A Step towards Dark Energy and Dark Matter

Sebastiano Tosto

Retired Scientist Formerly at ENEA Casaccia, Roma, Italy.

DOI: 10.4236/jamp.2025.1312238                

Abstract

The paper proposes an “ab initio” theoretical contribution to the understanding of dark matter and dark energy. The model has quantum basis and correlates the Einstein cosmological constant to the vacuum energy density; the cosmological parameters are expressed as a self-consistent set of data as a function of age and size of the universe, in agreement with the values of the fundamental constants of nature. The MOND equation is inferred from first principles, the $a_0$ interpolation constant is calculated from cosmological data.

Keywords

Cosmology, Dark Matter, Dark Energy

1. Introduction

Indeterminism and non-locality, key concepts of the quantum world, imply an agnostic way to redefine the reality. The basic idea is that any experiment is in fact a form of interaction between observer and observed object, which implies that the reality we detect is the perturbed state of a previous unperturbed and unknown state prior the measurement process. A reasonable way of modeling the physical reality as we know it after the observation process should someway simulate this idea, i.e. to introduce first the indeterminacy of the physical parameters of interest and next extract information about the perturbed state through those parameters. This consideration has stimulated the idea of a possible way to formulate the physical models based on an extended concept of quantum uncertainty implementing the relationship $\delta x\delta p_x=n\hslash =\delta \epsilon \delta t$ : the number $n$ of quantum states allowed for a system of particles is formulated via arbitrary ranges of fundamental dynamical variables, as a reasonable alternative to the deterministic trust in their local values. The agnostic assumption inherent the uncertainty represents the unperturbed reality to be observed, the way to extract all possible information form the uncertainty ranges is the conceptual equivalent of the observation process.

On the one hand, this approach to understand the reality regards the local values of the physical observables as precursors of the actual values we can detect with the experiment; on the other hand, this conceptual perspective bypasses in a natural way the determinism of input local values.

Once introducing as a sole source of information energy and time ranges linked to conjugate space and momentum ranges, then the space time reality becomes mere combination of dynamical parameters that translate their symbolic meaning into actual information to be confirmed by comparison with the measurement process.

For this reason, this way to formulate any physical problem pays crucial attention to the physical definition and physical dimensions of the dynamical variables, rather than to their representative specificity in a prefixed theoretical frame. For clarity, these ideas, highlighted in the next sections, are exemplified here. Write $\delta ϵ/\delta p_x=v_x=\delta x/\delta t$ ; it defines $v_x$ without specifying what is moving and why. But if one specifies in particular $\delta \epsilon =\delta \left(\hslash \omega \right)$ and $\delta p_x=\delta \left(h/\lambda \right)$ , one trivial step only is enough to define $v_x\equiv v_{gx}=\delta \omega /\delta \kappa$ with $\kappa =2\pi /\lambda$ ; in this particular case $v_x$ is the $x$ -component of the group velocity of a wave packet. Also, $\delta p=-h\delta \lambda /{\lambda }^2\equiv -\left(h/c^2\right)\left(c^2/{\lambda }^2\right)=-\left(h\nu /c^2\right)\left(\nu \delta \lambda \right)=-\left(ϵ\delta v/c^2\right)$ ; it yields the relativistic momentum $p=ϵv/c^2$ , with $\delta p$ expressed as a function of $\delta v=\nu \delta \lambda$ and $\delta \lambda$ . The minus sign is obvious because $p$ increases with $v$ but decreases with $\lambda$ . This result merges quantum and relativistic definitions of momentum. With this kind of approach, attention is due to the concepts, not to the formulas: here we see that velocity is not mere ratio of space and time intervals, it inherently implies also energy and momentum changes. The next two sections exemplify how the dynamical variables take themselves an autonomous physical meaning regardless of the specific physical problem; it opens the way to a variety of interesting outcomes. The previous example shows how to proceed: the task was not to calculate purposely the group velocity, but to extract information on $v$ from its own physical meaning via the uncertainty.

These general considerations are the reading key of the theoretical model proposed in this paper.

2. Preliminary Considerations: The Π Probability Function

In this introductory section consider again the concept of velocity $v$ in a more systematic way introducing the function $\Pi \left(v\right)=v^2/c^2$ and regarding $v$ as a dynamical variable characterized by its own physical properties in an arbitrary reference system, where are also defined its change $\delta v$ and acceleration $\delta v/\delta t$ . The quantum uncertainty is not implemented, to show that the ideas exposed in the section 1 have also classical validity. Give $\Pi$ the meaning of probability: thus let be by definition $v\le c$ in order that $0\le \Pi \le 1$ , i.e.

$\frac{\delta \Pi }{\Pi }=\frac{2v\delta v}{v^2}=2\delta \log \left(v\right)=2\log \frac{v}{c}\Rightarrow \frac{\delta \Pi }{\Pi }=\log \left(\frac{v^2}{c^2}\right)=\log \Pi .$ (2.1)

Let us show that $\log \Pi <0$ is consistent with the statistical definition of entropy. Think a set of $N$ particles each one with its own set of allowed $v_j$ ; in this example $v$ takes specific physical meaning if regarded as $j$ -th dynamical state $v_{ji}$ of the $i$ -th particle among the many in a complex system of particles. As no hypothesis is necessary about $v_{ji}$ , the step from (2.1) to (2.2) holds regardless of whether the $j$ -th state results in a set of interacting or non-interacting particles: it is enough to refer each $v_{ji}$ to the probability of an allowed state of a system to infer statistical information about the whole system. Summing first over $j$ one finds

$\delta {\Pi }_{ji}={\Pi }_{ji}\log {\Pi }_{ji}\Rightarrow {\displaystyle \sum }_{j=1}^{j_i}\delta {\Pi }_{ji}=\delta {\Pi }_i={\displaystyle \sum }_{j=1}^{j_i}\left({\Pi }_{ji}\log {\Pi }_{ji}\right)=-S_i,$ (2.2)

being $-S_i>0$ because $v<c$ implies ${\Pi }_{ji}<1$ ; moreover define also

$\overline{\delta {\Pi }_i}=X_i^{-1}{\displaystyle {\int }_0^{X_i}{\Pi }_{ji}\log {\Pi }_{ji}\text{d}{\Pi }_{ji}},$ (2.3)

where $X_i$ is the parameter to calculate average value of $\delta {\Pi }_{ji}$ of the $i$ -th particle. Since ${\displaystyle \int \left(x\log x\right)\text{d}x}=x^2\log \left(x\right)/2-x^2/4+const$ , where the right hand side does not diverge even for $x\to 0$ , summing as before over all $j$ -th allowed states, the result of (2.3) yields

$\overline{\delta {\Pi }_i}=\frac{1}{2}{X}_i\log X_i-\frac{1}{4}{X}_i+\frac{const}{X_i}\Rightarrow \overline{\delta {\Pi }_i}\ge \frac{1}{2}{X}_i\log X_i-\frac{1}{4}{X}_i+const{X}_i\le 1;$

owing to (2.2), $S$ and $\overline{\delta \Pi }$ obtained summing $-S_i$ and $\overline{\delta {\Pi }_i}$ over the number of particles, reads

$S\ge \pm Y\pm ZS={\displaystyle \sum }_{i=1}^N-S_{i}Y=\frac{1}{2}{\displaystyle \sum }_{i=1}^N{X}_i\log X_i+C_{y}Z=C_z-\frac{1}{4}{\displaystyle \sum }_{i=1}^N{X}_i,$ (2.4)

where $C_y$ and $C_z$ are arbitrary constants resulting from $C_y+C_z=N\times const$ ; in principle $C_y$ and $C_z$ justify the double signs of $Y$ and $Z$ . However, being positive the left hand side of (2.4), at least one addend at the right hand side must be positive. As the double signs of $Y$ and $Z$ are independent, this condition is easily identified: $Z$ is positive simply defining $C_z>{\displaystyle \sum X_i/2}$ , in which case $S=\pm Y+Z$ . As concerns the sign of $Y$ , in principle nothing excludes that $Y⋛Z$ with $-Y$ ; the realistic and unforeseeable chance of negative sign of $Y$ due to $C_y$ requires considering in particular $S\ge -Y+Z$ . Thus a general notation to express (2.4) including both signs $\pm Y$ allowed for a given value of $\left|Y\right|$ is

$S\ge Y+Z\Rightarrow E_{0}S\ge E_{0}Y+E_{0}Z,$

with both sides multiplied by a positive constant energy $E_0$ to convert the relationship between dimensionless quantities into a relationship between different forms of energy. This result appears more familiar rewriting $E_0=k_{B}T$ and calculating $\delta \left(\right)$ of the functions to eliminate the constants at the right hand side; i.e. one finds

$k_{B}T\delta S\ge \delta U+\delta WU=k_{B}T{\displaystyle \sum }_{i=1}^N{X}_{i}W=k_{B}T{\displaystyle \sum }_{i=1}^N{X}_i\log X_i\delta S=S-cons{t}_s$ (2.5)

which does not need further comments; the constants remind that in fact any energy is defined an arbitrary constant apart, which can be put equal to zero. The importance of the first law, here obtained as a corollary of $\Pi$ , has full statistical meaning: it does not depend upon the single $X_i$ but upon sums over a total number of initial $\overline{\delta {\Pi }_{ji}}$-th terms.

Return now to (2.2) where $v$ has in general mere meaning of dynamical variable, regardless of any specific reference to a particular physical system. Consider the following steps

$\delta \Pi =\frac{2v\delta v}{c^2}=\frac{2v\delta v\delta t}{c^2\delta t}=\frac{2a\delta r}{c^2}a=\frac{\delta v}{\delta t}\delta r=v\delta t,$ (2.6)

An approximate solution of (2.6) is found by replacing the finite differentials with

$\frac{\partial \Pi }{\partial r}\approx \frac{2a}{c^2}$ (2.7)

and then integrating with respect to $r$ . Let hold for $a$ the same considerations introduced for $v$ , i.e. regard $a$ as a mere dynamical parameter defined by the change rate $dv$ during the arbitrary time lapse $dt$ as a function of a space parameter $r$ . To prove that (2.7) is sensible, define

$a=a\left(r\right)=\frac{v_0^2}{r}{v}_0=const.$ (2.8)

Replacing in (2.7) one finds $d\left(v^2/2v_0^2\right)=dr/r$ , whose solution is $v^2/2v_0^2+C=\text{log}\left(r/r_0\right)$ and then

$\frac{r}{r_0}=\text{exp}\left(\frac{v^2/2+C{v}_0^2}{v_0^2}\right)\Rightarrow r^{\prime }=\frac{{r^{\prime }}_0}{\text{exp}\left(\frac{v^2/2+C{v}_0^2}{v_0^2}\right)\mp 1}\frac{r}{r_0}=\frac{{r^{\prime }}_0\pm r^{\prime }}{r^{\prime }}{r^{\prime }}_{-}\le {r^{\prime }}_{0-};$ (2.9)

the last inequality is required for the minus sign in the expression of $r/r_0$ , which corresponds to the positive sign in $r^{\prime }$ . Divide both sides of (2.9) by an arbitrary length ${\ell }_0$ and acknowledge in the exponential the ratio of different energies; this result with negative value of the integration constant $C$ reads

$N^{\prime }=\frac{{N^{\prime }}_0}{\text{exp}\left(\frac{ϵ-{ϵ}_F}{{ϵ}_0}\right)\mp 1}{N^{\prime }}_0=\frac{{r^{\prime }}_0}{{\ell }_0}{N}^{\prime }=\frac{r^{\prime }}{{\ell }_0}ϵ=\frac{m{v}^2}{2}{ϵ}_F=m{v}_0^{2}C{ϵ}_0=m{v}_0^2,$ (2.10)

whereas (2.9) becomes

$\frac{r^{\prime }}{{r^{\prime }}_0}\le 1N^{\prime }\le {N^{\prime }}_0.$ (2.11)

Let ${N^{\prime }}_0$ be integer number, which is possible with an appropriate ${\ell }_0$ : taking in particular the minimum value ${N^{\prime }}_0=1$ , the plus sign in (2.9) implies $N^{\prime }=0,1$ for any state defined by its own quantum numbers; this restrictive condition is not required for the minus sign of $N^{\prime }$ in (2.9). The physical meaning of (2.10) is well known, for brevity further considerations are omitted.

Rather it is significant to generalize this result obtained form the definition of $\Pi$ in one particular case only, where $a$ was specifically identified by (2.8). To this aim, calculate (2.7) writing in general

$a\left(r\right)={\displaystyle \sum }_j\frac{{\xi }_j}{r^j}r>0;$ (2.12)

the inequality excludes the divergence for $r=0$ . This chance could be a series expansion of $a$ or, more rationally, a linear combination of possible definitions of $a_j\left(r\right)$ via appropriate coefficients ${\xi }_j$ sharing the physical dimensions of (2.6). In quantum physics a linear combination of allowed states of a system via complex coefficients defines a possible state of a macroscopic system. This idea is here extended thinking that various parts of a macroscopic system are describable by several classical states $a_1={\xi }_1/r$ or $a_2={\xi }_2/r^2$ and so on; the real coefficients mean that, for example, various parts of a whole system are described by different dynamical states. Although (2.12) is in fact a particular way to concern $a$ as a function of various $r^{-j}$ , short considerations exemplify the outcomes of these concepts. Write thus (2.16) reads

$\Pi =\frac{2}{c^2}{\displaystyle \int a\text{d}r}=\frac{2}{c^2}{\displaystyle \sum _j{\xi }_j{\displaystyle \int a_j\text{d}r}}$ (2.13)

the sole requirement being that anyway $a_j=length/tim{e}^2$ ; e.g. $a_1=v_0^2/r$ or $a_2=mG/r^2$ or $a_3=r_{0}mG/r^3$ and so on. Since $\Pi$ is dimensionless, define thus dimensionless coefficients ${\xi }_j$ as follows

$\Pi ={\displaystyle \sum _j{\Pi }_j}=C+\frac{2}{c^2}{\xi }_1{v}_0^2{\displaystyle \int \frac{\text{d}r}{r}}+\frac{2}{c^2}{\xi }_{2}mG{\displaystyle \int \frac{\text{d}r}{r^2}}+\frac{2}{c^2}{\xi }_{3}mG{r^{\prime }}_0{\displaystyle \int \frac{\text{d}r}{r^3}}+\frac{2}{c^2}{\xi }_4{\displaystyle \int \frac{\text{d}r}{r^4}}+\cdots ,$ (2.14)

being $C$ the integration constant. Seemingly this reasoning is a conceptual loop: starting from $\Pi$ one defines its differential $\delta \Pi$ and next integrates a linear combination of $\delta {\Pi }_j$ to concern the respective $a_j$ via the approximation (2.7) of (2.6), which in turn bring back again to ${\Pi }_j$ . But precisely the principle of combination of several classical states yields physical information about the various $a_j$ of the possible $j_{th}$ states, some of which are exemplified at the right hand side of (2.14) and read

$\Pi =C+{\xi }_1\frac{v_0^2}{c^2}\log \left(\frac{r^2}{r_0^2}\right)-{\xi }_2\frac{r_{bh}}{r}-{\xi }_3\frac{4{r^{\prime }}_0}{m^{\prime }{c}^2}{F}_N+\frac{2}{3c^2}\frac{{\xi }_4}{r^3}+\cdots r_{bh}=\frac{2mG}{c^2}{F}_N=G\frac{m^{\prime }m}{r^2}.$ (2.15)

The probabilistic meaning of $\Pi$ corresponds to the various ${\Pi }_j$ -th states definable with values between 0 and 1 by normalizing appropriately the coefficients ${\xi }_j$ .

The various addends ${\Pi }_j$ of (2.15) concurring to $\Pi =v^2/c^2$ are now considered separately in order to examine their content of physical information.

-The first term of (2.15) reads

${\Pi }_1=\frac{v_1^2}{c^2}=C+{\xi }_{1}const\log \left(\frac{r^2}{r_0^2}\right)const=\frac{v_0^2}{c^2}\Rightarrow \frac{r^2}{r_0^2}=\text{exp}\left(-\frac{C}{{\xi }_{1}const}\right)\text{exp}\left(\frac{v_1^2/c^2}{{\xi }_{1}const}\right).$

In this example $v_1^2/c^2=a_1{r}_1/c^2=r_1/r^{\prime }$ , having put $a_1=v_1^2/r_1$ and $r^{\prime }=c^2/a_1$ by dimensional reasons; then, depending on the actual sign of ${\xi }_1$ in the first (2.15), write

$\frac{r^2}{r_0^2}=U_0\exp \left(\pm \frac{r_1/{\xi }_0}{r^{\prime }}\right)U_0=\exp \left(-\frac{C}{{\xi }_0}\right){\xi }_0=\left|{\xi }_1\right|const;$

taking the minus sign of ${\xi }_1$ to avoid divergent exponential, this result reads

$\frac{r}{r_0}=\pm {U^{\prime }}_0\exp \left(-\frac{r_1/{\xi }_0}{2r^{\prime }}\right){U^{\prime }}_0=\exp \left(\frac{C}{2{\xi }_0}\right).$ (2.16)

In this equation there are three independent variables $r,r^{\prime },r_1$ , because the initial $v_1$ was arbitrary; it implies that the chance of two links between them is still compatible with one freedom degree, e.g. for $r$ at the left hand side, to fulfill the exponential of $r_1$ and $r^{\prime }$ at the right hand side. Let us concern the way to express these links in the case where the exponential takes form $\text{exp}\left(-r_1/constant\right)$ simply taking $r^{\prime }$ constant, which means $2r^{\prime }{\xi }_0={r^{\prime }}_0$ and thus $a_1=c^2/r^{\prime }$ constant too; this particular condition aims to obtain quickly recognizable results. Indeed dividing both sides of (2.16) by $r_1$ one finds

$\frac{r}{r_0}=\pm {U^{\prime }}_0\exp \left(-\frac{r_1}{{r^{\prime }}_0}\right)\Rightarrow \frac{r/r_0}{r_1}=\pm \frac{{U^{\prime }}_0}{r_1}\exp \left(-\frac{r_1}{{r^{\prime }}_0}\right),$ (2.17)

which reads

$U_{{}_Y}=\pm \frac{{U^{\prime }}_0}{r_1}\text{exp}\left(-\frac{r_1}{{r^{\prime }}_0}\right)$ (2.18)

Moreover, multiplying both sides of (2.17) by $r_1^2/{r^{\prime }}_0$ , one finds

$\frac{r_1}{{r^{\prime }}_0}\frac{r}{r_0}=\frac{{r^{″}}^2}{{r^{\prime }}_0{r}_0}=\pm {U^{\prime }}_0\frac{r_1}{{r^{\prime }}_0}\text{exp}\left(-\frac{r_1}{{r^{\prime }}_0}\right){r^{″}}^2=r_{1}r;$

owing to the fact that $r_1$ and $r$ are independent variables, ${r^{″}}^2$ is in principle a new variable. The right hand side has a maximum at $r_{1_{\max }}={r^{\prime }}_0$ , thus

${\frac{{r^{″}}^2}{r_0{r^{\prime }}_0}|}_{\max }={U^{\prime }}_0\exp \left(-1\right)\Rightarrow \frac{1}{{r^{″}}^2}=\frac{Constant}{r_0{r^{\prime }}_0};$ (2.19)

i.e. the right hand side explains the meaning of the left hand side.

-With the minus sign, (2.18) has the form of the Yukawa potential; the independent $r$ and $r_1$ define in fact via the proportionality factor $r_0$ a new variable taking the values calculable at the right hand side as a function of $r_1$ .

