Proportional Equation and Multiplicative Superposition: New Way to Understand Reality

1. Introduction

We believe that, in the universe, there is a quantitative order that can be explained by physics. Where does such a quantitative order come from? Regardless of whether it is right or not, an answer to this question relies on what model of space we select, because all physical phenomena occur in space. Roughly speaking, there have been two types of space models: a) 0-based models interpreting the space as nothing to which the number of zero can be assigned and b) 1-based models opposing this interpretation. The currently most-widely accepted model is the 0-based model called the atomism, in which the space is interpreted as an empty (i.e., zero-assigned) void between atoms. However, in the case that zero, the very special number, is assigned to the space, the space itself can no longer be treated as a quantitative entity and is completely forgotten in a theorization process. Only by assuming additional constraints, such as the conservation laws, the least action principle, or the Einstein field equation (EFE), the space could be incorporated into theories. Nevertheless, this process enables the space to have some quantitative properties that are limitedly given by such constraints, at best.

Despite the great achievements accomplished based on the atomism, there are still difficulties in understanding the inflationary and accelerating expansions of the universe[1,2,3], the missing mass problem[4,5,6,7,8], and the origins of the four fundamental forces and the quantum phenomena[9,10]. Thus, in this article, we will consider a new 1-based space model, in which the whole universe is regarded as a single kind of real entity itself (hereinafter, "Mach fluid") completely filling a 3-dimensional space and the apparent difference between matter and space is regarded as a consequence of a difference in (1-dimensional) density of the Mach fluid (hereinafter, “(1D) Mach density”). It should be noted that, in this model, the space has inherent quantitative properties, not properties limitedly given by the constraints. In particular, the reader should pay attention to the fact that, due to such inherent quantitative properties, the Mach density of the space can be changed not only in space, but also in time. We will show that the temporal variation of the 1D Mach density is related to most of the above problems.

Before we go on, let us first define some terms to be used in this article. Two observers, Murphy and Cooper, will be mentioned in this article. Murphy is an actual observer (e.g., in the laboratory) in the present epoch of the expanding universe, observing a target object (hereinafter, “TARS”). Cooper is an imaginary observer who is located at the same point in spacetime as TARS, is instantaneously at rest with respect to Murphy, not to TARS, and is accelerated with the same acceleration as TARS. It is obvious that the above requirements for Cooper cannot be satisfied by a single observer, and this means that Cooper should be interpreted as a theoretical element for parameterization or a group of observers. TARS, i.e., the target object, may be an electron, a positron, a star, or a galaxy, depending on what system we select, but in a sense, it corresponds to a free-falling observer in the general theory of relativity. In this article, we will pay attention to physical descriptions made by Cooper, not by TARS (i.e., the free-falling observer). This means that we will develop our theory in a manner similar, but different, to the general theory of relativity (see Appendix A and Appendix B). A distribution factor $\Lambda (\mathbf{r},t)$, which is a scalar function defined at every point $(\mathbf{r},t)$ in spacetime, will be used to represent a variation in space and time of the 1D Mach density. It should be remembered that $\Lambda (\mathbf{r},t)$ should be a non-negative real number (hereinafter, “numerical restriction”) and has an observer-independent property, because the Mach fluid is supposed as the real entity.

The article is organized as follows. In Sec. 2, we introduce three postulates and derive two equations of motion, Eqs. (8) and (9), which are valid for Cooper and Murphy, respectively. In the first postulate, we propose a mathematical way of expressing the superposition in a many body system. In the second postulate, we introduce a quantity, $\epsilon$, which will be used as a constant of motion for Cooper. In the third postulate, we introduce calibration factors, which define ratios between spatial/temporal lengths in Cooper and Murphy’s frames and are used as parts of the metric tensor. The equation of motion, Eq. (8), which is valid for Cooper, is derived from the constant of motion $\epsilon$, and the equation of motion, Eq. (9), which is valid for Murphy, is obtained by projecting Eq. (8) onto the Murphy’s frame. The calibration factors in the postulate 3 are used for this projection. In Sec. 3.1, a system of $\Lambda =\Lambda \left(t\right)$ will be discussed to explain our results on the inflationary and accelerating expansions of the universe. In Sec. 3.2, a spherically symmetric static system will be discussed to determine a distribution factor $\Lambda \left(r\right)$ producing an inverse square field. In Sec. 3.3, we show that the constant of motion, $\epsilon$, is a quantity associated with the energy conservation law. In Sec. 3.4, we define a pair of distribution factors, ${\Lambda }_{\pm }\left(r\right)$, for proton and electron. In Sec. 3.5, we discuss implications of the distribution factors ${\Lambda }_{\pm }\left(r\right)$. In Sec. 3.6, we briefly explain how to understand the quantum phenomena in our approach and then derive an equation resembling the Schrödinger’s equation. In Sec. 3.7, we explain how to understand the lensing mass distribution, which is one of observations associated with the missing mass problem, based on the first postulate. In Sec. 3.8, we show that, in the case that the Mach density is changed with time (e.g., by the expansion of the universe), the electric force can be split into eight distinct forces. In Sec. 3.9, we review interesting and perplexing aspects in the results of Sec. 3.8. In Sec. 3.10, we mention our vague ideas to alleviate the perplexing problems. In Sec. 4, we propose experiments to test our approach.

2. Theory

In this article, $d{X}^0(=dp)$ and $d{X}^i$ will be used to represent temporal and spatial lengths in Cooper’s frame, and $d{x}^0(=dt)$ and $d{x}^i$ ($i=1,2,3$) will be used to represent those in Murphy’s frame. A calibration factor $h_{\mu }(\mathbf{r},t)$ will be used to define a ratio between the lengths $d{X}^{\mu }$ and $d{x}^{\mu }$, where $\mathbf{r}$ is a position of TARS or Cooper at time t in Murphy’s frame and $\mu =0,1,2,3$. Retarded time is not considered in this article, for simplicity.

2.1. Postulate 1 - Mach Principle

The superposition in a many body system is represented by multiplication of distribution factors of objects constituting the system, not by summation, as follows:

$$\Lambda (\mathbf{r},t)={\Lambda }_M(\mathbf{r},t)\prod _{i=1}^n{\Lambda }_i\left(\mathbf{r}-{\mathbf{r}}_i,t\right),$$

(1)

where n is the number of source objects (i.e., matters) constituting a chosen source system, ${\Lambda }_i$ is a distribution factor of an i-th source object whose characteristic point (e.g., center) is located at ${\mathbf{r}}_i$, and ${\Lambda }_M$ is a factor representing contributions from all other objects in the universe, except the chosen source system, on the distribution factor $\Lambda$.

