# Static State of a Black Hole Supported by Dark Matter

## Abstract

**:**

^{rr}(r) changes sign twice. The behavior of the gravitational field and material fields in the vicinity of these two Schwarzschild radii were studied in detail. The equality of the energy–momentum tensors of the scalar and longitudinal vector fields at the interface supports the phase equilibrium of a black hole and dark matter. Considering the gravitating scalar field as an example, a possible internal structure of a black hole and its influence on the dark matter at the periphery of a galaxy are clarified. In particular, the speed on the plateau of a galaxy rotation curve as a function of a black hole’s mass is determined.

## 1. Introduction

^{6}M

_{ʘ}, and its radius to be less than 0.002 light years [1]. The mass of the Sun is M

_{ʘ}= 1.989 × 10

^{33}g. If the mass of an object in the center of a galaxy is six orders of magnitude greater than the maximum equilibrium mass of a neutron star, then it is natural to consider black holes as the most likely candidates for super-massive objects in the centers of galaxies.

## 2. Space-Time, Strongly Curved Up to Changing the Signature of Metric Tensor

_{0}is a constant of integration, and ${g}^{rr}\left(r\right)$ is a regular function, provided that the integral in Equation (5) converges. The convergence of this integral means that the mass within the layer $\left({r}_{0},r\right)$,

^{−3}as $r\to \infty $. Looking at Equation (3), one can see that, in the region of changed signature ${g}^{rr}>0$, the derivative (g

^{rr})′ becomes negative when (1 + g

^{rr})/r exceeds $\kappa r{T}_{0}^{0}$ with growing r. It means, that there may exist (and does exist, see below) a solution where, with increasing r, the metric component ${g}^{rr}\left(r\right)$ gets into the region of violated signature, passes through a maximum, and returns back into the region of the Galilean signature. ${g}^{rr}\left(r\right)$ intersects the x-axis twice.

**in a vacuum**with a broken signature of the metric tensor. Nevertheless, one cannot exclude a possibility that the event horizon ${r}_{h}$ takes place on the interface between two gravitating objects, for instance, if the inside pressure of a black hole is balanced by the pressure of dark matter from outside.

## 3. Gravitating Scalar Field behind the Horizon

^{*}satisfy the Klein–Gordon equation.

^{−2}. It is related to the mass of the quantum m: $\partial U/\partial {\left|\psi \right|}^{2}={\left(mc/\hslash \right)}^{2}$.

#### 3.1. Regular Gravitational Radius

#### 3.2. Horizon

## 4. Numerical Analysis

#### 4.1. Strip of Regular Solutions

#### 4.2. Example of a Regular Solution behind the Horizon

#### 4.3. In the Vicinity of the Upper Boundary

#### 4.4. Core of a Black Hole

#### 4.5. Super-Heavy Black Hole

_{ʘ}= 2 × 10

^{33}g, the component of the energy–momentum tensor ${T}_{r}^{r}\left({x}_{h}\right)\approx 4\times {10}^{36}$ g/(cm × sec

^{2}), that is, $4\times {10}^{30}$ atmospheres. A static gravitating scalar field with a very high uncompensated pressure on the interface with a vacuum cannot exist. But this is without dark matter. The presence of dark matter outside the gravitating scalar field makes it possible to ensure a pressure balance at the interface of these two media.

## 5. Longitudinal Vector Field

## 6. Galaxy Rotation Curve: Dependence of Plateau Velocity on the Mass of a Black Hole

^{−16}cm for a quantum with the rest mass ~ 100 GeV/c

^{2}). The argument of covariant divergence ${\phi}_{;M}^{M}$ and other functions in Equations (48)–(51) is $y=(\mu \text{\hspace{0.05em}}c/\hslash )r$. On the galactic scale, a natural unit of length is the de Broglie wavelength $\hslash /\mu \text{\hspace{0.05em}}c$ of a quantum of the longitudinal vector field. According to damped oscillations on the plateau of galaxy rotation curve (see Figure 11), $\hslash /\mu \text{\hspace{0.05em}}c~$ 10 kpc ≈ 3 × 10

^{22}cm.

^{2}/r gives

^{22}cm, the value ${\phi}_{;M}^{M}\left({r}_{h}\right)$ can be used as a boundary condition at $(\mu \text{\hspace{0.05em}}c/\hslash )r<<1.$ The solution to Equations (61) and (62) is