-As concerns (2.19), the notation emphasizes the physical meaning of $r^{″}$ . Let ${r^{″}}^{-2}$ be the Gauss curvature ${ℛ}_G$ , as it is in principle reasonable, and let ${\left(r_0{r^{\prime }}_0\right)}^{-1}$ consist of two constant values of this curvature. These latter can be regarded as characteristic values of any other ${ℛ}_G$ calculable via the arbitrary ${\left({r^{\prime }}_0\right)}^{-1}$ and ${\left(r_0\right)}^{-1}$ ; this idea suggests that the specific ${r^{\prime }}_0^{-1}$ and $r_0^{-1}$ are the characteristic minimum and maximum curvatures related to the contextual ${r^{″}}^{-2}$ . Clearly all of this agrees not only with the interpretation of the constant ${\left({r^{\prime }}_0{r}_0\right)|}_{\max }$ but also with the Gauss “theorema egregium”, which appears itself as a natural corollary. Therefore the Yukawa potential is relevant and effective not only because of its short range character but also because it is linked to the space time curvature.

-The second addend ${\Pi }_2$ of (2.15) introduces the relevant result $r_{bh}$ further concerned later; it is remarkable that precisely $r_{bh}$ is obtained through the condition (2.12) of non-divergence. Take in this context the negative sign of the coefficient ${\xi }_2$ and consider the cases of integration constant equal to or different from 0. On the one hand this term has anyway the meaning of $\left(squarevelocity\right)/c^2$ , which implies, taking ${\xi }_2<0$ and the integration constant equal to 0, the kink

$\frac{v^2}{c^2}\iff {\xi }_2\frac{r_{bh}}{r}{r}_{bh}=\frac{2mG}{c^2}:$ (2.20)

the definition of ${\Pi }_2$ makes $v^2/c^2$ expressible via the ratio $r_{bh}/r$ .

On the other hand it is possible to write this term again with ${\xi }_2<0$ also as

${\Pi }_2={\xi }_2{r}_{bh}\left(\frac{1}{r}+\frac{1}{r_0}\right)$ (2.21)

being now $1/r_0$ integration constant of (2.15). This result involves the classical Young-Laplace curvature radius $ℛ$ , which suggests that ${{\Pi }^{\prime }}_2$ should have itself a curvature meaning. Write then

${{\Pi }^{\prime }}_2={\xi }_2{r}_{bh}\left(\frac{1}{r}+\frac{1}{r_0}\right)={\xi }_2{r}_{bh}ℛ$ (2.22)

and note that

$\underset{r\to r_0}{\text{lim}}{{\Pi }^{\prime }}_2=\frac{2r_{bh}}{r_0}=\frac{4mG/c^2}{r_0}.$ (2.23)

To be short: it is well known that $4mG/c^2{r}_0$ is the deflection angle $\delta \varphi \equiv {{\Pi }^{\prime }}_2$ of a light beam at a given distance $r_0$ from the source $m$ of a gravity field.

-The third term ${\Pi }_3$ of (2.15) is regarded along with the higher order terms to obtain

$F_N^{\ast gen}=G\frac{m^{\prime }m}{r^2}+\frac{2}{3c^2}\frac{{\xi }_4}{r^3}+\cdots {\xi }_3=\frac{m^{\prime }{c}^2}{4{r^{\prime }}_0}:$ (2.24)

the notation emphasizes that this is a generalized form of the classical Newton law $F_N^{\text{*}}$ , which in principle can fit experimental values with any numerical accuracy. It is significant that the coefficient ${\xi }_3$ introduces the energy $m^{\prime }{c}^2$ . Also, putting ${\xi }_4={{\xi }^{\prime }}_4{r^{\prime }}_0$ in (2.24), one finds

$\frac{1}{{r^{\prime }}_0}=\frac{{{\xi }^{\prime }}_4}{m^{\prime }{c}^2}F=G\frac{m^{\prime }m}{r^2}+\frac{2}{3c^2}\frac{{{\xi }^{\prime }}_4}{r^3}+\cdots ;$ (2.25)

It is interesting to acknowledge that, without introducing additional hypotheses, $F$ is related via the constant length $r_{0\text{'}}$ to the rest energy $m^{\prime }{c}^2$ of $m^{\prime }$ that generates the gravity field. Eventually it is worth noticing in particular that $\delta r=v\delta t$ of (2.6) suggests by analogy $r=v^{\prime }t+const$ and thus yields

$F_N=G\frac{m^{\prime }m}{{\left(v^{\prime }t+const\right)}^2}.$ (2.26)

This form of Newton’s law is more sensible than $F_N^{\text{*}}$ of (2.15): the latter corresponds to an instantaneous action at a distance $r$ , which worried Newton himself, the former implies the propagation rate $v^{\prime }$ of the gravitational propagation corresponding to the distance between interacting masses. Despite the Newton formula is anyway approximate once neglecting the higher order terms introduced as corollaries from the initial (2.12), (2.26) is physically rational. Nevertheless, as $F_N$ results from $a_3$ of (2.14) only, trivial considerations on a further term ${\xi }_4{a}_4\left(r\right)$ of (2.14) would give

$\Pi =\cdots +F_N+\frac{{\xi }_4}{{\left(v^{\prime }t+cons{t}^{\prime }\right)}^3}+\cdots$ (2.27)

and so on for further terms of the linear combination.

The fact that the early Newton-like (2.24) includes actually further higher order terms, means that the third and fourth allowed states of (2.12) concur to the gravitational interaction state, which in turn contributes to the total probability function $\Pi$ . This obvious consideration implies defining the corresponding potential energy directly related to (2.24)

$U=U_0-\frac{{\xi }_3}{r}+\frac{{\xi }_4}{r^2}+\cdots$ (2.28)

In conclusion, even in lack of input information about what is moving and its specific dynamics, these outcomes consist of well known and recognizable results; in particular it is singular that the probabilistic approach yields $r_{bh}$ in the frame of a non-diverging requirement. These short notes have shown that significant results are achieved simply introducing the function $\Pi$ and implementing $m$ and $v$ as dynamical parameters regardless of their specific reference to a predefined physical problem. The extrapolation of these preliminary results to the cosmological frame is evident, e.g. to the MOND problem concerned later. Note in this respect that an immediate extension of (2.12) is obtained rewriting the coefficients ${\xi }_j$ without loss of generality as ${\xi }_2={\xi }_1^2{{\xi }^{\prime }}_2$ and so on. It follows

$a=\frac{{\xi }_1}{r}+\frac{{\xi }_1^2{{\xi }^{\prime }}_2}{r^2}+\frac{{\xi }_1^3{{\xi }^{\prime }}_3}{r^3}+\cdots \Rightarrow a=a_1+{{\xi }^{\prime }}_2{a}_1^2+{{\xi }^{\prime }}_3{a}_1^3+\cdots :$ (2.29)

the first replica of (2.12) merely renames the coefficients, the second one truncated at the second order emphasizes the standard form of the MOND model as a corollary of this section.

To this purpose consider the last (2.29); neglecting for simplicity the higher order terms, rewrite (2.12) more expressively as

$a=a_1+\frac{{{\xi }^{″}}_2}{a_0}{a}_1^2+\cdots {{\xi }^{\prime }}_2\equiv \frac{{{\xi }^{″}}_2}{a_0},$ (2.30)

being $a_0$ a constant acceleration.

On the one hand, regarding reasonably $a_1$ as the standard Newtonian acceleration of a rotating gravitational system, $a_N=v_1^2/r$ once more, this result reads

$a=\frac{v_1^2}{r}+{{\xi }^{″}}_2\frac{{\left(v_1^2/r\right)}^2}{a_0}+\cdots :$ (2.31)

clearly at large $r$ the first term overcomes the second one, whereas the opposite holds at small $r$ . Note that owing to the first (2.29) even $v_1=const$ is acceptable. On the other hand, regarding consequently $a$ as due to a circular motion of any mass in the field of a source mass $M$ , (2.31) implies

$a=\frac{MG}{r^2}=\frac{v_1^2}{r}+{{\xi }^{″}}_2\frac{{\left(v_1^2/r\right)}^2}{a_0}+\cdots$ (2.32)

First of all rewrite identically (2.32) splitting the second addend as follows

$\frac{MG}{r^2}=\frac{v_1^2}{r}+{\zeta }_1\frac{{\left(v_1^2/r\right)}^2}{a_0}+{\zeta }_2\frac{{\left(v_1^2/r\right)}^2}{a_0}+\cdots {\zeta }_1+{\zeta }_2={{\xi }^{″}}_2,$ (2.33)

which in turn implies two equations

$\frac{MG}{r^2}-{\zeta }_1\frac{{\left(v_1^2/r\right)}^2}{a_0}=\frac{v_1^2}{r}+{\zeta }_2\frac{{\left(v_1^2/r\right)}^2}{a_0}+\cdots {\zeta }_1+{\zeta }_2={{\xi }^{″}}_2.$ (2.34)

This result reads $\left(MG-{\zeta }_1{v}_1^4/a_0\right)/r^2=v_1^2/r+{\zeta }_2{v}_1^4/\left(a_0{r}^2\right)+\cdots$ . The left hand side yields $v_1$ , whereas the right hand side is calculated implementing $v_1$ just found. The necessity of splitting the coefficient $\xi \text{'}{\text{'}}_2$ into ${\zeta }_1$ and ${\zeta }_2$ of (2.33) is evident: writing directly (2.32) in the form $MG-{{\xi }^{″}}_2{v}_1^4/a_0=0$ would have implied $v_1^2/r=0$ , which holds at $r=\infty$ only. So, neglecting the higher order terms, trivial calculations yield

$v_1={\left(\frac{MG{a}_0}{{\zeta }_1}\right)}^{1/4}a=\frac{\sqrt{a_{0}MG/{\zeta }_1}}{r}+\frac{2MG}{r^2}+\frac{{\zeta }_{2}MG}{{\zeta }_1{r}^2},$ (2.35)

i.e. $a$ has the form $cons{t}_1/r+cons{t}_2/r^2+\cdots$ expected in agreement with the first (2.29) written with these coefficients: very large $r$ imply constant $v_1$ and acceleration decreasing like $r^{-1}$ , the last two terms have instead mere Newtonian character prevalent at short $r$ .

Equations (2.30) to (2.35) anticipate the outcomes on the MOND theory of the next section 5.2. It is worth noticing that these results are not mere corollaries able to explain the specific problem of rotational behavior of galaxies; the concept of galaxy is totally out of the conceptual frame hitherto introduced, although it is evident that the space scale of a galaxy is appropriate to make knowledgeable the astrophysical relevance of (MD7). All terms of (2.12) have their own physical meaning: e.g. (2.28) extends the physical meaning of potential energy beyond the Newtonian $r^{-1}$ form, (2.29) is one among a variety of results already exposed above and further extended later.

All of this is interesting for two reasons:

-$U$ of (2.28) includes the expected first order Newtonian potential $U_N=r^{-1}$ plus a second order term $r^{-2}$ and higher order terms. At the first order, (2.28) and (2.31) yield $m^{\prime }{v}^2=-m^{\prime }mG/r+const$ , which fulfills $2\langle \text{T}\rangle =\langle U\rangle$ ; also, whatever $m$ might be, $v=r/t$ in (2.28) yields $r^3/t^2\approx const$ . Eventually $v^2/r=mG/r^2$ , centripetal acceleration of mass $m$ in the force field $F_N$ of $m^{\prime }$ , yields $\nabla \cdot field=-mG\nabla \cdot r/r^3=-3Gm/r^3$ i.e. $\varphi =F/m^{\prime }$ and $\nabla \cdot \varphi =-4\pi G\rho$ with $\rho =m/V$ and $V=\left(4/3\right)\pi r^3$ . These cornerstones of Newtonian physics, although approximated, deserve however being sketched in view of the MOND approach concerned later.

-The most interesting term of (2.28) is surely $r^{-2}$ . It is known [1] that the Newtonian potential cannot calculate properly the perihelion precession of planets, the correct calculation requires a second order potential term. However the Newton potential cannot account itself for this additional $r^{-2}$ dependence, which instead is justified in the frame of the general relativity. Nevertheless this second order term also appears in (2.28) in a natural way owing to the initial (2.12) and (2.24): this approach does not postulate “a priori” the Newton law, which instead is inferred itself as a first order approximation as shown in (2.15). In fact, the higher order potential terms that in turn imply the correct perihelion precession, which therefore can be regarded as a further corollary of (2.13). For sake of brevity further details on this point, well explained in the quoted textbook, are omitted.

It is instead worth emphasizing that all previous considerations do not necessarily hold for $a=const/r^j$ only or combination of various $a_j$ ; rather, it is significant to generalize the basic idea of regarding the dynamical variables via their own physical meaning proposing a further definition of acceleration. After having generalized (2.8) via (2.12), examine now the additional chance to this purpose implementing $a=r^2{\omega }^3/c$ to explain the Einstein formula on the gravitational orbital energy loss contextual to the emission of gravitational waves. For brevity, start from the well known formula

$-\frac{\delta E}{\delta t}=\frac{32G}{5c^5}{\left(\frac{m_1{m}_2}{m_1+m_2}\right)}^2{r}^4{\omega }^6;$ (2.36)

a simple elaboration is useful to better highlight the physical meaning of the equation: for reasons clear shortly, rewrite the integration factor as $32/5\approx 2\pi$ , the deviation being a few % only. Also, rewrite the term in parenthesis as ${\mu }^{-1}=1/m_1+1/m_2$ and note that

$a=\frac{r^2{\omega }^3}{c}$ (2.37)

has physical dimension of acceleration. Recalling eventually that $\delta t=\hslash /\delta ϵ$ , trivial manipulations of this formula yield $-\delta E\delta ϵ={\left(\mu a{\ell }_{Pl}\right)}^2$ , being ${\ell }_{Pl}^2=hG/c^3$ , consistent with both signs of $a$ . Therefore this result must be intended as

$-\delta E=\mu \left|a\right|{\ell }_{Pl}\delta ϵ=\mu \left|a\right|{\ell }_{Pl}\text{or}\delta E=\mu \left|a\right|{\ell }_{Pl}-\delta ϵ=\mu \left|a\right|{\ell }_{Pl}$ (2.38)

in agreement with the sign of (2.36). Thus two energy changes, $\delta E$ and $\delta ϵ$ with alternate meaning of source and dissipation terms, are related to the effects energy amounts $\mu \left|a\right|{\ell }_{Pl}$ that propagate as waves; this pint is better clarified in the next section 4 with the help of the quantum uncertainty. Instead appears contextually evident the meaning of $a$ : it is clear that implementing the particular $a$ of (2.37) into the definition of energy resulting from the classical definition reducedmass × acceleration × Plancklength, the reverse path from (2.38) to (2.36) brings back via (2.37) to the Einstein formula.

Eventually note how physical information is gained merging the general definitions of velocity $v=\delta s/\delta t$ and acceleration $\dot{v}=\delta v/\delta t$ even without hypothesizing any analytical form of acceleration; the steps consider two components $v_x$ and $v_y$ of $v$ as follows

$\frac{\delta v_x}{\delta t}=\frac{\delta }{\delta t}\frac{\delta x}{\delta t}=\frac{{\delta }^{2}x}{\delta t^2}{v}_y=\frac{\delta y}{\delta t}\delta t^2=\frac{\delta y^2}{v_y^2}\frac{\delta v_x}{\delta t}=\frac{{\delta }^{2}x}{\delta y^2/v_y^2}\Rightarrow 0=\frac{{\delta }^{2}x}{\delta t^2}-v_y^2\frac{{\delta }^{2}x}{\delta y^2}:$ (2.39)

the first step defines $\delta v_x$ , the second and third $\delta v_y$ , next replacing $\delta t^2$ and subtracting side by side one finds the d’Alembert wave equation.

This section has outlined some preliminary considerations introducing the basic essence of the physical model proposed in the following of this paper. This exposition however cannot be considered closed without two further remarks about how to regard the mass in this conceptual frame rooted on $\Pi$ and how to show that even $1-\Pi$ completes itself the probabilistic physical meaning of the theoretical approach hitherto configured.

Consider the constant $e^2/G$ noting that its physical dimensions are $mas{s}^2$ . Define then this mass as ${ℳ}_0^2=e^2/G$ and divide both sides by ${ℳ}_0{c}^2$ ; the result is

$\frac{{ℳ}_{0}G}{c^2}={\ell }_0=\frac{e^2}{{ℳ}_0{c}^2}$ (2.40)

The left hand side has physical dimensions of $length$ , the result at the right hand side has a well known form; both ${ℳ}_0$ and ${\ell }_0$ are fixed values. Multiply now both sides by a proportionality coefficient $\xi$ and rewrite (2.40) as $r_e=e^2/m_e{c}^2$ with $m_e={ℳ}_0/\xi$ : identifying in particular ${ℳ}_0/\xi =m_e$ with the electron mass, the right hand side corresponds to the classical electron radius. On the one hand is reasonable the attempt to extend this result to the case of the proton. On the other hand, however, there is no reason to assume that (2.40) still holds for the proton too: the electron is an elementary particle, the proton does not. Hence rises the question: is this approach extensible to the proton and possibly even to neutral particles too, e.g. the neutron?

To generalize (2.40) define ${ℳ}_{kin}={ℳ}_0/\beta$ , being $\beta$ an appropriate function that appears here as a proportionality factor replacing the aforesaid constant $\xi$ . Repeating the steps with ${ℳ}_{kin}$ , the result is now

$\frac{{ℳ}_{kin}G}{c^2}=r=\frac{e^2}{{ℳ}_{kin}{c}^2}{ℳ}_{kin}=\frac{{ℳ}_0}{\beta }{\beta }^2=1-\Pi .$ (2.41)

Thus, whatever the initial ${ℳ}_0$ and $\xi$ might be, write for electron and proton

$r_{el}=\frac{e^2}{{ℳ}_{el}{c}^2}{ℳ}_{el}=\frac{{ℳ}_{0el}}{{\beta }_{el}}{\beta }_{el}=\beta \left(v_{el}\right)r_{pr}=\frac{e^2}{{ℳ}_{pr}{c}^2}{ℳ}_{pr}=\frac{{ℳ}_{0pr}}{{\beta }_{pr}}{\beta }_{pr}=\beta \left(v_{pr}\right),$ (2.42)

reminding once more that according to the previous considerations $v_{el}$ and $v_{pr}$ must be regarded as kinetic parameters having contextualized meaning, and not as properties of something really displacing for any reason to be specified. It follows thus

$\frac{r_{el}}{r_{pr}}=\frac{{ℳ}_{kin,el}}{{ℳ}_{kin,pr}}=\frac{{ℳ}_{0el}}{{ℳ}_{0pr}}\frac{{\beta }_{pr}}{{\beta }_{el}}.$ (2.43)

This result suggests that the ratio $r_{el}/r_{pr}$ does not depend upon the dimensional mass ratio only, it is affected by the factor ${\beta }_{pr}/{\beta }_{el}$ . Moreover, regarding now $e^2/c^2$ as mere numerical factor, appears clear the chance of calculating (2.43) for the neutron too via $M_{0ne}$ and ${\beta }_{ne}$ exactly as done here

$\frac{r_{ne}}{r_{pr}}=\frac{{ℳ}_{kin,ne}}{{ℳ}_{kin,pr}}=\frac{{ℳ}_{0ne}}{{ℳ}_{0pr}}\frac{{\beta }_{pr}}{{\beta }_{ne}}.$ (2.44)

It is known that $r_{el}/r_{pr}\approx 3.2$ whereas $r_{ne}/r_{pr}\approx 1$ , in agreement with the same orders of magnitude reasonably expected for the right hand sides of (2.43) and (2.44) and with the number of elementary constituents of proton and neutron; this guess, and not the real masses themselves, might be a possible key to explain these ratios. This topic is outside the purposes of this paper; although further considerations about $m_{el}/m_{pr}$ and ${\beta }_{pr}/{\beta }_{el}$ are omitted for brevity, it appears clear in principle the conceptual worth of this approach to describe the particle classical radii. Instead note that

(2.45)

which eventually implies and thus

(2.46)

so that

$av=\frac{\partial \left(v^2/2\right)}{\partial t}=-\frac{\partial \phi }{\partial t}\Rightarrow a_r=-\frac{\partial \phi }{\partial r}\text{d}r=v\text{d}t.$ (2.47)

It is worth stressing that $ℳ$ is mere dimensional mass, not a real measurable mass; $P$ and $E$ are momentum and energy corresponding to the virtual $ℳ$ , analogous to the respective functions of $m$ . Any classical mass can be measured assuming its position somewhere, this is however nonsensical for $ℳ$ . Yet these elementary considerations have shown not only that the virtual mass $ℳ$ can be identified with and behaves like the ordinary mass, but also that ${ℳ}_{kin}$ allows an immediate explanation about why in general the classical radius of composite particles implies a more sophisticated description than that of the elementary particles.