Here, $\Lambda (\mathbf{r},t)$ in the left-hand side can be understood as a distribution factor representing a background space for TARS, and in the same context, ${\Lambda }_M(\mathbf{r},t)$ can be understood as a distribution factor representing a background space (hereinafter, ‘Machian space’) for the chosen source system. In this article, a term ’Machian objects’ will be used to refer to all other objects in the universe, except the chosen source system.

2.2. Postulate 2 - Constant of Motion

For an observer (e.g., Cooper) at the same point in spacetime as TARS, the following proportional equation is valid.

$$\begin{array}{cc}\hfill \epsilon (X^i,\mathbf{V},p)& \equiv {\psi }_0\frac{\gamma \left(\mathbf{V}\right)}{\Lambda (X^i,p)}=\mathrm{constant},\mathrm{and}\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \gamma \left(\mathbf{V}\right)& ={\left(1-\frac{V^2}{c^2}\right)}^{-1/2},\hfill \end{array}$$

(3)

where $\mathbf{V}$ is a three velocity of TARS measured by Cooper: that is, $\mathbf{V}={\sum }_{i=1,2,3}\frac{d{X}^i}{dp}{\widehat{\mathbf{X}}}_i$ and ${\widehat{\mathbf{X}}}_i$ is a three basis vector in Cooper’s frame. The equation (3) is the Lorentz factor, which is given as a function of the velocity $\mathbf{V}$. It should be noted that Eq. (2) is valid for Cooper, not for Murphy, and thus, that the quantity $\epsilon$ in Eq. (2) serves as a constant of motion for only Cooper. A further detailed comment on the postulate 2 can be found in Appendix A. Meantime, in some topics, we will consider a case, in which TARS and Cooper are located at the same point in spacetime as Murphy (hereinafter, ‘coincidence condition’). Note that, under such a coincidence condition, $\mathbf{V}$ becomes equal to a velocity, $\mathbf{v}={\sum }_{i=1,2,3}\frac{d{x}^i}{dt}{\widehat{\mathbf{x}}}_i$, measured by Murphy; that is, Cooper’s description becomes equal to Murphy’s description. The quantity ${\psi }_0$ is introduced to represent a physical quantity associated with TARS, and it will be regarded as a constant unless otherwise noted.

2.3. Postulate 3 - Metric Tensor

Calibration factors $h_{\mu }$ are given as follows:

$$\begin{array}{cc}\hfill d{X}^i& =h_i(\mathbf{r},t)d{x}^i,\mathrm{and}\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill cdp& =cdt/h_0(\mathbf{r},t),\hfill \end{array}$$

(5)

where c is the speed of light, and for a spherically symmetric system, the calibration factors are given as follows:

$$h_0=h_1=\Lambda (r,t),h_2=r,h_3=r\sin \theta$$

(6)

In this article, we will restrict our attention to only three systems of $\Lambda \left(t\right)$, $\Lambda \left(r\right)$, and $\Lambda (r,t)$. The gist of the postulate 3 is that, in such three systems, the metric tensor can be diagonalized and the diagonalized components can be expressed in terms of the distribution factor, as stated by Eq. (6). One might think that the postulate 3 is reasonable, but in a sense, it is true that it is not rigorous or complete. A further comment on this issue can be found in Appendix B.

2.4. Equation of Motion

Using $\partial \gamma /\partial V^i=({\gamma }^3/c^2)V^i$ and $V^2=\mathbf{V}·\mathbf{V}$, the total derivative of the constant $\epsilon$ in Eq. (2) with respect to Cooper’s time p gives:

$$0=-\frac{c^2}{\mu {\gamma }^2}\mathbf{V}·\left(\frac{\partial \mu }{\partial p}\frac{\mathbf{V}}{V^2}+\tilde{\nabla }\mu -\frac{\mu {\gamma }^2}{c^2}\mathbf{A}\right),$$

(7)

where $\mu =\Lambda (\mathbf{r},t)/{\psi }_0,\mathbf{A}=d\mathbf{V}/dp$, and $\tilde{\nabla }\mu$ is a gradient operator in Cooper’s frame (i.e., $\frac{\partial \mu }{\partial X^i}{\widehat{\mathbf{X}}}_i$). In general, $\mathbf{V}$ is not a zero vector, so Eq. (7) is valid only when the bracketed quantity is equal to $\mathbf{0}$ and/or is perpendicular to $\mathbf{V}$. In the former case, we can obtain the following equation (8) from Eq. (7). For the time being, we will continue our discussion on the basis of Eq. (8), but the reader should remember that Eq. (7), not Eq. (8), is the equation valid for general cases.

$$\mathbf{A}=\frac{c^2}{\mu {\gamma }^2}\left(\frac{\partial \mu }{\partial p}\frac{\mathbf{V}}{V^2}+\tilde{\nabla }\mu \right).$$

(8)

According to the postulate 2, the trajectory of TARS described by Eq. (8) is valid for Cooper, but not for Murphy in general. This means that if Murphy uses Eq. (8) as it is, she will observe an anomalous result. This is because Murphy’s ruler and clock are not calibrated to have validity at a region of Murphy, not TARS, and this is the gist of the postulate 2 (see Appendix B). According to the postulate 3, such a calibration can be achieved by using Eqs. (4) and (5). For example, by transforming Eq. (8) to Murphy’s frame using Eqs. (4) and (5)[11], we obtain the equation of motion, which is valid for Murphy, as follows (see Appendix C):

$$\begin{array}{c}\hfill \frac{d^2{x}^k}{d{t}^2}=\frac{c^2}{{\gamma }^2}\left[\left(\frac{1}{\mu }\frac{\partial \mu }{\partial t}\right)\frac{v^k}{V^2}+\frac{1}{h_0^2{h}_k^2}\left(\frac{1}{\mu }\frac{\partial \mu }{\partial x^k}\right)\right]-\frac{d{x}^k}{dt}\left(\frac{1}{h_0}\frac{d{h}_0}{dt}\right)-{\Gamma }_{j\beta }^k\frac{d{x}^j}{dt}\frac{d{x}^{\beta }}{dt},\end{array}$$

