# Guessing the Riddle of a Black Hole

## Abstract

**:**

## 1. Introduction

#### 1.1. Einstein’s Hypothesis

**We assume that this is nowhere to be found**. In this case, det ${g}_{ik}$ cannot change its sign; we will accept, in accordance with the special theory of relativity, that det ${g}_{ik}$

**always has a finite and negative value.**This assumption is a hypothesis about the physical nature of the considered continuum and, at the same time, a rule concerning the choice of a coordinate system.” (see [2], below Equation (18a)).

**nowhere to be found**”, people use the condition $det{g}_{ik}<0$ unhesitatingly and everywhere. In the Schwarzschild metric, ${g}^{rr}$ is positive at $r<{r}_{g}$, and there is an inevitable singularity in the center $r=0$. At a regular center, the ratio of the circumference to diameter tends to $\pi $ as $r\to 0$; wherein ${g}^{rr}\left(0\right)=-1.$ The assumption “nowhere to be found” is violated in the center. For a real physical object of arbitrary high mass with a Schwarzschild gravitational field outside and a regular center, at least one more gravitational radius should exist inside the gravitating body.

#### 1.2. Gravitating Bose–Einstein Condensate

#### 1.3. Regularity Instead of det ${g}_{ik}<0$

**instead of the requirement**$det{g}_{ik}<0$

**, a weaker condition of regularity is applied**. Allowing $det{g}_{ik}$ to change sign, I reconsidered the equilibrium structure of a spherically symmetric gravitating Bose–Einstein condensate [16]. By not using the representation ${g}^{rr}=-{e}^{-\lambda}$ that fixes the sign, I avoid the trouble of the incompleteness of Schwarzschild’s coordinate system [17,18,19]; see also [20] §103 and [21] §14.

## 2. Behind the Horizon

^{2}/g × cm, $k=6.67\times {10}^{-8}$ cm

^{3}/g × s

^{2}. Relation:

## 3. Gravitating Scalar Field

## 4. In the Vicinity of a Gravitational Radius

#### 4.1. Regular Gravitational Radius

#### 4.2. Event Horizon

## 5. Regular Static Solutions with No Restriction on Mass

#### 5.1. Upper Boundary of the Regularity Strip: Simple Analytic Solution

#### 5.2. An Illustrative Example of a Regular Static Solution

#### 5.3. Total Mass as a Function of Density in the Center

#### 5.4. Black Holes and Dark Matter

#### 5.5. Superheavy Black Hole

## 6. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A nontrivial solution (41) to Equations (21)–(24) for ${u}_{0}^{2}={u}_{g}^{2}=0.1,$ ${\left(\frac{{x}_{g}}{{x}_{g\text{}\mathrm{max}}}\right)}^{2}=1.$ A logarithmic scale along the x axis is used. The dashed horizontal line is the level $ln2,$ where $g\left(x\right)=-1+\frac{1}{3}{u}_{0}^{2}{x}^{2}=0.$

**Figure 2.**The blue growing curve $ln\left(1+\frac{1}{3}{u}_{0}^{2}{x}^{2}\right)$ is the same as in Figure 1: ${u}_{g}^{2}=0.1,$ $\Lambda {u}_{g}^{2}=-2/3,$ ${\left(\frac{{x}_{g}}{{x}_{g\text{}\mathrm{max}}}\right)}^{2}=1.$ The red curve with the parameters ${u}_{g}^{2}=0.1,$ $\Lambda {u}_{g}^{2}=-2/3,$ ${\left(\frac{{x}_{g}}{{x}_{g\text{}\mathrm{max}}}\right)}^{2}=0.99995$ practically coincides with the blue one on the growing left side. However, even with a so small decrease of ${x}_{g}$, the growing part of $g\left(x\right)$ changes to a decreasing one, and a second gravitational radius ${x}_{h}$ appears.

**Figure 3.**Dependence of the event horizon radius ${x}_{h}$ on the condensate density ${u}_{g}^{2}$ in dimensionless units. The logarithmic scale is used along the vertical axis. Three dashed horizontal lines correspond to the masses of the Earth, the Sun, and the black hole in the center of the Milky Way, respectfully.

**Figure 4.**The observed acceleration (vertical axis) as a function of the expected one from the distribution of baryons (horizontal axis) for 240 galaxies [34,35]. Logarithmic scales along both axes. Colored dots correspond to galaxies of different morphological types. The straight 1:1 line shows where the observed and the expected Newtonian accelerations would coincide without dark matter.

**Figure 5.**Solution to Equations (21)–(24) corresponding to the superheavy black hole located in the center of the Milky Way Galaxy. ${u}_{g}^{2}=22,$ and ${x}_{g}^{2}=2.99985{u}_{g}^{-2}$. The red line is $ln\left(g\left(x\right)+2\right),$ and the green line is ${u}^{2}\left(x\right).$ The blue line shows the analytical solution $g\left(x\right)=-1+{u}_{0}^{2}{x}^{2}/3$ (41) on the upper border ${x}_{g}^{2}=3{u}_{g}^{-2},$ ${u}_{g}^{2}=22$ of the regularity string (40).

**Figure 6.**Solution to Equations (21)–(24) corresponding to the superheavy black hole located in the center of the Milky Way Galaxy. ${u}_{g}^{2}=22,$ and ${x}_{g}^{2}=2.99985{u}_{g}^{-2}$. The green line is $u\left(x\right)/{u}_{g},$ and the brown line is $h\left(x\right).$ The horizon ${x}_{h}$ $\left(ln{x}_{h}=63.96\right)$ is far outside this graph.

**Figure 7.**Fragment of oscillating $u\left(x\right)$ in the interval $24.3<lnx<24.5$ derived analytically (54) (red line) and found numerically (blue line). The angularity of the blue line is due to the fact that the condensing oscillations in the huge interval (${x}_{m},{x}_{h}$) are obtained at the limit of computer accuracy.

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Meierovich, B.E.
Guessing the Riddle of a Black Hole. *Universe* **2020**, *6*, 113.
https://doi.org/10.3390/universe6080113

**AMA Style**

Meierovich BE.
Guessing the Riddle of a Black Hole. *Universe*. 2020; 6(8):113.
https://doi.org/10.3390/universe6080113

**Chicago/Turabian Style**

Meierovich, Boris E.
2020. "Guessing the Riddle of a Black Hole" *Universe* 6, no. 8: 113.
https://doi.org/10.3390/universe6080113