Black holes: a prediction of theory or phantasy?
A.A.\
Logunov*
M.A.\
Mestverishvili*
%
We argue for black holes do not represent a strict consequence of general relativity.
gr-qc0412058.tex
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\begin{document}
%\begin{flushright}
%\underline{27 ������ 2004 �.}
%\end{flushright}
\title[\small Black holes: a prediction of theory or phantasy?]
{Black holes: a prediction of theory or phantasy?}
\author{A.A.\,Logunov*, M.A.\,Mestverishvili*, %}
%\author{
V.V.\,Kiselev*}%\dag}%\ddag}
%%%%\fax{}
\address{*%\dag
\ Russian State Research Center ``Institute for High Energy
Physics'', %\\
Pobeda 1, Protvino, Moscow Region, 142281, Russia}
%\address{\dag\ %\ddag\
%Moscow Institute of Physics and Technology, Institutskii per. 9,
%Dolgoprudnyi Moscow Region, 141700, Russia}
%%%\date{}
\ead{kiselev@th1.ihep.su}
\begin{abstract}
We argue for black holes do not represent a strict consequence of
general relativity.
\end{abstract}

In Einstein general relativity a gravity is described by a metric
tensor $g_{\mu\nu}$ of pseudo-Riemannian space-time. An interval
of space-time has a generic form \be
ds^2=g_{\mu\nu}dx^\mu dx^\nu,\qquad\textrm {with the signature}\;(- 2).%(1)
\ee
 As usual, let us isolate time-like and space-like parts in (1)
 \be
ds^2=c^2d\tau^2-d\ell^2\,,%(2)
\ee
 with
 \be d\tau=\f{\,1\,}{\,c\,}
\f{g_{0\nu}dx^\nu}{\sqrt{g_{00}}}%(3)
\ee
 representing a physical time, while the expression
\be
d\ell^2=\varkappa_{ik}dx^i dx^k,\quad \varkappa_{ik}
=\f{g_{0i}g_{0k}}{g_{00}}-g_{ik}%(4}
\ee
 gives a square of space distance between two infinitely close
 points. The metric coefficients $g_{\mu\nu}$ have to satisfy well-known inequalities
 by Hilbert [1, 2].

A.Einstein especially noted [3]: ``Therefore, there is no way
except to recognize all thinkable\footnote{We will not concern for
some constraints caused by requirements of unambiguity %certainty
and continuity. (A.Einstein)} coordinate systems to be, in
principle, equivalent for the description of nature''. We will
further follow this statement by A.Einstein.

A motion of probe bodies takes place on time-like geodesics. In
the pseudo-Riemannian space-time, a time-like geodesic curve, by
definition, cannot become isotropic at a point. From the basic
statement of theory on the pseudo-Riemannian geometry of
space-time one unavoidably deduces that a physical solution of
gravitation equations at any time-like geodesic curve should
satisfy the following inequality:
 \be
v^2=\Bigl(\f{d\ell}{d\tau}\Bigr)^2<1\,.%(5)
\ee
 It means that the physical velocity cannot be equal to the speed
 of light at any point of trajectory (including those at
 infinity). The equality $v^2=1$ implies the interval becomes
 isotropic, i.e. it equals zero. \textbf{For a physical solution}
 of gravitation equations the time-like interval does not permit
 the physical velocity equal to the speed of light. The
 inequality of (5) should be valid in any system of coordinates.
 If it is broken in a system of coordinates (or reference frame)
 for a time-like geodesic curve even in a single point, then the
 whole solution contradicts to primary principles of the theory.
 The solution has the physical sense in the validity region of (5), only.

Let us consider the Schwarzschild solution. K.\,Schwarzschild
found the following exact spherically symmetric solution of
gravitation equation in January, 1916:
 \be ds^2=\Bigl(\f{r-\alpha}{r}\Bigr)dt^2
-\Bigl(\f{r}{r-\alpha}\Bigr)dr^2
-r^2(d\theta^2+\sin^2\theta d\phi^2)\,,%(6)
 \ee
where
 \be
\alpha=\f{2GM}{c^2}\,.%(7)
\ee
 In addition to the singularity at $r=0$ the solution has the
 singularity at $r=\alpha$. This singularity was called
 ``the Schwarzschild singularity'' or ``Hadamard catastrophe''.


