Title
Title
We briefly outline several main results concerning various new physically relevant features found in gravity -- both ordinary Einstein or $f(R)=R+R^2$ % (Starobinsky) gravity in the first-order formalism, coupled to a special kind of nonlinear electrodynamics containing a square-root of the standard Maxwell Lagrangian and known to produce charge confinement in flat spacetime.
BJP-grav+NED.tex
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\title{Dynamical Couplings and Charge Confinement/Deconfinement from Gravity 
Coupled to Nonlinear Gauge Fields \thanks{Talk at Second Bulgarian National 
Congress in Physics, Sept. 2013}}

\begin{start}
\author{E.~Guendelman}{1}, \coauthor{A.~Kaganovich}{1},
\coauthor{E.~Nissimov}{2}, \coauthor{S.~Pacheva}{2}

\address{Department of Physics, Ben-Gurion Univ. of the Negev,
Beer-Sheva 84105, Israel}{1}

\address{Institute of Nuclear Research and Nuclear Energy,
Bulg. Acad. Sci., Sofia 1784, Bulgaria}{2}

\runningheads{E.~Guendelman, A.~Kaganovich, E.~Nissimov, S.~Pacheva}{Dynamical 
Couplings and Charge Confinement/Deconfinement $\ldots$}
% from Gravity Coupled to  Nonlinear Gauge Fields}

\received{}

\begin{Abstract}
We briefly outline several main results concerning various new physically relevant
features found in gravity -- both ordinary Einstein or $f(R)=R+R^2$ % (Starobinsky) 
gravity in the first-order formalism, coupled to a special kind of nonlinear 
electrodynamics containing a square-root of the standard Maxwell Lagrangian and known 
to produce charge confinement in flat spacetime.
\end{Abstract}

\PACS{04.50.-h,04.70.Bw,11.25.-w}
%%%%%%%%%%%%%%%%%%%%%%
% 11.25.-w: strings and branes
% 04.70.Bw: classical black holes
% 04.50.-h: higher-dimensional gravity and other theories of gravity
%%%%%%%%%%%%%%%%%%%%%%
% 11.27.+d: extended classical solutions; cosmic strings, domain walls, texture
% 04.70.Bw: classical black holes
% 04.30.Nk: wave propagation & interactions
%%%%%%%%%%%%%%%%%%%%%%

\end{start}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section[]{Introduction}

G. 't Hooft \ct{thooft} has shown that in any effective quantum gauge theory, which is 
able to describe QCD-like charge confinement phenomena, the energy density of 
electrostatic field configurations should be a linear function of the electric 
displacement field in the infrared region (the latter appearing as a quantum
``infrared counterterm''). The simplest way to realize these ideas in flat spacetime
is to incorporate into the full gauge field action an additional term being a 
square-root of the standard Maxwell (or Yang-Mills) gauge field Lagrangian 
\ct{GG-1,GG-2,GG-3}:
\br
S = \int d^4 x L(F^2) \quad ,\quad
L(F^2) = -\frac{1}{4} F^2 - \frac{f_0}{2} \sqrt{-F^2} \; ,
\lab{GG-flat} \\
F^2 \equiv F_{\m\n} F^{\m\n} \quad ,\quad 
F_{\m\n} = \pa_\m A_\n - \pa_\n A_\m  \; .
\nonu
\er
The ``square-root'' Maxwell term is naturally produced as a result of 
spontaneous breakdown of scale symmetry in the standard gauge theory \ct{GG-1}.
Moreover, the (dimensionfull) coupling constant $f_0$ measures the strength
of the effective confining potential among quantized fermions produced by
\rf{GG-flat} \ct{GG-2}.

In a series of recent papers \ct{our-main-1,our-main-2} we have studied in detail
physically more 
interesting models where the above nonlinear ``square-root electrodynamics'' 
couples to gravity --� either standard Einstein gravity or generalized 
$f(R)$-gravity ($f(R)$ being a nonlinear function of the scalar curvature of 
space-time), as well as coupled to scalar dilaton field. Let us recall that 
$f(R)$-gravity models are attracting a lot of interest in modern cosmology as 
possible candidates to cure problems in the standard cosmological scenarios 
related to dark matter and dark energy. For a recent review, see 
\textsl{e.g.} \ct{f(R)-grav} and references therein. The first $R+R^2$-model 
(in the second-order formalism) which was also the first inflationary model, 
was proposed by Starobinsky in \ct{starobinsky}.

Here we will briefly describe some of our main results \ct{our-main-1,our-main-2} 
concerning the new physically relevant features we uncovered in the coupled 
gravity/``square-root'' nonlinear gauge field/dilaton system (defined in 
Eq.\rf{f-gravity+GG+D} below):

(i) Appearance of dynamical effective gauge couplings and confinement-deconfinement 
transition effect as functions of the dilaton vacuum expectation value, in
particular due to appearance of ``flat'' region of the effective dilaton
potential.

(ii) New mechanism for dynamical generation of cosmological constant.

