corollaries of (1.2) and (1.1), which are the conceptual root of both relativistic and quantum physics. Moreover this result supports the present way to regard the concept of derivative as a ratio of uncertainty ranges. Follow now two checks of the present way of reasoning.

(i) The classical Newton law inferred in the section 2.2 seemingly does not account for the finite propagation rate of any perturbation or interaction. Actually such information is explicitly available writing

$\begin{array}{l}\delta \stackrel{\dot{}}{p}=-\frac{n\hslash}{\delta {x}^{2}}\delta \stackrel{\dot{}}{x}=-\frac{\delta \epsilon \delta t}{\delta {x}^{2}}\delta \stackrel{\dot{}}{x}=-\frac{\delta \epsilon}{\delta {x}^{2}}\delta t\delta \stackrel{\dot{}}{x}=-\frac{\delta \epsilon \delta \stackrel{\dot{}}{x}}{\delta {x}^{2}c}c\delta t=-\frac{\delta \epsilon}{\delta {x}^{2}c}\delta s\delta \stackrel{\dot{}}{x},\\ \delta s=c\delta t.\end{array}$ (4.8)

The force defined in this way is related to an energy
${\u03f5}^{\prime}\le \u03f5\le {\u03f5}^{\u2033}$, vanishes with a
$\delta {x}^{-2}$ law, is proportional to the deformation rate
$\delta \stackrel{\dot{}}{x}\ne 0$ of the space time range
$\delta x$, is positive or negative depending on whether
$\delta x$ swells or shrinks as a function of time and vanishes for
$\delta s\to 0$ ; *i.e.* the force is defined within
$\delta s\ne 0$. It reasonably means that a time range
$\delta t$ is necessary in order to allow its propagation at distance
$\delta s$, outside which the force in null. The fact that
$\delta s$ has been defined via *c* means the carrier of the force must be a virtual photon or a graviton or anyway a massless particle propagating at speed *c*. Note that instead the classical
$G{m}_{1}{m}_{2}/{\mathcal{l}}^{2}$ has the form of a force propagating instantaneously because it is based only on (2.9) and (2.10) without implementing (1.2). Actually (2.11) avoids itself this error because it is expressed via the uncertainty range
$\delta \mathcal{l}$, not via the deterministic
$\mathcal{l}$ ; since (1.2) involve inherently
$\delta t$, (2.11) could have been written itself as
$G{m}_{1}{m}_{2}\delta {p}_{\mathcal{l}}/\left(\delta {\u03f5}_{\mathcal{l}}\delta {t}_{\mathcal{l}}\right)$ thus involving anyway the time range
$\delta {t}_{\mathcal{l}}$ governing its propagation. This holds of course for any force. As concerns the gravity note that also now it is possible to repeat for (4.8) the considerations introduced for (2.11):
$\delta s\delta \epsilon =mass\times lengt{h}^{3}\times tim{e}^{-2}$, *i.e.* from a dimensional point of view
$\delta \epsilon \delta s\delta \stackrel{\dot{}}{x}{c}^{-1}=mas{s}^{2}\times G$. Hence, whatever the actual form of the function
$\delta \epsilon \delta s\delta \stackrel{\dot{}}{x}{c}^{-1}$ might be, it is reasonable to regard its series expansion whose first order term is a constant; if so, then neglecting for the moment the higher order terms, it is possible to write

$\delta \stackrel{\dot{}}{p}=\pm G\frac{{m}^{2}}{\delta {x}^{2}}+\cdot \cdot \cdot =\pm G\frac{{m}_{1}{m}_{2}}{\delta {x}^{2}}+\cdot \cdot \cdot .$ (4.9)

This formula is formally similar to (2.11), yet it incorporates the idea of a non instantaneous long term force that worried Newton himself. The form of the higher order terms will be concerned later, see next (5.114).

(ii) Consider eventually that (2.12) yields with the help of (1.2) and (4.2)

$\begin{array}{c}\stackrel{\dot{}}{v}=\frac{\delta \stackrel{\dot{}}{\epsilon}\delta p-\delta \epsilon \delta \stackrel{\dot{}}{p}}{\delta {p}^{2}}=\frac{\delta \stackrel{\dot{}}{\epsilon}}{\delta p}-\frac{vF}{\delta p}=-\frac{n\hslash}{\delta p\delta {t}^{2}}-\frac{vF}{\delta p}=-\frac{\delta x}{\delta {t}^{2}}\left(1+\frac{vF\delta {t}^{2}}{n\hslash}\right)\\ =-\frac{\delta x}{\delta {t}^{2}}\left(1+\frac{F\delta \mathcal{l}}{\delta \epsilon}\right)=-\frac{\delta x}{\delta {t}^{2}}\left(1+\frac{\delta {\epsilon}_{\mathcal{l}}}{\delta \epsilon}\right)\end{array}$

where

$F=\delta \stackrel{\dot{}}{p},\text{\hspace{1em}}\delta \mathcal{l}=v\delta t,\text{\hspace{1em}}\delta {\epsilon}_{\mathcal{l}}=F\delta \mathcal{l},\text{\hspace{1em}}\delta \epsilon =\frac{n\hslash}{\delta t}.$ (4.10)

In summary it is possible to write this result as

$\stackrel{\dot{}}{v}=-\frac{\delta \phi}{\delta x},\text{\hspace{1em}}\delta \phi =\frac{\delta {x}^{2}}{\delta {t}^{2}}\left(1+\frac{\delta {\epsilon}_{\mathcal{l}}}{\delta \epsilon}\right),$ (4.11)

where
$\phi $ is a new function having physical dimensions *velocity*^{2}; with vector notation the first equation reads

$\stackrel{\dot{}}{v}=-\nabla \phi \mathrm{.}$ (4.12)

This definition, inferred here as a corollary, was taken in [13] as a basis to infer special and general relativity; the sign of $\delta \phi $ depends on whether $\delta {\epsilon}_{\mathcal{l}}/\delta \epsilon \u22db-1$. Moreover (4.11) also reads according to (2.8)

$\stackrel{\dot{}}{v}=\frac{\epsilon c}{\hslash}=-{a}_{x}\left(1+\frac{\delta {\epsilon}_{\mathcal{l}}}{\delta \epsilon}\right),\text{\hspace{1em}}{a}_{x}=\frac{\delta x}{\delta {t}^{2}},$ (4.13)

where ${a}_{x}$ has physical dimensions of acceleration. It is immediate to acknowledge that $\phi $ of (4.11) is the definition gravitational potential [13], which will be more specifically concerned in the next subsection 5.5. Also, $\stackrel{\dot{}}{v}$ is not simply ${a}_{x}$ but includes a further addend $-{a}_{x}\delta {\epsilon}_{\mathcal{l}}/\delta \epsilon $.

These concepts, more systematically examined in the next sections, have been preliminarily introduced to show the validity of the definition (1.1) of space time, which will be implemented next according to (1.3).

Space Time Curvature

Consider (3.20) under the particular condition where the velocity ${v}^{2}$ can be expressed as follows

${v}^{2}=a\delta \mathcal{l},\text{\hspace{1em}}v=v\left(\delta \mathcal{l},\delta t\right),\text{\hspace{1em}}\delta \mathcal{l}=\delta \mathcal{l}\left(\delta t\right),$ (4.14)

being *a* acceleration by dimensional reasons. Hence (3.20) yields

$\delta \left(\frac{\hslash G}{{v}^{2}}\right)=\frac{\hslash G}{a}\delta \left(\frac{1}{\delta \mathcal{l}}\right)-\frac{\hslash G}{{a}^{2}\delta \mathcal{l}}\delta a=\frac{\hslash G}{a}\delta \left(\frac{1}{\delta \mathcal{l}}\right)-\frac{\hslash G}{{v}^{2}}\frac{\delta a}{a}=\frac{\hslash G}{a}\left(\delta \left(\frac{1}{\delta \mathcal{l}}\right)-\frac{\delta a}{{v}^{2}}\right).$

Summarizing therefore this result as

$\delta \left(\frac{\hslash G}{{v}^{2}}\right)=\frac{\hslash G}{a}\left(\frac{1}{\delta {\mathcal{l}}_{1}}-\frac{1}{\delta {\mathcal{l}}_{2}}-\frac{\delta a}{{v}^{2}}\right),\text{\hspace{1em}}\delta \left(\frac{1}{\delta \mathcal{l}}\right)=\frac{1}{\delta {\mathcal{l}}_{1}}-\frac{1}{\delta {\mathcal{l}}_{2}},\text{\hspace{0.05em}}$ (4.15)

by definition of uncertainty range ${\left(\delta \mathcal{l}\right)}^{-1}$, there are in principle three chances. The first one is that

$\delta \left(\frac{\hslash G}{{v}^{2}}\right)=\frac{\hslash G}{{a}_{0}}\left(\frac{1}{\delta {\mathcal{l}}_{1}}-\frac{1}{\delta {\mathcal{l}}_{2}}\right),\text{\hspace{1em}}a\equiv {a}_{0},\text{\hspace{1em}}\delta a=0,$ (4.16)

with notation emphasizing that $a$ is a constant. Moreover are also possible for (4.15)

$\delta \left(\frac{\hslash G}{{v}^{2}}\right)=-\frac{\hslash G}{a}\frac{1}{\delta {\mathcal{l}}_{2}},\text{\hspace{1em}}\frac{1}{\delta {\mathcal{l}}_{1}}=\frac{\delta a}{{v}^{2}}$

and

$\delta \left(\frac{\hslash G}{{v}^{2}}\right)=\frac{\hslash G}{a}\frac{1}{\delta {\mathcal{l}}_{1}},\text{\hspace{1em}}\frac{1}{\delta {\mathcal{l}}_{2}}=-\frac{\delta a}{{v}^{2}}.$

Clearly $a$ is the acceleration describing the change of space time swelling rate as a function of time; since the time is inherent the physical dimension of $\hslash G$ appear natural the positions (4.14). In the particular case (4.16) where $a$ is constant, it is possible to write

${v}^{*}=\frac{{a}_{0}}{{v}^{*}}\delta \mathcal{l},\text{\hspace{1em}}{v}^{*}=\frac{\delta \mathcal{l}}{\delta {t}^{*}},\text{\hspace{1em}}\delta {t}^{*}=\frac{{v}^{*}}{{a}_{0}},$ (4.17)

whereas (4.15) becomes

$\delta \left(\frac{\hslash G}{{v}^{2}}\right)=\frac{\hslash G}{{a}_{0}}\left(\frac{1}{\delta {\mathcal{l}}_{1}}-\frac{1}{\delta {\mathcal{l}}_{2}}\right).$ (4.18)

Both $\delta {\mathcal{l}}_{1}$ and $\delta {\mathcal{l}}_{2}$ are arbitrary, in particular these boundaries of the range ${\left(\delta \mathcal{l}\right)}^{-1}$ can be positive or negative; the resulting sign of mean that for any physical reason $\delta \mathcal{l}$ expands or shrinks with constant acceleration ${a}_{0}$ as a function of time. Implementing again the arbitrariness of range boundaries, the last equation reads

$\delta \left(\frac{1}{\delta \mathcal{l}}\right)=\pm \mathcal{R},\text{\hspace{1em}}\mathcal{R}=\frac{1}{\delta {\mathcal{l}}_{2}}\pm \frac{1}{\delta {\mathcal{l}}_{1}},$ (4.19)

being $\mathcal{R}$ the Laplace-like curvature radius of space time with principal curvature radii $\pm \delta {\mathcal{l}}_{1}$ and $\pm \delta {\mathcal{l}}_{2}$. In general the signs of these radii depend on the specific problem [14], e.g.: for a liquid droplet in a gaseous environment are both positive, for a gas bubble in a liquid environment both negative, for a liquid meniscus between solid cylinders with saddle-like geometry one positive and one negative. So it is not surprising that in principle all chances have been found in the present general approach starting from first principles as concerns the space time swelling.

It is significant anyway that the concept of space time curvature is definable in a natural way even in the present quantum/relativistic context through the concept of uncertainty range. It is instructive in this respect the crucial role of (1.2) in linking quantum and relativistic points of view. Consider two remarks.

(i) Consider (1.2) to express in particular radial range size $\delta r$ and conjugate radial momentum range size

$\delta r=\frac{n\hslash}{\delta {p}_{r}}.$ (4.20)

Specify this equation as done in (2.24) and (2.25) *i.e.* implementing the De Broglie definition
${p}_{r}=h/{\lambda}_{r}$ of radial momentum, corollary itself of (1.2) [15]. Accordingly write

$2\pi \delta r=\frac{nh}{\delta {p}_{r}}=\frac{n}{\delta \left({p}_{r}/h\right)}=\frac{n}{\delta \left(1/{\lambda}_{r}\right)};$ (4.21)

then also consider that by definition $\delta {\lambda}_{r}^{-1}$ is nothing else but ${{\lambda}^{\prime}}_{r}^{-1}-{{\lambda}^{\u2033}}_{r}^{-1}$, whatever the range boundaries ${{\lambda}^{\prime}}_{r}$ and ${{\lambda}^{\u2033}}_{r}$ might be. Hence (4.21) turns into

$2\pi \delta r=\frac{nh}{\delta {p}_{r}}=\frac{n}{1/{{\lambda}^{\prime}}_{r}-1/{{\lambda}^{\u2033}}_{r}}=\frac{n}{\mathcal{R}},\text{\hspace{1em}}\mathcal{R}=\frac{1}{{{\lambda}^{\prime}}_{r}}-\frac{1}{{{\lambda}^{\u2033}}_{r}},$ (4.22)

with notation emphasizing again that $\mathcal{R}$ is clearly the Laplace curvature radius according to the reasoning carried out to infer (4.19); this in turn supports the condition (4.14).

(ii) In general the space time size change rate $\delta \stackrel{\dot{}}{x}$, as defined in (4.2), is related to the rising of a force field $\delta \stackrel{\dot{}}{p}$ within $\delta x$ itself; so write via (4.10)

$\delta \stackrel{\dot{}}{x}\u22da0,\text{\hspace{1em}}\delta \stackrel{\dot{}}{p}=\frac{\delta}{\delta t}\delta p\u22db\mathrm{0:}$ (4.23)

the sign of the force field $\delta \stackrel{\dot{}}{p}$ depends on whether $\delta x$ swells or shrinks for any physical reason, whereas the force field is null if $\delta \stackrel{\dot{}}{x}=const$. Then, combining this result with the second (2.9) one finds

$\frac{\delta}{\delta t}\delta x=-\frac{n\hslash}{\delta {p}^{2}}\frac{m\u03f5c}{\hslash}=-\frac{nm\u03f5c}{\delta {p}^{2}}=-\frac{m\u03f5c}{n{\hslash}^{2}}\delta {x}^{2}=-force\frac{\delta {x}^{2}}{n\hslash}$ (4.24)

that in turn reads

$\begin{array}{l}force=-\frac{n\hslash}{\delta t}\frac{\delta \left(\delta x\right)}{\delta {x}^{2}}=-\delta \epsilon \left(\frac{\delta {x}_{2}}{\delta {x}^{2}}-\frac{\delta {x}_{1}}{\delta {x}^{2}}\right)=-\delta \epsilon \left(\frac{1}{\delta {x}^{\prime}}-\frac{1}{\delta {x}^{\u2033}}\right)=-\delta \epsilon \mathcal{R},\\ \delta \epsilon \mathrm{=}\frac{n\hslash}{\delta t}\mathrm{;}\end{array}$ (4.25)

the last equality is legitimated in analogy with (4.19) once having defined by dimensional reasons
$\delta {{x}^{\prime}}^{-1}=\delta {x}_{2}/\delta {x}^{2}$ and
$\delta {{x}^{\u2033}}^{-1}=\delta {x}_{1}/\delta {x}^{2}$ whatever
$\delta {x}_{1}$ and
$\delta {x}_{2}$ might be. The interesting fact is that in the second (2.9) the concept of force was directly related to that of acceleration
$\u03f5c/\hslash $, here the same force is related to the concept of curvature via the space time ranges
$\delta {x}^{\prime}$ and
$\delta {x}^{\u2033}$ replacing the acceleration. The only possible conclusion is that Newtonian concept of force and relativistic concept of space time curvature are equivalent in describing the concept of *force*. The Einstein intuition becomes corollary of the quantum uncertainty.

5. The General Relativity as a Corollary

Some relevant concepts of general relativity are quoted in this section to show how to generalize the approach hitherto followed for the special relativity. Are examined in particular further significant implications of the quantum uncertainty ranges, to show how both special and general relativity contextually merge in a unique non-local and non-real conceptual frame. Some hints in this respect have been early examined in [16]; further topics are here reminded along with new considerations just to point out what have to do these typical concepts of quantum theory with the gravitational field. Indeed the problem of quantum gravity involves non only the quantization of this field according to the distinctive concept of superposition of quantum states, but also the inherent concepts non-reality and non-locality.

5.1. The So-Called “EPR Paradox”

It has been shown in the subsection 2.8 that the wave formalism is a corollary of (1.2) together with the relativistic properties inferred in subsections 2.5, 3.1 and 3.2; these results make in principle the present model compatible with the standard answer of wave quantum mechanics to the paradox. Yet, although it would be legitimate to skip additional comments to the ample literature already existing on this topic, it is instructive to emphasize the distinctive contribution provided to the paradox by the present model based on the quantum uncertainty. Deserve attention the following crucial points of this theoretical framework: (i) the concept of uncertainty ranges replacing the local dynamical variables is in principle compatible with the concept of entanglement; (ii) the difficulty of superluminal distance is bypassed, because the deterministic concept of distance between physical objects is unphysical; (iii) the concept of non-locality reduces to that of unpredictable randomness of particles confined and delocalized in quantum uncertainty ranges and excludes any kind of local information; (iv) by consequence of (iii), the concept of “non-locality” is strictly related to that of “non-reality”.

