A Fundamental Solution of the Hubble Tension

Abstract

Einstein derived the expansion of space ever since the Big Bang started and introduced the possible cosmological constant $\Lambda$. The expansion of space and the present-day expansion rate $H_0$, the Hubble constant, has been discovered by Hubble. Perlmutter discovered the positive value of $\Lambda$, and Zeldovich showed that $\Lambda$ corresponds to the energy density $u_{DE}$ of space. Lamb and Retherford as well as Casimir provided evidence for the idea that $u_{DE}$ might be based on quanta, and Riess et al. provided evidence that $H_0$ is an idealization. In this paper, using the hypothetico-deductive method with very founded hypotheses, these two pieces of evidence are confirmed in a very founded and precise manner. Thereby, neither a fit is executed, nor a postulate, nor an unfounded hypothesis is proposed.

1. Introduction

The properties of space are essential for theories, see, e.g., Newton [1], Maxwell [2], Einstein [3], Minkowski [4], Einstein [5], Hilbert [6], Einstein [7], Hubble [8], Perlmutter [9], Zeldovich [10], Lamb and Retherford [11], Casimir [12], Riess et al. [13]. Thereby, outer space is a relatively ideal form of space in the universe, as outer space is hardly disturbed by molecules or other objects. Hereby, the universe consists of space and of the objects in space.

Moreover, the properties of space are important for applications such as space navigation, see, for instance, Soffel [14], telecommunication, quantum cryptography or quantum computer networks.

In order to understand space, it is essential to understand that space has a nonzero energy density $u_{DE}$ or $u_{\Lambda }$. It has essentially been proposed by [7] with his cosmological constant $\Lambda$. Its value has been measured by Perlmutter et al. [9], and later by Riess and others [15], and with a different method by Smoot [16].

Furthermore, space was not given in advance. Instead, space expanded from the Big Bang until today, and space will expand in the future; see Friedmann [17], Wirtz [18], and more completely by Lemaître [19], and with convincing observation by Hubble [8], and in a textbook by Hobson et al. [20], and from an elementary particle point of view by Workman and others [21], and by very precise observation of early universe structures by Planck Collaboration [22]. Thereby, the measurable present-day rate $H_0$ of increase of the space exhibits a problem, the $H_0$-tension: measurements provide different values of $H_0$ at the $5\sigma$ confidence level, for very precise observation of late universe structures, see Riess and others [13].

This problem is explained here at the level of first principles and a founded and critically reflected scientific method; see Section 1.1 and Section 1.1.1.

Firstly, an idealization of space is identified with help of a paradox. The idealization is that in present-day science space is a single entity. With the help of a founded derivation, this idealization is overcome: homogeneous space is a stochastic average of indivisible volume portions.

Secondly, a volume dynamics is derived for these volume portions.

Thirdly, the quantum postulates, gravity, and the Local Formation of Volume (LFV) are derived from this volume dynamics.

Fourthly, based on these founded and derived dynamical processes, and for the case of the homogeneous universe, the energy density of volume is deduced. For that case, this energy density is called $u_{vol}$. It is in accurate concordance with the observed value of the early universe, which was very homogeneous. Moreover, for this homogeneous case, the corresponding $H_0$ value is identified, in precise accordance with observation.

Of course, the universe is heterogeneous. Accordingly, the heterogeneity is analyzed in addition to the homogeneous universe. Consequently, the $H_0$ value of the late universe, which is very heterogeneous, is derived. This result is in accurate concordance with observation.

These results provide very convincing evidence for the concept that homogeneous space is a stochastic average of indivisible volume portions. Moreover, these results represent a very clear evidence that the $H_0$ tension is explained by the gradual evolution of heterogeneity in the universe. These two fundamental insights improve the present-day model of cosmology, the $\Lambda$CDM model. Additionally, these findings can be applied to space navigation and exploration; see Carmesin [23,24].

1.1. Hypothetico-Deductive Method

In this paper, the results are obtained by the hypothetico-deductive method. That method is very founded and broadly accepted; see original research by Popper [25,26], and see a general epistemological view by Niiniluoto, Sintonen and Wolenski [27]. The used hypotheses have already been tested by many experiments and scientists. This is shown by corresponding citations. Therefore, these very founded hypotheses will not be tested again in the present investigation. These very founded hypotheses are used as a basis for derivations in the hypothetico-deductive method. Moreover, these hypotheses are explained with their mutual logical connections. For these reasons, these hypotheses are explained in paragraphs, and these hypotheses are not expressed as usual hypotheses that have to be tested in the future. Hereby, the following very founded hypotheses are used:

1.1.1. Used Very Founded Hypotheses

In this section, the hypotheses are presented that are used in the hypothetico-deductive method. Thereby, these hypotheses are very founded. Therefore, the risk of failure is very small.

(1)

The space that can be observed in the whole volume V, ranging from Earth to the light horizon, is isotropic and homogeneous at this universal scale. Local heterogeneities are possible, for instance near a mass M; see Schwarzschild [28], see observation of space by Dyson, Eddington and Davidson [29], and see observation at Earth’s ground by [30]. This has been observed in a very precise manner; see Planck Collaboration [22].

(1.1)

Moreover, there is natural space that is very homogeneous and isotropic at small scales. For instance, natural space is very homogeneous and isotropic in a void; see Zeldovich, Einasto and Shandarin [31], Contarini et al. [32], and natural space was very homogeneous and isotropic in the early universe; see Planck Collaboration [22].

(1.2)

Furthermore, in the heterogeneous universe, natural space can exhibit slight heterogeneity, additionally. For instance, in the heterogeneous universe, Abbott et al. [33] observed the process of merging of two black holes. For that observation, gravitational waves had been utilized. These gravitational waves can be interpreted as coherent states that cannot be emitted in a natural homogeneous universe without heterogeneity, as only an appropriate heterogeneity can emit coherent states.

(2)

Additionally, the space in (1) has a positive energy density $u_{DE}$. Hereby, the subscript $DE$ abbreviates dark energy. It describes the energy E of a respective volume V in part (1), divided by this volume:

$$u_{DE}={\displaystyle \frac{E}{V}}.$$

(1)

This energy density $u_{DE}$ has been discovered by Perlmutter et al. [9], later by Riess and others [15], and with help of the CMB by Smoot [16].

(3)

In general, for each object, including the space in (1), the energy-speed relation of special relativity theory (SRT) holds, see Einstein [3], Hobson et al. [20]: In general, a body that has an energy E, can have a velocity $\overrightarrow{v}$ relative to a mass $m_{ref}$, which is used as a reference. Its absolute value is called speed $v=|\overrightarrow{v}|$. In particular, at $v=0$, the energy is called rest energy $E_0$. The energy speed relation of SRT is as follows:

$$E_0^2=E^2·\left(1-{\displaystyle \frac{v^2}{c^2}}\right)$$

(2)

In the case $v<c$, the relation has the following equivalent form:

$$E^2={\displaystyle \frac{E_0^2}{1-\frac{v^2}{c^2}}}$$

(3)

Hereby, and in the following, the velocity is determined relative to an adequate coordinate system of relativity theory, for details, see Section 1.1.5.

(4)

Moreover, each volume or volume portion $\Delta V$ of space in (1) has zero rest energy $E_0$, and it has zero rest mass $m_0=E_0/c^2$.

$$m_0\left(V\right)=0=E_0\left(V\right).$$

(4)

This is shown in Section 1.1.2.

(5)

In a process of increase in volume or space in (1), the dark energy density $u_{DE}$ in (2) is a nonzero constant, whereby only very small variations might occur. This approximate constancy has been observed for the expansion of the universe, see an early universe observation by Planck Collaboration [22], and a late universe observation by Riess et al. [13]. Additionally, that constancy has been proposed by general relativity theory and cosmology, see a first analysis by Einstein [7], a follow-up theoretical study by Friedmann [17], a more general theory by Lemaître [19], and a textbook by Hobson et al. [20]. Furthermore, the value of $u_{DE}$ is derived here, and the results are additional evidence for this approximate constancy.

The above very founded hypotheses (1)–(5) will be used for deductions in this paper.

1.1.2. Why Volume Has No Rest Mass

In this section, it is shown that volume has zero rest mass $m_0$.

Near a mass M, there occurs additional volume $\delta V$, see Section 1.1.4 or Figure 1. If that additional volume would have a rest energy $m_0\left(\delta V\right)>0$, then there would be an additional rest mass $m_0$ near each mass M. Such an additional $m_0$ has never been observed. This can be confirmed in a textbook by Landau and Lifschitz [34], or by reported research Workman et al. [21], Planck Collaboration [22], Zogg [35]. For instance, if there were such an additional rest mass $m_0$, then this $m_0$ would modify the orbits of the GPS satellites, and this would have been observed, but this has not been observed. Consequently, additional volume $\delta V$ has no rest mass.

Moreover, the additional volume near a mass is the same type of volume as common volume that occurs without any mass. This is confirmed by the observation that only one type of volume exists, it is documented in Workman et al. [21], or also in Planck Collaboration [22], Casimir [12], Zeldovich [10], Perlmutter et al. [9]. Therefore, in general, volume has no rest mass.

1.1.3. Mass Causes an Increase in Radial Light Travel Distance

In this section, it is shown that a mass M causes an increase in the radial distance R to M. This increase occurs at each radial difference $\Delta R$; see Figure 1.

For instance, near a mass M, an original (for the case of $M=0$) radial difference $\Delta R$ is increased by $\sqrt{g_{RR}}$. Note that it is the root of the tensor element $g_{RR}$. The increased value is $\Delta L$, see Figure 1,

$$\Delta L=\Delta R·\sqrt{g_{RR}}.$$

(5)

Hereby, the value $\Delta L$ can be measured as a light travel distance $d_{LT}$. It is described by Hobson et al. [20]. Similarly, the original value $\Delta R$ is measurable. For it, a pair of hand leads can be used. That distance measure is named gravitational parallax distance (in the context of two hand leads), see Carmesin [23,36].

As a consequence of general relativity, see Einstein [5], Hilbert [6], Hobson et al. [20], at each radial coordinate (or gravitational parallax distance) R, the radial element $g_{RR}$ of the metric tensor is a function of the Schwarzschild [28] radius $R_S$:

$$g_{RR}={\displaystyle \frac{1}{1-\frac{R_S}{R}}},\mathrm{with}{R}_S={\displaystyle \frac{2G·M}{c^2}}$$

(6)

These relations are in accurate concordance with observed values. This is documented by Dyson, Eddington and Davidson [29], and for the case of an observation on Earth’s ground by Pound and Rebka [30], or in an overview by Will [37].

According to the Hacking [38] criterion of reality, this increase in radial light travel distance $\Delta L$ is real, as it can be manipulated as follows: The mass M can be increased, and this causes an increase in radial light travel distance $\Delta L$, see Equations (5) and (6).

Moreover, the gravitational parallax distance $\Delta R$ is real, as it can be manipulated as well: For instance, in Figure 1, $\Delta R$ is the spatial shortest difference between two locations A and B. If B moves towards A, then $\Delta R$ decreases. Next, the corresponding volume is analyzed:

1.1.4. Mass Causes an Increase in Volume

In this section, the following is shown: Nearby a mass M, the increase in a radial length $\Delta L$ causes an enhanced volume.

Without loss of generality, a portion of volume $\Delta V_R=\Delta R·\Delta x·\Delta y$ is used with $|\Delta R|=|\Delta x|$ and $|\Delta R|=|\Delta y|$.

Near a mass M, $\Delta R$ (radial distance) is enhanced to the respective light travel distance $\Delta L$ by the factor $\sqrt{g_{RR}}$, while $\Delta x$ and $\Delta y$ are not changed by M. Consequently, the volume $\Delta V_R$ is increased to the respective volume $\Delta V_L$ by a factor $\sqrt{g_{RR}}$, see Equation (5):

$$\Delta V_L=\Delta L·\Delta x·\Delta y=\Delta R·\Delta x·\Delta y·\sqrt{g_{RR}}=\Delta V_R·\sqrt{g_{RR}}.$$

(7)

The difference of the volume $\Delta V_R$ measured with the gravitational parallax distance and the volume $\Delta V_L$ measured with the light travel distance is called additional volume $\delta V$:

$$\delta V:=\Delta V_L-\Delta V_R.$$

(8)

The ratio $\delta V/\Delta V_L$ is called relative additional volume ${\epsilon }_L$:

$${\epsilon }_L:={\displaystyle \frac{\delta V}{\Delta V_L}}.$$

(9)

1.1.5. On the Adequate Coordinate Systems

In the theory of relativity, a time interval $t_{sa{t}_E}$ of a clock onboard a satellite E and a time interval $t_{sa{t}_F}$ of a clock onboard a satellite F exhibit the phenomenon of kinematic time dilation. Relativity theory states: Between two events A and B, a clock in a laboratory coordinate system exhibits a time $t_{lab}$, a clock onboard the satellite E exhibits a time $t_{sa{t}_E}$, and a clock onboard the satellite F exhibits a time $t_{sa{t}_F}$ as follows:

$$t_{lab}=t_{sa{t}_E}·{\displaystyle \frac{1}{\sqrt{1-v_{E,lab}^2/c^2}}}\mathrm{and}{t}_{lab}=t_{sa{t}_F}·{\displaystyle \frac{1}{\sqrt{1-v_{F,lab}^2/c^2}}}.$$

(10)

Hereby, $v_{E,lab}=\left|{\overrightarrow{v}}_{E,lab}\right|$ is the absolute value of the velocity (speed) of the satellite E with respect to the laboratory coordinate system (CS), and $v_{F,lab}=\left|{\overrightarrow{v}}_{F,lab}\right|$ is the absolute value of the velocity (speed) of the satellite F with respect to the CS. However, the International Astronomical Union (IAU) realized that relativity theory does not provide information about the choice of an adequate laboratory coordinate system or laboratory frame. Such information about the frame is essential for predicting the times shown by the above clocks, and it is important for space navigation. The IAU called this lack of information the problem of finding an adequate coordinate system (ACS), see Soffel [14].

The problem of finding an adequate coordinate system (ACS) has been solved [23,24,39]. Thereby, for each point P in the universe, the adequate coordinate system (ACS) is characterized by the following derived properties:

(1)

An ACS exists.

(2)

The ACS has a uniquely determined velocity ${\overrightarrow{v}}_{ACS,CS}$, relative to an arbitrarily chosen coordinate system $CS$.

(3)

${\overrightarrow{v}}_{ACS,CS}$ can be measured, and procedures of measurement are provided.

(4)

${\overrightarrow{v}}_{ACS,CS}$ can be predicted and calculated, and procedures for it are provided.

(5)

The equations of time dilation in Equation (10), as well as the energy speed relation in Equations (2) and (3) hold, when the ACS is used. Moreover, the typical results of relativity hold, when the ACS is used.

(6)

There exists a universal null of the fractional kinematic difference of time

$$\delta t_{kin,frac}:={\displaystyle \frac{t_{sa{t}_E}-t_{lab}}{t_{lab}}}\le 0,$$

(11)

whereby the chosen laboratory coordinate system is equal to the ACS.

In this paper, the ACS is used. Therefore, the derived results have clearly defined conditions.

1.2. Observed Energy Density of Volume

In astrophysical space, Perlmutter et al. [9], and later Riess et al. [15] observed the acceleration of the universe’s expansion rate. As a consequence of the Friedmann Lemaître equation (FLE), empty space has a positive energy density. This is documented in Friedmann [17], together with Lemaître [19], and in the textbook Hobson et al. [20], and with a detailed analysis in Carmesin [40]. It is called density $u_{\Lambda }$, corresponding to the cosmological constant $\Lambda$, or to the density $u_{DE}$ of the dark energy density [7,10,20,41]. An actual value has been observed on the basis of the CMB (Cosmic Microwave Background), documented by [22,23,40]:

$$u_{\Lambda ,obs}=5.{133}_{-0.2432}^{+0.2432}·{10}^{-10}{\displaystyle \frac{\mathrm{J}}{{\mathrm{m}}^3}}=u_{DE,obs}.$$

(12)

2. Problem of the Hubble Tension

Here, we summarize a very interesting problem of modern physics. In particular, this problem is a discrepancy between two observed values that should be equal according to the $\Lambda$CDM model of present-day physics, see, e.g., Hobson et al. [20]. Consequently, this discrepancy falsifies the $\Lambda$CDM model. In general, according to the hypothetico-deductive method, a solution to such a falsification problem provides the chance to improve concepts of physics [25,26]. In this paper, we will use this problem in order to develop founded and essential improvements of present-day physics.

Hubble [8] discovered the present-day expansion rate of space, the Hubble constant $H_0$. In general, the Hubble constant is measured by utilizing signals emitted at an instant of time $t_{em}$ or a respective redshift $z_{em}$, documented in Hobson et al. [20].

Penzias and Wilson [42] observed the CMB (Cosmic Microwave Background). It has been emitted at $t_{em}=\mathrm{380,000}\mathrm{years}$ (after the Big Bang). Utilizing that radiation, the Planck Collaboration [22] observed the following:

$$H_{0,obs,CMB}=66.{88}_{-0.92}^{+0.92}{\displaystyle \frac{\mathrm{km}}{\mathrm{s}·\mathrm{Mpc}}}$$

(13)

Hereby, the redshift of emission is $z_{em}=z_{CMB}=1090.3\pm 0.41$. In the measurement, the so-called T-T correlation (T abbreviates temperature) is utilized [22] ([table 2, column 1]).

As another example, Riess et al. [13] utilized signals from nearby galaxies (with Ia supernovae) emitted at ca. $z=0.055$. and observed

$$H_{0,obs,near,Ia}=73.{04}_{-1.01}^{+1.01}{\displaystyle \frac{\mathrm{km}}{\mathrm{s}·\mathrm{Mpc}}}.$$

(14)

Thereby, many galaxies with different redshift values $z_{em}$ have been observed. Accordingly, the redshift $z_{em}$ of emission is described by the average of the redshift values of the observed [13] galaxies, $\langle z\rangle$ is equal to = 0.055.

Riess et al. [13] discovered a difference $H_{0,obs,near,Ia}-H_{0,obs,CMB}$. It is named Hubble tension or $H_0$ tension, documented in Planck Collaboration [22]. That discrepancy is at the five $\sigma$ confidence level. Consequently, that discrepancy can be regarded and named as a scientific result [13]. This result falsifies the $\Lambda$CDM model, as this model states that the value of $H_0$ should not depend on the redshift value of observed objects. Based on such a falsification of a model, that model can be interpreted as an idealization, see Shech [43], Song et al. [44]. Consequently, the idealization of the $\Lambda$CDM model is identified:

The Hubble tension falsifies the $\Lambda$CDM model. Accordingly, the $\Lambda$CDM model is an idealization.

Therefore, the Hubble tension or $H_0$ tension is a founded scientific problem. A fundamental solution to this problem is derived in this paper. Hereby, no fit is executed, no unfounded hypothesis is proposed, and a precise accordance with observation is achieved, and predictions are derived.

Moreover, the value $H_{0,obs,CMB}$ describes a nearly homogeneous universe, since the early universe was nearly homogeneous [22]. In contrast, $H_{0,obs,near,Ia}$ is a $H_0$-value of the late universe, as the observed supernovae occurred in the late universe. Additionally, $H_{0,obs,near,Ia}$ is a value of the heterogeneous and late universe [22,45,46].

As the Hubble constant describes the expansion rate, we analyze the properties of space in more detail. For this, in Section 3, we show that the present-day view of space includes a paradox, the so-called space paradox. Moreover, we derive a solution to that space paradox. With it, we will explain and derive the $H_0$ tension. This solves the respective $H_0$-tension problem.

3. Space Paradox

In general, a paradox provides the chance to develop a deep insight [47]. In this section, we will derive a paradox, and we will use it in order to develop a deep insight.

In the following, the space, that can be observed in the whole volume V ranging from Earth to the light horizon, is considered. For instance, this space has been observed [22].

In physical concepts that are commonly used in the present-day, the space with its volume V is considered as a single entity, see, e.g., Newton [1], Maxwell [2], Einstein [5,17], Lemaître [19] and Planck Collaboration [22].

That space has a positive energy density $u_{DE}$, see Equation (1):

$$u_{DE}={\displaystyle \frac{E}{V}}.$$

(15)

As a consequence of time dilation, the volume V with its energy E of space, velocity $\overrightarrow{v}$ and speed $|\overrightarrow{v}|=v$, obey the energy–speed relationship, see Equations (2) and (3)

$$E^2={\displaystyle \frac{E_0^2}{1-\frac{v^2}{c^2}}}\mathrm{or}{E}^2·\left(1-{\displaystyle \frac{v^2}{c^2}}\right)=E_0^2.$$

(16)

As the energy density is nonzero, the energy E is nonzero. Thus, the above relationship can be divided by $E^2$. Therefore, the following form of the energy–momentum relationship holds:

$$1-{\displaystyle \frac{v^2}{c^2}}={\displaystyle \frac{E_0^2}{E^2}}.$$

(17)

As a consequence of the zero rest energy $E_0$ of V, see Equation (4), the term ${\displaystyle \frac{E_0^2}{E^2}}$ in Equation (17) is zero. Consequently, the volume V with the energy E has velocity c.

$$v=c=v\left(V\right)=v\left(E\right).$$

(18)

3.1. The Paradox

In physical concepts that are commonly used in present-day, the volume is considered as a single entity, see, e.g., Newton [1], Maxwell [2], Einstein [5], Friedmann [17], Lemaître [19], Hobson et al. [20], Planck Collaboration [22]. In such a concept of volume, the whole volume would move parallel to some unit direction vector ${\overrightarrow{e}}_v$ and with the speed of light, $\overrightarrow{v}=c·{\overrightarrow{e}}_v$, see Equation (18).

However, that velocity $\overrightarrow{v}$ would break the isotropy of space observed at a universal scale [22]. This is a contradiction. In general, a paradox is a contradiction, whereby the solution creates a fundamental insight [47]. The above contradiction is called the space paradox. It is solved next:

3.2. A Solution of the Space Paradox in Natural Homogeneous Space

In this section, the paradox is solved in three separate steps: The source of the paradox is identified in item (1). Therefrom, a solution of the paradox is developed in item (2). And the validity of the solution is confirmed in item (3).

(1) The space paradox has five premises: There are four founded premises about space and its volume V, see Section 1.1, parts (1)–(4), part (5) is not used here: isotropy, the positive dark energy density, the energy–speed relation of SRT (which is correct in the used ACS, see Section 1.1.5), and the zero rest energy $E_0$. In addition, there is one hardly founded premise (see the beginning of Section 3.1): the concept of space as a single entity. Therefore, space is not a single entity.

As space is not a single object, it must consist of many parts of the volume V. These parts $\delta V$ are analyzed next. Thereby, a part of the space with its volume V is called a volume portion (VP).

Hereby, parts of the volume with at least one local maximum of volume are analyzed. For each part, one such maximum is used in order to localize that part $\delta V$. Such a part is called a localized VP.

Since the energy and volume of space have the speed $v=c$, the above parts $\delta V$ must have the same speed. For instance, each part $\delta V_j$ has its speed

$$v_j=c.$$

(19)

In general, a part of volume is called volume portion.

(2) Next, the following question is analyzed (for a homogeneous universe and common density ${\rho }_{hom}$: Can such a part $\delta V_j$ with $v_j=c$ consist of smaller parts $\delta V_k$? Hereby, a usual density ${\rho }_{hom}$ is a density below an ultrahigh critical density ${\rho }_{cr.,symmetrybreaking}$ at which the symmetry of space breaks in a phase transition. This can occur in a phase transition from space to matter in the Higgs [48] mechanism. Moreover, this can take place in a dimensional phase transition [36,40,49,50]. If such a part $\delta V_j$ with $v_j=c$ of space would consist of smaller parts $\delta V_k$, then the following would be implied:

(2.1)

Each smaller part $\delta V_k$ would have the speed $v_k=c$, as the energy and volume of space have the speed $v=c$.

