# Do Fractals Confirm the General Theory of Relativity?

## Abstract

**:**

## 1. Introduction

- Main sequence stars obey a mass-luminosity relation, $L~{m}^{\alpha}$, $\alpha \approx 3.52$;
- Period-luminosity relation for pulsating variable stars, $L\sim {P}^{\alpha}$, for classical Cepheids $\alpha \approx 1.15$;
- Luminosity functions for stars, galaxies, and quasars, $\Phi ~{L}^{\alpha}$, $-2.6<\alpha <-0.6$;
- Faber-Jackson relation for luminosity and their central stellar velocity dispersion of stars of elliptical galaxies, $L~{\sigma}^{4}$;
- Tully-Fisher relation for the luminosity and rotation rate of spiral galaxies, $L~{v}^{3}$;
- Brightness distribution in the images of bright elliptical galaxies, $I~{r}^{-\alpha}{\left(1+{\left(\frac{r}{{r}_{0}}\right)}^{\beta}\right)}^{\gamma}$, $-1.37<\alpha <0.85$, $0.28<\beta <1$, $-7.57<\gamma <-0.18$;
- Power-law spectra of the radio emission of the jets from active galactic nuclei, $F~{\nu}^{-\alpha}$, $0.6<\alpha <0.7$ (jets are composed of plasma clouds with a magnetic field);
- Baldwin’s ratio in the active galactic nucleus for the equivalent width of the emission lines and the luminosity of its galaxy, $EW(CaII)~{L}^{-\alpha}$, $\alpha >1$;
- Red shift distribution of absorption lines in quasar spectra, $\frac{dN}{dz}~{\left(1+z\right)}^{\alpha}$, $1.67<\alpha <2.09$;
- Spatial correlation function of galaxies in the clusters of galaxies and clusters in superclusters, $\xi (r)~{r}^{-\alpha}$, $1.6<\alpha <2.2$.

## 2. Exact Geometrical Solution of the Lagrange and Einstein Equations for a Complex Scalar Field with Symmetry U (1)

#### 2.1. Selection of the Christoffel’s Symbols and Metric Tensor

#### 2.2. Algebraic form of the Einstein’s Equations

## 3. Spatial Variations of the Hubble Constant and Local Gravitational Perturbations

## 4. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Hubble constant (data [13]).

Galaxy | ${\mathit{H}}_{0}^{\mathit{H}\mathit{S}\mathit{T}}$ (km/s/Mpc) | $\mathit{i}$ |
---|---|---|

M101 | 68.39 | 2 |

N1015 | 80.09 | 9 |

N1309 | 70.24 | 3 |

N1365 | 68.39 | 2 |

N1448 | 77.77 | 8 |

N2442 | 73.42 | 5 |

N3021 | 63.94 | −1 |

N3370 | 76.00 | 7 |

N3447 | 74.37 | 6 |

N3972 | 77.98 | 8 |

N3982 | 64.80 | −1 |

N4038 | 79.69 | 9 |

N4258 | 72.25 | 5 |

N4424 | 63.97 | −1 |

N4536 | 71.48 | 4 |

N4639 | 77.98 | 8 |

N5584 | 78.67 | 9 |

N5917 | 72.75 | 5 |

N7250 | 74.75 | 6 |

U9391 | 66.53 | 1 |

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**MDPI and ACS Style**

Rozgacheva, I.
Do Fractals Confirm the General Theory of Relativity? *Symmetry* **2019**, *11*, 740.
https://doi.org/10.3390/sym11060740

**AMA Style**

Rozgacheva I.
Do Fractals Confirm the General Theory of Relativity? *Symmetry*. 2019; 11(6):740.
https://doi.org/10.3390/sym11060740

**Chicago/Turabian Style**

Rozgacheva, Irina.
2019. "Do Fractals Confirm the General Theory of Relativity?" *Symmetry* 11, no. 6: 740.
https://doi.org/10.3390/sym11060740