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Critical issue on Black-Holes physics
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Black Holes; Schwarzschild solution; Singularity.
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\begin{document}
\title{\textbf{An alternative to the Schwarzschild solution}}
\author{\textbf{Christian Corda }}
\maketitle
\lyxaddress{\begin{center}
Institute for Basic Research, P. O. Box 1577, Palm Harbor, FL 34682,
USA and Associazione Galileo Galilei, Via Buozzi 47, 59100 PRATO,
Italy
\par\end{center}}
\begin{center}
\textit{E-mail addresses:} \textcolor{blue}{cordac.galilei@gmail.com}
\par\end{center}
\begin{abstract}
We propose an alternative to the famous Schwarzschild solution to
Einstein field equations which arises from a different physical hypothesis.
In this solution, which works for the external geometry of a spherical
static star and circumnavigates the Birkhoff Theorem, Black-Hole Singularities
are removed at the classical level, i.e. without arguments of quantum
gravity.
\end{abstract}
\textbf{Keywords:} Black Holes; Schwarzschild solution; Singularity.
\textbf{PACS numbers:} 04.70.-s; 04.70.Bw
\section{Introduction}
The concept of Black-Hole (BH) has been considered very fascinating
by scientists even before the introduction of general relativity (see
\cite{key-1} for an historical review). However, an unsolved problem
concerning such objects is the presence of a space-time singularity
in their core. Such a problem was present starting by the firsts historical
papers concerning BHs \cite{key-2,key-3,key-4}. It is a common opinion
that this problem could be solved when a correct quantum gravity theory
will be, finally, obtained, see \cite{key-5} for recent developments.
On the other hand, fundamental issues which dominate the question
about the existence or non-existence of BH horizons and singularities
and some ways to avoid the development of BH singularities at a classical
level, which does not require the need for a quantum gravity theory,
have been discussed by various authors in the literature, see references
from \cite{key-6} to \cite{key-19}.
In general relativity the Einstein field equations connect the curvature
tensor of space-time on the left hand side to the energy-momentum
tensor in space-time on the right hand side \cite{key-1}. A key point
is that the \emph{strong principle of equivalence} (SPOE) implies
that special relativity has to hold locally for all of the laws of
physics in all of space-time as seen by time-like observers \cite{key-20}).
Then, the frame of reference of co-moving observers within a gravitationally
collapsing object have to always be able to be connected to the frame
of reference of stationary observers by special relativistic transformations
with physical velocities which are less than the speed of light. Recently,
various arguments and computations have been introduced to support
the idea that physically acceptable solutions to the Einstein field
equations will only be the ones preserving the SPOE as \emph{a law
of nature} in the universe \cite{key-6,key-7,key-8}. As consequence,
the compact objects which emerge from the process of gravitational
collapse could not have event horizons (EHs). In fact, the existence
of EHs would prevent co-moving observers within a gravitationally
collapsing object from being able to be connected to the frame of
reference of stationary observers by special relativistic transformations
with physical velocities which are less than the speed of light. Hence,
a result of the SPOE is that objects with non-zero mass having EHs
could be physically prohibited \cite{key-6,key-7,key-8}. The preservation
of the SPOE in the field equations would put an overall constraint
on the nature of the non-gravitational physical elements which go
into the energy-momentum tensor on the right hand side of the field
equations. This constraint does not determine in a unique way the
specific form of the non-gravitational dynamics of the energy-momentum
tensor \cite{key-7,key-8,key-9}. Various different theories have
been constructed (e.g. \emph{eternally collapsing objects} (ECO),
\emph{magneto-spheric eternally collapsing objects} (MECO), \emph{non-linear
electrodynamics} (NLED) \emph{objects}), which preserve the SPOE and
hence can generate highly redshifted compact objects without EHs and
Singularities \cite{key-7,key-8}, \cite{key-9}-\cite{key-14}. Since
each of these different SPOE preserving theories have unique observational
predictions associated with the interaction of their non-gravitational
components with the environment of their highly redshifted compact
objects without EHs and Singularities, the specific one chosen by
Nature can only be determined by astrophysical observations which
test these predictions.
Let us emphasize another fundamental point \cite{key-15}-\cite{key-19}.
The basis of general relativity are 10 non-linear coupled equations.