Consider two particles, whose delocalization is in principle possible either in their own independent uncertainty ranges or in one shared uncertainty range. In the first case the particles in
$\delta {x}_{1}$ and
$\delta {x}_{2}$ are in general non-interacting, e.g. any physical reason that deforms
$\delta {x}_{1}$ like in (4.23) does not necessarily affect
$\delta {x}_{2}$, so that the force field
$\delta {\stackrel{\dot{}}{p}}_{1}$ in
$\delta {x}_{1}$ does not imply
$\delta {\stackrel{\dot{}}{p}}_{2}$ in
$\delta {x}_{2}$ too: the particle in
$\delta {x}_{1}$ experiences a force, whereas that in
$\delta {x}_{2}$ does not, *i.e.* the particles do not interact. In the second case a unique delocalization range
$\delta \stackrel{\dot{}}{x}$ also results from the way the particles interact, even if no external perturbation causes or affects
$\delta \stackrel{\dot{}}{p}$ : this is the typical case of (3.12) where pairs of virtual particles with opposite charges and spins are generated by vacuum energy fluctuation. First of all, the modulus *v* of velocity of any particle in a given point of space time cannot be specifically local velocity because are missing by definition both the space coordinates
${x}_{1}$ and
${x}_{2}$ from which to which a particle moves and the time coordinates
${t}_{1}$ and
${t}_{2}$ defining the displacement time lapse; it follows that it is unphysical to define velocity and distance and thus superluminal distances. In fact two particles confined in
$\delta x$ are neither far away nor close each other, they simply *are* in
$\delta x$. This agrees with the Aharonov-Bohm effect [17] simply acknowledging that one particle is neither “here” or “there”, rather it is simply everywhere. This holds even though the particles are delocalized in different
$\delta {x}_{1}$ and
$\delta {x}_{2}$ : as the boundary coordinates of uncertainty ranges are arbitrary, certainly the impossibility of determining distances and velocities holds identically also for two particles in their own uncertainty ranges. Hence do not exist “spooking actions at a distance” but rather “actions at a spooky distance”: once having renounced to the classical determinism and accepted (1.2) there is no way to distinguish the behavior of particles far apart or close each other confined in a given delocalization range, whatever their interaction mechanism might be. Particularly interesting is the former case of two entangled particles born within a unique uncertainty range where, for example when one
$\gamma $ photon decays by interacting with a nucleus or via vacuum fluctuation, e.g. [18] [19]; in fact the latter chance found in (3.9) reasonably agrees with (2.2). Accordingly, in the conceptual frame based on (1.1) and (1.2) the EPR paradox shouldn’t even be formulated: the present model is inherently non-local by definition. Moreover the agnosticism of (1.2), not purposely invoked here but assumed since the beginning as the unique leading idea of the present physical model, implies a conceptual gap in (3.1) between elusiveness of (1.1) and reality of (3.9); the former is mere dimensional definition of the framework allowed for latent events, the latter made feasible by the measure process breaks the latency of possible events. Since nothing is “a priori” known about
${m}_{\text{v}}$ and
${m}_{\text{v}}^{\mathrm{*}}$, e.g. number of pairs or energy and lifetime of pairs and so on, the present model is inherently also non-real by definition. In other words the physical agnosticism implied by the concept of uncertainty as hitherto exposed, corresponds to the non real essence of the quantum world before the experiment; hence one must accept the idea that also the relativistic properties hitherto inferred are subjected to the same non-weird but logical consequences of (1.2) without need of postulating any “collapse” of wave function into a well defined quantum state. It means that *n* introduced in (1.2) and next appearing in (3.9) remains arbitrary and undefinable until when the measurement converts it into a specific
${n}_{obs}$ ; in turn, the wave formalism allows calculating the probabilities inherent the superposition of allowed states. Consider now the orientation of the possible spins of the particles with respect to an arbitrary direction. When measured, their spin orientation must yield a total angular momentum equal to zero like that of the empty space time (1.1) before the vacuum fluctuation (3.11). Physical information in this respect is provided only by the angular momentum conservation law, which however presupposes a measurement process. In general this is a perturbation action that affects the quantum state of any particle. In particular, being both particles in the same
$\delta x$, the measure process perturbs the system of entangled particles wherever they might be, not either particle only. If for any physical reason the shared
$\delta x$ is modified, then the consequent
$\delta \stackrel{\dot{}}{x}$ implies
$\delta \stackrel{\dot{}}{p}$ and thus a force field in
$\delta x$ that in turn perturbs the couple of particles. Is clear at this point the connection of the present reasoning with the possible spins of
${m}_{\text{v}}$ and
${m}_{\text{v}}^{\mathrm{*}}$ introduced in (3.9); in a certain sense the concept of entanglement is here stronger than usually intended once having removed the idea of “superluminal” or relativistic “luminal” distance, no longer conceived as separate and mutually excluding distinguishable situations. With the language of wave formalism, the quantum state of two entangled particles is a superposition of luminal and superluminal states. Nonetheless the conclusion is the same: the most controversial premise of the entanglement, the simultaneous perturbation linking particles infinitely apart via spooky action, is here automatically removed. The answer provided by the total agnosticism of (1.2) can bypass also the simultaneity inherent inherent entangled pairs, as proposed by the mere wave formalism, and suggests a further implication. The EPR paradox, conceived to demonstrate the inadequacy of wave mechanics, demonstrates instead the inadequacy of the deterministic metrics that fail explaining via tensor calculus the correlations between entangled particles; the experimental data show indeed that the relativity needs the “external” contribution of the wave formalism introducing the concept of entanglement. The present model explains both relativistic results and wave formalism while removing in principle the paradox itself. This is because the total agnosticism of (1.2) makes the relativity non-local and non-real itself. On the one hand the concept of non-locality, unpredictable randomness of particles delocalized in quantum uncertainty ranges excluding any kind of local information, is in turn related to that of “non-reality”: in fact this idea does not violate any relativistic principle, apart from its out of place local determinism. On the other hand it is not crucial whether the spins of entangled particles are actually aligned or counter-aligned inside
$\delta x$ before the correlation experiment, rather it is only required that they are in fact measured counter-aligned after the correlation experiment. In this sense the experiment creates the reality fulfilling the angular momentum conservation although starting from any undefined and undefinable state, be it wave/corpuscle duality or dead/alive states of Schrödinger’s cat or luminal/superluminal distance. Eventually appears clear the task of the present section: to find relativistic results without starting from a deterministic metrics, whatever it might be. To confirm that all of these considerations hold also for the relativity, the next subsections concern a few selected topics purposely chosen to emphasize the role of the quantum uncertainty in the general relativity: the latter is in fact a corollary of the former. The most important point in this respect is the equivalence principle, which is soon examined first in the section below.

5.2. The Equivalence Principle

Two relevant results previously obtained, Equations (2.9) to (4.25), address directly to Einstein’s equivalence principle, as it has been explained through the simple reasoning early concerned in [9]: the reasoning is so crucial and short to deserve being sketched here for completeness. Think a space time uncertainty range
$\delta x={x}_{2}-{x}_{1}$ with time dependent size, and two observers sitting on the boundary coordinates of this range. Let for example the lower boundary
${x}_{1}={x}_{1}\left(t\right)$ be defined with respect to the origin of an arbitrary reference system *R*, *i.e.* it defines the “position” of
$\delta x$ in *R* at a given time, whereas
${x}_{2}$ is a fixed coordinate that defines the “size” of
$\delta x$. Although neither information is actually definable and accessible, it appears in principle that if
${x}_{1}\left(t\right)$ is subjected to change as a function of time for any physical reason, then the size change rate
$\delta \stackrel{\dot{}}{x}$ of
$\delta x$ is related to the rising of a force acting on a particle possibly delocalized in
$\delta x$ ; indeed (4.23) and (4.10) predict a local force field
$\stackrel{\dot{}}{p}$ whose strength falls within a range of forces
$\delta \stackrel{\dot{}}{p}={\stackrel{\dot{}}{p}}_{2}-{\stackrel{\dot{}}{p}}_{1}$. More specifically, is interesting the following chain of equations inferred with the help of (4.2) and (2.9)

$\delta \stackrel{\dot{}}{x}=-\frac{n\hslash}{\delta {p}^{2}}\delta \stackrel{\dot{}}{p}=-\frac{\delta x}{\delta p}\frac{m\epsilon c}{\hslash}=-\frac{\delta x}{\delta p}ma=-\frac{\delta {x}^{2}}{n{\hslash}^{2}}m\u03f5c=-\frac{\delta {x}^{2}}{n\hslash}force:$ (5.1)

hence, owing to (4.2),

$force=n\hslash \frac{\delta \stackrel{\dot{}}{x}}{\delta {x}^{2}}=-n\hslash \frac{\delta}{\delta t}\left(\frac{1}{\delta x}\right)=-\frac{\delta}{\delta t}\delta p,$ (5.2)

*i.e.* the force field
$\delta \stackrel{\dot{}}{p}=m\u03f5c/\hslash $ due to the space time deformation rate
$\delta \stackrel{\dot{}}{x}$ is repulsive or null or attractive depending on
$\delta \stackrel{\dot{}}{x}\u22da0$ respectively. The chances of
$\delta x\left(t\right)$ are swelling, shrinking, remaining constant. Also, as

$m\frac{\u03f5c}{\hslash}=-n\hslash \frac{\delta}{\delta t}\left(\frac{1}{\delta x}\right),$ (5.3)

(5.1) yields with the help of (1.2)

$-\hslash \delta \left(\frac{1}{\delta x}\right)=\frac{m\u03f5c\delta t}{n\hslash}=\frac{m\u03f5c}{\delta \u03f5}=\frac{\u03f5\delta t}{n{\lambda}_{m}}=\frac{\u03f5}{\lambda}\delta t,\text{\hspace{1em}}\u019b=n{\u019b}_{m};$

the last position has been explained about (2.22) when commenting the eq (4.23). The fact that *force* of (5.2) results equal to
$n\hslash $ times the left hand side of the chain shows the quantum nature of this force. Eventually the right hand side of (5.3) shows the geometrical essence of an attractive force, indeed

$\frac{\hslash}{\delta t}\mathcal{R}=\frac{-\u03f5}{\lambda},\text{\hspace{1em}}\mathcal{R}=\delta \left(\frac{1}{\delta x}\right)=\frac{1}{\delta {x}_{2}}-\frac{1}{\delta {x}_{1}}$

in agreement with the second (2.19). Clearly $\mathcal{R}$ is the space time Laplace-like curvature radius corresponding to the attractive gravity force $-\u03f5$ of (2.11), as explained in (4.19).

Once having expressed the deformation of space time in terms of range size change rate
$\delta \stackrel{\dot{}}{x}$, return now to the Einstein equivalence principle considering for simplicity the change of
${x}_{1}\left(t\right)$ only with constant
${x}_{2}$ ; this is enough to account for the rising of a force field inside
$\delta x$ and highlight the reasonable conclusions of two independent Newtonian observers sitting on either boundary of
$\delta x={x}_{1}\left(t\right)-{x}_{2}$. The key points are: (i) the observer 1 sitting on
${x}_{1}$ experiences an acceleration since his variable coordinate is defined with respect to the origin of *R*, *i.e.* this observer moves far from or towards to the origin of *R* during the deformation of
$\delta x$ ; (ii) the observer 2 feels anyway a force field inside
$\delta x$ although he is at rest in *R*.

Therefore the observer 2 concludes that an external field is acting on
$\delta x$, whereas the observer 1 acknowledges an acceleration as if his position in *R* would be perturbed by the force field in
$\delta x$. Once more the consistent conclusion is that in fact the space time deformation rate
$\delta \stackrel{\dot{}}{x}$ causes itself the rising of a force field and that an accelerated reference frame is equivalent to such a force field. Only for
$\delta x\to 0$ the force field appears as a local classical force. It is immediately evident the role of the quantum uncertainty in this explanation of the concept of force, required by the physical equivalence of the boundary coordinates in lack of any discriminating information about their behavior: indeed
$\delta \stackrel{\dot{}}{p}$ is nothing else but a corollary of
$\delta \stackrel{\dot{}}{x}$ via (1.2), whereas the conclusions of the two observers are equally valid. External gravity force and space time deformation driven force are indistinguishable because the properties of space boundaries
${x}_{1}$ and
${x}_{2}$ of
$\delta x$ are conceptually arbitrary and unknowable by definition. Note that other forces of nature are directly related the interactions between particles, e.g. the electromagnetic interaction; the gravity force is instead inferred as a property of the space time that manifests under deformation of the uncertainty range sizes. This short discussion allows explaining what have to do the Equations (1.3) and (1.4) via (1.2) with relativistic and quantum physics: the Einstein intuition and thought experiment are now corollaries of the quantum uncertainty.

5.3. Quantum Angular Momentum

This topic has been concerned in [11] [16] [20]. Here are sketched for completeness some selected reminds only, useful later. By definition the component of angular momentum along an arbitrary direction defined by the unit vector
$z$ is
${\text{M}}_{z}=r\times p\cdot z$, which reads in the present conceptual frame as
${\text{M}}_{z}=\delta r\times \delta p\cdot z$ *i.e.*
${\text{M}}_{z}=\delta p\cdot \left(z\times \delta r\right)=\delta p\cdot \delta s$ where
$\delta s=z\times \delta r$. Hence owing to (1.2) the unit vector
$\delta {s}_{z}=\delta s/\left|\delta s\right|$ yields

${\text{M}}_{z}=\delta p\cdot \delta s=\delta p\cdot \frac{\delta s}{\left|\delta s\right|}\left|\delta s\right|=\delta p\cdot \delta {s}_{z}\left|\delta s\right|=\delta {p}_{s}\delta s=l\hslash ;$ (5.4)

here *n* of (1.2) has been replaced by *l* according to the usual notation
$l\hslash $ to express the component
${\text{M}}_{z}$ of *M*. The problem is now that the direction of
$z$ is arbitrary and unknown; so repeating the reasoning with a different
${z}^{\prime}$ would be physically insignificant, as it does not add anything conceptually new to the given result. The only information available is that *l* is an integer number
$\u22db0$ depending on the scalar
$\delta p\cdot \delta {s}_{z}$.

Let us sketch some properties of quantum angular momentum, which will be useful in the next subsection, assuming that
$-L\le l\le L$ ; *i.e.* *l* ranges between two allows values −*L* and *L*, of course arbitrary, whereas (5.4) holds for any *L*, exactly as done in (1.9). The following considerations emphasize the reasoning carried out in [11], although here the steps to calculate M^{2} differ slightly from that therein exposed: consider here that if
${\text{M}}_{z}$ is the only component knowable, then M^{2} must be somehow related to
${\text{M}}_{z}$ only. Note that

$\underset{l=-L}{\overset{L}{\sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}l=0,\text{\hspace{1em}}{\displaystyle \underset{l=-L}{\overset{L}{\sum}}}\left|l\right|=2{\displaystyle \underset{0}{\overset{L}{\sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}l=L\left(L+1\right),\text{\hspace{1em}}-L\le l\le L.$ (5.5)

The first equality follows by symmetry between the given limits of *l*, while the second equality is straightforward consequence of the first one; this explains why the second equation computes all *l*-th states of the given component as twice the sum from
$l=0$ to
$l=L$. Since the angular quantum number *l* is actually a number of allowed quantum states likewise *n* of (1.2), the idea is now that M^{2} should be defined by its own quantum angular number of *l* states and that in turn this latter is related to the sum of all *l*-th states allowed to its unique definable component
${\text{M}}_{z}$. In other words (5.4) suggests counting all quantum states of
${\text{M}}_{z}$ included in the range
$-L\to L$, *i.e.* summing all positive quantum numbers
$\left|l\right|$ of allowed states as done in (5.5); this result should be reasonably related to M^{2}. To verify this reasoning consider the *j*-th component
${\text{M}}_{j}$ of M, which owing to (5.4) and first (5.5) reads

$\frac{{\text{M}}_{j}^{2}}{{\hslash}^{2}}={\displaystyle \underset{{l}_{j}=-{L}_{j}}{\overset{{L}_{j}}{\sum}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{l}_{j}^{2}=\left(2{L}_{j}+1\right)\frac{{L}_{j}\left({L}_{j}+1\right)}{3}.$ (5.6)

Since the number of states from $-{L}_{j}$ to ${L}_{j}$ is $2{L}_{j}+1$, it is possible to calculate the average $\langle {\text{M}}_{j}^{2}\rangle $ as

$\frac{\langle {\text{M}}_{j}^{2}\rangle}{{\hslash}^{2}}={\left(2{L}_{j}+1\right)}^{-1}\frac{{L}_{j}\left({L}_{j}+1\right)}{3}$ (5.7)

Although one component only of M is calculable, assume reasonably that in a isotropic space time

$\langle {\text{M}}_{x}^{2}\rangle =\langle {\text{M}}_{y}^{2}\rangle =\langle {\text{M}}_{z}^{2}\rangle \mathrm{;}$ (5.8)

taking these averages, (5.7) with equal ${L}_{j}$ for all three components yields, as done in [11],

$\underset{j=1}{\overset{3}{\sum}}}\frac{\langle {\text{M}}_{j}^{2}\rangle}{{\hslash}^{2}}=3\frac{\langle {\text{M}}_{j}^{2}\rangle}{{\hslash}^{2}}=\frac{{\text{M}}^{2}}{{\hslash}^{2}}=L\left(L+1\right),\text{\hspace{1em}}{L}_{j}\equiv L\mathrm{.$ (5.9)

In effect, once having written ${\text{M}}^{2}=3\langle {\text{M}}_{j}^{2}\rangle $, the knowledge of the three components (5.6) reduces in fact to that of one component only; hence it is natural that this result coincides with that of (5.5) expressed in ${\hslash}^{2}$ units and confirms (5.8). Follow now three important corollaries.