(9)

where ${\Gamma }_{j\beta }^k=\frac{{\partial }^2{X}^i}{\partial x^j\partial x^{\beta }}\frac{\partial x^k}{\partial X^i}$ and the repeated indices, except the calibration factors $h_k$, mean the summation. Rewriting Eq. (9) using Eq. (6), we can obtain the equations of motion for a spherically symmetric system, as follows (see Appendix D):

$$\begin{array}{cc}\hfill & a_r=\left[\frac{c^2}{V^2}-3\right]\left[\frac{1}{\Lambda }\frac{\partial \Lambda }{\partial t}\dot{r}+\frac{V^2}{{\Lambda }^4}\frac{1}{\Lambda }\frac{\partial \Lambda }{\partial r}\right]+\left[\frac{2}{\Lambda }\frac{\partial \Lambda }{\partial r}+\frac{1}{r}\right]\frac{r^2{\dot{\Omega }}^2}{{\Lambda }^2},\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & a_{\theta }=\left[\frac{c^2}{V^2}-2\right]\frac{1}{\Lambda }\frac{\partial \Lambda }{\partial t}\dot{\theta }-\frac{1}{\Lambda }\frac{\partial \Lambda }{\partial r}\dot{r}\dot{\theta }-\frac{2}{r}\dot{r}\dot{\theta }+\cos \theta \sin \theta {\dot{\phi }}^2,\mathrm{and}\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & a_{\phi }=\left[\frac{c^2}{V^2}-2\right]\frac{1}{\Lambda }\frac{\partial \Lambda }{\partial t}\dot{\phi }-\frac{1}{\Lambda }\frac{\partial \Lambda }{\partial r}\dot{r}\dot{\phi }-\frac{2}{r}\dot{r}\dot{\phi }-2\cot \theta \dot{\theta }\dot{\phi },\hfill \end{array}$$

(12)

where $V^2={\Lambda }^4\left({\dot{r}}^2+r^2{\dot{\Omega }}^2/{\Lambda }^2\right)$ and $\dot{\Omega }=\dot{\theta }+\sin \theta \dot{\phi }$.

3. Application and Discussion

3.1. System of $\Lambda =\Lambda \left(t\right)$: Inflationary and Accelerating Expansions of the Universe

Here, we discuss the inflationary and accelerating expansions of the universe, based on Eq. (10). For brevity, we will restrict our attention to a one-dimensional motion of TARS, under the cosmological principle (i.e., ${\partial }_x\Lambda =0$). Then, we can describe this system using the position-independent distribution factor $\Lambda =\Lambda \left(t\right)$ and can neglect any angular component in Eq. (10). That is, it is enough to consider only a difference in $\Lambda \left(t\right)$ between epochs, in which Cooper and Murphy live. To take such a difference into consideration, suppose that Murphy and Cooper are two observers in the present and past epochs, $t_0$ and $t<t_0$, respectively. Then, we can rewrite Eq. (10), as follows:

$$\ddot{d}\left(t\right)=\left(\frac{1}{\Lambda {\left(t\right)}^4}\frac{c^2}{\dot{d}{\left(t\right)}^2}-3\right)\left(\frac{1}{\Lambda \left(t\right)}\frac{\partial \Lambda \left(t\right)}{\partial t}\right)\dot{d}\left(t\right),$$

(13)

where $\dot{d}\left(t\right)$ is a projection speed that is obtained by projecting a one-dimensional speed of TARS, measured by Cooper in the past epoch t, onto the Murphy’s frame in the present epoch $t_0$.

Meanwhile, in this system of $\Lambda =\Lambda \left(t\right)$, we can freely choose the positions of Cooper and TARS, without the constraint of the sameness in space, although Cooper and TARS should be still chosen to be in the same epoch. Thus, we can define a distance, $d\left(t\right)$, from Cooper to TARS, even when TARS is very far from Cooper. However, if, as stated by the Hubble’s law, the expansion rate of the universe (i.e., the speed $\dot{d}\left(t\right)$) depends on the distance $d\left(t\right)$, the acceleration $\ddot{d}\left(t\right)$ in the above equation should also vary depending on a distance $d\left(t\right)$ to a chosen object (i.e., TARS). We think that the accelerating expansion of the universe can be explained based on this property of the acceleration $\ddot{d}\left(t\right)$.

In detail, let us introduce the scale factor $S\left(t\right)=d\left(t\right)/d_c$, where $d_c$ is a co-moving distance between Cooper and TARS (e.g., a distant supernova). Owing to the cosmological principle, we can freely place Cooper at the same position as Murphy, and in this case, Murphy has the same co-moving distance. By definition, the distribution factor is inversely proportional to the scale factor: that is, $\Lambda \left(t\right)\propto 1/S\left(t\right)$. Using this, we can define a deceleration parameter $q\left(t\right)$, from Eq. (13), and a time-dependent Hubble parameter $H\left(t\right)$, as follows:

$$\begin{array}{cc}\hfill q\left(t\right)& =-\frac{S\left(t\right)\ddot{S}\left(t\right)}{\dot{S{\left(t\right)}^2}}\propto \left(\frac{S{\left(t\right)}^2{c}^2}{H{\left(t\right)}^2{d}_c^2}-3\right),\mathrm{and}\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill H\left(t\right)& =\frac{\dot{S}\left(t\right)}{S\left(t\right)}=-\frac{1}{\Lambda \left(t\right)}\frac{\partial \Lambda \left(t\right)}{\partial t}.\hfill \end{array}$$

(15)

It should be noted that the sign of the deceleration parameter $q\left(t\right)$ relies on the co-moving distance $d_c$ to the chosen supernova. Since, by definition, $S\left(t_0\right)=1$, Eq. (14) states that if we observe a distant object farther than $c/\sqrt{3}H$, we will find a positive value of acceleration. If the Hubble parameter $H\left(t_0\right)$ at the present epoch (hereinafter, $H_0$) is selected as 70 km/s/Mpc[12], the co-moving distance to such a turning point is $7.6\times {10}^{25}$ m or 8.1 billion ly[13]. Furthermore, Eq. (15) states that the Hubble parameter $H\left(t\right)$ can vary depending on the distribution factor $\Lambda \left(t\right)$, and this may be a clue to the discrepancy between two Hubble parameters independently obtained from the cosmic microwave background (CMB) and supernovae observations[12].

Now, let us discuss the inflationary expansion of the universe. An expansion of the universe means that $\dot{d}>0$ and ${\partial }_t\Lambda \left(t\right)<0$, and, due to the numerical restriction, $\Lambda \left(t\right)$ should be a non-negative real number. Thus, the acceleration $\ddot{d}$ in Eq. (13) can be positive, only when

$$\dot{d}\left(t\right)>\frac{c}{\sqrt{3}}\frac{1}{\Lambda {\left(t\right)}^2}\stackrel{\mathrm{def}}{=}{v}_c(\Lambda ).$$

(16)

According to this condition, if $0<\dot{d}<v_c$, there is no lasting expansion, and just a small vibration is allowed. However, if $\dot{d}>v_c$, the inflationary expansion or the big bang can be triggered. At the initial stage of the big bang, $\Lambda$ would be a very large value, and this means that the inflationary expansion could be triggered by even a very small value of $\dot{d}$. The inflationary expansion would lead to a fast decrease of $\Lambda$ and a fast increase of $\dot{d}$. Nevertheless, during this inflation era, the evolution of the universe would be substantially dominated by the decrease of ${\Lambda }^4$ in Eq. (13), not by the increase of ${\dot{d}}^2$, because ${\Lambda }^4$ would be still extremely high and $c^2/{\dot{d}}^2$ would be substantially a bounded value. Thereafter, when $\Lambda$ decreased to a sufficiently low value (e.g., roughly of the order of one), the system could be naturally escaped from the condition of Eq. (16).

