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From: Leonid Verozub l.verozub@googlemail.com
To: gcnews@aoc.nrao.edu
Subject: submit NCverozub1.tex, Il Nouvo Cimento, submitted
% arXiv:0806.3744
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\title{On compact super-massive objects without event horizon}
\author{L.~Verozub\from{ins:x}}%\ETC}
%J.~Grey\from{ins:x},
%H.~McCoy\from{ins:x},
%R.~Drake\from{ins:x}\\
%W.~Worthington\from{ins:x}
% \atque
%Mr.~M\from{ins:evil}\thanks{The bad fellow.}}
\instlist{\inst{ins:x} Kharkov National University - Kharkov, Ukraine\\
E-mail: leonid.v.verozub@univer.kharkov.ua}
\PACSes{\PACSit{04.20.20.Cv}
\PACSit{04.50.+h}}%{\ldots}}
\begin{document}
\begin{abstract}
This paper aims to show a possibility of the existence of super-massive
compact objects with radii less than the Schwarzschild ones, which is one
of the principal consequences of the author's geodesic-invariant gravitation
equations. [Ann. Phys. (Berlin) , v. 17, p. 28 (2008)]. The physical
interpretation of solutions of the equations is based on the conclusion that
only
an aggregate "space-time geometry + used reference frame" has a physical
sense.
\end{abstract}
\maketitle
\section{Introduction}
\label{intro}
In Einstein's theory of gravitation space-time is relative in the sense that
the metric depends on the distribution of matter. However, long before the
Einstein theory Poincar\'{e} showed that geometry of space depends also
on the properties of measuring instruments. Only an aggregate "geometry +
measuring instruments" has a physical meaning, verifiable by experience.
After Minkowski it can be also said about geometry of space-time.
Some results of an attempt to actualize these ideas, and a generalization
of the vacuum Einstein's gravitation equations are considered in
\cite{Verozub08a}.
Such approach allows to consider gravitation both as a field in flat
space-time and as space-time curvature.
%In Einstein's theory of gravitation space-time is relative in the sense
that the metric depends on the distribution of matter. But owing to
Poincare we know that the geometry of space also depends on the properties
of measuring instruments, and, therefore, it has no meaning in itself. Only
a combination of "geometry + measuring instruments" has the physical sense,
verifiable from the experience.
The equations do not contradict the existing observations data. However, the
physical consequences resulting from them are radically different from the
ones of general relativity at distances of the order of the Schwarzschild
radius or less than that from a dot mass. It is very important that this
fact provides a natural explanation of modern data of the Universe
expansion.
However, this fact leads also to another important physical consequence,
which still has no confirmation.
% It is considered in this paper.
Observations give evidences for the existence of supermassive compact cold
objects in galactic centers \cite{Genzel}. Standard conditions of the
equilibrium of selfgravitating degenerate Fermi-gas forbid the existence
of very massive objects. For this reason they are usually identified with
black holes. However lack of the evidence of the existence of
an event horizon admits also other explanations of the nature of such
objects.
In \cite{Verozub96} the possibility of the existence of supermassive
equilibrium configurations of the degenerate Fermi-gas with radii less
than the Schwarzschild ones has been considered.
Such objects have no event horizon and are an alternative to the
hypothesis of the existence of black holes.
It is later, in \cite{Verozub06a}, some observable consequences of the
existence of such object in the Galaxy center have been considered.
However, it is seems that a theoretical justification of the existence of
such objects is insufficiently convincing for observers as it is based on
conclusions resulting from the author's gravitation equations considered
only in a brief note \cite{Verozub91}. In the present paper, being based on
the recent paper \cite{Verozub08a}
a simple and a clear justification of possibility of the existence of
such objects is given .
\section{Gravitation equations}
\label{sec:1}
%and \cite{RefJ}
%\subsection{Subsection title}
%\label{sec:2}
Unlike electrodynamics, Einstein's equations are not invariant with respect
to
a wide class of transformations of field variables in use
( Christoffel symbols or metric tensor) leaving equation of motion of test
particles (geodesic
lines) invariant \cite{Petrov}.
