# Relativistic Fractional-Dimension Gravity

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Theory for Spaces with Non-Integer Dimension and NFDG

## 3. Euler-Lagrange Equations for Spaces with Non-Integer Dimension

#### 3.1. Rectangular Coordinates

#### 3.2. Spherical Coordinates

#### 3.3. Cylindrical Coordinates

## 4. Relativistic Equations for Spaces with Non-Integer Dimension

#### 4.1. RFDG Field Equations

#### 4.2. Cosmology and RFDG

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. RFDG Tensors for the FLRW Metric

## Notes

1 | SI units will be used throughout this paper, unless otherwise noted. |

2 | |

3 | Dimensionless coordinates, such as ${w}^{i}={x}^{i}/{l}_{0}$, ${w}_{r}=r/{l}_{0}$, etc., should be used in most equations in this section and in the following ones. For simplicity’s sake, in this paper we left standard coordinates (${x}^{i}$, r, R, etc.) in most equations, without transforming them into dimensionless, rescaled ones. |

4 | |

5 | Even using a combined weight, ${v}_{t}\left(t\right){v}_{r}\left(r\right)=a\left(t\right){v}_{r}\left(r\right)$, does not seem to yield fully isotropic Friedmann equations for the cosmological problem. A more detailed study of cosmological weights, including possible radial factors or even direct modifications to the FLRW metric in terms of variable space-time dimensions, will be done in a future publication. |

6 | In RFDG, the connection with open ($\kappa <0$), flat ($\kappa =0$), and closed ($\kappa >0$) universes is not simply related to the density parameter $\Omega \u2a8b1$ as in standard cosmology, due to the presence of the additional $\beta $ term in Equation (41). |

7 | Mathematica, Version 12.2.0.0, Wolfram Research Inc. |

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**Figure 1.**Expansion histories for different values of ${\Omega}_{M0}$, ${\Omega}_{\Lambda 0}$, and of the RFDG parameter ${\alpha}_{t}$. Three notable cases from standard cosmology (red, blue, and green solid curves) are compared with RFDG results for similar ${\Omega}_{M0}$, ${\Omega}_{\Lambda 0}$ parameters, but with variable ${\alpha}_{t}>0$. RFDG curves for ${\alpha}_{t}=0.01$, $0.50$ (dotted and dashed curves) are only slightly different from their respective standard cosmology solid curves.

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Varieschi, G.U.
Relativistic Fractional-Dimension Gravity. *Universe* **2021**, *7*, 387.
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**AMA Style**

Varieschi GU.
Relativistic Fractional-Dimension Gravity. *Universe*. 2021; 7(10):387.
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**Chicago/Turabian Style**

Varieschi, Gabriele U.
2021. "Relativistic Fractional-Dimension Gravity" *Universe* 7, no. 10: 387.
https://doi.org/10.3390/universe7100387