Until now, we have discussed the inflationary and accelerating expansions of the universe only in terms of acceleration based on Eq. (10)). Thus, in order to see if the above results are compatible with the existing results, it is necessary to further study these topics on thermodynamics and the like.

3.2. System of $\Lambda =\Lambda \left(r\right)$ : Determining $\Lambda \left(r\right)$

We will here determine a distribution factor $\Lambda$ of a source object producing an inverse square field (e.g., the electric or gravitational field). To do this, we will substitute $k/r^2\widehat{\mathbf{r}}$ into the left-hand side (LHS) of Eq. (8), where k is $k_{e}Q$ and $-GM$ for the electric and gravitational fields, respectively, $k_e$ and G are the Coulomb and Newton constants, Q and M are an electric charge and a mass of the source object, and r is the radial distance from the center of the source object. Because of $\Lambda =\Lambda \left(r\right)$, the right-hand side (RHS) of Eq. (8) has only the radial component of the gradient term. Thus, using the relation $\gamma =\epsilon \Lambda /{\psi }_0$ of Eq. (2), Eq. (8) can be written as follows:

$$\frac{k}{r^2}\widehat{\mathbf{r}}=\frac{1}{\chi }\frac{c^2}{{\Lambda }^3}\frac{d\Lambda }{dr}\widehat{\mathbf{r}},$$

(17)

where $\chi ={\epsilon }^2/{\psi }_0^2$. The solutions of this differential equation are:

$$\Lambda \left(r\right)=\pm {\left[\frac{2k\chi }{c^2}\left(\frac{1}{r}-\frac{1}{r_0}\right)+\frac{1}{{\Lambda }_0^2}\right]}^{-1/2},$$

(18)

where $r_0$ is a reference position and ${\Lambda }_0$ is a reference value of $\Lambda$ defined at $r_0$. By the numerical restriction on $\Lambda$, the negative solution will be excluded from our discussion. If the references $r_0$ and ${\Lambda }_0$ are choose as ∞ and 1, respectively, the positive solution can be written as follows:

$$\Lambda \left(r\right)={\left(1+\frac{2k\chi }{r{c}^2}\right)}^{-1/2}.$$

(19)

Meanwhile, by the postulate 2, the formula $\gamma =\epsilon \Lambda /{\psi }_0$ and Eq. (8) are valid in only the Cooper’s frame. Thus, Eq. (17) is true under the coincidence condition. Nevertheless, due to its observer-independent property, $\Lambda$ has the same physical meaning, regardless of who is chosen to determine it. By analogy, even when a balloon has a size varying with height, it is possible to uniquely determine the geometry of a mountain, if a specific balloon (e.g., at a specific height) is solely used. Mathematically, this means that a relationship between ${\Lambda }_c$ and ${\Lambda }_m$, described by Cooper and Murphy, respectively, is given by ${\Lambda }_c(\mathbf{r},t)=c_1{\Lambda }_m(\mathbf{r},t)$, where $c_1$ is a ratio specifically given based on their positions in spacetime. Here, since $c_1$ is canceled in each term in RHS of Eq. (8), it does not affect the actual calculation. Thus, even if $\Lambda$ is obtained under the coincidence condition, it can be validly used for in our discussion.

3.3. Physical Meanings of the Constant of Motion $\epsilon$

Here, we will discuss the physical meanings of the constant of motion $\epsilon$, based on the above results. Using Eqs. (2) and (19), we can obtain the following formula.

$$\frac{\epsilon }{{\psi }_0}=\frac{\gamma }{\Lambda }=\sqrt{\chi }=\pm {\left[1-\frac{V^2}{c^2}-\frac{2k}{r{c}^2}\right]}^{-1/2}.$$

(20)

Here, we will again neglect the negative solution1 of Eq. (20). Multiplying both sides of the positive solution of Eq. (20) by $m_0{c}^2$ and expanding the RHS in series, we find

$$E=\frac{1}{2}{m}_0{V}^2+\frac{k{m}_0}{r}+O(V^4,\frac{1}{r^2}),$$

(21)

where $E=m_0{c}^2(\sqrt{\chi }-1)$. This result shows that we can say that, just as $\gamma$ is a dimensionless number associated with a relativistic kinetic energy of an object with unit mass, $\epsilon$ and $\Lambda$ are dimensionless numbers associated with a relativistic total energy (including a rest energy, $m_0{c}^2$) and a relativistic potential energy, respectively, for a system of ${\psi }_0=1$. That is, we can say that Eq. (2) or (20) is another expression of the energy conservation law.

3.4. Distribution Factors of Proton and Electron

If $\chi =1$ and $k=-GM$, the RHS of Eq. (19) is the very factor characterizing the Schwarzschild solution[14], and thus, we can immediately construct the Schwarzschild metric from Eqs. (4), (5), (6), and (19). Moreover, we also find that, in the case of ${\partial }_t\Lambda =0$, the equations of motion (10) to (12) are similar to, but slightly different from, those for the Schwarzschild system in the general relativity[15]. However, we will trace the route of $k=k_{e}Q$, instead of $k=-GM$, because one of ${\partial }_t\Lambda$-containing terms in Eq. (10) exhibits gravity-like behaviors, as will be described below. If this route is right, it implies that the ${\partial }_t\Lambda$ factor (e.g., caused by the expansion of the universe), not a usually-expected ${\partial }_x\Lambda$ factor, is responsible for the gravity.