It is now well known that while such equations may yield various solutions,
most of them could have no bearing with physical reality. Let us consider
the Reissner-Nordström solution, originally derived as vacuum space-time
exterior to a spherically symmetric charged fluid \cite{key-1}. It
is not only a correct solution, but obviously, a solution with a strong
physical meaning. This solution is usually interpreted as a charged
BH solution which forces the charged fluid with radius $R_{0}$ to
be contracted to a geometrical point, $R_{0}\rightarrow0$ \cite{key-1}.
From a purely mathematical point of view it is granted that such a
process could be possible because mathematics allows it \cite{key-1}.
But, from a physical point of view, let us ask {}``\emph{do physics
allow a charged fluid to be contracted into a geometrical non-dimensional
point?}'' New ideas of mathematical physicists are going beyond this
purely mathematical {}``\emph{take for granted}''\cite{key-15}-\cite{key-19}.
The authors of \cite{key-19} claim that a charged sphere may collapse
to a quasi BH rather than to an exact BH; i.e., no trapped surfaces,
no EH, while $R_{0}$ will eventually hover over the running mathematical
EH.
On the other hand, the famous Schwarzschild solution is obtained for
a neutral mass point and the BH mass came from the integration constant
$\alpha_{0}=2GM_{0}/c^{2}$ \cite{key-1}. From both of physical and
mathematical points of view, integrations constants do not need always
to be finite or arbitrary. Even though as algebra symbols, they may
appear to assume arbitrary value. Considering the generalization given
by the Birkhoff theorem, the same solution also represents the vacuum
space-time exterior to a static neutral spherical fluid. In this case,
the corresponding general integration constant $\alpha=2GM/c^{2}$
indeed gives gravitational mass of the fluid having a radius $R_{0}$,
i.e. $M=\intop_{0}^{R_{0}}4\pi\rho R^{2}dR.$
Let us ask: \emph{how does the value of M would change if we would
allow $R_{0}\rightarrow0?$ Is there any guarantee that $M$ must
remain finite as $R_{0}\rightarrow0?$} If this would be seen as an
adiabatic gravitational collapse without emission of radiation, one
would say, yes, one must have $M=M_{0}$ (no loss of mass-energy).
But in \cite{key-16} it has been shown that an adiabatic collapse
could not exist in the physical reality. In that case, all those so-called
adiabatic collapses could be only respectable looking mathematical
exercises with respect a different, non-singular physical reality.
In \cite{key-18} it has been proposed that, like a consequence, when
the fluid sphere would be assumed to have contracted to a point,
\begin{equation}
M\rightarrow M_{0}=Lim_{R_{0}\rightarrow0}\intop_{0}^{R_{0}}4\pi\rho R^{2}dR=0.\label{eq: 5}\end{equation}
In other words, the mass of a Schwarzschild BH that arises from a
real, non adiabatic collapse could be zero.
In this paper, we propose an alternative to the Schwarzschild solution
to Einstein field equations which starts from a different physical
hypothesis. In this solution, which works for the external geometry
of a spherical static source (a star) and circumnavigates the Birkhoff
Theorem, Black-Hole Singularities are removed at the classical level.
Arguments of quantum gravity are not taken into account.
Note that we are \emph{not} claiming that our solution is correct
and the Schwarzschild solution is wrong. Actually, we will see that
the two solutions are incompatible as they arises from \emph{different
physical hypotheses}. Only the Nature can chose which is the correct
one among the physical hypotheses and, consequently, which is the
correct solution for the external space-time geometry of a spherical
static star.
\section{The solution}
Following \cite{key-21} and working with $G=1$, $c=1$ and $\hbar=1$
(natural units), the more general line-element which respects central
symmetry is
\begin{equation}
ds^{2}=h(r,t)dr^{2}+k(r,t)(\sin^{2}\theta d\varphi^{2}+d\theta^{2})+l(r,t)dt^{2}+a(r,t)drdt,\label{eq: generale}\end{equation}
where $r\geq0,$ $0\leq\theta\leq\pi$ $0\leq\varphi\leq2\pi$.
We search a line-element solution in which the metric is spatially
symmetric with respect to the origin of the coordinate system, i.e.
that we find again the same solution when spatial coordinates are
subjected to a orthogonal transformations and rotations and it is
asymptotically flat at infinity \cite{key-1}. In order to obtain
the standard Schwarzschild solution to Einstein field equations in
vacuum one uses transformations of the type \cite{key-21}
\begin{equation}
\begin{array}{ccc}
r=f_{1}(r',t'), & & t=f_{2}(r',t')\end{array},\label{eq: transformations}\end{equation}
where $f_{1}$ and $f_{1}$ are arbitrary functions of the new coordinates
$r'$ and $t'$. At this point, if one wants the standard Schwarzschild
solution, $r$ and $t$ have to be chosen in a way that $a(r,t)=0$
and $k(r,t)=-r^{2}$ \cite{key-21}. In particular, the second condition
implies that the standard Schwarzschild radius is determined in a
way which guarantees that the length of the circumference centred
in the origin of the coordinate system is $2\pi r$ \cite{key-21}.
