# The Ambiguity in the Definition and Behavior of the Gravitational and Cosmological ‘Coupling Constants’ in the Theory of Induced Gravity

## Abstract

**:**

## 1. Introduction

#### 1.1. Introduction to the Original Theory

**Remark**

**1.**

**Remark**

**2.**

#### 1.2. Different Types of Solutions

## 2. Cosmological Solutions

#### 2.1. Cosmological Vacuum Solutions. Y = const

#### 2.2. Cosmological Solutions with Matter

#### 2.3. The Case without Quadratic Terms

#### 2.4. Oscillating Solutions. Influence of Quadratic Terms

**The Purusha Universe:**${k}_{2}=0$. Starting from the description sequence, first we consider the exotic case ${k}_{2}=0$. A slightly comic name for this model is due to the properties of the solutions, about which, in brief, will be discussed below. The peculiarity of this case (in our opinion is not contrary to the observational data) associated with the fact that the Equations (70) and (71) for $Z={Z}_{0}$ allow any constant solutions $b(x)=const$. The initial conditions ${\gamma}_{0}:{Z}_{m}=Z({x}_{0}),\phantom{\rule{4pt}{0ex}}{b}_{m}=b({x}_{0}),\phantom{\rule{4pt}{0ex}}{P}_{bm}\equiv {\dot{b}}_{m}=\dot{b}({x}_{0})$ that violate this initial state ($Z=\forall {Z}_{0}$, $b(x)=\forall const$), initiate solutions $\gamma :Z=Z(x,{Z}_{m}),\phantom{\rule{4pt}{0ex}}b=b(x,{b}_{m})$ which are related to the initial conditions as some perturbations. The resulting solutions are of stochastic (random) character.The stochasticity is as follows: if from the resulting solution $\gamma $ we choose different point ${\gamma}_{1}:{Z}_{n}=Z({x}_{1}),\phantom{\rule{4pt}{0ex}}{b}_{n}=b({x}_{1}),\phantom{\rule{4pt}{0ex}}{p}_{Zn}=\dot{Z}({x}_{1}),\phantom{\rule{4pt}{0ex}}{p}_{bn}=\dot{b}({x}_{1})$ (corresponding to the moment of time ${x}_{1}$) as new initial conditions for the same equations, then the new solutions ${\gamma}_{1}$ won’t match $\gamma $. This assertion follows from our computer studies and means that the uniqueness condition for the Cauchy problem is violated.

**Consideration of the case ${k}_{2}\ne 0$**leads to the models similar to $\mathsf{\Lambda}CDM$ in the general scenario of evolution, which are additionally accompanied by fluctuations. Figure 13, Figure 14 and Figure 15 show graphs of numerical solutions of equations with initial conditions (76), for the case with parameters: ${k}_{0}=0,\phantom{\rule{4pt}{0ex}}{k}_{1}=-0.4$, $\phantom{\rule{4pt}{0ex}}{B}_{1}=144.517022939$, ${Z}_{m}=1.0041022218\dots $, ${\rho}_{p}=50.7042299920\dots $, 0.95[0.95]${\rho}_{r}=0.1950162692\dots $ (dots after numbers mean that we give only the first ten numbers after the decimal point). The main feature of these solutions is the presence of fluctuations that lead to alternation of accelerated with decelerated expansion passing into a contraction state (at certain values of the parameters).

## 3. Centrally Symmetric Solutions

#### 3.1. Analysis of Equations in Centrally Symmetric Space

#### 3.2. Numerical Solution of Equations for the “Conditional Sun” Model

#### 3.3. Numerical Solution of Equations for the Model of “Galaxy”

#### 3.4. General Characteristics of the Models. Attempt to Reconcile

## 4. Conclusions

The MTIG model is proposed for a macroscopic description of gravity and cosmology, which (possibly) is capable of solving problems 1–3, given at the beginning of the article, and motivating to further experiments. We propose the working hypothesis according to which the physical parameters associated with gravitation, such as the gravitational and cosmological “constants”—G and ${\mathsf{\Lambda}}_{eff}$, the Hubble “constant” H, in addition to monotonic evolution, fluctuate about their mean values. Because of the implementation of the two branches of solutions, these fluctuations can contain elements of stochasticity. This hypothesis is realized in the mathematical model considered in this article.