To go this route, let us first consider the cases of $Q=\pm e$ ($e=1.602\times {10}^{-19}C$). Then, Eq. (19) is written as follows:

$${\Lambda }_{\pm }\left(r\right)={\left(1\pm \frac{2k_{e}e\chi }{r{c}^2}\right)}^{-1/2}.$$

(22)

The reader can find that the plotting curves of ${\Lambda }_{\pm }\left(r\right)$ are asymmetric about the reference value ${\Lambda }_0$: i.e., $|{\Lambda }_{-}\left(r\right)-1|\ne |{\Lambda }_{+}\left(r\right)-1|$. This is analogous to the fact that there is a difference in mass between proton and electron that have opposite electric charges. Also, ${\Lambda }_{-}\left(r\right)$ has a singularity at $r_s=2k_{e}e\chi /c^2$ and has an imaginary value in the inner region ($r<r_s$), but ${\Lambda }_{+}\left(r\right)$ does not. Considering that the numerical restriction prohibits an imaginary value of $\Lambda$, it looks as if an internal structure is required to prevent the imaginary value of ${\Lambda }_{-}\left(r\right)$. Furthermore, if the inverse square field is natural in the 3-dimensional space, it will be natural to regard Eq. (22) obtained therefrom as the most fundamental distribution factors (e.g., for the stable particles)2. For these reasons, we will regard ${\Lambda }_{-}\left(r\right)$ and ${\Lambda }_{+}\left(r\right)$ as the distribution factors of proton and electron and will interpret the electric charge (Q) and the mass (m) as numbers associated with the sign and value of the quantity $(\Lambda -1)$.

3.5. Implications of Equation (22)

The above interpretation forces us to regard the proton or electron as a 3D ubiquitous object existing throughout the universe, not as a point-particle. Indeed, this is just the very result required by our space model. According to this picture, each object interacts with all other objects in the universe, because it has not only an observable part (i.e., matter) but also an unobservable prevailing part. Although this picture may look very weird in the classical point of view, it seems to be compatible with the quantum mechanical point of view. Rather, this picture may alleviate the fundamental difficulty in understanding the quantum phenomena, such as the wave-particle duality (especially, the self-interaction of single electrons in the double slit experiment[16]). Furthermore, this picture allows us to avoid the unnatural interpretation, which states that a 0-dimensional point particle has ’intrinsic properties’ such as mass, spin, and charge, although such an interpretation was inevitable under the existing atomism. For example, according to this picture, each object is a 3D ubiquitous structure with a very large (in fact, infinite) degree of freedom, and this allows us to correlate a number of physical properties in each object with a 3-dimensional structure given by Eq. (22), instead of relying on an unnatural expression of ’intrinsic’. In this case, it would be natural to interpret some fundamental quantities, such as mass, charge, and spin, as being related to structural stability of each particle under the cosmic-scale interaction with all other objects.

Considering the cosmic-scale interaction of each object, behaviors of each object could be completely comprehended, only when the object is described as a whole, not just an observable part (i.e., matter). In this context, it is clear that ${\psi }_0$ in Eq. (2) is an oversimplified description for TARS and that such a particle-based description, which describes only the observable part of the object using a constant ${\psi }_0$, cannot be true, although it is inevitable. In order to correctly describe the motion of TARS as a whole, it would be necessary to restore ${\psi }_0$ to $\psi (\mathbf{r},t)$. Although, as of now, it is unclear whether $\psi (\mathbf{r},t)$ corresponds to the distribution factor of TARS, we will assume that such a restoration gives a correct result, and the topic of quantum mechanics below will be discussed, under this assumption.

3.6. Quantum Phenomena : Implications of Equation (7)

The reader will know that our results on the expansions of the universe are closely related to the quantity of ${\Lambda }^{-1}{\partial }_t\Lambda$ in Eq. (7). In a sense, such discussions are related to the collective or time-averaged behavior of ${\Lambda }^{-1}{\partial }_t\Lambda$ in a many body system, in that an astronomical object is made of many elementary particles. Here, we will discuss the instantaneous behaviors of ${\Lambda }^{-1}{\partial }_t\Lambda$ of an elementary particle. To do this, we will assume that the restoration ${\psi }_0\to \psi (\mathbf{r},t)$ gives the more correct result, as mentioned above, and, for brevity, the source system is composed of a single object. Furthermore, we will discuss this topic under the coincidence condition, because what we want to know here is just the physical meanings of Eq. (7), not exact calculation. Using Eqs. (1) and (2) under the coincidence condition, we can rewrite Eq. (7), as follow:

$$\left(\frac{1}{{\Lambda }_s}\frac{\partial {\Lambda }_s}{\partial t}-\frac{1}{\psi }\frac{\partial \psi }{\partial t}+\frac{1}{{\Lambda }_M}\frac{\partial {\Lambda }_M}{\partial t}\right)=\mathbf{v}·\left(\frac{{\gamma }^2}{c^2}\mathbf{a}-\frac{1}{\mu }\nabla \mu \right),$$

(23)

where ${\Lambda }_s$ is a distribution factor of the source system and ∇ is the gradient operator. From Eq. (15), we know that the LHS of Eq. (23) (especially, the third term) will be affected by the expansion of the universe. In addition to an effect by such a cosmic expansion, according to Eq. (1), the first to third terms in LHS of Eq. (23) will also be affected by each of changes in position and distribution factor of the source, target (TARS), and Machian objects. For example, a change in position of any object having a non-uniform distribution factor leads to a change in the total distribution factor. This means that, in general, the LHS of Eq. (23) is neither zero nor constant: that is, there is a ceaseless fluctuation in every point of space. This fluctuation in the value of the LHS of Eq. (23) will be used as an argument to explain an aspect of the gravitational force.

Meanwhile, for some reasons, it seems that Eq. (7) or (23) is related to the quantum phenomena. For example, since this equation is a direction-independent scalar equation, it allows for a certain freedom in direction of motion of TARS, unexpected in the classical physics, as long as the motion satisfies the scalar condition given by Eq. (23) or the energy conservation required by Eq. (2) or (20). In classical physics, $\mathbf{v}$, $\mathbf{a}$, and $\nabla \mu$ are causally-connected vector quantities, and thus, for a free-falling object, if the LHS of Eq. (23) is zero, the cancellation of the common factor $[\mathbf{v}·]$ in Eq. (23) may be allowed, as in Eq. (8). However, if the LHS of Eq. (23) is not zero as stated above, such a cancellation will not be allowed in general. This means that the scalar equation (7), not the vector equation (8), should be used to more correctly describe the universe. Furthermore, this seems to imply that the causality in direction may be an illusory notion (e.g., resulting from our experience in the macroscopic world or the collective objects) and that the difficulties in understanding the quantum phenomena may perhaps be due, at least in part, to our belief on causality in direction.