Having considered a radial motion of probe particle on a time-like
geodesic curve in the Schwarzschild metric, in 1917 D.\,Hilbert
has obtained the equation [1]
 \be
\f{d^2r}{dt^2}-\f{3\alpha}{2r(r-\alpha)} \Bigl(\f{dr}{dt}\Bigr)^2
+\f{\alpha(r-\alpha)}{2r^3}=0\,,%(8)
\ee
 and found the integral of motion $A$,
  \be \Bigl(\f{dr}{dt}\Bigr)^2
=\Bigl(\f{r-\alpha}{r}\Bigr)^2
+A\Bigl(\f{r-\alpha}{r}\Bigr)^3\,,%(9)
\ee
 where $A$ is a constant determined by initial data.
 By (4) and (6) the physical velocity of particle is equal to
 \be v^2=\f{r^2}{(r-\alpha)^2}
\Bigl(\f{dr}{dt}\Bigr)^2\,.%(10)
\ee
 Substituting it to equation (9), we get
\be
v^2=1+A\f{r-\alpha}{r}\,.%(11)
\ee
 For the time-like geodesic, the constant
 $A$ can take values in the region
\be
-\f{r_0}{r_0-\alpha}\le A<0\,,%(12)
\ee
 where $r_0>\alpha$ or $r_0<0$, and $A=0$ corresponds to the light.

 From (11) we can see that on the radial time-like geodesics at $r=\alpha$
 the physical velocity equals the speed of light. Thus, the
 Schwarzschild solution at $r=\alpha$ breaks inequality (5),
 converting the \textbf{time-like} curve into the \textbf{isotropic} one.
 However, this fact is not consistent with the general relativity.
 Therefore such the solution is valid at $r>\alpha$, only.

 The analogous fact can be observed, if by the transformation
\be
t=\tau+2\sqrt{\alpha r}
+\alpha \ln\Bigl|\f{\sqrt{r}-\sqrt{\alpha}}{\sqrt{r}+\sqrt{\alpha}}\Bigr|%(13)
\ee
 we arrive to the interval in the form of Painlev\'e [4]
\be ds^2=\Bigl(\f{r-\alpha}{r}\Bigr)d\tau^2
+2\sqrt{\f{\,\alpha\,}{\,r\,}}\,dr d\tau - dr^2
-r^2(d\theta^2+\sin^2\theta d\phi^2)\,.%(14)
\ee
 Then, the physical velocity equals
\be
v^2=
\f{r^2\Bigl(\ds\f{dr}{d\tau}\Bigr)^2}
{\Bigl[(r-\alpha)+\sqrt{\alpha r}\ds\f{dr}{d\tau}\Bigr]^2}\,,%(15)
\ee
 so that, for instance, at $A=-1$ we get on the radial geodesic
curve
 \be \f{dr}{d\tau}=\sqrt{\f{\,\alpha\,}{\,r\,}},\quad
\f{dr}{d\tau}=-\sqrt{\f{\,\alpha\,}{\,r\,}}\Bigl(\f{r-\alpha}{r+\alpha}\Bigr)\,.%(16)
\ee
 The physical velocity of (15) becomes equal to
\be
v^2=\f{\,\alpha\,}{\,r\,}\,.%(17)
\ee
 It is again equal to the speed of light at $r=\alpha$, hence, the
 solution in this form contradicts to the primary fundamental
 statement of the theory in the form of (5).

In discussion devoted to the Schwarzschild singularity in 1939
A.\,Einstein wrote in [4]: ``\textit{The basic result of study is
the distinct understanding that there are no "Schwarzschild
singularities" in the real world. Thought the offered theory
considers only such systems, in which particles move by circle
trajectories, we have to hardly doubt that the investigation of
generic case will lead to the same results. The Schwarzschild
singularity is absent, because the matter cannot concentrated in
an arbitrary way; otherwise particles composing the clump will
reach the speed of light}''.

We have been convinced for this fact by having got eqs. (11) and
(17) for the physical velocity. By the opinion of A.\,Einstein,
such the singularity is not admissible in the vacuum, since it
contradicts the general relativity. However, if it does appear,
then the solution itself has a restricted or constrained sense. It
is valid at $r>\alpha$, only. The above caution by A.\,Einstein
remained with no due attention and analysis.

A further analysis of Schwarzschild solution went in the way of
searching for such coordinates, in which the metric coefficients
would be regular at $r=\alpha$. Making use of singular
transformations the corresponding procedure was performed. So, for
example, transforming to new coordinates $\tau, R$ by formulae of
[2, 6]
 \be \tau=t+\int\f{\sqrt{r}dr}{r-1},\quad
R=t+\int\f{\sqrt{r}\,r}{r-1}dr\,,%(18)
\ee
 (here we put \textbf{units} of $\ds{2GM}/{c^2}=1$)
interval (6) takes the form
 \be ds^2=d\tau^2
-\f{\,1\,}{\,r\,}(dR)^2
-r^2(d\theta^2+\sin^2\theta d\phi^2)\,,%(19)
\ee
 where
\be
r=\Bigl[\f{\,3\,}{\,2\,}(R-\tau)\Bigr]^{2/3}\,.%(20)
\ee
 In coordinates $\tau, R$, the singularity of metric coefficients
 is absent at $r=1$. The singularity is present at $r=0$.