(i) Implement now these quantum results considering arbitrary numbers *n* of states to describe also properties of orbiting systems. Rewrite thus (5.7) as

$\frac{{\text{M}}^{2}}{{\hslash}^{2}}=3\frac{\langle {\text{M}}_{j}^{2}\rangle}{{\hslash}^{2}}=\left(2n+1\right)n\left(n+1\right),\text{\hspace{1em}}2n+1=\frac{{\text{M}}^{2}/{\hslash}^{2}}{n\mathrm{(}n+\mathrm{1)}}\mathrm{;}$ (5.10)

*i.e.*
${\text{M}}^{2}/{\hslash}^{2}$ consists of
$2n+1$ states, from −*L* to *L*. An immediate corollary of the third (2.9) reads

$\frac{m\epsilon c\delta \mathcal{l}}{\hslash}=ma\delta \mathcal{l}=energy$

and thus

$\frac{\epsilon \delta \mathcal{l}}{\hslash c}=\frac{energy}{m{c}^{2}}\ge \mathrm{1;}$ (5.11)

indeed
$m{c}^{2}$ is rest energy, whereas *energy* denotes in general the dynamic energy of a physical system. Thus, since according to (2.11)

${\epsilon}^{2}={\epsilon}_{G}^{2}={\left(G\frac{{m}^{2}}{2\delta \mathcal{l}}\right)}^{2},\text{\hspace{1em}}{\hslash}^{2}=\frac{{\text{M}}^{2}}{L\left(L+1\right)}\mathrm{,}$

(5.11) squared reads

$\frac{{\left({m}^{2}G/2\right)}^{2}L\left(L+1\right)}{{\text{M}}^{2}{c}^{2}}\ge 1$

so that

${m}^{2}\ge {\left(\frac{\left|M\right|c}{mG}\right)}^{2}\frac{4}{L\left(L+1\right)}\mathrm{.}$ (5.12)

As the minimum non-zero value of $L\left(L+1\right)$ is 2, it is possible to infer

${m}^{2}>{\left(\frac{\left|M\right|c}{mG}\right)}^{2}\mathrm{.}$ (5.13)

Eventually, since it is possible to regard
$\delta \mathcal{l}=\delta {\mathcal{l}}_{m}+\delta {\mathcal{l}}_{q}$ as done in (3.25), being *q* is electric charge, one identifies (5.13) as a spin effect; in an analogous way
$\delta {\mathcal{l}}_{q}$ as a charge effect, indeed the analytical form of the Coulomb law is analogous to that of the Newton law. So
$\u03f5={\u03f5}_{m}+{\u03f5}_{q}$ yields a new addend
${\u03f5}_{q}\delta {\mathcal{l}}_{q}={q}^{2}$. Replace thus
${\text{M}}_{z}c$ with
${q}^{2}$, as both have physical dimensions
$energy\times length$ ; the same reasoning yields now

${\left(\frac{\left|M\right|c}{mG}\right)}^{2}+{\left(\frac{{q}^{2}}{mG}\right)}^{2}\le {m}^{2}\mathrm{.}$ (5.14)

(ii) A further corollary concerns the spin of particles and the Pauli principle. Since in ${\hslash}^{2}$ units $L\left(L+1\right)\equiv {\left(L+1/2\right)}^{2}-1/4$, it is possible to write

${\text{M}}^{2}+{\left(\frac{1}{2}\hslash \right)}^{2}+\left(L+\frac{1}{2}\right){\hslash}^{2}=\left(L+\frac{1}{2}\right)\left(L+\frac{1}{2}+1\right){\hslash}^{2}$ (5.15)

*i.e. *

${\mathcal{M}}^{2}=J\left(J+1\right){\hslash}^{2},\text{\hspace{1em}}J=L+\frac{1}{2},\text{\hspace{1em}}{\mathcal{M}}^{2}={\text{M}}^{2}+{\left(\frac{1}{2}\hslash \right)}^{2}+\left(L+\frac{1}{2}\right){\hslash}^{2}\mathrm{.}$ (5.16)

Note that the left hand side of (5.15) defines an angular momentum in fact allowed, so
$\mathcal{M}$ in (5.16) is a half integer angular momentum due to
$\hslash /2$, which is clearly by analogy with
$l\hslash $ the component along an arbitrary direction of a new half integer angular momentum. In [9] is concerned the spin of particles more in detail starting from (5.15). Here this topic is not further concerned for brevity, e.g. to show why actually
$J=L\pm S$ ; it is interesting instead to remark that the Pauli principle follows as a corollary of (5.16) [20]. This interpretation of the Pauli principle is a crucial consequence of the fact that*l* and *L* are not mere quantum numbers, but numbers of allowed quantum states likewise *n* of (1.2).

(iii) Consider the following definition of ${\text{M}}_{z}$, which reads

$\hslash =\zeta mv\delta \mathcal{l}=\zeta {m}_{0}c\delta {\mathcal{l}}_{0}\mathrm{:}$ (5.17)

both equalities express actually the component of angular momentum with $\zeta $ accounting for the sin and cos factors. Then owing to (2.35) and (3.3) the second equality reads

$\hslash =\zeta {m}_{0}c\delta {\mathcal{l}}_{0}=\zeta \frac{{m}_{0}{c}^{2}}{\beta}\frac{\beta \delta {\mathcal{l}}_{0}}{c}=\zeta \epsilon \frac{\delta {\mathcal{l}}^{\prime}}{c},\text{\hspace{1em}}\delta {\mathcal{l}}^{\prime}=\beta \delta {\mathcal{l}}_{0},\text{\hspace{1em}}\epsilon =\frac{{m}_{0}{c}^{2}}{\beta},$

in agreement with (3.3). Hence

$\epsilon =\frac{\hslash c/\zeta}{\delta {\mathcal{l}}^{\prime}}=\frac{{e}^{2}}{\delta {\mathcal{l}}^{\prime}}=\frac{{e}^{2}}{\beta \delta \mathcal{l}},\text{\hspace{1em}}\zeta =\frac{\hslash c}{{e}^{2}}={\alpha}^{-1}$ (5.18)

and thus

$\zeta \epsilon =\frac{\hslash c}{\delta {\mathcal{l}}^{\prime}}=\frac{{e}^{2}\zeta}{\delta {\mathcal{l}}^{\prime}}=\frac{{e}_{P}^{2}}{\delta {\mathcal{l}}^{\prime}},\text{\hspace{1em}}{e}_{P}=\frac{e}{\sqrt{\alpha}}$ (5.19)

by definition of Planck charge. The first equality (5.17) yields

$\hslash =\zeta mv\delta \mathcal{l}=\hslash \frac{m{v}^{2}}{e}\frac{c\delta \mathcal{l}}{ev}$

so that

$\mathcal{A}=\frac{ev}{c\delta {\mathcal{l}}^{\u2033}}=\frac{m{v}^{2}}{e},\text{\hspace{1em}}\delta {\mathcal{l}}^{\u2033}=\zeta \delta \mathcal{l},$ (5.20)

having merged $\zeta $ with the arbitrary range size of $\delta \mathcal{l}$. This simple reasoning has defined via ${\text{M}}_{z}$ the fine structure constant, the Coulomb law, the magnetic potential $\mathcal{A}$ and the definition of Planck charge.

5.4. Black Hole

Consider (3.2) rewritten according to (2.20) as

$\frac{Gm}{\delta \mathcal{l}}={v}^{2},\text{\hspace{1em}}\delta \mathcal{l}=n{\u019b}_{m}=\frac{n\hslash}{mc},\text{\hspace{1em}}{v}^{2}=\frac{{c}^{2}\hslash}{\xi},$ (5.21)

where *v* is velocity by dimensional reasons. Let us define now a dimensionless parameter
$\zeta $ such that

$\frac{{v}^{2}}{{c}^{2}}=1-\zeta \frac{{v}^{2}}{{c}^{2}};$ (5.22)

then, dividing both sides of (5.21) by ${c}^{2}$ one finds

$\frac{Gm}{{c}^{2}\delta \mathcal{l}}=\frac{{v}^{2}}{{c}^{2}},\text{\hspace{1em}}\frac{{v}^{2}}{{c}^{2}}=\frac{1}{\zeta +1},\text{\hspace{1em}}v=v\left(\zeta \right),\text{\hspace{1em}}0<\zeta <\infty .\text{\hspace{0.05em}}$ (5.23)

To show the physical meaning of (5.22), consider first the particular case ${v}^{*}=v\left(\zeta =1\right)$ ; so (5.21) rewritten with pertinent notation $m={m}^{*}$ and $\delta \mathcal{l}=\delta {\mathcal{l}}^{*}$ reads

$\frac{G{m}^{*}}{{c}^{2}\delta {\mathcal{l}}^{*}}=1-\frac{{v}^{*2}}{{c}^{2}}=\frac{{v}^{*2}}{{c}^{2}}=\frac{1}{2},\text{\hspace{1em}}{v}_{\pm}^{*}=\pm \frac{c}{\sqrt{2}},\text{\hspace{1em}}{v}_{-}^{*2}+{v}_{+}^{*2}={c}^{2},\text{\hspace{1em}}\zeta =1,$ (5.24)

and yields

$\delta {\mathcal{l}}^{*}=\frac{2{m}^{*}G}{{c}^{2}}$ (5.25)

in agreement with (3.15) and (3.16). Note that (5.24) corresponds to (3.24) already introduced to infer (3.25); in particular one implements here ${v}_{s}^{2}={v}_{\mathcal{l}}^{2}$, so the implications of (5.22) are in fact related to and confirm (3.27). In fact (5.24) allows defining as a function of a unique reference length $\delta {\mathcal{l}}_{0}$ two internal and external surfaces corresponding to ${v}_{\pm}^{\mathrm{*}}$ that fulfill (5.25); indeed $\delta {t}^{\mathrm{*}}$ implies

$\delta {\mathcal{l}}_{\pm}^{*}=\delta {\mathcal{l}}_{0}\pm \frac{c\delta {t}^{*}}{\sqrt{2}},$

*i.e.* two characteristic lengths
$\delta {\mathcal{l}}_{+}^{\mathrm{*}}$ and
$\delta {\mathcal{l}}_{-}^{\mathrm{*}}$ are definable as

$\delta {\mathcal{l}}_{+}^{*}=\delta {\mathcal{l}}_{0}+\frac{c}{\sqrt{2}}\delta {t}^{*},\text{\hspace{1em}}\delta {\mathcal{l}}_{-}^{*}=\delta {\mathcal{l}}_{0}-\frac{c}{\sqrt{2}}\delta {t}^{*}.$ (5.26)

Assuming that $\delta {\mathcal{l}}_{+}^{\mathrm{*}}\equiv \delta {\mathcal{l}}^{\mathrm{*}}$ of (5.25), then it is possible to calculate $\delta {\mathcal{l}}_{0}$ ; hence (5.26) yields

$\delta {\mathcal{l}}_{+}^{*}=\frac{2{m}^{*}G}{{c}^{2}},\text{\hspace{1em}}\delta {\mathcal{l}}_{-}^{*}=\frac{2{m}^{*}G}{{c}^{2}}-\sqrt{2}c\delta {t}^{*},\text{\hspace{1em}}\delta {t}^{*}\le \frac{\sqrt{2}{m}^{*}G}{{c}^{3}},$ (5.27)

the third result being due to $\delta {\mathcal{l}}_{-}^{\mathrm{*}}\ge 0$. The result (5.25) agrees with that already found in (3.15), which shows that the particular ${v}^{\mathrm{*}}$ defined by (5.24) has really a specific physical meaning. Moreover (5.27) shows the existence an inner event horizon at distance $\delta {\mathcal{l}}_{-}^{\mathrm{*}}$ with respect to the gravity center of ${m}^{\mathrm{*}}$ compatible with the outer one of (5.25). The event horizon appears here as an outer shell of the black hole of finite thickness, rather than an ideal two dimensional boundary surface. The fact of having implemented uncertainty ranges instead of deterministic metrics excludes in the present model the rising of divergence at the gravity center of the black hole, which is local inner boundary coordinate of the radii $\delta {\mathcal{l}}_{+}$ and $\delta {\mathcal{l}}_{-}$ and then unphysical.

Encouraged by the particular result (5.27), let us generalize the condition (5.24) by considering instead $\zeta \ne 1$ in (5.23): two different chances of generalization concern reasonably $0<\zeta <1$ and $\zeta >1$.

On the one hand, considering $\zeta <1$ and putting now ${v}^{*2}=\zeta {v}^{2}$, (5.22) and (5.23) yield

$\frac{G{m}^{*}}{\left(1-{v}^{*2}/{c}^{2}\right)\delta {\mathcal{l}}^{*}}=\frac{G{m}_{o}}{\delta {\mathcal{l}}_{o}}={c}^{2},\text{\hspace{1em}}{m}_{o}=\frac{{m}^{*}}{\sqrt{1-{v}^{*2}/{c}^{2}}},\text{\hspace{1em}}\delta {\mathcal{l}}_{o}=\delta {\mathcal{l}}^{*}\sqrt{1-{v}^{*2}/{c}^{2}}:$ (5.28)

these equations have well known meaning consistent with (2.36) regarding

${m}^{*}={m}_{rest},\text{\hspace{1em}}\delta {\mathcal{l}}^{*}=\delta {\mathcal{l}}_{rest},$ (5.29)

*i.e.*
${m}_{o}$ and
$\delta {\mathcal{l}}_{o}$ are defined in a reference system
${R}_{o}$ moving at constant rate
${v}_{o}$ with respect to *R* where
${m}^{\mathrm{*}}$ and
$\delta {\mathcal{l}}^{\mathrm{*}}$ are at rest. These results agree with and confirm the different reasoning exposed to infer (3.3).

On the other hand, also follows from (5.28) the chance of defining via $\delta {\mathcal{l}}^{\mathrm{*}}$ the time range $\delta {t}_{o}$ such that

$\frac{\delta {\mathcal{l}}_{o}}{c}=\frac{{m}_{o}G}{{c}^{3}}=\frac{G{m}^{*}}{{c}^{3}\sqrt{1-{v}^{*2}/{c}^{2}}}=\frac{\delta {t}^{*}}{\sqrt{1-{v}^{*2}/{c}^{2}}},\text{\hspace{1em}}\delta {t}^{*}=\frac{G{m}^{*}}{{c}^{3}}:$

as expected owing to (3.28),
$\delta {\mathcal{l}}_{o}/c=\delta {t}_{o}$ agrees with Lorentz transformation of proper time lapse
$\delta t$ *i.e.*

$\delta {t}_{o}=\frac{\delta {t}^{\mathrm{*}}}{\sqrt{1-{v}^{*2}/{c}^{2}}}$ (5.30)

with the same physical meaning of (5.28). So the second (5.28) is the well known velocity dependent dynamic mass law with respect to the rest mass, the third one yields the Lorentz contraction of the proper length $\delta {\mathcal{l}}_{rest}$. Although (5.24) concerns in particular $v={v}_{\pm}^{*}$ only, the velocity ${v}_{\pm}^{\mathrm{*}}$ shows the condition under which hold (5.28) and the following (5.25) that describe a law of nature; the same holds for (3.11) implied by (3.3).

Follows now an interesting implication of (5.25). Consider

$\delta {\mathcal{l}}^{*2}=\frac{4{m}^{*2}{G}^{2}}{{c}^{4}}=4{m}^{*}G\frac{{m}^{*}G}{{c}^{4}},$

which dividing both sides by ${c}^{2}$ yields

$\frac{\delta {\mathcal{l}}^{*}}{{c}^{2}}=\frac{4{m}^{*}G}{{c}^{2}\delta {\mathcal{l}}^{*}}\frac{{m}^{*}}{{F}_{P}},\text{\hspace{1em}}{F}_{P}=\frac{{c}^{4}}{G}$ (5.31)

and thus

$\frac{\delta {\mathcal{l}}^{*}}{{c}^{2}}=\frac{1}{{a}^{*}}=\delta \varphi \frac{{m}^{*}}{{F}_{P}},\text{\hspace{1em}}\delta \varphi =\frac{4{m}^{*}G}{{c}^{2}\delta {\mathcal{l}}^{*}},$ (5.32)

being by dimensional reasons ${a}^{\mathrm{*}}$ acceleration. This way of rearranging (5.31) is significant as, owing to (2.8),

$\frac{{\u03f5}^{*}c}{\hslash}=\frac{{c}^{2}}{\delta {\mathcal{l}}^{*}}.$ (5.33)

On the one hand the first equality (5.32) does not conflict with (2.8) as this definition yields

${\u03f5}^{*}=\frac{\hslash c}{\delta {\mathcal{l}}^{*}}=\frac{\hslash {c}^{3}}{2{m}^{*}G}=\frac{{p}_{P}^{2}}{2{m}^{*}},$ (5.34)

being ${p}_{P}$ the Planck momentum, whereas the second equality reads ${F}_{P}={m}^{*}{a}^{*}\delta \varphi $ ; hence $\delta \varphi $ is consistent with the concept of angle inherent more in general ${F}_{P}={m}^{*}{a}^{*}$. On the other hand (5.32) yields via (2.9)

${F}_{P}=\frac{{m}^{*}{c}^{2}}{\delta {\mathcal{l}}^{*}}\delta \varphi =\frac{{m}^{*}{c}^{2}}{\delta {\mathcal{l}}^{*}}\frac{\delta s}{\delta s}\delta \varphi ,$

being $\delta s$ an arbitrary length range; hence

${F}_{P}\delta s={m}^{*}{c}^{2}\delta {\varphi}^{\prime}\delta \varphi ,\text{\hspace{1em}}\delta {\varphi}^{\prime}=\frac{\delta s}{\delta {\mathcal{l}}^{*}}.$

Regard
$\delta s$ as the length of arc of circumference of the circle osculating the true path that defines the angular deviation
$\delta \varphi $ ; so
${F}_{P}\delta s$ calculates the work done by the force component *F* along a curved trajectory. Then

$\begin{array}{l}F\delta s={m}^{*}{c}^{2}\delta \varphi ={m}^{*}\frac{{\u03f5}^{*}c}{\hslash}\delta s={\u03f5}^{*}\frac{\delta s}{{\u019b}_{m}^{*}},\text{\hspace{1em}}\delta \varphi =\frac{4{m}^{*}G}{{c}^{2}\delta {\mathcal{l}}^{*}}=\frac{F\delta s}{{m}^{*}{c}^{2}},\\ F=\frac{{F}_{P}}{\delta {\varphi}^{\prime}},\text{\hspace{1em}}{\u019b}_{m}^{*}=\frac{\hslash}{{m}^{*}c},\end{array}$ (5.35)

having specified
$\delta \mathcal{l}=\delta s$ in the third (2.9) for obvious reasons. The second equation is the Einstein result of light beam bending in the gravity field of
${m}^{\mathrm{*}}$ ; the deviation angle
$\delta \varphi $ is also equal the ratio between the work
$F\delta s$ done by the field along the photon path
$\delta s$ and the rest energy
${m}^{\mathrm{*}}{c}^{2}$ of the field source. Moreover it appears that
${\u03f5}^{*}=F{\u019b}_{m}^{*}$, *i.e.* the energy
${\u03f5}^{\mathrm{*}}$ corresponding to *F* involves the Compton length of
${m}^{\mathrm{*}}$. The mathematical approximation inherent the Einstein result corresponds here to having assumed
$\delta s$ as a circumference arc instead of a more general curved arc of the actual photon trajectory.