For example, all
Christoffel symbols $\overline{\Gamma}_{\beta\gamma}^{\alpha}(x) $
obtained by the transformations
\begin{equation}
\overline{\Gamma}_{\beta\gamma}^{\alpha}(x)=\Gamma_{\beta\gamma}^{\alpha
}(x)+\delta_{\beta}^{\alpha}\ \phi_{\gamma}(x)+\delta_{\gamma}^{\alpha}%
\phi_{\beta}(x),
\label{TransformChristoffels}%
\end{equation}
where $\phi_{\beta}(x)$ are an arbitrary differentiable vector-function,
describe the same gravitational field because the geodesic equations%
\begin{equation}
\ddot{x}^{\alpha}+(\Gamma_{\beta\gamma}^{\alpha}-c^{-1}\Gamma_{\beta\gamma
}^{0}\dot{x}^{\alpha})\dot{x}^{\beta}\dot{x}^{\gamma}=0\label{GeodeicEqs}%
\end{equation}
remain invariant under transformation (\ref{TransformChristoffels}) in any
given coordinate system. (The points denote differentiation with respect to
$t=x^{0}/c$, $c$ is speed of light ).
However, Einstein's equations are not invariant under such transformations
because the Ricci tensor is transformed under (\ref{TransformChristoffels})
as
follows
\begin{equation}
\overline{R}_{\alpha\beta}=R_{\alpha\beta}-\phi_{\alpha\beta},
\end{equation}
where $\phi_{\alpha\beta}=\phi_{\alpha;\beta}-\phi_{\alpha}\phi_{\beta}$,
and
$\phi_{\alpha;\beta}$ is a covariant derivative with respect to
$x^{\alpha}$.
Each transformation (\ref{TransformChristoffels})
induces some mapping $g_{\alpha\beta} \rightarrow
\overline{g}_{\alpha\beta}$ of metric tensor $g_{\alpha\beta}$ which
can be obtained by solving some partial differential equation
\cite{Eisenhart}, \cite{Verozub08a}.
Such mappings leave the equations of motion of test particles invariant,
and, consequently, all metric tensors resulting from a given
$g_{\alpha\beta}$ by a geodesic transformation describe the same
gravitational field.
Thus, such transformations are gauge transformations of the tensor
$g_{\alpha\beta}$.
%This means that all tensors resulting from a given $g_{\alpha\beta}$ by a
geodesic transformation describe the same gravitational field.
%( Similarly, as well as 4-potential in classical electrodynamics).
%This means the the tensor $g_{\alpha\beta}$, as a
%is defined up to arbitrary geodesic transformations . (The same as
4-potentail in classical electrodynamics. is defined up to arbitrary
gradient transformations).
For this reason, one can suppose that in any gravitational theory, based
on Einstein's hypothesis of the motion of test particles along geodesic
lines, only geodesic-invariant objects can have a physical sense.
The simplest geodesic-invariant tensor $B^{\alpha}_{\beta\gamma}$ can be
formed as follows \cite{Verozub08a}:
\begin{equation}
B_{\beta\gamma}^{\alpha}=\Pi_{\beta\gamma}^{\alpha}-\overset{\circ}{\Pi
}_{\beta\gamma}^{\alpha}%
\end{equation}
where
\begin{equation}
\Pi_{\alpha\beta}^{\gamma}=\Gamma_{\alpha\beta}^{\gamma}-(n+1)^{-1}\left[
\delta_{\alpha}^{\gamma}\Gamma_{\beta}+\delta_{\beta}^{\gamma}\Gamma_{\alpha
}\right] ,\;
\end{equation}
and
\begin{equation}
\overset{\circ}{\Pi}_{\alpha\beta}^{\gamma}=\overset{\circ}{\Gamma}%
_{\alpha\beta}^{\gamma}-(n+1)^{-1}\left[ \delta_{\alpha}^{\gamma}%
\overset{\circ}{\Gamma}_{\beta}+\delta_{\beta}^{\gamma}\overset{\circ}{\Gamma
}_{\alpha}\right],
\end{equation}
are the Thomas symbols for the Riemannian
space-time $V_{4}$ and for the Minkowski space-time $E_{4}$
in an used coordinate system, respectively.
The Thomas symbols are formed
by the Christoffel symbols $\Gamma_{\alpha\beta}^{\gamma}$ and $\overset
{\circ}{\Gamma}_{\alpha\beta}^{\beta}$ of $V_{4}$ and $E_{4}$, respectively,
$\Gamma_{\alpha}=\Gamma_{\alpha\beta}^{\beta}$, and $\overset{\circ}{\Gamma
}_{\alpha}=\overset{\circ}{\Gamma}_{\alpha\beta}^{\beta}$.
In paper \cite{Verozub08a} a theory in which geodesic mappings play a role
of gauge transformations is considered. An investigation of the problem
leads
to the conclusion that is of interest to explore physical consequences
from the following bimetric geodesic-invariant generalization of Einstein's
vacuum equations \footnote{Greek indexes run from $0$ to $3$} :
\begin{equation}
\nabla_{\alpha}B_{\beta\gamma}^{\alpha}-B_{\beta\delta}^{\epsilon}%
B_{\epsilon\gamma}^{\delta}=0.