For interval (19) the physical velocity is calculated by the
formula
 \be
v^2=\f{\,1\,}{\,r\,}\Bigl(\f{dR}{d\tau}\Bigr)^2\,.%(21)
\ee
 On the basis of (9), let us choose the radial motion
 at $A=-1$, when the velocity at the infinity is equal to zero:
\be
\f{dr}{dt}=\f{1}{\sqrt{r}}\Bigl(\f{r-1}{r}\Bigr)\,.%(22)
\ee
 By (18) and (22), we get
\[
\f{dt}{d\tau}=\f{r}{r+1}\,.
\]
 Using the above equality and (22), we find
\be
\f{dr}{d\tau}=\f{1}{\sqrt{r}}\Bigl(\f{r-1}{r+1}\Bigr)\,.%(23)
\ee
 Then, by (20) we derive
\be
\f{dR}{d\tau}=\f{2r}{r+1}\,.%(24)
\ee
 Substituting (24) into (21), for interval (19) we get the physical velocity

\be
v^2=\f{4r}{(r+1)^2}\,.%(25)
\ee
 Therefore, we see that in terms of $\tau, R$ the physical velocity
 at $r=1$ is equal to the speed of light, too, i.e. the time-like
 geodesic curve becomes isotropic at this point. Then,  by transformation
 (18) the Schwarzschild singularity at $r=1$ disappears in the
 geodesic radial motion
\[
\f{dr}{dt}=-\f{1}{\sqrt{r}}\Bigl(\f{r-1}{r}\Bigr)\,,
\]
 but it still remains on the geodesic curve
\[
\f{dr}{dt}=\f{1}{\sqrt{r}}\Bigl(\f{r-1}{r}\Bigr)\,.
\]
 It is absolutely evident that by singular transformations it
 could be removed from $r=1$, nevertheless, it will unavoidably
 appear at another point.

 Although in this case all of metric coefficients in (19) are
 regular outside the source, nevertheless, inequality (5) is broken
 on some geodesic curves. This fact witnesses that the Schwarzschild solution
 has a constrained sense in the general relativity, therefore, its
 continuation to other regions has a formal mathematical
 character. Then, black holes invented about fifty years ago in
 scientific articles represent a product of pure phantasy, since they have
 no solid theoretical ground.

Now consider singular transformation by Kruskal. The Kruskal
variables [7] are the followings:
 \ba
&&U=\sqrt{r-1}\,e^{r/2}\cosh\f{\,t\,}{\,2\,}\,,\nonumber\\[-4mm]
\label{26}\\
&&V=\sqrt{r-1}\,e^{r/2}\sinh\f{\,t\,}{\,2\,}\,.\nonumber
 \ea
 In the case of radial motion, interval (6) takes the form
  \be
ds^2=\f{\,4\,}{\,r\,}e^{-r}(dV^2-dU^2)\,.%(27)
\ee
 In the Schwarzschild interval the component $g_{00}$ is equal to
\[
g_{00}=\f{r-1}{r}\,,
\]
which gives its nullification at $r=1$, while in the Kruskal
coordinates of (27) the component $g_{00}$ equals
\[
g_{00}=\f{\,4\,}{\,r\,}e^{-r}
\]
 and it tends to zero at $r\to\infty$. For the radial motion on
 the time-like geodesic curve at $A=-1$
\be
\f{dr}{dt}=\f{1}{\sqrt{r}}\Bigl(\f{r-1}{r}\Bigr)\,.%(28)
\ee In terms of $U$ and $V$ the physical velocity equals
 \be
\f{dU}{dV}=\f{V\sqrt{r}+U}
{U\sqrt{r}+V}\,.%{29)
\ee
 In accordance with (28) after the integration we get, for
 instance,
\be
t=\beta(r)
+\ln\f{\sqrt{r}-1}{\sqrt{r}+1},\quad
\beta(r)=\f{\,2\,}{\,3\,}r^{3/2}+2\sqrt{r}\,.%(30)
\ee
 By (26) and (30) we deduce
\be
e^t=
e^\beta\Bigl(\f{\sqrt{r}-1}{\sqrt{r}+1}\Bigr)
=\f{U+V}{U-V}\,,%(31)
\ee
wherefrom we obtain
 \be \f{V}{U}= \f{e^\beta(\sqrt{r}-1)
-(\sqrt{r}+1)}
{e^\beta(\sqrt{r}-1) +(\sqrt{r}+1)}\,.%(32)
\ee
 Substituting (32) into (29), we find
  \be \f{dU}{dV}
=\f{\ds e^{\beta(r)}-1}
{\ds e^{\beta(r)}+1}\,.%(33)
\ee
 For the other radial geodesic motion
\be
\f{dr}{dt}=-\f{1}{\sqrt{r}}\Bigl(\f{r-1}{r}\Bigr)\,.%(34)
\ee
 In the same way we easily get the following expression for the
 physical velocity:
  \be \f{dU}{dV} =-\f{\ds
e^{\beta(r)}-1}
{\ds e^{\beta(r)}+1}\,.%(35)
\ee
 From (33) and (35) we deduce that at $r\to\infty$ the physical velocity
  on the geodesic time-like curve becomes equal to the speed of
  light, and the \textbf{time-like} geodesic curve becomes \textbf{isotropic}.