Now note eventually an interesting corollary of (5.33) that reads ${\u03f5}^{\mathrm{*}}=\hslash c/\delta {\mathcal{l}}^{\mathrm{*}}$ and is identically rewritten as

${\u03f5}^{\mathrm{*}}=\frac{\hslash c}{\delta {\mathcal{l}}^{\mathrm{*}}}=\frac{\hslash c{\text{z}}^{2}}{\delta {\mathcal{l}}^{\mathrm{*}}{\text{z}}^{2}}=\frac{\hslash c}{{\text{z}}^{2}}\frac{{\text{z}}^{2}}{\delta {\mathcal{l}}^{\mathrm{*}}}=\frac{1}{\alpha}\frac{{\text{z}}^{2}}{\delta {\mathcal{l}}^{\mathrm{*}}},\text{\hspace{1em}}\alpha =\frac{{\text{z}}^{2}}{\hslash c}\mathrm{,}$ (5.36)

where z is a new arbitrary parameter not yet introduced in the present model to be appropriately defined in agreement with (5.18). Simply renaming z as $\text{z}=\pm e$, where both signs are compatible with ${\text{z}}^{2}={e}^{2}$, it follows that $\pm e$ is the electric charge, whereas the proportionality factor $\alpha $ linking the Coulomb law hidden in ${\u03f5}^{\mathrm{*}}$ via $\hslash c$ is actually the fine structure constant. This last result is closely related to the results from (5.17) to (5.20) previously found. Emphasize now that the particular condition (5.24) is sensible, although it has been introduced preliminarily just in order for (5.23) to match (3.17) and not as a consequence of a fundamental requirement; yet (5.24) can be generalized while regarding (5.28) as mere particular case. The key point is to replace (5.23) via a function $\mathcal{F}$ of $\zeta $ and $\delta {r}_{bh}$ defined as follows

${\left(\delta {r}_{bh}-\frac{{m}^{\mathrm{*}}G}{{c}^{2}}\right)}^{2}-{\left(\frac{{m}^{\mathrm{*}}G}{{c}^{2}}\right)}^{2}=\mathcal{F}\left(\zeta \right)=\mathcal{F}\left({\zeta}^{\prime}\right)+\mathcal{F}\left({\zeta}^{\u2033}\right),\text{\hspace{1em}}\mathcal{F}\left(\zeta =1\right)=\mathrm{0;}$ (5.37)

The condition on $\zeta =1$ corresponds to (5.24), because then this equation admits solutions $\delta {r}_{bh}=0$ and again $\delta {r}_{bh}=2{m}^{\mathrm{*}}G/{c}^{2}$. Discarding the null solution, $\delta {r}_{bh}$ is just that expected according to (MJM) pertinent to $\zeta =1$, in agreement with (5.25). To generalize (5.24) it is enough to consider (5.37) with $\mathcal{F}\left({\zeta}^{\prime}\right)=-\mathcal{F}\left({\zeta}^{\u2033}\right)$ for $\zeta =1$ and with any $\mathcal{F}\left({\zeta}^{\prime}\right)\ne -\mathcal{F}\left({\zeta}^{\u2033}\right)$ for $\zeta \ne 1$, which in principle it is possible with both $0<{\zeta}^{\prime}<1$ and $1<{\zeta}^{\u2033}<\infty $. This generalized equation can be rewritten putting respectively

$\mathcal{F}\left({\zeta}^{\prime}\right)={\left(\delta {{r}^{\prime}}_{bh}-\frac{{m}^{*}G}{{c}^{2}}\right)}^{2}-{\left(\frac{{m}^{*}G}{{c}^{2}}\right)}^{2},\text{\hspace{1em}}\mathcal{F}\left({\zeta}^{\u2033}\right)={\left(\delta {{{r}^{\prime}}^{\prime}}_{bh}-\frac{{m}^{*}G}{{c}^{2}}\right)}^{2}-{\left(\frac{{m}^{*}G}{{c}^{2}}\right)}^{2}.$ (5.38)

On the one hand it is reasonable to assume that these equations concern two different properties of $\delta {r}_{bh}$ ; on the other hand it is also reasonable to guess that two properties of the mass ${m}^{\mathrm{*}}$ can be charge and angular momentum due to its possible angular spinning or to its possible spin or both. Anyway, since the uncertainty ranges at left hand side represent square lengths, it is immediate to conclude that the same holds for the right hand size terms; in other words, to include the charge terms it is enough to express the space range sizes that appear in the Coulomb law of (5.36). So, in Planck units, ${Q}^{2}/{\mathcal{l}}^{2}={c}^{4}/G$ yields $\mathcal{F}\left({\zeta}^{\prime}\right)={\mathcal{l}}_{Q}^{2}={Q}^{2}G/{c}^{4}$ according to (5.36). An analogous reasoning for the rotation of ${m}^{\mathrm{*}}$ via the angular momentum $J$ reads $\mathcal{F}\left({\zeta}^{\u2033}\right)={\mathcal{l}}_{J}^{2}={J}^{2}/{\left({m}^{*}c\right)}^{2}$. Replacing both contributions (5.38) into (5.37), whose left hand side is <0 because ${\left({m}^{*}G/{c}^{2}\right)}^{2}>{\left({m}^{*}G/{c}^{2}-\delta {r}_{bh}\right)}^{2}$, one finds thus

${\left(\delta {r}_{bh}-\frac{{m}^{*}G}{{c}^{2}}\right)}^{2}-{\left(\frac{{m}^{*}G}{{c}^{2}}\right)}^{2}=-\frac{{Q}^{2}G}{{c}^{4}}-\frac{{J}^{2}}{{\left({m}^{*}c\right)}^{2}},$

with appropriate signs at the right hand side. This yields the well known result consistent with (5.14)

$\delta {r}_{bh}=\frac{G}{{c}^{2}}\left({m}^{*}+\sqrt{{m}^{{*}^{2}}-\frac{{Q}^{2}}{G}-\frac{{J}^{2}{c}^{2}}{{\left({m}^{*}G\right)}^{2}}}\right).$ (5.39)

5.5. From Special to General Relativity

Rewrite the first (3.32) as

$2nc\frac{\delta t}{\delta r}=\delta \left(\frac{{v}^{2}}{{c}^{2}}\right)=\frac{{v}_{1}^{2}}{{c}^{2}}-\frac{{v}_{0}^{2}}{{c}^{2}}$ (5.40)

having simply implemented at the right hand side the definition of
$range=\delta \left(any\text{\hspace{0.17em}}function\right)$. Note that the possible chance
$n\to \infty $ requires
$\delta t\to 0$ or
$\delta r\to \infty $ or both, because the right hand side of (5.40) is anyway finite being
${v}_{1}\le c$ and
${v}_{0}\le c$. This problem requires defining appropriately
$\delta r$ and
$\delta t$ as done with (3.33) to infer the relativistic property (3.38) of *c*. Consider that any range size tending to zero implies in fact the classical determinism for the concerned variable: *i.e**.*
${t}_{0}\le t\le {t}_{1}$ means that
$t\to specific\text{\hspace{0.17em}}local\text{\hspace{0.17em}}value$ for
${t}_{1}\to {t}_{0}$, whatever
${t}_{0}$ and
${t}_{1}$ might be. This conclusion holds for all quantum variables and ensures here that (5.40) is definable for any *n* implementing the positions (3.35), *i.e.* replacing
$\delta t$ and
$\delta r$ as follows

$2c\frac{\delta {t}_{n}}{\delta {r}_{n}}=\delta \left(\frac{{v}^{2}}{{c}^{2}}\right),\text{\hspace{1em}}\delta {r}_{n}=\frac{\delta r}{\sqrt{n}},\text{\hspace{1em}}\delta {t}_{n}=\delta t\sqrt{n};$ (5.41)

as of course
$\delta {t}_{n}$ and
$\delta {r}_{n}$ are still arbitrary, it is possible to implement likewise (5.41) or (5.40) examining some possible cases where
${v}_{0}$ and/or
${v}_{1}$ are in particular constant or more in general any functions of *n*. Anyway note that (5.41) fulfill the invariant condition of special relativity already concerned in (2.47)

$\delta {r}_{n}\delta {t}_{n}=\delta r\delta t\mathrm{:}$ (5.42)

in fact, being *n* arbitrary likewise
$\delta r$ and
$\delta t$, it is possible to regard the left and right hand sides as if they would refer to different inertial reference systems. This point will be further clarified below.

1) ${v}_{0}$ and ${v}_{1}$ both constants.

In this case write (5.40) as

$2nc\frac{\delta t}{\delta r}=const-\frac{{v}_{0}^{2}}{{c}^{2}}=const\left(1-\frac{{v}_{0}^{2}/const}{{c}^{2}}\right)=const\left(1-\frac{{v}_{0}^{*2}}{{c}^{2}}\right)=const{\beta}^{*2}$ (5.43)

where

$const=\frac{{v}_{1}^{2}}{{c}^{2}},\text{\hspace{1em}}{v}_{0}^{*}=\frac{{v}_{0}}{\sqrt{const}},\text{\hspace{1em}}{\beta}^{*}=\sqrt{1-\frac{{v}_{0}^{*2}}{{c}^{2}}},$ (5.44)

so that (5.43) reads

$\left(2n\frac{c}{const}\right)\frac{\delta t/{\beta}^{*}}{{\beta}^{*}\delta r}=1=\left(2n\frac{c}{const}\right)\frac{\delta {t}^{\prime}}{\delta {r}^{\prime}},\text{\hspace{1em}}\delta {t}^{\prime}=\frac{\delta t}{{\beta}^{*}},\text{\hspace{1em}}\delta {r}^{\prime}={\beta}^{*}\delta r;$ (5.45)

thus, apart from the factor in parenthesis appearing at both sizes and thus irrelevant, (5.45) implies again

$\delta {r}^{\prime}\delta {t}^{\prime}=\delta r\delta t\mathrm{,}$ (5.46)

in agreement with (5.42) and the result of the Appendix B. Regarding the primed and unprimed quantities in the respective inertial reference systems *R* and *R’* reciprocally moving at constant rate
${v}_{0}^{\mathrm{*}}$, appear again the Lorentz space contraction and time dilation together with the invariant behavior of the product
$\delta \left(space\right)\times \delta \left(time\right)$ of uncertainty ranges. This conclusion is not trivial: (5.46) holds regardless of *n* in parenthesis of (5.45) and justifies the chance of regarding the uncertainty ranges likewise the local coordinates of relativity, while remarking however that the former only and not the latter fulfill the Heisenberg principle.

2) ${v}_{0}={v}_{0}\left(n\right)$ and ${v}_{1}={v}_{1}\left(n\right)$.

In this case it is possible to start from (5.41) for further considerations about (5.40). Rewrite (5.41) as

$2\frac{\delta {\mathcal{l}}_{n}}{\delta {r}_{n}}=\delta \left(\frac{{v}^{2}}{{c}^{2}}\right)=\delta \left(1-{\beta}^{2}\right)=-\delta \left({\beta}^{2}\right),\text{\hspace{1em}}\delta {\mathcal{l}}_{n}=c\delta {t}_{n};$

merging the equalities at the right hand side one finds

$\begin{array}{c}4\frac{\delta {\mathcal{l}}_{n}^{2}}{\delta {r}_{n}^{2}}=-2\delta \left(1-{\beta}^{2}\right)\beta \delta \beta =-2\delta \left(1-{\beta}^{2}\right){\beta}^{2}\frac{\delta \beta}{\beta}\\ =-2\beta \delta \left(1-{\beta}^{2}\right)\beta \delta \left(\mathrm{log}\beta \right)=-2\beta {\beta}_{0}\delta \left(1-{\beta}^{2}\right)\frac{\beta}{{\beta}_{0}}\delta \left(\mathrm{log}\frac{\beta}{{\beta}_{0}}\right),\end{array}$

being ${\beta}_{0}$ an arbitrary constant, and thus eventually

$4\frac{\delta {\mathcal{l}}_{n}^{2}}{\delta {r}_{n}^{2}}=-2\beta {\beta}_{0}\delta \left(1-{\beta}^{2}\right)w\delta \mathrm{log}w,\text{\hspace{1em}}w=\frac{\beta}{{\beta}_{0}}.$

This result is interesting because it takes the final form

$\frac{2}{Q}\frac{\delta {\mathcal{l}}_{n}^{2}}{\delta {r}_{n}^{2}}=-w\delta \mathrm{log}w=-w\left(\mathrm{log}w-\mathrm{log}{w}_{0}\right),\text{\hspace{1em}}Q=Q\left(\beta \right)=\beta {\beta}_{0}\delta \left(1-{\beta}^{2}\right),$

having simply replaced $\delta \mathrm{log}\left(w\right)$ with the usual notation of uncertainty range $\mathrm{log}\left(w\right)-\mathrm{log}\left({w}_{0}\right)$ as in (1.7). Since the range boundaries are arbitrary and is reasonable to expect that ${w}_{0}$ is defined by $\beta ={\beta}_{0}$, then ${w}_{0}=1$ yields

$\frac{\delta {\mathcal{l}}_{n}^{o2}}{\delta {r}_{n}^{o2}}=-w\mathrm{log}w,\text{\hspace{1em}}\frac{\delta {\mathcal{l}}_{n}^{o2}}{\delta {r}_{n}^{o2}}=\frac{2}{Q}\frac{\delta {\mathcal{l}}_{n}^{2}}{\delta {r}_{n}^{2}}:$ (5.47)

the interesting fact is that at the left hand side appears the ratio of the square lengths
$\delta {\mathcal{l}}_{n}^{o2}$ and
$\delta {r}_{n}^{o2}$ whatever the value of the dimensionless factor *Q* might be, at the right hand side appears the entropy like term
$-w\mathrm{log}w$. Since
$\beta <1$ by definition, *w* can be regarded as probabilities via an appropriate choice of the arbitrary constant
${\beta}_{0}$. It is evident that the arbitrariness of the quantum ranges plays a fundamental role in this respect; indeed, being the range sizes arbitrary by definition, the left hand size ratio is indistinguishable from and thus identically readable as the initial
$\delta {\mathcal{l}}_{n}$ and
$\delta {r}_{n}$ in turn reducible to the respective lengths of (5.40). This result is helpful for its implications in the next section 5.7

3) Virial theorem.

It is possible to assess now (5.40), which is useful to obtain information whenever the quantization condition of the dynamical variables is explicitly required. Since the boundaries of any uncertainty range are arbitrary by definition, examine the two possible cases where
${v}_{0}$ is in particular a constant or it is in general any function of *n*. Write (5.40) with the help of (2.35) as

$\begin{array}{l}2nc\frac{\delta t}{\delta r}=2nc\frac{\delta t}{n\hslash /\delta {p}_{r}}=2\delta \left({p}_{r}c\right)\frac{\delta t}{\hslash}=\frac{\delta \left({p}_{r}c\right)}{{\u03f5}_{r}}\omega \delta t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{{v}_{r}}{c}\frac{\delta \left({p}_{r}c\right)}{{p}_{r}c}\omega \delta t=\delta \left(\frac{{v}^{2}}{{c}^{2}}\right)=\frac{{v}_{1}^{2}-{v}_{0}^{2}}{{c}^{2}},\\ {\u03f5}_{r}=\frac{\hslash \omega}{2}\end{array}$

having regarded
${\u03f5}_{r}$ as a zero point energy. So, multiplying by *m* the fourth equality, one finds

$\frac{m{v}_{r}}{{c}^{2}}\frac{\delta \left({p}_{r}c\right)}{{p}_{r}}\omega \delta t=\frac{{\beta}_{r}}{{c}^{2}}\delta \left({p}_{r}c\right)\omega \delta t=m\delta \left(\frac{{v}^{2}}{{c}^{2}}\right),\text{\hspace{1em}}m{v}_{r}={p}_{r}{\beta}_{r},$

whereas the second equality reads

${\beta}_{r}\delta \left({p}_{r}c\right)\omega \delta t=m\delta \left({v}^{2}\right).$ (5.48)

Here is useful the classical approximation ${\beta}_{r}\approx 1$ assuming for simplicity ${v}_{r}\ll c$ ; (5.48) simplifies to

$\delta \left({p}_{r}c\right)\omega \delta t=m\delta \left({v}^{2}\right)=\delta \left(m{v}^{2}\right).$ (5.49)

Even (5.49) contains the same classical approximation of (5.48), because (3.3) shows that the dynamic mass becomes in fact a constant. Both steps clarify the physical meaning of $\delta {v}^{2}$ and ${p}_{r}c$, as (5.49) reads

$\omega \delta t=-\frac{2\delta \text{T}}{\delta \text{U}},\text{\hspace{1em}}\text{T}=\frac{m{v}^{2}}{2},\text{\hspace{1em}}\text{U}=-{p}_{r}c\mathrm{,}$ (5.50)

so that the formal position ${p}_{r}c=-\text{U}$ yields

$\omega \delta t=-\frac{2\delta \text{T}}{\delta \text{U}}=1,\text{\hspace{1em}}2\text{T}=-\text{U}\mathrm{.}$ (5.51)

The positions (5.50) allow to find the expected result $\omega =\delta {t}^{-1}$, which fits the classical virial theorem $2\text{T}=-\text{U}$ consistent with the second equation; the classical limit of (5.40) has sensible physical meaning.