\label{MyVacuumEqs}%
\end{equation}
The symbol $\nabla_{\alpha}$ denotes a covariant derivative in $E_{4}$ with
respect to $x^{\alpha}$.
%It should be expected that in any completely correct geometrical theory of
%gravitation based on the hypothesis of motion of test particles along
geodesic
%ines, mappings (\ref{TransformChristoffels}) (named usually geodesic
%mappings) are transformations which in any given coordinate system
%should play a role of
%gauge transformations.
%, and differential equations for the field
%variables of gravitational field must be invariant with respect to such
%mappings.
%It seems from this point of view that
%the Einstein equations rather describe gravitation at some specific
gauge
%condition, and do not show all set of possible solutions in a given
%coordinate system.
The derivation of basic equations, they physical interpretations, and a
correct utilization are possible only owing to reconsider of a deep problem
of space-time relativity with respect to measuring instruments, going back
to Poincar\'{e} fundamental ideas. According to \cite{Verozub08a} both the
space-time, $E_{4}$ and $V_{4}$, have physical meaning since only
aggregate ``space-time geometry + frame of reference'' have a physical
meaning.
We can consider space-time as the Minkowski one in an inertial reference
frames \footnote{By inertial reference frame we mean the frame in which
Newton's first and second laws are obeyed globally.}
, and -- is a Riemannian with curvature other than zero -- in so-called
proper reference frames of the field, the body of reference of which is
formed by particles moved in this gravitational field. For purpose of this
paper it is essentially only the important conclusion that an observer,
which explores gravitational field of a distant massive compact object in an
inertial reference frame, can consider global space-time as the Minkowski
one.
In order to see better the relationship between (\ref{MyVacuumEqs}) and the
Einstein vacuum equation, it should be noted that we can select a gauge
condition for the Christoffel symbols as follows:
\begin{equation}
Q_{\alpha}=\Gamma_{\alpha\beta}^{\beta}-\overset{\circ}{\Gamma^{\beta}%
}_{\alpha\beta}=0.\label{AdditionalConditions}%
\end{equation}
At such covariant gauge condition eqs. (\ref{MyVacuumEqs}) coincide with
the vacuum Einstein's equations.
( At least locally).
Therefore,
the observer which is located in an inertial reference frame far away from
a compact object can describe gravity of the object field
by the spherically-symmetric solution of the vacuum
Einstein equations in the Minkowski space-time (in which
$g_{\alpha\beta}(x)$
is simply a tensor field) at the additional condition $Q_{\alpha}=0$ .
Such equations somewhat resemble the equations of classical electrodynamics
for 4-potential together with some specific gauge conditions.
It is that will be used in the next section.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Gravitational Energy of a matter sphere }
\label{sec:2}
If a matter sphere is considered as formed by means of consecutive injection
of thin spherically-symmetric layers of the mater from infinity
\cite{Landau}, an equation for finding of the
gravitational energy of the sphere can be found by using
the vacuum solution of the gravitational equations \cite{Verozub08a}
for a dot source.
%Starting from the peculiarities of motion of test particles in the above
theory, we give here
%a simple argumentation of the possibility of the existence of compact
supermassive objects without event horizon
%which perhaps can be identified with the ones in the galactic centres.
The differential equations of the motion of a test particle in the
spherically-symmetric gravitational field of a dot mass $M$ from the point
of view of a distant observer
can be found from the Lagrangian
\begin{equation}
L=-m\,c\,[A\dot{r}^{2}+B(\dot{\theta}^{2}+\sin^{2}\theta\ \dot{\varphi}%
^{2})-c^{2}C]^{1/2} \label{LagrTestParticls},%
\end{equation}
where $(t,r,\varphi)$ are the spherical coordinates,
$m$ is the particle mass,
$A$, $B$ and $C$ are the functions of the radial coordinate $r$.
The equations of the motion of a test particle in the field of a point mass
$M$, based on the
solution of the Einstein equations under the condition $Q_{\alpha}=0$ in
Minkowski space-time, are of the form \cite{Verozub08a}
\begin{equation}
{\dot{r}}^{2}=(c^{2}C/A)[1-(C/\overline{E}^{2})(1+r_{g}^{2}\overline{J}%
^{2}/f^{2})], , \label{EqsMotionTestPart1}%
\end{equation}%
\begin{equation}
\dot{\varphi}=c\;C\overline{J}/r_{g} f^{2}\overline{E} ,
\label{EqsMotionTestPart2}%
\end{equation}
where $f=(r_{g}^{3}%
+r^{3})^{1/3}$, $C=1-rg/f$ , $\dot{r}=dr/dt$, $\dot{\varphi}=d\varphi/dt$ ,
$\overline{E}=E/mc^{2}$, $\overline{J}=J/r_{g}mc$, $E$ and $J$ are the
energy
and angular momentum of the particle, $r_{g}=2 G M/c^{2}$ is the
Schwarzschild radius of the central object.,
$G$ is the gravitational constant.