The solutions of the MTIG equations for the case of a centrally symmetric gravitational field, in addition to the Schwarzschild-de Sitter solutions (for $Z=const$), contain solutions that lead to new physical effects at large distances from the center. For distances greater than a certain critical value, the following effects can appear (depending on the value of the parameters of the theory): deviation from the law of gravitational interaction of general relativity and its Newtonian approximation, antigravity, absence of asymptotic flatness at infinity. The responsibility for these effects (in the first place) carries not an integration constant that corresponds to the mass of a physical body, but some other (dimensionless) charge ${Z}_{2}$, which is brought into the theory as the boundary value of the field $Z(r)$ at the center of symmetry ($r=0$), different from the (vacuum) mean value (${Z}_{0}=1$) of field Z. The mass affects the value of the critical radius, the charge ${Z}_{2}$ affects the amplitude of the deviation, ${B}_{n}$ affects the oscillation frequency. We consider the hypothesis that the parameter ${B}_{n}$ is the same for all objects, and the charge ${Z}_{2}$ is different for local objects (stars, galaxies, clusters…).

Thus, the flat asymptotics (the asymptotics of the field at large distances $r>{r}_{cr}$ in the Newton and Schwarzschild theory) becomes unstable and can enter the oscillatory regime with the elements of of chaotic behavior. The mean oscillation frequency depends on the value of ${B}_{n}$, which depends on the vacuum polarization energy density. Near the center, the influence of mass predominates and the laws of general relativity and Newton’s law of universal gravitation (for the weak field) are approximately fulfilled. At sufficiently large distances (greater than critical) these solutions pass into other solutions, where the influence of ${Z}_{2}$ and ${B}_{n}$ prevails. (Because of the nonlinearity of the theory, such allocation of roles of the parameters, based on computer experiments, is conditional (approximate)).

## 5. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

DOAJ | Directory of open access journals |

TLA | Three letter acronym |

LD | linear dichroism |

MTIG | Modified Theory of Induced Gravity |

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**Figure 1.**Oscillatory solutions for flat cosmological model. $x=t/{t}_{m}$, ${\tilde{B}}_{n}=144.517\dots $, ${k}_{2}=-0.2$. Border conditions: (76), ${Z}_{m}=1.0040965.$

**Figure 2.**The dependence comparison ${g}_{00}$ (the metric component) for the centrally symmetric space with the Schwarzschild de Sitter’s solution (dashed line); r is given in au. ${B}_{n}=0.0059986\dots $, ${k}_{1}$ = −9.263854653 · 10${}^{-31}$ au${}^{-2}$, $GM=G{M}_{\u2a00}$. Border conditions: (97).

**Figure 3.**Plot of $Z=Y/{Y}_{0}$ and $b(t)=a(t)/{a}_{0}$: $b=1$—current age of the universe. After reaching the point A, the function Y (t) branches. It may evolve on the straight line d or the curve ${d}_{1}$.

**Figure 4.**Plot of $Z=Y/{Y}_{0}$ and $b(t)=a(t)/{a}_{0}$: $b=1$—current age of the universe. Double transition: from RS into ES and back from ES into RS.

**Figure 5.**Relation of $Z=Z(x)$, where $Z(1)={Z}_{m}=1.40908675530186$: $x=t/{t}_{m}$—current age of the Universe, ${\tilde{B}}_{n}=1.44517022939$, ${k}_{2}=0$.

**Figure 6.**Relation of $b=b(x)=a(x)/{a}_{m}$, $b(1)=1$—the current value of the scale factor. ${\tilde{B}}_{n}=1.44517022939$, ${k}_{2}=0$.

**Figure 7.**Relation of $H(x)=\dot{b}(x)/b(x)$: ${H}_{m}(1)=1$—the current value of the Hubble constant. ${\tilde{B}}_{n}=1.44517022939$, ${k}_{2}=0$.

**Figure 8.**Relation of $Z=Y/{Y}_{0}$, $Z(1)={Z}_{m}=1.040908675530$: $x=t/{t}_{m}$—current age of the Universe. ${\tilde{B}}_{n}=14.4517022939$, ${k}_{2}=0$.

**Figure 9.**Relation of $b(x)=a(x)/{a}_{m}$: $b(1)=1$—the current value of the scale factor. ${\tilde{B}}_{n}=14.4517022939$, ${k}_{2}=0$.

**Figure 10.**Relation of $Z=Y/{Y}_{0}$, $Z(1)={Z}_{m}=1.04072460$: $x=t/{t}_{m}$—current age of the Universe. ${\tilde{B}}_{n}=14.4517022939$, ${k}_{2}=0$.

**Figure 11.**Relation of $b(x)=a(x)/{a}_{m}$: $b(1)=1$—the current value of the scale factor. ${\tilde{B}}_{n}=14.4517022939$, ${k}_{2}=0$.

**Figure 12.**Relation of $Z=Y/{Y}_{0}$, $Z(1)={Z}_{m}=1.0000407246$: $x=t/{t}_{m}$—current age of the Universe. ${\tilde{B}}_{n}=14517.022939$, ${k}_{2}=0$.

**Figure 13.**Relation of $Z=Y/{Y}_{0}$, $Z(1)={Z}_{m}=1.0041022218\dots $: $x=t/{t}_{m}$—current age of the Universe. ${\tilde{B}}_{n}=144.517022939$, ${k}_{2}=-0.4$.