If Eq. (23) is correct, we might be able to derive the equations of quantum mechanics from Eq. (23), because the quantum mechanics gives more correct results for such a microscopic object. We can find one clue for this, from Eqs. (20) and (21). That is, changing ${\psi }_0$ to $\psi (\mathbf{r},t)$, using the formula $\chi =q/m_0$ for TARS, and using the operators (i.e., $i\hslash {\partial }_t=E$ and $-i\hslash {\partial }_x=P_x$) for a plane wave, we can get the following equation, from Eqs. (20) and (21), under the coincidence condition.

$$i\hslash \frac{\partial }{\partial t}\psi (\mathbf{r},t)=\left(-\frac{{\hslash }^2}{2m_0}{\nabla }^2+\frac{q}{\chi }\frac{k}{r}\right)\psi (\mathbf{r},t).$$

(24)

Although this equation resembles the Schrödinger’s equation[21], it is true that there are some vague aspects associated with physical meanings of $\psi$ and $\chi$ and so further study is needed.

3.7. Missing Mass Problem: Lensing Mass Distribution

This problem is related to two different types of observations: 1) the galaxy rotation anomaly[4,5] and 2) the lensing mass distribution[6,7,8]. Here, we will first discuss the lensing mass distribution, based on the postulate 1, and then discuss the galaxy rotation anomaly in the issue [A3] of Sec. 3.9. According to the conventional (i.e., sum-based) representation of superposition, the total gravitational potential of a galaxy is expressed by $V_G={\sum }_{i=1}^n{V}_i$, where $V_i=-k_i/|\mathbf{r}-{\mathbf{r}}_i|$. This means that objects (e.g., stars or atoms) in the galaxy contribute independently to the total gravitational potential $V_G$. By contrast, according to the multiplicative representation of superposition required by the postulate 1, the total distribution factor of the galaxy is expressed by ${\Lambda }_G={\prod }_{i=1}^n{\Lambda }_i$, if ${\Lambda }_M=1$ in Eq. (1) for brevity. This implies that each object reveals its presence (i.e., ${\Lambda }_i$) as a ratio to a background distribution factor made by all other objects constituting the galaxy.

Note that $({\Lambda }_G-1)c^2-\left|V_G\right|>0$ always. This means that there is a boosting effect. It should be noted that, according to our multiplicative representation, the greater the background distribution factor at a position of each object, the larger the overall boosting effect. This means that a high-$\Lambda$ region (e.g., in the vicinity of a black hole or in a region of densely-crowded stars) serves as a boosting seed causing a significant increase of ${\Lambda }_G$. By contrast, for a dilute gas structure, even if it has a large total mass, a boosting effect caused thereby is negligible, because a distance between atoms is too large. This model including the boosting effect allows us to explain why the lensing effect in the bullet cluster[6] is overlapped with the galaxies, not with the dilute hot gas, and probably why there is a large difference in dark matter amount between Dragonfly 44 and NGC 1052-DF2[7,8] (if they have a large difference in compactness).

3.8. System of $\Lambda =\Lambda (r,t)$: Eight Separate Forces

We now calculate a radial acceleration (i.e., Eq. (10)), which is exerted on one electron or positron located near one proton, in the present epoch in the expanding universe. Note that, in our 1-based model, this system should be treated as a three-body system including the Machian space, in addition to proton and electron.

To calculate Eq. (10), we should know ${\Lambda }^{-1}{\partial }_r\Lambda ,{\Lambda }^{-1}{\partial }_t\Lambda$, and ${\Lambda }^2$. To simplify this calculation, let us place the source object (i.e., proton) at the origin of a spherical coordinate system. Then, Eq. (1) for this system can be written as $\Lambda (\mathbf{r},t)={\Lambda }_M(\mathbf{r},t){\Lambda }_p(\mathbf{r},t)$, where ${\Lambda }_M(\mathbf{r},t)$ and ${\Lambda }_p(\mathbf{r},t)$ are time-dependent distribution factors of the Machian space and the proton.

Next, to reduce the difficulty in calculating Eq. (10), we will make some assumptions on ${\Lambda }_M$ and ${\Lambda }_p$. First, suppose that each of ${\Lambda }_M$ and ${\Lambda }_p$ can be separated into spatial and temporal parts (i.e., ${\Lambda }_M=S_M\left(\mathbf{r}\right)T_M\left(t\right)$ and ${\Lambda }_p=S_p\left(\mathbf{r}\right)T_p\left(t\right)$). Second, suppose that the spatial parts of the Machian space and the proton are homogeneous and spherically symmetric, respectively (i.e., $S_M\left(\mathbf{r}\right)=1$ and $S_p\left(\mathbf{r}\right)=S_p\left(r\right)$). Then, we find $\Lambda =T_M\left(t\right)T_p\left(t\right)S_p\left(r\right)$ and ${\Lambda }^{-1}{\partial }_r\Lambda =S_p^{-1}{\partial }_r{S}_p$.

Our next goal is to determine ${\Lambda }^{-1}{\partial }_t\Lambda$. From Eq. (15), we know $T_M^{-1}{\partial }_t{T}_M=-H_0$ under the cosmological principle, but strictly speaking, we don’t know instantaneous values of $T_M^{-1}{\partial }_t{T}_M$ and $T_p^{-1}{\partial }_t{T}_p$. Thus, let us suppose that ${\Lambda }^{-1}{\partial }_t\Lambda$ is in proportion to the Hubble constant $H_0$ (i.e., ${\Lambda }^{-1}{\partial }_t\Lambda =\alpha H_0$). Here, it should be noted that, due to the ceaselessly changing property of the LHS of Eq. (23) stated above, the parameter $\alpha$ should also be a ceaselessly changing value.

Lastly, in the present epoch, $T_M\left(t_0\right)=T_p\left(t_0\right)=1$, and thus, we can say that ${\Lambda (r,t)|}_{t=t_0}=S_p\left(r\right)$. The spatial part of the source system, $S_p\left(r\right)$, can be expressed using ${\Lambda }_{-}\left(r\right)$ of Eq. (22), and thus, the RHS of Eq. (10) can be split into eight terms, as follows:

$$\begin{array}{cc}\hfill a_r& =\frac{\tilde{k}}{r^2}+\frac{2{\tilde{k}}^2}{r^3{c}^2}+c^2\alpha H_0{W}_r+\alpha H_0{W}_r\frac{4\tilde{k}}{r}+\alpha H_0{W}_r\frac{4{\tilde{k}}^2}{r^2{c}^2}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -3\alpha H_0{v}_r-3\frac{\tilde{k}{v}_r^2}{r^2{c}^2}{S}_p^2+\frac{v_a^2}{r}{S}_p^{-2},\mathrm{and}\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill W_r& =\frac{v_r}{v_r^2+v_a^2{S}_p^{-2}},\hfill \end{array}$$

(26)

where $\tilde{k}=k_{e}e\chi$ and $v_r$ and $v_a$ are radial and angular velocities, respectively, of TARS (i.e., electron or positron). From now, the eight terms in RHS will be referred to as $a_1\mathrm{to}{a}_8$, respectively, in the listed order, for brevity.