Actually $\Pi$ and $\beta$ are the conceptual bases of a self contained physical model, rather than mere intuitions proposed because of their encouraging results. These ideas, in particular (2.47), are further concerned in the next sections aimed to discuss dark matter, dark energy and MOND dynamics. Here however it is worth mentioning an implication of (2.45) considering in particular (2.47) that yields

$-\frac{\text{d}\phi }{c^2}=\frac{\text{d}r}{\text{d}t}\frac{\text{d}{v}_r}{c^2}=\frac{v_r}{c^2}\text{d}{v}_r=\frac{v_r^2}{c^2}\frac{\text{d}{v}_r}{v_r}=\frac{v_r^2}{c^2}\frac{\text{d}{v}_r}{r_0}\frac{r_0}{v_r}\Rightarrow -\frac{\text{d}\phi }{c^2}=\frac{v_r^2}{c^2}\frac{\text{d}\nu }{\nu }\nu =\frac{v_r}{r_0};$ (2.48)

it is interesting to calculate the limit of the result for $v_r\to c$ , which reads

$\underset{v_r\to c}{\text{lim}}\frac{\text{d}\phi }{c^2}=-\frac{\text{d}\nu }{\nu }$ (2.49)

This result will be obtained again in the next section, without any concern to $ℳ$ ; this strategy aims to emphasize that the similarity of (2.45) with the properties of real observable matter is not accidental. For this reason (2.47) is found later. It is anticipated here that $\phi$ is the gravitational potential and in fact (2.49) is well known, it is the red/blue shift $\leftrightarrow$ energy loss/gain equation of a photon moving radially in the field of a gravitational real mass in vacuum.

This paper concerns a quantum model and implements quantum approach. The strategy of the paper is to introduce a conceptual basis general enough to infer the necessary information on which is rooted the cosmological nature of the universe. The notation $\delta x_i^2$ means ${\left(\delta x_i\right)}^2={\left(x_i-x_0\right)}^2$ , whereas $\delta \left(x_i^2\right)=x_i^2-x_0^2$ . The text organized in order to be as self-contained as possible.

3. Preliminary Considerations: The $\beta$ Function

After having examined $\Pi$ , this section concerns $1-\Pi$ i.e., owing to (2.41), $\beta$ . It is known [2] that

$\nabla \phi =-\dot{v}\dot{v}=\frac{\delta v}{\delta t},$ (3.1)

being $\dot{v}$ acceleration and $\phi =\phi \left(r,t\right)$ a function of space coordinates and time; e.g., if $\phi$ is defined by the classical Lagrangian as $ℒ=m{v}^2/2-m\phi$ , then (3.1) introduces the gravitational potential per unit mass $m$ subjected to attractive field of another mass $M$ . So it is also possible to define by consequence

$v\cdot \nabla \phi =-v\cdot \dot{v}$ (3.2)

and then

$v\cdot \dot{v}=\frac{1}{2}\frac{\delta \left(v\cdot v\right)}{\delta t}=\frac{\delta \left(v^2/2\right)}{\delta t}=-\dot{\phi }\phi =-\frac{v^2}{2}+const.$ (3.3)

Note that in fact all quantities of (3.1) to (3.3) are functions of $x,y,z,t$ , as suggested by $\nabla$ ; however the second (3.1) emphasizes the $t$ -dependence only, which shortens and simplifies the notation $\phi =\phi \left(x,t\right)$ . Also note that (3.1) has been already found in the more general (2.47), whose sign has not been specified without information about $\partial v/\partial t$ . The rest of this section deals with the possible implications of (3.1) to (3.3). First of all (3.3) implies

$\frac{\delta \phi }{c^2}=-\frac{1}{2}\delta \frac{v^2}{c^2}=\frac{1}{2}\delta \left(1-\frac{v^2}{c^2}\right),$ (3.4)

which evidences the connection of this section with the section 1, whence

$\frac{\delta \phi }{c^2}=\frac{1}{2}\delta \left({\beta }^2\right)=\beta \delta \beta \Rightarrow \frac{\phi }{c^2}=\frac{{\beta }^2}{2}+cons{t}^{\prime }\beta =\pm \sqrt{1-\frac{v^2}{c^2}};$ (3.5)

it also implies

$\nabla \frac{\phi }{c^2}=\beta \nabla \beta =-\frac{1}{2}\nabla \frac{v^2}{c^2},$ (3.6)

while the connection with the results of the previous section is given by

${\beta }^2=1+\frac{2\phi }{c^2}+const=1-\Pi \Rightarrow 2\beta \delta \beta =2\frac{\delta \phi }{c^2}=-\delta \Pi .$ (3.7)

Since ${\left(c\beta \right)}^2=c^2-v^2$ owing to (3.5), multiplying both sides by $\delta t^2$ one finds

${\left(c\beta \delta t\right)}^2=\delta {\ell }_{inv}^2=c^2\delta t^2-\delta x^2\delta x=v\delta t.$ (3.8)

It is shown in [2] that this invariant is enough to infer the special relativity. The fact of having inferred (3.8) assures in principle that any result of special relativity could be regarded and implemented via the conceptual frame hitherto introduced; it holds of course also for the basic assumptions of the special relativity, e.g. the value of $c$ constant and independent on the motion speed of a light source. Nevertheless the next subsection shows how (3.1) directly brings itself to some more general implications. An example is given here writing

${\beta }^2=\frac{c^2-v^2}{c^2}=\frac{{\ell }_{+}{\ell }_{-}}{c^2\delta t^2}{\ell }_{+}=c\delta t+\delta x{\ell }_{-}=c\delta t-\delta x\delta \ell =c\delta t\delta x=v\delta t$

whence

${\left(c\beta \delta t\right)}^2={\ell }_{+}{\ell }_{-};$ (3.9)

as $\left(c\beta \delta t\right)$ of (3.8) is invariant, these results imply

${\ell }_{+}{\ell }_{-}={\ell }_{0+}{\ell }_{0-}={\ell }_{0+}\beta \frac{{\ell }_{0-}}{\beta }\Rightarrow {\ell }_{+}={\ell }_{0+}\beta {\ell }_{-}=\frac{{\ell }_{0-}}{\beta }.$ (3.10)

In particular note that ${\ell }_{+}/c=time$ , so that it is possible to write

$\frac{{\ell }_{-}}{c}=t=\frac{{\ell }_{0-}/c}{\beta }=\frac{t_0}{\beta }.$ (3.11)

Equations (3.10) and (3.11) introduce space contraction and time dilation with respect to the respective proper values; if in particular $v=const$ in (3.5), then these results are known as the Lorentz transformations in different inertial reference systems moving a relative velocity $v=const$ .

Also note that $\beta$ implies

$-\frac{\beta \delta \beta }{\delta v}=\frac{v}{c^2}=\frac{\left(m/\beta \right)v}{\left(m/\beta \right)c^2}=\frac{p}{ϵ}p=\frac{mv}{\beta }ϵ=\frac{m{c}^2}{\beta },$ (3.12)

being

$ϵ\beta ={ϵ}_0=m{c}^2\beta \delta ϵ=-ϵ\delta \beta .$ (3.13)

Let ${ϵ}_0$ be the energy in a reference system where one particle is at rest, whereas ${ϵ}_0/\beta$ is the kinetic energy in a different reference system; then, owing to (3.8) and (3.13), one finds contextually

${\left(m{c}^2\right)}^2={ϵ}^2{\beta }^2={ϵ}^2-{ϵ}^2\frac{v^2}{c^2}\Rightarrow {ϵ}^2={ϵ}^2\frac{v^2}{c^2}+{\left(m{c}^2\right)}^2={\left(pc\right)}^2+{\left(m{c}^2\right)}^{2}p=\frac{ϵv}{c^2}$ (3.14)

Consider now any velocity $v_b>v$ ; defining thus $v_b/\beta =v$ write

$v_b=\frac{v}{\beta }=\frac{v}{\sqrt{1-v^2/c^2}}\Rightarrow v=\frac{v_b}{\sqrt{1+\frac{v_b^2}{c^2}}}{v}_b>v,$ (3.15)

having merely solved the first equation with respect to $v$ . The result (3.15) reads, in the particular case of constant acceleration,

$v=\frac{a_{b}t}{\sqrt{1+\frac{{\left(a_{b}t\right)}^2}{c^2}}}{v}_v=a_{b}t.$ (3.16)

To infer further information on $\phi /c^2$ , define the dimensional relationship $ϵv=\hslash \dot{v}+const$ and express $\dot{v}$ via (3.1), which yields $ϵv-const=\hslash \dot{v}=-\hslash \nabla \phi$ ; the arbitrary vector $const$ has been added to $\dot{v}$ in order that $ϵ$ does not vanish for $v=const$ . Then one finds

$\epsilon v=\hslash \left(a_0+\dot{v}\right)=\hslash \left(a_0-\nabla \phi \right).$ (3.17)

Three simple considerations show that (3.17) is sensible.

(i) The right hand side of (3.17) defines the scalars

$\epsilon \frac{v^2}{c^2}=\hslash v\cdot \frac{a_0-\nabla \phi }{c^2}=\hslash \omega \omega =v\cdot \frac{a_0-\nabla \phi }{c^2}.$ (3.18)

On the one hand this result reads

$p=\epsilon \frac{v}{c^2}=\hslash u\cdot \frac{a_0-\nabla \phi }{c^2}=\frac{\hslash \omega }{v}u=\frac{v}{v}$ (3.19)

and thus also

$p=\frac{\epsilon v}{c^2}\Rightarrow p=\frac{h\nu }{v}=\frac{h}{\lambda }v=\lambda \nu ;$ (3.20)

on the other hand it implies by dimensional reasons

${\omega }_0=v\cdot \frac{a_0}{c^2}$ (3.21)

so that (3.17) yields

$\omega -{\omega }_0=-v\cdot \frac{\nabla \phi }{c^2}=-{\omega }^{\prime }\frac{\delta \phi }{c^2}.$ (3.22)

Indeed, regarding one component of the scalar, e.g. $\left(\delta x/\delta t\right)\left(\delta \phi /\delta x\right)$ , the last (3.22) reads $\delta \omega \approx -{\omega }^{\prime }\delta \phi /c^2$ with ${\omega }^{\prime }=1/\delta t$ , i.e.

$\frac{\delta \omega }{{\omega }^{\prime }}=-\frac{\delta \phi }{c^2}.$ (3.23)

in agreement with (2.49) and related considerations about $ℳ$ .

(ii) It is also immediate to extend this result to the case of any massive particle, or photon moving radially at $v<c$ in a medium with refractive index $\text{n}$ . It is convenient to write first

$t=\frac{t_p}{\beta }\frac{h}{t_p}={ϵ}_0=\frac{h}{t\beta }=ϵ\beta ϵ{\beta }^2=\frac{h}{t}ϵ=\frac{{ϵ}_0}{\beta },$ (3.24)

whence

$\frac{h\beta }{t_p}=\frac{h}{t}ϵ={ϵ}^2{\beta }^2={ϵ}^2-{ϵ}^2\frac{v^2}{c^2}={ϵ}^2-{\left(pc\right)}^2={\left(m{c}^2\right)}^2.$ (3.25)

Thus

$\delta t=\frac{\delta t_p}{\beta }-\frac{t_p}{{\beta }^2}\delta \beta =\frac{\hslash }{\delta \epsilon }\Rightarrow \delta \beta =\frac{\delta t_p}{t_p}\beta -\frac{\hslash /t_p}{\delta \epsilon }{\beta }^2=\frac{\delta t_p}{t_p}\beta -\frac{\hslash /t}{\delta \epsilon }\beta$

yields

${\beta }^2=\frac{h/ϵ}{t}\frac{\delta \beta }{\beta }=\frac{\delta t_p}{t_p}-\frac{\epsilon }{\delta \epsilon }\Rightarrow \frac{\delta \phi }{c^2}={\beta }^2\frac{\delta \beta }{\beta }=-{\beta }^2\left(\frac{\epsilon }{\delta \epsilon }-\frac{\delta t_p}{t_p}\right)=-\frac{h}{ϵ}\left(\frac{\epsilon }{t\delta \epsilon }-\frac{\delta t_p}{t_{p}t}\right),$

which eventually identifies by dimensional reasons

$\frac{\delta \phi }{c^2}=-\frac{\nu -{\nu }^{\prime }}{{\nu }^{″}}\nu =\frac{\epsilon /\delta \epsilon }{t}{\nu }^{\prime }=\frac{\delta t_p/t_p}{t}{\nu }^{″}=\frac{ϵ}{h}$ (3.26)

analogous to that already found. Here the frequency $\nu =v/\lambda$ refers to the wavelike behavior of massive particles having momentum $h/\lambda$ . Note that implementing the uncertainty equation $\delta ϵ=n\hslash /\delta t$ the actual result would be $\nu =n\epsilon /\left(t\delta \epsilon \right)$ ; however, even without introducing the quantization, an acceptable result has been obtained merely considering the dimensional relationship $\epsilon =h/t$ .

(iii) Equation (3.17) yields contextually

$-\hslash \frac{\nabla \phi }{c^2}=p-constp=\frac{ϵv}{c^2}const=\frac{\hslash a_0}{c^2},$ (3.27)

while being also

$v\cdot \nabla \frac{\phi }{c^2}=\frac{\xi }{\delta t}\Rightarrow ϵ=\hslash v\cdot \nabla \frac{\phi }{c^2}p=\hslash \left(v\cdot \nabla \frac{\phi }{c^2}\right)\frac{v}{c^2}.$ (3.28)

At this point is also interesting the fact that (3.1) implies itself

$\dot{v}\cdot \nabla \phi =-{\dot{v}}^2\Rightarrow {\hslash }^2\dot{v}\cdot \nabla \frac{\phi }{c^2}=-{\hslash }^2\frac{{\dot{v}}^2}{c^2}=-{\left(\hslash \omega \right)}^2\omega =\frac{\dot{v}}{c};$ (3.29)

thus, taking the square root of this result, one finds

$\hslash \frac{\delta \left(v/c\right)}{\delta t}=\pm ϵϵ=\hslash \omega .$ (3.30)

On the one hand owing to (3.27) the second (3.29) implies

$p=\frac{ϵ}{c^2/v}=\frac{\hslash \omega }{c^2/v}=\frac{h\nu }{c^2/v}=\frac{h}{\lambda }\lambda =\frac{c^2}{v\nu }=\text{n}\frac{c}{\nu }\text{n}=\frac{c}{v},$ (3.31)

where the definition of $\lambda$ is justified in turn putting $\text{n}=c/v$ . As a check, implementing this wave formulation of momentum (3.31), one finds again (3.23)

$-\frac{\delta \phi }{c^2}=\frac{\delta p}{p}=\frac{h\delta \left(c/c\lambda \right)}{p}=\frac{h\delta \left(\nu /c\right)}{p}=\frac{\delta \left(h\nu /c\right)}{h/\lambda }=\frac{\delta \nu }{c/\lambda }=\frac{\delta \nu }{\nu }.$ (3.32)

Implement these results to calculate $\delta \left(pc\right)$ and $\delta ϵ$ , noting that $\delta ϵ/\delta p$ has physical dimensions of velocity, say $v_g$ . Merging (3.30) and (3.31), one finds

$\frac{c}{v_g}=\frac{\delta \left(pc\right)}{\delta ϵ}=\frac{h\delta \left(c/\lambda \right)}{h\delta \nu }=\frac{h\delta \left(c/v\right)\left(v/\lambda \right)}{h\delta \nu }=\frac{\delta \left(\text{n}\nu \right)}{\delta \nu }\nu =\frac{v}{\lambda }$

whence

$v_g=\frac{c}{\delta \left(\text{n}\nu \right)/\delta \nu },$ (3.33)

which agrees with the result introduced in the section 1.

On the other hand, look now for a more general expression of $v\left(t\right)$ solving the first (3.17) with respect to $\dot{v}$ and $v$ of (3.27) to calculate the components $v=v\cdot u$ and $\dot{v}=\dot{v}\cdot u$ of $v$ and $\dot{v}$ along an arbitrary direction defined by the constant unit vector $u$ : multiplying both sides of the first equality (3.17) by $u$ , elementary calculations yield

$\begin{array}{l}\hslash \frac{\delta v}{\delta t}=ϵv-{\epsilon }_0{v}_0ϵ=ϵ\left(x,t\right)\\ \Rightarrow v\left(x,t\right)=\left(-\frac{{\epsilon }_0{v}_0}{\hslash }{\displaystyle \int {\text{e}}^{-\zeta \left(t\right)}\text{d}t}+{v^{\prime }}_0\right){\text{e}}^{\zeta \left(t\right)}\zeta \left(t\right)={\displaystyle \int \frac{ϵ\left(t\right)}{\hslash }\text{d}t},\end{array}$ (3.34)

being ${v^{\prime }}_0$ integration constant. Moreover (3.34) rewritten as

$\frac{\delta v}{\delta t}=\frac{ϵ\left(t\right)}{\hslash }\left(-\frac{{\epsilon }_0{v}_0}{\hslash }{\displaystyle \int {\text{e}}^{-\zeta \left(t\right)}\text{d}t}+{v^{\prime }}_0\right){\text{e}}^{\zeta \left(t\right)}-\frac{{ϵ}_0}{\hslash }{v}_0$ (3.35)

is quantum equation of acceleration. Check for the validity of these last results noting that neither $v_0$ nor $\zeta$ are “a priori” determined; rather in principle $v$ and its time derivative are determined by $ϵ\left(x,y,z,t\right)$ , which is a free parameter as a function of which are calculable further results of physical interest as corollaries. In particular, nothing hinders to think that (3.34) can fit the standard expression of relativistic acceleration, e.g. [2], or even $v_g$ itself of (3.33), via an appropriate best fit function $ϵ\left(x,y,z,t\right)$ . The notation emphasizes the time dependence of $\zeta \left(x,y,z,t\right)$ necessary to calculate the time integral coming from (3.3); of course the time integration constant $v_0=v_0\left(x,y,z\right)$ can actually be function of the coordinates likewise ${v^{\prime }}_0={v^{\prime }}_0\left(x,y,z\right)$ .

In this respect it is worth calculating the general result (3.34) in the following particular cases.

(i) The first case concerns $ϵ\left(t\right)=const$ , thus (3.34) becomes

$\begin{array}{l}ϵ\left(t\right)\equiv -{ϵ}_0\Rightarrow \zeta =\frac{-{ϵ}_{0}t}{\hslash }\\ \Rightarrow v\left(t\right)=v_0\exp \left(-{ϵ}_{0}t/\hslash \right)+v_0\frac{\hslash /t_0}{{ϵ}_0}\dot{v}\left(t\right)=-\frac{v_0{ϵ}_0}{\hslash }\exp \left(-{ϵ}_{0}t/\hslash \right);\end{array}$ (3.36)

i.e. $v\left(t\right)$ does not diverge, it changes in the interval $v\left(t=0\right)={v^{\prime }}_0+\hslash const/{ϵ}_0$ to $v\left(t=\infty \right)=\hslash const/{ϵ}_0$ . This result shows that defining appropriately the constants in (3.36), at least this one case does in fact exist that implies reasonably

$0\le v\left(t\right)\le c;$ (3.37)

moreover (3.36) also shows the necessity of taking the minus sign of (3.30) to avoid divergent $v\left(t\right)$ . Actually this conclusion could be expected in general via (3.4) and (3.5) itself. Consider indeed two arbitrary velocities $v$ and $v^{\prime }$ such that $\delta v^{\prime }\le \delta v$ and write the identity

$\delta v\equiv \frac{{\beta }^2\delta v}{{\beta }^2}=\frac{\delta v^{\prime }}{{\beta }^2}\delta v=v_2-v_1\delta v^{\prime }={v^{\prime }}_2-{v^{\prime }}_1=v_2{\beta }^2-v_1{\beta }^2;$

putting $v^2={v^{\prime }}_1{v^{\prime }}_2$ , these definitions of ${v^{\prime }}_2$ and ${v^{\prime }}_1$ yield

$\delta v=\frac{{v^{\prime }}_1-{v^{\prime }}_2}{1-{v^{\prime }}_1{v^{\prime }}_2/c^2}:$ (3.38)

reasonable conditions on ${v^{\prime }}_1$ and ${v^{\prime }}_2$ are enough to make $\beta$ compliant with a well known formula of addition of velocities. If either ${v^{\prime }}_1=c$ or ${v^{\prime }}_2=c$ , then $\delta v=\pm c$ ; of course this holds even though $\delta v\equiv v$ when in particular $v_1=0,v_2=c$ or $v_1=c,v_2=0$ .