4) Quantum gravity.

Multiply now both sides of (5.40) by $\hslash /\delta t$ in order to find

$\frac{n\hslash c}{\delta r}+\frac{\hslash {v}_{0}^{2}}{2{c}^{2}\delta t}=\frac{\hslash {v}_{1}^{2}}{2{c}^{2}\delta t};$ (5.52)

a further result is now available implementing the definitions

$\omega =\frac{c}{\delta r},\text{\hspace{1em}}\omega =\frac{{v}_{0}^{2}}{{c}^{2}\delta t}$ (5.53)

that yield

$n\hslash \omega +\frac{1}{2}\hslash \omega =\frac{1}{2}\frac{\hslash {v}_{1}^{2}}{{c}^{2}\delta t}=\frac{1}{2}\frac{\hslash {v}_{1}^{2}}{c\delta \mathcal{l}},\text{\hspace{1em}}\delta \mathcal{l}=c\delta t$ (5.54)

and merge themselves into

$\delta r=\frac{{c}^{3}}{{v}_{0}^{2}}\delta t=\frac{{c}^{2}}{{v}_{0}^{2}}\delta \mathcal{l}.$ (5.55)

The left hand side of (5.54) diverges for $n\to \infty $ ; however this is not a problem, being allowed by $\delta t\to 0$.

The first (5.53) is sensible, as it reads
$2\pi \nu \delta r=c$, *i.e.*

$2\pi \delta r=\frac{c}{\nu}=\frac{n{v}_{r}}{\nu}=n{\lambda}_{r},\text{\hspace{1em}}{v}_{r}=\frac{c}{n},\text{\hspace{1em}}{\lambda}_{r}=\frac{{v}_{r}}{\nu};$ (5.56)

the given definition of
${v}_{r}$, quantized likewise the number *n* of respective
${\lambda}_{r}$, is compatible with
${v}_{r}\le c$ and with (5.41). Thus one finds once again the key quantum equation (2.22) and accordingly also

$\begin{array}{l}\omega =\frac{n{v}_{r}}{\delta r},\text{\hspace{1em}}energy=\hslash \omega =n\u03f5,\text{\hspace{1em}}\u03f5=\hslash \frac{{v}_{r}}{\delta r}=\hslash \frac{{v}_{r}}{{v}_{r}\delta t}=\frac{\hslash}{\delta t},\\ \delta r={v}_{r}\delta t,\text{\hspace{1em}}\frac{h}{{\lambda}_{r}}=\frac{n\hslash}{\delta r}=\delta {p}_{r}\end{array}$ (5.57)

and then, owing to the second (5.53), also

$\frac{n{v}_{r}}{\delta r}=\frac{{v}_{0}^{2}}{{c}^{2}\delta t}.$

Since according to (5.41)

$\frac{\delta r}{\delta t}=n\frac{\delta {r}_{n}}{\delta {t}_{n}},$

then

${v}_{r}=\frac{{v}_{0}^{2}}{{c}^{2}}\frac{\delta {r}_{n}}{\delta {t}_{n}},\text{\hspace{1em}}\omega =\frac{{v}_{rn}}{\delta {r}_{n}},\text{\hspace{1em}}{v}_{rn}=\frac{{v}_{r}}{\sqrt{n}};$

the last position is coherent with that of (3.35). Eventually (5.41) yields

$\frac{{v}_{1}^{2}}{{c}^{2}}=\frac{{v}_{0}^{2}}{{c}^{2}}+2c\frac{\delta {t}_{n}}{\delta {r}_{n}}=\left({v}_{r}+2c\right)\frac{\delta {t}_{n}}{\delta {r}_{n}};$

in this way both positions (5.53) are consistent with (5.41): *i.e.* defining appropriately
$\delta r/\delta t$ via
$\delta {r}_{n}/\delta {t}_{n}$ there is no divergence of velocity in (5.40) and (5.54) for
$n\to \infty $. A further corollary of the positions (5.53) is found merging them by eliminating *c*: replacing
$c=\omega \delta r$ in the second one, two interesting results are

${\omega}^{3}\delta {r}^{2}=\frac{{v}_{0}^{2}}{\delta t},\text{\hspace{1em}}\frac{{\omega}^{3}\delta {r}^{2}}{c}=\frac{{v}_{0}^{2}}{c\delta t}=a.$ (5.58)

The familiar form at left hand side of the first equation will be useful below, see next (5.71) and (5.72); the second equation provides a further definition of acceleration. Both equations support the results just obtained. Consider now (5.54) noting that the physical dimensions at the right hand side are

$\frac{\hslash {v}_{1}^{2}}{2c}=mass\times lengt{h}^{3}\times tim{e}^{-2},$ (5.59)

whereas at the left hand side appears instead the expression of vibrational harmonic system along with its zero point energy; this suggests that
$\hslash {v}_{1}^{2}/c$ should be itself proportional to
${m}^{2}G$ by dimensional reasons, *i.e.*
$\hslash {v}_{1}^{2}/c=\xi {m}^{2}G$, being
$\xi $ the proportionality constant. Defining therefore without loss of generality the dimensional square mass as
${m}^{2}={m}_{1}{m}_{2}$ because *m* is in fact arbitrary likewise as
${m}_{1}$ and
${m}_{2}$ themselves, write for this distinctive reading of the energy defined by (5.53) in agreement with (2.11)

$n\hslash \omega +\frac{1}{2}\hslash \omega =\frac{\xi}{2}G\frac{{m}_{1}{m}_{2}}{\delta \mathcal{l}}=\xi \left(G\frac{{m}_{1}{m}_{2}}{\delta \mathcal{l}}-\frac{1}{2}G\frac{{m}_{1}{m}_{2}}{\delta \mathcal{l}}\right),\text{\hspace{0.05em}}$ (5.60)

where it is possible to identify at the right hand side

$G\frac{{m}_{1}{m}_{2}}{\delta \mathcal{l}}=-U,\text{\hspace{1em}}-\frac{1}{2}G\frac{{m}_{1}{m}_{2}}{\delta \mathcal{l}}=\frac{U}{2}=-T$ (5.61)

with the help of the quantum virial theorem inferred in (5.51). Then

$n\hslash \omega +\frac{1}{2}\hslash \omega =\xi \left(-U-T\right),\text{\hspace{1em}}2T=-U,$

so that, putting $\xi =1$, the result is

$n\hslash \omega +\frac{1}{2}\hslash \omega =-H,\text{\hspace{1em}}\xi =1,\text{\hspace{1em}}H=U+T<0,\text{\hspace{1em}}{\omega}^{2}=\frac{{k}_{f}}{m}.$ (5.62)

The right hand side is the Hamiltonian of the orbiting system, *i.e.* Newtonian binding energy
$-{\u03f5}_{G}$, which agrees with the idea of harmonic oscillator as a bound system itself. The minus sign of
${\u03f5}_{G}$ means that the force constant
${k}_{f}$ defining the quantum oscillator frequency implies an attractive energy between two orbiting masses; a repulsive energy would be instead inconsistent with steady quantum oscillations. As expected, whatever
${m}_{1}$ and
${m}_{2}$ might be, an appropriate *n* shows the actual quantization of orbital motion: for large masses, *n* is so large that the quantization is hidden by the values
$n\approx n+1$, being
$\delta \mathcal{l}$ the circular orbit radius. So the steady harmonic oscillations are the quantum equivalent of the steady orbital motion of a Newtonian system via the circular frequency
$\omega $ : the force constant
${k}_{f}$ governing the one dimensional linear oscillation of the former turns into the behavior of *m* in a central force field governed by *G* and seemingly non-quantized. The last equation (5.62) is direct consequence of (5.53) because

${\omega}^{2}=\frac{{c}^{2}}{\delta {r}^{2}}=\frac{m{c}^{2}}{m\delta {r}^{2}}=\frac{\u03f5/\delta {r}^{2}}{m}=\frac{{k}_{f}}{m}:$ (5.63)

*i.e.* classically
${\omega}^{2}$ is proportional to
${m}^{-1}$ via the factor
${k}_{f}$. Moreover, the link between a one dimensional oscillation and a two dimensional system orbiting on an arbitrary plane implies 2 degenerate states, as the clockwise and counter clockwise rotation are both allowed and in principle indistinguishable; this also holds in the quantum world, indeed *l* takes all values
$-n\le l\le n$ identically to
$-l$. So the macroscopic measurable orbiting energy (2.11) is twice that
${m}_{1}{m}_{2}G/2\delta \mathcal{l}$ of (5.60). This degeneracy can be also regarded as a statement of equivalence between inertial and gravitational mass: the degenerate energies concern now the systems where *m*_{1} moves around *m*_{2} or *m*_{2} around *m*_{1}, depending on either reference system *R*_{1} or *R*_{2} where the respective mass is at rest. Without the equivalence principle, *R*_{1} and *R*_{2} would not interchangeable, as instead it is true according to the quantum (2.41). The macroscopic Newton law inferred from an oscillating quantum system reveals and requires the sought equivalence.

5) Implications of harmonic oscillations.

As a closing remark consider now the following dimensional definitions

${m}_{\omega}=\xi \frac{{c}^{3}}{\omega G},\text{\hspace{1em}}{\rho}_{\omega}=\xi \frac{{\omega}^{2}}{G},$ (5.64)

being $\xi $ an appropriate proportionality factor plausible in any dimensional equations and thus introduced here; the subscript emphasizes that these definitions implement explicitly the frequency $\omega $. These positions aim to calculate the force constant ${k}_{f}$ of the harmonic oscillator implied by (5.54)

${k}_{f}={m}_{\omega}{\omega}^{2}=\xi \frac{{c}^{3}\omega}{G}.$ (5.65)

In fact (5.64) are alternative to (2.1) and (2.2) in defining via $\omega $ mass, density and energy density in an arbitrary volume $\delta {\mathcal{l}}^{3}$. The connection with these equations is given by

${\omega}^{2}=\frac{{\rho}_{\omega}G}{\xi}=\frac{{m}_{\omega}}{\delta {\mathcal{l}}^{3}}\frac{G}{\xi},\text{\hspace{1em}}{\rho}_{\omega}=\frac{{m}_{\omega}}{\delta {\mathcal{l}}^{3}}$ (5.66)

and yields

${k}_{f}=\frac{{m}_{\omega}^{2}G}{\xi \delta {\mathcal{l}}^{3}}=\frac{{\omega}^{4}\delta {\mathcal{l}}^{3}\xi}{G}=-\frac{{\u03f5}_{G}}{\xi \delta {\mathcal{l}}^{2}}$ (5.67)

so that

${\u03f5}_{G}=-G\frac{{m}_{\omega}^{2}}{\delta \mathcal{l}}.$ (5.68)

Moreover (5.64) defines

${\eta}_{\omega}=\xi \frac{{\left(c\omega \right)}^{2}}{G}.$ (5.69)

The physical meaning of these results will be highlighted by calculating their numerical values in section 6.

6) Gravitational waves.

This subsection aims to sketch that the gravitational waves are actually quantized and fit the result inferred in (5.60); details on the physical model and results have been already concerned in a paper [16] on this topic. To add further considerations in this respect and highlight this point, let us start from the Einstein formula

$-\frac{dE}{dt}=\frac{32G}{5{c}^{5}}{\left(\frac{{m}_{1}{m}_{2}}{{m}_{1}+{m}_{2}}\right)}^{2}{r}^{4}{\omega}^{6},\text{\hspace{1em}}{\omega}^{2}{r}^{3}=\left({m}_{1}+{m}_{2}\right)G$ (5.70)

where *r* and
$\omega $ are the deterministic radius and angular frequency of an orbiting body in a circular orbit; in this case holds the second equation, which is direct consequence of the third Kepler law. The explanation of these formulas is reported in [13]; here the priority is remarking how to inspect the energy loss *dE* by emission of gravitational wave energy. Rewrite (5.70) in order to replace the deterministic orbit parameters *r* and
$\omega $ to highlight their quantum meaning hidden in the given formulas and replace the integration factor 32/5 = 6.4 with
$2\pi $, which differ by about 1.8% only; this numerical replacement allows highlighting conveniently the following considerations. Regard then the original Einstein result as

$-\frac{\delta E}{\delta t}=\frac{2\pi}{{W}_{p}}{\left(\frac{{m}_{1}{m}_{2}}{{m}_{1}+{m}_{2}}\right)}^{2}\delta {r}^{4}{\omega}^{6},\text{\hspace{1em}}\omega =\frac{2\pi}{\delta t},\text{\hspace{1em}}{W}_{P}=\frac{{c}^{5}}{G};$ (5.71)

all quantities with notation $\delta $ are now uncertainty ranges, $\delta t$ is the time lapse to complete one orbit.

First of all rewrite identically the first equation with the help of the second (5.70) and (5.60) itself as follows

$-\frac{\delta E}{\delta t}=\frac{2\pi}{{W}_{p}}{\left(\frac{{\u03f5}_{G}\delta r}{{\omega}^{2}\delta {r}^{3}}\right)}^{2}\delta {r}^{4}{\omega}^{6}=\frac{2\pi}{{W}_{p}}{\left({\u03f5}_{G}\omega \right)}^{2};$ (5.72)

moreover replace once more $\delta t=n\hslash /\delta \epsilon $ according to (1.2), so that

$-\delta E\delta \epsilon =n\hslash \frac{2\pi}{{W}_{p}}{\left({\epsilon}_{G}\omega \right)}^{2}=nh{W}_{P}{\left({t}_{G}\omega \right)}^{2},\text{\hspace{1em}}{t}_{G}=\frac{{\epsilon}_{G}}{{W}_{P}}$

with notation emphasizing that $energy/{W}_{P}$ has physical dimensions of time. Also, this equation becomes

$-\delta E\frac{\delta \u03f5}{{W}_{P}}=-\delta E\delta {t}_{G}=nh{\left({t}_{G}\omega \right)}^{2},\text{\hspace{1em}}\delta {t}_{G}=\frac{\delta \u03f5}{{W}_{P}}$

and eventually

$-\delta E=nh{\nu}_{G}{\left({t}_{G}\omega \right)}^{2},\text{\hspace{1em}}{\nu}_{G}=\frac{1}{\delta {t}_{G}},$ (5.73)

with notation emphasizing that the energy loss $-dE$ can be nothing else but loss of some quanta $h{\nu}_{G}$. Now it is possible to introduce the last step and compare this result to (5.60), according which $-{\u03f5}_{G}=nh\nu +h\nu /2$ ; therefore $dE=\delta {\u03f5}_{G}=h\nu \delta n$ yields

$-h\nu \delta n=nh{\nu}_{G}{\left({t}_{G}\omega \right)}^{2}.$

Just this conclusion is the key to guess the dimensionless ${t}_{G}\omega $ that appears to be just a correction factor: being $\delta n$ integer, $-\delta E$ can be nothing else but something like ${n}^{\prime}h\nu $ with ${n}^{\prime}$ integer in order to fit (5.60) [21]. Also, as ${t}_{G}$ is introduced via $\delta \epsilon $ and thus arbitrary, put then ${\left({t}_{G}\omega \right)}^{2}={n}^{\prime}/n$ so that

$-\delta E={n}^{\prime}h{\nu}_{G}\mathrm{.}$ (5.74)

Otherwise stated, ${t}_{G}\omega $ has been defined in order that (5.74) is consistent with (5.62). In synthesis, the initial Einstein formula, deterministic, becomes here a very simple quantum result, showing at the right hand side the number ${n}^{\prime}$ of gravitational energy quanta lost. Also here $-\delta E$ expresses the fact that ${n}^{\prime}$ must be intended as ${n}^{\u2033}\le {n}^{\prime}\le {n}^{\u2034}$, with ${n}^{\u2033}$ and ${n}^{\u2034}$ of course arbitrary, once more according to (1.9).

Although for brevity this result has been introduced here as mere elaboration of Einstein’s early achievement, reversing the steps from (5.74) to (5.70) one could find the initial $\delta E$ whose quantization is however hidden. The paper [9] concerns instead an “ab initio” model, where are also described further implications of this result. The Einstein formula is actually a quantum of gravitational energy dissipated by an orbiting system. In this quoted paper, published before the experimental evidence of the gravitational waves, it is remarked that not necessarily the gravitational system must collapse; rather both signs possible for $\delta n$ describe the exchange of gravitational quanta between orbiting systems, possibly the so called gravitons, could be regarded in principle in analogy with electromagnetic excitation and decay of atoms by exchange of photons. This supports the idea of gravitons inherent the gravity propagation rate (4.8).