3.9. Interesting and Perplexing Aspects of Eqs. (25) and (26)

There are interesting and perplexing aspects in Eqs. (25) and (26). To begin with, note that the $W_r$-including terms in Eq. (25) cause an infinite acceleration for TARS, which is at rest with respect to the source object, because the $W_r$ factor of Eq. (26) includes a velocity in denominator. If a small velocity is neglected as usual, this infinite acceleration is problematic. However, we can find some phenomena, which hinder TARS from having a zero velocity, in the physical world. Examples are 1) Hubble velocity $v_H$, 2) thermal velocity $v_T$, and 3) orbital and rotation velocities $v_O$ and $v_R$ of an object or its part. Here, note that the Hubble velocity is special in that, unlike other velocities, it has an everlastingly non-zero value as long as the universe expands.

Thus, it is necessary to find a way of incorporating the Hubble velocity into the theory. If $\Lambda$ is spatially flat, an effect caused by the Hubble expansion[17] will be canceled somehow; otherwise, it will remain anisotropically. Then, can such an effect be formulated by Hubble’s law? For the present, this is somewhat unclear, so we will assume that such an effect can be described in terms of $v_H=\beta H_{0}r$ and $v_r=\dot{r}+v_H$, where $\beta$ is an unknown parameter, $v_r$ and $\dot{r}$ are resultant and coordinate velocities, respectively, of TARS in the radial direction, and r is a center-to-center distance between TARS and the source (i.e., the proton here) of the anisotropy. Here, $\dot{r}$ will be given as a sum of $v_T$, $v_O$, $v_R$, and so forth.

Now, let us consider Figure 1 and Figure 2, obtained by plotting Eq. (25). The physical constants and parameters used in the plotting are as follows: $c=3.0\times {10}^{8}m/s$, $k_e=8.987\times {10}^{9}N{m}^2{C}^{-2}$, $e=1.602\times {10}^{-19}C$, $H_0=2.2\times {10}^{-18}{s}^{-1}$, $\chi =1.759\times {10}^{11}C/kg$, $\alpha =-0.1$, $\beta =1$, $\dot{r}=c/1000$ (in Figure 1) and $0m/s$ (in Figure 2), and $v_a=0$. Mathematica codes, which were used for this plotting, can be found in Appendix E and Appendix F.

From Figure 1 and Figure 2, we can find some interesting aspects, as follows:

[A1]

Based on three intersection points at $r_1\sim 5.6\times {10}^{-15}m$, $r_2\sim 1.4\times {10}^{3}m$, and $r_3\sim 7.6\times {10}^{25}m$, the whole space can be divided into four distinct shells S-I to S-IV, which are governed by different terms $a_2,a_1,a_3$, and $a_6$, respectively.

[A2]

The total acceleration in the shell S-IV is positive, as discussed in Sec. 3.1. Meanwhile, using the mathematica codes, one can find that, in order to obtain such a positive acceleration and the distance of 8.1 billion ly (calculated in Sec. 3.1) as the value of $r_3$, the parameter $\beta$ should be a value close to +1, as assumed for the plotting. Thus, hereinafter, we will refer to the value of $\beta$ as $+1$. In addition, since, except for the value of $r_3$, most of the results discussed in this article are less sensitive to the value of $\beta$, we will suppose that this choice is valid. This means that we will use the Hubble’s law as it is.

[A3]

The total acceleration in the shell S-III has an almost constant negative value. According to Newtonian physics, this gives the result of $v_{star}\sim \sqrt{r}$, which may allow us to explain the galaxy rotation anomaly[18], without invoking the dark matter hypothesis[19]. Meanwhile, the above-stated boosting effect will make the gradient of the distribution factor steeper, and thus, this effect may also be partly responsible for the galaxy rotation anomaly.

[A4]

At $r={10}^{-16}m$, a ratio between strengths of the curves 2, 1, 7, and 5 is about $1:{10}^{-2}:{10}^{-8}:{10}^{-38}$ that is similar to that between the known four fundamental forces. Based on this clue, hereinafter, we will examine whether $a_2,a_1,a_7$, and $a_5$ can be interpreted as the strong, electric, weak, and gravitational terms.

[A5]

In particular, the term $a_5$ obeys the law of inverse square and has a direction independent of the sign of electric charge, because it is written in the form of ${\tilde{k}}^2/r^2$. Especially, it should be noted that, despite the fact that the Newton constant is not used to plot the curve 5, the curve 5 is plotted along and very close to a reference line G showing the gravitational field of the proton, as shown in Figure 1. In this sense, we can strongly suspect that the term $a_5$ is responsible for the gravitational force.

However, we can also find some perplexing aspects, as follows:

[B1]

In the shell S-III, the term $a_3$ is excessively high, compared with the reasonable value (e.g., of about ${10}^{-10}m/s$ in the MOND theory[20]). Similarly, there is a slight discrepancy in strength between the curves 5 and G.

[B2]

The point $r_1$ is slightly different from the known proton radius $0.8\times {10}^{-15}m$.

[B3]

There is an issue in direction of each of the terms $a_2$, $a_3$, and $a_5$. In detail, the term $a_2$ is always repulsive, and thus, it is clear that the strong force cannot be explained by the term $a_2$ alone. Also, the terms $a_3$ and $a_5$ are attractive or repulsive, depending on a sign of $\alpha W_r$, although they are independent of the sign of electric charge. This means that there are difficulties in explaining the galaxy rotation anomaly and the gravitational force through the terms $a_3$ and $a_5$.

[B4]

At the moment, it is unclear that the term $a_7$ is responsible for the weak force.

3.10. Some Possibilities for Perplexing Aspects

With regard to the issue [B1], the mentioned differences can be reduced or eliminated by adjusting the values of $\alpha$ and $\dot{r}$, as exemplarily shown in Figure 3. If so, why should they be such values? Our ideas on this topic will be mentioned in the following discussion on the issue [B3].

With regard to the issue [B2], we have not considered the substructure (i.e., quarks) of the proton. This means that there is a chance of solving this issue and that further research is needed.