(2.2)

Consequently, each part $\delta V_k$ would have a velocity ${\overrightarrow{v}}_k=c·{\overrightarrow{e}}_k$, with a direction vector ${\overrightarrow{e}}_k$ of norm one.

(2.3)

As the considered universe is homogeneous (see, part (2)), there is no source that could provide a uniform direction for the direction vectors ${\overrightarrow{e}}_k$.

(2.4)

Hence, the velocity ${\overrightarrow{v}}_j$ of the considered part $\delta V_j$ with $v_j=c$ would be an average of the velocities ${\overrightarrow{v}}_k$. Thereby, as a consequence of the different direction vectors ${\overrightarrow{e}}_k$ in part (2.3), the velocity ${\overrightarrow{v}}_j$ would have an absolute value smaller than c, i.e., $|{\overrightarrow{v}}_j|=v_j<c$. This would contradict the speed $v_j=c$ in Equation (19).

(2.5)

Therefore, the parts $\delta V_j$ with speeds $v_j=c$ cannot consist of smaller parts. This is the answer to the question in (2). This implies that the parts $\delta V_j$ with $v_j=c$ are indivisible. Such a part $\delta V_j$ with its speed $v_j=c$ is called indivisible volume portion, indivisible VP.

(3) Next, it is shown how the indivisible VPs solve the space paradox:

(3.1)

The energy and volume of space have the speed c, and the energy of space obeys $E^2·\left(1-{\displaystyle \frac{v^2}{c^2}}\right)=E_0^2=0$, as space consists of indivisible VPs $\delta V_j$ with the speed $v_j=c$.

(3.2)

Moreover, the velocities ${\overrightarrow{v}}_j=c·{\overrightarrow{e}}_j$ of the indivisible VPs have stochastic direction vectors ${\overrightarrow{e}}_j$, that are distributed isotropically. Hence, these velocities average to zero.

(3.3)

As a consequence, space is isotropic at a universal scale, see Figure 2. In this manner, the space paradox is solved.

Therefore, we obtain the following deep insight, the stochastic property of space: In an isotropic homogeneous universe, space represents a stochastic average of many indivisible volume portions $\delta V_j$, each with the speed $v_j=c$. Hereby, the velocity vectors ${\overrightarrow{v}}_j$ of the $\delta V_j$ average to zero.

(4) Next, the solution is generalized to heterogeneous space: The early universe was very homogeneous and isotropic [22]. In the universe, the heterogeneity of the locations of mass increased gradually, and thereby the heterogeneity of space evolved gradually [23,36,46,51,52]. Therefore, the natural heterogeneous space is a slight variation of the natural homogeneous and isotropic space. Details of that variation will be derived in the solution of the Hubble tension below.

3.3. Examples of Parts of Space

In this section, the essential parts of natural space with volume V are analyzed.

(1)

Additional volume near a mass M: Near a mass M, there occurs additional volume $\delta V$, see Section 1.1.4. It is at rest relative to M. Moreover, in the vicinity of M, the ACS (see Section 1.1.5) [23,24,39] is nearly at rest at M. Consequently, the speed of the additional volume nearby a mass is clearly smaller than c, i.e., $v<c$. Each additional volume with a speed $v<c$ must be a stochastic average of the indivisible VPs $\delta V_j$ with their speeds $v_j=c$, as space with its volume V consists of indivisible VPs $\delta V_j$ with their speeds $v_j=c$. This is the case for homogeneous and isotropic space and for the case of heterogeneous space which is a slight variation of homogeneous and isotropic space.

Altogether, additional volume that is at rest in the vicinity of a mass M is a stochastic average of indivisible VPs.

(2)

Relative additional volume ${\epsilon }_L$: In general, a VP $\delta V$ is located within an underlying VP $\Delta V_L$. The relative additional volume ${\epsilon }_L$ is the ratio of the VP $\delta V$ with respect to its underlying VP $\Delta V_L$, see Section 1.1.4:

$$\begin{array}{c}{\epsilon }_L:={\displaystyle \frac{\delta V}{\Delta V_L}}={\displaystyle \frac{(\Delta L-\Delta R)·\Delta x·\Delta y}{\Delta L·(\Delta x·\Delta y)}}={\displaystyle \frac{\Delta L-\Delta R}{\Delta L}}={\displaystyle \frac{\delta L}{\Delta L}},\\ \mathrm{with}\delta L:=\Delta L-\Delta R.\end{array}$$

(20)

Thereby, the ratio ${\displaystyle \frac{\delta L}{\Delta L}}$ is usually interpreted as a tensor element, whereby the L-direction is the z-direction or the 3-direction:

$${\epsilon }_{zz}:={\displaystyle \frac{\delta z}{\Delta z}}={\epsilon }_{jj},\mathrm{with}j=3.$$

(21)

In general, a volume portion VP $\delta V$ represents a change in an underlying VP $\Delta V$, and this change represents a tensor element or tensor. Thereby, typically, the change in each VP $\Delta V$ has a quadrupolar structure. A dipolar structure is excluded, as volume cannot be negative. Therefore, each change in an underlying VP $\Delta V$ can be described by a tensor of rank two. It is called change tensor [23,36] ${\epsilon }_{ij}$. In general, an element ${\epsilon }_{ij}$ of a change tensor is the ratio of the change $\delta L_i$ and the underlying length $\Delta L_j$:

$${\epsilon }_{ij}:={\displaystyle \frac{\delta L_i}{\Delta L_j}}.$$

(22)

Hereby, for each normalized direction vector ${\overrightarrow{e}}_j$, the underlying length $\Delta L_j$ is the sum of the original length $\Delta R_j$ and the change $\delta L_j$:

$$\Delta L_j:=\Delta R_j+\delta L_j.$$

(23)

In general, indivisible VPs can have the structure of a change tensor as well, see Figure 3.

(3)

Gravitational wave: Without loss of generality, a gravitational wave (GW) can be described via [53]: It has an angular frequency $\omega$. At a location, a GW has one propagation unit vector ${\overrightarrow{e}}_z$, and two transverse unit vectors ${\overrightarrow{e}}_x$ and ${\overrightarrow{e}}_y$. There are two possible modes. In the first mode, the elongations are ${\epsilon }_{xx}·cos\left(\omega t\right)$ and ${\epsilon }_{yy}·cos\left(\omega t\right)$, with ${\epsilon }_{xx}=-{\epsilon }_{yy}$, see Figure 3. The second mode is equal to the first mode rotated by 45° around ${\overrightarrow{e}}_z$.

The elongations and the velocity of the gravitational wave can be measured with the help of a Michelson interferometer. For instance, Abbott et al. [33] measured the amplitude ${\epsilon }_{xx,max}=1·{10}^{-21}$.

In the theory of waves, the above-observed gravitational wave is a wave with a coherence length that amounts to several wavelengths. Accordingly, a gravitational wave could be interpreted as a coherent state in the framework of quantum physics, if quantum physics is applicable.

Figure 3. Volume portions (VPs) $\delta V_j$ (marked by dotted lines or by wiggly lines with arrow) with speed $v_j=c$ with different velocities ${\overrightarrow{v}}_j=c·{\overrightarrow{e}}_j$: In general, each VP can include a change as marked on the left. In principle, each VP $\delta V_j$ can be characterized very precisely by that change. For instance, the change in a VP $\delta V_j$ moves with its velocity ${\overrightarrow{v}}_j$, and typically, it has a quadrupolar structure (dotted lines). A dipolar structure is excluded, as volume cannot be negative. On the right, these changes are indicated by wiggly lines, for simplicity. The average of the velocities ${\overrightarrow{v}}_j=c·{\overrightarrow{e}}_j$ is zero. This causes global isotropy of space consisting of volume portions with speeds $v_j=c$. Therefore, this solves the space paradox.

Figure 3. Volume portions (VPs) $\delta V_j$ (marked by dotted lines or by wiggly lines with arrow) with speed $v_j=c$ with different velocities ${\overrightarrow{v}}_j=c·{\overrightarrow{e}}_j$: In general, each VP can include a change as marked on the left. In principle, each VP $\delta V_j$ can be characterized very precisely by that change. For instance, the change in a VP $\delta V_j$ moves with its velocity ${\overrightarrow{v}}_j$, and typically, it has a quadrupolar structure (dotted lines). A dipolar structure is excluded, as volume cannot be negative. On the right, these changes are indicated by wiggly lines, for simplicity. The average of the velocities ${\overrightarrow{v}}_j=c·{\overrightarrow{e}}_j$ is zero. This causes global isotropy of space consisting of volume portions with speeds $v_j=c$. Therefore, this solves the space paradox.

3.4. On Quantum Properties of an Indivisible VP

The indivisible VP in homogeneous space is very founded. According to this indivisibility, a minimal extension of an indivisible VP is estimated with the help of the Heisenberg [54] uncertainty relation in quantum physics:

In general, an indivisible VP can have very different forms, and matter or radiation can be placed in the VP. As an example, and as a very simple model, an indivisible VP is analyzed, that has the form of a ball with a radius $\Delta L$, and contains no radiation or matter.

Consequently, the volume is

$$\Delta V_L={\displaystyle \frac{4\pi }{3}}\Delta L^3.$$

(24)

Thus, the VP has the following energy:

$$\Delta E_L={\displaystyle \frac{4\pi }{3}}\Delta L^3·u_{DE}.$$

(25)

As the VP has the speed $v=c$, the energy momentum relation of special relativity theory (SRT) provides the following momentum:

$$\Delta p_L={\displaystyle \frac{\Delta E_L}{c}}={\displaystyle \frac{4\pi }{3}}\Delta L^3·{\displaystyle \frac{u_{DE}}{c}}.$$

(26)

In the rest system of the VP, the momentum is zero, so that the standard deviation alias uncertainty of momentum is estimated by $\Delta p_L$. Similarly, the spatial standard deviation alias uncertainty of the indivisible VP is estimated by

$${\sigma }_L=\Delta L.$$

(27)

The Heisenberg [54] uncertainty relation for these standard deviations is as follows:

$$\Delta p_L·\Delta L \ge {\displaystyle \frac{\hslash }{2}}.$$

(28)

The estimated standard deviations in Equations (26) and (27) are inserted into the above uncertainty relation:

$${\displaystyle \frac{4\pi }{3}}\Delta L^4·{\displaystyle \frac{u_{DE}}{c}} \ge {\displaystyle \frac{\hslash }{2}}.$$

(29)

This uncertainty relation is solved for the spatial uncertainty $\Delta L$:

$$\Delta L\ge \sqrt[4]{{\displaystyle \frac{3\hslash ·c}{8\pi ·u_{DE}}}}=3.957·{10}^{-5}\mathrm{m}.$$

(30)

Altogether, the very particularly considered indivisible VP, which does not contain any matter or radiation, has a minimal spatial uncertainty of $\Delta L\approx 0.03957\mathrm{mm}.$

3.5. Empirical Evidence for Indivisible Volume Portions

The derivation of indivisible VPs is based on isotropy at small scales. Accordingly, empirical evidence for indivisible VPs is of interest. Such evidence is presented in this section.

3.5.1. Estimated Time Uncertainty of Indivisible Volume Portions

In four-dimensional spacetime, the length $\Delta L$ of an indivisible VP is estimated by Equation (30). It is the standard deviation according to the Heisenberg [54] uncertainty relation. In four-dimensional spacetime, the corresponding time uncertainty $\Delta t$ of an indivisible volume portion is estimated by the ratio of $\Delta L$ to c:

$$\Delta t_{indivisibleVP}={\displaystyle \frac{\Delta L}{c}}\ge {\displaystyle \frac{3.957·{10}^{-5}\mathrm{m}}{c}}=1.319·{10}^{-13}\mathrm{s}.$$

(31)

For instance, for the case of an average time of one second, ${\tau }_{av}=1\mathrm{s}$, the fractional uncertainty is estimated as follows:

$$\Delta t_{indivisibleVP,frac}\ge \approx 1.319·{10}^{-13}.$$

(32)

3.5.2. Time Uncertainty of an Atomic Clock Using Gas

Many atomic clocks use gas atoms [55,56]. As these atoms move in natural space, and as an atom is extremely small relative to the estimated extension of an indivisible VP, the clock’s uncertainty is greater than or equal to the uncertainty of an indivisible VP in homogeneous natural space, in general. Similarly, if an object moves within a ship, then the object’s uncertainty is larger or equal to the ship’s uncertainty, in general.

For the case of natural space consisting of indivisible VPs, the uncertainty of space is estimated by $\Delta t_{indivisibleVP,frac}$ in Equation (32). Therefore, in the case of averaging one second,

$$\Delta t_{atomicgasclock,frac}\ge \Delta t_{indivisibleVP,frac}\ge \approx 1.319·{10}^{-13}\mathrm{s}.$$

(33)

In general, the uncertainty of a clock can be reduced by averaging during time ${\tau }_{av}$. Thereby, according to statistics and according to observation, the uncertainty decreases by the factor $1/\sqrt{{\tau }_{av}}$ [55,57]. Empirical data show the following [55]: Atomic clocks that use gas atoms, which are not confined to any particular location or trajectory, and are averaged at most one second, ${\tau }_{av}\le 1\mathrm{s}$, exhibit time uncertainties larger than the estimated uncertainty of indivisible VPs in Equation (32). This provides evidence for the existence of indivisible VPs. Moreover, a control experiment is available:

3.5.3. Time Uncertainty of an Atomic Lattice Clock

In an atomic lattice clock, the used atoms are confined to the potential minima of a standing wave of laser light. Hereby, the standing wave, and the device that generates that wave, are large compared to the estimated size of an indivisible VP. As a consequence, these atoms do not move freely in an indivisible VP of homogeneous natural space, but in artificially generated potential minima. Therefore, the atomic lattice clock’s uncertainty can be smaller than the estimated uncertainty of indivisble VPs in Equation (32). In fact, measurements show that this is clearly the case. There is even a local maximum of the uncertainty depending on the time ${\tau }_{av}$ of averaging [58]:

$$\Delta t_{atomiclatticeclock,frac}\le \Delta t_{maximumatomiclatticeclock,frac}\approx 3·{10}^{-16}.$$

(34)

This control experiment provides evidence for the following: The independence of the uncertainty of natural space and of indivisible VPs can provide a very small uncertainty of an atomic lattice clock.

$$\Delta t_{atomiclatticeclock,frac}\ll \Delta t_{indivisibleVP,frac}$$

(35)

4. Dynamics of Volume Portions

Here, a very valuable and insightful description of VPs is developed. With it, a very general differential equation (DEQ) of the dynamics of VPs is derived.

Here and in the following, mathematically, a VP could be arbitrarily small. Physically, the Heisenberg [54] relation of uncertainty is a lower limit of the standard deviation of a VP. A particular example is analyzed in Section 3.4.

4.1. Additional Volume

Here, the increase $\delta V$ of the volume $\Delta V_R$ of a VP is analyzed. For it, the difference of the increased volume $\Delta V_L$ and the original volume $\Delta V_R$ of the VP is used:

$$\delta V:=\Delta V_L-\Delta V_R$$

(36)

Hereby and in the following, a light travel distance $\Delta L$ is greater than or equal to the corresponding gravitational parallax distance $\Delta R$, i.e., $\Delta L\ge \Delta R$, so the particular case without curvature is included. The above difference $\delta V$ is called additional volume, see Section 1.1.4.

Similarly, as in Section 1.1.3, according to the Hacking [38] criterion, the enlarged volume $\Delta V_L$ and the additional volume $\delta V$ are real, as they can be manipulated. In contrast, the original volume $\Delta V_R$ is an idealization corresponding to the idealization of flat space:

Each VP has a real additional volume $\delta V$ and a real enlarged volume $\Delta V_L$, according to the Hacking [38] criterion. In order to derive general laws of physics, this difference is normalized in the next section:

4.2. Relative Additional Volume

In general, for the purpose of a derivation of general laws of physics, it is valuable to use intensive quantities rather than extensive quantities [59]. In the present case of a VP, it is useful to utilize the ratio of its real additional volume $\delta V$ to its real enlarged volume $\Delta V_L$. Such ratios are formed and analyzed in detail in this section:

That ratio is called relative additional volume ${\epsilon }_L$, see Equation (9):

$${\epsilon }_L:={\displaystyle \frac{\delta V}{\Delta V_L}}.$$

(37)

The relation to the metric tensor is achieved by inserting Equations (7) and (36) into Equation (37):

$${\epsilon }_L:={\displaystyle \frac{\Delta V_L-\Delta V_R}{\Delta V_L}}=1-{\displaystyle \frac{\Delta V_R}{\Delta V_L}}=1-{\displaystyle \frac{1}{\sqrt{g_{RR}}}}.$$

(38)

This is an example of the fact that curvature is describable by a metric tensor or, likewise, by the relative additional volume. Moreover, relative additional volume of a VP is real, as it is a ratio of real quantities of that VP: Each VP has a real relative additional volume ${\epsilon }_L$.

Furthermore, the description with the additional volume is compatible with the volume portions that become necessary for the solution of the space paradox. In contrast, the metric tensor does not include the concept of volume portions, and it does not provide the dynamics of the volume portions. As a consequence, the description with the volume portions provides the additional possibility to derive the dynamics of volume portions. Of course, the metric tensor provides the description of time dilation directly. Volume portions can be described with the help of the Schwarzschild [28] metric. Thereby, a tensor element of the time $g_{tt}={\displaystyle \frac{1}{g_{RR}}}$ is utilized. With it, an interval of coordinate time $dt$ is transformed to an interval of own time $d\tau =dt·\sqrt{g_{tt}}$ [20]. With help of the volume portions, the dynamics of volume portions is derived next:

4.3. Derivation of the Dynamics of Volume Portions

In this section, the dynamics of an arbitrary VP is described by its relative additional volume. The result is achieved with the help of calculus or standard analysis only. As a consequence, the dynamics of VPs derived here is very general.

Firstly, the applicability of calculus or standard analysis is investigated: Calculus or analysis is a mathematical tool, such as algebra and stochastics. Thereby, calculus or analysis does not include a prejudice about the continuity of space and time. The reason is that, in the present investigation, space is a stochastic average of VPs. Hereby, VPs will be characterized by eigenvalues with a spectrum of eigenvalues. This spectrum can turn out to be discrete, continuous or partially discrete and partially continuous. As calculus or analysis is used in this study, these fields of mathematics do not necessarily include a prejudice about the discreteness or continuity of space.

In contrast, an exclusion of calculus and analysis would include a prejudice about the discreteness or continuity of space.

In fact, the space paradox and the implied structure of space as a stochastic average of VPs already deduce an essential degree of discreteness of space. And it is important to investigate the amount of discreteness and continuity of these VPs in the following.

Moreover, in this paper, the physical reality of the used and deduced concepts is permanently tested with the help of the Hacking [38] criterion.

Secondly, the differential equation, DEQ, of VPs is derived: Each localized VP (see Section 3.2, part (1)) has a local maximum of its relative additional volume ${\epsilon }_L$. It depends on the position vector $\overrightarrow{L}$ as well as on time $\tau$. An illustration is Figure 4, whereby the three-dimensional position vector is summarized by a position L. Altogether, a localized volume portion is describable by a term ${\epsilon }_L(\tau ,\overrightarrow{L})$ with a local maximum. If there should be several local maxima or a set of local maxima, then one local maximum can be chosen as a convention.

Calculus alias analysis imply: The local maximum of ${\epsilon }_L(\tau ,\overrightarrow{L})$ has zero slope. Consequently,

$$d{\epsilon }_L(\tau ,\overrightarrow{L}(\tau ))=0.$$

(39)

Here, partial derivatives are utilized:

$$d{\epsilon }_L(\tau ,\overrightarrow{L}(\tau ))={\displaystyle \frac{𝜕}{𝜕\tau }}{\epsilon }_L(\tau ,\overrightarrow{L})·d\tau +{\displaystyle \frac{𝜕}{𝜕\overrightarrow{L}}}{\epsilon }_L(\tau ,\overrightarrow{L})·d\overrightarrow{L}=0.$$

(40)

The VP moves parallel to a corresponding direction unit vector ${\overrightarrow{e}}_v$ of its velocity $\overrightarrow{v}={\displaystyle \frac{𝜕\overrightarrow{L}}{𝜕\tau }}$. Therefore, during a time $d\tau$, the vector $\overrightarrow{L}$ changes by the amount $d\overrightarrow{L}$ = $v·d\tau ·{\overrightarrow{e}}_v$. With it, the total derivative in Equation (40) is

$${\displaystyle \frac{𝜕}{𝜕\tau }}{\epsilon }_L(\tau ,\overrightarrow{L})·d\tau +v·{\overrightarrow{e}}_v·{\displaystyle \frac{𝜕}{𝜕\overrightarrow{L}}}{\epsilon }_L(\tau ,\overrightarrow{L})·d\tau =0.$$

(41)

Next, the derived formula is divided by $d\tau$. Then, the formula is solved for ${\displaystyle \frac{𝜕}{𝜕\tau }}{\epsilon }_L$:

$$\begin{array}{c}\hfill {\displaystyle \frac{𝜕}{𝜕\tau }}{\epsilon }_L(\tau ,\overrightarrow{L})=-v·{\overrightarrow{e}}_v·{\displaystyle \frac{𝜕}{𝜕\overrightarrow{L}}}{\epsilon }_L(\tau ,\overrightarrow{L}),\mathrm{each}\mathrm{indivisible}\mathrm{VP}\mathrm{has}v=c\end{array}$$

(42)

The above expression is the differential Equation, DEQ, of VPs or of volume dynamics (VD). Next, the DEQ of VD is transformed to a Lorentz invariant expression. For it, the expression Equation (42) is squared, and the right hand side is subtracted,

$${\left({\displaystyle \frac{𝜕{\epsilon }_L}{𝜕\tau }}\right)}^2-v^2·{\left({\displaystyle \frac{𝜕{\epsilon }_L}{𝜕\overrightarrow{L}}}\right)}^2=0,\mathrm{each}\mathrm{indivisible}\mathrm{VP}\mathrm{has}v=c$$

(43)

Altogether, each localized VP fulfills the volume dynamics in Equation (42) or in the Lorentz-invariant Equation (43). Thereby, for the following reasons, the derived dynamics is very well founded and insightful: The derivation needs only the presence of a local maximum. Only founded observation is used, see Figure 5. For comparison, general relativity postulates the Einstein Hilbert action or the Einstein field equation [5,6]. Also, quantum physics utilizes postulates [60,61,62]. In the same manner, gravity is based on postulates, via relativity [5], or via Newton [1], or via the graviton [63].

Indeed, we will show later that the differential Equation of the volume dynamics implies gravity, curved space and quantum postulates.

5. Schrödinger Equation: Deduction

Here, we show that each VP fulfills a generalized Schrödinger equation (GSEQ), which implies the usual Schrödinger equation (SEQ), which holds for a slow ($v/c\ll 1$) mass M.

5.1. Volume Dynamics Implies the GSEQ

In this section, it is shown that VPs with their VD imply the GSEQ. The differential equation of volume dynamics is applied to indivisible volume portions with help of three essential steps:

(1)

In a first investigation, the geometry of relative additional volume is analyzed: For each VP, the DEQ (42) of VD is fulfilled. Thereby, the VP moves in the direction ${\overrightarrow{e}}_v$ of its velocity, see Figure 4.

In general, a relative additional volume with an increase $\delta L_3$ with the normalized direction vector ${\overrightarrow{e}}_3$ and with a propagation of indivisible VPs in the same direction, and with an underlying length $\Delta L_3$, is called unidirectional relative additional volume ${\epsilon }_{L,33}$, see Equations (21)–(23). More information about such tensors can be found in the literature [23,36,53,64,65].