If the radial distance $r$ from the dot massive object is many larger than
the
Schwarzschild radius $r_{g}$ of the object , physical consequences following
from the (\ref{MyVacuumEqs}) are very close to the ones resulting from
Einstein's equations. However, they are very different when $r$ is of the
order of $r_{g}$, or less than that.
From point of view of a distant observer free-falling particles move up to
the center.
The spherically-symmetric solution has no event horizon \cite{Verozub08a},
and this fact it is not a consequence of some specific coordinate system.
It follows from eq. (\ref{EqsMotionTestPart1}) that at $J=0$ the energy
of a rest particle at the distance $r$ from the center is :
\begin{equation}
E=mc^{2}\sqrt{C}.
\end{equation}
At the condition $\overset{\_}{r}=r/r_{g}\ll1,$ it yields
\begin{equation}
E\simeq mc^{2}-\frac{GMm}{r}.
\end{equation}
If $\overset{\_}{r}=r/r_{g}\gg1,$we obtain in first order to $\overset{\_}%
{r}^{-1}:$%
\begin{equation}
E\approx \frac{mc^{2}r^{3/2}}{2\sqrt{6}G^{3/2}M^{3/2}}.
\end{equation}
In this case the gravitational energy of the test particle tends to zero
when $r\rightarrow 0$.
Consequently, the difference between gravitational energy of the thin
matter layer of mass $\delta m$ at the distance $r$ from the center and
the one at infinity is
$\delta m\, c^{2}\sqrt{C}-\delta m\, c^{2}$. The
gravitational energy of a material sphere, formed by means of consecutive
addition of thin spherically-symmetric layers of the mater \cite{Landau}
is equal to
\begin{equation}
\mathcal{E}_{g}=-\int(1-\sqrt{C})dm,\label{Egr_integral}%
\end{equation}
where $dm=4\pi\rho r^{2}dr,$and $\rho$ is the mass-energy density.
If the matter sphere is an isentropic ideal fluid in equilibrium state,
(\ref{Egr_integral}) can be transformed to more useful form. The
equilibrium
condition of the ideal isentropic fluid is of the form \cite{Verozub08b}
\begin{equation}
(\varkappa^{2}C)^{\prime}=0,
\end{equation}
where prime denotes differentiation with respect to $r$, $\varkappa=w/\rho
_{0}c,$, $\rho_{0}\approx m_{n} n$ is matter rest-density, where $m_{n}$ is
the neutron mass and $n$ is the particles number density,
$w$ is the enthalpy per unit
volume. Therefore, we can use the following equation of the sphere
equilibrium:%
\begin{equation}
\varkappa^{2}C=C_{0},\label{EquilibriumEquationSphere}%
\end{equation}
where
\begin{equation}
\varkappa=1+\frac{P}{\rho_{0}c^{2}},
\end{equation}
$P$ is the presure,%
\begin{equation}
C_{0}=1-\frac{r_{g}}{^{(r_{g}^{3}+R^{3})^{1/3}}},
\end{equation}
where $r_{g}=2GM/R$ is the Schwarzschild radius of the sphere.
By using (\ref{EquilibriumEquationSphere}) eq. (\ref{Egr_integral}) can be
written as follows:%
\begin{equation}
\mathcal{E}_{g}=-Mc^{2}+c^{2}\sqrt{C_{0}}\int\varkappa^{-1}dm.
\end{equation}
For estimates we can use it in the more simple form%
\begin{equation}
\mathcal{E}_{g}=-Mc^{2}+Mc^{2}\overset{\_}{\varkappa}^{-1}\sqrt{C_{0}},
\end{equation}
where $\overset{\_}{\varkappa}$ is an averaged value of $\varkappa$ over the
sphere.
As $r_{g}/R\ll1$б the functions $C_{0}\approx 1- r_{g}/R$, and
$\varkappa^{-1} \approx 1-P/\rho_{0} c^{2}$ because usually
$P/\rho_{0}c^{2}\ll1$
By using a polytropic equation of the fluid state we obtain that
the gravitational energy of
the object with the radius $R\gg r_{g}$ is
\begin{equation}
\mathcal{E}_{g}=-\frac{GM^{2}}{R}-\int PdV,
\end{equation}
where $dV$ is a volume element. The gravitational energy is very small as
compared with full energy $Mc^{2}$ of the object
For the object with the radius $R