#
Bose-Einstein Condensate in Synchronous Coordinates^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

_{11}. Thus, not caring about the presence or absence of a singularity, we exclude ${g}_{11}>0$ from consideration as non-physical. In this case, the coordinate system turns out to be incomplete [3,4,5]. A critical mass ${M}_{cr}$ arises (for neutron stars, it is of the order of the Sun mass), so that for $M>{M}_{cr}$, regular static solutions to Einstein equations do not exist [6,7,8]. Shwartzshild’s solution [1]

_{f}is the fermion mass, and k is the gravitational constant. For neutron stars (neutron rest mass ${m}_{f}=1.67\times {10}^{-24}$ g) the critical mass ${M}_{cr\hspace{0.33em}f}~{10}^{33}$ g is of the order of the Sun mass.

_{b}about 100 GeV/c

^{2}) the critical mass of the condensate is ${M}_{cr\hspace{0.17em}b}~{10}^{12}$ g. It is only about a million tons.

_{g}. Inside the sphere r < r

_{g}; the equation of state of the condensate is

_{cr}< M < ∞ [25].

## 2. In Synchronous Coordinates

_{g}, on which the component ${g}^{rr}\left({r}_{g}\right)=0$, is not a physical singularity in the Schwarzschild metric. In the problem 4 at the end of §100 in Ref. [22], the transformation of the Schwarzschild metric (2) to the conformal Euclidean form is given. The non-uniqueness of the solution to the system of Einstein and Klein-Gordon equations with boundary conditions exactly on the gravitational radii $r={r}_{g}$ and $r={r}_{h}$, where ${g}^{rr}\left(r\right)=0$ [25], is a feature of the Schwarzschild metric. In the conformal Euclidean form, ${g}^{rr}\left(r\right)$ does not vanish.

## 3. Dark Matter

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Meierovich, B.E.
Bose-Einstein Condensate in Synchronous Coordinates. *Phys. Sci. Forum* **2023**, *7*, 47.
https://doi.org/10.3390/ECU2023-14121

**AMA Style**

Meierovich BE.
Bose-Einstein Condensate in Synchronous Coordinates. *Physical Sciences Forum*. 2023; 7(1):47.
https://doi.org/10.3390/ECU2023-14121

**Chicago/Turabian Style**

Meierovich, Boris E.
2023. "Bose-Einstein Condensate in Synchronous Coordinates" *Physical Sciences Forum* 7, no. 1: 47.
https://doi.org/10.3390/ECU2023-14121