7) Quantum remarks on the Newton equation.

At this point some remarks on (5.40) and (5.60) deserve attention.

(i) According to (5.59), (5.60) and (5.62), write

$\frac{\hslash}{2c}=G\frac{{m}_{1}{m}_{2}}{{v}_{1}^{2}},$ (5.75)

which yields

${\u019b}_{{m}_{1}}=\frac{\hslash}{{m}_{1}c}=\frac{2{m}_{2}G}{{v}_{1}^{2}},\text{\hspace{1em}}{\u019b}_{{m}_{2}}=\frac{\hslash}{{m}_{2}c}=\frac{2{m}_{1}G}{{v}_{1}^{2}};$ (5.76)

so, taking the limit ${v}_{1}\to c$, one finds respectively

${\u019b}_{{m}_{1}}=\frac{2{m}_{2}G}{{c}^{2}},\text{\hspace{1em}}{\u019b}_{{m}_{2}}=\frac{2{m}_{1}G}{{c}^{2}}.$

The black hole radii are thus the limit of (5.40) for ${v}_{1}\to c$ suggest an interesting feature of a bound gravitational system where either mass is a black hole; merging of their masses occurs when the event horizon of the latter approaches the Compton length of the former.

(ii) It is usual to say that at the center of a black hole there is a space time singularity. Emphasizing that no singularity is explicitly required by or directly implied in the present conceptual frame, such a singularity is actually unknowable and thus unphysical: according to (1.2) and (2.41), by definition non-deterministic, no information is accessible about what happens inside an uncertainty range. Thus the concept of local singularity is merely an arbitrary extrapolation allowed in the classical world only; here instead the relativity is conceived in the quantum frame of (1.2).

(iii) Via (5.59) $\hslash {v}^{2}/2c={m}^{2}G$ yields an expression for the force constant ${k}_{f}$ of harmonic oscillations (5.62)

${k}_{f}={\omega}^{2}m=\frac{{\omega}^{2}\hslash {v}^{2}}{2Gmc}=\frac{1}{2}\frac{{\u019b}_{m}}{G}{a}^{2},\text{\hspace{1em}}a=\omega v$ (5.77)

Where*
$a$ * is a further definition of acceleration being
$v\omega =\delta x/\delta {t}^{2}$. This result, which clearly plugs the force constant of the quantum oscillator into the frame of the general relativity, will be further considered in the next subsection.

(iv) The reduced Compton length ${\u019b}_{m1}$ of ${m}_{1}$ defines in (5.76) the Schwarzschild radius of the interacting mass ${m}_{2}$ as a limit case. This result follows the link between the energy of a two body gravitational system of masses and the energy of a quantum oscillator with binding force constant ${k}_{f}$ ; in turn this link suggests that the quantum relationship between ${m}_{1}$ and ${m}_{2}$ makes their event horizons equal to the respective Compton lengths; these latter correspond to the minimum approaching distance below which the masses merge into a unique black hole of total mass ${m}_{1}+{m}_{2}$ with event horizon $2\left({m}_{1}+{m}_{2}\right)G/{c}^{2}$. Indeed

${\u019b}_{m1}+{\u019b}_{m2}={\u019b}_{\mu}=\frac{\hslash}{\mu c},\text{\hspace{1em}}\mu =\frac{{m}_{1}{m}_{2}}{{m}_{tot}},\text{\hspace{1em}}\frac{1}{2}\mu {c}^{2}=G\frac{{m}_{1}{m}_{2}}{{\u019b}_{\mu}}=-U,\text{\hspace{1em}}U<0.$ (5.78)

(v) The fact that (5.60) is related to an arbitrary number *n* of quantum states allowed to a harmonic oscillator shows that even the gravity equation at the right hand side can be expressed as a superposition of states corresponding to and defined by the respective *n*. Start from (5.60) written according to (5.53) as

$\frac{G{m}_{1}{m}_{2}}{2\delta \mathcal{l}}-\frac{1}{2}\hslash \omega =n\hslash \omega =\left({n}^{\prime}-{n}^{\u2033}\right)\hslash \omega ,\text{\hspace{1em}}\omega =\frac{{v}_{0}^{2}}{{c}^{2}\delta t},\text{\hspace{1em}}n={n}^{\prime}-{n}^{\u2033};$ (5.79)

then, with the last position where ${n}^{\prime}$ and ${n}^{\u2033}$ are arbitrary integers too, one finds via (5.61) and (1.9)

$\delta {\epsilon}_{G}=\delta {\epsilon}_{\psi},\text{\hspace{1em}}\delta {\epsilon}_{G}=T-\frac{1}{2}\hslash \omega ,\text{\hspace{1em}}\delta {\epsilon}_{\psi}={n}^{\prime}\hslash \omega -{n}^{\u2033}\hslash \omega \text{\hspace{0.05em}}$ (5.80)

In this way the average kinetic energy of Newton orbital motion identified in (5.61) defines the range size
$\delta {\epsilon}_{G}$ equal to that
$\delta {\epsilon}_{\psi}$ of the quantized energies corresponding to the wave functions
${{\psi}^{\prime}}_{\mathcal{l}}^{\mathrm{*}}$ and
${{\psi}^{\u2033}}_{\mathcal{l}}^{\mathrm{*}}$ defined by (2.58). As concerns *U*, write according to (5.61)

$-U=\frac{G{m}_{1}{m}_{2}}{\delta \mathcal{l}}=\left(2n+1\right)\hslash \omega =\frac{{\text{M}}_{j}^{2}/{\hslash}^{2}}{n\left(n+1\right)}\hslash \omega \mathrm{.}$

So the average potential energy of the gravitational system is equal to
$\hslash \omega $ times the number
$2n+1$ of allowed states of angular momentum; in other words the left hand side consists of
$2n+1$ degenerate states
$\hslash \omega $. At the left hand side still appears the gravitational energy of the same masses now
$\delta \mathcal{l}$ apart, at the right hand side the energy
$\hslash {\omega}_{\mathcal{l}\mathrm{,}n}$ with the same coefficient
$2n+1$ due to the degeneracy of
${l}_{j}$ states consistent with *n*. Thus Newton’s equation is equivalent to a superposition of *l* states of energy
$\hslash {\omega}_{\mathcal{l}\mathrm{,}n}$ having quantum origin.

(vii) Consider (5.25) and write

$\frac{2{m}_{bh}G}{{c}^{2}}=\frac{2{m}_{bh}{c}^{2}G}{{c}^{4}}=\frac{2h{\nu}_{bh}}{{F}_{P}}=\delta {\mathcal{l}}_{bh},\text{\hspace{1em}}{m}_{bh}=\frac{h{\nu}_{bh}}{{c}^{2}},\text{\hspace{1em}}h{\nu}_{bh}=\frac{1}{2}{F}_{P}\delta {\mathcal{l}}_{bh};$

via the Planck force ${F}_{P}={c}^{4}/G$. Recalling (5.54), it is possible to regard the last term as a zero point energy and thus to define by analogy a more general energy given by

${E}_{n}=nh{\nu}_{bh}+\frac{1}{2}h{\nu}_{bh}=\left(2n+1\right){F}_{P}\delta {\mathcal{l}}_{bh}.$

To interpreter this result, think a set of *n* non-interacting harmonic oscillators consisting of point masses
${m}_{bh}$ that displace with frequency
${\nu}_{bh}$ by a length
$\delta \mathcal{l}$ with respect to their equilibrium positions: the energy of such a system results averaging the mechanical work of
${F}_{P}$ to displace all point masses and their oscillation energy
$h{\nu}_{bh}$. This average
${E}_{n}/n\left(n+1\right)$ concerns clearly the corpuscle/wave behavior of
${m}_{bh}$.

8) The invariance in quantum special relativity.

The starting key equations are now (3.29), which yield

$p=\frac{\epsilon v}{{c}^{2}},\text{\hspace{1em}}\delta p=\frac{\delta \left(\epsilon v\right)}{{c}^{2}};$

multiplying side by side these equations

$p\delta p=\frac{\left(\epsilon v\right)\delta \left(\epsilon v\right)}{{c}^{4}}$

one finds

$\frac{1}{2}\delta {\left(pc\right)}^{2}=\frac{1}{2}\frac{\delta {\left(\epsilon v\right)}^{2}}{{c}^{2}}$ (5.81)

and thus

$\delta {\left(\epsilon v/c\right)}^{2}-\delta {\left(pc\right)}^{2}=\mathrm{0,}$ (5.82)

being of course by definition

$\delta {\left(\epsilon v/c\right)}^{2}={\left({\epsilon}_{1}{v}_{1}/c\right)}^{2}-{\left({\epsilon}_{2}{v}_{2}/c\right)}^{2},\text{\hspace{1em}}\delta {\left(pc\right)}^{2}={\left({p}_{1}c\right)}^{2}-{\left({p}_{2}c\right)}^{2}.\text{\hspace{0.05em}}$ (5.83)

Clearly (5.82) reads identically ${\left(\epsilon v/c\right)}^{2}-{\left(pc\right)}^{2}=0$, which of course is fulfilled by $p=\u03f5v/{c}^{2}$ ; yet, considering the ranges (5.83) of dynamical variables instead of their local values (2.32), (5.82) is also fulfilled by

${\left(\epsilon v/c\right)}^{2}-{\left(pc\right)}^{2}=const$ (5.84)

because $\delta \left(const\right)=0$. This equation yields thus

${\left(\epsilon v/c\right)}^{2}={\left(pc\right)}^{2}+const.$ (5.85)

Owing to (3.29), if in particular
$const=0$ and
$v\ne c$ then (5.85) reduces trivially to an identity. The fact that (3.29) holds even for
$v=c$, whereas (2.35) do not, is the key to understand and verify the next step: although both *p* and
$\epsilon $ diverge for
$v=c$ according to (2.35), their ranges
$\delta p$ and
$\delta \epsilon $ defined in (5.82) do not. Even for
$v=c$ they take the form
$\infty -\infty $ that in principle could admit finite limits uniquely defined by (5.84). In effect it has been already remarked that have physical meaning the uncertainty ranges and not the local dynamical variables, random unknown and unknowable according to (1.2). Then, even implementing the particular position
$v=c$, (5.85) resulting from (5.82) yields for
$const\ne 0$ the well known result

${\epsilon}^{2}=const+{\left(pc\right)}^{2},\text{\hspace{1em}}const={\left({m}_{0}{c}^{2}\right)}^{2},\text{\hspace{1em}}v=c,$ (5.86)

being ${m}_{0}$ a rest mass according to (2.33). This well known result of special relativity together with (WHF) defines the energy and the Lorentz transformations already found in (2.35) and (3.28); the fact of having replaced the local values with the respective uncertainty ranges makes plausible the step from (5.85) to (5.86). The quantum uncertainty is thus essential to generalize (5.85): eq (5.86) has been concerned here although already obtained in (2.34) just to emphasize the link between uncertainty and special relativity and the physical importance of the uncertainty ranges. This is the reason of having repeated the result (2.34).

Let us show further that however (5.86) is not itself the most useful result for the next purposes just because its deterministic local values. Start therefore directly from (1.2); squaring both sides of $\delta \epsilon =v\delta p$, being of course $v=\delta x/\delta t$, one finds owing to (2.35)

${\left(\delta \epsilon \right)}^{2}={\left(v/c\right)}^{2}{\left(\delta \left(pc\right)\right)}^{2}={\left(\delta \left(pc\right)\right)}^{2}-{\beta}^{2}{\left(\delta \left(pc\right)\right)}^{2},\text{\hspace{1em}}\beta =\sqrt{1-{v}^{2}/{c}^{2}}.$ (5.87)

Next, calculating $\delta p$ as $\delta p={p}_{2}-{p}_{1}={m}_{02}v/\beta -{m}_{01}v/\beta $ via (2.35), one finds

$\beta \delta \left(pc\right)=\beta \left({m}_{02}vc/\beta \right)-\beta \left({m}_{01}vc/\beta \right)=\left({m}_{02}-{m}_{01}\right)vc;$ (5.88)

as expected from previous considerations, there is no difficulty to calculate this result of (5.87) for $v=c$. So

${\left(\delta \epsilon \right)}^{2}-{\left(\delta \left(pc\right)\right)}^{2}=-{\left(\delta \left({m}_{0}{c}^{2}\right)\right)}^{2},\text{\hspace{1em}}\delta {m}_{0}={m}_{02}-{m}_{01},$ (5.89)

whence it is possible to infer

$\begin{array}{c}{\left(\delta \epsilon \right)}^{2}-{\left(\delta \left(pc\right)\right)}^{2}={\left(n\hslash \right)}^{2}\left(\frac{1}{{\left(\delta t\right)}^{2}}-\frac{{c}^{2}}{{\left(\delta x\right)}^{2}}\right)\\ ={\left(n\hslash \right)}^{2}\frac{{\left(\delta x\right)}^{2}-{c}^{2}{\left(\delta t\right)}^{2}}{{\left(\delta t\delta x\right)}^{2}}=-{\left(\delta {m}_{0}{c}^{2}\right)}^{2}.\end{array}$ (5.90)

Clearly the right hand side is a constant, thus invariant by definition; hence the ratio at the left hand side is an invariant as well. It is known indeed that this ratio is defined by two invariant quantities. Since the first (5.46) demonstrates that $\delta x\delta t$ is a relativistic invariant, it follows that the numerator is also invariant as well itself. The ranges at the left hand side of the last equality correspond thus to the $\delta {m}_{0}={m}_{02}-{m}_{01}$ at the right hand side. The invariant interval $\delta {x}^{2}-{c}^{2}\delta {t}^{2}$, in particular, has been stated in [13] as the conceptual foundation of the special relativity; just for this reason it is remarkable the fact that in the present model i(5.46) and (5.90) are actually straightforward corollaries of the quantum uncertainty. The crucial difference between (5.90) and (5.86) is that now $\epsilon $ and $pc$ appear through their uncertainty ranges and not as deterministic values. This result not only demonstrates the link between special relativity and quantum physics, but also allows further important steps concerning directly the general relativity. Although this point has been examined in several previous papers, see e.g. [4] [11] [12], the next section reports some relevant considerations just on this topic. Consider once more (5.83) for ${v}_{1}={v}_{2}=c$, as already done to infer (5.86); the reasoning is still that already highlighted, but now extended to find a further interesting result. Write explicitly (5.83) with the help of (5.34) as follows

$\delta \left({\epsilon}^{2}\right)={\epsilon}^{2}-{\left({m}_{0}{c}^{2}\right)}^{2},\text{\hspace{1em}}\delta {\left(pc\right)}^{2}={\left(pc\right)}^{2}-\frac{{p}^{2}}{2{m}_{1}}\epsilon ,$ (5.91)

being ${m}_{1}$ a new arbitrary mass; here we have simply expressed also the right hand side in the form of a range of square energies. Then (5.82) for $v=c$ reads

${\epsilon}^{2}-{\left({m}_{0}{c}^{2}\right)}^{2}={\left(pc\right)}^{2}-\frac{{p}^{2}}{2{m}_{1}}\epsilon .$ (5.92)

Clearly this equation reduces for ${m}_{1}\to \infty $ to the standard form (5.86) of Einstein’s special relativity. In fact the additional term in (5.92), more general than (5.86), is a known result of quantum gravity that helps solve three cosmological paradoxes [22]. More details about (5.92) are reported in [12].

5.6. Red Shift and Time Dilation

Starting from (2.10) and (2.9) consider $\stackrel{\dot{}}{v}=\u03f5c/\hslash =-\nabla \phi $ [13], being $-\nabla \phi $ a force per unit mass related to $\stackrel{\dot{}}{v}$ due to the gravitational potential $\phi $, and write in the present one-dimensional approach

$\frac{\u03f5c}{\hslash}=-\frac{\delta \phi}{\delta x}$

whence

$\frac{\u03f5\delta x}{\hslash c}=-\frac{\delta \phi}{{c}^{2}}.$

Let us show now that

$-\frac{\delta \phi}{{c}^{2}}=\frac{\u03f5\delta x}{\hslash c}=\frac{\delta \omega}{\omega}.$

Being by definition
$\delta x={x}_{2}-{x}_{1}$, regard the second equality considering that
$\u03f5{x}_{1}/\hslash c={\omega}_{1}/\omega $ *i.e.*
$\u03f5{x}_{1}\omega /c=\hslash {\omega}_{1}$ : indeed this result reads
$\u03f5{v}_{1}/c=\hslash {\omega}_{1}$ because
$\omega {x}_{1}={v}_{1}$ by dimensional reasons. Owing to (3.29) in turn this means
${p}_{1}c=\hslash {\omega}_{1}$ at the coordinate
${x}_{1}$ where the gravitational potential
$\phi $ reads
$\phi ={\phi}_{1}$ ; as consistently to
$\delta x$ corresponds
$\delta \phi ={\phi}_{2}-{\phi}_{1}$, while an analogous reasoning holds of course for
$\epsilon {x}_{2}/\hslash c={\omega}_{2}/\omega $, one finds

$\frac{\delta \omega}{\omega}=\frac{\delta \left(-\phi \right)}{{c}^{2}},\text{\hspace{1em}}\delta \omega ={\omega}_{2}-{\omega}_{1},\text{\hspace{1em}}\delta \left(-\phi \right)=-{\phi}_{2}-\left(-{\phi}_{1}\right);$ (5.93)

this result is the red shift of a photon moving radially in the attractive gravitational potential
$\phi <0$ because
${\omega}_{2}>{\omega}_{1}$ implies
$\left|{\phi}_{2}\right|>\left|{\phi}_{1}\right|$, *i.e.* lower frequency at the coordinate where the central gravitational potential is weaker. A corollary of this result is found replacing reasonably
$\omega \to {t}^{-1}$ ; (5.93) yields

$\frac{\delta \omega}{\omega}=\frac{\delta \left(1/t\right)}{1/t}=-\frac{\delta t}{t}=\frac{\delta \left(-\phi \right)}{{c}^{2}}=-\frac{\left|\delta \phi \right|}{{c}^{2}}$

*i.e.* summarizing

$\frac{t-{t}_{0}}{t}=\frac{\left|\delta \phi \right|}{{c}^{2}},\text{\hspace{1em}}\delta t=t-{t}_{0}.$ (5.94)

It is sensible to regard
${t}_{0}$ as a proper time, with respect to which is defined *t* determining
$\delta t$ ; as the right hand side describes gravitational potential rising from 0 to
$\phi $ with
$\phi -0<0$ (5.94) reads
$1-{t}_{0}/t=\left|\phi \right|/{c}^{2}$ *i.e.*

${t}_{0}=\left(1-\frac{\left|\phi \right|}{{c}^{2}}\right)t.$ (5.95)

Owing to proper time ${t}_{0}<t$, this result yields time dilation $\delta t$ due to gravity field with respect to field null.