_{pl},

## 7. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Gillessen, S.; Eisenhauer, F.; Trippe, S. Monitoring Stellar Orbits around the Massive Black Hole in the Galactic Center. Astrophys. J.
**2009**, 692, 1075. [Google Scholar] [CrossRef] - Landau, L.D.; Lifshitz, E.M. Statistical Physics. Part 1; Nauka-Fizmatlit: Moscow, Russian, 1995. [Google Scholar]
- Meierovich, B.E. Macroscopic Theory of Dark Sector. J. Gravity
**2014**, 586958. [Google Scholar] [CrossRef] - Landau, L.D.; Lifshitz, E.M. Statistical Physics. Part 2: Theory of the Condensed State; Fizmatlit: Moscow, Russian, 2002. [Google Scholar]
- Meierovich, B.E. On the Equilibrium State of a Gravitating Bose--Einstein Condensate. J. Exp. Theor. Phys.
**2018**, 127, 889–902. [Google Scholar] [CrossRef] - Landau, L.D.; Lifshitz, E.M. The Classical Theory of Fields; Nauka: Moscow, Russian, 1973. [Google Scholar]
- Schwarzschild, K. Uber das Gravitationsfeld Eines Massenpunktes Nach Einsteinschen Theorie; Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften: Berlin, Germany, 1916; pp. 189–196. [Google Scholar]
- Colpi, M.; Shapiro, S.L.; Wasserman, I. Boson stars: Gravitational equilibria of self-interacting scalar fields. Phys. Rev. Lett.
**1986**, 57, 2485. [Google Scholar] [CrossRef] [PubMed] - Andreev, A.F. Macroscopic bodies with zero rest mass. J. Exp.Theor. Phys.
**1974**, 38, 648. [Google Scholar] - Meierovich, B.E. Galaxy rotation curves driven by massive vector fields: Key to the theory of the dark sector. Phys. Rev. D Part. Fields Gravit. Cosmol.
**2013**, 87, 103510. [Google Scholar] [CrossRef] - Brownstein, J.R.; Moat, J.W. Galaxy rotation curves without non-baryonic dark matter. Astrophys. J.
**2006**, 636, 721. [Google Scholar] [CrossRef]

**Figure 1.**The blue line ${u}_{g}^{2}={x}_{g}^{-2}$ in the plane of parameters ${x}_{g},{u}_{g}^{2}$ (Equation (36)) is the lower boundary of the area of existence of regular solutions with finite mass M. The denominator of ${\nu}^{\prime}({x}_{g})$ (Equation (32)) is zero on the blue line. The red line ${u}_{g}^{2}=2{x}_{g}^{-2}$ is the upper limit. At ${x}_{g}^{2}{u}_{g}^{2}\to 2$, the total mass $M\to \infty $. ${\nu}^{\prime}({x}_{g})$ (Equation (32)) is zero on the red line.

**Figure 2.**An example of a regular solution to the set of Equations (24)–(27) with boundary conditions in Equation (33). ${x}_{g}=1,\text{\hspace{0.17em}}{u}_{g}=\sqrt{2}-0.01$. In accordance with Equation (31), ${h}_{g}=0.492853.$ The horizon radius ${x}_{h}=13.78581985$ is found numerically.

**Figure 3.**The same functions $u(x),w(x),g(x),$ and h(x), as in Figure 2, near ${x}_{h}$ (left graph), and very close to the horizon ${x}_{h}$ (right graph).

**Figure 4.**Functions u(x) green, w(x) blue, g(x) red, and h(x) brown in the vicinity of the upper border of the strip in Equation (36).

**Figure 5.**Functions $w(x),g(x),h(x)$ on the right boundary ${x}_{g}={x}_{g\text{\hspace{0.17em}}\mathrm{max}}=1.4285$ of the regularity domain in Equation (36). ${x}_{g}{u}_{g}=\sqrt{2}$.

**Figure 6.**Dashed green line is $x\times {u}^{\prime}(x)$ (

**a**). Dependence $C({x}_{g}{u}_{g})$ (Equation (37)) (

**b**).

**Figure 7.**Red lines in graphs

**A**,

**B**, and

**C**show that, with a decrease of ${x}_{g}$ (and a simultaneous increase of ${u}_{g}$ along the path in Equation (40)), the metric function $g(x)$ gets closer to −1 in the center. Values of ${x}_{g}$ are 0.1 (

**A**), 0.01 (

**B**), and 0.001 (

**C**). The blue line is $w(x)$, and the brown line is $h(x)$.

**Figure 8.**Dependence $\mathrm{ln}[{x}_{h}({x}_{g})]$ (

**a**) along the dashed line ${x}_{g}{u}_{g}=\sqrt{2}-0.1$ (

**b**).

**Figure 10.**Function h(x) in the interval (${x}_{g}$,${x}_{h}$) of broken signature. ${x}_{g}=0.55$, ${x}_{g}{u}_{g}=\sqrt{2}-0.0005$.

**Figure 11.**Rotation curve of the spiral galaxy NGC 3769 in the Ursa Major cluster. The vertical axis is the speed in km/sec. The horizontal axis is the distance from the center of the galaxy in kiloparsecs. The solid line is fitting by Equation (67).

© 2019 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**

Meierovich, B.E.
Static State of a Black Hole Supported by Dark Matter. *Universe* **2019**, *5*, 198.
https://doi.org/10.3390/universe5090198

**AMA Style**

Meierovich BE.
Static State of a Black Hole Supported by Dark Matter. *Universe*. 2019; 5(9):198.
https://doi.org/10.3390/universe5090198

**Chicago/Turabian Style**

Meierovich, Boris E.
2019. "Static State of a Black Hole Supported by Dark Matter" *Universe* 5, no. 9: 198.
https://doi.org/10.3390/universe5090198