(2)

Next, each VP is described by the DEQ of VD. In that DEQ, we utilize v is equal to c. Additionally, we apply the derivative with respect to time, so that the relative additional volume ${\epsilon }_{L,jj}$ becomes ${\displaystyle \frac{𝜕}{𝜕\tau }}{\epsilon }_{L,jj}$, which is usually expressed by ${\dot{\epsilon }}_{L,jj}$. The differential Equation of volume dynamics implies

$${\displaystyle \frac{𝜕}{𝜕\tau }}{\dot{\epsilon }}_{L,jj}=-c·{\overrightarrow{e}}_v·{\displaystyle \frac{𝜕}{𝜕\overrightarrow{L}}}{\dot{\epsilon }}_{L,jj}.$$

(44)

(3)

Next, the expression is multiplied by $i\hslash$.

$$i\hslash ·{\displaystyle \frac{𝜕}{𝜕\tau }}{\dot{\epsilon }}_{L,jj}=c·{\overrightarrow{e}}_v·\left[-i\hslash ·{\displaystyle \frac{𝜕}{𝜕\overrightarrow{L}}}\right]{\dot{\epsilon }}_{L,jj}$$

(45)

Moreover, for the case of an indivisible VP, this indivisibility corresponds to the quantum property with its universal unit of quantization, the Planck [66] constant h or its reduced version $\hslash :={\displaystyle \frac{h}{2\pi }}$.

Accordingly, the correspondence principle is applicable [67]. Moreover, the momentum operator is derived in the Appendix A. On that basis, that operator is identified; it is the above bracket with right angles:

$$i\hslash ·{\displaystyle \frac{𝜕}{𝜕\tau }}{\dot{\epsilon }}_{L,jj}={\overrightarrow{e}}_v·c·\widehat{\overrightarrow{p}}·{\dot{\epsilon }}_{L,jj}$$

(46)

In this DEQ, ${\dot{\epsilon }}_{L,jj}$ represents a rate. That rate is normalized by a factor $t_n$. In physics, $t_n$ has the dimension time and the unit second. Consequently, the normalized rate $t_n·{\dot{\epsilon }}_{L,jj}$ is dimensionless and has the unit one. It will be shown that the normalized rate has the role of a wave function in the above Equation. Accordingly, $t_n·{\dot{\epsilon }}_{L,jj}$ is named wave function $\Psi$. Of course, this can be regarded as an abbreviation, if desired. Therefore, the differential Equation of volume dynamics implies the following DEQ:

$$i\hslash ·{\displaystyle \frac{𝜕}{𝜕\tau }}\Psi =c·{\overrightarrow{e}}_v·\widehat{\overrightarrow{p}}·\Psi$$

(47)

$$\Psi :=t_n·{\dot{\epsilon }}_{L,jj}$$

(48)

The product of the momentum operator and the direction vector of propagation, (${\overrightarrow{e}}_v·\widehat{\overrightarrow{p}}$), is the quantum mechanical operator of the measurable value of the absolute value of the momentum in physics $\widehat{p}$:

$$i\hslash ·{\displaystyle \frac{𝜕}{𝜕\tau }}\Psi =(c·\widehat{p})·\Psi$$

(49)

In the present case of a VP that propagates with the speed c, special relativity implies that the above bracket is equal to the energy E. Consequently, according to the correspondence principle [67], pages 673–674, the bracket is equal to the energy operator:

$$i\hslash ·{\displaystyle \frac{𝜕}{𝜕\tau }}\Psi =\widehat{E}·\Psi$$

(50)

The above equation is identified as the Schrödinger equation (SEQ). The SEQ holds for slow ($v/c\ll 1$) objects. This differential Equation holds for fast objects too, and it implies the differential Equation for slow objects, see next Section 5.2. Accordingly, this differential Equation is a generalized SEQ, it is abbreviated with GSEQ. Moreover, ${\dot{\epsilon }}_{L,rr}$ is real, as it is the derivative of the real ${\epsilon }_{L,rr}$. As a consequence, the wave function $\Psi$ is real, as it is the normalized form of the real ${\dot{\epsilon }}_{L,rr}$.

Each VP fulfills the GSEQ (50) and has a physically real wave function $\Psi \propto {\dot{\epsilon }}_{L,rr}$ in Equation (48). Thereby, according to the Hacking [38] criterion, a wave function has an extraordinary physical reality, as it can be manipulated even so that nonlocal phenomena occur [68].

5.2. The Volume Dynamics Implies the SEQ

In this section, it is shown that volume portions and their volume dynamics imply the Schrödinger equation. The following is derived for localizable quantum objects with $v\le c$ The following derivation steps are essential:

(1) In a first case, entities with $p^2{c}^2\gg m_0^2{c}^4$ [21] are analyzed. Accordingly, following (linear) approximation is applied to $E=\sqrt{p^2{c}^2+m_0^2{c}^4}$, whereby ${\displaystyle \frac{m_0^2{c}^4}{p^2·c^2}}\ll 1$ is considered:

$$E\dot{=}p·c+{\displaystyle \frac{m_0^2{c}^4}{2·p·c}}$$

(51)

As usual, $\dot{=}$ represents an approximation of order one, for small ${\displaystyle \frac{m_0^2{c}^4}{p^2·c^2}}$. With it, the GSEQ takes the following form:

$$i\hslash {\displaystyle \frac{𝜕}{𝜕\tau }}\Psi \dot{=}\left(\widehat{p}·c+{\displaystyle \frac{m_0^2{c}^4}{2·\widehat{p}·c}}\right)·\Psi$$

(52)

(2) A second case is determined by $p^2{c}^2\ll m_0^2{c}^4=:E_0^2$. This implies the items below:

(2a) For each eigenvalue of energy, $E=\sqrt{p^2{c}^2+m_0^2{c}^4}$ is approximated with ${\displaystyle \frac{p^2·c^2}{E_0^2}}\ll 1$:

$$E\dot{=}{E}_0+{\displaystyle \frac{p^2}{2·m_0}}$$

(53)

(2b) In the differential equation, each eigenvalue p is substituted by $\widehat{p}$.

$$i\hslash ·{\displaystyle \frac{𝜕}{𝜕\tau }}{\Psi }_{E_0}\dot{=}\left(E_0+{\displaystyle \frac{{\widehat{p}}^2}{2·m_0}}\right)·{\Psi }_{E_0}$$

(54)

In the above Equation, the rest energy $E_0$ is inherent to ${\Psi }_{E_0}$. As a consequence of linear algebra,

$${\widehat{p}}^2={\widehat{\overrightarrow{p}}}^2.$$

(55)

(2c) An Ansatz of factorization yields

$${\Psi }_{E_0}=\Psi ·exp\left({\displaystyle \frac{E_0·\tau }{i\hslash }}\right).$$

(56)

In Equation (54), the derivative yields

$$E_0{\Psi }_{E_0}+exp\left({\displaystyle \frac{E_0\tau }{i\hslash }}\right)i\hslash {\displaystyle \frac{𝜕\Psi }{𝜕\tau }}\dot{=}\left(E_0+{\displaystyle \frac{{\widehat{p}}^2}{2m_0}}\right)·{\Psi }_{E_0}$$

(57)

(2d) In the above DEQ, the terms $E_0{\Psi }_{E_0}$ compensate each other. Then, the terms $exp\left({\displaystyle \frac{E_0\tau }{i\hslash }}\right)$ compensate each other. Therefore, the common Schrödinger [69] Equation is deduced from the VD:

$$i\hslash ·{\displaystyle \frac{𝜕}{𝜕\tau }}\Psi \dot{=}{\displaystyle \frac{{\widehat{p}}^2}{2·m_0}}·\Psi =\widehat{H}\Psi$$

(58)

In general, a term $E_{pot}$ is added:

$$i\hslash ·{\displaystyle \frac{𝜕}{𝜕\tau }}\Psi \dot{=}{\displaystyle \frac{{\widehat{p}}^2}{2·m_0}}\Psi +E_{pot}\Psi =\widehat{H}\Psi$$

(59)

5.3. Interpretation of VPs and Masses

In this section, the relationship between VPs and masses is elaborated on the basis of the above dynamics of the VD, the GSEQ and the SEQ, and of the Higgs [48] mechanism: Higgs [48] partially deduced and proposed that some elementary particles form their mass from vacuum or from the volume portions by the process of a phase transition [21,70,71,72]:

In physics [12,67,73], empty space is called vacuum, and its energy density is the density $u_{DE}$ of dark energy [9,41]. In the above phase transition, the vacuum transforms to a mass.

As shown by the space paradox, the vacuum is a stochastic average of indivisible VPs. As a consequence, the above phase transition transforms indivisible volume portions to a mass. Thereby, in a typical phase transition, objects change from one phase to another phase. Thereby, these objects are described by the same fundamental dynamics in both phases of the transition, and in the process of the transition. An example is the phase transition of condensation [74,75].

In particular, in the present case, such fundamental dynamics is derived that holds in both phases, and during transition. It is the differential equation of volume dynamics. It holds for VPs in the first phase, and it implies the SEQ that holds for the masses in the second phase. Phase transitions with vol. portions can explain elementary particles [72]. Hereby, of course, the question remains how a mass can emit its wave function. This will be derived in the next section: In part (Section 6), it is shown that the VPs provide gravitational interaction and curved space as a byproduct. It has already been shown that a mass causes additional volume and VPs, see Section 1.1.3 and Section 4.1. In Section 7, the process of local formation of volume (LFV) and of VPs is described, and a corresponding DEQ of the LFV is derived. These VPs formed by a mass M represent the wave function of that mass. In part (Section 10), for the homogen. univ., it is shown that this DEQ describing LFV provides an emergence of the volume of our universe and of its $u_{vol}$ (energy density) value. More generally, with matter fluctuations, $u_{DE}$ (energy dens.) and the difference of $u_{DE}-u_{vol}$ increase gradually with the heterogeneity of the universe, and this difference remained relatively small, even at the present-day universe.

Both results are in accurate concordance with empirical records. Therefore, these findings provide clear evidence for this present theory of VPs. Thereby, each mass M fulfills the SEQ, Equation (59) implied by the GSEQ. M can emerge from vol. portions via a phase change [48,50,72].

6. Gravitation and Curved Space

Nearby a mass M, the phenomena of curved space, an exact gravitational potential and field emerge, see Figure 1. These phenomena are explained and implied by the differential equation (DEQ) of volume dynamics

6.1. Exact Generalized and Gravitational Potential and Field

Here, an exact potential and field are deduced from the volume portions.

6.1.1. Generalized Potential and Field

Near a mass M (Figure 6), the unidirectional radial rel. add. vol. ${\epsilon }_{L,rr}$ has the following properties. In Figure 6, spherical polar coordinates are utilized, and these are applied in the following.

(1)

For each indivisible VP $\delta V_j$, the DEQ of VD describes the relative additional volume ${\epsilon }_{L,rr,j}(\tau ,\overrightarrow{L})$. This is elaborated step by step and progressively in the following paragraphs.

(1.1)

That DEQ is expanded by the factor c:

$$c·{\displaystyle \frac{𝜕}{𝜕\tau }}{\epsilon }_{L,rr,j}(\tau ,\overrightarrow{L})={\overrightarrow{e}}_v·{\displaystyle \frac{𝜕}{𝜕\overrightarrow{L}}}\left(-c^2·{\epsilon }_{L,rr,j}(\tau ,\overrightarrow{L})\right)$$

(60)

(1.2)

At an event $(\tau ,\overrightarrow{L})$, including the set $\mathcal{S}$ of its surrounding events in an interval of time ${\tau }_i\in [\tau -\Delta \tau ,\tau +\Delta \tau ]$ and in some ball $|{\overrightarrow{L}}_i-\overrightarrow{L}| \le \Delta L$, with $\overrightarrow{L}$ near the mass M, the sum of the rel. add. vol. of all indivisible VPs $\delta V_j$ is applied:

$$c·{\displaystyle \frac{𝜕}{𝜕\tau }}\sum _{jin\mathcal{S}}{\epsilon }_{L,rr,j}(\tau ,\overrightarrow{L})={\overrightarrow{e}}_v·{\displaystyle \frac{𝜕}{𝜕\overrightarrow{L}}}\left(-c^2·\sum _{jin\mathcal{S}}{\epsilon }_{L,rr,j}(\tau ,\overrightarrow{L})\right)$$

(61)

(1.3)

In the above diff. equation, the gradient is applied to the bracket. The bracket essentially represents the scalar field of rel add. volume. These facts show that the bracket represents a generalized potential ${\Phi }_{gen}(\tau ,\overrightarrow{L})$

$${\Phi }_{gen}(\tau ,\overrightarrow{L}):=-c^2·\sum _{jin\mathcal{S}}{\epsilon }_{L,rr,j}$$

(62)

Moreover, the special potential must be regarded as a generalized potential, as it does not describe a static scalar function. Instead, it describes very fast indivisible volume portions, as ${\epsilon }_{L,rr,j}$ describes the rel. add. vol. of indivisible vol. portions that move rapidly in various directions. In fact, an appropriate average of that generalized volume will turn out to be the common gravitational potential.

(1.4)

In principle, in calculus, the gradient (multiplied by minus one) of that generalized potential would be the respective generalized field ${\overrightarrow{G}}_{gen}$, see Figure 6. In order to apply that gradient, the potential must be a differentiable function. In the present case of VPs, differentiability emerges only in a sufficiently smooth average. Algebraically, the gradient operator can be applied in principle. Under these conditions, the gradient is applied:

$${\overrightarrow{G}}_{gen}(\tau ,\overrightarrow{L}):=-{\displaystyle \frac{𝜕}{𝜕\overrightarrow{L}}}\left(-c^2·\sum _{jin\mathcal{S}}{\epsilon }_{L,rr,j}\right)=-{\displaystyle \frac{𝜕}{𝜕\overrightarrow{L}}}{\Phi }_{gen}(\tau ,\overrightarrow{L})$$

(63)

(1.5)

Therefore, the DEQ of VD in Equation (60), under the conditions in item (1.4), transforms to the following form of the rate gravity relation:

$$c·{\displaystyle \frac{𝜕}{𝜕\tau }}\sum _{jin\mathcal{S}}{\epsilon }_{L,rr,j}={\overrightarrow{e}}_v·{\displaystyle \frac{𝜕}{𝜕\overrightarrow{L}}}{\Phi }_{gen}(\tau ,\overrightarrow{L})=-{\overrightarrow{e}}_v{\overrightarrow{G}}_{gen}(\tau ,\overrightarrow{L})$$

(64)

The potential and field in the above equation are generalized to the respective quantities for one indivisible VP $\delta V_j$. Thereby, the potential is

$${\Phi }_{gen,j}(\tau ,\overrightarrow{L})=-c^2{\epsilon }_{L,rr,j}(\tau ,\overrightarrow{L}),$$

(65)

and the field is as follows:

$${\overrightarrow{G}}_{gen,j}(\tau ,\overrightarrow{L})=-{\displaystyle \frac{𝜕}{𝜕\overrightarrow{L}}}{\Phi }_{gen,j}(\tau ,\overrightarrow{L})=c^2{\displaystyle \frac{𝜕}{𝜕\overrightarrow{L}}}{\epsilon }_{L,rr,j}.$$

(66)

These relations are results of the VD. Formally, these results can be obtained by an application of the above average with one indivisible VP only.

For comparison, expectation values of the field have been derived from the VD as follows: Firstly, a quantum field theory has been derived from the VD. Secondly, the expectation value of a field has been derived from the quantum field theory [23,65].

(2)

For these indivisible VPs, and for each event $(\tau ,\overrightarrow{L})$, a very essential new scalar is identified and introduced: It is the rate gravity scalar $RG{S}_{gen}$, and it is equal to zero. This is the rate gravity relation. It can be written in various useful forms:

$$RG{S}_{gen}:={\left(c·{\displaystyle \frac{𝜕}{𝜕\tau }}\sum _{jin\mathcal{S}}{\epsilon }_{L,rr,j}\right)}^2-{\overrightarrow{G}}_{gen}^2,\mathrm{thus}$$

(67)

$$RG{S}_{gen}={\left(c·{\displaystyle \frac{𝜕}{𝜕\tau }}\sum _{jin\mathcal{S}}{\epsilon }_{L,rr,j}\right)}^2-\sum _{k=1}^{dimensionD}{G}_{gen,k}^2\mathrm{and}$$

(68)

$$RG{S}_{gen}={\left(c·{\displaystyle \frac{𝜕}{𝜕\tau }}\sum _{jin\mathcal{S}}{\epsilon }_{L,rr,j}\right)}^2-{\left(c{\displaystyle \frac{𝜕}{𝜕\overrightarrow{L}}}{\Phi }_{gen}\right)}^2\mathrm{and}$$

(69)

$$RG{S}_{gen}=0$$

(70)

(3)

For each underlying volume $\Delta V_L$, and for each event $(\tau ,\overrightarrow{L})$, which is at a very useful distance $d_{GP}$ from M ($d_{GP}$ abbreviates the gravit. par. dist.), a respective generalized field is proportional to ${\displaystyle \frac{1}{R^2}}$:

$$|{\overrightarrow{G}}_{gen}|\propto {\displaystyle \frac{1}{R^2}},\mathrm{for}D=3,\mathrm{and}\mathrm{for}\Delta V_L$$

(71)

This result is derived as follows:

(3.1)

Equation (64) is multiplied by ${\overrightarrow{e}}_v$. Hence, the field is as follows:

$${\overrightarrow{e}}_v·c·{\displaystyle \frac{𝜕}{𝜕\tau }}\sum _{jin\mathcal{S}}{\epsilon }_{L,rr,j}=-{\overrightarrow{G}}_{gen}(\tau ,\overrightarrow{L})$$

(72)

(3.2)

Integration with respect to $\tau$ from zero to a time $\delta \tau$ yields:

$${\overrightarrow{e}}_v·c·\sum _{jin\mathcal{S}}{\epsilon }_{L,rr,j}(\delta \tau )=-{\int }_0^{\delta \tau }{\overrightarrow{G}}_{gen}(\tau ,\overrightarrow{L})d\tau$$

(73)

For it, a very small time $\delta \tau$ is used. In particular, that time is sufficiently small, so that the above integral in the above equation is approximated by $\delta \tau ·{\overrightarrow{G}}_{gen}$. Thus, the field is as follows:

$${\overrightarrow{e}}_v·c·\sum _{jin\mathcal{S}}{\epsilon }_{L,rr,j}(\delta \tau )=-\delta \tau ·{\overrightarrow{G}}_{gen}$$

(74)

(3.3)

The definition ${\epsilon }_{L,rr,j}={\displaystyle \frac{\delta V_{L,rr,j}}{d{V}_L}}$ is utilized. In the whole emerging gravity of this theory, it is essential that the energy density $u_{vol}$ of volume is positive. (it is approximately the same as $u_{DE}$, which is positive [9,15,16]). Therefore, the indivisible VP $\delta V_{L,rr,j}$ is proportional to the corresponding energy $\delta E_j$. The latter is proportional to the momentum $\delta p_j=\delta E_j/c$, as each VP propagates with the speed c, that is: $\delta V_{L,rr,j}={\displaystyle \frac{\delta E_j}{u_{vol}}}={\displaystyle \frac{c·\delta p_j}{u_{vol}}}$.

(3.4)

An underlying volume $d{V}_L$ is considered. It is very useful to apply the shell that has its center at the field generating mass M. That volume is $d{V}_L=A_D·R^{D-1}·dL$. Consequently, for $D=3$, the field is as follows:

$$-{\overrightarrow{e}}_v·{\displaystyle \frac{c^2}{u_{vol}·A_D·dL}}·{\displaystyle \frac{{\sum }_{jin\mathcal{S}}\delta p_j}{\delta \tau }}·{\displaystyle \frac{1}{R^2}}={\overrightarrow{G}}_{gen}$$

(75)

Hereby, the thickness $dL$ of each shell is chosen constant, so that it does not depend on R. In that case, in the above equation the left ratio is constant. The middle ratio is identified with the momentum current, which propagates through each of the shells. This momentum current does not change, since none of the shells causes any momentum. Consequently, ${\displaystyle \frac{1}{R^2}}\propto \left|{\overrightarrow{G}}_{gen}\right|$. Thus, it is proved. A version of the corresponding result for a dimension $D>3$ is shown in Carmesin [65] ([THM 9]).

(4)

Moreover, according to the Galileo equivalence principle, the mass M is proportional to the field (the mass may be interpreted as the source of the field) ${\overrightarrow{G}}_{gen}(R)$: $\left|{\overrightarrow{G}}_{gen}(R)\right|\propto {\displaystyle \frac{M}{R^2}}$. Consequently, there is a proportionality factor (it is interpreted as a universal constant) $G_{gen}$:

$$\left|{\overrightarrow{G}}_{gen}(|R|)\right|={\displaystyle \frac{G_{gen}M}{R^2}},\mathrm{for}D=3.$$

(76)

In general, the value of a universal constant, such as $G_{gen}$, must be obtained from observation. Accordingly, $G_{gen}$ is obtained in the next section. Additionally, the curvature of space is analyzed:

6.1.2. Spin, Statistics and the Additive Structure of VPs, Potentials and Fields

In general, a VP has the tensor property of a quadrupolar structure, see Section 3. Consequently, it is represented by a tensor of rank two.

Therefore, at the level of quantum physics, an indivisible VP has an integer valued spin, see Landau and Lifschitz [76], §58, Carmesin [77]:

$$S=n·\hslash ,\mathrm{with}\mathrm{a}\mathrm{natural}\mathrm{number}n.$$

(77)

As a consequence, at the level of quantum physics, an indivisible VP is a boson, see Landau and Lifschitz [76], §64, Carmesin [77]. Consequently, at the level of quantum physics, an indivisible VP obeys the Bose [78] statistics, alias Bose-Einstein statistics, see Landau and Lifschitz [76], §64, Sakurai and Napolitano [61], Section 7.2.

The Bose-Einstein statistics imply that several bosons can exist at the same place. Consequently, the additional volumes ${\epsilon }_{L,rr,j}$ caused by different masses $m_j$ can exist simultaneously at each point P in the universe, and these ${\epsilon }_{L,rr,j}$ add up at P. This, in turn, implies that the potential ${\Phi }_{gen}\left(P\right)$ at P is the sum of these additional volumes ${\epsilon }_{L,rr,j}\left(P\right)$ multiplied by $-c^2$. This founds Equation (62) at the level of quantum physics. As a consequence of this potential, the generalized field ${\overrightarrow{G}}_{gen}\left(P\right)=-{\displaystyle \frac{𝜕}{𝜕\overrightarrow{L}}}{\Phi }_{gen}\left(P\right)$ has the same additive structure.