**Figure 14.**Relation of $b(x)=a(x)/{a}_{m}$, $Z(1)={Z}_{m}=1.0041022218\dots $: $x=t/{t}_{m}$—current age of the Universe. ${\tilde{B}}_{n}=144.517022939$, ${k}_{2}=-0.4$.

**Figure 15.**Relation of $H(x)=\dot{b}(x)/b(x)$: ${H}_{m}(1)=1$—the current value of the Hubble constant. ${\tilde{B}}_{n}=144.517022939$, ${k}_{2}=-0.4$.

**Figure 16.**Graph of $Z(r)-1$. r is measured in astronomical units. ${B}_{n}=0.0059986\dots $, ${k}_{1}=-9.263854653\xb7{10}^{-31}a{u}^{-2},\phantom{\rule{4pt}{0ex}}GM=G{M}_{\u2a00}$.

**Figure 17.**The “additional acceleration” $\Delta A(r)$ graph. r is measured in astronomical units. ${B}_{n}=0.0059986\dots $, ${k}_{1}=-9.263854653\xb7{10}^{-31}a{u}^{-2},\phantom{\rule{4pt}{0ex}}GM=G{M}_{\u2a00}$.

**Figure 18.**The graph of the function $\Delta \lambda (r)=\alpha (r)$. r is measured in astronomical units. ${B}_{n}=0.0059986\dots $, ${k}_{1}=-9.263854653\xb7{10}^{-31}a{u}^{-2},\phantom{\rule{4pt}{0ex}}GM=G{M}_{\u2a00}$.

**Figure 19.**The deviation graph of $\Delta \nu (r)=F(r)-\alpha (r)$. r in au. ${B}_{n}=0.0059986\dots $, ${k}_{1}=-9.263854653\xb7{10}^{-31}a{u}^{-2},\phantom{\rule{4pt}{0ex}}GM=G{M}_{\u2a00}$.

**Figure 20.**A comparison of the numerical solution ${g}^{11}$ (solid line) with the Schwarzschild-de Sitter solution (dashed line); r in au.

**Figure 23.**The “gravity” (“in”) and “antigravity” (“out”) bands formation scheme with different directions of acceleration in relation to the center.

**Figure 24.**“Observed circular rotational velocities” vs. distance circled in pencil, signs-(bottom) mean “antigravity”; $r\in (100,700)$ $au$.

**Figure 25.**The field Z deviation vs. distance. r is measured in kiloparsec (kpc). ${B}_{n}=0.2665\dots $, ${k}_{1}=-4.035487823\xb7{10}^{-14},\phantom{\rule{4pt}{0ex}}GM=G{M}_{\u2a00}\xb7{10}^{10}$.

**Figure 26.**“Observed circular rotational velocities” vs. distance; $r\in (0.5,25)kpc$. ${B}_{n}=0.2665\dots $, ${k}_{1}=-4.035487823\xb7{10}^{-14},\phantom{\rule{4pt}{0ex}}GM=G{M}_{\u2a00}\xb7{10}^{10}$.

**Figure 30.**A comparison of the numerical solution ${g}_{00}$ (solid line) with the Schwarzschild-de Sitter solution (dashed line); r—in the parsecs. ${B}_{n}=1.599\xb7{10}^{-6}(p{c}^{-2})$. The solid graph is obtained for the boundary condition: $Z(0)$ = 1.00009833247, $p(0)$ = 0, $\alpha (0)$ = 0, $F(0)$ = 0.

**Figure 31.**Test body acceleration $A(r)$ vs. distance; r—in the parsecs. For the case of solar mass. ${B}_{n}=1.599\xb7{10}^{-6}(p{c}^{-2})$, $Z(0)$ = 1.00009833247, $p(0)$ = 0, $\alpha (0)$ = 0, $F(0)$ = 0.

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**MDPI and ACS Style**

Zaripov, F.
The Ambiguity in the Definition and Behavior of the Gravitational and Cosmological ‘Coupling Constants’ in the Theory of Induced Gravity. *Symmetry* **2019**, *11*, 81.
https://doi.org/10.3390/sym11010081

**AMA Style**

Zaripov F.
The Ambiguity in the Definition and Behavior of the Gravitational and Cosmological ‘Coupling Constants’ in the Theory of Induced Gravity. *Symmetry*. 2019; 11(1):81.
https://doi.org/10.3390/sym11010081

**Chicago/Turabian Style**

Zaripov, Farkhat.
2019. "The Ambiguity in the Definition and Behavior of the Gravitational and Cosmological ‘Coupling Constants’ in the Theory of Induced Gravity" *Symmetry* 11, no. 1: 81.
https://doi.org/10.3390/sym11010081