With regard to the issue [B3], although, as stated above, the always-repulsive second term $a_2$ alone cannot explain the strong force, this problem can be alleviated, if the repulsive term $a_2$ is overwhelmed by any other attractive term. Fortunately, there seems to be such a term. For example, the repulsive term $a_2$ is overwhelmed by the attractive term $a_5$, if $\alpha <-1/2$ for $\dot{r}=v_a=0$ (see Figure 2) or $\alpha <0$ for $\dot{r}=-H_{0}r$ and $v_a=0$. This overwhelming condition can be satisfied for a short time period (although not always), if the neglected velocities $v_T$, $v_O$, and $v_R$ are considered, but if $-\alpha$ is sufficiently large in the shell S-I, the term $a_5$ can suppress the term $a_2$ in a time-averaged manner. For this reason, we conjecture that the strong force or the asymptotic freedom in the shell S-I would be related to a time-averaged behavior of the combined terms $a_5$ and $a_2$.

Meanwhile, in order to explain the galaxy rotation anomaly in the shell S-III, the parameter $\alpha$ is expected to be a small value (e.g., of about $-0.01$) (see Figure 3). This means that the parameter $\alpha$ has different values in the shells S-I and S-III; that is, in order to explain both of the strong force in S-I and the galaxy rotation anomaly in S-III, the parameter $\alpha$ should depend on distance. In this regard, it would be worth noting the facts that a difference in $\Lambda$-value between the primordial and present epochs is similar to that between the shells S-I and S-III and that there is a difference in ${\Lambda }^{-1}{\partial }_t\Lambda$ or $\alpha$ between the primordial and present epochs (see Eq. (15)). Based on these similarities, we conjecture that there will be a difference in ${\Lambda }^{-1}{\partial }_t\Lambda$ or $\alpha$ between the shells S-I and S-III. This means that the dependence of $\alpha$ on distance could be true, but it is also true that further research is needed.

Next, in order to make the terms $a_3$ and $a_5$ attractive, $\alpha /(\dot{r}+H_{0}r)$ (for $v_a=0$) or $\alpha W_r$ (for general cases) should be negative (see Eqs. (25) and (26)) 3. This means that the parameter $\alpha$ should have dependence on the velocity of TARS or time. In this regard, it should be noted that the LHS of Eq. (23) is generally neither zero nor constant. This shows that the parameter $\alpha$ can be changed with time. Thus, if we found a solution for $\alpha$ from Eq. (7) or (23), we could further examine whether there is a story capable of explaining the gravitational and strong forces. As of now, we can just conjecture that the Hubble expansion and the quantum phenomena, which affect the LHS of Eq. (23), are closely related to the gravitational and strong forces.

With regard to the directions of the terms $a_3$ and $a_5$, there is another possibility, even when the parameter $\alpha$ has a fixed sign. Due to the Hubble expansion effect, the $W_r$ factor of Eq. (26) is symmetric about a line of $\dot{r}=-H_{0}r$, which is offset from an intuitively natural line of $\dot{r}=0$. In the $W_r$-containing terms $a_3$ to $a_5$, symmetric effects about the line of $\dot{r}=0$ will be canceled, and only asymmetric effects will be left as a non-vanishing acceleration. Due to such a remnant asymmetric acceleration effect by the $W_r$ factor, the term $a_5$ would have a fixed direction in a time-averaged behavior, although not in an instantaneous behavior, if the parameter $\alpha$ has a fixed sign. However, if this scenario is true, it means that we have to abandon our belief that a gravitational force exerted on a single elementary particle has a fixed direction always; in other words, the direction of the gravitational force may be a time-averaged property of a collective object, which is composed of a number of particles. Of course, in order to know whether this scenario is true, it is necessary to formulate the time-averaged or collective behavior of the remnant asymmetric effect in the $W_r$-containing terms $a_3$ to $a_5$. Although I have failed to formulate this, this scenario seems plausible, at least until now, in that there has been no gravity experiment for a single particle[22]. In addition, this scenario may be somewhat related to the emergent gravity theories[23,24].

Meanwhile, there is an elusive, but, interesting aspect on the infinite acceleration at $\dot{r}=-H_{0}r$. For example, it seems to require a step of kicking an object with the infinite acceleration, in a velocity-changing process. This means that the velocity is changed in a stepwise manner. Presumably, the kicking step of the infinite acceleration may be an underlying cause of the wave nature of matter, which is the most fundamental difficulty in understanding the quantum phenomena, and/or of the bosenova phenomenon[25].

With regard to the issue [B4], due to its $S_p{\left(r\right)}^2$ (i.e., ${\Lambda }_{-}{\left(r\right)}^2$) factor, the term $a_7$ is stronger than any other terms near the point $r_1$ and is attractive and repulsive inside and outside the point $r_1$, as shown in the small box of Figure 1. Here, due to the fact that the term $a_7$ has opposite directions inside and outside the point $r_1$, we can say that the term $a_7$ serves as an outward pulling force on the point $r_1$ and leads to a reduction in stability of the object, under a certain condition. Also, due to the fact that the term $a_7$ is stronger than any other terms near the point $r_1$, we can say that the term $a_7$ possesses the ability to cause a decay of the object, beyond the reduction in stability, under a certain condition. In this sense, we conjecture that if we take the chirality of a particle, as the mentioned condition, into consideration, we might find something related to the beta decay or the weak interaction.

4. Proposal of Experimental Tests

Now, we propose experimental tests to verify our approach. First, according to our results, in the shell S-III beyond $r_2$, the electric field (i.e., $a_1$) is overwhelmed by the term $a_3$ mentioned to explain the galaxy rotation curve (see Figure 1). Although it depends on the parameters, the presence of such an inflection point $r_2$ has not been expected in most of the fundamental theories, except for the MOND theory, and thus, our approach can be decisively tested by examining whether $r_2$ is present or not. Second, the terms $a_3$ to $a_5$ give very large accelerations at a very small velocity (i.e., low temperature), because of the $W_r$ factor. Especially, such accelerations would be changed abruptly from $+\infty$ to $-\infty$ or vice versa, at a special velocity of $\dot{r}=-H_{0}r$ and $v_a=0$ or at a corresponding temperature, if the parameter $\alpha$ has a fixed sign. This large acceleration or its abrupt change could be easily measured. Perhaps, this process may be related to the Bosenova phenomenon[25], mentioned above.

5. Conclusions

Until now, we have tried to convince the reader that our approach based on the proportional equation and the multiplicative superposition can provide new clues to some fundamental problems. For example, by considering the temporal variation of the distribution factor, we have shown that there are meaningful clues to the cosmological problems, such as the inflationary and accelerating expansions of the universe and the missing mass problem, and that the electric force can be split into eight distinct forces. However, it is true that there are still some unclear aspects in our approach on gravity and quantum physics. Thus, in order to know whether they are true clues, it is necessary to conduct further research in the context of the existing physics, such as thermodynamics and field theories, in addition to the proposed experimental tests. We think that understanding Eq. (7) or (23) would be important in this process.