5.7. Black Hole Entropy

Define the ratio of Planck length and mass, ${\mathcal{l}}_{P}=\sqrt{\hslash G/{c}^{3}}$ and ${m}_{P}=\sqrt{\hslash c/G}$, which reads ${\mathcal{l}}_{P}/{m}_{P}=G/{c}^{2}$ ; as (5.25) reads $\delta {\mathcal{l}}^{\mathrm{*}}/2{m}^{\mathrm{*}}=G/{c}^{2}$, the starting point of this section is

$\frac{\delta {\mathcal{l}}^{*}}{{m}^{*}}=\frac{2G}{{c}^{2}}=\frac{2{\mathcal{l}}_{P}}{{m}_{P}}.$ (5.96)

This section is based on the ideas exposed about (5.47).

(i) Surface entropy.

Assume that the surface entropy is an extensive property that increases proportionally to the black hole surface. Squaring both sides of (5.96), it is possible to define a function ${S}^{\mathrm{*}}$ as follows

${S}^{*}={\left(\frac{{m}^{*}}{{m}_{P}}\right)}^{2}=\frac{\delta {\mathcal{l}}^{*2}}{4{\mathcal{l}}_{P}^{2}}=\frac{1}{4\pi}\frac{4\pi \delta {\mathcal{l}}^{*2}}{4{\mathcal{l}}_{P}^{2}}={S}_{0}\frac{{A}^{*}}{4{\mathcal{l}}_{P}^{2}},\text{\hspace{1em}}{A}^{*}=4\pi \delta {\mathcal{l}}^{*2},\text{\hspace{1em}}{S}_{0}=\frac{1}{4\pi}$ (5.97)

*i.e.* such that

${S}_{sur}^{*}=\frac{{S}^{*}}{{S}_{0}}=\frac{{A}^{*}}{4{\mathcal{l}}_{P}^{2}}.$ (5.98)

In effect the given ratio of squared masses in (5.97) is equal to the ratio of square lengths that, owing to (5.47), has physical meaning of surface entropy; so ${S}_{sur}^{\mathrm{*}}$ is the sought function that increases linearly with the surface ${A}^{\mathrm{*}}$ defined by the black body length $\delta {\mathcal{l}}^{\mathrm{*}}$ via the proportionality constant ${S}_{0}$. Since accordingly

${S}_{sur}^{*}=4\pi {S}^{*}=4\pi {\left(\frac{{m}^{*}}{{m}_{P}}\right)}^{2}=4\pi \frac{{m}^{*2}G}{\hslash c},$

the factor $4\pi $ reminds the Gauss theorem and suggests its link to the flux ${\Phi}^{\mathrm{*}}$ of an appropriate function ${F}^{\mathrm{*}}$ through a surface element $\delta {A}^{\mathrm{*}}$ of ${A}^{\mathrm{*}}$. So write

$\delta {\Phi}^{*}={F}^{*}\cdot \delta {A}^{*},\text{\hspace{1em}}{F}^{*}={F}^{*}\frac{r}{{r}^{3}},\text{\hspace{1em}}{F}^{*}=-{m}^{*2}G,\text{\hspace{1em}}\delta {A}^{*}=n\delta {A}^{*},$ (5.99)

where r is unit vector directed inside the flux surface $\delta {A}^{\mathrm{*}}$ whereas n is a unit vector oriented outside the surface $\delta {A}^{\mathrm{*}}$. Hence

$\Phi ={\displaystyle \int}\left({F}^{*}\frac{r\cdot n}{\delta {r}^{3}}\delta {A}^{*}\right)={\displaystyle \int}\left({m}^{*2}G\frac{\delta {A}^{*}}{\delta {r}^{2}}\right)={\displaystyle \int}\left({m}^{*2}G\delta \Omega \right)=4\pi {m}^{*2}G$ (5.100)

and then the last (5.97) yields

${S}_{0}=\frac{{m}^{*2}G}{\hslash c}=\frac{\hslash G/{c}^{2}}{c{\u019b}_{{m}^{*}}^{2}}=\frac{{\mathcal{l}}_{P}^{2}}{{\u019b}_{{m}^{*}}^{2}},\text{\hspace{1em}}{S}^{*}=\frac{{A}^{*}}{4{\u019b}_{{m}^{*}}^{2}}.$ (5.101)

With
${F}^{\mathrm{*}}$ related to the classical Newton law, the given definition (5.97) of
${S}_{0}$ makes
${S}_{sur}^{\mathrm{*}}$ proportional to the incoming flux
$\Phi $ of gravity force at the black hole surface expressed in
$\hslash c$ units; so
${S}_{sur}^{\mathrm{*}}$ does not depend explicitly on
${m}^{\mathrm{*}}$ but on
$\left|\Phi \right|$ it generates. The presence of
$\hslash $ and *c* in (5.101) shows the link between black body matter and usual matter inherent the standard definition (5.47) of entropy, so merging quantum and relativistic concepts. This definition of
${S}_{0}$ is sensible, as it results to be a ratio of square lengths too, whose physical meaning of entropy agrees with that of
${S}_{sur}^{\mathrm{*}}$.

Equation (5.98) is the famous Hawking-Bekenstein surface entropy of a spherical non-rotating black body.

(ii) Volume entropy

In an analogous way it is possible to calculate the volume entropy. Once knowing that ${\left({m}^{\mathrm{*}}/{m}_{P}\right)}^{2}$ is related to ${\left(\delta {\mathcal{l}}^{\mathrm{*}}/{\mathcal{l}}_{P}\right)}^{2}$, which has physical meaning of entropy, it is reasonable to guess that now the expected ${\left(\delta {\mathcal{l}}^{\mathrm{*}}/{\mathcal{l}}_{P}\right)}^{3}$ related to ${\left({m}^{\mathrm{*}}/{m}_{P}\right)}^{3}$ should yield a result ${m}^{\mathrm{*}}/{m}_{P}$ times that previously obtained. Write thus

${{S}^{\prime}}^{*}={\left(\frac{{m}^{*}}{{m}_{P}}\right)}^{3}=\frac{4\pi \delta {\mathcal{l}}^{*3}/3}{4\pi {\left(2{\mathcal{l}}_{P}\right)}^{3}/3}={{S}^{\prime}}_{0}\frac{{V}^{*}}{8{\mathcal{l}}_{P}^{3}},\text{\hspace{1em}}{{S}^{\prime}}_{0}={\left(\frac{4\pi}{3}\right)}^{-1},\text{\hspace{1em}}{V}^{*}=\frac{4\pi}{3}\delta {\mathcal{l}}^{*3}$ (5.102)

and then, as before,

${S}_{vol}^{*}=\frac{{{S}^{\prime}}^{*}}{{{S}^{\prime}}_{0}^{*}}=\frac{{V}^{*}}{8{\mathcal{l}}_{P}^{3}};$ (5.103)

this result corresponds to (5.98). Moreover (5.101) becomes

${S}_{vol}^{*}=\frac{4\pi}{3}{\left(\frac{{m}^{*}}{{m}_{P}}\right)}^{3}=\frac{1}{3}\frac{{m}^{*}}{{m}_{P}}{S}_{sur}^{*},$ (5.104)

where now the mas ${m}^{\mathrm{*}}$ appears explicitly; also, owing to (5.101), (5.104) should reasonably yield the volume entropy of a spherical black body with an analogous meaning of ${{S}^{\prime}}_{0}=3{S}_{0}$. Also, since

$\frac{{V}^{*}}{8{\mathcal{l}}_{P}^{3}}=\frac{{m}^{*}}{{m}_{P}}\frac{{m}^{*2}G}{\hslash c},$

the flux ratio $\left|\Phi \right|/\hslash c$ multiplied by the huge ratio ${m}^{\mathrm{*}}/{m}_{P}$ corresponds to the expected ${V}^{\mathrm{*}}/{\mathcal{l}}_{P}^{3}\gg {A}^{\mathrm{*}}/{\mathcal{l}}_{P}^{2}$.

5.8. Perihelion Precession

Consider the square ranges ${\left(\delta \epsilon \right)}^{2}$ and ${\left(\delta \left(pc\right)\right)}^{2}$ of (5.89), now defined explicitly as

${\left({\epsilon}_{sr}-{\epsilon}_{\zeta}\right)}^{2}={\left(c{p}_{sr}-c{p}_{\zeta}\right)}^{2}+{\left({m}_{0}{c}^{2}\right)}^{2},\text{\hspace{1em}}{m}_{0}={m}_{sr}-{m}_{\zeta};$ (5.105)

the subscript *sr* stands for “*special* *relativity*, whereas
${\epsilon}_{\zeta}$ and
$c{p}_{\zeta}$ are appropriate boundary energies to be defined. This choice is possible because the boundaries of uncertainty ranges are arbitrary. Once having defined
${\epsilon}_{sr}$ consistently with the special relativity energy (5.86),
$c{p}_{\zeta}$ and
${\epsilon}_{\zeta}$ represent instead the actual momentum and energy in a gravitational system congruently to the respective quantities of the special relativity. In other words the boundary energies and momenta of
${\left(\delta \epsilon \right)}^{2}$ and
${\left(\delta \left(pc\right)\right)}^{2}$ of (5.89) are chosen in (5.105) in order to generalize the corresponding values of special relativity to the case of a gravitational system just taking advantage of the arbitrariness and agnosticism inherent the uncertainty. Write therefore

${\epsilon}_{sr}^{2}+{\epsilon}_{\zeta}^{2}-2{\epsilon}_{sr}{\epsilon}_{\zeta}={\left(c{p}_{sr}\right)}^{2}+{\left(c{p}_{\zeta}\right)}^{2}-2{c}^{2}{p}_{sr}{p}_{\zeta}+{\left({m}_{0}{c}^{2}\right)}^{2}$

that fulfills by definition

${\epsilon}_{sr}^{2}={\left(c{p}_{sr}\right)}^{2}+{\left({m}_{sr}{c}^{2}\right)}^{2},$ (5.106)

so that

${\epsilon}_{\zeta}^{2}-2{\epsilon}_{sr}{\epsilon}_{\zeta}={\left(c{p}_{\zeta}\right)}^{2}-2{c}^{2}{p}_{sr}{p}_{\zeta}+{\left({m}_{\zeta}{c}^{2}\right)}^{2}-2{m}_{sr}{m}_{\zeta}{c}^{4};$ (5.107)

then, putting

${\epsilon}_{\xi}^{2}=2\left({\epsilon}_{sr}{\epsilon}_{\zeta}-{c}^{2}{p}_{sr}{p}_{\zeta}-{m}_{sr}{m}_{\zeta}{c}^{4}\right),$ (5.108)

the right hand side reads

${\epsilon}_{\zeta}^{2}={\left(c{p}_{\zeta}\right)}^{2}+{\left({m}_{\zeta}{c}^{2}\right)}^{2}+{\epsilon}_{\xi}^{2}$ (5.109)

where

${\epsilon}_{\xi}^{2}=2{\epsilon}_{sr}{\epsilon}_{\zeta}\left(1-\frac{{c}^{2}{p}_{sr}{p}_{\zeta}}{{\epsilon}_{sr}{\epsilon}_{\zeta}}-\frac{{m}_{sr}{m}_{\zeta}{c}^{4}}{{\epsilon}_{sr}{\epsilon}_{\zeta}}\right)=2{\u03f5}_{\xi}^{2}\left(1-\frac{{v}_{sr}{p}_{\zeta}}{{\epsilon}_{\zeta}}-\frac{{m}_{sr}{m}_{\zeta}{c}^{4}}{{\epsilon}_{sr}{\epsilon}_{\zeta}}\right),\text{\hspace{1em}}{\u03f5}_{\xi}^{2}={\epsilon}_{sr}{\epsilon}_{\zeta}.$

Note that now are used notations like ${p}_{\zeta}$, and not $\delta {p}_{\zeta}$, because are implemented boundaries of uncertainty ranges that of course are not deterministic values. From a formal point of view the Equation (5.109) is similar to (2.34), inferred through a quantum approach, apart from the additional term ${\epsilon}_{\xi}^{2}$ ; this is clearly consistent with the fact that by definition the subscript $\zeta $ refers to dynamical variables of general relativity introduced in (5.105). On the one hand (5.109) confirms the quantum gravity result (5.92), where in effect the corrective term $\u03f5{p}^{2}/m$ also appears with respect to the Einstein energy equation of the standard special relativity; this comparison suggests that even ${\epsilon}_{\xi}^{2}$ should be someway reducible to that in (5.92), which in fact has been also inferred reasoning on the boundaries of energy uncertainty ranges. On the other hand, ${\u03f5}_{\xi}^{2}$ represents the sought generalization of (2.34) or (5.86). Eventually rewrite (5.109) as

$\frac{{\epsilon}_{\zeta}^{2}-{\left({m}_{\zeta}{c}^{2}\right)}^{2}}{{\left(c{p}_{\zeta}\right)}^{2}}=1+\frac{{\epsilon}_{\xi}^{2}}{{\left(c{p}_{\zeta}\right)}^{2}}.$ (5.110)

This equation is interesting; it depends upon how is defined ${\epsilon}_{\xi}$ at the right hand side. For example specify in particular ${\epsilon}_{\xi}$, in principle arbitrary itself, according to (2.11); in this way it is possible to finalize (5.110) to the purpose of generalizing energy and momentum of the special relativity to the corresponding dynamical variables of the general relativity. In this specific case, possible and reasonable, regard ${\epsilon}_{\xi}$ and ${p}_{\zeta}$ defining

${\left(\frac{{\epsilon}_{\xi}}{c{p}_{\zeta}}\right)}^{2}={\left(-\frac{G{m}_{1}{m}_{2}}{\chi c\text{M}}\right)}^{2},\text{\hspace{1em}}{\epsilon}_{\xi}=-\frac{G{m}_{1}{m}_{2}}{{\mathcal{l}}_{\zeta}},\text{\hspace{1em}}{\mathcal{l}}_{\zeta}{p}_{\zeta}=\chi \text{M}\mathrm{,}$

where $\text{M}=\left|M\right|$ has physical dimensions of modulus of angular momentum and $\chi $ is regarded as a proportionality factor linking ${\mathcal{l}}_{\zeta}{p}_{\zeta}$ to M to be defined. Since the last equation has mere dimensional basis, $\chi $ can be defined conveniently as the factor that converts ${\mathcal{l}}_{\zeta}{p}_{\zeta}$ into an average value $\text{M}=\langle \left|M\right|\rangle $ of the modulus of the orbiting angular momentum $M$. This result turns next (5.110) into

$\frac{{\epsilon}_{\zeta}-{m}_{\zeta}{c}^{2}}{c{p}_{\zeta}}\frac{{\epsilon}_{\zeta}+{m}_{\zeta}{c}^{2}}{c{p}_{\zeta}}=1+{\left(\frac{G{m}_{1}{m}_{2}}{\chi c\text{M}}\right)}^{2}\mathrm{,}$

which reads eventually

$\delta \varphi =\frac{1}{k}+\frac{1}{k{\chi}^{2}}{\left(\frac{G{m}_{1}{m}_{2}}{c\text{M}}\right)}^{2},\text{\hspace{1em}}\delta \varphi =\frac{{\epsilon}_{\zeta}-{m}_{\zeta}{c}^{2}}{c{p}_{\zeta}},\text{\hspace{1em}}k=\frac{{\epsilon}_{\zeta}+{m}_{\zeta}{c}^{2}}{c{p}_{\zeta}}\mathrm{.}$ (5.111)

Note that if
$G=0$ then
$\delta \varphi ={k}^{-1}$ ; *i.e.* actually the value of *k* is not essential as concerns
$\delta \varphi $, it is simply a reference value with respect to which is defined the change of
$\varphi $ with respect to a value
${\varphi}_{0}$ in the absence of field. The essential quantity is instead the range size
$\delta \varphi $, *i.e.* the shift of
${\epsilon}_{\zeta}$ with respect to the rest energy
${m}_{\zeta}{c}^{2}$. Hence it is sensible to introduce
$\Delta \varphi =\delta \varphi -{k}^{-1}$ in order to account for the change
$\varphi -{\varphi}_{0}$ the gravitational effect in parenthesis only. Rewrite then (5.111) as

$\Delta \varphi =const{\left(\frac{G{m}_{1}{m}_{2}}{c\text{M}}\right)}^{2},\text{\hspace{1em}}const=\frac{1}{k{\chi}^{2}}\mathrm{:}$ (5.112)

the second position regards
$k{\chi}^{2}$ as mere proportionality constant of the gravitational term in parenthesis, assumed to be the one physically relevant because in fact it concerns the parameters *G* and M that govern the orbital behavior of
${m}_{1}$ and
${m}_{2}$. Remind now (2.26) that refers to a circular orbit of
${m}_{2}$ in the gravity field of
${m}_{1}$ ; in the case of an elliptic orbit one expects that the early steady condition consistent with
$\lambda ={\lambda}_{circ}$ is reasonably to be replaced by a different wavelength
${\lambda}_{ell}$. Let be
${\lambda}_{ell}>{\lambda}_{circ}$, although still being
$n{\lambda}_{ell}/\delta {r}_{ell}=const$ in order to generalize (2.26)
$n{\lambda}_{circ}/\delta {r}_{circ}=2\pi $ while fulfilling the same kind of equation; with this assumption (2.26) is simply a particular case for
$\delta {r}_{ell}\to \delta {r}_{circ}$. Clearly
$\delta {r}_{ell}$ is now an “effective” radius, taking into account that the perimeter *C* of ellipse is actually a function of its semi axes *a* and *b*; an approximate formula is for example
$C\approx \pi \left[3\left(a+b\right)-\sqrt{\left(3a+b\right)\left(a+3b\right)}\right]$, which is reliable for the present purposes because for
$a=b$ it reduces to
$C=2\pi a$. If
$b>a$ then
$C\approx \pi \left[6a+3\left(b-a\right)-\sqrt{\left(3a+b\right)\left(a+3b\right)}\right]$ reduces to
$C\approx 6a\pi $ if the difference
$3\left(b-a\right)-\sqrt{\left(3a+b\right)\left(a+3b\right)}$ of positive numbers mutually self eliding becomes negligible with respect to 6*a*; so
$\delta {r}_{ell}\approx 6a$. With this numerical approximation, clearly
$2\pi \delta {r}_{circ}=n{\lambda}_{circ}$ is to be replaced by
$6\pi \delta {r}_{ell}=n{\lambda}_{ell}$ with identical physical meaning; so, merging (5.112) with the quantum condition
$const=n{\lambda}_{ell}/\delta {r}_{ell}=6\pi $ yields eventually

$\Delta \varphi \approx 6\pi {\left(\frac{G{m}_{1}{m}_{2}}{c\text{M}}\right)}^{2},\text{\hspace{1em}}const=6\pi \mathrm{.}$ (5.113)

Are worth noticing in this respect two remarks on the “Kepler problem” exposed in the textbooks [13] [21].