6.2. Curvature near a Mass

In fact, the DEQ of VD can be used to deduce the curvature near a mass M. This is done step by step and progressively in the following paragraphs:

(1)

As shown above, a shell is considered, which has its center at M, and has a gravitational parallax radius R, the relative additional volume ${\epsilon }_{L,rr}$ is as follows:

$${\epsilon }_{L,rr}={\displaystyle \frac{d{V}_L-d{V}_R}{d{V}_L}}=1-{\displaystyle \frac{d{V}_R}{d{V}_L}}=1-{\displaystyle \frac{4\pi R^{2}dR}{4\pi R^{2}dL}}=1-{\displaystyle \frac{dR}{dL}}.$$

(78)

The resulting fraction ${\displaystyle \frac{dR}{dL}}$ is called position factor [65]:

$${\epsilon }_E\left(R\right):={\displaystyle \frac{dR}{dL}}=1-{\epsilon }_{L,rr}$$

(79)

Hereby, the ratio ${\displaystyle \frac{dL}{dR}}$ is equal to $\sqrt{g_{rr}}$:

$$\sqrt{g_{rr}}:={\displaystyle \frac{dL}{dR}}={\displaystyle \frac{1}{{\epsilon }_E\left(R\right)}}.$$

(80)

More generally, at each R, or at each event $(\tau ,\overrightarrow{L})$ with $|\overrightarrow{L}|\ge R$, the curvature can be generalized for the case of a single indivisible VP $\delta V_j$, with the relative additional volume ${\epsilon }_{L,rr,j}$, as follows:

$$g_{rr,j}:={\displaystyle \frac{1}{{\epsilon }_{E,j}^2}}:={\displaystyle \frac{1}{{(1-{\epsilon }_{L,rr,j})}^2}}.$$

(81)

(2)

A DEQ for the position factor is derived: For it, ${\displaystyle \frac{𝜕}{𝜕L}}$ is applied to Equation (79). This implies:

$${\displaystyle \frac{𝜕}{𝜕L}}{\epsilon }_E={\displaystyle \frac{𝜕R}{𝜕L}}{\displaystyle \frac{𝜕}{𝜕R}}{\epsilon }_E={\epsilon }_E{\displaystyle \frac{𝜕}{𝜕R}}{\epsilon }_E=-{\displaystyle \frac{𝜕{\epsilon }_{L,rr}}{𝜕L}}$$

(82)

The potential ${\Phi }_{gen}=-c^2{\epsilon }_{L,rr}$ is solved for ${\epsilon }_{L,rr}$, and this is inserted into Equation (82):

$${\epsilon }_E\left(R\right){\displaystyle \frac{𝜕{\epsilon }_E}{𝜕R}}={\displaystyle \frac{𝜕{\Phi }_{gen}/c^2}{𝜕L}}$$

(83)

The gradient ${\displaystyle \frac{𝜕}{𝜕L}}={\overrightarrow{e}}_v{\displaystyle \frac{𝜕}{𝜕\overrightarrow{L}}}$ is applied:

$${\epsilon }_E\left(R\right){\displaystyle \frac{𝜕{\epsilon }_E}{𝜕R}}={\overrightarrow{e}}_v{\displaystyle \frac{𝜕{\Phi }_{gen}/c^2}{𝜕\overrightarrow{L}}}$$

(84)

The field $-{\displaystyle \frac{𝜕{\Phi }_{gen}}{𝜕\overrightarrow{L}}}={\overrightarrow{G}}_{gen}$ is identified:

$$-{\overrightarrow{e}}_v{\displaystyle \frac{1}{c^2}}{\overrightarrow{G}}_{gen}={\epsilon }_E\left(R\right)·{\displaystyle \frac{𝜕{\epsilon }_E}{𝜕R}}$$

(85)

${\overrightarrow{G}}_{gen}$ and ${\overrightarrow{e}}_v$ have opposite directions. Consequently, the field is as follows:

$${\displaystyle \frac{1}{c^2}}|{\overrightarrow{G}}_{gen}|={\epsilon }_E\left(R\right){\displaystyle \frac{𝜕{\epsilon }_E}{𝜕R}}$$

(86)

Equation (76) is applied:

$${\displaystyle \frac{G_{gen}M}{R^2{c}^2}}={\epsilon }_E\left(R\right){\displaystyle \frac{𝜕{\epsilon }_E}{𝜕R}}$$

(87)

(3)

The above diff. Equation is solved:

$$\int {\displaystyle \frac{G_{gen}M}{R^2{c}^2}}dR=\int {\epsilon }_E\left(R\right)d{\epsilon }_E$$

(88)

Integration and a respective K yield

$$K-{\displaystyle \frac{G_{gen}M}{R{c}^2}}={\displaystyle \frac{1}{2}}{\epsilon }_E^2.$$

(89)

The solution with respect to ${\epsilon }_E$ provides

$${\epsilon }_E=\sqrt{2K-{\displaystyle \frac{2G_{gen}M}{c^2}}{\displaystyle \frac{1}{R}}}.$$

(90)

There is zero curvature at $R\to \infty$. Consequently, ${\epsilon }_E={\displaystyle \frac{dR}{dL}}=1.0$. As a consequence, $K=1/2$:

$${\epsilon }_E=\sqrt{1-{\displaystyle \frac{2G_{gen}M}{c^2}}{\displaystyle \frac{1}{R}}}$$

(91)

(4)

${\epsilon }_E$ in Equation (91) is related to observation:

$${\epsilon }_E=\sqrt{1-{\displaystyle \frac{2G_{gen}M}{c^2}}{\displaystyle \frac{1}{R}}}={\displaystyle \frac{1}{\sqrt{g_{rr}}}}$$

(92)

Empirical records show [20,79]:

$$\sqrt{1-{\displaystyle \frac{2GM}{c^2}}{\displaystyle \frac{1}{R}}}={\displaystyle \frac{1}{\sqrt{g_{rr}}}}=\sqrt{1-{\displaystyle \frac{R_S}{R}}}.$$

(93)

Next, the Schwarzschild radius $R_S:={\displaystyle \frac{2GM}{c^2}}$ is related to the two tensor elements in Equations (92) and (93). The result is $G_{gen}$ = G (the universal constant of gravity $G=6.67430\left(15\right)·{10}^{-11}{\displaystyle \frac{{\mathrm{m}}^3}{\mathrm{kg}·{\mathrm{s}}^2}}$). Consequently, ${\overrightarrow{G}}_{gen}$ = ${\overrightarrow{G}}^{*}$. The respective relation holds for the potentials. The derived quantities, terms and relations are exact, since they have been derived exactly. Therefore, in the vicinity of a mass, the indivisible VPs cause a generalized potential, a generalized field, their exact field and potential of the gravitational interaction, and curved space as well as time. The respective Equations are derived exactly.

6.3. A Discussion of Gravity and Curvature

Near a mass M, the volume dyn. determines exact quantities, terms and relations: ${\epsilon }_{L,rr}\left(r\right)$, which provides ${\Phi }_{exact}=-c^2·{\epsilon }_{L,rr}\left(r\right)$, which provides ${\overrightarrow{G}}_{exact}^{*}=-{\displaystyle \frac{𝜕}{𝜕\overrightarrow{L}}}{\Phi }_{exact}$. Next, these valuable and insightful properties are analyzed and interpreted:

6.3.1. Transmission of the Potential and Field

The indivisible VPs $\delta V_j$ with their rel. add. vol. ${\epsilon }_{L,rr,j}\left(r\right)$ cause a net outward propagation. The indivisible vol. portions transfer momentum, as $u_{vol}>0$. Consequently, the indivisible vol. portions transfer the interaction of gravity. It confirms the graviton idea proposed by Blokhintsev and Galperin [63].

Additionally, these indivisible vol. portions explain and imply curved space. Thereby, curvature is explained with the help of the sum of indivisible VPs. More generally, the curvature of a single indivisible VP has been generalized, see Equations (80) and (81).

6.3.2. On the Exactness of the Potential and Field

The generalized potential and field derived here are exact. In contrast, the gravitational potential is only an approximation in Newton’s theory. For instance, correction terms depending on the gravitational parallax distance R and on the velocity $\overrightarrow{v}$ have been elaborated in Post-Newtonian approximations [80] (see Equation (2.49) in [81]). Accordingly, the following question arises: How is the exactness achieved here?

Essentially, this is achieved here by the application of especially useful distance measures and coordinate systems:

Firstly, for each point P in the universe, there is an adequate coordinate system (ACS), so that an object at rest in the ACS has an absolute zero of velocity, see Section 1.1.5 or Carmesin [23,24,39]. This ACS is used here. As a consequence, the velocity terms in the Post-Newtonian approximation are zero.

Secondly, two measurable distance measures are used, see Figure 1: $d_{LT}$ [82] (light travel distance), and $d_{GP}$ [23,36,40] (gravitational parallax distance, it can also be measured as a circumferential distance [83] (see Section (2.6) in [84]).

Hereby, according to the Hacking [38] criterion, the light travel distance $d_{LT}$ to a field-generating mass M is real, as an increase of M can change that distance $d_{LT}$. In contrast, the $d_{GP}$ of an object to a field generating mass M is idealized, since it represents the limit M to zero of the light travel distance:

$$d_{GP}=\underset{M\to 0}{lim}{d}_{LT}.$$

(94)

This is a typical idealization based on a limit [44]. Though the gravitational parallax distance $d_{GP}$ is idealized and not real, that distance $d_{GP}$ can be measured, and that distance $d_{GP}$ provides the ${\displaystyle \frac{1}{R^2}}$ law, see Figure 6.

Thirdly, the exact potential and field are determined as follows: For the case of a point P, the idealized gravitational parallax distance $d_{GP}$ from P to the field-generating mass M is measured or determined by other means. With it, ${\Phi }_{exact}$ and $\overrightarrow{G}{*}_{exact}$ are functions of $d_{GP}=R$. The result is exact, as no approximation has been used in the derivation of these equations.

But in the theory proposed by Newton, the flatness of space has been introduced as a postulate. Accordingly, a user might use that postulate and the light travel distance $d_{LT}\ne dR$, and the equations for potentials and fields in Newtonian mechanics. In such a postulate-based determination of the potential or field, the result is an approximation only.

6.3.3. Advantage of the Exact Potential and Field

Indivisible vol. portions provide exactly the transmission, potential, and field of gravity, as well as curved space, in an impartible form. Thereby, by using the adequate coordinate system and the measurable idealized gravitational parallax distance, relatively short, highly structured, and clarifying equations can be used. Moreover, the results are part of the unification of relativity, gravity and quanta. This unification uses the relative difference ${\epsilon }_{L,rr}$, which is based on both distance measures: the real $d_{LT}$ and the idealized $d_{GP}$.

7. Local Formation of Volume in Nature

Here, the dynamics of local formation of volume (LFV) is derived.

Near a mass M, add. vol. $\delta V$ propagates outwards. Hereby, the amount of add. vol. increases. As a consequence, locally formed volume, LFV, emerges $\underline{\delta }V$ or $\underline{\delta }{V}_{rr}$ in radial orientation r. Hereby, at the following, an underlined variable marks the formed volume or the process of formation of volume. Such LFV can be caused via a mass M.

In detail, LFV forms at a gradient of rel. add. vol. ${\epsilon }_{L,jj}$, or of the potential ${\Phi }_L=-c^2·{\epsilon }_{L,jj}$, or at a gravitational field ${\overrightarrow{G}}^{*}$. Consequently, a gravitational field causes LFV.

7.1. Definition of Locally Formed Volume

Add. vol. $\underline{\delta }{V}_{jj}$, which emerges in $d{V}_L$, in a direction ${\overrightarrow{e}}_j$, and in a time interval $\underline{\delta }\tau$, is described by

$${\underline{\dot{\epsilon }}}_{L,jj}:={\displaystyle \frac{\underline{\delta }{V}_{jj}}{\underline{\delta }\tau ·d{V}_L}}$$

(95)

It is the named normalized rate of unidirectional LFV.

7.2. Law of Locally Formed Volume

Here, the process of LFV is explained and formulated.

Near a mass M (it can also be an effective mass $M_{eff}$), at a $d_{GP}$ named R from M or $M_{eff}$, the normalized rate is as derived step by step and progressively in the following paragraphs:

(1)

A far distance approximation, FDA, is introduced, in which $R_S/R\ll 1$. Hereby, at order ${(R_S/R)}^1$, the normalized rate is

$${\underline{\dot{\epsilon }}}_{L,rr}^2·c^2=G_{gen,r}^2.$$

(96)

Thereby, $G_{gen,r}$ denotes the r-component (parallel to ${\overrightarrow{e}}_r$) of that field. $G_{gen,r}$ causes ${\underline{\dot{\epsilon }}}_{L,rr}$. In general, ${\overrightarrow{G}}_{gen,j}$ causes ${\underline{\dot{\epsilon }}}_{L,jj}$.

$${\underline{\dot{\epsilon }}}_{L,jj}^2·c^2=G_{gen,j}^2={\left({\overrightarrow{G}}_j^{*}\right)}^2\mathrm{with}{\overrightarrow{G}}_{gen}=-G_{gen,j}·{\overrightarrow{e}}_j$$

(97)

(2)

Consequently, non-diagonal components ${\underline{\dot{\epsilon }}}_{L,ij,i\ne j}$ cause no LFV.

(3)

The rate ${\underline{\dot{\epsilon }}}_{L,rr}$ has the exact value

$${\underline{\dot{\epsilon }}}_{L,rr}=c·{\displaystyle \frac{2}{R}}·{\epsilon }_E·\left(1.0-{\epsilon }_E-{\displaystyle \frac{1}{4{\epsilon }_E^2·\frac{R}{R_S}}}\right).$$

(98)

(4)

In general, ${\displaystyle \frac{dR}{dL}}$ is abbreviated ${\epsilon }_F:={\displaystyle \frac{dR}{dL}}$, it is a function that has to be determined. As a matter of definition, ${\underline{\dot{\epsilon }}}_{L,rr}$ = ${\displaystyle \frac{\frac{𝜕}{𝜕\tau }\delta V\underline{\delta }\tau }{d{V}_L\underline{\delta }\tau }}$. The substitution ${\displaystyle \frac{𝜕}{𝜕\tau }}$ = ${\displaystyle \frac{𝜕L}{𝜕\tau }}·{\displaystyle \frac{𝜕}{𝜕L}}$ = $c·{\displaystyle \frac{𝜕}{𝜕L}}$ provides ${\underline{\dot{\epsilon }}}_{L,rr}$ = $c{\displaystyle \frac{\frac{𝜕}{𝜕L}\delta V}{d{V}_L}}$. The substitution ${\displaystyle \frac{𝜕}{𝜕L}}$ = ${\displaystyle \frac{𝜕R}{𝜕L}}{\displaystyle \frac{𝜕}{𝜕R}}$ = ${\epsilon }_F·{\displaystyle \frac{𝜕}{𝜕R}}$ yields ${\underline{\dot{\epsilon }}}_{L,rr}$ = ${\epsilon }_{F}c{\displaystyle \frac{\frac{𝜕}{𝜕R}\delta V}{d{V}_L}}$. Using $d{V}_L=4\pi R^{2}dL=4\pi R^{2}dR/{\epsilon }_F$, and $\delta V=4\pi R^2(dL-dR)$ = $4\pi R^{2}dR(1/{\epsilon }_F-1)$ yields: ${\underline{\dot{\epsilon }}}_{L,rr}$ = ${\epsilon }_F^{2}c{\displaystyle \frac{4\pi dR\frac{𝜕}{𝜕R}{R}^2(1/{\epsilon }_F-1)}{4\pi R^{2}dR}}$ = ${\epsilon }_F^{2}c{\displaystyle \frac{\frac{𝜕}{𝜕R}{R}^2(1/{\epsilon }_F-1)}{R^2}}$. Evaluation of the derivative yields ${\underline{\dot{\epsilon }}}_{L,rr}$ = ${\epsilon }_F^2·c·{\displaystyle \frac{2R(1/{\epsilon }_F-1)-R^2·{\epsilon }_F^{\prime }/{\epsilon }_F^2}{R^2}}$ = ${\epsilon }_F·c·{\displaystyle \frac{2(1.0-{\epsilon }_F)}{R}}$ − $c·{\epsilon }_F^{\prime }$. ${\epsilon }_L$ = ${\displaystyle \frac{dL-dR}{dL}}=1-{\epsilon }_F$ and ${\epsilon }_F^{\prime }=-{\epsilon }_L^{\prime }$ imply

$${\underline{\dot{\epsilon }}}_{L,rr}={\displaystyle \frac{2c}{R}}·{\epsilon }_L·(1.0-{\epsilon }_L)+c·{\epsilon }_L^{\prime }.$$

(99)

(5)

If desired, Equation (99) can be studied in a computer simulation. It provides a metric, typically with rotation symmetry:

$${\displaystyle \frac{dR}{dL}}=1.0-{\epsilon }_L\mathrm{or}{\displaystyle \frac{dL}{dR}}={\displaystyle \frac{1.0}{1.0-{\epsilon }_L}}=\sqrt{|g_{rr}|}={\displaystyle \frac{1.0}{\sqrt{|g_{tt}|}}}$$

(100)

7.3. Derivation of the Law of LFV

Here, the process and the law of LFV are deduced.

Part (1): Shells with their center at M and $d_{GP}$-based radius R are studied: One shell has a thickness $dL$. It has the investigated volume $d{V}_L$. A second shell represents the additional shell-volume $\delta V$. It has the thickness $\delta R$, and shell-volume $\delta V=4\pi R^2\delta R$. A third shell represents the LFV $\underline{\delta }V$. It has the thickness $\underline{\delta }R$, and the shell-volume $\underline{\delta }V=4\pi R^2\underline{\delta }R$.

It is sufficient to analyze the LFV at order one (exponent one) of $dR$. The result is exact, since $dR$ can be chosen arbitrarily small. If the additional shell-volume $\delta V\left(R\right)$ increases as a function of R, formed volume $\underline{\delta }{V}_{rr}$ emerges (and the direction r is radial).

In general, observable additional volume $\delta V$ is a function of R. Consequently, the LFV is the change in $\delta V$ as a function of R. Therefore, the change $\underline{\delta }{V}_{rr}$ of volume in a shell with thickness $\underline{\delta }R$ is equal to the derivative of the observable additional volume ${\displaystyle \frac{𝜕}{𝜕R}}\delta V$ multiplied by that thickness $\underline{\delta }R$:

$$\underline{\delta }{V}_{rr}:=\underline{\delta }R{\displaystyle \frac{𝜕}{𝜕R}}\delta V$$

(101)

Thereby, $\delta V$ is as follows:

$$\delta V=d{V}_L-d{V}_R=4\pi R^2·(dL-dR)=4\pi R^{2}dR·({\epsilon }_E^{-1}-1)$$

(102)

The FDA is utilized at order one, ${\displaystyle \frac{R_S}{R}}<<1$, consequently

$${\epsilon }_E^{-1}\dot{=}1+0.5·{\displaystyle \frac{R_S}{R}}.$$

(103)

As a consequence,

$${\displaystyle \frac{𝜕}{𝜕R}}\delta V={\displaystyle \frac{𝜕}{𝜕R}}\left(4\pi R^{2}dR·{\displaystyle \frac{1}{2}}·{\displaystyle \frac{R_S}{R}}\right)=2\pi ·dR·R_S$$

(104)

As a consequence, Equation (101) becomes

$$\underline{\delta }{V}_{rr}\dot{=}2·\pi ·dR·R_S·\underline{\delta }R.$$

(105)

Using $\underline{\delta }\tau ={\epsilon }_E·\underline{\delta }t={\epsilon }_E·\underline{\delta }R/c$, and $dL=dR/{\epsilon }_E$, we derive

$${\underline{\dot{\epsilon }}}_{L,rr}={\displaystyle \frac{\underline{\delta }{V}_{rr}}{\underline{\delta }\tau ·d{V}_L}}={\displaystyle \frac{2\pi ·dR·R_S·\underline{\delta }R}{\underline{\delta }R/c·4\pi R^{2}dR}}={\displaystyle \frac{R_S·c}{2R^2}}$$

(106)

Utilizing $R_S=2GM/c^2$, we deduce

$$c·{\underline{\dot{\epsilon }}}_{L,rr}\dot{=}{\displaystyle \frac{G·M}{R^2}}.$$

(107)

Application of the field yields

$${\overrightarrow{G}}^{*}={\overrightarrow{G}}_{gen}=-{\displaystyle \frac{G·M}{R^2}}·{\overrightarrow{e}}_r=-G_{gen,r}·{\overrightarrow{e}}_r.$$

(108)

Part (2): Application of the square yields Equation (97):

$${\underline{\dot{\epsilon }}}_{L,rr}^2·c^2={\overrightarrow{G}}_{gen,r}^2$$

(109)

In general, near an effective mass, no invariance related to rotation exists. Then, each direction is analyzed [84] ([chapter 9]). Each field ${\overrightarrow{G}}_{gen}$ parallel ${\overrightarrow{e}}_j$ causes the unidirectional rate ${\underline{\dot{\epsilon }}}_{L,jj}$, see Equation (97).

Part (3): The rate in Equation (95) is studied:

$${\underline{\dot{\epsilon }}}_{L,rr}={\displaystyle \frac{\underline{\delta }{V}_{rr}}{\underline{\delta }\tau }}·{\displaystyle \frac{1}{d{V}_L}}$$

(110)

$\underline{\delta }{V}_{rr}$ is the difference of two values of $\delta V_{rr}$

$${\underline{\dot{\epsilon }}}_{L,rr}={\displaystyle \frac{1}{d{V}_L}}·{\displaystyle \frac{\delta V_{rr}({\tau }_0+\underline{\delta }\tau )-\delta V_{rr}\left({\tau }_0\right)}{\underline{\delta }\tau }}$$

(111)

Since $\underline{\delta }\tau$ can be chosen arbitrarily small, order one in $\underline{\delta }\tau$ is sufficient:

$${\underline{\dot{\epsilon }}}_{L,rr}={\displaystyle \frac{1}{d{V}_L}}·{\displaystyle \frac{\frac{𝜕}{𝜕\tau }\delta V_{rr}·\underline{\delta }\tau }{\underline{\delta }\tau }}={\displaystyle \frac{1}{d{V}_L}}·{\displaystyle \frac{𝜕}{𝜕\tau }}\delta V_{rr}={\displaystyle \frac{1}{d{V}_L}}·\underset{c}{\underbrace{{\displaystyle \frac{𝜕L}{𝜕\tau }}}}·{\displaystyle \frac{𝜕}{𝜕L}}\delta V_{rr}$$

(112)

Substitution of L by R yields:

$${\underline{\dot{\epsilon }}}_{L,rr}={\displaystyle \frac{c}{d{V}_L}}·\underset{{\epsilon }_E}{\underbrace{{\displaystyle \frac{𝜕R}{𝜕L}}}}·{\displaystyle \frac{𝜕}{𝜕R}}\delta V_{rr}={\displaystyle \frac{{\epsilon }_{E}c}{d{V}_L}}·{\displaystyle \frac{𝜕}{𝜕R}}\delta V_{rr}$$

(113)

Next, $\delta V_{rr}=4\pi R^2\delta R=4\pi R^2(dL-dR)=4\pi R^{2}dR({\epsilon }_E^{-1}-1)$ is used, as well as $d{V}_L=4\pi R^{2}dR/{\epsilon }_E$:

$${\underline{\dot{\epsilon }}}_{L,rr}={\displaystyle \frac{{\epsilon }_E·c}{4\pi R^{2}dR/{\epsilon }_E}}·4\pi dR·{\displaystyle \frac{𝜕}{𝜕R}}\left[R^2·({\epsilon }_E^{-1}-1)\right]$$

(114)

The derivative is evaluated:

$${\underline{\dot{\epsilon }}}_{L,rr}={\displaystyle \frac{{\epsilon }_E^2·c}{R^2}}·\left[2R·({\epsilon }_E^{-1}-1)-{\epsilon }_E^{-3}·{\displaystyle \frac{R_S}{2}}\right]\mathrm{or}$$

(115)

$${\underline{\dot{\epsilon }}}_{L,rr}={\displaystyle \frac{2·{\epsilon }_E·c}{R}}·\left[1-{\epsilon }_E-{\displaystyle \frac{R_S}{4·{\epsilon }_E^2·R}}\right].$$

(116)

Therefore, each mass and each effective mass cause a gravitational field. At each location, a gravity field causes LFV. The respective rates are expressed in Equations (96), (98), (99) and (107).

8. Energy Density of a Gravitational Field

Here, a field’s energy density $u_f$ is introduced and derived.

8.1. Measurement of the Energy Density of the Field

$u_f$ can be measured as follows: The process described in Figure 7 is performed: A mass M is lifted by a radial difference $\Delta R$. Thereby, the required energy $\Delta E_M$ is measured. This energy is located in the field within the shell with center M, thickness $\Delta R$ and radius R, see Figure 7. That shell has the volume

$$\Delta V=4\pi R^2\Delta R.$$

(117)

Consequently, the absolute value of the measured energy density is

$$|u_f|={\displaystyle \frac{\Delta E_M}{\Delta V}}.$$

(118)

The positive required energy compensates the field’s energy in the shell, as the field in the shell with volume $\Delta V$ is eliminated in the process of lifting the mass. Consequently, the sign of the field is negative:

$$u_f=-{\displaystyle \frac{\Delta E_M}{\Delta V}}$$

(119)

Alternatively, this measured energy can be calculated:

Figure 7. At radii R and $R+\Delta R$, there are two very thin shells with a common center. Initially, in the inner shell, there is a mass M. M is lifted gradually in parts $dM$ to $R+\Delta R$. During that process, at each time, there remains a mass $M_{rest}$ in the inner shell. Between these shells, there is a field ${\overrightarrow{G}}_{gen}={\overrightarrow{G}}^{*}$, which gradually becomes zero. The dotted line symbolizes the surroundings, the fields of which add up to zero in a homogeneous universe. This has already been derived by Newton.