In our approach, we will suppose again that $a(r,t)=0,$ but, differently
from the standard analysis, we will assume that the length of the
circumference centred in the origin of the coordinate system \emph{is
not} $2\pi r$. We release \emph{a different physical assumption},
i.e. that arches of circumference \emph{are not Euclidean}, but they
are deformed by the presence of the mass of the central body $M.$
Note that this different physical hypothesis permits to circumnavigate
the Birkhoff Theorem \cite{key-3} which leads to the Schwarzschild
solution \cite{key-1,key-3}. In fact, the demonstration of the Birkhoff
Theorem \emph{starts} from a line element \emph{in which $k(r,t)=-r^{2}$
has been chosen}, see the discussion in paragraph 32.2 of \cite{key-1}
and, in particular, look at Eq. (32.2) of such a paragraph.
Then, we proceed assuming $k=-mr^{2},$ where $m$ is a generic function
to be determined in order to obtain that the length of circumferences
centred in the origin of the coordinate system\emph{ }are not Euclidean.
In other words, $m$ represents \emph{a measure of the non-Euclidean
behaviour} of circumferences centred in the origin of the coordinate
system.
The line element (\ref{eq: generale}) becomes
\begin{equation}
ds^{2}=hdr^{2}-mr^{2}(\sin^{2}\theta d\varphi^{2}+d\theta^{2})+ldt^{2}.\label{eq: generale 2}\end{equation}
One puts \begin{equation}
\begin{array}{c}
X\equiv\frac{1}{3}r^{3}\\
\\Y\equiv-\cos\theta\\
\\Z\equiv\varphi.\end{array}\label{eq: trasforma}\end{equation}
In the $X,Y,Z$ coordinates the line-element (\ref{eq: generale 2})
reads
\begin{equation}
ds^{2}=ldt^{2}+\frac{h}{^{r^{4}}}dX^{2}-mr^{2}[\frac{dY^{2}}{1-Y^{2}}+dZ^{2}(1-Y^{2})].\label{eq: alternative}\end{equation}
Let us consider three functions
\label{eq: nuove funzioni}\begin{equation}
\begin{array}{c}
A\equiv-\frac{h}{^{r^{4}}}\\
\\B\equiv mr^{2}\\
\\C\equiv l\end{array}\label{eq: functions}\end{equation}
which satisfy the conditions \begin{equation}
\begin{array}{c}
X\rightarrow\infty\mbox{ }implies\mbox{ }A\rightarrow\frac{1}{r^{4}}=\frac{1}{(3X)^{\frac{4}{3}}},\mbox{ }B\rightarrow r^{2}=3X^{\frac{2}{3}},\mbox{ }C\rightarrow1\\
\\normalization\mbox{ }condition\mbox{ }AB^{2}C=1.\end{array}\label{eq: condizioni}\end{equation}
The line-element (\ref{eq: alternative}) becomes
\begin{equation}
ds^{2}=Cdt^{2}-AdX^{2}-B\frac{dY^{2}}{1-Y^{2}}-BdZ^{2}(1-Y^{2}).\label{eq: alternative 2}\end{equation}
From the metric (\ref{eq: alternative 2}) one gets the Christoffell
coefficients like (only the non zero elements will be written down)
\begin{equation}
\begin{array}{ccc}
\Gamma_{tX}^{t}=-\frac{1}{2C}\frac{\partial C}{\partial X} & & \Gamma_{XX}^{X}=-\frac{1}{2A}\frac{\partial A}{\partial X}\\
\\\Gamma_{YY}^{X}=\frac{1}{2A}\frac{\partial B}{\partial X}\frac{1}{1-Y^{2}} & & \Gamma_{ZZ}^{X}=\frac{1}{2A}\frac{\partial B}{\partial X}(1-Y^{2})\\
\\\Gamma_{tt}^{X}=-\frac{1}{2A}\frac{\partial C}{\partial X} & & \Gamma_{YX}^{Y}=-\frac{1}{2B}\frac{\partial B}{\partial X}\\
\\\Gamma_{YY}^{Y}=\frac{-Y}{1-Y^{2}} & & \Gamma_{ZZ}^{Y}=-Y(1-Y^{2})\\
\\\Gamma_{ZX}^{Z}=-\frac{1}{2B}\frac{\partial B}{\partial X} & & \Gamma_{ZX}^{Z}=\frac{Y}{1-Y^{2}}.\\
\\\end{array}\label{eq: connessioni}\end{equation}
By using the equation for the components of the Ricci tensor, the
components of Einstein field equation in vacuum are \cite{key-21}