(ii) imaginary $\zeta$ .

Calculate $\zeta \left(x,t\right)$ of (3.34) regarding $ϵ\left(x,t\right)$ as an imaginary energy with either possible chance $\pm iϵ\left(x,t\right)$ . It is immediate to show by direct calculation that even this replacement can be interpreted as a sensible solution the first (3.34); the result of this replacement is indeed the following equation to be fulfilled

$\pm \hslash i\frac{\delta v\left(x,t\right)}{\delta t}=vϵ\left(x,t\right)\pm i{v}_0{ϵ}_0.$ (3.39)

It is instructive to make now an approximation concerning this result: put $v_0=0$ , which means $a_0=0$ in (3.17). Replace next $\delta \to \partial$ , to obtain the following differential equations corresponding to either sign of (3.39)

$\hslash i\frac{\partial v\left(x,t\right)}{\partial t}=vϵ\left(x,t\right)-\hslash i\frac{\partial v\left(x,t\right)}{\partial t}=vϵ\left(x,t\right).$

Now multiply both sides of the first equation by an appropriate $f=f\left(x\right)$ to define the scalar $v\cdot f=\psi$ ; the result is

$\hslash i\frac{\partial \psi }{\partial t}=ϵ\left(x,t\right)\psi .$ (3.40)

Proceed analogously with the second equation; (3.34) and (3.39) yield

$-\hslash i\frac{\delta \left(v\left(x,t\right)/c^2\right)}{\delta t}=\frac{v\left(x,t\right)ϵ\left(x,t\right)}{c^2}\Rightarrow -\hslash i\frac{\delta \left(v\left(x,t\right)/c\right)}{c\delta t}=\frac{v\left(x,t\right)}{c}\frac{ϵ\left(x,t\right)}{c};$

now multiplying again $v/c$ by $f$ at both sides, one finds

$\frac{\hslash }{i}\frac{\partial \psi }{\partial \ell }=p\left(\ell ,t\right)\psi \delta \ell =c\delta tp\left(x,t\right)\approx \frac{ϵ\left(x,t\right)}{c}.$ (3.41)

Clearly (3.40) and (3.41) are inherently approximate, being obtained replacing $\delta \to \partial$ ; moreover the last (3.41) is justified by (3.14), as ${ϵ}^2\approx {\left(pc\right)}^2$ once neglecting ${\left(m{c}^2\right)}^2$ and expressing the arbitrary $x$ space coordinate via $ct$ in order that $\psi \left(x,t\right)\to \psi \left(\ell ,t\right)$ .

The results (3.40) and (3.41) are well known and confirm the general validity of (3.34).

(iii) Quantum meaning of $\beta$ .

As concerns the quantum implications of the $\beta$ function, consider the following steps to infer further corollaries. Write

$\begin{array}{l}{\beta }^2=1-\frac{v^2}{c^2}=1-\frac{v^2\delta t^2}{c^2\delta t^2}=1-\frac{\delta r^2}{\delta {\ell }^2}=\frac{\delta {\ell }^2-\delta r^2}{\delta {\ell }^2}=\frac{\delta \ell -\delta r}{\delta \ell }\frac{\delta \ell +\delta r}{\delta \ell }\\ \Rightarrow {\beta }^2=\frac{\delta \ell -\delta r}{\delta \ell }\left(1+\frac{\delta r}{\delta \ell }\right),\end{array}$

which in turn yields

${\beta }^2=\frac{\delta \ell -\delta r}{\delta \ell }+\frac{\delta \ell -\delta r}{\delta \ell }\frac{\delta r}{\delta \ell }\delta \ell =c\delta t\delta r=v\delta t.$ (3.42)

Let $\delta \ell ={\ell }_2-{\ell }_1$ be the total size of the uncertainty range in an arbitrary reference system where is delocalized one quantum particle, and let $\delta \ell$ consist of two ideal sub-ranges $\delta \ell -\delta r$ and $\delta r=r_2-r_1$ : the first addend of (3.42) is the probability for the particle to be found in $\delta \ell -\delta r$ only, the second addend is the probability that the particle is simultaneously in $\delta \ell -\delta r$ and in $\delta r$ . Is interesting the fact that ${\beta }^2$ introduces separate probabilities for a particle to be found in either sub-range or in both sub-ranges simultaneously. The latter is intuitively evident: if one particle is delocalized in the total $\delta \ell$ , then it must be somewhere in both sub-ranges by definition; yet is interesting that also appears the probability for the particle to be in either of them as a specific event separately definable. The physical meaning of this reasoning is clear: it suggests the chance that the particle is a corpuscle, located somewhere in $\delta \ell$ , but also a wave that spreads through and pervades all available $\delta \ell$ : a wave is everywhere, the corpuscle is here or there.

On the one hand these probabilities imply their zero point energies ${ϵ}_{\left(\ell -r\right)}=\delta p_{\left(\ell -r\right)}^2/2m$ and ${ϵ}_r=\delta p_r^2/2m$ being $m$ the mass of the corpuscle: indeed the uncertainty principle links $\delta p_r$ and $\delta p_{\left(\ell -r\right)}$ to the respective $\delta r$ and $\delta \ell -\delta r$ .

On the other hand it happens regardless of any additional information, i.e. this would hold even though there is a potential wall of value ${ϵ}_{thr}$ somewhere inside $\delta \ell$ and regardless of the actual energy of the particle, e.g. located at the boundary between $\delta \ell -\delta r$ and $\delta r$ .

Clearly this reasoning has to do with the tunnel effect: e.g. if the arbitrary size of $\delta r$ tends to zero, the local delocalization energy of the corpuscle diverges. In other words (3.42) is explainable if ${ϵ}_r>{ϵ}_{thr}$ , which in fact is allowed by an appropriate $\delta r$ even in the presence of a threshold energy ${ϵ}_{thr}$ between $\delta \ell -\delta r$ and $\delta r$ . In turn this conclusion means that the corpuscle, owing to its own zero point energy, has a non-null probability of being found in both sub-regions of $\delta \ell$ regardless of its initial energy before penetrating from $\delta \ell -\delta r$ into $\delta r$ . As it holds even regardless of any information about $ϵ$ , this result is the well known quantum tunneling of particles overcoming finite potential barrier, here inferred from as a straightforward corollary of the typical relativistic function $0\le \beta \le 1$ regarded via its probabilistic meaning without any “ad hoc” hypothesis.

Eventually it is also worth mentioning what kind of additional information can be deducted from $\phi /c^2$ : as it is physically significant to examine not only $v\cdot \nabla$ but also $v\times \nabla$ , write then the identity $v\times \nabla \phi \equiv ev\times \nabla \phi /e$ . The physical dimension of $\nabla \phi$ is acceleration by definition, i.e. $m\nabla \phi$ defines force; in turn, as force/charge and force/mass define field, one finds

$v\times \nabla \phi \equiv \frac{ev}{m}\times \nabla \frac{m\phi }{e}=\frac{ev}{m}\times field\Rightarrow mv\times \nabla \phi =ev\times field.$

Since dividing both sides by $c$ , the left hand side has physical dimensions of force, this result is therefore acknowledged as the Lorentz force

$F=e\frac{v}{c}\times H.$ (3.43)

Define now by dimensional reasons $v/c\times H=F/e=er/r^3$ , in principle possible with appropriate values at the right side; the physical motivation of this step is the chance of defining

$\frac{F}{e}=\frac{er}{r^3}=\begin{array}{l}\nearrow \hfill \\ \searrow \hfill \end{array}\begin{array}{l}F\cdot r=ℰ=\frac{e^2}{r}\hfill \\ \text{}\text{}\hfill \\ E=e\frac{r}{r^3}\hfill \end{array},$ (3.44)

i.e. a new field $E=F/e$ with $r$ oriented like $\left(v/c\right)\times H$ and thus $H\cdot E=0$ , e.g. like the orthogonal fields of an e.m. wave.

Consider (3.43) to calculate $\nabla \cdot E$ of (3.43); as $E=er/r^3$ , then $\nabla \cdot r/r^3=3/r^3$ yields the Gauss result

$\nabla \cdot E=4\pi \left(e/V\right)=4\pi \rho V=\frac{4\pi r^3}{3}.$ (3.45)

Implement now the shortened notation $\nabla \cdot E\propto \rho$ to skip factors like $4\pi ,{ϵ}_0,{\mu }_0$ of the SI system equations and consider now (3.43) to calculate $\nabla \times E\propto \nabla \times \left(v\times H\right)=\left(\nabla \cdot H\right)v-\left(\nabla \cdot v\right)H$ , having skipped again the proportionality constant $e^2/c$ . Assuming $\nabla \cdot H=0$ and noting the addends of the scalar $\nabla \cdot v$ have the form $\left(\partial /\partial x_i\right)\left(\partial x_i/\partial t\right)=\partial /\partial t$ , one infers $\nabla \times E\propto -H$ . As concerns (3.43), owing to $H\cdot E=0$ , note that multiplying (3.43) by an appropriate vector operator $X$ one finds $X\cdot E\propto X\cdot \left(v\times H\right)=v\cdot \left(X\times H\right)$ . Let be $X=\nabla$ : then this result reads $\nabla \cdot E=v\cdot \left(\nabla \times H\right)$ . Next, multiplying again by $v^{\prime }$ one finds $J={\rho }^{\prime }{v}^{\prime }=\left(v\cdot \left(\nabla \times H\right)\right)v^{\prime }$ . The fact that appear in this result two velocities, in fact no reason requires that $v^{\prime }$ is equal $v$ , suggests that $v$ defines $\rho$ via the scalar $v\cdot \left(\nabla \times H\right)$ as already found, whereas $v^{\prime }$ defines $J$ via $\rho v^{\prime }$ : it is enough to introduce ${\rho }_{tot}=\rho +{\rho }^{\prime }$ and split the two addends of the total charge into the respective displacement rates related to two displacement mechanisms. As a result one finds starting from (3.43)

$\nabla \cdot E\propto \rho \nabla \cdot H=0\nabla \times E\propto -\dot{H}\nabla \times H\propto J+\dot{E}$ (3.46)

i.e. the Maxwell equations. In this respect it is worth emphasizing that a recent paper has introduced an interaction mechanism involving the quantum vacuum, as a consequence of which even magnetic monopoles can exist at an appropriate threshold energy [3].

To complete the list of corollaries of (3.1) note the chance that

$\dot{v}=\frac{\delta v}{\delta t}=\frac{\delta }{\delta t}\frac{\delta x}{\delta t}=\frac{{\delta }^{2}x}{\delta t^2}=-\nabla \phi ,$

whence

$\frac{\dot{v}}{c^2}=-\nabla \frac{\phi }{c^2}=-\frac{{\delta }^{2}x}{\delta {\ell }^2}\delta \ell =c\delta t\Rightarrow -c^2\frac{{\delta }^{2}x}{\delta {\ell }^2}=-\frac{{\delta }^{2}x}{\delta t^2}.$ (3.47)

In general, given any function $f=f\left(x\right)$ with $x=x\left(t\right)$ , note that owing to (3.3)

$\frac{\delta f}{\delta t}=\frac{\delta f}{\delta x}\frac{\delta x}{\delta t}=v_x\frac{\delta f}{\delta x}$

which implies

$\frac{{\delta }^{2}f}{\delta t^2}=\frac{\delta \left(v_x\frac{\delta f}{\delta x}\right)}{\delta t}=v_x\frac{\delta \left(v_x\frac{\delta f}{\delta x}\right)}{\delta x}=v_x^2\frac{\delta \left(\frac{\delta f}{\delta x}\right)}{\delta x}+v_x\frac{\delta v_x}{\delta x}\frac{\delta f}{\delta x}=v_x^2\frac{{\delta }^{2}f}{\delta x^2}-\frac{\delta f}{\delta x}\frac{\delta \phi }{\delta x}$

and thus yields

$\frac{{\delta }^{2}f}{\delta t^2}=v_x^2\frac{{\delta }^{2}f}{\delta x^2}-\frac{\delta f}{\delta x}\frac{\delta \phi }{\delta x};$ (3.48)

if in particular $v_x$ does not depend upon $x$ , e.g. vibrating homogeneous spring or e.m. wave propagating in a homogeneous dispersive medium, then the second addend at the right hand side vanishes. This particular the case implies the D’Alembert equation

$\frac{{\delta }^{2}f\left(x,t\right)}{\delta t^2}=v_x^2\frac{{\delta }^{2}f\left(x,t\right)}{\delta x^2},$ (3.49)

which generalizes (3.47) to any propagation rate $v_x$ of the wave.

Note that all information hitherto inferred has been extracted uniquely from the mere definition (3.1) of gravitational potential $\phi$ only, without additional hypotheses to explain the link between classical physics, quantum physics, electromagnetism and relativity.

The link of this preliminary section with the introductory section 1 is summarized by (3.7). The essential points of this section are:

-All considerations hitherto introduced generalize the concept of Lorentz transformations, according which $v/c$ constant concerns the displacement rate of inertial reference systems only.

-The actual analytical forms of $v\left(x,y,z,t\right)$ and $\dot{v}\left(x,y,z,t\right)$ are specified by $ϵ\left(x,y,z,t\right)$ , i.e. the results hold for $v$ and $\dot{v}$ governed by an appropriate $ϵ\left(x,y,z,t\right)$ , arbitrary input.

-The first step (3.2) defines $\phi$ in (3.3), i.e. $v\cdot \dot{v}$ , deserves more attention; this point is again concerned later.

-The physical meaning of (3.4), and thus of (3.10) and (3.11), are further considered in the next sections to explain why stretching/shrinking of the space time extends these preliminary results directly to the cosmological conceptual frame.

4. Further Considerations on $\beta$ : Quantum Uncertainty

The results of the section 2 suggest that the physical meaning of the $\beta$ function of (3.5) is well beyond that of mere Lorentz factor of special relativity, where in particular $v=constant$ accounts for inertial reference systems in relative motion. This section concerns three corollaries of $\beta$ aimed to investigate this point with the help of the quantum uncertainty equation inferred in [4]

$\delta x\delta p_x=n\hslash =\delta \epsilon \delta t.$ (4.1)

To show (4.1) for completeness, consider the Planck constant $n$ times, e.g. for $n\hslash /t$ to describe the huge energy of a macroscopic system. Since by definition $\hslash$ is defined by the Planck energy and time, $\hslash ={\epsilon }_{{}_{Pl}}{t}_{{}_{Pl}}$ , it follows $n\hslash =n{\epsilon }_{{}_{Pl}}{t}_{{}_{Pl}}$ . Yet at the right hand side $n$ multiplies two quantities; so write $n\hslash =n\text{'}{\epsilon }_{{}_{Pl}}n\text{'}\text{'}{t}_{{}_{Pl}}$ with the condition $n\text{'}n\text{'}\text{'}=n$ . By definition $n$ is integer, whereas $n\text{'}$ and $n\text{'}\text{'}$ do not. Rewrite thus this result as $n\hslash =\delta \epsilon \delta t$ , with $\delta \epsilon =n\text{'}{\epsilon }_{{}_{Pl}}$ and $\delta t=n\text{'}\text{'}{t}_{{}_{Pl}}$ ; i.e., simply admitting that $n\text{'}$ and $n\text{'}\text{'}$ are defined by arbitrary boundary values as $n_{1\text{'}}\le n\text{'}\le n_{2\text{'}}$ and $n_{1\text{'}\text{'}}\le n\text{'}\text{'}\le n_{2\text{'}\text{'}}$ , the result is $\delta \epsilon \delta t=n\hslash$ . Moreover it is also true that $\hslash =x_{{}_{Pl}}{p}_{{}_{Pl}}$ via Planck length and momentum; so an identical reasoning yields $n\hslash =\delta x\delta p_x$ . These results merge into (4.1).

Actually the number of possible space extra dimensions $\delta x_j$ and conjugate momenta $p_{xj}$ concerned by the scalar is in principle arbitrary; in this paper the number of space-momentum components is the usual 3, however also the space extra-dimensions, if any, fulfill the time product at the right hand side. It is also worth emphasizing that any uncertainty range is not related to a specific reference frame because neither boundary coordinate is known or knowable by fundamental assumption. Eventually (4.1) shows that the time is inherently linked to any calculation involving explicitly space and momenta coordinates only: (4.1) implies by corollary the concept of space time, whatever the number of space coordinates at the left hand side might be. In lack of information about size and position of any range with the respect to the origin of a system of coordinates $R$ , once having waived the local values of dynamical variables it follows that any other $R^{\prime }$ is indistinguishable from $R$ . In other words, $\delta x$ and $\delta p_x$ enclosing any $x,p_x$ in a given reference system $R$ imply different $\delta x^{\prime }$ and $\delta {p^{\prime }}_x$ enclosing $x^{\prime },{p^{\prime }}_x$ in $R^{\prime }$ while being $\delta x\delta p_x=n\hslash$ and $\delta x^{\prime }\delta {p^{\prime }}_x=n^{\prime }\hslash$ ; however $R$ and $R^{\prime }$ are in fact indistinguishable, because by definition the numbers $n$ and $n^{\prime }$ of allowed states are by definition sequences of arbitrary integers and not specific values identifiable for any $R$ . Hence the physical meaning of quantum curvature radius $\delta r^{-1}$ is conceptually different from the deterministic $r^{-1}$ of the classical physics. In the classical physics both $x$ and $x^{\prime }$ are exactly known and subjected to the Lorentz transformation in different inertial reference systems in the special relativity. Of course here all of this still holds, but in fact $\delta r^{-1}$ and $\delta {r^{\prime }}^{-1}$ are indistinguishable, because nothing in known about them: this a conceptual requirement not a simplifying approximation for numerical purposes. In fact different reference systems are themselves not distinguishable “per se”, once waiving the local values of dynamical variables because of indistinguishable $n$ and $n^{\prime }$ . Once acknowledging the uncertainty

$\begin{array}{l}\text{space time background}\Rightarrow \text{momentum change and space gap}\\ \leftrightarrow \text{energy change and time lapse}\end{array}$

are contextually configured by the number $n$ of allowed states of the physical system. According to the quantum uncertainty: (i) no instantaneous changes, $\partial /\partial$ , but ratios of finite ranges $\delta /\delta$ of dynamical variables are introduced by the uncertainty; (ii) unspecified reference system in lack of information about the range boundaries.

Equation (4.1) has classical, quantum and relativistic implications.

(i) Classical corollary.

Multiply $\delta x\delta p_x=\delta ϵ\delta t$ by $v_x$ ; as $v_x=\delta x/\delta t$ and $v_x\delta p_x=m{v}_x\delta v_x=\delta \left(m{v}_x^2\right)/2$ by definition, then

$v_x\delta x\delta p_x=v_x\delta ϵ\delta t\Rightarrow \delta \left(m{v}_x^2\right)/2=\delta ϵ\delta x=v_x\delta t\Rightarrow \frac{1}{2}m{v}_x^2+const=ϵ.$

(ii) Quantum corollary.