Figure 7. At radii R and $R+\Delta R$, there are two very thin shells with a common center. Initially, in the inner shell, there is a mass M. M is lifted gradually in parts $dM$ to $R+\Delta R$. During that process, at each time, there remains a mass $M_{rest}$ in the inner shell. Between these shells, there is a field ${\overrightarrow{G}}_{gen}={\overrightarrow{G}}^{*}$, which gradually becomes zero. The dotted line symbolizes the surroundings, the fields of which add up to zero in a homogeneous universe. This has already been derived by Newton.

8.2. Derivation of the Field’s Energy Density

The required energy $\Delta E_M$ can be calculated [65]:

$$\Delta E_M={\displaystyle \frac{G·M^2\Delta R}{2R^2}}.$$

(120)

The measured energy density in Equation (119) can be calculated by inserting Equations (120) and (117) into Equation (119):

$$u_f=-{\displaystyle \frac{G·M^2}{8\pi R^4}}=-{\displaystyle \frac{{({\overrightarrow{G}}^{*})}^2}{8\pi G}}$$

(121)

For comparison, Peters [85] derived the same formula for the energy density

$$u_f=-{\displaystyle \frac{{({\overrightarrow{G}}^{*})}^2}{8\pi G}}.$$

(122)

Peters [85] derived this result in a Newtonian approximation.

In contrast, the exact energy density $u_f$ is derived here. This is achieved with help of the exact field, since no approximation is used here. This exactness is achieved here with the help of the distinction between the two distance measures: the $d_{GP}$ distance $\Delta R$ and the corresponding $d_{LT}$ distance $\Delta L$.

8.3. Compensation of the Negative Field

The RGS in Equations (67) and (70) is divided by $8\pi G$, and ${\overrightarrow{G}}_{gen}={\overrightarrow{G}}^{*}$ is used:

$$0={\displaystyle \frac{{\left(c·\frac{𝜕}{𝜕\tau }{\sum }_{jin\mathcal{S}}{\epsilon }_{L,rr,j}\right)}^2}{8\pi G}}-{\displaystyle \frac{{({\overrightarrow{G}}^{*})}^2}{8\pi G}}$$

(123)

The sum in Equation (123) is the summed rel. add. vol. of the indivisible VPs:

$${\epsilon }_{L,rr,indivisibleVPs}:=\sum _{jin\mathcal{S}}{\epsilon }_{L,rr,j}$$

(124)

With it, the RGS in Equation (123) is

$$0={\displaystyle \frac{c^2·{\dot{\epsilon }}_{L,rr,indivisibleVPs}^2}{8\pi G}}+{\displaystyle \frac{-{({\overrightarrow{G}}^{*})}^2}{8\pi G}}.$$

(125)

The second fraction in the above Equation is $u_f$.

$$0={\displaystyle \frac{c^2·{\dot{\epsilon }}_{L,rr,indivisibleVPs}^2}{8\pi G}}+u_f$$

(126)

As a consequence, the fraction in the above Equation is an energy density u. That fraction describes indivisible VPs $\delta V_j$ that cause the field. Therefore, that fraction describes the energy density of the indivisible VPs $\delta V_j$:

$$u_{indivisibleVPs}={\displaystyle \frac{c^2·{\dot{\epsilon }}_{L,rr,indivisibleVPs}^2}{8\pi G}}.$$

(127)

Consequently, the sum of these two energy densities is

$$u_{indivisibleVPs}+u_f=0.$$

(128)

The energy density of one indivisible VP is obtained by using ${\overrightarrow{G}}_{gen,j}={\overrightarrow{G}}_j^{*}$ and by inserting the field in Equation (66) into Equation (122):

$$u_{f,j}=-{\displaystyle \frac{{({\overrightarrow{G}}_j^{*})}^2}{8\pi G}}.$$

(129)

Next, the rate gravity relation (see Equations (67) and (70)) is generalized to one indivisible VP:

$$c^2{\dot{\epsilon }}_{L,rr,indivisibleVP,j}^2={({\overrightarrow{G}}_j^{*})}^2.$$

(130)

Similarly, Equation (128) is generalized to one indivisible VP:

$$u_{indivisibleVP,j}+u_{f,j}=0.$$

(131)

With these generalizations, the energy density in Equation (129) implies the energy density $u_{indivisibleVP,j}$ of one indivisible VP $\delta V_j$ as follows:

$$u_{indivisibleVP,j}=-u_{f,j}={\displaystyle \frac{c^2·{\dot{\epsilon }}_{L,rr,indivisibleVP,j}^2}{8\pi G}}.$$

(132)

These relations are results of the VD.

Each indivisible VP has the energy density $u_{indivisibleVP,j}$ based on its rate. Moreover, each indivisible VP causes the energy density $u_{f,j}$ based on its gravity. Furthermore, each indivisible VP propagates at the speed c. Accordingly, such an indivisible VP can be called rate gravity wave, RGW. In an advanced theory of the VD, expectation values of the field have been derived from the VD as follows, see [23,65].

9. Derivation of the Quantum Postulates

The VD, the dynamics of the indivisible VPs, implies the SEQ, see Section 5.2, the fundamental deterministic dynamics of quantum physics. Accordingly, the following question arises: Does the VD imply the stochastic dynamics of quantum systems and the complete system of quantum postulates?

In order to answer this question, the deterministic time evolution is summarized in a postulate (see Section 9.1), and mathematical consequences (see Section 9.2 and Section 9.3) are derived first.

Then, the stochastic dynamics (see Section 9.4) and the complete system (see Section 9.5) of the quantum postulates are derived.

Additional rules about mixed states and about entanglement [62] ([p. 46]) have been derived from the VD [65].

9.1. Postulate About the Deterministic Time Evolution

The VD implies the postulate concerning deterministic time evolution of quanta [86] ([p. 170]):

• Postulate about the deterministic time evolution

Kumar writes literally [86] ([p. 170]): ’The time evolution of the state vector is governed by the time-dependent Schrödinger equation, SEQ [69] ([Equation (4”)]):

$$\begin{array}{c}\hfill i\hslash {𝜕}_t|\psi \rangle =\widehat{H}|\psi \rangle ,\end{array}$$

(133)

where $\widehat{H}$ is the Hamilton operator corresponding to the total energy of the system.

More generally, the VD implies the GSEQ, see Section 5.2.

9.2. On Hilbert Space

In this section, the solution spaces of the SEQ and of the GSEQ are analyzed, in order to derive quantum postulates in later sections. In Section 5.2 it is shown, that indivisible VPs can be described by the DEQ of VD, and that indivisible VPs as well as objects at the speed $v=c$ are described by the quantum physical GSEQ. In particular, objects with relatively small momentum, ${\displaystyle \frac{p^2}{m_0^2{c}^2}}<<1$, in leading order in ${\displaystyle \frac{p^2}{m_0^2{c}^2}}$, are described by the SEQ.

In this section, it is shown that the solutions of the SEQ, of the GSEQ and of the DEQ of VD form Hilbert spaces: As usual, Dirac’s notation is used: A wave function (WF) $\Psi$ is expressed by a so-called ket $|\Psi \rangle$. Moreover, for two WFs, there is a scalar product

$$\begin{array}{c}\hfill \langle {\Psi }_1|{\Psi }_2\rangle =\int d^{3}L{\Psi }_1^{*}(\tau ,\overrightarrow{L})·{\Psi }_2(\tau ,\overrightarrow{L})\end{array}$$

(134)

The superscript ∗ indicates the complex conjugate [62,86,87,88]. Based on the scalar product, a normalization factor $t_n$ is as follows:

$$\begin{array}{c}\hfill \langle \Psi ·t_n|\Psi ·t_n\rangle =\int d^{3}r{\Psi }^{*}(\tau ,\overrightarrow{L})·\Psi (\tau ,\overrightarrow{L})·{\left|t_n\right|}^2=1\end{array}$$

(135)

It is always useful and insightful to analyze the used vector space. It is the space of the solutions of the respective linear differential equation. With the above scalar product, it is a Hilbert space $\mathcal{H}$ [89] ([p. 47]) or Sakurai and Napolitano [61] ([p. 57]).

For the case of the SEQ, the GSEQ, and the VD, the solution space is a Hilbert space. These DEQs and their respective Hilbert spaces, can describe VPs, indivisible VPs, matter, and radiation (the case of radiation is analyzed [65]).

9.3. On Measurements, Operators, and Possible Outcomes

In this section, the modeling of measurements is elaborated, by using respective Hilbert spaces of the solutions of the deterministic dynamics of the GSEQ or of the SEQ. This will be used in later sections in order to derive quantum postulates.

Firstly, the possible outcomes of a single measurement process are derived.

Secondly, the necessity of an additional dynamics is identified.

(1) In physics, in general, a measurement is described as follows:

(1.1) A measurable physical quantity A of an object is a function f of the mathematical description of that object.

(1.2) Thereby, a change in the state should be as small as possible.

(1.3) In general, the function f can be described in linear order by an operator $\widehat{A}$ in Hilbert space, and by a Taylor series thereof: $f={\sum }_{k=0}^{\infty }{c}_k{\widehat{A}}^k$. Thereby, the zeroth order is not essential in physics. Thus, $f={\sum }_{k=1}^{\infty }{c}_k{\widehat{A}}^k$. Therefore, a measurable physical quantity A of an object is characterized via a linear operator $\widehat{A}$, which acts upon the states in the respective Hilbert space.

(1.4) In the present case, the mathematical description is a state $|\Psi \rangle$ in a Hilbert space.

(1.5) In linear order in $\widehat{A}$, the function applied to the state is described by

$$\begin{array}{c}\hfill f_{linear}\left(\right|\Psi \rangle )=\widehat{A}|\Psi \rangle ,\mathrm{in}\mathrm{linear}\mathrm{order}.\end{array}$$

(136)

(1.6) In general, each state $|\Psi \rangle$ is described by a linear combination of eigenvectors $|{\Psi }_{A,n}\rangle$ with eigenvalues $a_n$. Here, the case of discrete and different eigenvalues $a_n$, each with one eigenvector, is analyzed in detail, as other cases can be analyzed analogously:

$$\begin{array}{c}\hfill f_{linear}(|\Psi \rangle )=\widehat{A}|\sum _n{\Psi }_{A,n}\rangle =\sum _n{a}_n|{\Psi }_{A,n}\rangle \end{array}$$

(137)

(1.7) In order to keep the state unchanged in a single measurement process, see part (1.2), that process must act upon an eigenvector:

$$\begin{array}{c}\hfill f_{linear}\left(\right|{\Psi }_{A,n}\rangle )=\widehat{A}|{\Psi }_{A,n}\rangle =a_n|{\Psi }_{A,n}\rangle ,\mathrm{single}\mathrm{measurement}\mathrm{process}.\end{array}$$

(138)

Therefore, a possible result of a single measurement process of a quantity A of an object, is one of the eigenvalues of the operator $\widehat{A}$ corresponding to A.

(2) In general, a state is a linear combination of eigenvectors. But a possible result of each single measurement process is an eigenvalue. As a consequence, there must be a second dynamics that provides the choice of the eigenvalue that occurs in the measurement.

(2.1) The derivation of the deterministic dynamics of indivisible VPs is completely general, as only analysis alias calculus has been used in the derivation. As a consequence, if a second dynamics would be deterministic, it would reduce the generality of the deterministic dynamics.

(2.2) Therefore, a general second dynamics must be a stochastic dynamics. This stochastic dynamics is derived next:

9.4. On the Stochastic Dynamics

In this section, the stochastic dynamics is derived from the properties of the indivisible VPs step by step in a progressive manner in the following paragraphs:

(1)

For the case of a single indivisible VP $\delta V_j$, the probability to measure an indivisible VP at an event $(\tau ,\overrightarrow{L})$ is proportional to the VP’s energy density $u_{indivisibleVP,j}$, at that event $(\tau ,\overrightarrow{L})$.

(2)

The energy density $u_{indivisibleVP,j}$ is related to the wave function as follows:

(2.1)

For the case of a single indivisible VP $\delta V_j$, the WF is the time derivative of its relative additional volume, multiplied by a normalization factor $t_n$, see Section 5.2:

$$\begin{array}{c}\hfill |{\Psi }_{indivisibleVPj}\rangle =t_n·{\dot{\epsilon }}_{indivisibleVPj}.\end{array}$$

(139)

(2.2)

The absolute square is applied to the above equation:

$$\begin{array}{c}\hfill |{\Psi }_{indivisibleVPj}^2|=\langle {\Psi }_{indivisibleVPj}|{\Psi }_{indivisibleVPj}\rangle =t_n^2·{\dot{\epsilon }}_{indivisibleVPj}^2.\end{array}$$

(140)

(2.3)

The above equation is multiplied by $1={\displaystyle \frac{8\pi G}{c^2}}·{\displaystyle \frac{c^2}{8\pi G}}$:

$$\begin{array}{c}\hfill |{\Psi }_{indivisibleVPj}^2|={\displaystyle \frac{8\pi G}{c^2}}{t}_n^2·{\displaystyle \frac{c^2{\dot{\epsilon }}_{indivisibleVPj}^2}{8\pi G}}.\end{array}$$

(141)

(2.4)

In the above equation, the second fraction is the energy density of the indivisible VP $\delta V_j$:

$$\begin{array}{c}\hfill |{\Psi }_{indivisibleVPj}^2|={\displaystyle \frac{8\pi G}{c^2}}{t}_n^2·u_{indivisibleVPj}.\end{array}$$

(142)

(3)

As a consequence, the probability $P_{indivisibleVPj}$ to measure an indivisible VP at an event $(\tau ,\overrightarrow{L})$ is proportional to the WF’s absolute square:

$$\begin{array}{c}\hfill P_{indivisibleVPj}\propto |{\Psi }_{indivisibleVPj}^2|.\end{array}$$

(143)

(4)

For each measurable quantity A, and for the corresponding operator $\widehat{A}$, the probability to measure an eigenvalue $a_n$ of an eigenvector $|{\Psi }_{A,n}\rangle$ has been derived from the result in part (3) or in Equation (143) [65].

Therefore, the result in part (3) or in Equation (143) is the fundamental stochastic dynamics of quantum physics.

(5)

As a consequence, the VD implies both, deterministic time evolution as well as the stochastic dynamics of quantum physics. Consequently, the VD implies the full dynamics of quantum physics. This is the case, as for their respective systems, quantum field theory and the Dirac theory can both be derived with help of the above deterministic and stochastic dynamics [23,61,65,84,90].

Using the above mathematical results as well as the deterministic and stochastic dynamics, the postulates of quantum physics are derived next:

9.5. Derivation of the Postulates

In this section, it is shown how the quantum postulates [86] are implied by the VD:

(1) The postulate about the deterministic time evolution has been derived in Section 5.2 and Section 9.1.

(2) The postulate about probabilistic outcomes of measurements is stated literally by Kumar [86] (pp. 169–170): If a measurement of an observable A is made in a state $|\Psi (t)\rangle$ of the quantum mechanical system, then, the following holds:

[1] The probability of obtaining one of the non-degenerate discrete eigenvalues $a_j$ of the corresponding operator $\widehat{A}$ is given by

$$\begin{array}{c}\hfill P\left(a_j\right)={\displaystyle \frac{|\langle {\varphi }_j|\Psi \rangle {|}^2}{\langle \Psi |\Psi \rangle }},\end{array}$$

(144)

where $|{\varphi }_j\rangle$ is the eigenfunction of $\widehat{A}$ with the eigenvalue $a_j$. If the state vector is normalized to unity, $P\left(a_j\right)={\left|\langle {\varphi }_j|\Psi \rangle \right|}^2$.

[2] If the eigenvalue $a_j$ is $m_j$-fold degenerate, this probability is given by

$$\begin{array}{c}\hfill P\left(a_j\right)={\displaystyle \frac{{\mathrm{\Sigma }}_{i=1}^{m_j}{\left|\langle {\varphi }_{j,i}|\Psi \rangle \right|}^2}{\langle \Psi |\Psi \rangle }},\end{array}$$

(145)

[3] If the operator $\widehat{A}$ possesses a continuous eigenspectrum $\left\{a\right\}$, the probability that the result of a measurement will yield a value between a and $a+da$ is given by

$$\begin{array}{c}\hfill P\left(a\right)={\displaystyle \frac{{\left|\langle \varphi \left(a\right)|\Psi \rangle \right|}^2}{\langle \Psi |\Psi \rangle }}·da={\displaystyle \frac{{\left|\langle \varphi \left(a\right)|\Psi \rangle \right|}^2}{{\int }_{-\infty }^{\infty }{|\Psi \left(a^{\prime }\right)|}^{2}d{a}^{\prime }}}·da\end{array}$$

(146)

This postulate can be derived from the stochastic dynamics in Section 9.4. The derivation is presented in Carmesin [65,90,91].

(3) The postulate about Hilbert space is stated literally by Kumar [86] (p. 168): ‘The state of a quantum mechanical system, at a given instant of time, is described by a vector $|\Psi (t)\rangle$, in the abstract Hilbert space $\mathcal{H}$ of the system.’

For each given quantum state, a respective full dynamics should be determined. It is the deterministic and stochastic dynamics in the above postulates (1) and (2). It has been shown here, that the states of the respective Hilbert space provide the full information to derive the deterministic time evolution (with the SEQ or with the GSEQ, more generally), and to derive the correct probabilities for the probabilistic outcomes. Therefore, for each time, a quantum state is characterized as a state inside a respective Hilbert space.

Moreover, it is insightful to realize that the correct probabilistic outcomes are a consequence of the gravitational properties of the VD. Therefore, it is enlightening to understand that the basis of the quantum postulates is a quantum gravitational foundation. As a consequence, a deep and fundamental understanding of quantum physics without understanding gravity and quantum gravity is hardly possible.

(4) The postulate about the relation between observables and operators is stated literally by Kumar [86] (p. 169): ‘A measurable physical quantity A (called an observable or dynamic physical quantity), is represented by a linear and hermitian operator $\widehat{A}$ acting in the Hilbert space of state vectors.’

This postulate is founded by the VD, as described in Section 9.3.

(5) The postulate about the relation between outcomes of a measurement and eigenvalues is stated literally by Kumar [86] (p. 169): ‘The measurement of an observable A in a given state may be represented formally by the action of an operator $\widehat{A}$ on the state vector $|\Psi (t)\rangle$. The only possible outcome of such a measurement is one of the eigenvalues, $\left\{a_j\right\},j=1,2,3,\dots ,\mathrm{of}\widehat{A}$.’

This postulate is founded by the VD, as described in Section 9.3.

10. Emergence of Volume in Nature and of Its Energy Density

Here, the following results are derived: In a homogeneous and empty universe, the process of GFV (global formation of volume) of the universe is elaborated on the basis of LFV (local formation of volume), everywhere in the universe. Moreover, it is shown that this process provides exactly the complete volume of the universe. Furthermore, it is shown that this process provides an energy density of volume, $u_{vol}={lim}_{heterogeneity\to 0}{u}_{DE}$. Additionally, it is shown that this $u_{vol}$ is in accurate concordance with empirical values of the dark energy density $u_{DE}$ or $u_{\Lambda }$. These results are very insightful, as they show how the complete and global volume of the universe emerges from a local process: LFV.

10.1. On the Development and Improvements of the Lambda-CDM Model

Here, the development of the basic cosmology model ($\Lambda$CDM), is summarized and related to the new results derived in this paper.

Einstein [7] realized that general relativity theory (GRT) implies an expansion of space since a Big Bang, in general. Thereby, he introduced the cosmological constant $\Lambda$. Zeldovich [10] showed that corresponds to an energy density of space:

$$u_{\Lambda }={\displaystyle \frac{\Lambda c^4}{8\pi G}}$$

(147)

Hubble [8] discovered the expansion of space and its rate H at the present-day time $t_0$ (with $t_{BigBang}=0$). That present-day rate $H\left(t_0\right)=H_0$ is called Hubble constant. Thereby, the expansion is modeled by homogeneous and isotropic universe. Hereby, the expansion is described with help of a uniform scaling, characterized by a scale radius R:

$$H:={\displaystyle \frac{\dot{R}}{R}}.$$

(148)

Based on GRT, Friedmann [17] and Lemaître [19] derived the dynamics of that expansion, depending on $\rho ={\displaystyle \frac{u}{c^2}}$ and a global curvature parameter k in the form of the Friedmann Lemaître equation (FLE):

$$H^2={\displaystyle \frac{8\pi G}{3}}·\rho -{\displaystyle \frac{k·c^2}{R^2}}.$$

(149)

Thereby, $k=0$, based on empirical data [22], as well as on the basis of derivation [84,92]. A corresponding $\rho$ value is called critical density:

$$H^2={\displaystyle \frac{8\pi G}{3}}·{\rho }_{cr},\mathrm{and}{H}_0^2={\displaystyle \frac{8\pi G}{3}}·{\rho }_{cr,0}.$$

(150)

Hereby and in general, present-day values are marked by a subscript zero. A good approximation of the universe’s age $t_0$ is the Hubble time $t_{H_0}:=1/H_0$ [20,40] and Equation (150):

$$t_0\approx t_{H_0}:={\displaystyle \frac{1}{H_0}}=\sqrt{{\displaystyle \frac{3}{8\pi G·{\rho }_{cr,0}}}}.$$

(151)

There is a FLE-insight: This Equation shows that $t_{H_0}$ and $t_0$ cannot be chosen arbitrarily, these are results of the dynamic density $\rho$ and of the dynamics of the FLE.

Therefore, the time evolution and dynamics of the density are essential. For its analysis, the density is a sum of parts with particular dynamical properties, $\rho ={\rho }_r+{\rho }_m+{\rho }_{\Lambda }$:

(1)

${\rho }_r$ is the dynamic density of radiation with ${\rho }_r\propto {(R_0/R)}^4$,

(2)

${\rho }_m$ describes ${\displaystyle \frac{m}{V}}$, with CDM [21,22], and with ${\rho }_m\propto {(R_0/R)}^3$

(3)

${\rho }_{\Lambda }={\displaystyle \frac{u_{\Lambda }}{c^2}}$ represents a dynamic density, corresponding to $\Lambda$, with ${\rho }_{\Lambda }\propto {(R_0/R)}^0$. That density has been discovered [9,15,16].

Therefore, $\rho$ depends on the scale radius R and the density parameters ${\Omega }_j:={\rho }_j/{\rho }_{cr,0}$:

$$\rho ={\rho }_{r,0}·{(R_0/R)}^4+{\rho }_{m,0}·{(R_0/R)}^3+{\rho }_{\Lambda }={\rho }_{cr,0}·({\Omega }_r{(R_0/R)}^4+{\Omega }_m·{(R_0/R)}^3+{\Omega }_{\Lambda })$$

(152)

There is a $\Lambda$CDM-insight: The above components of the density are underlying the model’s name $\Lambda$CDM, it includes visible densities of radiation and of visible matter, as well as the dark densities that are not visible with help of electromagnetic radiation, cold dark matter and dark energy $u_{\Lambda }$ alias $u_{DE}$. Additionally, the $\Lambda$CDM model uses the following idealizations:

(1)

a homogeneous and isotropic space,

(2)

the classical GRT without using the quantum dynamics,

(3)

a space described as one entity without using volume portions.

In this paper, these three idealizations are overcome. Thereby, precise accordance with observation is achieved, and a high predictive power is provided. Hereby, the volume’s energy density is deduced from first principles as well as in three-dimensional space for the first time with this method.

