# Dark Matter as a Result of Field Oscillations in the Modified Theory of Induced Gravity

## Abstract

**:**

## 1. Introduction

## 2. Centrally Symmetric Solutions

**Remark**

**1.**

#### Numerical Solutions for Geodesic Lines

## 3. Conclusions and Discussion

## Funding

## Conflicts of Interest

## Abbreviations

GR | General relativity |

MTIG | Modified Theory of Induced Gravity |

## References

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**Figure 1.**Potential energy for mass $GM=3.95\xb7{10}^{6}G{M}_{\odot}$, ${L}_{\varphi}=0$; (

**a**) $r\in (0.09,\phantom{\rule{4pt}{0ex}}1)$; (

**b**) $r\in (0.05,\phantom{\rule{4pt}{0ex}}55)$.

**Figure 2.**Radial acceleration, with zero angular momentum, $GM=3.95\xb7{10}^{6}G{M}_{\odot}$; figure (

**a**) for $r\in (0.025,\phantom{\rule{4pt}{0ex}}6.8)$ shows a comparison with the Schwarzschild solution (dashed line) and the transition to oscillations, figure (

**b**) for $r\in (0.1,\phantom{\rule{4pt}{0ex}}55)$.

**Figure 3.**Galaxy rotation curve (${v}_{L}$ in $\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}/\mathrm{s}$, for different central masses: (

**a**) $GM=3.95\xb7{10}^{6}G{M}_{\odot}$; (

**b**) $GM={10}^{10}G{M}_{\odot}$; (

**c**) $GM=4.5\xb7{10}^{10}G{M}_{\odot}$; (

**d**) $GM={10}^{11}G{M}_{\odot}$. Dashed Lines—Keplerian speeds.

**Figure 4.**Geodesic curves, for initial conditions: (

**a**) ${r}_{2}\approx 3.89\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$, $\phantom{\rule{4pt}{0ex}}{v}_{20}=0\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}/\mathrm{s}$, ${L}_{\varphi}\approx 2\xb7{10}^{-5}\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$; (

**b**) ${r}_{2}\approx 0.7\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$, $\phantom{\rule{4pt}{0ex}}{v}_{20}=304.05\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}/\mathrm{s}$, ${L}_{\varphi}\approx 4.203\xb7{10}^{-4}\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$; (

**c**) ${r}_{2}\approx 1.9\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$, ${v}_{20}=-130\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}/\mathrm{s}$, ${L}_{\varphi}=0.001457675\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$; (

**d**) ${r}_{2}\approx 3.778358\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$, $\phantom{\rule{4pt}{0ex}}{v}_{20}=0\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}/\mathrm{s}$, ${L}_{\varphi}\approx 0.001103125149\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$.

**Figure 5.**(

**a**) geodesic curve for initial values: ${r}_{2}\approx 4\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$, $\phantom{\rule{4pt}{0ex}}{v}_{20}=0\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}/\mathrm{s}$, ${L}_{\varphi}=0.0026685\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$; (

**b**) radial velocity versus radius.

**Figure 6.**Geodesic curves, for initial conditions: (

**a**) ${r}_{2}\approx 5\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$, $\phantom{\rule{4pt}{0ex}}{v}_{20}=0\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}/\mathrm{s}$, ${L}_{\varphi}=6.6712819\xb7{10}^{-12}\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$; (

**b**) ${r}_{2}\approx 10.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$, $\phantom{\rule{4pt}{0ex}}{v}_{20}=0\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}/\mathrm{s}$, ${L}_{\varphi}=6.671281904\xb7{10}^{-12}\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$.

**Figure 7.**Geodesic curves, for initial conditions: (

**a**) $GM=3.95\xb7{10}^{6}G{M}_{\odot}$, ${r}_{2}=8\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$, $\phantom{\rule{4pt}{0ex}}{v}_{20}=-15\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}/\mathrm{s}$, ${L}_{\varphi}=0.00547\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$; (

**b**) $GM={10}^{10}G{M}_{\odot}$, ${r}_{2}=8\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$, $\phantom{\rule{4pt}{0ex}}{v}_{20}=-15\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}/\mathrm{s}$, ${L}_{\varphi}=0.006584\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$.

**Figure 8.**(

**a**) geodesic curve and (

**b**) radial velocity depending on radius for initial values: $GM={10}^{10}G{M}_{\odot}$, ${r}_{2}=8\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$, $\phantom{\rule{4pt}{0ex}}{v}_{20}=-15\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}/\mathrm{s}$, ${L}_{\varphi}\approx 0.00480693\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$.

**Figure 9.**Geodesic curves, $GM=3.95\xb7{10}^{6}G{M}_{\odot}$ for initial conditions: (

**a**) ${r}_{2}=5.34998\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$, $\phantom{\rule{4pt}{0ex}}{v}_{20}=0$, ${L}_{\varphi}\approx 0.001935333888\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$; (

**b**) ${r}_{2}=5.355\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$, $\phantom{\rule{4pt}{0ex}}{v}_{20}==0$, ${L}_{\varphi}\approx 0.001927466767693\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$.

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Zaripov, F.
Dark Matter as a Result of Field Oscillations in the Modified Theory of Induced Gravity. *Symmetry* **2020**, *12*, 41.
https://doi.org/10.3390/sym12010041

**AMA Style**

Zaripov F.
Dark Matter as a Result of Field Oscillations in the Modified Theory of Induced Gravity. *Symmetry*. 2020; 12(1):41.
https://doi.org/10.3390/sym12010041

**Chicago/Turabian Style**

Zaripov, Farkhat.
2020. "Dark Matter as a Result of Field Oscillations in the Modified Theory of Induced Gravity" *Symmetry* 12, no. 1: 41.
https://doi.org/10.3390/sym12010041