Multiply $n\hslash =\delta ϵ\delta t$ by $\delta t^{-1}$ ; thus one finds

$\frac{n\hslash }{\delta t}=\delta ϵ\Rightarrow \delta \frac{n\hslash }{\delta t}-\hslash \frac{\delta n}{\delta t}=\delta ϵ;$

if $n$ does not depend upon $\delta t$ , then $\delta \left(n\hslash \right)/\delta t=\delta ϵ$ reads $n\hslash \omega +const=ϵ$ . This well known energy quantization is actually approximate, it holds if $n$ is time independent.

(iii) Relativistic corollary.

Identify $ϵ=m\phi$ and calculate according to (3.3) $\delta ϵ={ϵ}_2-{ϵ}_1=-m{v}_2^2/2+m{v}_1^2/2$ ; thus, owing to the attractive nature of the gravitational field, one expects that $\delta ϵ=-\delta \phi$ . Write then $\delta ϵ=-\left(m{c}^2\right)\delta \left(\phi /c^2\right)$ , i.e.

$-\frac{\delta ϵ}{{ϵ}_0}=\delta \frac{\phi }{c^2}\to -\frac{\delta \hslash \omega }{\hslash {\omega }_0}=\frac{\delta \phi }{c^2}{ϵ}_0=m{c}^2\Rightarrow -\frac{\delta \omega }{{\omega }_0}=\frac{\delta \phi }{c^2}:$

this result, agrees with (3.23) previously found, does not depend on $m$ because $\delta ϵ/{ϵ}_0$ turns into $\delta \omega /{\omega }_0$ and thus still holds even for $v\to c$ , i.e. for a photon displacing radially in the vacuum with respect to a gravitational source. Owing to the general considerations carried out in the section 1, the aforesaid three points are the natural consequence of the theoretical approach followed in this paper.

Note that the scalar $\delta x\delta p_x$ of (4.1) does not exclude and thus implies also the vector forms

$\delta x\cdot \delta p=n\hslash =\delta \epsilon \delta t$ (4.2)

and

$u\cdot \delta x\times \delta p=n\hslash =\delta \epsilon \delta t,$ (4.3)

being $u$ unit vector. If (4.1) has physical meaning, the same must hold even for both (4.2) and (4.3). In fact (4.2) and (4.3), although conceptually identical to (4.1) with mere vector notation, deserve further attention.

Regard the classical definition $M=r\times p$ according to the uncertainty, i.e. replacing the deterministic $r$ and $p$ as $\delta r$ and $\delta p$ ; thus $\delta M=\delta r\times \delta p$ is the range of $M$ corresponding to all $r$ and all $p$ falling in the respective $\delta r$ and $\delta p$ . Starting from the component ${\text{M}}_u=M\cdot u$ of $M$ , write ${\text{M}}_u=u\cdot \left(\delta r\times \delta p\right)\equiv \delta p\cdot \left(u\times \delta r\right)$ ; then $\delta k=u\times \delta r$ yields ${\text{M}}_u=\delta p\cdot \delta k$ , i.e.

${\text{M}}_u=\delta p\cdot \delta k=\pm \delta p_k\delta k=\pm n\hslash \delta k=\left|\delta k\right|n=0,integer.$ (4.4)

It is also immediate to calculate ${\text{M}}^2={\text{M}}_x^2+{\text{M}}_y^2+{\text{M}}_z^2$ assuming $\langle {\text{M}}_x^2\rangle =\langle {\text{M}}_y^2\rangle =\langle {\text{M}}_z^2\rangle$ in an isotropic space. If $-N\le n\le N$ , with $N$ arbitrary boundary allowed to $n$ , then the average of each square component is

${\displaystyle \sum _{-N}^N{\text{M}}_i^2}=N\left(N+1\right)\left(2N+1\right)/3$ (4.5)

in ${\hslash }^2$ units; so the average of the $2N+1$ terms of each sum is $\langle {\text{M}}_i^2\rangle =N\left(N+1\right)/3$ for each square component. Eventually, summing also over the three components the result is

$\langle {\text{M}}^2\rangle =N\left(N+1\right){\hslash }^2.$ (4.6)

Eventually note that $N\left(N+1\right)\equiv {\left(N+1/2\right)}^2-1/4$ reads $\langle {\text{M}}^2\rangle ={\left(2N+1\right)}^2{\hslash }^2/4-{\hslash }^2/4$ , i.e. the right hand side is square angular momentum; the fractional notation has a meaning analogous to that of (4.5). Then, summing $\left(N+1/2\right){\hslash }^2$ to both sides of $\langle {\text{M}}^2\rangle +{\hslash }^2/4={\left(N+1/2\right)}^2{\hslash }^2$ , the result is

$\langle {\text{M}}^2\rangle +{\left(\frac{1}{2}\right)}^2+\left(L+\frac{1}{2}\right)\equiv \left(L+\frac{1}{2}\right)\left(L+\frac{1}{2}+1\right)={ℳ}_{LS}\left({ℳ}_{LS}+1\right){ℳ}_{LS}=L\pm \frac{1}{2}.$ (4.7)

It is crucial to note that (2.38) was previously obtained implementing $\delta t=\hslash /\delta ϵ$ ; implementing instead $\delta t=n\hslash /\delta ϵ$ , i.e. accounting for the uncertainty, the actual result reads now

$-\delta E=n\mu \left|a\right|{\ell }_{Pl}\delta ϵ=\mu \left|a\right|{\ell }_{Pl}\text{or}\delta E=\mu \left|a\right|{\ell }_{Pl}-\delta ϵ=n\mu \left|a\right|{\ell }_{Pl};$ (4.8)

the amounts of energy $\mu \left|a\right|{\ell }_{Pl}$ become now quanta $n\mu \left|a\right|{\ell }_{Pl}$ of energy released to the space time. The quantization of the original (2.36) avoids that the fate of any orbiting system is, sooner or later, the collapse. Similarly to electromagnetic field, the energy exchanges occur via quantized wave packets: the decay of one gravitational system, i.e. the change of its $n$ , can excite another resonant system, which in turn changes its own actual $n\text{'}$ . The universe becomes more interconnected than predicted by the original Einstein formula. Follow now some corollaries of the quantum uncertainty.

4.1. Corollary 1

Let $r=r\left(t\right)$ be the radius of a spherical zone of the space time and assume that the size of this space region can expand as a function of time, e.g. because the whole universe expands itself. The function describing the expansion of the range $\delta r$ from its initial size $r$ during $\delta t$ at rate $\delta r/\delta t$ reads

$H\left(r,t\right)=\frac{\delta v_r}{r}=\frac{\delta r/r}{\delta t}.$ (4.9)

Actually $H$ is described in three ways, pertinent to radius, surface and volume of the growing sphere: since volume and boundary surface of this environment increase along with the enclosed diametric size, the right hand side of (4.9) yields

$\frac{1}{2\pi r}\frac{\delta \left(2\pi r\right)}{\delta t}\to \frac{1}{r}\frac{\delta r}{\delta t}=H.$ (4.10)

Moreover for the spherical surface of this zone

$\frac{1}{4\pi r^2}\frac{\delta \left(4\pi r^2\right)}{\delta t}\to \frac{2r}{r^2}\frac{\delta r}{\delta t}=2\frac{1}{r}\frac{\delta r}{\delta t}=2H.$ (4.11)

Eventually an identical reasoning for a spherical volume implies also

$\frac{1}{\left(4\pi /3\right)r^3}\frac{\delta \left(\left(4\pi /3\right)r^3\right)}{\delta t}\to \frac{3r^2}{r^3}\frac{\delta r}{\delta t}=3\frac{1}{r}\frac{\delta r}{\delta t}=3H.$ (4.12)

These results define the function

$\frac{1}{r}\frac{\delta r}{H\delta t}=kk=1,2,3$

compatible with the driving force $F$ and energy $ϵ$ of the growth process via $P=F/S$ and $\eta =ϵ/V$ having the same physical dimensions. Since $P/\eta$ is dimensionless like $k$ , let be $F/S\propto ϵ/V$ . As the energy $PV$ cannot overcome $ϵ$ , by definition enclosed in $V$ , let be ${\left(PV\right)}_{\max }=ϵ$ i.e. $PV=\left(k/k_{\max }\right)ϵ$ ; thus

$\frac{F}{S}=\frac{k}{3}\frac{ϵ}{V}\Rightarrow P=\frac{k}{3}\eta .$ (4.13)

This well known result and (HH7) support the validity of this reasoning. It is implicitly evident the usefulness of considering not only (4.10) but also (4.11) and (4.12), of course consistent with both chances (4.15).

To proceed further, note that $r^{-1}\delta r/\delta t$ of (4.9) consists of two functions $r_2/\left(r\delta t\right)$ and $r_1/\left(r\delta t\right)$ once expressing explicitly $\delta r=r_2-r_1$ ; if so, then it is convenient to consider $\delta r$ as

$\delta r=r_2-r_1\equiv r_2-r^{\prime }+r^{\prime }-r_1=\left(r_2-r^{\prime }\right)+\left(r^{\prime }-r_1\right)$ (4.14)

in order to obtain from $\delta r$ two terms $\delta r_1=r_2-r^{\prime }$ and $\delta r_1=r^{\prime }-r_1$ formally analogous to the ones just considered in (4.10) to (4.12). Consider that in principle even $r^{\prime }$ could be regarded as a range $\delta r^{\prime }=r^{\prime }-0$ , i.e. with its lower boundary centered on the origin of its own reference system $R$ . This emphasizes why the deterministic coordinate $r^{\prime }$ requires specifying its own $R$ , whereas $\delta r$ does not; $\delta r$ could be defined as $r-{r^{\prime }}_0$ with ${r^{\prime }}_0$ related to its own $R\text{'}$ , or as $r-{r^{″}}_0$ with ${r^{″}}_0$ related to $R^{″}$ , and so on. However, if ${r^{\prime }}_0$ and ${r^{″}}_0$ are undefined, then the respective links to $R^{\prime }$ and $R^{″}$ are undefinable as well. The same holds for the upper boundary $r$ . Clearly the lack of any information about both range boundaries implies the lack of information about their related $R$ . The same holds for the range sizes: in fact $n\hslash$ in $R$ is indistinguishable from any $n^{\prime }\hslash$ in $R^{\prime }$ , because the integers $n$ and $n^{\prime }$ symbolize sequences of numbers of states rather than specific allowed states, i.e. $\delta x\delta p_x$ and $\delta x^{\prime }\delta {p^{\prime }}_x$ are in fact indistinguishable. The step (4.14) aims to convert $r_2/\left(r\delta t\right)$ and $r_1/\left(r\delta t\right)$ into $\delta r_2/\left(r\delta t\right)$ and $\delta r_1/\left(r\delta t\right)$ analogous to that defining $H$ of the initial (4.9), in agreement with the formalism of (4.1). Therefore rewrite identically the first (4.9) according to the last (4.14)

$H_2=\frac{1}{r}\frac{\delta r_2}{\delta t}{H}_1=\frac{1}{r}\frac{\delta r_1}{\delta t}\delta r_2=r_2-r^{\prime }\delta r_1=r^{\prime }-r_1,$ (4.15)

so that (4.15) yields owing to (4.14)

$\frac{1}{r}\frac{\delta r}{\delta t}=H=H_2+H_1:$ (4.16)

in other words, the splitting of $r^{-1}\delta r/\delta t$ into two levels $H_1$ and $H_2$ rewrites the initial $r_2/\left(r\delta t\right)$ and $r_1/\left(r\delta t\right)$ in order to preserve the form (4.15) at the right hand side. Thus two values are allowed to $H$ . Note that if $r_1<r^{\prime }<r_2$ , i.e. $r^{\prime }$ is inside $\delta r$ , then both $H_1$ and $H_2$ are positive. If instead $r^{\prime }<r_1$ , i.e. $r^{\prime }$ is outside $\delta r$ , then in principle (4.14) still holds with $H=H_2-H_1$ and $H_1>0$ ; in fact it means that $H$ splits into two values $H_2$ and $H_1$ greater and smaller than the initial $H$ . In summary, simply considering the possible ways (H09) or (4.14) to regard $r^{-1}\delta r$ , from the initial function $H\left(r,t\right)$ of (4.9) have been inferred from (4.1) without additional hypotheses two functions $H_2$ and $H_1$ of (4.15) formally equivalent to (H00) and concurring to $H_u$ of (4.16).

4.2. Corollary 2

Equation (4.11) reeads

$\frac{\delta A}{A}=2\frac{\delta r}{r}\Rightarrow \frac{A\delta A}{A_0^2}=2\frac{A^2}{A_0^2}\frac{\delta r}{r}$

being $A$ the arbitrary surface concerned by $4\pi r^2$ ; the second equation is inferred via trivial algebraic steps after having multiplied both sides of the first equation by $A^2/A_0^2$ , with $A_0$ constant surface. Since $A\delta A=\delta \left(A^2\right)/2$ it follows next

$\delta \frac{A^2}{4A_0^2}=\frac{A^2}{A_0^2}\text{log}\frac{r}{r_0}\Rightarrow \delta \frac{A^2}{4A_0^2}=\frac{1}{2}\frac{A^2}{A_0^2}\text{log}\frac{r^2}{r_0^2}=\frac{1}{2}\frac{A^2}{A_0^2}\text{log}\frac{A}{A_0}.$

Define $\delta \left(A^2\right)$ as $A^2-A_0^2$ , so that the left hand side becomes $A^2/A_0^2-const$ ; thus the last result reads

$\frac{A^2}{4A_0^2}-const=\frac{1}{2}\frac{A^2}{A_0^2}\text{log}\frac{A}{A_0}.$

If $const\ll A^2/4A_0^2$ , then one finds

$\frac{A^2}{4A_0^2}\approx \frac{A}{A_0}\frac{S}{2}\text{i}.\text{e}.\frac{A}{2A_0}\approx SS=\frac{A}{A_0}\text{log}\frac{A}{A_0}.$ (4.17)

The last equation identifies $S$ , whatever the initial $\delta r/r$ and $A_0$ might be. In the particular case where $r\equiv r_{bh}$ and $A_0$ is of the order of the Plank length², then (4.17) is compatible with the Hawking entropy

$\frac{A}{4{\ell }_{Pl}^2}=S{A}_0=2{\ell }_{Pl}^2.$

Although this topic is outside the purposes of this paper, (4.17) has been shortly sketched owing to its conceptual importance. For sake of brevity is also waived the chance of implementing (4.12) in an analogous way, as strongly suggested by this result; exploring the problem of black body volume entropy is certainly an amazing extension of the Hawking result. It would be interesting to extend this reasoning to (4.12) too.

4.3. Corollary 3

From (4.2) in radial form write $\delta r\delta p_r=\delta \epsilon \delta t$ via (3.3) and (3.12)

$\delta r\delta \frac{m{v}_r}{\beta }=n\hslash =\delta \epsilon \delta t\iff v_r\delta \frac{m{v}_r}{\beta }=n\hslash \omega =\delta \epsilon v_r=\frac{\delta r}{\delta t}$ (4.18)

Next note that the differential $m{v}_r\delta \left(v_r/\beta \right)=\delta \left(m{c}^2/\beta \right)$ , so that (3.23) yields

$\delta \frac{m{c}^2}{\beta }=\delta \epsilon \Rightarrow \frac{m{c}^2}{\beta }=\epsilon +const.$ (4.19)

This result, coherent with (3.12), identifies $\epsilon$ as the energy corresponding to the given definition of momentum $p_r$ simply linking the respective $\delta p_r$ and $\delta \epsilon$ . Regard now this preliminary reasoning about the correlation between $p_r$ and $\epsilon$ as a significant hint to infer a new and more interesting result: the correlation between kinetic energy $\epsilon$ and gravitational potential $\phi$ , due to the position of a body in a gravitational field. In principle this task is reasonable as $\epsilon$ and $\phi$ are two forms of energy a body can possess. Try thus to relate $m\phi$ to the analytical form of the gravitational field acting on the body replacing $p_r$ with $m\phi f$ , where $f$ is proportionality factor allowing $\epsilon$ to be expressed as a function of $m\phi$ . In other words, the previous correlation $p_r\leftrightarrow \epsilon$ via $\delta p_r$ and $\delta \epsilon$ reads now $\epsilon \leftrightarrow m\phi f$ via $\delta \epsilon$ and $\delta \left(m\phi f\right)$ . Repeating the same steps of (4.18) and (4.19), rewrite

$\delta \frac{m{c}^2}{\beta }=\frac{n\hslash }{\delta t}=\delta \left(m\phi f\right);$

next, since $\phi ={\beta }^2{c}^2/2+const$ owing to (3.5), explain $\delta \left(m{c}^2/\beta \right)=\delta \left(m\phi f\right)$ as

$\frac{m{c}^2}{\beta }+cons{t}^{\prime }=\frac{m{c}^2{\beta }^{2}f}{2}\Rightarrow f=\frac{2}{{\beta }^3}+\frac{2cons{t}^{\prime }}{{\beta }^{2}m{c}^2}.$ (4.20)

To check the validity of this reasoning, it is useful to expand $f$ in series of $v$ around $v=0$ ; owing to (3.3) trivial calculations yield

$f=k_0+k_1{v}^2+k_2{v}^4+\cdots \Rightarrow m\phi f=U={\xi }_0-{\xi }_1{v}^2+{\xi }_2{v}^4+{\xi }_3{v}^6+\cdots ,$ (4.21)

being ${\xi }_j$ the coefficients of the series. This result is more conveniently expressed in the equivalent form

$\frac{U_r}{F_{Pl}}=cons{t}_0+r_0\left(-\frac{{\xi }_1}{r}+\frac{{\xi }_2}{r^2}+\cdots \right)F_{Pl}=\frac{c^4}{G}{v}^2=\frac{mG}{r},$ (4.22)

as in fact it has been already found in (2.28).

4.4. Corollary 4

Consider (3.17)

$p-p_0=\hslash \nabla \frac{\phi }{c^2}\Rightarrow \frac{\delta p}{\delta t}=-F=-\frac{\hslash }{\delta t}\nabla \frac{\phi }{c^2}=\nabla U:$

since $\nabla$ operates on the space coordinates only, one would expect classically $F=-\nabla U$ with $U=\hslash v^2/2c^2\delta t$ an arbitrary constant apart. Thus $U$ should be defined by $v^2$ only, which however is inconsistent with (2.31). In fact $\delta t$ must be regarded as $\delta t=\delta x\delta p_x/\delta \epsilon$ , so that this last step is wrong; in principle this disagreement excludes the idea of absolute time, independent upon the coordinate system where are defined the space coordinates. In other words it is true that not necessarily holds $F=-\nabla U$ in general; however, even though this condition is actually fulfilled in a conservative force field, $\delta t^{-1}\nabla \phi \ne \nabla \left(\phi /\delta t\right)$ does not contradict $U$ of (2.31).

Before commenting these results, note eventually the further chance of defining $\beta$ via (3.3) and (2.15) as

${\beta }^2=const+\frac{2\phi }{c^2}=const-\frac{2mG}{c^{2}r}=const-\frac{r_{bh}}{r}{r}_{bh}=\frac{2mG}{c^2}$ (4.23)

which implies according to (3.11) and (2.15)

$\beta =\frac{t_0}{t^{\prime }}=\sqrt{1-\frac{r_{bh}}{r}};$ (4.24)

As $t^{\prime }$ is equal to $t_0$ at $r\to \infty$ and at $m\to 0$ , whereas $t^{\prime }$ is the time expressed at finite distance from $m$ , this result accounts for the gravitational time dilation when replacing $v$ as a function of (4.22). It also appears that the local time at $r=r_{bh}$ is $t_0=0$ .

