# Excluded Volume for Flat Galaxy Rotation Curves in Newtonian Gravity and General Relativity

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Vacuum Solutions and Averaging Processes

_{m}, C

_{m}, A

_{l}and B

_{l}are coefficients, and P(l,m) are the Associated Legendre polynomials. The J

_{l}are the Bessel functions, and k is a constant. The quantities q and a are also constants. To explain the flat galaxy rotation curves, normally the vacuum terms are not added, rather, a dark matter density distribution falling off with range as 1/r

^{2}or similar function is added to the luminous matter distribution, as this contribution results in a field going as 1/r, and a potential going as the logarithm.

_{HOMO}(8) is due to Zhang in cylindrical coordinates with κ a constant [15].

- The process is constrained to match the experimental observation that a logarithmic potential is required.
- Weighting need not be equal for all orientations.
- Averaging need not utilize the entirety of the surface.
- Processed solutions for 3D cannot obey the original 3D gravitational vacuum equations.
- Solutions are built from manipulations of the original vacuum solutions.

_{o}/r

^{2}, and again, not the needed functional form. The divergence of the field associated with (12) is zero, as so the 3D average remains a vacuum solution in 3D from which it was constructed.

- What is the scale factor of the gravitational vacuum, and is it the same as the scale factor of the universe with matter, radiation, dark energy, and curvature?
- Is there more than one vacuum, one for each type of particle field, with its own cosmological constant?
- Can multiple scales in the vacuum be simultaneously active, such that one experiences a sum of potentials at any given position?

_{0}= 0, and ζ

_{o}is a constant. The upper limit s is taken to be the scale factor of the universe today.

_{enc}, a simple approximation for a star at the periphery of a galaxy, with a spherically symmetric matter distribution, and the vacuum term, one finds in the orbital plane,

^{1/2}for large r, and the rotation curve is therefore flat. Note that neither mass nor G explicitly/directly enter q.

^{2}, and the Equivalence Principle to first order. For sufficiently large radii, the red (or blue) shift is,

^{2}, so q/c

^{2}~ 10

^{−7}.

## 3. Excluded Volume with Assumed Gravitational Property of the Vacuum

_{vac}associated with it, per (24).

_{q}in an infinite domain Ω then provides a vector cancellation at all points R per (25),

_{S}, why the rotary charge coincides with the galactic center, and why it would become important only on galactic scales. If the latter is the case, Equations (29) and (30) imply q = ρ

_{q}R

_{S}

^{3}∝ ρ

_{q}(GM

_{BH}/c

^{2})

^{3}, where R

_{S}is the Schwarzschild radius, and M

_{BH}is the black hole mass. The galaxy circular velocity in the flat region is then from (21) v ∝ M

_{BH}

^{3/2}. This volume estimation is simplistic, as it assumes that the density of rotary charge in the vacuum is constant beneath the event horizon, while neglecting aspects concerning the physical volume due to the metric, and unknowns with how the vacuum interacts with it.

_{S}from interacting, and in this case, it just happens to be collapsed mass with G and M

_{BH}entering indirectly. An excellent overview of void-based astrophysics and cosmology may be found in [16], and other recent work in [17]. Though the references do not deal directly with the impact of collapsed matter on the vacuum, the implication is that voids are relevant for understanding the nature of the vacuum, and dark gravity. The voids in the cases of [16,17] represent regions of space in which dark matter is minimal, and dark energy would dominate.

## 4. Modified General Relativity with Excluded Volume

^{μν}. The exclusion is enforced in the sense that it does not rely solely on the gravitational field arising from a matter distribution. The challenge is to find the right formulation for the exclusion, producing a conserved charge, and not in conflict with any observations. The boundary conditions will be developed in future work, with S taken to be beneath the event horizon of a black hole. Five schemes will be initially sketched below, and they involve either the total contribution to the action from a void being zero, or, the variation of the action in a void being zero, or a field in the void going to zero, or a field becoming discontinuous at the boundary of S.

^{μ}, along the lines of,

^{l}are zero on the boundary of S. The Lagrangian must display the symmetry, and may be non-unique.

_{S}and the Schwarzschild radius R

_{S},

_{S}, so Equation (34a,b) are no longer applicable.

_{S}βφ, the division being that the scalar field becomes zero (or some other value proportional to ε) beneath the event horizon,

^{μν}and φ of (44) yield,

_{vac}= Cφ, an initial motivator for examining this form, as this term could play the role of the Cosmological Constant. A simple way to exclude volume of the field φ would be to decompose it as φ=f(r)$\mathcal{O}$, yielding R

_{vac}= C f(r)$\mathcal{O}$. If the partitioning is per (40), there is then a gradient in the vacuum energy, and effectively, there would be an added force. The vacuum (or dark) energy R

_{vac}links the source of dark energy to the alternative explanation for dark matter. The radius of partitioning in (40), and any resulting horizon must be found self-consistently, and will not generally be equal to the usual definition of the Schwarzschild Radius of 2GM/c

^{2}.

^{(φ}

^{)}

_{μν}= 0, and so the scalar field does not act effectively like dark matter—that is, adding to the energy momentum tensor of matter. Yet, one may introduce an intentional non-equality between R and κT along with a metric with sufficient adjustability that leads to a non-constant scalar field satisfying (46) to (49) by self-consistency. The analogy is the use of the many possible forms of Equation (7a–c) for the unsourced Newtonian vacuum potential to fit data.

## 5. Discussion

## 6. Conclusions

## Funding

## Conflicts of Interest

## Appendix A. Rotary Vector Potential in Cylindrical Coordinates

**Figure A1.**Vector potentials f of Equation (A2c) plotted in Cartesian coordinates. (

**Left**) Top solution. (

**Right**) Bottom solution. The rotary charge q = 1, axis units are arbitrary. The scalar field ξ is zero.

## Appendix B. Comment on Gravitomagnetic Field

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Dannenberg, R.
Excluded Volume for Flat Galaxy Rotation Curves in Newtonian Gravity and General Relativity. *Symmetry* **2020**, *12*, 398.
https://doi.org/10.3390/sym12030398

**AMA Style**

Dannenberg R.
Excluded Volume for Flat Galaxy Rotation Curves in Newtonian Gravity and General Relativity. *Symmetry*. 2020; 12(3):398.
https://doi.org/10.3390/sym12030398

**Chicago/Turabian Style**

Dannenberg, Rand.
2020. "Excluded Volume for Flat Galaxy Rotation Curves in Newtonian Gravity and General Relativity" *Symmetry* 12, no. 3: 398.
https://doi.org/10.3390/sym12030398