(i) The first one introduces the condition
$2\pi \text{\hspace{0.05em}}j/i$ in a non-relativistic approach, where *i* and *j* are integer numbers to get a steady closed trajectory via a rational fraction of
$2\pi $ ; in effect even in a classical model the perihelion precession is still possible, although insufficient to explain the astronomical observations. Here this condition is replaced by the quantum condition of an integer number of wavelengths in an elliptic orbit.

(ii) The second one lucidly shows step by step how to infer classically this famous Einstein formula of Mercury perihelion simply comparing two forms of potential energy of orbiting system; the mathematical formulation introduces first the mere Newtonian potential
${U}_{N}={\beta}_{1}/r$ and then also assumes an extended potential form
${U}_{\zeta}={\beta}_{2}/{r}^{2}$, where
${\beta}_{1}$ and
${\beta}_{2}$ are appropriate constants. Of course the plain Newton law does not justify
${U}_{\zeta}$, so that the approach shown in the book had mere speculative/didactic character. Nevertheless the procedure therein reported, very instructive and significant in principle, has actual interest here because the present theoretical frame does admit in fact higher order potential terms like
${U}_{\zeta}$ besides
${U}_{N}$ once replacing the deterministic *r* with
$\delta r$. Indeed (4.23) and (4.3) yield according to (1.2) and (2.12)

$\begin{array}{c}force=n\hslash \frac{\delta \stackrel{\dot{}}{x}}{\delta {x}^{2}}=-n\hslash \frac{\delta}{\delta t}\left(\frac{1}{\delta x}\right)=-\frac{n\hslash}{\delta t}\delta \left(\frac{1}{\delta x}\right)\\ =-\delta \epsilon \delta \left(\frac{1}{\delta x}\right)=-\frac{n\hslash v}{\delta x}\delta \left(\frac{1}{\delta x}\right)=\frac{n\hslash v}{\delta {x}^{3}}\delta \left(\delta x\right):\end{array}$ (5.114)

the notation emphasizes that the time derivative $\delta \stackrel{\dot{}}{x}$ is actually regarded as ratio of ranges $\delta \left({x}^{-1}\right)$ and $\delta t$ as previously explained in the sections 2.6 and 2.7. The former is in particular relevant: it reminds the curvature of space time, as explained in (4.15) and (4.16). Since the last equality of the chain yields in turn

$\begin{array}{l}force=\frac{n\hslash v}{\delta {x}^{3}}\delta \left(\delta x\right)=\frac{n\hslash v\delta {t}^{\prime}}{\delta {x}^{3}}\frac{\delta}{\delta {t}^{\prime}}\delta x=\frac{n\hslash \delta {x}^{\prime}}{\delta {x}^{3}}\frac{\delta}{\delta {t}^{\prime}}\delta x=\frac{n\hslash \delta {x}^{\prime}}{\delta {x}^{3}}\delta {\stackrel{\dot{}}{x}}^{\prime},\\ \delta {x}^{\prime}=v\delta {t}^{\prime}=\frac{\delta \epsilon \delta {t}^{\prime}}{n\hslash},\end{array}$ (5.115)

is clear the implication here: requiring that $\delta {x}^{\prime}=cons{t}^{\prime}=\delta {x}_{0}$ with an appropriate choice of $\delta {t}^{\prime}$, the result due to the space time curvature reads

$force=\frac{n\hslash \delta {x}_{0}}{\delta x}\frac{\delta {\stackrel{\dot{}}{x}}^{\u2033}}{\delta {x}^{2}},\text{\hspace{1em}}\delta {\stackrel{\dot{}}{x}}^{\u2033}=\frac{\delta}{\delta {t}^{\prime}}\delta x.$ (5.116)

It is evident that the reasoning from (5.114) to (5.116) just shown can be repeated, thus obtaining higher power potential terms. So the simple fact of having justified via (1.2) the potential (5.116) allows obtaining with elementary methods of classical mechanics the sought result, the perihelion precession tentatively exemplified classically in the quoted textbook. Here has been proposed an alternative derivation just to show that (5.105) is enough to obtain a crucial result of general relativity. Now, after having introduced the conceptual frame outlined by (1.1) and (1.2), it is sensible to proceed with calculations implementing (1.5).

6. Cosmological Calculations

$\frac{\hslash G}{{c}^{2}}=7.8\times {10}^{-62}\text{\hspace{0.17em}}{\text{m}}^{3}\cdot {\text{s}}^{-1}$ (6.1)

In this section are calculated the numerical values of some relevant formulas inferred in the previous sections; after having outlined the theoretical frame, the aim to show how the values (1.5) at today’s time fit the concepts hitherto introduced. The strategy of the calculation scheme is that of implementing the values (1.5) more than once in various equations, whose global self-consistency supports the validity of the single results and outlines a unique conceptual frame. The fact of having inferred relativistic concepts via (1.2), see for example (5.93) (5.95) and (5.113) and the results in [16], shows that in fact even cosmological information should be sensibly accessible despite the agnosticism of the quantum uncertainty. The following calculations are carried out assuming the value of $\pi $ approximately equal to that of the flat Euclidean space. The paper sketches also a few results already published in order to be as self-contained as possible. For sake of clarity the calculations are listed one by one in the various points below.

1) Examining the numerical values quoted in (1.5), it appears that $\Lambda \approx 4{H}_{u}^{2}$. This numerical evidence suggests that actually the true relationship between the literature estimates of ${H}_{u}^{2}$ and $\Lambda $ is reasonably

$\Lambda =\frac{4}{3}\pi {H}_{u}^{2}\mathrm{.}$ (6.2)

This fact is interesting because an energy density
$\eta $ is calculable via (2.2) as
${\left(c/\tau \right)}^{2}{G}^{-1}$ according to (1.4); in particular the
$tim{e}^{-2}$ dependence inherent
$\Lambda $ is
$4\pi /3$ times that calculated via
${H}_{u}^{2}$ because, despite the same physical dimensions, the former contains geometrical information with respect to the latter. Since
$\eta $ is anyway *energy*/*volume*, (6.19) calculated as a function of
${H}_{u}^{2}$ and
$\Lambda $ read

$\frac{{c}^{2}{H}_{u}^{2}}{G}=\frac{{c}^{2}\Lambda}{G4\pi /3}=\frac{\u03f5}{\left(4\pi /3\right)\delta {\mathcal{l}}^{3}}\mathrm{:}$ (6.3)

hence according to the first equality the energy density $\u03f5/{V}_{\eta}$ involving ${H}_{u}^{2}$ implies the total energy $\u03f5$ calculated in the total volume ${V}_{\eta}=\left(4\pi /3\right)\delta {\mathcal{l}}^{3}$, whereas according to the second equality involving $\Lambda $ the volume pertinent $\u03f5$ is in fact expressed via $V=\delta {\mathcal{l}}^{3}$. In other words ${H}_{u}$ requires explicitly the space time curvature implied by spherical geometry of mass containing universe, $\Lambda $ does not. For this reason $\sqrt{\Lambda}$ and ${H}_{u}$ are numerically interchangeable via a coefficient ~2 apart. Note now that

$2\pi \frac{c}{{r}_{u}}={\omega}_{u}=4.33\times {10}^{-18}\text{\hspace{0.17em}}{\text{s}}^{-1},\text{\hspace{1em}}\sqrt{\Lambda}=4.36\times {10}^{-18}\text{\hspace{0.17em}}{\text{s}}^{-1},\text{\hspace{1em}}\frac{c}{{r}_{u}}={\nu}_{u}\mathrm{,}\text{\hspace{0.05em}}$ (6.4)

which shows the link between the cosmological constant and the estimated universe radius. This link is further confirmed considering that $\sqrt{{c}^{2}/\Lambda}$ and $c/{H}_{u}$ are lengths; then calculate

${\mathcal{l}}_{\Lambda}=\frac{c}{\sqrt{\Lambda}}=6.88\times {10}^{25}\text{\hspace{0.17em}}\text{m},\text{\hspace{1em}}{\mathcal{l}}_{H}=\frac{c}{{H}_{u}}=1.36\times {10}^{26}\text{\hspace{0.17em}}\text{m}$ (6.5)

and implement the quantum definition $2\pi \delta r=n\lambda $ of (2.26), whose right hand side reads $n/\mathcal{R}$ in (4.22) after having replaced $\lambda $ with the reciprocal curvature radius $\mathcal{R}={{\lambda}^{\prime}}_{r}^{-1}-{{\lambda}^{\u2033}}_{r}^{-1}$ via the De Broglie momentum wavelengths. Taking appropriately $\delta r$ as the universe radius in the estimates (1.5), the numerical result is

$\begin{array}{cccccc}2\pi \delta r& =& n{\mathcal{R}}^{-1}& & & \\ \Updownarrow & & \Updownarrow & & & \\ 2\pi \sqrt{\frac{{c}^{2}}{\Lambda}}& =& n{r}_{u}& & \delta r=\sqrt{{c}^{2}{\Lambda}^{-1}}& {r}_{u}={\mathcal{R}}^{-1}\\ \Updownarrow & & \Updownarrow & & & \\ 4.32\times {10}^{26}\text{\hspace{0.17em}}\text{m}& =& n4.35\times {10}^{26}\text{\hspace{0.17em}}\text{m}& & & \end{array}$ (6.6)

With $n=1$ the agreement of ${r}_{u}$ with the estimates (1.5) is surprisingly decent. On the one hand this result highlights the link of Einstein cosmological constant $\Lambda $ and today’s radius of Universe via the early quantum condition (2.23) in agreement with (6.4): as expected the estimated value ${r}_{u}$ is related to the reciprocal curvature radius $\mathcal{R}$ of the Universe, here defined by the range of wavelengths $\delta {\lambda}^{-1}$ corresponding to $\delta r$.

On the other hand it is possible to match the dimensional definition of space time by defining the functions ${\left(c/\sqrt{\Lambda}\right)}^{3}{H}_{u}$ and ${\left(c/{H}_{u}\right)}^{3}\sqrt{\Lambda}$ along with ${r}_{u}^{3}/{t}_{u}$ that share the dimensional property $lengt{h}^{3}\times tim{e}^{-1}$. A reasonable chance to merge these definitions in a self consistent way is in fact owing to (6.2)

$\sqrt{{H}_{u}{\left(\frac{c}{\sqrt{\Lambda}}\right)}^{3}\frac{{r}_{u}^{3}}{{t}_{u}}}=1.16\times {10}^{61}\text{\hspace{0.17em}}{\text{m}}^{3}\cdot {\text{s}}^{-1},\text{\hspace{1em}}\sqrt{\Lambda}{\left(\frac{c}{{H}_{u}}\right)}^{3}=1.10\times {10}^{61}\text{\hspace{0.17em}}{\text{m}}^{3}{}^{3}\cdot {\text{s}}^{-1}\mathrm{.}$ (6.7)

It is worth noticing that

$\frac{\hslash G}{{c}^{2}}\sqrt{\Lambda}{\left(\frac{c}{{H}_{u}}\right)}^{3}\approx 1\text{\hspace{0.17em}}{\left({\text{m}}^{3}\cdot {\text{s}}^{-1}\right)}^{2}\mathrm{.}$ (6.8)

Combining together these equations thanks to their numerical values nicely coincident, one finds via $\Lambda $ the link between the fundamental cosmological parameters

$\sqrt{{H}_{u}{\left(\frac{c}{\sqrt{\Lambda}}\right)}^{3}\frac{{r}_{u}^{3}}{{t}_{u}}}=\sqrt{\Lambda}{\left(\frac{c}{{H}_{u}}\right)}^{3},\text{\hspace{1em}}i\mathrm{.}e\mathrm{.}\text{\hspace{1em}}\frac{{r}_{u}^{3}}{{t}_{u}}=\frac{{c}^{3}{\Lambda}^{5/2}}{{H}_{u}^{7}}=1.70\times {10}^{62}\text{\hspace{0.17em}}{\text{m}}^{3}\cdot {\text{s}}^{-1}\mathrm{;}$ (6.9)

it appears in particular the link between the estimated volume of of universe, proportional to ${r}_{u}^{3}$, and its estimated age ${t}_{u}$. Owing to the physical dimensions of $\Lambda $ and ${H}_{u}$ and their time dependence, one expects

$\frac{{r}_{u}^{3}}{{t}_{u}}\propto function\text{\hspace{0.17em}}of\text{\hspace{0.17em}}\left(tim{e}^{2}\right)\mathrm{.}$ (6.10)

Whatever this time function
$X=X\left(\delta {t}^{2}\right)$ might be, the fact that the dependence of the ratio at the left hand side involves
$\delta {t}^{2}$ means that
$\delta t$ can be identically positive or negative, *i.e.* the time can run in principle away from or back towards the initial big bang. It is worth noticing a numerical accident, *i.e.* both today’s values (6.7) are reciprocal of
$7.83\times {10}^{-62}{\text{m}}^{\text{3}}/\text{s}$ calculable directly with the fundamental constants of (1.1).

2) In (6.4) $\Lambda $ was related to ${\omega}_{u}$ pertinent to ${r}_{u}$ ; now, to extend further the link between theoretical results and (1.5), implement ${H}_{u}$ too. Are relevant in this respect the Equations (5.64) to (5.65) that define mass, density and energy density as a function of the frequency $\omega $ appearing in (5.62) and (5.63). Indeed via ${k}_{f}$ it is possible to identify further mass density and mass directly related to the gravitational effect of matter in the universe, which clarify the meaning of such $\omega $ ; it has also to do with the geometrical implication of matter on the space time curvature, introduced in (6.2) and emphasized in (6.6). So ${\omega}_{u}$ is a property of the space time, as it results in (6.4) regardless of any mass, instead $\omega $ is related to the gravitational effect of matter in the universe. To clarify this point, let us introduce the following definitions

${r}_{u}=\xi \frac{2c}{\omega},\text{\hspace{1em}}\omega =\frac{{H}_{u}}{2}$ (6.11)

The first definition is nothing else but a way to express $length=c/frequency$ via the proportionality constant $2\xi $, the second expresses $\omega $ as a function of ${H}_{u}$ via the coefficient 1/2; in this way $\hslash {H}_{u}/2$ takes the meaning of zero point energy of the universe, which of course is sensible once having introduced (5.60), (5.77) and (5.79). These positions are useful to obtain via the first and second (5.64)

$\xi =\frac{{r}_{u}{H}_{u}}{4c}=0.8,\text{\hspace{1em}}{\rho}_{\omega}=\xi \frac{{\omega}^{2}}{G}=\frac{{r}_{u}{H}_{u}^{3}}{16cG}=14.4\times {10}^{-27}\frac{\text{kg}}{{\text{m}}^{\text{3}}}\mathrm{;}$ (6.12)

as reasonably expected $\xi \sim 1$, whereas instead a proportionality constant very different from 1 would have suggested that something important was neglected when formulating the pertinent dimensional definition. However the value of $\xi $ is not the only reason to justify the definitions (6.11). Indeed note that

${m}_{\omega}=\xi \frac{{c}^{3}}{\omega G}=\frac{{r}_{u}{H}_{u}}{4c}\frac{2{c}^{3}}{{H}_{u}G}=\frac{{r}_{u}{c}^{2}}{2G}=2.93\times {10}^{53}$