Hitherto $v$ and $c$ defining $\beta$ have been regarded separately, e.g. to obtain (4.24). Regard now the ratio $c/v$ with the physical meaning of refraction index $\text{n}$ of an e.m. wave propagating at speed $v<c$ through a transparent medium. In this case $v/c\le 1$ is concerned by $1/\text{n}\ge 1$ , while $\beta$ reads

$\beta =\sqrt{1-\frac{v^2}{c^2}}=\sqrt{1-{\sin }^2\theta }=\cos \theta \sin \theta =\frac{v}{c}:$

i.e. $0\le v\le c$ for $0\le \theta \le \pi /2$ . Therefore $\text{n}=c/v$ implies for two different media 1 and 2 through which the beam propagates

$\text{n}\sin \theta =1\Rightarrow {\text{n}}_1\sin {\theta }_1={\text{n}}_2\sin {\theta }_2.$

Appears in this result the Snell refraction law, which can be extended to the reflection law if ${\text{n}}_1={\text{n}}_2$ , as it is known. Try thus to express $\beta$ as a function of this physical parameter, the refractive index $\text{n}=c/v\ge 1$ ; if so, express then $\phi /c^2=-1/2{\text{n}}^2=-\beta /2\sqrt{{\text{n}}^2-1}$ and $\beta =\sqrt{{\text{n}}^2-1}/\text{n}$ . Even the basic laws of geometrical optics fit the conceptual frame of the previous sections.

It is worth emphasizing at this point that the Equations (3.22), (4.24), and (4.22) go well beyond the mere domain of the special relativity; the freedom of introducing and implementing $v\left(x,y,z,t\right)$ not necessarily constant to define $\beta$ implies results typical of the general relativity, in agreement with the ability of $\beta$ to account for the space time deformation evidenced in (CCC) and (4.29) and (4.32). In this respects appear significant the steps from (3.13) to (3.14). In this respect are significant the next three subsections.

4.5. Corollary 5

As concerns the uncertainty ranges of (4.1) note that hold the notations

$\delta \sqrt{ϵ}=\frac{\delta ϵ}{2\sqrt{ϵ}}\delta \sqrt{x}=\frac{\delta x}{2\sqrt{x}}$

and so on for the other ranges. Then trivial algebraic manipulations yield identically the following notations

$\delta x\delta p_x=n\hslash =\delta ϵ\delta t\iff \delta \sqrt{x}=\frac{\delta x}{2\sqrt{x}},\delta \sqrt{ϵ}=\frac{\delta ϵ}{2\sqrt{ϵ}}2\sqrt{x}\delta \left(\sqrt{x}\right)\delta p_x=n\hslash =2\sqrt{ϵ}\delta \left(\sqrt{ϵ}\right)\delta t$

which in turn imply

$\delta p_x=\frac{n\hslash }{\delta x}=\frac{\delta \left(\sqrt{ϵ}\right)}{\delta \left(\sqrt{x}\right)}\sqrt{\frac{ϵ}{x}}\delta t.$

Let $ϵ$ be defined as the energy of two charges at rest; then

$ϵ=\frac{e^2}{x}\sqrt{\frac{ϵ}{x}}=\sqrt{\frac{e^2}{x^2}}=\pm \frac{e}{x}\Rightarrow \delta p_x=\frac{n\hslash }{\delta x}=\pm \frac{\delta \left(\sqrt{e^2/x}\right)}{\delta \left(\sqrt{x}\right)}\frac{e}{x}\delta t.$

So the replacement $\delta p_x\to -\delta p_x$ implies $\delta x\to -\delta x$ . This does not require $x\to -x$ , but $x_2-x_1\to x_1-x_2$ only; i.e. the mirror image of $\delta x$ implies the way of regarding the choice of the upper and lower range boundaries, not the signs of the respective coordinates. Regard thus the last equation as

$\frac{n\hslash }{\delta x}=\pm e\left(\frac{\delta \left(\sqrt{e^2/x}\right)}{\delta \left(\sqrt{x}\right)}\frac{1}{x}\right)\delta t;$ (4.25)

Consider this relationship with either sign, e.g. positive; then $\delta x\to -\delta x$ requires $e\to -e$ and concurrently $\delta t\to -\delta t$ to leave unchanged the whole equation. This is the quantum CPT theorem.

4.6. Corollary 6

Introduce a surface element $\text{d}A=\text{d}{x}^{\prime }\text{d}x$ defined by two infinitesimal lengths $\text{d}x$ and $\text{d}{x}^{\prime }$ in an arbitrary reference system $R$ ; this value of $\text{d}A$ is uniquely determined at any time $t$ with respect to an initial time $t_0$ . Implement now a non-deterministic quantum approach, where is definable a finite interval $\delta A$ of surface values where $\text{d}x$ and $\text{d}{x}^{\prime }$ are replaced by finite ranges $\delta x$ and $\delta x^{\prime }$ non-uniquely fixed: it is enough to consider an arbitrary number of elementary surface elements $\text{d}{A}_j=\text{d}{x}_j\text{d}{x^{\prime }}_j$ defining $\delta x={\displaystyle {\sum }_i\text{d}{x}_j}$ and $\delta x^{\prime }={\displaystyle {\sum }_j\text{d}{x^{\prime }}_j}$ such that $\delta A={\displaystyle {\sum }_j\text{d}{x}_j\text{d}{x^{\prime }}_j}=\delta x\delta x^{\prime }$ at the current time range $\delta t=t-t_0$ . Write anyway

$\frac{\delta A}{\delta x^{\prime }}=\delta x.$ (4.26)

Consider now the further chance that, for any reason, $\delta x$ and $\delta x^{\prime }$ are themselves time dependent: e.g. because the universe expands, so that any length linking two space coordinates is subjected to change itself as a function of time. This implies the necessity of introducing $\delta \left(\delta A\right)$ to account for the time dependence of variable $\delta A\left(t\right)$ . For simplicity, but without loss of generality, it is enough to express the time dependence of $\delta A$ via either $\delta x^{\prime }$ or $\delta x$ only; also, in this general framework, it is possible to assume the initial $\delta x=\delta x^{\prime }$ at $t=t_0$ .

These considerations plug $\delta A$ into a general context compliant with the formalism of the uncertainty, which implies the necessity of introducing $\delta \left(\delta A\right)$ and $\delta \left(\delta x\right)$ to account for the time dependence of $\delta A$ and $\delta x$ upon $\delta t$ once having introduced variable range sizes; via $\delta x=n\hslash /\delta p_x$ and $\delta t=n\hslash /\delta \epsilon$ calculate thus

$\delta {\dot{x^{\prime }}}_{\pm }=\pm \left|\frac{{\delta x^{\prime }|}_{t=t_2}-{\delta x^{\prime }|}_{t=t_1}}{\delta t}\right|\delta t=t_2-t_1\Rightarrow \delta \left(\delta x^{\prime }\right)={\delta x^{\prime }|}_{t=t_2}-{\delta x^{\prime }|}_{t=t_1}=\delta {\dot{x^{\prime }}}_{\pm }\delta t$(4.27)

where $\delta \dot{x^{\prime }}$ is the rate at which changes $\delta x^{\prime }$ during $\delta t$ . In an analogous way define the deformation rate $\delta \left(\delta A\right)$ of $\delta A$ during the time lapse $\delta t$ as $\delta \left(\delta A\right)=\left(\delta t\delta \dot{x^{\prime }}\right)\delta x$ having put $\delta x^{\prime }=\delta t\delta \dot{x^{\prime }}$ . The notation emphasizes that in fact the sign of $\delta \left(\delta x^{\prime }\right)$ depends on whether ${\delta x^{\prime }|}_{t=t_2}⋛{\delta x^{\prime }|}_{t=t_1}$ , in principle possible because ${\delta x^{\prime }|}_{t=t_2}$ and ${\delta x^{\prime }|}_{t=t_1}$ are both arbitrary. So

$\delta A=\delta x\delta x^{\prime }=\frac{{\left(n\hslash \right)}^2\delta \dot{x^{\prime }}}{\delta p_x\delta \epsilon }$ (4.28)

yields

$\delta \left(\delta A\right)=\delta x\delta {\dot{x^{\prime }}}_{\pm }\delta t\delta \left(\delta A\right)⋛0.$ (4.29)

So $\delta \left(\delta A\right)$ is related to and justified by the change of energy and momentum ranges corresponding to $\delta \left(\delta x^{\prime }\right)$ during $\delta t$ . This means that the change $\delta \left(\delta A\right)$ of $\delta A$ describes in general local stretching or shrinking of the initial surface range $\delta A$ , both in principle intuitively acknowledged and physically allowed by the uncertainty.

In general (4.26) and (4.29) yield

$\delta x=\frac{\delta \left(\delta A\right)}{\delta t\delta {\dot{x^{\prime }}}_{\pm }}=\frac{\delta }{\delta t}\frac{\delta A}{\delta {\dot{x^{\prime }}}_{\pm }}\delta x=\frac{\delta A}{\delta x^{\prime }}$ (4.30)

As the left hand side of this result takes both signs according to either stretching or shrinking rate $\delta \dot{x^{\prime }}$ of $\delta x^{\prime }$ , the two chances are

$\frac{\delta }{\delta t}\frac{\delta A_{+}}{\delta \dot{x^{\prime }}}=\frac{\delta A_{+}}{\delta x^{\prime }}\iff -\frac{\delta }{\delta t}\frac{\delta A_{-}}{\delta \dot{x^{\prime }}}=\frac{\delta A_{-}}{\delta x^{\prime }}.$ (4.31)

Compare eventually these results multiplying both sizes by a constant $m_0/t_0^2$ : the respective equations read in general as linear combinations of energies after having converted $A_{{}_{+}}$ into $ℒ=m_0{A}_{+}/t_0^2$ and $A_{-}$ into $\mathscr{H}=m_0{A}_{-}/t_0^2$ , i.e.

$\frac{\delta }{\delta t}\frac{\delta ℒ}{\delta \dot{x^{\prime }}}=\frac{\delta ℒ}{\delta x^{\prime }}\iff \frac{\delta }{\delta t}\frac{\delta \mathscr{H}}{\delta \dot{x^{\prime }}}+\frac{\delta \mathscr{H}}{\delta x^{\prime }}=0.$ (4.32)

So $ℒ$ and $\mathscr{H}$ differ by the sign of the potential energy related to the kinetic energy; the fact that changing the sign of the potential term the equations coincide, $\delta \mathscr{H}/\delta x=U=-\delta ℒ/\delta x$ , suggests that the linear combination of energies is consistent with $ℒ=T-U$ and $\mathscr{H}=T+U$ . In other words, once having defined $T$ and $U$ , (4.32) introduces the appropriate formulas to correlate $\delta \left(\delta x\right)$ and $\delta \left(\delta A\right)$ to $\delta ϵ$ of the uncertainty equation.

The crucial point is that these results have been contextually obtained by deforming the space time, as evidenced by $\delta \left(\delta x^{\prime }\right)$ and $\delta \left(\delta A\right)$ of (4.27) and (4.29).

4.7. Corollary 7

For the purposes of this section, remind first that (3.14) reads

${\beta }^2{ϵ}^2={ϵ}^2-{\left(pc\right)}^2={\left(m{c}^2\right)}^{2}p=\frac{ϵv}{c^2}ϵ=\frac{m{c}^2}{\beta }\beta =\sqrt{1-\frac{v^2}{c^2}}{p}_{0}c=m{c}^2=\beta ϵ.$ (4.33)

and that

${\left(c\beta \delta t\right)}^2={\left(c\delta t\right)}^2-\delta r^2\delta r=v\delta t.$ (4.34)

These are the standard equation and the invariant interval of the special relativity. Repeat now an identical reasoning but considering a different $\beta$ . The same steps yield

${\epsilon }^2={{\beta }^{\prime }}^2{\left(m{c}^2\right)}^2{\beta }_{\pm }=\sqrt{1-\frac{v^2}{c^2}\pm \frac{{v^{\prime }}^2}{c^2}}\Rightarrow {\epsilon }^2={\left(pc\right)}^2+{\left(m{c}^2\right)}^2\pm \frac{{v^{\prime }}^2}{c^2}{\left(m{c}^2\right)}^2$ (4.35)

and thus, subtracting side by side (4.36) and (4.33),

${\epsilon }^2={ϵ}^2\pm \frac{{v^{\prime }}^2}{c^2}{\left(m{c}^2\right)}^2.$ (4.36)

The reason of having quoted (4.33) and (4.36) is that (3.17), (3.1) and (3.14) yield

$p=-\hslash \nabla \frac{\phi }{c^2}-const=\hslash \frac{\dot{v}}{c^2}-p_{0}const=\frac{\hslash }{c^2}{a}_0=p_0\Rightarrow p+p_0=\hslash \frac{\dot{v}}{c^2},$ (4.37)

whence, squaring both sides of the last equation,

${\hslash }^2\frac{{\dot{v}}^2}{c^4}=p^2+p_0^2+2p_0\cdot p\Rightarrow {\hslash }^2\frac{{\dot{v}}^2}{c^2}={\left(pc\right)}^2+{\left(p_{0}c\right)}^2+2p_0\cdot p{c}^2$ (4.38)

and then

${\epsilon }^2={\left(pc\right)}^2+{\left(m{c}^2\right)}^2+2p_{0}p{c}^2\cos \theta {\epsilon }^2={\hslash }^2\frac{{\dot{v}}^2}{c^2}m{c}^2=p_{0}c:$ (4.39)

i.e. one finds again the standard (4.33) plus a correction term $2p_{0}p{c}^2\cos \theta =2m{c}^{2}pc\cos \theta$ likewise as in (4.36).

The first step to guess $\cos \theta$ is the rational definition

$\cos \theta =\pm \frac{\epsilon p}{{\epsilon }_{\theta }{p}_{\theta }+\epsilon p}{\epsilon }_{\theta }{p}_{\theta }=\frac{{\hslash }^2}{{\tau }_{\theta }{\ell }_{\theta }},$ (4.40)

which reproduces the expected range of values for $\epsilon p\to 0$ and $\epsilon p\to \infty$ ; the second equation defines the space and time scales of $\cos \theta$ , e.g. $\cos \theta \to 0$ for ${\ell }_{\theta }\to 0$ or ${\tau }_{\theta }\to 0$ , in which case (4.39) tends to the standard (4.33) of the special relativity. Then, assuming ${\tau }_{\theta }{\ell }_{\theta }=const$ i.e. a constant scale factor of the correction term, one finds

$2p_{0}p{c}^2\cos \theta =\pm 2p_0{C}_{\theta }\frac{\epsilon {\left(pc\right)}^2}{1+\epsilon p{C}_{\theta }}{C}_{\theta }=\frac{{\tau }_{\theta }{\ell }_{\theta }}{{\hslash }^2};$ (4.41)

thus replacing in (4.39) the result is

${\epsilon }^2={\left(pc\right)}^2+{\left(m{c}^2\right)}^2\pm 2p_0{C}_{\theta }\frac{\epsilon {\left(pc\right)}^2}{1+\epsilon p{C}_{\theta }}.$ (4.42)

In conclusion (4.39) reads

${\epsilon }^2-{\left(p^{\prime }c\right)}^2={\left(m{c}^2\right)}^2{p}^{\prime }=p{f}_{\pm }=p\sqrt{1\pm 2p_0{C}_{\theta }\frac{\epsilon }{1+\epsilon p{C}_{\theta }}}$ (4.43)

Here the crucial parameter to infer the standard (4.33) from the modified (4.39) is $C_{\theta }$ : if $C_{\theta }\to 0$ , then $f_{\pm }\to 1$ so that $p^{\prime }\to p$ implies $\epsilon \to ϵ$ , which in turn requires ${\beta }_{\pm }\to \beta$ in (4.35) i.e. $v^{\prime }\to 0$ in (4.36). In other words the crucial parameter to link (4.33) and (4.39), the scalar $p\cdot p_0$ , is actually the scale factor $C_{\theta }$ via the space and time factors ${\ell }_{\theta }$ and ${\tau }_{\theta }$ ; the fact that reasonably the correction factor of the special relativity should be small suggests the quantum scale of $C_{\theta }$ and thus that ${\ell }_{\theta }~{\ell }_{Pl}$ and ${\tau }_{\theta }~t_{Pl}$ . The endpoint of this reasoning is that $ϵ$ of the standard special relativity splits because of ${\beta }_{\pm }\ne 0$ into two energy levels (4.36) due to $v^{\prime }\ne 0$ , i.e.

${\epsilon }^2=\begin{array}{l}\nearrow \hfill \\ \searrow \hfill \end{array}\begin{array}{l}{ϵ}^2+\frac{v{\text{'}}^2}{c^2}{\left(m{c}^2\right)}^2\hfill \\ \text{}\text{}\hfill \\ {ϵ}^2-\frac{v{\text{'}}^2}{c^2}{\left(m{c}^2\right)}^2\hfill \end{array}.$ (4.44)

On the one hand is evident the formal analogy between (4.33) of the special relativity and (4.43): in fact their difference is due to $\beta \ne {\beta }_{\pm }$ only, in agreement with the hint of the corollary 1: here it appears again that the mere fact of modifying $\beta$ implies the transition from special relativity to a more general context that can be nothing but general relativity. In effect (4.42) is known in the literature [5], it is acknowledged as a relativistic equation of quantum gravity having cosmological relevance.

One must conclude that in fact (4.33) is an equation of Euclidean flat space of the special relativity, whereas (4.42) is an equation of a non-Euclidean curved space of the general relativity. Note that, owing to (3.5), $v$ defining $\beta$ is arbitrary; however, whatever its initial value might be, is crucial the change $\delta \beta ={\beta }_{\pm }-\beta$ . In other words one must conclude that changing the initial $\beta$ to another value ${\beta }_{\pm }$ the gap $v\leftrightarrow v^{\prime }$ reflects the changing of curvature of the space time. The question arises therefore about the physical meaning of (4.35) in generalizing the special relativity: this is concerned in the next subsection.

4.8. Corollary 8

The previous results rise the question: do $\nabla \beta$ and $\delta \beta /\delta r$ have really to do with the space time curvature? The answer is positive. Consider the geometrical definition of radius of curvature $r_c={\left|v\right|}^3/\left|v\times \dot{v}\right|$ , whence, expressing $\dot{v}$ via (3.1) and (3.17),

$ℛ=\frac{\left|v\times \dot{v}\right|}{{\left|v\right|}^3}\epsilon v=\hslash \left(a_0+\dot{v}\right)\Rightarrow ℛ=\frac{\left|v\times \dot{v}\right|}{{\left|v\right|}^3}=\frac{\left|\left(a_0+\dot{v}\right)\hslash /ϵ\times \dot{v}\right|}{{\left|\left(a_0+\dot{v}\right)\hslash /ϵ\right|}^3}=\frac{\left|-a_0\times \nabla {\left(c\beta \right)}^2\right|}{{\left|a_0-\nabla {\left(c\beta \right)}^2\right|}^3}\frac{{ϵ}^2}{{\hslash }^2}.$(4.45)

So (4.45) is the answer to the previous question: changing $\beta$ means changing $ℛ$ . Regard now $\nabla \left({\beta }^2\right)$ as $\delta \left({\beta }^2\right)/\delta r$ and thus with finite $\delta \left({\beta }^2\right)$ given by ${\beta }^2-{{\beta }^{\prime }}^2$ across the finite uncertainty range $\delta r$ . Of course $v$ is not deterministic ratio of local infinitesimal $\partial s/\partial t$ , but ratio of finite uncertainty ranges $\delta s/\delta t$ not related to any specific reference system. Without need of calculations, (4.45) shows that changing $\beta$ means changing the local curvature $ℛ$ of the space time. Is crucial the fact that $a_0\ne 0$ , otherwise $ℛ=0$ would mean flat space time. The reason is clear: dividing by $c^2$ the second (4.45) reads $p=p_0+ϵ\delta v/c^2$ , being $p_0=\hslash a_0/c^2$ and $ϵ=\hslash /\delta t$ . So $a_0\ne 0$ allows defining $\delta p$ and then writing $\delta p=m\delta v/\beta$ , with $m/\beta =ϵ/c^2$ .

