# Lorentz Symmetry Group, Retardation, Intergalactic Mass Depletion and Mechanisms Leading to Galactic Rotation Curves

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. General Relativity

**mass**density. We remind the reader that lowering and raising indices is done through the metric ${g}_{\mu \nu}$ and inverse metric ${g}^{\mu \nu}$, such that ${u}_{\mu}={g}_{\mu \nu}{u}^{\nu}$. The same metric serves to calculate s:

## 3. Linear Approximation of GR

**then**defines the quantity:

## 4. Beyond the Newtonian Approximation

## 5. Rotation Curves

#### 5.1. General Considerations

#### 5.2. M33 Density Profile

#### 5.3. M33 Rotation Curve

**current**estimation of the second derivative of mass; we make no claim about the past or future values of $\ddot{M}$, nor is there any claim in this section on the value of $\dot{M}$ at any time or the value of M in the past or the future. It is obvious that such questions involve an understanding of the mass exchange between the galaxy and the intergalactic medium, as described in Section 6.

## 6. A Dynamical Model

## 7. The Mass Formula

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Zwicky, F. On a New Cluster of Nebulae in Pisces. Proc. Natl. Acad. Sci. USA
**1937**, 23, 251–256. [Google Scholar] [CrossRef][Green Version] - Volders, L.M.J.S. Neutral Hydrogen in M33 and M101. Bull. Astr. Inst. Netherl.
**1959**, 14, 323. [Google Scholar] - Rubin, V.C.; Ford, W.K., Jr. Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions. Astrophys. J.
**1970**, 159, 379. [Google Scholar] [CrossRef] - Rubin, V.C.; Ford, W.K., Jr.; Thonnard, N. Rotational Properties of 21 Sc Galaxies with a Large Range of Luminosities and Radii from NGC 4605 (R = 4 kpc) to UGC 2885 (R = 122 kpc). Astrophys. J.
**1980**, 238, 471. [Google Scholar] [CrossRef] - Yahalom, A. The effect of Retardation on Galactic Rotation Curves. J. Phys. Conf. Ser.
**2019**, 1239, 012006. [Google Scholar] [CrossRef] - Yahalom, A. Retardation Effects in Electromagnetism and Gravitation. In Proceedings of the Material Technologies and Modeling the Tenth International Conference, Ariel, Israel, 20–24 August 2018. [Google Scholar]
- Yahalom, A. Dark Matter: Reality or a Relativistic Illusion? In Proceedings of the Eighteenth Israeli-Russian Bi-National Workshop 2019, The Optimization of Composition, Structure and Properties of Metals, Oxides, Composites, Nano and Amorphous Materials, Ein Bokek, Israel, 17–22 February 2019. [Google Scholar]
- Wagman, M. Retardation Theory in Galaxies. Ph.D. Thesis, Senate of Ariel University, West Bank, Israeli, 23 September 2019. [Google Scholar]
- Wagman, M.; Horwitz, L.P.; Yahalom, A. Retardation Theory and galactic Rotation Curves. Astrophys. J.
**2020**, in press. [Google Scholar] - Narlikar, J.V. Introduction to Cosmology, 2nd ed.; Cambridge University Press: Cambridge, UK, 1993. [Google Scholar]
- Eddington, A.S. The Mathematical Theory of Relativity; Cambridge University Press: Cambridge, UK, 1923. [Google Scholar]
- Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 1972. [Google Scholar]
- Misner, C.W.; Thorne, K.S.; Wheeler, J.A. Gravitation; W.H. Freeman & Company: New York, NY, USA, 1973. [Google Scholar]
- Jackson, J.D. Classical Electrodynamics, 3rd ed.; Wiley: New York, NY, USA, 1999. [Google Scholar]
- Asher, Y. The Geometrical Meaning of Time. Found. Phys.
**2008**, 38, 489–497. [Google Scholar] - Yahalom, A. The Gravitational Origin of the Distinction between Space and Time. Int. J. Mod. Phys. D
**2009**, 18, 2155–2158. [Google Scholar] [CrossRef] - Corbelli, E. Dark matter and visible baryons in M33. Mon. Not. R. Astron. Soc.
**2003**, 342, 199–207. [Google Scholar] [CrossRef][Green Version] - Rega, M.W.; Vogel, S.N. The near-infrared structure of M33. Astrophys. J.
**1994**, 434, 536–545. [Google Scholar] [CrossRef] - Van Dokkum, P.; Danieli, S.; Cohen, Y.; Merritt, A.; Romanowsky, A.J.; Abraham, R.; Brodie, J.; Conroy, C.; Lokhorst, D.; Mowla, L.; et al. A galaxy lacking dark matter. Nature
**2018**, 555, 629–632. [Google Scholar] [CrossRef] - Fodera-Serio, G.; Indorato, L.; Nastasi, P. Hodierna’s Observations of Nebulae and his Cosmology. J. Hist. Astron.
**1985**, 16, 1–36. [Google Scholar] [CrossRef][Green Version] - Van den Bergh, S. The Galaxies of the Local Group; Cambridge Astrophysics Series 35; Cambridge University Press: Cambridge, UK, 2000; p. 72. ISBN 978-0-521-65181-3. [Google Scholar]
- Einstein, A. Näherungsweise Integration der Feldgleichungen der Gravitation. In Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin; Part 1; The Prusssian Academy of Sciences: Berlin, Germany, 1916; pp. 688–696. [Google Scholar]
- Nobel Prize, A. Press Release The Royal Swedish Academy of Sciences; The Royal Swedish Academy of Sciences: Stockholm, Sweden, 1993. [Google Scholar]
- Castelvecchi, D.; Witze, W. Einstein’s gravitational waves found at last. Nat. News
**2016**. [Google Scholar] [CrossRef] - Binney, J.; Tremaine, S. Galactic Dynamics; Princeton University Press: Princeton, NJ, USA, 1987. [Google Scholar]
- Corbelli, E.; Salucci, P. The extended rotation curve and the dark matter halo of M33. Mon. Not. R. Astron. Soc.
**2000**, 311, 441–447. [Google Scholar] [CrossRef][Green Version] - Milgrom, M. A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. Astrophys. J.
**1983**, 270, 365–370. [Google Scholar] [CrossRef] - Mannheim, P.D. Linear Potentials and Galactic Rotation Curves. Astrophys. J.
**1993**, 149, 150. [Google Scholar] - Mannheim, P.D. Are Galactic Rotation Curves Really Flat? Astrophys. J.
**1997**, 479, 659. [Google Scholar] [CrossRef][Green Version] - Mannheim, P.D.; Kazanas, D. Exact vacuum solution to conformal Weyl gravity and galactic rotation curves. Astrophys. J.
**1989**, 342, 635. [Google Scholar] [CrossRef] - Tuval, M.; Yahalom, A. Newton’s Third Law in the Framework of Special Relativity. Eur. Phys. J. Plus
**2014**, 129, 240. [Google Scholar] [CrossRef][Green Version] - Tuval, M.; Yahalom, A. Momentum Conservation in a Relativistic Engine. Eur. Phys. J. Plus
**2016**, 131, 374. [Google Scholar] [CrossRef] - Yahalom, A. Retardation in Special Relativity and the Design of a Relativistic Motor. Acta Phys. Pol. A
**2017**, 131, 1285–1288. [Google Scholar] [CrossRef]