\begin{equation}
R_{ik}=\frac{\partial\Gamma_{ik}^{l}}{dx^{l}}-\frac{\partial\Gamma_{il}^{l}}{dx^{k}}-\Gamma_{ik}^{l}\Gamma_{lm}^{m}-\Gamma_{il}^{m}\Gamma_{km}^{l}=0.\label{eq: ricci}\end{equation}
By inserting Eqs. (\ref{eq: connessioni}) in Eqs. (\ref{eq: ricci})
one gets only three independent relations
\begin{equation}
\frac{\partial}{\partial X}(\frac{1}{A}\frac{\partial B}{\partial X})-2-\frac{1}{AB}(\frac{\partial B}{\partial X})^{2}=0\label{eq: einstein 2}\end{equation}
\begin{equation}
\frac{\partial}{\partial X}(\frac{1}{A}\frac{\partial A}{\partial X})-\frac{1}{2}(\frac{1}{A}\frac{\partial A}{\partial X})^{2}-(\frac{1}{B}\frac{\partial B}{\partial X})^{2}-\frac{1}{2}(\frac{1}{C}\frac{\partial C}{\partial X})^{2}=0\label{eq: einstein 1}\end{equation}
\begin{equation}
\frac{\partial}{\partial X}(\frac{1}{A}\frac{\partial C}{\partial X})-\frac{1}{AC}(\frac{\partial C}{\partial X})^{2}=0.\label{eq: einstein 3}\end{equation}
From the second of Eqs. (\ref{eq: condizioni}) (normalization condition)
one gets also
\begin{equation}
\frac{1}{A}\frac{\partial A}{\partial X}+\frac{2}{B}\frac{\partial B}{\partial X}+\frac{1}{C}\frac{\partial C}{\partial X}=0.\label{eq: normalizzata}\end{equation}
Eq. (\ref{eq: einstein 3}) can be rewritten like
\begin{equation}
\frac{\partial}{\partial X}(\frac{1}{C}\frac{\partial C}{\partial X})=\frac{1}{AC}\frac{\partial A}{\partial X}\frac{\partial C}{\partial X},\label{eq: einstein 3.1}\end{equation}
which can be integrated, giving
\begin{equation}
\frac{1}{C}\frac{\partial C}{\partial X}=aA,\label{eq: integra 1}\end{equation}
where $a$ is an integration constant.
By adding Eq. (\ref{eq: einstein 1}) to Eq. (\ref{eq: einstein 3.1})
one gets
\begin{equation}
\frac{\partial}{\partial X}(\frac{1}{A}\frac{\partial A}{\partial X}+\frac{1}{C}\frac{\partial C}{\partial X})=(\frac{1}{B}\frac{\partial B}{\partial X})^{2}+\frac{1}{2}(\frac{1}{A}\frac{\partial A}{\partial X}+\frac{1}{C}\frac{\partial C}{\partial X})^{2}\label{eq: einstein 1.1}\end{equation}
Considering Eq. (\ref{eq: normalizzata}) we obtain
\begin{equation}
2\frac{\partial}{\partial X}(\frac{1}{B}\frac{\partial B}{\partial X})=-3(\frac{1}{B}\frac{\partial B}{\partial X})^{2},\label{eq: einstein 2.1}\end{equation}
which can be integrated, giving
\begin{equation}
\frac{1}{B}\frac{\partial B}{\partial X}=\frac{2}{3X+b},\label{eq: integra 2}\end{equation}
where $b$ is an integration constant. A second integration gives
\begin{equation}
B=d(3X+b)^{\frac{2}{3}}.\label{eq: integra 3}\end{equation}
where $d$ is an integration constant. But the first of Eqs. (\ref{eq: condizioni})
implies $d=1,$ thus \begin{equation}
B=(3X+b)^{\frac{2}{3}}.\label{eq: integrata}\end{equation}
By using Eqs. (\ref{eq: integra 1}) and (\ref{eq: normalizzata})
we obtain \[
\frac{\partial C}{\partial X}=aAC=\frac{a}{B^{2}}=a(3X+b)^{-\frac{4}{3}}.\]
By integrating and considering the first of Eqs. (\ref{eq: condizioni})
one gets
\begin{equation}
C=1-a(3X+b)^{-\frac{1}{3}}\label{eq: C}\end{equation}
Then, from Eq. (\ref{eq: normalizzata}) one obtains
\begin{equation}
A=\frac{(3X+b)^{-\frac{4}{3}}}{1-a(3X+b)^{-\frac{1}{3}}}.\label{eq: singular}\end{equation}
By putting Eqs. (\ref{eq: singular}) and (\ref{eq: integrata}) in
Eq. (\ref{eq: einstein 2}) one immediately sees that this last equation
is automatically satisfied.