Here, VPs are utilized to deduce the dark energy’s density $u_{DE}$. Thereby, as a first step, a homogeneous universe is modeled. Therefore, $u_{DE}$ is derived for the case with zero heterogeneity. The result is an idealized value $u_{vol}={lim}_{heterogeneity\to 0}{u}_{DE}$ of the density of space. Accordingly, the derived result is compared with an observation that is based on radiation emitted at a highly homogeneous universe, which is very young. This radiation represents the cosmic microwave background (CMB). The empirical data utilizing the CMB, provides a value of the energy density [22] as follows:

$$u_{\Lambda ,obs}=5.{133}_{-0.2432}^{+0.2432}·{10}^{-10}{\displaystyle \frac{\mathrm{J}}{{\mathrm{m}}^3}}$$

(153)

The emergence of the volume of the universe and of its energy density in Equation (153) are derived and explained in this Section 10.

10.2. Process of Local Formation of Volume in the Universe

Here, a fundamental process for the locally emerging volume that provides the complete volume of the universe is described. This description makes possible a test of the completeness of the formed volume. Moreover, this description makes possible a deduction of the volume’s energy density $u_{vol}$. Additionally, the deduced value of $u_{vol}$ is compared with respective empirically based values in Equation (153).

Firstly, that process uses the global flatness of space, the curvature parameter is zero, see Equations (149)–(151): Secondly, the description of that process uses a probe volume: Based on the invariance with respect to shifts (translation), the LFV in the universe is investigated by building up the volume $d{V}_0$ in Figure 8. Thereby, $d{V}_0$ describes a fixed amount of three-dimensional extension, which has filled by volume of the universe in the time interval $t_{H_0}$:

$$d{V}_0\mathrm{has}\mathrm{been}\mathrm{filled}\mathrm{with}\mathrm{volume}\mathrm{of}\mathrm{nature}\mathrm{during}{t}_{H_0}$$

(154)

$t_{H_0}$ is named Hubble time.

Figure 8. A probe volume $d{V}_0$ is in the middle of a shell with radius R. In that shell, there is $d{M}_j=d{E}_j/c^2$ (black dot), which causes rates $d{\dot{\underline{\epsilon }}}_{rr}$, that propagate (dashed arrows) in the form of VPs or RGWs (freely propagating waves according to the VD) in all directions (dashed circle and dashed arrows) in the same manner.

Figure 8. A probe volume $d{V}_0$ is in the middle of a shell with radius R. In that shell, there is $d{M}_j=d{E}_j/c^2$ (black dot), which causes rates $d{\dot{\underline{\epsilon }}}_{rr}$, that propagate (dashed arrows) in the form of VPs or RGWs (freely propagating waves according to the VD) in all directions (dashed circle and dashed arrows) in the same manner.

Thirdly, that process is described step by step in the following paragraphs:

(1)

Local origin of the rate of LFV at $d{V}_0$: Empty space is investigated here. Consequently, LFV is caused at $d{V}_0$ by the volume increments or VPs $d{V}_j$, see Figure 8. The corresponding energy is described by the energy $d{E}_j$ or equivalently by the dynamic mass $d{M}_j=d{E}_j/c^2$. Each $d{M}_j$ causes a ${\underline{\dot{\epsilon }}}_{L,rr,atd{V}_0}\left(d{M}_j\right)$ within $d{V}_0$, according to the law of LFV.

$$d{M}_j\mathrm{causes}\mathrm{the}\mathrm{rate}{\underline{\dot{\epsilon }}}_{L,rr,atd{V}_0}\left(d{M}_j\right)$$

(155)

(2)

Complete rate of LFV at $d{V}_0$:

(2.1)

The complete ${\underline{\dot{\epsilon }}}_{L,rr,atd{V}_0}$ within $d{V}_0$ represents the sum of all rates in the above part (1).

(2.2)

The complete rate is identified with the Hubble constant $H_0$: ${\underline{\dot{\epsilon }}}_{L,rr,atd{V}_0}\left(t_0\right)$ within $d{V}_0$ is equal to ${\underline{\dot{\epsilon }}}_{L,iso,atd{V}_0}/3$:

$${\underline{\dot{\epsilon }}}_{L,rr,atd{V}_0}={\displaystyle \frac{1}{3}}·{\underline{\dot{\epsilon }}}_{L,iso,atd{V}_0}={\displaystyle \frac{1}{3}}·\left({\underline{\dot{\epsilon }}}_{L,xx,atd{V}_0}+{\underline{\dot{\epsilon }}}_{L,yy,atd{V}_0}+{\underline{\dot{\epsilon }}}_{L,zz,atd{V}_0}\right).$$

(156)

As a matter of calculus, $H_0$ represents the following [65]:

$$H_0=H\left(t_0\right)={\displaystyle \frac{1}{3}}·{\displaystyle \frac{\dot{V}}{V}}.$$

(157)

${\displaystyle \frac{\dot{V}}{V}}$ as well as ${\underline{\dot{\epsilon }}}_{L,iso,atd{V}_0}$ represent identical emerging isotropic volume within $d{V}_0$ [65]:

$$H_0={\displaystyle \frac{\dot{V}}{3V}}={\displaystyle \frac{{\underline{\dot{\epsilon }}}_{L,iso,atd{V}_0}}{3}}={\underline{\dot{\epsilon }}}_{L,rr,atd{V}_0}={\displaystyle \frac{\dot{R}}{R}}.$$

(158)

(3)

The complete rate provides the complete volume of nature in $d{V}_0$: By definition, ${\underline{\dot{\epsilon }}}_{L,rr,atd{V}_0}$ represents the volume $\underline{\delta }{V}_{rr}$ that forms in a considered volume $d{V}_0$ during a time $\underline{\delta }t$:

$${\underline{\dot{\epsilon }}}_{L,rr,atd{V}_0}={\displaystyle \frac{\underline{\delta }{V}_{rr}}{\underline{\delta }t·d{V}_0}}$$

(159)

For the purpose of a deduction of the complete volume formed in $d{V}_0$ during the complete Hubble time $t_{H_0}$, this Equation is solved for the formed volume, the rate ${\underline{\dot{\epsilon }}}_{L,rr,atd{V}_0}=H_0$ is used, and the complete time $\underline{\delta }t=t_{H_o}$ is utilized:

$$\underline{\delta }{V}_{rr}=\underset{1}{\underbrace{H_0·t_{H_0}}}·d{V}_0=d{V}_0.$$

(160)

Therefore, the complete probe volume $d{V}_0$ is filled with LFV. Translation invariance is the basis for the fact that the probe volume represents the complete volume of the universe. Consequently, the complete volume of the universe forms during the Hubble time. Altogether, this is the above introduced first test of the process: The described process shows how the LFV forms exactly the GFV. Thereby, the LFV takes place in the complete space inside the Hubble radius and during the complete Hubble time. This result provides a great additional evidence for the theory of VPs derived above.

10.3. Derivation of the Volume’s Energy Density

Here, the formation of LFV in Section 10.2 of volume on the basis of the LFV-dynamics is elaborated in more quantitative detail. Thereby, the energy density $u_{vol}={lim}_{heterogeneity\to 0}{u}_{DE}$ of volume is derived, and it is shown that $u_{vol}$ is in precise accordance with observation.

10.3.1. Rate of Formation of Volume by a Volume Portion

A VP $d{V}_j$ with $d{M}_j$ in Figure 8 is as follows:

$$\begin{array}{c}\hfill d{M}_j=d{V}_j·{\rho }_{vol}\end{array}$$

(161)

Exact gravity provides, see the section about gravity or [65]

$$\begin{array}{c}\hfill |d{\overrightarrow{G}}_j^{*}|\left(R\right)={\displaystyle \frac{G·d{M}_j}{R^2}}.\end{array}$$

(162)

The propagation of the above gravity is explained and deduced in the framework of the RGW, see also Figure 8. Hhereby, RGWs caused by $d{M}_j$ show no destructive interference. This is shown in part eight of Section 10.2.

Equations (96) and (107) imply for each $d{M}_j$

$$\begin{array}{c}\hfill d{\dot{\underline{\epsilon }}}_{L,rr,atd{V}_0}(R,d{M}_j)={\displaystyle \frac{|d{\overrightarrow{G}}_j^{*}|\left(R\right)}{c}}={\displaystyle \frac{G·d{M}_j}{R^2·c}}.\end{array}$$

(163)

10.3.2. Rate of Formation of Volume by One Shell

$d{M}_j$ in Figure 8 causes LFV as follows:

• Shell with center $d{V}_0$:

The $d{M}_j$ in Figure 8 cause portions of LFV that add up in a shell as follows

$$\begin{array}{c}\hfill dM(R,dR)=\sum _{j,R_j\in \mathrm{shell}}d{M}_j\end{array}$$

(164)

• Rate caused by above shell:

Equations (163) and (164) imply

$$\begin{array}{c}\hfill \sum _{j,R_j\in \mathrm{shell}}d{\dot{\underline{\epsilon }}}_{L,rr,atd{V}_0}(R,d{M}_j)={\displaystyle \frac{G·{\sum }_{j,R_j\in \mathrm{shell}}d{M}_j}{R^2·c}}.\end{array}$$

(165)

The sum in that Equation is equal to $dM(R,dR)$ in Equation (164):

$$\begin{array}{c}\hfill d{\dot{\underline{\epsilon }}}_{L,rr,atd{V}_0}(R,dR)={\displaystyle \frac{G·dM(R,dR)}{R^2·c}}\end{array}$$

(166)

• Using ${\rho }_{vol}$:

Geometrically, $dM(R,dR)=$ ${\rho }_{vol}·dV$ with the shell’s $dV=4\pi ·R^2·dR$:

$$dM(R,dR)={\rho }_{vol}·4\pi ·R^2·dR$$

(167)

Consequently, the rate in Equation (166) becomes

$$\begin{array}{c}\hfill d{\dot{\underline{\epsilon }}}_{L,rr,atd{V}_0}(R,dR)={\displaystyle \frac{4\pi ·G·{\rho }_{vol}·dR·R^2}{R^2·c}}.\end{array}$$

(168)

The terms $R^2$ cancel:

$$\begin{array}{c}\hfill d{\dot{\underline{\epsilon }}}_{L,rr,atd{V}_0}\left(dR\right)={\displaystyle \frac{4\pi ·G·{\rho }_{vol}·dR}{c}}\end{array}$$

(169)

10.3.3. Light-Travel Time of One Shell

Here, indivisible VPs emitted within the shell in Figure 8 are characterized by the time $t_{LT}$, in which they travel to the probe volume $d{V}_0$.

A VP travels with the speed $v=c$. Therefore, a VP travels through the shell with the thickness $dR$ during the light travel time $d{t}_{LT}=dR/c$:

$$\begin{array}{c}\hfill d{t}_{LT}={\displaystyle \frac{dR}{c}}\end{array}$$

(170)

Accordingly, $dR/c$ in Equation (169) is substituted by $d{t}_{LT}$:

$$\begin{array}{c}\hfill d{\dot{\underline{\epsilon }}}_{L,rr,atd{V}_0}\left(d{t}_{LT}\right)=4\pi ·G·{\rho }_{vol}·d{t}_{LT}\end{array}$$

(171)

Thereby, since the Big Bang until today, the light travel times range from $d{t}_{LT}=0$ to the Hubble time $d{t}_{LT}=t_{H_0}$:

10.3.4. Spatial Integration of the Shells

Here, shells within Figure 8 are integrated: Thereby, all VPs $d{V}_j$ that emit volume portions that can propagate to the probe volume are included. Therefore, the integration includes exactly the sources in space that provide a rate of LFV at the probe volume $d{V}_0$. Practically, this complete spatial integral is executed by the integration variable time according to the substitution in Section 10.3.3.

Consequently, integration intervals are $[0,t_{H_0}]$, as well as $[0,{\dot{\underline{\epsilon }}}_{L,rr,atd{V}_0}]$. Therefore, Equation (171) implies:

$$\begin{array}{c}\hfill {\int }_0^{{\dot{\underline{\epsilon }}}_{L,rr,atd{V}_0}}d{\dot{\underline{\epsilon }}}_{L,rr,atd{V}_0}=4\pi ·G·{\rho }_{vol}·{\int }_0^{t_{H_0}}d{t}_{LT}\end{array}$$

(172)

The above integral is evaluated:

$$\begin{array}{c}\hfill {\dot{\underline{\epsilon }}}_{L,rr,atd{V}_0}=4·\pi ·G·{\rho }_{vol}·t_{H_0}\end{array}$$

(173)

That rate represents $H_0$, see Equation (158):

$$\begin{array}{c}\hfill H_0=4\pi ·G·{\rho }_{vol}·t_{H_0}\end{array}$$

(174)

The above Equation is divided by $t_{H_0}$:

$$\begin{array}{c}\hfill {\displaystyle \frac{H_0}{t_{H_0}}}={\rho }_{vol}·4\pi ·G\end{array}$$

(175)

According to the FLE-insight in Section 10.1, the values of $H_0$ and $t_{H_0}$ are functions of the density. Therefore, the result in Equation (175) is valid and derived in our universe.

Next, Equation (175) is solved for ${\rho }_{vol}=u_{vol}/c^2$:

$$\begin{array}{c}\hfill {\rho }_{vol}={\displaystyle \frac{H_0}{t_{H_0}}}·{\displaystyle \frac{1}{4\pi ·G}}\end{array}$$

(176)

$H_0=1/t_{H_0}$ is utilized (Equations (175) and (176)). Moreover, the equality of rates ${\underline{\dot{\epsilon }}}_{L,rr,atd{V}_0}=H_0$ in Equation (158) is used:

$$\begin{array}{c}\hfill {\rho }_{vol}={\displaystyle \frac{H_0^2}{4\pi ·G}}\mathrm{or}{u}_{vol}={\displaystyle \frac{H_0^2{c}^2}{4\pi ·G}}\end{array}$$

(177)

10.3.5. Derivation for a Comparison with Observation

Equation (177) shows:

$$u_{vol}={\displaystyle \frac{c^2·H_{0,empty}^2}{4·\pi ·G}}$$

(178)

$H_{0,empty}$ represents the respective value for empty space. Empirical data show

$$\begin{array}{ccc}\hfill u_{vol}& =& {\displaystyle \frac{c^2{H}_{0,empty}^2}{4\pi G}}=5.034(\pm 0.1395)·{10}^{-10}{\displaystyle \frac{\mathrm{J}}{{\mathrm{m}}^3}},\mathrm{with}\hfill \\ \hfill H_{0,empty}& =& 66.88(\pm 0.92){\displaystyle \frac{\mathrm{km}}{\mathrm{s}·\mathrm{Mpc}}}\hfill \end{array}$$

(179)

For $H_0$ see [22]. $H_0$ is related to our universe’s age $t_0\approx 1/H_0$. An accurate Equation is deduced in Carmesin [40]. The empirical quantity $u_{\Lambda ,obs}$ is, see [22]:

$$u_{\Lambda ,obs}=5.{133}_{-0.2432}^{+0.2432}·{10}^{-10}{\displaystyle \frac{\mathrm{J}}{{\mathrm{m}}^3}}$$

(180)

Hereby, it is important that $H_0$ and $u_{\Lambda ,obs}$ are observed in a very homogeneous state of the universe. Such a homogeneous state corresponds to the young universe, observable via the CMB. For it, very accurate data have been obtained by [22].

Discussion of the results of this Section 10.1: The deduced value $u_{vol}$ corresponds accurately to the empirical data $u_{\Lambda ,obs}$, within the error of measurement $\pm 0.2432$. Therefore, the deduced $u_{vol}$ is a second test of the LFV in this Section 10.1. Together with the first test of the exact amount of formed volume, both tests provide a clear evidence for that process of formation of volume and for the theory of VPs. It is a new result that this founded, comprehensive (see Figure 5) and deduced theory based on derived discrete volume portions provides the functional relation for $u_{vol}$, see Equation (179).

11. Emergence of Volume in a Homogeneous Universe

Here, the above density $u_{vol}$ has been derived for the case of an empty universe. In this Section 11, the density $u_{vol}$ of volume of a homogeneous non-empty univ. is deduced. Therefore, that above model used in Section 10 is supplemented by a homogeneous density ${\rho }_{hom}$ of radiation and matter, whereby the heterogeneity of radiation and matter is not included. For it, it is essential to summarize the concepts of averaging in cosmology first, see Section 11.1:

11.1. Spatial Averages in Cosmology

Spatial averaging in cosmology uses regions. For instance, Figure 9 shows such regions. Thereby, space is partitioned into boxes with volume $L_{box}^3$, and with length L. For each box or window, a density is measured or determined [36,45,93,94]. The following terms are used:

Boxes: Each box has a length $L_{box}$, with a basic length as follows [36,93,95]:

$$L_{box}=8h^{-1}\mathrm{Mpc}\mathrm{with}h=H_0·{\displaystyle \frac{1}{100}}{\displaystyle \frac{\mathrm{km}}{\mathrm{s}·\mathrm{Mpc}}}$$

(181)

Local density: At each time t, each box is at a place $\overrightarrow{r}$, and has a density

$$\mathrm{local}\mathrm{density}{\rho }_m(t,\overrightarrow{r}).$$

(182)

Global density: The homogeneous density is the density’s global spatial average

$$\mathrm{global}\mathrm{density}{\rho }_{m,hom}(t)={\langle {\rho }_m(t,\overrightarrow{r})\rangle }_{\overrightarrow{r}}$$

(183)

Density of heterogeneity: The following difference is named density of heterogeneity:

$${\rho }_{m,het}(t,\overrightarrow{r}):={\rho }_m(t,\overrightarrow{r})-{\rho }_{m,hom}(t).$$

(184)

Overdensity: the following fractional difference is named overdensity:

$$\mathrm{overdensity}\delta (t,\overrightarrow{r}):={\displaystyle \frac{{\rho }_m(t,\overrightarrow{r})-{\rho }_{m,hom}(t)}{{\rho }_{m,hom}(t)}}.$$

(185)

The constituents of $\rho$ are

$$\rho ={\rho }_{m,hom}(t)+{\rho }_{m,het}(t,\overrightarrow{r})+{\rho }_r+{\rho }_{\Lambda }.$$

(186)

Averages of Fluctuations

In cosmology, spatial averages are as follows [36,45,93]. In general, function $f(\overrightarrow{r})$ occurs in a volume $V_{window}$ or $V_{win}$ of averaging:

$${\langle f(\overrightarrow{r})\rangle }_{V_{win}}={\displaystyle \frac{{\int }_{V_{win}}f(\overrightarrow{r})d{r}^3}{{\int }_{V_{win}}1d{r}^3}}$$

(187)

Fluctuations can be described via the standard deviation ${\sigma }_{V_{win}}$:

$${\sigma }_{V_{win}}^2={\langle {[f(\overrightarrow{r})-{\langle f(\overrightarrow{r})\rangle }_{V_{win}}]}^2\rangle }_{V_{win}}={\langle f^2(\overrightarrow{r})\rangle }_{V_{win}}-{\langle f(\overrightarrow{r})\rangle }_{V_{win}}^2$$

(188)

In particular, for the case of the matter, the standard deviation is called standard deviation of matter fluctuations, ${\sigma }_8(t)$ or $\delta (t)$:

$${\sigma }_8(t)=\delta (t)={\sigma }_{\delta ,V_{win}}(t)\mathrm{with}{\sigma }_{8,0}={\sigma }_8(t_0)$$

(189)

11.2. Expectation Values of Fields

In order to observe densities in cosmology, more local observations are averaged as outlined in Section 11.1. The corresponding theoretical values are expectation values. Those expectation values are investigated in this section, which are essential for the LFV at the probe volume $d{V}_0$. That LFV is based on Equation (97):

$${\underline{\dot{\epsilon }}}_{L,ii}^2·c^2={({\overrightarrow{G}}_i^{*})}^2$$

(190)

Therefore, for various objects, the important expectation values are derived in respective paragraphs, and an overview is presented in

Table 1. Expectation values of fields and the standard deviation or uncertainty of the field $\sigma :=\langle \Psi |{\overrightarrow{G}}_j^2|\Psi \rangle -\langle \Psi |{\overrightarrow{G}}_j^2{|\Psi \rangle }^2$ of the field. Hereby, the expectation value $\langle \Psi |{\overrightarrow{G}}_j|\Psi \rangle$ is abbreviated by $\langle {\overrightarrow{G}}_j\rangle$.

Object	State	$\langle {\overrightarrow{\mathit{G}}}_{\mathit{j}}\rangle$	$\mathit{\sigma }$	$\frac{1}{\mathit{N}}{\sum }_{\mathit{j}}^{\mathit{N}}\langle {\overrightarrow{\mathit{G}}}_{\mathit{j}}\rangle$	$\frac{1}{\mathit{N}}{\sum }_{\mathit{j}}^{\mathit{N}}{\langle {\overrightarrow{\mathit{G}}}_{\mathit{j}}\rangle }^2$	$\langle {\dot{\underline{\mathit{\epsilon }}}}_{\mathit{L},\mathit{rr},\mathit{at}{\mathit{dV}}_0}^2\rangle$
$m_j$	coherent	$\ne 0$	$\approx 0$	-	-	$\ne 0$
$V{P}_{indivisible}$	number	0	$\ne 0$	0	0	$\ne 0$
${\rho }_{\mathrm{m},\mathrm{het}}$	coherent	$\ne 0$	$\approx 0$	0	$\ne 0$	$\ne 0$
${\rho }_{\mathrm{m},\mathrm{hom}}$	coherent	$\ne 0$	$\approx 0$	0	0	0

:

(1)

One separate mass $m_j$ has a gravitational field, which is classical. Consequently, $m_j$ is characterized in terms of a coherent state [62,65,96,97]. Consequently, the field’s expectation value $\langle \Psi |{\overrightarrow{G}}_j|\Psi \rangle$ cannot be zero, and the standard deviation $\sigma =\langle \Psi |{\overrightarrow{G}}_j^2|\Psi \rangle -\langle \Psi |{\overrightarrow{G}}_j{|\Psi \rangle }^2$ of the field is nearly zero.

In this study, the classical expectation values for one mass are not essential (see columns five and six in Table 1):

(1.1)

In empty space, there is no mass.

(1.2)

In the non-empty homogeneous universe, the fields of masses are completely averaged to zero.

(1.3)

In the case of a heterogeneous density, there are many masses, whereby their fields average to zero, and the standard deviations of the fields remain.

Consequently, the nonzero expectation value of the field is not averaged out to zero. Therefore, such a mass contributes to the LFV at the probe mass (see column seven in Table 1. Such a single mass is not essential at the large scales of several Megaparsecs considered here. This case is included for comparison only.

(2)

For the case of an indivisible VP, the gravitational field is not classical. Accordingly, the VP is characterized in terms of a number state [62,65,96,97]. Thence, the expectation value $\langle \Psi |{\overrightarrow{G}}_j|\Psi \rangle$ of the field is zero, but the standard deviation is nonzero, $\sigma =\langle \Psi |{\overrightarrow{G}}_j^2|\Psi \rangle -0\ne 0$. The classical expectation values are not essential for an indivisible VP, as no classical fields can cancel out. Similarly, the quanta of electromagnetic radiation propagate from our home star to our home planet, without cancellation—otherwise it would be dark at Earth at night and at day. Consequently, the nonzero standard deviation contributes to the LFV at the probe mass (see column seven in Table 1).

(3)

For the case of the heterogeneous density, radiation is not essential. The reason is, that in our universe, heterogeneity developed at a time, at which the density of radiation was already negligible. Accordingly, the heterogeneous density is analyzed for masses only. Similarly, the homogeneous density is analyzed for masses only. In the case of the heterogeneous density, gravitational fields are classical. Consequently, they are described by coherent states [62,65,77,96,97]. Thus, the expectation value $\langle \Psi |{\overrightarrow{G}}_j|\Psi \rangle$ of the field of one mass $m_j$ is nonzero, and the standard deviation is negligible. The classical expectation value of all fields is zero. But the classical expectation value of all squared fields is nonzero. Therefore, the heterogeneous density contributes to the LFV at the probe mass (see column seven in Table 1).