Note that $\nabla {\left(\beta c\right)}^2=2\nabla \phi$ since $\phi =-v^2/2$ and $c^2{\beta }^2=c^2-v^2$ ; so the right hand side of the last $ℛ$ yields

$ℛ=\frac{\left|-2a_0\times \nabla \phi \right|}{{\left|a_0-2\nabla \phi \right|}^3}\frac{{ϵ}^2}{{\hslash }^2}=\frac{\left|-a_0\times \nabla /2m\phi \right|}{{\left|m{a}_0-\nabla \left(2m\phi \right)\right|}^3}\frac{{ϵ}^2}{{\hslash }^2}.$

In particular $\dot{\phi }$ is related via $H_u$ to the universe expansion and thus to the change of its time dependent curvature, which in turn governs gravitational effects. The following step provides an intuitive explanation of $ℛ$ :

$ℛ=\frac{{\left|vt\right|}^3}{\left|vt\times \dot{v}{t}^2\right|}=\frac{{\left|s\right|}^3}{\left|s\times s^{\prime }\right|}=\frac{V}{A}.$

The conceptual background hitherto exposed justifies the attempt to address two cosmological problems of crucial importance, the dark matter and dark energy.

5. Cosmological Data

In this paper are implemented cosmological data introduced in [6], which in turn have been inferred from the estimates quoted in [7] and reported for convenience in Table 1.

$t_u=8.7\times {10}^{17}\text{s}{r}_u=4.35\times {10}^{26}\text{m}{M}_u=5.9\times {10}^{53}\text{kg}{m}_{ob}=3\times {10}^{52}\text{kg}$ (5.1)

being $t_u$ and $r_u$ universe’s age and radius and $M_u$ universe total mass; $m_{ob}$ counts only the mass of observable stars, whose light has reached our point of

Table 1. Estimated cosmological data reported in [7] of today’s universe.

Property of present day observable universe	Approximate number of Planck units	Equivalents
Age	8.08 × 1060 tp	4.35 × 1017 s or 1.38 × 1010 years
Diameter	5.4 × 1061 Ip	8.7 × 1026 m or 9.2 × 1010 light-years
Mass	approx. 1060 mp	3 × 1052 kg or 1.5 × 1022 solar masses (only counting stars) 10 80 protons (sometimes known as the Eddington number)
Density	1.8 × 10−123 mp∙lp−3	9.9 × 10−27 kg∙m−3
Temperature	1.9 × 10−32 Tp	2.725 K temperature of the cosmic microwave background radiation
Cosmological constant	10−122 lp−2	10−52 m−2
Hubble constant	10−61 tp−1	10−18 s−1 102 (km/s)/Mpc

sight, whereas $M_u$ is calculated through $r_u$ as

$r_u=\frac{M_{u}G}{c^2}.$ (5.2)

Note that these values are slightly different from the mere literature estimates reported in Table 1,whereas in particular $t_u$ is about twice the value usually regarded as today’s age of universe; this discrepancy, explained in the quoted paper, agrees with the value calculated in [8]. Also,

$\Lambda =1.6\times {10}^{-53}{\text{m}}^{-2}{\Lambda }_t=c^2\Lambda =1.4\times {10}^{-36}{\text{s}}^{-2},$ (5.3)

being $\Lambda$ the Einstein cosmological constant. Eventually, are useful the density ${\rho }_{vac}$ and energy density ${\eta }_{vac}$ of quantum vacuum calculated in [6]

${\rho }_{vac}=58\times {10}^{-28} \frac{\text{kg}}{{\text{m}}^{\text{3}}}{\eta }_{vac}=5.2\times {10}^{-10}\frac{\text{J}}{{\text{m}}^{\text{3}}}$ (5.4)

in agreement with literature data [9]

${\rho }_{vac}=\left(60.3\pm 1.3\right)\times {10}^{-31}\frac{\text{g}}{{\text{cm}}^{\text{3}}}{\eta }_{vac}=\left(5.4\pm 0.1\right)\times {10}^{-9}\frac{\text{erg}}{{\text{cm}}^{\text{3}}}.$ (5.5)

Is attracting the chance that somehow these cosmological parameters can be rationally organized by means of and in agreement with the fundamental constants of nature. In fact it is easy to verify that all these data result in a coherent set of self-consistent cosmological parameters: from a mere numerical point of view, (5.1) and (5.3) fulfill the following numerical relationships:

$\Lambda =\frac{1}{{\left(c{t}_u\right)}^2}=1.5\times {10}^{-53}{\text{m}}^{-2}$ (5.6)

and the correlations

$\sqrt{{\Lambda }_t}=\sqrt{\Lambda c^2}=1.2\times {10}^{-18}{\text{s}}^{-1}\sqrt{{\Lambda }_t}=\frac{1}{t_u}=1.1\times {10}^{-18}{\text{s}}^{-1}.$ (5.7)

Moreover, the importance of (3.2) appears considering the following numerical coincidences that correlate the cosmological parameters directly to $t_u$ ; (5.1) and (5.3) yield

$\left|v\cdot v\right|=\left|\dot{\phi }\right|\Rightarrow \frac{1}{t_u^3\Lambda }=0.1,\frac{c^2}{t_u}=0.1,c^3\sqrt{\Lambda }=0.11,\frac{c}{t_u^2\sqrt{\Lambda }}=0.1\frac{{\text{m}}^2}{{\text{s}}^3}.$ (5.8)

These values cannot be merely accidental, rather they reveal significant links with $\Lambda$ , which becomes fundamental feature of the universe. If this is true, then these values must be someway involve $G$ . Indeed, as concerns (5.1) and (5.3) it is worth noticing that owing to (5.6)

$r_u=\frac{M_{u}G}{c^2}=M_{u}G\Lambda t_u^2\Rightarrow G=\frac{r_u/\Lambda }{M_u{t}_u^2}=\frac{lengt{h}^3}{mass\times tim{e}^2}=6.1\times {10}^{-11},$ (5.9)

as in fact

$M_{u}G\Lambda t_u^2=4.7\times {10}^{26}\text{m}\leftrightarrow r_u$ (5.10)

with the proposed age of the universe; i.e. the numerical value of $G$ is nothing else but the combination of cosmological parameters which in fact define the value of the gravity constant according to its own dimensional meaning.

Also, (5.6) suggests how to combine the cosmological data in order to define the following pure number

$\frac{\left(M_{u}G\right)\left(\Lambda t_u^2\right)}{r_u}=1.1:$ (5.11)

this result is interesting as it includes total mass, radius and age of the universe via $\Lambda G$ . It appears in (5.7) too.

Note that the set of data (5.5) and (5.3) should be optimized to allow a better self-consistency of (5.9) and (5.11): nevertheless the data concerned here are deliberately still those already reported in the previous paper [6] to show the continuity and consistency of the new results obtained here with the results therein exposed. This holds particularly for $t_u$ : this age of the universe, although considerably different from what is commonly acknowledged, fits several correlations with other cosmological parameters. The values (5.8) to (5.11) are relevant as they link reasonably $t_u$ , the cosmological Einstein parameter $\Lambda$ and the change rate $\dot{\phi }$ of the gravitational potential $\phi$ . Moreover the pure number (5.11) are also consistent with the aforesaid value of $M_u$ , calculated simply via the gravitational radius introduced in (5.2), which fulfills the left hand side equation: i.e. the value of $G$ is nothing else but the combination of cosmological parameters linked by the gravity constant according to their own dimensional meaning. In other words is reasonably true that in (5.9) the cosmological parameters surrogate the meaning of the respective physical dimensions. Note in this respect that

${\eta }_{vac}G=a_0^2\leftrightarrow a_0=1.9\times {10}^{-10}\frac{\text{m}}{{\text{s}}^2}:$ (5.12)

also this result, of interest for the MOND discussed in the next section, fits the age of the universe with the characteristic acceleration $a_0$ .

The validity of (5.2) is verifiable in two ways. Since $M_u$ is total mass, calculate the total surface and volume of the universe via $r_u$ as

$S_u=4\pi r_u^2{V}_u^{ru}=\frac{4\pi }{3}{r}_u^3$

and implement (4.13). Owing to (5.2), one finds

$\frac{M_{u}G}{r_u^2}=\frac{c^2}{r_u}=a_0=2.1\times {10}^{-10}\frac{\text{m}}{{\text{s}}^{\text{2}}}$

and

$\frac{1}{3}\frac{M_u{c}^2}{\left(4\pi /3\right)r_u^3}=\frac{{\eta }_u}{3}=P_u=\frac{F_u}{4\pi r_u^2}\Rightarrow \frac{M_u{c}^2}{r_u}=1.2\times {10}^{44}\frac{\text{N}}{{\text{m}}^{\text{2}}}\text{i}.\text{e}.F_u=F_{Pl}.$ (5.13)

The fact of having implemented the coefficient 1/3 will be explained later. Anyway, it is crucial the fact that both results are recognizable in the global frame of (5.1) to (5.5). Is interesting the fact that ${\eta }_u=1.5\times {10}^{-10}\text{J}/{\text{m}}^{\text{3}}$ is of the order of but considerably smaller than ${\eta }_{vac}$ of (5.4): this result is explained in the next subsection.

6. Dark Matter and Dark Energy

The visible mass $m_{ob}$ , as it results in Table 1 counting the stars only, amounts to $m_{ob}=3\times {10}^{52}\text{kg}$ about. As this value concerns by definition the stars whose light reaches our point of sight, calculate the total star mass $M_{ob}$ considering the radius $r_u=4.35\times {10}^{26}\text{m}$ at the time $t_u=8.7\times {10}^{17}\text{s}$ inferred in [6]. This age of universe is reasonable because it fits the other cosmological parameters (5.8). This result implies an interesting conclusion about the total amount of standard matter in the universe. As the current radius $r_u$ is due to the expansion rate of the universe from the big bang to today’s time $t_u$ around an arbitrarily small initial size, the assumption of statistically homogeneous universe implies a total observable mass $M_{ob}=m_{ob}{\left(r_u/c{t}_u\right)}^3$ . This extrapolated value of observable mass, appropriately definable as ordinary matter, means that a light beam starting from the growth nucleus of universe, wherever it might actually be, reaches the distance $c{t}_u$ smaller than the actual boundary $r_u$ of the universe; indeed $c{t}_u/r_u=0.6$ . So $M_{ob}$ scales up the estimate $m_{ob}$ we can afford by observing and counting the number of stars whose light has in fact reached us. The value of $m_{ob}$ yields

$M_{ob}=1.4\times {10}^{53}\text{kg};$ (6.1)

thus the total mass of the universe $M_u=5.9\times {10}^{53}\text{kg}$ , inferred from the gravitational radius $r_u{c}^2/G$ in agreement with (5.11), still implies a residual mass $M_u-M_{ob}=4.5\times {10}^{53}\text{kg}$ . Try now to calculate the ratios of the masses concerned in this reasoning normalized to the total $M_u$ ; one finds that

$\frac{m_{ob}}{M_u}\frac{M_{ob}}{M_u}=\frac{m_{ob}{\left(r_u/c{t}_u\right)}^3}{M_u}\frac{M_u-\left(m_{ob}+M_{ob}\right)}{M_u}$

yield

$\begin{array}{l}\frac{m_{ob}}{M_u}=\frac{3\times {10}^{52}}{5.9\times {10}^{53}}=0.05\frac{M_{ob}}{M_u}=\frac{1.4\times {10}^{53}}{5.9\times {10}^{53}}=0.24\\ \frac{M_u-M_{ob}-m_{ob}}{M_u}=\frac{4.2\times {10}^{53}}{5.9\times {10}^{53}}=0.71\end{array}$ (6.2)

In fact these ratios are well known: they fit reasonably the current literature estimates that regard however the three ratios as corresponding to different masses: in particular $m_{ob}$ is related to the amount of visible ordinary matter, $M_{ob}$ corresponds reasonably to the value expected for the so called “dark mass” $M_d$ , whereas the last equation should be related by difference to the dark energy $E_{de}/c^2=M_u-M_{ob}-m_{ob}$ . Actually, however, the physical meaning of the ratios appears to be completely different. More simply, $M_{ob}$ is mere extrapolation of $m_{ob}$ : in principle the ratio 0.24 includes the invisible matter of a universe too large to be fully observable, instead of including a new kind of matter whose identification is in fact beyond our current understanding. If this reasoning is true, then one should more appropriately write

$\frac{M_{ob}}{M_u}=\frac{1.4\times {10}^{53}}{5.9\times {10}^{53}}=0.24\frac{M_u-M_{ob}}{M_u}=\frac{4.5\times {10}^{53}}{5.9\times {10}^{53}}=0.76,$ (6.3)

while being

${ℳ}_{de}=M_u-M_{ob}=4.5\times {10}^{53}\text{kg},$ (6.4)

with notation emphasizing that is concerned here not a true mass but $E_{de}/c^2$ .

-On the one hand, (6.3) still supports the validity of total mass of the universe $M_u$ simply inferred from the concept of gravitational radius $r_m=M_{u}G/c^2$ in [6] according to (5.11); anyway (6.3) acknowledges $M_{ob}<M_u$ and implies

$M_{ob}{\Lambda }^{1.5}=9\times {10}^{-27}{a}_0=F_{Pl}/M_u=2\times {10}^{-10}\frac{\text{m}}{{\text{s}}^{\text{2}}}.$ (6.5)

-On the other hand the second equation calculates an excess mass inherent $M_u$ with respect to the ordinary matter mass $M_{ob}$ in the universe; in other words, the concept of dark energy $E_{de}/c^2=M_u-M_{ob}$ is still actual even in the whole universe assumed homogeneous. True physical meaning has therefore the dark energy here expressed as $E_{de}/c^2$ . Is obvious the question about where $E_{de}$ comes from. This point has been already concerned in [6] and shortly summarized below for completeness. Nevertheless $E_{de}$ is not mere residual mass, as it could seem in (6.3), it has its own physical meaning highlighted in the following equations. In agreement with the physical dimensions of $G$ write, in agreement with (5.11),

$\frac{{\Lambda }^{-1.5}}{{ℳ}_{de}{\Lambda }_t^{-1}}=\frac{c^2}{{ℳ}_{de}\sqrt{\Lambda }}=5\times {10}^{-11}\frac{c^4}{G}={ℳ}_{de}{a}_0;$ (6.6)

moreover

$\sqrt{{\Lambda }^{-1}}=2.5\times {10}^{26}\text{m}c{t}_u=2.6\times {10}^{26}\text{m}.$ (6.7)

Also, compare the quantum vacuum energy density (5.4) with that calculable via (6.6) and the universe volume of (V8V)

${\eta }_{vac}=5.2\times {10}^{-10}\frac{{ℳ}_{de}{c}^2}{V_u}=\frac{c^5{t}_u}{V_{u}G}=4.9\times {10}^{-10}\frac{\text{J}}{{\text{m}}^{\text{3}}}:$ (6.8)

this result supports the idea that the dark energy density is in fact the quantum vacuum energy density.

This explains the discrepancy with respect to ${\eta }_u$ of (5.13): i.e. ${\eta }_{vac}$ is not referable to $M_u{c}^2$ , mass energy, but to the vacuum energy having quantum meaning. Therefore the volume to calculate the mass density ${\rho }_u$ as $M_u/\left(4\pi r_u^3/3\right)$ is different fro that to calculate ${\rho }_{de}$ . In fact it is possible to define the volume via $\Lambda$ recalling its physical dimensions: i.e. as $V_u^{\Lambda }=\left(4\pi /3\right){\Lambda }^{-1.5}=6.5\times {10}^{79}{\text{m}}^3$ . So

$\frac{{ℳ}_{de}{c}^2}{V_u^{\Lambda }}=6.2\times {10}^{-10}\frac{\text{J}}{{\text{m}}^{\text{3}}},$ (6.9)

which fits reasonably ${\eta }_{vac}$ ; moreover, implementing (4.13), calculate

$\frac{1}{3}{\eta }_{vac}=\frac{F_{de}}{4\pi /\Lambda }\Rightarrow F_{de}=\frac{4\pi }{3}\frac{{\eta }_{vac}}{\Lambda }=1.4\times {10}^{44}.$ (6.10)

Thus once more $F_{de}=F_{Pl}$ like in (5.13). It is known in general that in (4.13) the coefficients $k=1$ or $k=2$ depend on whether or not the matter in the volume $V$ where is defined $\eta$ bounces back or not at the enclosing surface $S$ . In principle such a conservative situation could be fulfilled in a black hole universe. Unfortunately this does not seem the case. Calculating

$r^{\text{*}}=\frac{2M_{ob}G}{c^2}=2.1\times {10}^{26}\text{m};$ (6.11)

one concludes that the actual $r_u>r^{\text{*}}$ makes $M_{ob}$ insufficient to meet the threshold value to fulfill $r^{\text{*}}\equiv r_{bh}$ ; i.e. $M_{ob}$ should be twice its actual value. In effect, (6.10) and (5.13) are consistent assuming both $k=1$ . The coefficient $k=1$ of (4.13) accounts for the lack of any particle bouncing back at the boundary of the volume $V_u^{de}$ . It appears that the immaterial length $\sqrt{{\Lambda }^{-1}}$ implies the internal pressure $P_{de}$ acting inside $V_u^{ct}$ and for the vacuum energy density ${\eta }_{vac}$ . If so, then the fact of having assumed ${\eta }_{vac}\equiv {\eta }_{de}$ implies that $P_{de}\equiv P_{vac}$ keeps swelled the universe; also, the value of the active force inside $V_u^{\text{Λ}}$ corresponds to the Planck force. Note that four consequences are implied by these achievements:

-The dark matter does not exist, it is ordinary matter $M_{ob}$ that extrapolates the visible $m_{ob}$ , whereas the dark energy is nothing else but vacuum energy.

-the MOND option becomes now essential to explain the rotation of galaxies: this conceptual frame is credible if able to infer and explain the MOND approach too, which in effect has been already introduced in (2.31).

-the numerical value of ${ℳ}_{de}$ , initially introduced as the mere difference in (6.2), can be identified itself in a self-consistent way.

-Is crucial for these conclusions the given value $t_u$ of universe age.

These results are further concerned in the next sections. . The Hubble tension It has been shown in [4] that actually the Hubble factor consists of two separated values around an approximate average $\overline{H_u}$ ; this point has been shortly explained in (4.16) of section 3.1; for completeness here follows a short remark on the two allowed values of the Hubble function. Two equations are necessary to calculate $H_1$ and $H_2$ ; assuming by definition $H_2>H_1$ , consider the uncertainty equation (4.1) at the today time $t=t_u$ of (5.1). Since $H_u$ has physical dimensions of $tim{e}^{-1}$ , it is reasonable to define $n\hslash =\delta t_u\delta {\epsilon }_u$ putting $\delta {\epsilon }_u=h\delta H_u$ with $h\delta H_u\delta t_u=n\hslash$ and $\delta t_u=t_u-t_{bb}$ , being $t_{bb}$ the big bang time. It is of interest here to put $\delta H_u=H_2-H_1$ ; moreover, as $t_{bb}$ is in fact the beginning of space and time, put without loss of generality $t_{bb}=0$ with respect to which is defined $t_u$ . The uncertainty takes the form

$H_2-H_1=\frac{n}{2\pi t_u}.$ (6.12)

The second equation is guessed thinking that at the cosmological scale $H\approx t^{-1}$ , having approximated $1/\delta t\approx 1/t$ . Then the second equation has the form of a boundary condition for $H_1^{-1}$ and $H_2^{-1}$ , i.e.

$\frac{1}{H_1}+\frac{1}{H_2}=t_u.$ (6.13)

For $n=1$ these two equations yield

$t_u=8.7\times {10}^{17}\text{s}\Rightarrow H_1=2.21\times {10}^{-18}{H}_2=2.39\times {10}^{-18}{\text{s}}^{-1},$

i.e. 68.2 (Km/s)/Mpc and 73.8 (Km/s)/Mpc with conversion factor 1 (Km/s)/Mpc = 3.24 × 10⁻²⁰ s⁻¹. The observed values are 74.3 ± 2 and 68.2 ± 0.6 Mpc [10]: in fact there in the literature are several error ranges around different central values, yet the agreement is reasonable. Once more is evident the reliability of the value of $t_u$ proposed in (5.1).