We note that the function $A$ results singular for values $a(3X+b)^{-\frac{1}{3}}=1.$
However, this is a mathematical singularity due to the particular
coordinates $t,X,Y,Z$ defined by the transformation (\ref{eq: trasforma}).
In fact, by assuming that such a singularity is located at $X=0$
we get
\begin{equation}
b=a^{3},\label{eq: ba}\end{equation}
i.e. we find a relation between the two integration constants $b$
and $a$.
At the end we obtain
\begin{equation}
\begin{array}{c}
A=(r^{3}+a^{3})^{-\frac{4}{3}}[1-a(r^{3}+a^{3})^{-\frac{1}{3}}]^{-1}\\
\\B=(r^{3}+a^{3})^{\frac{2}{3}}\\
\\C=1-a(r^{3}+a^{3})^{-\frac{1}{3}}.\end{array}\label{eq: soluzioni}\end{equation}
By inserting the functions (\ref{eq: soluzioni}) in Eq. (\ref{eq: alternative 2})
and using Eqs. (\ref{eq: functions}) and (\ref{eq: trasforma}) to
return to the standard polar coordinates the line-element solution
reads
\begin{equation}
ds^{2}=(1-\frac{a}{(r^{3}+a^{3})^{\frac{1}{3}}})dt^{2}-(r^{3}+a^{3})^{\frac{2}{3}}(\sin^{2}\theta d\varphi^{2}+d\theta^{2})+\frac{d[(r^{3}+a^{3})^{\frac{2}{3}}]}{1-\frac{a}{(r^{3}+a^{3})^{\frac{1}{3}}}}.\label{eq: sol. generale}\end{equation}
Hence, we understand that the assumption to locate the mathematical
singularity of the function $A$ at $X=0$ coincides with the physical
condition that the length of the circumference centred in the origin
of the coordinate system is $2\pi(r^{3}+a^{3})^{\frac{1}{3}}$, which
is different from the Euclidean value $2\pi r.$ This is the fundamental
physical difference between the proposed solution and the standard
Schwarzschild solution. The value of the generic function $m$ which
permits that the length of circumferences centred in the origin of
the coordinate system\emph{ }are not Euclidean is
\begin{equation}
m=\frac{(r^{3}+a^{3})^{\frac{2}{3}}}{r^{2}}.\label{eq: funzione m}\end{equation}
On the other hand, in order to determinate the value of the constant
$a$, by following \cite{key-21} one can use the weak field approximation
which implies $g_{00}\cong1+2\varphi$ at large distances, where $g_{00}=(1-\frac{a}{(r^{3}+a^{3})^{\frac{1}{3}}})$
in Eq. (\ref{eq: sol. generale}) and $\varphi\equiv\frac{-M}{r}$
is the Newtonian potential. Thus, for $a\ll r,$ we immediately obtain:
$a=2M=r_{g},$ i.e. $a$ results exactly the gravitational radius
\cite{key-1,key-21}.