(iii) Non-standard black hole solutions with constant vacuum radial electric field 
with Reissner-Nordstr{\"o}m-(anti)de-Sitter or Schwarzschild-(anti)de-Sitter type 
geometry and with non-asymptotically flat ``hedgehog''-type spacetime asymptotics.
Let us stress that constant vacuum radial electric fields do not exist as
solutions of ordinary Maxwell electrodynamics.

(iv) The above non-standard black holes obey the first law of black hole 
thermodynamics.

(v) New ``tube-like universe''� solutions of Levi-Civita-Bertotti-Robinson tye
\ct{LC-BR};

(vi) Coupling to {\em lightlike} branes produces {\em �charge-hiding}� and 
{\em charge-confining} ``thin-shell'' wormhole solutions displaying QCD-like 
charge confinement (see also the previous talk \ct{BJP-LL-branes} at this conference).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \section{ Gravity Coupled to Confining Nonlinear Gauge Field}
\section{$R+R^2$-Gravity Coupled to Confining Nonlinear Gauge Field}

Let us consider coupling of $f(R)= R + \a R^2$ gravity (possibly
with a bare cosmological constant $\L_0$) to a ``dilaton'' $\phi$ and
the nonlinear gauge field system containing $\sqrt{-F^2}$ \rf{GG-flat} (we
are using units with the Newton constant $G_N=1$):
\br
S = \int d^4 x \sqrt{-g} \Bigl\lb \frac{1}{16\pi} 
\Bigl( f\bigl(R(g,\G)\bigr) - 2\L_0 \Bigr) + L(F^2(g)) + L_D (\phi,g) \Bigr\rb \; ,
\lab{f-gravity+GG+D} \\
f\bigl(R(g,\G)\bigr) = R(g,\G) + \a R^2(g,\G) \quad ,\quad 
R(g,\G) = R_{\m\n}(\G) g^{\m\n} \; ,
\lab{f-gravity} \\
L(F^2(g)) = - \frac{1}{4e^2} F^2(g) - \frac{f_0}{2} \sqrt{- F^2(g)} \; ,
\lab{GG-g} \\
F^2(g) \equiv F_{\k\l} F_{\m\n} g^{\k\m} g^{\l\n} \;\; ,\;\;
F_{\m\n} = \pa_\m A_\n - \pa_\n A_\m \;
\lab{F2-g} \\
L_D (\phi,g) = -\h g^{\m\n}\pa_\m \phi \pa_\n \phi - V(\phi) \; .
\lab{L-dilaton}
\er
$R_{\m\n}(\G)$ is the Ricci curvature in the first order (Palatini) formalism, 
\textsl{i.e.}, the space-time metric $g_{\m\n}$ and the affine connection 
$\G^\m_{\n\l}$ are \textsl{a priori} independent variables. The solution to the
corresponding equation of motion w.r.t. $\G^\m_{\n\l}$ --
$\nabla_\l \(\sqrt{-g} f^{\pr}_R g^{\m\n}\) = 0$ -- implements transition 
to the ``physical'' Einstein-frame metrics $h_{\m\n}$ via conformal rescaling of 
the original metric $g_{\m\n}$:
\be
g_{\m\n} = \frac{1}{f^{\pr}_R} h_{\m\n} \quad ,\quad
\G^\m_{\n\l} = \h h^{\m\k} \(\pa_\n h_{\l\k} + \pa_\l h_{\n\k} - \pa_\k h_{\n\l}\)
\; .
\lab{einstein-frame}
\ee
Here $f^{\pr}_R \equiv \frac{df(R)}{dR} = 1 + 2\a R(g,\G)$.

As shown in \ct{our-main-2}, using \rf{einstein-frame} the original $R+R^2$-gravity 
equations of motion resulting from \rf{f-gravity+GG+D} can be
rewritten in the form of {\em standard} Einstein equations:
\be
R^\m_\n (h) = 8\pi \({T_{\rm eff}}^\m_\n (h) - \h \d^\m_\n {T_{\rm eff}}^\l_\l (h)\)
\lab{einstein-h-eqs}
\ee
with effective energy-momentum tensor of the following form:
\be
{T_{\rm eff}}_{\m\n} (h) = h_{\m\n} L_{\rm eff} (h) 
- 2 \partder{L_{\rm eff}}{h^{\m\n}} \; .
\lab{T-h-eff}
\ee
The pertinent effective ``Einstein-frame'' matter Lagrangian reads:
\br
L_{\rm eff} (h) = - \frac{1}{4 e_{\rm eff}^2 (\phi)} F^2(h) 
- \h f_{\rm eff} (\phi) \sqrt{- F^2(h)} 
\nonu \\
+ \frac{X(\phi,h)\bigl(1+16\pi\a X(\phi,h)\bigr) - V(\phi) -\L_0/8\pi
}{1+8\a\( 8\pi V(\phi)+\L_0\)}
\lab{L-eff-h}
\er
with the following dynamical $\phi$-dependent couplings:
\br
\frac{1}{e_{\rm eff}^2 (\phi)} = \frac{1}{e^2} + 
\frac{16\pi\a f_0^2}{1 + 8\a \(8\pi V(\phi) + \L_0 \)} \; ,
\lab{e-eff} \\
f_{\rm eff}(\phi)=f_0 \frac{1+32\pi\a X(\phi,h)}{1 + 8\a\(8\pi V(\phi)+\L_0\)}
\; .
\lab{f-eff}
\er
The dilaton kinetic term $X(\phi,h) \equiv -\h h^{\m\n}\pa_\m \phi \pa_n \phi$ 
will be ignored in the sequel.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Dynamical Couplings and Confinement/Deconfinement in $R+R^2$ 
Gravity}