**Figure 3.**The $\psi $ function of M33, the function converges to one for large distances as expected from Equation (44).

**Figure 5.**The $\chi $ function of M33; the function converges to one for large distances, as expected from Equation (42).

**Figure 6.**Rotation curve for M33. The observational points were supplied by Dr. Michal Wagman, a former PhD student at Ariel University, under my supervision, using [17]; the full line describes the complete rotation curve, which is the sum of the dotted line, describing the retardation contribution, and the dashed line, which is the Newtonian contribution.

**Figure 7.**An idealized cylindrical galaxy from different perspectives. (

**a**) From above; (

**b**) tilted edge perspective.

**Figure 9.**An initial density profile outside the galactic plane, in which ${\rho}_{0}={\rho}_{1}+{\rho}_{2},\frac{{\rho}_{2}}{{\rho}_{1}}=-0.2$ and ${z}_{i}=5$ (kpc) and $k=0.32\phantom{\rule{4pt}{0ex}}\left({\mathrm{kpc}}^{-1}\right)$.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yahalom, A. Lorentz Symmetry Group, Retardation, Intergalactic Mass Depletion and Mechanisms Leading to Galactic Rotation Curves. *Symmetry* **2020**, *12*, 1693.
https://doi.org/10.3390/sym12101693

**AMA Style**

Yahalom A. Lorentz Symmetry Group, Retardation, Intergalactic Mass Depletion and Mechanisms Leading to Galactic Rotation Curves. *Symmetry*. 2020; 12(10):1693.
https://doi.org/10.3390/sym12101693

**Chicago/Turabian Style**

Yahalom, Asher. 2020. "Lorentz Symmetry Group, Retardation, Intergalactic Mass Depletion and Mechanisms Leading to Galactic Rotation Curves" *Symmetry* 12, no. 10: 1693.
https://doi.org/10.3390/sym12101693