The MOND Approach

This section has clearly classical character; it updates the section 3, where the Newton law has been introduced along with non-Newtonian potential terms. Differentiate (3.12) expressed in radial coordinates with respect to $v_r$ and calculate the ratio $\delta p_r/\delta t$ via the uncertainty equation; as $v_r=\delta r/\delta t$ and $\delta \left(v/\beta \right)=\delta v/{\beta }^3$ owing to (3.5) and (3.17), one finds

$F=\frac{\delta p}{\delta t}=\frac{m}{{\beta }^3}\frac{\delta v}{\delta t}\beta =\beta \left(v\right)\Rightarrow \frac{\hslash }{{\beta }^3}\frac{\delta v}{\delta t}=ϵv-\hslash a_0.$ (6.14)

Approximating $\beta \approx 1$ for $v\ll c$ , this result is nothing else but the Newton law; thus (6.14) is a reasonable starting point to concern the MOND approach. Implementing these equations one finds

$\frac{F}{m}=\frac{1}{{\beta }^3}\frac{\delta v}{\delta t}=\frac{ϵv}{\hslash }-a_0{a}_N=\frac{\delta v}{\delta t}=\frac{m{c}^{2}v{\beta }^2}{\hslash }-a_0{\beta }^3$

so that

$\underset{v=0}{\text{lim}}{a}_N=-a_0\underset{v=c}{\text{lim}}{a}_N=0,$ (6.15)

being $a_N$ the Newtonian acceleration. The vanishing of $a_N$ for $v\to c$ is self evident recalling (3.16).

However (6.14) waives a key point in this respect: the number $n$ of allowed states of a gravitational quantum system, hidden in and implied by the uncertainty ranges of (6.14). To this purpose consider a spiral galaxy and introduce $\delta F_r=F_r-F_N$ , being $F_r$ and $F_N$ the radial force components at the periphery and near the center of the galaxy. Since $F_r=\delta p_r/\delta t$ , calculate first $\delta F_r=\delta \left(\delta p_r\right)=\delta \left(n\hslash /\delta r\right)$ ; it follows

$\begin{array}{c}\delta \left(\delta p_r\right)=\hslash \frac{\delta n}{\delta r}-n\hslash \frac{\delta \left(\delta r\right)}{\delta r^2}\Rightarrow \\ \frac{\delta \left(\delta p_r\right)}{\delta t}=\delta F_r=\delta ϵ\frac{\delta n/n}{\delta r}-n\hslash \frac{\delta \left(\delta r\right)/\delta t}{\delta r^2}\delta ϵ=\frac{n\hslash }{\delta t}:\end{array}$ (6.16)

the radial momentum gradient along the radial distance $\delta r$ from a gravity source introduces acceleration and force radial components. In general (6.16) reads

$\delta F_r=\delta ϵ\left(\frac{\delta n/n}{\delta r}-\frac{\delta \left(\delta r\right)}{\delta r^2}\right)\delta ϵ=\frac{n\hslash }{\delta t}\Rightarrow F_r=\delta ϵ\frac{\delta n/n}{\delta r}{F}_N=\delta ϵ\frac{\delta \left(\delta r\right)}{\delta r^2}:$ (6.17)

the non-Newtonian and Newtonian terms decrease according to $\delta r^{-1}$ and $\delta r^{-2}$ , i.e. at large distance from the gravity source the latter becomes negligible with respect to the former.

The notation $\delta F_r$ implied by $\delta \left(\delta p_r/\delta t\right)$ is coherent with the difference of two radial forces at the right hand side of (6.16), which combine the components $ϵ/\delta r$ and $ϵ\delta \left(\delta r\right)/\delta r^2$ in a unique interpolation formula governed by the Newtonian stretching rate $\delta \left(\delta r\right)/\delta t$ of the space range $\delta r$ and by the quantum change rate $\delta n/\delta t$ of the number $n$ of allowed states for the gravitational system described by $\delta F_r$ . Clearly the latter is a relativistic effect of space deformation $\delta \left(\delta r\right)$ due to the interacting masses, in fact the space curvature that stretches the Euclidean $\delta r$ of the special relativity, remind (4.26) and (4.29), the former is mere quantum effect. Regard thus separately these interpolation terms (6.17) of (6.16): the first addend in parenthesis dependent on the radial distance $\delta r^{-1}$ is governed by $\delta n/n$ , whereas the square radial distance $\delta r^{-2}$ is governed by $\delta \left(\delta r\right)$ . Even without explicit calculations, one infers the classical Newtonian and non-Newtonian trend at large and small $\delta r$ : depending on the relative values of $\delta n$ and $n$ , intermediate behaviors are also possible without additional hypotheses and exotic assumptions about “dark matter”. These considerations show that the interpolation terms have actually a profound foundation: relativistic meaning and quantum meaning.

The remainder of this section aims to emphasize the physical meaning of $\delta n$ and $\delta \left(\delta r\right)$ .

To explain why $n$ changes, consider first that the gravitational interaction propagates at rate ${v^{\prime }}_r$ between two masses $\delta r$ apart $m$ and $m^{\prime }$ ; thus (6.16) reads

$\delta F_r=\frac{n\hslash \delta r}{\delta t\delta r^2}\left(\frac{\delta n}{n}-\frac{\delta \left(\delta r\right)}{\delta r}\right)=\frac{n\hslash {v^{″}}_r}{\delta r^2}{v^{″}}_r={v^{\prime }}_r\left(\frac{\delta n}{n}-\frac{\delta \left(\delta r\right)}{\delta r}\right){v^{\prime }}_r=\frac{\delta r}{\delta t},$ (6.18)

i.e. the radial distance range $\delta r$ between the masses is defined by the propagation rate itself of the gravitational interaction. So, $\delta F_r$ related to ${v^{″}}_r$ requires anyway $\left|{v^{″}}_r\right|\le c$ . Therefore put ${v^{\prime }}_r=c$ and write

$\delta F_r=\frac{n\hslash {v^{″}}_r}{\delta r^2}{v^{″}}_r=c\left(\frac{\delta n}{n}-\frac{\delta \left(\delta r\right)}{\delta r}\right)\Rightarrow \left|\frac{\delta n}{n}-\frac{\delta \left(\delta r\right)}{\delta r}\right|\le 1,$ (6.19)

with gravitational interaction field propagating at rate $c$ as already found in (DVQ).

-On the one hand put in (6.18) $n\hslash {v^{″}}_r=\pm G{m}^{\prime }m$ by dimensional reasons, with $G$ proportionality constant, and $\delta r/\delta t=c$ to intend that the distance $\delta r$ between the interacting masses is compliant itself with the propagation rate $c$ of the gravitational interaction. Thus $\delta F_r$ fulfills the Newtonian form

$\delta F_r=\pm \frac{G{m}^{\prime }m}{\delta r^2}{v^{″}}_r=\pm \frac{G{m}^{\prime }m}{n\hslash }{a}_r=\frac{Gm}{\delta r^2},$ (6.20)

where however two different terms are hidden in and summarized by ${v^{″}}_r$ . So, if $m^{\prime }m$ is large enough so that $G{m}^{\prime }m/n\hslash >c$ for a given $n$ , then $n$ must increase to $n+n^{\prime }$ in order that $G{m}^{\prime }m/\left(n+n^{\prime }\right)\hslash <c$ is still compatible with a new $\left|{v^{‴}}_r\right|<c$ . Simply rewriting (6.14) as (6.20), the uncertainty introduces the quantization and removes the idea of instantaneous action at a distance once having introduced space ranges through which the gravitational interaction propagates at radial rate $c$ . Even without higher order potential terms of (4.22) or (2.27), the Newton law is approximate but not conceptually wrong like the deterministic classical form $G{m}^{\prime }m/r^2$ .

-On the other hand, an analogous conclusion holds also for the potential energy. Indeed (6.18) yields

$\begin{array}{l}c\left(\frac{\delta n}{n}-\frac{\delta \left(\delta r\right)}{\delta r}\right)=\pm \frac{G{m}^{\prime }m}{n\hslash }\\ \Rightarrow \left(\frac{\delta n}{n}-\frac{\delta \left(\delta r\right)}{\delta r}\right)\delta ϵ=\pm \frac{m^{\prime }mG}{\delta \ell }n\hslash c=\delta ϵ\delta \ell \delta \ell =c\delta t.\end{array}$ (6.21)

To understand the physical meaning of $\delta n$ remind (4.4) to (4.7) and note that $\delta p/\delta t=F$ whereas $\delta M/\delta t=\mathcal{T}$ , being $\mathcal{T}$ torque. The former has been concerned in (6.16) and implies $F_N$ of (FFF), the latter explains what has to do $F_r$ with the rotation of galaxies; the key point is $\delta n$ . Is interesting in particular (4.4), because the quantization of ${\text{M}}_u$ implies $\delta {\text{M}}_u=\hslash \delta n$ introduced in (FFF); it suggests that $\hslash \delta n/\delta t$ is thinkable as torque acting on a rotating quantized system. Since $\delta v_r=\delta \left(\delta r/\delta t\right)=\delta \left(\delta r\right)/\delta t$ during a fixed time lapse $\delta t$ , (6.16) yields

$\delta \frac{\delta p_r}{\delta t}=\frac{\mathcal{T}}{\delta r}-\frac{n\hslash }{\delta t}\frac{\delta \left(\delta r\right)}{\delta r^2}{\mathcal{T}}_u=\frac{\delta \left(n\hslash \right)}{\delta t}.$ (6.22)

In general the torque expresses the ability of a force to impart a rotation around an axis. Given a two body system, the uncertainty evidences that the radial component of a gravitational gravitational force acting on a mass determines properties like radial $a_r$ and $F_N$ components, whereas the change $\delta n$ of the number $n$ of allowed states of the system accounts for their orbiting behavior. However this latter is triggered by large orbiting systems like a huge galaxy, where reasonably peripheral $\delta r$ imply the $\delta r^{-2}$ Newtonian term negligible with respect to $\delta r^{-1}$ uncertainty driven term: in other words the masses and their distance determine whether the attractive force implies implies mere radial effect or also combined $\delta n$ driven orbiting motion to fulfill $\left|{v^{″}}_r\right|\le c$ in (6.19). A few considerations highlight further these notes paying attention to the specific system of a spiral galaxy, whose behavior is controlled not only by the relative values of radial ranges $\delta r^{-1}$ and $\delta r^{-2}$ but also by the finite $n$ and $\delta n$ . In fact the near field behavior of (6.21) is reasonably expected for stars with small average distance from the galaxy rotation center; the far field behavior prevails instead for stars very distant from the galaxy center. Anyway, the expected whole behavior of a large gravitational system, i.e. with variable number of allowed quantum states, cannot be in principle merely Newtonian. In practice, in order for the addend $\delta \left(\delta r\right)/\delta r$ to be negligible with respect to the non-Newtonian term $\delta n/n$ , the galaxy must be huge and massive; these conditions allow large $\delta r$ and $\delta n$ . Thus small galaxies should not exhibit the MOND behavior.

At this point, once having explained the physical meaning of the terms $r^{-1}$ and $r^{-2}$ , here merely replaced by the respective uncertainty ranges (6.17), the well known (2.35) does not need further comments. It is worth emphasizing that according to (6.20), putting $\delta r=r_u$ and $m^{\prime }=M_u$ , universe radius and mass, one finds

$a_r=\frac{M_{u}G}{r_u^2}=1.9\times {10}^{-10}\frac{\text{m}}{{\text{s}}^{\text{2}}},$ (6.23)

which agrees reasonably with $a_0=F_{Pl}/M_u=2\times {10}^{-10}\text{m}/{\text{s}}^{\text{2}}$ and $c^2/r_u=2\times {10}^{-10}\text{m}/{\text{s}}^{\text{2}}$ ; also, as concerns (2.35) note that ${\left(M_{u}G{a}_0\right)}^{1/4}=c$ so that $v_1=c{\zeta }^{-1/4}\text{m}/\text{s}$ . These results support the value of $M_u$ and suggests that the galaxies rotate consistently with centripetal acceleration whose order of magnitude is definable through parameters at the universe scale; in other words, this centripetal acceleration is not mere local feature but appears to have more general meaning. It appears that ${\rho }_{de}$ and ${\eta }_{de}$ calculated with $M_{de}{c}^2/V_u$ of (6.3) fit reasonably the literature values of ${\rho }_{vac}$ and ${\eta }_{vac}$ . Moreover note the dimensional relationship $G\times energydensity=acceleratio{n}^2$ ; indeed

$\sqrt{{\eta }_{de}G}=2\times {10}^{-10}\text{m}/{\text{s}}^2.$ (6.24)

in agreement with (6.23). The previous (2.34) had classical meaning; here the addends (6.17) and (6.22) have quantum meaning, the value of $a_0$ is calculated in (5.12) and (6.23) in the cosmological frame of parameters of the universe.

7. Discussion and Conclusions

One of main tasks of this paper was to regard the cosmological parameters not separately, i.e. as individual quantities that concur to calculate something, but as values collectively connected themselves by a rational link. Whit this strategy, any calculation confirms other calculations if the input data are a self-consistent basis to describe physical effects apparently independent. Thanks to this basic idea, it has been possible to ascertain by direct calculation that the total visible mass $M_{ob}$ extrapolated to the whole universe volume, beyond our boundary of sight due to the finite light speed propagation, is enough to include the so called “dark matter” as mere completion of the visible $m_{ob}$ . Moreover, once having calculated ${\eta }_{vac}$ in a previous paper, the numerical calculations hitherto carried out bring to the conclusion the connection between dark energy and quantum vacuum energy. Eventually, replacing systematically $\partial$ with $\delta$ not only aims to overcome the deterministic approach based on any physical variable, but also to bypass the idea of immediate cause-effect and instantaneous long range effect. The fact a finite time range is necessary for a finite change of velocity of the particle subjected to a force, implies in a natural way to define the distance $\delta r$ via the propagation rate of any perturbation.

As concerns ordinary matter existing in the universe, note interesting information coming from the wavelike and corpuscular definitions of $p$ ; the formalism is driven by (3.12), (3.30) and (3.31). By differentiating (3.20) one finds

$\delta p=-\frac{h}{{\lambda }^2}\delta \lambda =-\frac{h{c}^2}{{\lambda }^2{c}^2}\delta \lambda =-\frac{h\nu }{c^2}\nu \delta \lambda =-\frac{h\nu }{c^2}\delta v \delta v=\nu \delta \lambda .$

Now rewrite this result via corpuscular formalism; the wavelike to corpuscular correspondence reads

$\delta \left(\frac{h}{\lambda }\right)=-\frac{h\nu }{c^2}\delta v\iff \delta \left(\frac{mv}{\beta }\right)=-\frac{ϵ}{c^2\beta }\delta v,$ (7.1)

being $ϵ/\beta$ the amount of energy corresponding to $h\nu$ . Rewrite the right hand side of (7.1) as follows

$\frac{m{v}_2}{{\beta }_2}-\frac{m{v}_1}{{\beta }_1}=\frac{m}{\beta }{v}_1-\frac{m}{\beta }{v}_2\Rightarrow \frac{m{v}_1}{{\beta }_1}=\frac{ϵ/\beta }{c^2}{v}_2\frac{m{v}_2}{{\beta }_2}=\frac{ϵ/\beta }{c^2}{v}_1.$

Multiply side by side these two equations; then

$m^2\frac{v_1{v}_2}{{\beta }_1{\beta }_2}=\frac{{ϵ}^2}{c^4{\beta }^2}{v}_1{v}_2\Rightarrow \frac{{\left(m{c}^2\right)}^2}{{ϵ}^2}=\frac{{\beta }_1{\beta }_2}{{\beta }^2}\Rightarrow m{c}^2=\pm \frac{ϵ}{\beta }\sqrt{{\beta }_1{\beta }_2}=\pm h\nu \sqrt{{\beta }_1{\beta }_2}.$

The states of negative energy are well known since Dirac. The negative quantum energy, i.e. less than that of the quantum vacuum, is also known in the field theory; it is allowed to occur for a certain time and space ranges [11]. Once more, see e.g. (2.45), dimensional mass and material mass of (7.1) are self-consistent. In fact $m$ results here extracted from immaterial momentum $h/\lambda$ through the corpuscular momentum $mv/\beta$ via $h\nu \leftrightarrow m{c}^2/\beta$ : i.e. a massive particle corresponds to wavelike energy. The well known concept mass$\leftrightarrow$ energy still holds in the form mass$\leftrightarrow$ quantum wave energy; the last equation is in fact an energy balance. Further work on this subtle topic is in progress, in particular as concerns the physical meaning of ${\beta }_{eff}=\sqrt{{\beta }_1{\beta }_2}$ . Express the effective energy and momentum at the macroscopic and Planck scales as ${ϵ}_{eff}{\ell }_{eff}{v}_{eff}/c={ϵ}_{Pl}$ and , functions of $v_{eff}{\ell }_{eff}$ only, plus the obvious condition $m_{eff}{c}^2=h{v}_{eff}$ . Test a set of trial values ${\ell }_{eff}$ to identify ${\ell }_{eff}{v}_{eff}$ fulfilling both equations, and select the one with $v_{eff}$ satisfying also the given condition. It yields $v_{eff}$ = 3 × 10²⁵ s⁻¹ and $m_{eff}$ = 2.2 × 10²⁵ Kg: in fact $h{v}_{eff}=m_{eff}{c}^2$ = 125 GeV implies and ${ϵ}_{eff}=m_{eff}{\Lambda }_t/\Lambda$ too, which link Planck and cosmological scales.

It is worth noticing that the definition of (3.1) is in fact included in a more general context, e.g. noting that

$c^2-\frac{c^2}{\beta }=-\frac{1}{2}{v}^2-\frac{3}{8}\frac{v^4}{c^2}+\cdots \Rightarrow \frac{\phi }{c^2}=1-\frac{1}{\beta }\Rightarrow \frac{\delta \phi }{c^2}=\left(1-\frac{1}{\beta }\right);$

it could provide in principle a more fundamental basis to extend further the ideas proposed in this paper.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References

[1]	Landau, L. and Lifchitz, E. (1966) Mecanique. Ed MIR Moscow. (In French)
[2]	Landau, L. and Lifchitz, E. (1966) Theorie du Champ. Ed MIR Moscow. (In French)
[3]	Tosto, S. (2024) Maxwell Equations and Magnetic Monopoles. Journal of Applied Mathematics and Physics, 12, 737-763.[CrossRef]
[4]	Tosto, S. (2023) Dimensional Analysis for Relativistic and Quantum Outcomes. Journal of Applied Mathematics and Physics, 11, 1996-2052.[CrossRef]
[5]	Amelino-Camelia, G. (2002) Space-Time Quantum Solves Three Experimental Paradoxes. Physics Letters B, 528, 181-187.[CrossRef]
[6]	Tosto, S. (2024) Cosmological Implications of Diffusion: The Hubble Tension. Journal of Applied Mathematics and Physics, 12, 4077-4097.[CrossRef]
[7]	Rumble, J. (2022) CRC Handbook of Chemistry and Physics. CRC Press.
[8]	Gupta, R.P. (2023) JWST Early Universe Observations and λCDM Cosmology. arXiv: 2309.13100.
[9]	Prat, J., Hogan, C., Chang, C. and Frieman, J. (2022) Vacuum Energy Density Measured from Cosmological Data. Journal of Cosmology and Astroparticle Physics, 2022, Article 15.[CrossRef]
[10]	Hu, J.P. and Wang, F.Y. (2023) Hubble Tension: The Evidence of New Physics. arXiv: 2302.05709.
[11]	Grumiller, D., Parekh, P. and Riegler, M. (2019) Local Quantum Energy Conditions in Non-Lorentz-Invariant Quantum Field Theories. Physical Review Letters, 123, Article ID: 121602.[CrossRef] [PubMed]