Then, we can rewrite the solution (\ref{eq: sol. generale}) in an
ultimate way like
\begin{equation}
ds^{2}=(1-\frac{r_{g}}{(r^{3}+r_{g}^{3})^{\frac{1}{3}}})dt^{2}-(r^{3}+r_{g}^{3})^{\frac{2}{3}}(\sin^{2}\theta d\varphi^{2}+d\theta^{2})+\frac{d[(r^{3}+r_{g}^{3})^{\frac{2}{3}}]}{1-\frac{r_{g}}{(r^{3}+r_{g}^{3})^{\frac{1}{3}}}}.\label{eq: sol. gen.fin.}\end{equation}
Some comments are needed. By defining
\begin{equation}
\widehat{r}\equiv(r^{3}+r_{g}^{3})^{\frac{1}{3}},\label{eq: definizione}\end{equation}
eq. (\ref{eq: sol. generale}) becomes
\begin{equation}
ds^{2}=(1-\frac{r_{g}}{\widehat{r}})dt^{2}-\widehat{r}^{2}(\sin^{2}\theta d\varphi^{2}+d\theta^{2})+\frac{d\widehat{r}^{2}}{1-\frac{r_{g}}{\widehat{r}}}.\label{eq: sol. gen 2}\end{equation}
Eq. (\ref{eq: sol. gen 2}) looks \emph{formally} \emph{equal} to
standard Schwarzschild solution, but it is \emph{physically different.
}In fact, from Eq. (\ref{eq: definizione}) it is \emph{always} $\widehat{r}\geq r_{g}$
in Eq. (\ref{eq: sol. gen 2}) as it is $r\geq0$ in Eq. (\ref{eq: sol. gen.fin.}).
In this way, \emph{there are not physical singularities} in Eq. (\ref{eq: sol. gen 2}).
In fact, $r=0$ in Eq. (\ref{eq: sol. gen.fin.}) implies $\widehat{r}=r_{g}$
in Eq. (\ref{eq: sol. gen 2}) which corresponds to the mathematical
singularity at $X=0.$ This singularity is \emph{not physical} but
is due to the particular coordinates $t,X,Y,Z$ defined by the transformation
(\ref{eq: trasforma}).
We emphasize that the solution Eq. (\ref{eq: sol. gen 2}) and the
Schwarzschild solution \emph{are not} comparable as they arises from
\emph{different physical hypotheses. }As it is carefully explained
in \cite{key-21}, the standard Schwarzschild solution arises from
the hypothesis that the coordinates $r$ and $t$ of the two functions
(\ref{eq: transformations}) are chosen in order to guarantee that
the length of the circumference centred in the origin of the coordinate
system is $2\pi r$. Indeed, in the proposed solution, $r$ and $t$
are chosen in order to guarantee that the length of the circumference
centred in the origin of the coordinate system is \emph{not} $2\pi r$,
i.e. the circumferences centred in the origin of the coordinate system
are \emph{not} Euclidean. In particular, choosing to put the mathematical
singularity of the function $A$ at $X=0$ is equivalent to the physical
condition that the length of the circumference centred in the origin
of the coordinate system is $2\pi(r^{3}+r_{g}^{3})^{\frac{1}{3}}$.
Thus, again we emphasize that we are \emph{not} claiming that the
solution proposed in this paper is correct and the Schwarzschild solution
is wrong. Only the Nature can chose which is the correct one among
the physical hypotheses and, consequently which is the correct solution
for the external space-time geometry of a spherical static star. If
the Nature decides that the length of the circumference centred in
the origin of the coordinate system is $2\pi r$ the correct solution
is the Schwarzschild one. If the Nature decides that the length of
the circumference centred in the origin of the coordinate system is
$2\pi(r^{3}+r_{g}^{3})^{\frac{1}{3}}$ the correct solution is the
one of Eq. (\ref{eq: sol. gen.fin.}).
Notice that the space-time defined by the line element solution (\ref{eq: sol. gen.fin.})
is devoid of trapped surfaces. Thus, the Singularity Theorem in \cite{key-22}
\emph{cannot be applied.}
We also emphasize that a large distances, i.e. where $r_{g}\ll r,$
the proposed solution coincides with the standard Schwarzschild solution,
thus, both of the weak field approximation and the analysis of astrophysical
situations remain the same.
\section{Conclusion remarks}
In this paper an alternative to the famous Schwarzschild solution
to Einstein field equations has been analysed. The solution arises
from a different physical hypothesis, which assumes that that arches
of circumference centred in the origin of the coordinate system \emph{are
not Euclidean}. For the external geometry of a spherical static star
the Birkhoff Theorem is circumnavigated and the Black-Hole Singularities
are removed at the classical level, i.e. without arguments of quantum
gravity.
\subsubsection*{Acknowledgements }
I thank Herman Mosquera Cuesta for interesting discussions on Black
Hole physics. The Institute for Basic Research and the Associazione
Scientifica Galileo Galilei have to be thanked for supporting this
paper.
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