(4)

For the case of the homogeneous density, gravitational fields are classical. Consequently, they are described by coherent states [62,65,77,96,97]. Thus, the expectation value of the field of one mass $m_j$ is nonzero, and the standard deviation is negligible. But the classical expectation values of all fields and of all squared fields is zero. Therefore, the homogeneous density does not contribute to the LFV at the probe mass (see column seven in Table 1).

11.3. Energy Density of Volume

As the homogeneous density ${\rho }_{\mathrm{m},\mathrm{hom}}$ of matter does not contribute to the LFV at $d{V}_0$, the volume’s energy density $u_{vol}$ is equal to that of the empty universe in Equation (179).

In the case of the homogeneous universe, there is sufficient content, that the density parameter can be derived in a meaningful manner:

$${\Omega }_{vol}:={\displaystyle \frac{u_{vol}}{u_{cr,0}}}={\displaystyle \frac{{\rho }_{vol}}{{\rho }_{cr,0}}}$$

(191)

The respective terms in Equations (150) and (177) are inserted:

$${\Omega }_{vol}={\displaystyle \frac{\frac{H_0^2}{4\pi ·G}}{\frac{3H_0^2}{8\pi ·G}}}={\displaystyle \frac{2}{3}}$$

(192)

This is in accurate concordance with empirical data ([22], table 2, column 2), within errors of measurement:

$${\Omega }_{\Lambda }=0.679\pm 0.013$$

(193)

Therefore, this result of the homogeneous universe provides clear additional evidence for the present mechanism of formation and for the theory of volume portions.

12. Emergence of Volume in a Heterogeneous Universe

In the cases of the empty and of the homogeneous universe, the flatness of space can be used, see Section 10 and Section 11. On that basis, and by using adequate coordinate systems (ACS), it is possible to use a regular and constant evolution of time.

$u_{vol}=u_{DE,withheterogeneity0}$ differs from the corresponding value $u_{DE}$ in the heterogeneous universe. Space’s density or volume’s density in a heterogeneous universe and in present-day’s universe are marked by $u_{DE}$. The density of space or volume in a homogeneous univ. and in an extremely young univ. is denoted by $u_{vol}$. The cosm. const. $\Lambda$ and a corresponding energy density of space $u_{\Lambda }$ describe our present-day universe, since Einstein [7] proposed $\Lambda$ without any specification of the early universe or of a particularly homogeneous universe different from the present-day universe. In each case, a fraction of an energy’s density $u_j$ and the critical energy density $u_{cr.}={\rho }_{cr.}·c^2$ represents the respective density parameter:

$${\Omega }_j={\displaystyle \frac{u_j}{u_{cr.}}}.$$

(194)

Moreover, in the cases of empty and homogeneous universe, $u_{DE}$ is a constant, so that the essential density parameters ${\Omega }_m:={\displaystyle \frac{{\rho }_m}{{\rho }_{cr,0}}}$ of matter and ${\Omega }_{\Lambda }={\displaystyle \frac{{\rho }_{\Lambda }}{{\rho }_{cr,0}}}$ of the cosmological constant are essentially constant.

However, in the heterogeneous universe, ${\rho }_{\Lambda }$ and ${\Omega }_{\Lambda }$ increase. This causes an increase of $H_0$. Indeed, such an increase has been recorded empirically [13,22]. Consequently, the Hubble time $t_{H_0}$ decreases.

For these reasons, the changes of ${\Omega }_{\Lambda }$ and of $t_{H_0}$ are analyzed simultaneously in this section.

12.1. VPs Field and Rate Caused by Heterogeneity

In this section, based on the LFV in the homogeneous universe, the additional LFV is derived step by step in respective paragraphs, that additional LFV is caused by heterogeneity.

(1)

Overdensity ${\delta }_j$ of one $d{M}_j$: $d{M}_j$ is in its $dM(R,dR)$-shell in Figure 8. Its density is $d{M}_j/d{V}_j$. Consequently, its overdensity is

$${\delta }_j={\displaystyle \frac{d{M}_j/d{V}_j-{\rho }_{m,hom}}{{\rho }_{m,hom}}}$$

(195)

The above fraction is expanded by the shell’s volume $dV=4\pi R^2·R$:

$${\delta }_j={\displaystyle \frac{d{M}_j·\frac{dV}{d{V}_j}-dM(R,dR)}{dM(R,dR)}}\mathrm{with}dM(R,dR)=4\pi R^{2}dR·{\rho }_{m,hom}=dV{\rho }_{m,hom}$$

(196)

(2)

${\delta }_j$ causes a rate: The overdensity ${\delta }_j$ in Equation (196) multiplied by the mass $dM(R,dR)$ is the mass of heterogeneity. The field’s $1/R^2$ law (${\overrightarrow{G}}^{*}=-{\displaystyle \frac{GM}{R^2}}{\overrightarrow{e}}_R$) implies

$$d{\overrightarrow{G}}_{gen,j}=-G{\displaystyle \frac{d{M}_j·\frac{dV}{d{V}_j}-dM(R,dR)}{R^2}}{\overrightarrow{e}}_R=-G{\displaystyle \frac{{\delta }_j·dM(R,dR)}{R^2}}{\overrightarrow{e}}_R.$$

(197)

The shell in Figure 8 provides an average ${\langle \dots \rangle }_{s,R}$. It is applied to the field’s square in Equation (197):

$${\langle d{\overrightarrow{G}}_{gen,j}^2\rangle }_{s,R}={\left(G{\displaystyle \frac{dM(R,dR)}{R^2}}\right)}^2·{\langle {\delta }_j^2\rangle }_{s,R},\mathrm{with}$$

(198)

$${\langle {\delta }_j^2\rangle }_{s,R}={\displaystyle \frac{{\int }_{shell}{\delta }_j^2{d}^{3}x}{{\int }_{shell}1d^{3}x}}$$

(199)

Equation (190) implies that the squared field causes the following squared rate:

$$d{\overrightarrow{G}}_{gen,j}^2=c^{2}d{\dot{\underline{\epsilon }}}_{L,rr}^2,\mathrm{therefore},$$

(200)

$${\left(G·{\displaystyle \frac{dM(R,dR)}{R^2}}\right)}^2·{\langle {\delta }_j^2\rangle }_{s,R}=c^2·{\langle d{\dot{\underline{\epsilon }}}_{L,rr}^2\rangle }_{s,R}$$

(201)

Here, the power with exponent $0.5$ is applied. Equation (196) is used. Moreover, $dR=c·dt$ is utilized:

$$4·\pi ·G·dt{\rho }_{m,hom}·\sqrt{{\langle {\delta }_j^2\rangle }_{s,R}}=\sqrt{{\langle d{\dot{\underline{\epsilon }}}_{L,rr}^2\rangle }_{s,R}}$$

(202)

Here, standard deviations (or uncertainties such as $\Delta {\dot{\underline{\epsilon }}}_{het}$ or $d{\dot{\underline{\epsilon }}}_{het}$, in the language of physics) are utilized:

$${\sigma }_m:=\sqrt{{\langle {\delta }_j^2\rangle }_{s,R}}$$

(203)

$$d{\dot{\underline{\epsilon }}}_{het}:=\sqrt{{\langle d{\dot{\underline{\epsilon }}}_{L,rr}^2\rangle }_{s,R}}$$

(204)

Rate caused by heterogeneity: The rate in Equations (202) and (204) caused by ${\rho }_{m,het}$ is the standard deviation $d{\dot{\underline{\epsilon }}}_{het}$. It is written by using usual density parameters of homogeneous densities:

$${\Omega }_{vol}={\displaystyle \frac{{\rho }_{vol}}{{\rho }_{cr,0}}}$$

(205)

$${\Omega }_m={\displaystyle \frac{{\rho }_{m,hom}}{{\rho }_{cr,0}}}$$

(206)

Consequently, ${\rho }_{m,hom}$ is

$${\rho }_{m,hom}={\rho }_{vol}·{\displaystyle \frac{{\Omega }_m}{{\Omega }_{vol}}}.$$

(207)

With it, Equations (202)–(204) imply

$$4\pi Gdt{\rho }_{vol}·{\displaystyle \frac{{\Omega }_m}{{\Omega }_{vol}}}{\sigma }_m=d{\dot{\underline{\epsilon }}}_{het}.$$

(208)

In the evolution of the universe, gravity accumulated matter. As a consequence, the standard deviation of matter fluctuations ${\sigma }_m$ increased as a function of time. This increase is modeled next in part (3):

(3)

Linear growth theory: That theory implies $\sigma \left(\tilde{a}\right)$ or ${\sigma }_8\left(\tilde{a}\right)$ (standard deviation of matter fluctuations) depending on $\tilde{a}$ (scale radius) [36,46,94]:

$${\sigma }_m\left(\tilde{a}\right)={\sigma }_8·\tilde{a}$$

(209)

Essential for the growth of heterogeneity is the matter era. In that era, $\tilde{a}=$ ${\tilde{t}}^{2/3}$. This is documented in Karttunen et al. [98] ([Equation 19.33]). More details are provided in Carmesin [99] (p. 297):

$$\tilde{a}={\tilde{t}}^{q_a}\mathrm{with}{q}_a={\displaystyle \frac{2}{3}}$$

(210)

Application of Equation (209) implies

$${\sigma }_m={\sigma }_8·{\tilde{t}}^{2/3}.$$

(211)

(4)

Time integral of the rate caused by heterogeneity: The time evolution in Equation (211) is inserted into the rate $d{\dot{\underline{\epsilon }}}_{het}$ caused by heterogeneity in Equation (208) is as follows:

$$d{\dot{\underline{\epsilon }}}_{het}=4\pi G{\rho }_{vol}·{\displaystyle \frac{{\Omega }_m}{{\Omega }_{vol}}}·{\sigma }_8·{\tilde{t}}^{2/3}·dt$$

(212)

(4.1)

Different Hubble times: Since $t_{H_0}=1/H_0$, these times differ in the heterogeneous and homogeneous univ. $t_{H_{0,het}}=1/H_{0,het}$ ≠ $t_{H_{0,hom}}=1/H_{0,hom}$. The fraction is introduced:

$$q_t:={\displaystyle \frac{t_{H_{0,het}}}{t_{H_{0,hom}}}}$$

(213)

The value is derived by an iteration. As a first step, the value $0.881$ is chosen. That value has to be compared with the values of $t_{H_{0,hom}}$ and $t_{H_{0,het}}$ that will be derived:

$$q_t^{\left(1\right)}=0.881,\mathrm{as}\mathrm{a}\mathrm{first}\mathrm{step}.$$

(214)

(4.2)

Scaled time: $t_0$ (present-day time) is $t_{H_0}·$$I_0=0.9455$, see ([40], Equation (2.32))

$$t_0=t_{H_0}·I_0,\mathrm{with}{I}_0=0.9455.$$

(215)

The rate in Equation (212) and in general, the increment of time $dt$ is scaled by the Hubble time. The idealized value is $t_{H_{0,hom}}$. The realistic value is $t_{H_{0,het}}$, which must be multiplied by $I_0$. Consequently, the realistic scaled increment is as follows:

$$d\tilde{t}={\displaystyle \frac{dt}{t_{H_{0,het}}·I_0}}$$

(216)

This Equation is solved for $dt$, and the ratio in Equation (213) is used:

$$dt=d\tilde{t}·t_{H_{0,hom}}·I_0·q_1$$

(217)

This realistic scaling is applied to the rate in Equation (212):

$$d{\dot{\underline{\epsilon }}}_{het}=\left[4\pi G{t}_{H_{0,hom}}{\rho }_{vol}\right]·{\displaystyle \frac{{\Omega }_m{\sigma }_8}{{\Omega }_{vol}}}·q_1·I_0·{\tilde{t}}^{2/3}·d\tilde{t}$$

(218)

Hereby, the rectangular bracket is equal to the rate ${\dot{\underline{\epsilon }}}_{L,rr,atd{V}_0}$ at the homogeneous universe. This rate is abbreviated by ${\dot{\underline{\epsilon }}}_{hom}$. Therefore, the rate in the heterogeneous universe in Equation (218) has the following relation to the rate in the homogeneous universe:

$$d{\dot{\underline{\epsilon }}}_{het}={\dot{\underline{\epsilon }}}_{hom}·{\displaystyle \frac{{\Omega }_m{\sigma }_8}{{\Omega }_{vol}}}·q_1·I_0·{\tilde{t}}^{2/3}·d\tilde{t}$$

(219)

(4.3)

Integration: The integral is applied to the above Equation:

$${\int }_0^{\Delta {\dot{\underline{\epsilon }}}_{het}}d{\dot{\underline{\epsilon }}}_{het}={\dot{\underline{\epsilon }}}_{hom}·{\displaystyle \frac{{\Omega }_m·{\sigma }_8}{{\Omega }_{vol}}}·q_1·I_0·{\int }_0^{{\tilde{t}}_{em}}{\tilde{t}}^{2/3}·d\tilde{t}$$

(220)

The integrals are evaluated:

$$\Delta {\dot{\underline{\epsilon }}}_{het}={\dot{\underline{\epsilon }}}_{hom}·{\displaystyle \frac{{\Omega }_m·{\sigma }_8}{{\Omega }_{vol}}}·q_1·I_0·{\displaystyle \frac{3}{5}}·{\tilde{t}}_{em}^{5/3}$$

(221)

The ratio ${\displaystyle \frac{\Delta {\dot{\underline{\epsilon }}}_{het}}{{\dot{\underline{\epsilon }}}_{hom}}}$ of the rates of the heterogeneous and of the homogeneous universe is formed, and it is named ${\kappa }_{q_t}$. Moreover, the scaled time is expressed by the scaled scale radius with the relation $\tilde{t}={\tilde{a}}^{3/2}$ in Equation (210):

$$\begin{array}{c}\hfill {\kappa }_{q_t}({\tilde{t}}_{em}):={\displaystyle \frac{\Delta {\dot{\underline{\epsilon }}}_{het}({\tilde{t}}_{em})}{{\dot{\underline{\epsilon }}}_{hom}}}={\tilde{a}}_{em}^{5/2}·{\displaystyle \frac{{\Omega }_m{\sigma }_8}{2{\Omega }_{vol}}}·\underset{1}{\underbrace{{\displaystyle \frac{6q_t^{\left(1\right)}{I}_0}{5}}}}·{\displaystyle \frac{q_t^{\left(n\right)}}{q_t^{\left(1\right)}}}\end{array}$$

(222)

Hereby, $q_t^{\left(n\right)}$ is determined by an iteration, here a fixed point iteration will be performed. The fixed point iteration is a valid mathematical method to solve the physically based equations of this Section 12 simultaneously. Thereby, the first value for $q_t^{\left(n\right)}$ is $q_t^{\left(1\right)}$. With it and Equation (213), the Hubble times are obtained, and with these, the new value $q_t^{\left(2\right)}{\displaystyle \frac{t_{H_{0,het}}}{t_{H_{0,hom}}}}$ is calculated numerically. This procedure is repeated, until the n-th value $q_t^{\left(n\right)}$, at which $q_t^{\left(n\right)}$ and the following value $q_t^{(n+1)}$ are equal for the first time, up to an accepted numerical error. Next, the scaled scale radius is expressed by the redshift with the relation ${\tilde{a}}^{5/2}={\left({\displaystyle \frac{1}{1+z}}\right)}^{5/2}$:

$$\begin{array}{c}\hfill {\kappa }_{q_t}\left(z_{em}\right)={\displaystyle \frac{{\Omega }_m·{\sigma }_8}{2{\Omega }_{vol}{(1+z_{em})}^{2.5}}}·{\displaystyle \frac{q_t^{\left(n\right)}}{q_t^{\left(1\right)}}}\end{array}$$

(223)

12.2. Complete Rate in the Heterogeneous Universe

In this section, the rate of the homogeneous and of the heterogeneous univ. are added and studied.

(1)

Addition: The sum of $\Delta {\dot{\underline{\epsilon }}}_{het}\left({\tilde{\overline{t}}}_{em}\right)$ and ${\dot{\underline{\epsilon }}}_{hom}$ is forms the complete rate:

$${\dot{\underline{\epsilon }}}_{sum}:=\Delta {\dot{\underline{\epsilon }}}_{het}+{\dot{\underline{\epsilon }}}_{hom}={\dot{\underline{\epsilon }}}_{hom}·(1+\kappa )$$

(224)

Hereby and in the following, the ratio ${\kappa }_{q_t}$ is abbreviated by $\kappa$, as a short notation.

(2)

Rate of LFV for the sum of rates: The integrated rate ${\dot{\underline{\epsilon }}}_{L,rr,atd{V}_0}$, shortly called ${\dot{\underline{\epsilon }}}_{hom}$ in Equation (173) is used:

$$\begin{array}{c}\hfill {\dot{\underline{\epsilon }}}_{hom}=4\pi ·G·{\rho }_{vol}·t_{H_{0,hom}}\end{array}$$

(225)

The corresponding rate in the heterogeneous universe is the complete rate ${\dot{\underline{\epsilon }}}_{sum}$. The above relation (Equation (225)) is transferred to the complete rate by replacing corresponding terms: The rate ${\dot{\underline{\epsilon }}}_{hom}$ corresponds to ${\dot{\underline{\epsilon }}}_{sum}$. The density ${\rho }_{vol}=u_{vol}/c^2$ corresponds to ${\rho }_{\Lambda }=u_{\Lambda }/c^2$. The Hubble time $t_{H_{0,hom}}$ corresponds to $t_{H_{0,het}}$. Therefore, the above relation is transferred to the following rate:

$$\begin{array}{c}\hfill {\dot{\underline{\epsilon }}}_{sum}=4\pi ·G·{\rho }_{\Lambda }·t_{H_{0,het}}\end{array}$$

(226)

(3)

Ratio of rates: The ratio of the rates in Equations (225) and (226) is formed:

$$\begin{array}{c}\hfill \underset{1+\kappa }{\underbrace{{\displaystyle \frac{{\dot{\underline{\epsilon }}}_{sum}}{{\dot{\underline{\epsilon }}}_{hom}}}}}=\underset{{\displaystyle \frac{H_{0,hom}}{H_{0,het}}}}{\underbrace{{\displaystyle \frac{t_{H_{0,het}}}{t_{H_{0,hom}}}}}}·\underset{{(1+\kappa )}^{\xi \left(z\right)}}{\underbrace{{\displaystyle \frac{{\rho }_{\Lambda }}{{\rho }_{vol}}}}}\end{array}$$

(227)

Hereby, the first fraction is the rate ratio, it is implied by Equation (224). Similarly, the second ratio of Hubble times is equal to the corresponding ratio of Hubble constants, as $t_{H_0}=1/H_0$. For each redshift $z_{em}$ or z (in a short notation), the third ratio of dynamic densities is an unknown number, which depends on the redshift z:

$$\begin{array}{c}\hfill {\displaystyle \frac{{\rho }_{\Lambda }}{{\rho }_{vol}}}=\mathrm{unknown}\left(z\right)={(1+\kappa )}^{\xi },\mathrm{with}\xi =\xi \left(z\right)\mathrm{and}\kappa =\kappa \left(z\right).\end{array}$$

(228)

Hereby, without loss of generality and for each z, the unknown is expressed with help of another unknown $\xi$. In particular, the unknown$\left(z\right)$ is replaced by the term ${(1+\kappa \left(z\right))}^{\xi \left(z\right)}$.

(4)

Ratio of Hubble constants: As a consequence of the FLE at $k=0$ and of Equation (152), the squared Hubble parameter is

$$H^2={\displaystyle \frac{8·\pi ·G}{3}}·{\rho }_{cr,0}·\left({\Omega }_r·{\displaystyle \frac{R_0^4}{R^4}}+{\Omega }_m·{\displaystyle \frac{R_0^3}{R^3}}+{\Omega }_{\Lambda }\right).$$

(229)

Next, $H_0$ is represented at $t=t_0$ and $r=r_0$:

$$H_{0,het}^2=\underset{H_{0,hom}^2}{\underbrace{{\displaystyle \frac{8·\pi ·G}{3}}·{\rho }_{cr,0}}}·\left({\Omega }_r+{\Omega }_m+{\Omega }_{\Lambda }\right)$$

(230)

Hereby, $H_0$ of a homogeneous univ. in Equation (150) is identified in the underbrace in the above equation. That Equation is solved for the fraction of Hubble constants in the ratio in Equation (227):

$${\displaystyle \frac{H_{0,hom}}{H_{0,het}}}={\displaystyle \frac{1}{\sqrt{{\Omega }_r+{\Omega }_m+{\Omega }_{\Lambda }}}}\mathrm{or}{H}_{0,het}=H_{0,hom}\sqrt{{\Omega }_r+{\Omega }_m+{\Omega }_{\Lambda }}$$

(231)

Moreover, the ratio of dynamic densities,

$${\displaystyle \frac{{\rho }_{\Lambda }}{{\rho }_{vol}}}={(1+\kappa )}^{\xi },$$

(232)

is the same as the respective fraction ${\Omega }_j$:

$${\displaystyle \frac{{\Omega }_{\Lambda }}{{\Omega }_{vol}}}={(1+\kappa )}^{\xi },\mathrm{or}{\Omega }_{\Lambda }={\Omega }_{vol}·{(1+\kappa )}^{\xi }$$

(233)

These results are inserted in the rate ratio in Equation (227):

$$\begin{array}{c}\hfill 1+\kappa ={\displaystyle \frac{1}{\sqrt{{\Omega }_r+{\Omega }_m+{\Omega }_{\Lambda }}}}·{(1+\kappa )}^{\xi },\mathrm{or}\\ \hfill (1+\kappa )·\sqrt{{\Omega }_r+{\Omega }_m+{\Omega }_{vol}·{(1+\kappa )}^{\xi }}={(1+\kappa )}^{\xi }\end{array}$$

(234)

Here, a term for the exponent $\xi$ must be derived:

(5)

Deduction of $\xi$: In Equation (234), the square is applied. Additionally, the following abbreviations are introduced, $y:=1+\kappa \left(z_{\mathrm{em}}\right)$ and $w:=y^{\xi }$:

$$\begin{array}{c}\hfill y^2({\Omega }_m+{\Omega }_{vol}·w)=w^2\mathrm{or}\end{array}$$

(235)

$$\begin{array}{c}\hfill 0=w^2-w·{\Omega }_{vol}·y^2-{\Omega }_m·y^2\mathrm{with}\end{array}$$

(236)

$$\begin{array}{c}\hfill y=1+\kappa \left(z_{\mathrm{em}}\right)\mathrm{and}w=y^{\xi }\end{array}$$

(237)

In general, Equation (235) has a pair of solutions:

$$\begin{array}{c}\hfill w_{\pm }={\displaystyle \frac{{\Omega }_{vol}·y^2}{2}}·\left(1\pm \sqrt{1+{\displaystyle \frac{4{\Omega }_m}{{\Omega }_{vol}^2·y^2}}}\right)\end{array}$$

(238)

Since $w_{-}<0$, it provides no real exponent in Equation (237). Therefore, it is hardly physical. Consequently, $w_{+}$ is used:

$$\begin{array}{c}\hfill \to \xi ={\displaystyle \frac{ln\left(w_{+}\right)}{ln\left(y\right)}}\mathrm{with}y=1+\kappa \left(z_{\mathrm{em}}\right)\mathrm{and}\end{array}$$

(239)

$$\begin{array}{c}\hfill w_{+}={\displaystyle \frac{{\Omega }_{vol}·y^2}{2}}·\left(1+\sqrt{1+{\displaystyle \frac{4{\Omega }_m}{{\Omega }_{vol}^2·y^2}}}\right)\end{array}$$

(240)

12.3. Comparison with Observation

Here, values of $H_{0,het}\left(z\right)$ corresponding to a heterogeneous universe are studied, depending on the redshift z. Thereby, values deduced with the volume dynamics (VD) are compared with observed values in Figure 10. For a huge range of redshifts, corresponding to time ranging from $t=\mathrm{380,000}\mathrm{years}$ (after Big Bang) until $t_0$ (present-day), derived results are in precise accordance with the observed values. In this section, no fit is executed, no assumption is made, and no new hypothesis is proposed. This provides a clear evidence for the presented theory of volume portions.