The Schwarzschild singularity of metric coefficients at the sphere
$r=1$ is formally cancelled by the Kruskal transformation.
Nevertheless, the solution of Hilbert--Einstein equations does not
become physical, since in accordance to the solution the physical
velocity became equal to the speed of light at infinity, which is
unacceptable. In the Schwarzschild variables the physical velocity
at infinity of $r$ is always less than the speed of light, while
at $r=1$ it is equal to the speed of light. In the Kruskal
coordinates the situation is slightly different: the physical
velocity equals the speed of light at infinity of $r$, while it is
always less than the speed of light at $r=1$. The presence of
Schwarzschild singularity is \textbf{not consistent} with the
general relativity. Thus, the \textbf{exact} spherically symmetric
solution of gravitation equations is \textbf{physically
inconsistent} in both the Schwarzschild coordinates and Kruskal
ones due to the presence of Schwarzschild \textbf{singularity},
since the solution breaks constraint (5). But the \textbf{concept}
of ``black holes'' is based on such the solution. In other words,
the Schwarzschild \textbf{singularity} led to ``black holes''. On
another hand, the presence of such the singularity contradicts to
the general relativity. Therefore, it is not true to
\textbf{accept} ``black holes'' as a \textbf{consequence} of
general relativity. The above analysis has shown that the
Schwarzschild singularity cannot be removed by coordinate
transformations, since it is related with the nullification of
interval $ds^2$. The singularity can be removed from the metric
coefficients, but not from the interval. Thus, we see that the
notion of ``black holes'' is based on the presence of
Schwarzschild singularity, which is in contradiction with the
basics of general relativity, i.e. the pseudo--Riemannian geometry
of space-time.

\vspace*{3mm}
 Finally, authors express a sincere gratitude to S.S.\,Gershtein,
V.A.\,Petrov, and N.E.\,Tyurin for valuable discussions.

%\newpage
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\end{document}


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gr-qc0412058.extracted_bib
\begin{thebibliography}{99}
\bibitem{1}
Hilbert D., Nachr. Ges. Wiss.  G\"{o}ttingen, Math-Phys. Klasse.
1916. \textbf{23}.
\bibitem{2}
Landau L.D., Lifshitz E.M., ``Field Theory'', Moscow: Fizmatlit,
%������ �.�., ����� �.�. ����� ����. �.: �����⫨�,
2001, p. 536.
\bibitem{22}
Einstein A., Collection of scientific papers, Moscow: Nauka, 1966,
\textbf{I}, p. 459 (in Russian).
\bibitem{3}
Painlev\'{e} P., Comptes rendus, Acad\'{e}mie des sciences. 1921,
 \textbf{173} (17), pp. 667--680. \\
See also Gravity. 2001, \textbf{5} (1), pp. 16--27 (in Russian).
\bibitem{4}
Einstein A., Collection of scientific papers, Moscow: Nauka, 1966,
\textbf{II}, p. 878 (in Russian).\\
See also  Einstein A., Ann. Math. 1939, \textbf{40}, pp. 922--936.
\bibitem{5}
Rylov Yu.A., Zh.E.T.P., 1961, \textbf{40}, pp. 1755--1757 (in
Russian).
\bibitem{6}
Kruskal M.D., Phys. Rev. 1960, \textbf{119}, 1743
\end{thebibliography}