Additionally, the numerical study is outlined in detail for the case of the CMB (see part 1) and for the case of the redshift $z=0.055$ (see part 2). Also in these cases, the values are in precise concordance with empirical data.

(1)

Relation of $H_0\left(z_{CMB}\right)$ and $H_{0,hom}$ alias $H_{0,\Lambda CDM}$: At $z_{CMB}$, heterogeneity was very small. Therefore, ${\Omega }_{vol}={\displaystyle \frac{2}{3}}$ describes volume, and ${\Omega }_m=1.0-{\Omega }_{vol}={\displaystyle \frac{1.}{3.}}$, and l${\sigma }_{8,CMB}=2.34·{10}^{-4}$ [22] ([table 2]), and $H_0\left(z_{CMB}\right)=66.{88}_{-0.92}^{+0.92}{\displaystyle \frac{\mathrm{km}}{\mathrm{s}·\mathrm{Mpc}}}$ [22] ([table 2]). Hereby, a value $q_t=q_t^{\left(1\right)}$ is chosen at sufficient accuracy. Consequently, $\kappa \left(z_{\mathrm{CMB}}\right)$ in Equation (222) is

$$\kappa \left(z_{\mathrm{CMB}}\right)={\displaystyle \frac{0.000234·1/3}{2.·2/3·{(1+1090.3)}^{2.5}}}=1.5·{10}^{-12}$$

(241)

With it, we evaluate $\xi$, see Equations (239) and (240):

$$\begin{array}{c}\hfill {\xi }_{\mathrm{CMB}}=1.5\end{array}$$

(242)

Consequently, the Hubble constant $H_{0,het}\left(z_{CMB}\right)$ is deduced by using $\kappa \left(z_{\mathrm{CMB}}\right)=\mathcal{O}\left({10}^{-12}\right)$, see Equation (241):

$$H_{0,het}\left(z_{CMB}\right)=H_{0,hom}·(1+\mathcal{O}\left({10}^{-12}\right))\approx H_{0,hom}=:H_{0,\Lambda CDM}$$

(243)

That result is insightful: $H_{0,het}\left(z_{CMB}\right)\approx$ $H_{0,\Lambda CDM}$. This result confirms that the $\Lambda$CDM model is an idealization.

(2)

Hubble constant $H_{0,het}\left(z_{late}\right)$ at $z_{late}=0.055$ alias $z_{near}=0.055$: The redshift $z=0.055$, correspomds to the late universe, it is already relatively heterogeneous. Therefore, $H_0$ is deduced by utilizing Equation (231):

$$H_{0,het}=H_{0,\Lambda CDM}·\sqrt{{\Omega }_m+{\Omega }_{vol}·{(1+\kappa )}^{\xi }}$$

(244)

Further quantities in Equation (244): Empty space based density parameters: ${\Omega }_{vol}={\displaystyle \frac{.2}{3}}$, ${\Omega }_m=1.-{\Omega }_{vol}={\displaystyle \frac{1.}{3}}$, ${\sigma }_8=0.{8118}_{-0.0089}^{+0.0089}$ [22] ([table 2]). Therefore, $\kappa \left(z_{\mathrm{late}}\right)$ is:

$$\kappa \left(z_{\mathrm{late}}\right)={\displaystyle \frac{0.8118·1/3}{2·2/3·{(1+0.055)}^{2.5}}}·{\displaystyle \frac{q_t^{\left(n\right)}}{q_t^{\left(1\right)}}}=0.{1842}_{-0.003}^{+0.003}$$

(245)

Hereby, the numerical investigation provides the following values of $q_t^{\left(n\right)}$: $q_t^{\left(1\right)}=0.881$, $q_t^{\left(2\right)}=0.916$, $q_t^{\left(3\right)}=0.913$, $q_t^{\left(4\right)}=0.914$, $q_t^{\left(5\right)}=0.91398$, $q_t^{\left(6\right)}=0.91399$ = $q_t^{\left(7\right)}$. This value is used as a fixed point, at sufficient numerical accuracy.

With it, the idealized value ${\xi }_{CMB}=1.5$ is improved to the realistic value of $\xi$, see Equations (239) and (240):

$$\begin{array}{c}\hfill {\xi }_{\mathrm{late}}=1.532\end{array}$$

(246)

Therefore, the deduction provides this $H_0$ result:

$$\begin{array}{ccc}\hfill H_{0,het}\left(z_{\mathrm{near}}\right)& =& H_{0,\Lambda CDM}\sqrt{{\displaystyle \frac{1}{3}}+{\displaystyle \frac{2}{3}}·1.{1842}^{1.532}}\hfill \end{array}$$

(247)

$$\begin{array}{ccc}& =& 73.174\pm 1.08{\displaystyle \frac{\mathrm{km}}{\mathrm{s}·\mathrm{Mpc}}}\hfill \end{array}$$

(248)

As a consequence, the derived value is in accurate concordance with empirical data in Equation (14):

$$H_{0,obs}\left(z_{late}\right)=73.04(\pm 1.01){\displaystyle \frac{\mathrm{km}}{\mathrm{s}·\mathrm{Mpc}}}\mathrm{at}\langle z\rangle =0.055$$

(249)

The above is named baseline result [13] ([Sections 5.1 and 5.2]).

Empirical and deduced data have the fractional difference shown below:

$$\begin{array}{ccc}\hfill {\Delta }_{obs,theo}& =& {\displaystyle \frac{H_{0,obs}\left(z_{late}\right)-H_{0,het}\left(z_{\mathrm{late}}\right)}{H_{0,obs}\left(z_{late}\right)}}={\displaystyle \frac{73.174-73.04}{73.04}}=0.18\%\hfill \end{array}$$

(250)

The fractional difference is very small. This provides evidence for the present theory.

13. Conclusions

In the present paper, the hypothetic deductive method has been used, see Section 1.1. Achieved results are presented progressively in respective short paragraphs:

(1)

In a first step of that method, very reliable observations are characterized as a basis.

13.1. Deduced Results

As a second step in the hypothetic deductive method, results are derived.

(2)

From the above basic facts in step (1), the following results have been deduced:

(2.1)

The quantum postulates have been deduced, see Section 9.

(2.2)

Laws of gravity and curved space near a mass have been deduced, see Section 6.

(2.3)

For the case of the homogeneous universe, the energy density of dark energy has been derived, $u_{DE}$, this value is also called $u_{vol}$.

(2.4)

For values of the redshift z ranging from the redshift $z_{CMB}$ of the emission of the Cosmic Microwave Background (CMB) until to the present-day redshift $z_0=0$, the function $H_0\left(z\right)$ of the Hubble constant as a function of the redshift z is deduced. This function explains the Hubble tension.

13.2. Comparison with Empirical Findings

As a third step in the hypothetic deductive method, the deduced results are compared with respective empirical data.

(3)

In order to complete the hypothetic deductive method, the deduced results are compared with respective empirical findings:

(3.1)

The quantum postulates are in accordance with many experimental data. This is shown in standard text books of quantum physics.

(3.2)

The deduced laws of gravity and curved space near a mass are confirmed by many empirical findings. This is shown in standard text books of classical mechanics, gravity and relativity.

(3.3)

For the case of the homogeneous universe, the deduced value of the energy density of dark energy $u_{DE}$ or $u_{vol}$ is in precise accordance with empirical data within the errors of measurement, see Section 10.3.5 and the respective Equations (178)–(180).

(3.4)

The deduced function $H_0\left(z\right)$ of the Hubble constant as a function of the redshift z is in accurate concordance with empirical data within the errors of measurement, see Section 12.3, Figure 10 and Equations (243) and (248)–(250). This deduced function explains the Hubble tension.

Moreover, this function predicts the values $H_0\left(z\right)$ of the Hubble constant for those values of the redshift z, for which a measurement of the Hubble constant is not yet available.

13.3. Hypothetic Deductive Method II

The hypothetic deductive method and the used basis are described in Section 1.1. On that basis, results are deduced, and especially essential deduced results that are summarized in Section 13.1. These deduced results are compared with respective empirical data in Section 13.2. Thereby, a precise accordance with empirical data is achieved within the errors of measurement. Hereby, no fit is executed, and no unfounded hypothesis is proposed. Accordingly, as a special conclusion, a highly convincing evidence for the derived function $H_0\left(z\right)$ and for the derived explanation of the Hubble tension is achieved. As a consequence of the precise accordance elaborated above, this conclusion is supported by the deduced results.

Additionally, this precise accordance elaborated above completes the used hypothetic deductive method. Altogether, the used method is described in very high detail, especially essential results are presented in an especially clear and detailed manner, and the respective conclusions (including their support by these results) are worked out step by step in an especially detailed manner.

14. Discussion

14.1. The Problem Is Solved with Derived Indivisible Portions of Space

The problem of the Hubble-tension has been addressed and solved. Thereby, a critically discussed method is used, and an explanation is derived. The Hubble-tension is the following finding: Significantly different values have been measured for $H_0$.

In order to solve that problem, essential fundamental properties of space have been used: The energy density $u_{DE}$ of space is important, as it shows that space has a real energy. Moreover, the space has no rest mass. On that basis, a very enlightening and momentous result has been derived: A space paradox and its solution are derived: Space is not a single entity, but space is a stochastic average of volume portions propagating at $v=c$.

As a consequence, for the case of homogeneous space, by using a local isotropy, the indivisibility of volume portions with $v=c$ is derived. Additionally, empirical evidence has been provided for it. Therefore, these results are very founded.

14.2. Portions of Space Imply a Momentous Unification

The derived volume portions have many insightful and momentous implications: For mathematical reasons, these volume portions obey a differential equation of volume dynamics. In particular for the case of an indivisible volume portion $\delta V_j$, this differential equation implies the Schrödinger equation and a generalization thereof. Thereby, the corresponding wave function $\Psi$ is the time derivative of a rel. add. vol. ${\epsilon }_{L,rr,j}$ multiplied by a normalization constant $t_n$. According to the Higgs [48] mechanism, this derived Schrödinger equation includes the case of matter, as matter is formed from space via a phase transition.

Furthermore, the volume dynamics implies gravity. Moreover, on the basis of that implied gravity, the volume dynamics implies the quantum postulates. Altogether, this shows that the volume dynamics implies the deeply founded unification of quanta (including quantum postulates), gravitation (including exact fields and potentials related to volume portions), and relativity (including curvature explained by VPs). That foundation of the unification is very fundamental, as gravity is needed for the derivation of the quantum postulates. This provides the clarifying insight that quantum physics can hardly be understood without gravity at a fundamental level. Accordingly, the more general volume dynamics of indivisible volume portions is a fundamental unification of quanta and gravitation (quantum gravity). An additional consequence is the local formation of volume (LFV).

14.3. The Unification Implies Values for Homogeneous Space

On the basis of that unification, a homogeneous universe is analyzed first. In that ideal case, many basic and ideal results have been deduced: The GFV (global formation of volume) in the universe, from its start until today. The result is $u_{DE,hom}=u_{vol}$, and its value. These results are in accurate concordance with the corresponding empirical values, values of the very early universe, which was very homogeneous. The Hubble constant $H_{0,hom}$ represents a calendar date $t_0\approx t_{H_{0,hom}}={\displaystyle \frac{1}{H_{0,hom}}}$, therefore, $H_{0,hom}$ cannot be derived from fundamental physical constants only.

14.4. The Unification Implies Values for Heterogeneous Space

The heterogeneity in the universe increased gradually in the course of time. That time can be described in an especially robust manner depending on the cosmological redshift: The present-day universe has $z=0$, and the emission of the CMB in a very early universe occurred near $z=1090.3$.

On this basis founded by the above unification in the case of homogeneity, the results in Section 14.3 have been generalized by including heterogeneity. In this manner, the following results have been derived as a function of the redshift: the GFV (global formation of volume), the density of volume $u_{DE}$ and the Hubble constant $H_0\left(z\right)$. These results are in accurate concordance with the corresponding observed values in Figure 10. Moreover, the derived Hubble constant $H_0\left(z\right)$ as a function of z includes many values that have not yet been observed. These values are predicted.

14.5. Very Convincing Evidence Is Achieved

In this paper, all results have been derived without using any unfounded hypothesis, without proposing any postulate, and without executing any fit. Thereby, an accurate concordance with empirical data is achieved within measurement errors.

Therefore, the comparison of theory and observations provides a very convincing evidence for the developed and derived theory. That theory includes the following essential results: indivisible volume portions, volume portions, the volume dynamics, the derivation of the quantum postulates, the derivation of exact gravity, the derived unification of quantum physics, relativity and gravity, $u_{DE}$, $u_{DE,hom}=u_{vol}$ as an insightful ideal value, deduction of the Hubble tension by a gradual increase in heterogeneity, from the early universe until today.

14.6. Structure of the Derived Innovative Theory

The Hubble tension represents a fundamental problem about the dynamics of space in the universe. This assessment is affirmed by other hardly understood problems about space and time:

(1)

the International Astronomical Union (IAU) proclaimed the problem of finding an adequate coordinate system (ACS), see [14,24],

(2)

the unexplained nature and value of the energy density of dark energy, see [9,10,16,111],

(3)

the incompatibility of general relativity theory and quantum theory, see [68,112,113,114].

(4)

and the identification of the source of the Hubble tension, see [13].

The fundamental role of these problems is emphasized by the fact of the Nobel Prizes in Physics in 2006, see [16], in 2011, see [9,15], and in 2022, see [68,115,116].

Accordingly, an innovative and novel theory has been derived from founded observations, see the cognitive map in Figure 5:

As a first step, the problem of finding the ACS and its velocity ${\overrightarrow{v}}_{ACS,CS}$ relative to an arbitrary reference coordinate system (CS) has been solved. While the coordinate systems proposed by relativity theory are insufficient for space navigation, see [14,117], so that present-day relativity theory is incomplete, the derived ACS is in precise concordance with observation, see [23,24]. Together with the ACS, relativity theory can now be used in space flight: If a clock C is described in the ACS, then relativity theory can predict the time of the clock correctly, even under the circumstances of the precise measurements in space navigation. Moreover, the present fundamental derived unification of gravity, relativity and quanta can improve the precision of space navigation, this will be presented in a forthcoming paper. Furthermore, in the ACS, the velocity in the energy velocity relation and in the energy momentum relation is in accordance with a clock in the ACS. Accordingly, these relations are based on the Equation of time dilation and on the ACS in Figure 5. Therefore, the derived theory solves the first of the above problems and enables high precision space navigation.

In a second step, the space paradox is derived, which shows that space cannot be a single object ranging from one end of the light horizon to the other, and beyond, as it is modeled in relativity theory, see [3,5,6,20]. In contrast, the solution of the space paradox shows that homogeneous and isotropic space consists of indivisible volume portions, see Figure 5.

Thirdly, this raises the question, what the dynamics of these volume portions is. That dynamics has been derived without any additional assumption on a completely mathematical basis. As a consequence, that dynamics is very founded, it is the volume dynamics (VD), a differential equation, see Figure 5.

Fourthly, this VD explains and implies three fundamental physical dynamics: the local formation of volume (LFV), exact gravity, and the deterministic and stochastic dynamics of quanta.

Fifthly, this shows that gravity, an ACS-based relativity theory, and quantum physics are derived from the same dynamics, the VD. As a consequence, in this derived theory, the terms of these theories can be transformed to each other, and they can be used correctly in combination. For instance, the wave function in the Schrödinger equation is proportional to the time derivative of the relative additional volume. Consequently, the wave function has a physical meaning.

Hereby, the local formation of the relative additional volume explains and implies the correct value of the energy density $u_{DE}$ of dark energy. This value has neither been derived from relativity theory, nor from quantum theory. Another correctly combined application is the derivation of the Hubble constant $H_0$ as a function of the redshift z.

Therefore, the theory derived here essentially solves the problem (3) in the above list, the incompatibility of relativity theory and quanta. Moreover, the derived theory solves problem (4), the identification of the source of the Hubble tension. Furthermore, the presented theory provides the process of formation and the value of the dark energy density $u_{DE}$, so that the theory essentially solves problem (2) of the above list. Altogether, this shows that the equations in this theory have a new and innovative additional meaning, as they describe derived indivisible volume portions, and as they unify quanta with an ACS-based relativity.

Sixthly, the generality of the theory derived here is emphasized by the fact that the Dirac theory as well as quantum field theory can be developed on the basis of the Schrödinger equation, which has been developed here, see [61,65].

Seventh, based on the theory derived here, Hamilton’s principle and the principle of gauge invariance have been successfully used in order to solve problems in the physics of fundamental interactions and elementary particles, see [72]. This shows that the present theory is compatible with very valuable and insightful principles in physics.

Altogether, the theory presented here is derived from founded observations as well as from the equation of time dilation. As a consequence, the derived theory is a physically adequate unification. Moreover, that theory essentially solves the four problems presented above.

14.7. Used Cosmological Parameters

In the present derivation of the Hubble constant $H_0\left(z\right)$ as a function of the redshift z, the following parameters and measurement procedures are used and explained progressively in respective paragraphs:

(1)

The $\Lambda$CDM-model is valid in a homogeneous universe. Moreover, at the Cosmic Microwave backgrount (CMB), the universe was very homogeneous. Consequently, measurements based on the CMB can be evaluated with the $\Lambda$CDM-model in a precise and reliable manner.

(2)

The value $H_0\left(z_{CMB}\right)$ of the Hubble constant at the redshift $z_{CMB}$ of the CMB is used.

In general, one value for the Hubble constant must be measured, as the Hubble constant represents a calendar date, $t_{H_0}={\displaystyle \frac{1}{H_0}}$. As a calendar date is a specific value, it cannot be derived from general physical constants or laws.

Moreover, that value $H_0\left(z_{CMB}\right)$ is used, that is measured on the basis of the CMB and that is evaluated with the $\Lambda$CDM-model. As a consequence of item (1), that value $H_0\left(z_{CMB}\right)$ is precise and reliable.

(3)

The value ${\Omega }_m$ of the density parameter of matter is used. Thereby, that value is used here, that has been measured with help of CMB data that have been evaluated by application of the $\Lambda$CDM-model. As a consequence of item (1), that value of ${\Omega }_m\left(z_{CMB}\right)$ is precise and reliable in the homogeneous universe, characterized by $0\le {\sigma }_8\left(z\right)\ll 1$.

In addition, the value of ${\Omega }_m={\displaystyle \frac{1}{3}}$ has been derived for the case of the completely homogeneous universe. This is in accordance with the CMB—based measured value ${\Omega }_m=0.321\pm 0.013$.

In general, at a redshift z, in a heterogeneous universe, characterized by $0\le {\sigma }_8\left(z\right)$, the value of ${\Omega }_m\left(z\right)$ differs slightly from ${\Omega }_m\left(z_{CMB}\right)$. The difference can be expressed as a function of ${\sigma }_8\left(z\right)$, similarly as elaborated in the present theory for the case of $H_0$, see Figure 10. This example shows that the $\Lambda$CDM-model can be a useful effective theory for the parameter estimation.

(4)

The value ${\sigma }_8$ of the standard deviation of matter fluctuations is used. Thereby, that value is used here, that has been measured with help of CMB data that have been evaluated by application of the $\Lambda$CDM-model.

Similarly as in item (3), also in this case, the $\Lambda$CDM-model can be a useful effective theory for the parameter estimation. Altogether, these examples show, how the $\Lambda$CDM-model can be a useful effective theory for the parameter estimation.

14.8. Dynamical Effects of the Breaking of Translation Symmetry

In general, the breaking of translation symmetry has immense dynamical effects. These effects are elaborated progressively in respective paragraphs:

(1)

A breaking of translation symmetry changes time: A completely empty space could have the property of translation invariance. When a gravitational field ${\overrightarrow{G}}^{*}$ enters that space, this has momentous geometrical and dynamical consequences: ${\overrightarrow{G}}^{*}$ breaks translation invariance. Based on ${\overrightarrow{G}}^{*}$, each clock C in that space obtains a determined velocity ${\overrightarrow{v}}_{C,\underline{{\overrightarrow{G}}^{*}}}$ relative to ${\overrightarrow{G}}^{*}$, in an unscreenable manner. As a consequence, each such clock C has the kinematic fractional time difference $\delta t_{kin,frac,C}$ = $-{\displaystyle \frac{{\overrightarrow{v}}_{C,\underline{{\overrightarrow{G}}^{*}}}^2}{2c^2}}$. This has been confirmed by observation, and predictions for future observations are provided. Consequently, a coordinate system is an ACS, iff its velocity relative to ${\overrightarrow{G}}^{*}$ is zero, ${\overrightarrow{v}}_{ACS,\underline{{\overrightarrow{G}}^{*}}}=\overrightarrow{0}$. Therefore, ${\overrightarrow{G}}^{*}$ breaks the principle of relativity by causing local special coordinate systems, the Adequate Coordinate Systems (ACS).

(2)

A breaking of translation symmetry changes the Hubble rate $H_0$ of the expansion of the universe. A completely homogeneous universe could have the property of translation invariance. At the time $t_{CMB}$ of the formation of the CMB, the heterogeneity of our universe was very small. Without heterogeneity, space expanded according to the $\Lambda$CDM model. During the time evolution of our universe, gravity caused an increase in heterogeneity, described by the standard deviation of matter fluctuations ${\sigma }_8\left(t\right)$.

The $\Lambda$CDM model describes the dynamical effect of homogeneous matter, but the dynamical effect of ${\sigma }_8\left(t\right)$ is not included. More generally, the present theory derives and includes the dynamical effect of ${\sigma }_8\left(t\right)$. As a consequence, the function $H_0\left(z\right)$ is derived, the function $H_0\left(z\right)$ is in precise accordance with observation (see Figure 10, hereby, no fit is executed, and no unfounded hypothesis is proposed), the function $H_0\left(z\right)$ predicts future observations, and the Hubble tension is explained. This provides convincing evidence for including heterogeneity in the dynamics.

In both examples (1) and (2), translation symmetry is broken by a heterogeneity of matter density or of the gravitational field ${\overrightarrow{G}}^{*}$. In both cases, this has an immense effect upon the dynamics. Especially in (1), this causes a breaking of the principle of relativity, which causes a change in time, localized at the field ${\overrightarrow{G}}^{*}$.

In general, the similar paradigms of translation symmetry, of homogeneity, and of the principle of relativity can cause an immense idealization. This idealization can be overcome by a paradigm shift, which includes the dynamical effects of translation symmetry breaking, of heterogeneity, and of principle of relativity breaking. Here, the dynamical effect of heterogeneity in nature is described with help of the volume dynamics. It provides the formation of volume in nature, the energy density $u_{DE}$ of volume, the function $H_0\left(z\right)$, and the explanation of the Hubble tension. Moreover, the dynamical effect of local breaking of the principle of relativity is described with the Adequate Coordinate System (ACS), which provides improved space navigation. Furthermore, for the first time, the ACS also provides the proof that at each point of the universe, there exists an ACS, which guarentees the validity of the relativistic kinematic time difference.

Additionally, the finding of moving regions of an ACS and of time opens the question: What are the smallest moving portions of space. The answer is provided with help of the space paradox: space consists of stochastically moving indivisible volume portions. These imply a volume dynamics, the quantum postulates, emergent gravity, and emergent curvature of space. In this manner, that question opens the path to the founded and impartible unification of cosmological volume, including adequate coordinate systems, of curved space and time, of quanta and of gravity.

Altogether, the derived Fundamental Solution overcomes the idealized principle of relativity by accepting the omnipresent breaking of translation invariance in our world, and by deriving the immense dynamical consequences of that breaking of translation symmetry. This represents an important paradigm shift from a description restricted to translation symmetry to a new description including realistic and dynamically essential breaking of translation symmetry. Thereby, the equations of the Fundamental Solution include localized structures that are not included in the principle of relativity, and these equations of the Fundamental Solution are generally applicable everywhere in the universe.