In what follows we consider constant ``dilaton'' $\phi$ extremizing the effective
Lagrangian \rf{L-eff-h}. Here we observe an interesting feature of \rf{L-eff-h} --
the dynamical couplings and effective potential are extremized 
{\em simultaneously}, which is an explicit realization of the so called 
``least coupling principle'' of Damour-Polyakov \ct{damour-polyakov}:
\be
\partder{f_{\rm eff}}{\phi} = - 64\pi\a f_0 \partder{V_{\rm eff}}{\phi}
\;\; ,\;\; \partder{}{\phi}\frac{1}{e_{\rm eff}^2} =
-(32\pi\a f_0)^2 \partder{V_{\rm eff}}{\phi} 
\;\; \to \partder{L_{\rm eff}}{\phi} \sim \partder{V_{\rm eff}}{\phi} \; ,
\lab{f-e-extremize}
\ee
where:
\be
V_{\rm eff}(\phi) = 
\frac{V(\phi) + \frac{\L_0}{8\pi}}{1+8\a\(8\pi V(\phi)+\L_0\)} \; .
\lab{V-eff}
\ee
Thus, at a constant extremum of $L_{\rm eff}$ \rf{L-eff-h} $\phi$ must satisfy:
\be
\partder{V_{\rm eff}}{\phi} = 
\frac{V^{\pr}(\phi)}{\llb 1+8\a\( 8\pi V(\phi)+\L_0\)\rrb^2} = 0 \; .
\lab{V-extremum}
\ee
There are two generic cases:

(a) {\em Confining phase}: Eq.\rf{V-extremum} is satisfied for some 
finite-value $\phi_0$ extremizing the original ``bare'' dilaton potential $V(\phi)$: 
$V^{\pr}(\phi_0) = 0$.

(b) {\em Deconfinement phase}: For polynomial or exponentially growing ``bare'' 
potential $V(\phi)$, so that $V(\phi) \to \infty$ when $\phi \to \infty$, we have:
\be
\partder{V_{\rm eff}}{\phi} \to 0 \quad ,\quad 
V_{\rm eff} (\phi) \to \frac{1}{64\pi\a} = {\rm const} \quad {\rm when} \;\;
\phi \to \infty \; ,
\lab{flat-region}
\ee
\textsl{i.e.}, for sufficiently large values of $\phi$ we find a ``flat region''
in $V_{\rm eff}$. This ``flat region'' triggers a {\em transition from 
confining to deconfinement dynamics}.

Namely, in the ``flat-region'' phase (b) we have 
$f_{\rm eff} \to 0 \; ,\; e^2_{\rm eff} \to e^2$,
% \be
% f_{\rm eff} \to 0 \quad ,\quad e^2_{\rm eff} \to e^2
% \lab{deconfine}
% \ee
and the effective gauge field Lagrangian in \rf{L-eff-h} reduces to the ordinary
\textsl{non-confining} one (the ``square-root'' Maxwell term $\sqrt{-F^2}$ vanishes):
\be
L^{(0)}_{\rm eff} = -\frac{1}{4e^2} F^2(h) - \frac{1}{64\pi\a}
\lab{L-eff-h-0}
\ee
with an {\em induced} cosmological constant $\L_{\rm eff} = 1/8\a$, which is
{\em completely independent} of the bare cosmological constant $\L_0$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Non-Standard Black Holes and ``Tube-Like'' Universes}

In the {\em confining phase} (phase (a) above: 
$\phi_0$ -- generic minimum of the effective dilaton potential; 
$e_{\rm eff}(\phi))$, $f_{\rm eff}(\phi)$ as in \rf{e-eff}--\rf{f-eff})
we obtain several physically interesting 
solutions of the Einstein-frame gravity Eqs.\rf{einstein-h-eqs} due to the special 
form of the effective Einstein-frame matter Lagrangian \rf{L-eff-h}.

First, we find \textsl{non-standard} Reissner-Nordstr{\"o}m-(anti-)de-Sitter-type
black holes with metric:
% carrying an additional {\em constant vacuum radial electric field}
% apart from the Coulomb one: 
\br
ds^2 = - A(r) dt^2 + \frac{dr^2}{A(r)} + r^2 \bigl(d\th^2 + \sin^2 \th d\vp^2\bigr)
\; ,
\lab{spherical-static-2} \\
A(r) = 1 - \sqrt{8\pi}|Q|f_{\rm eff}(\phi_0) e_{\rm eff}(\phi_0) 
- \frac{2m}{r} + \frac{Q^2}{r^2} - \frac{\L_{\mathrm{eff}}(\phi_0)}{3} r^2 \; ,
\lab{CC-eff-2}
\er
where $\L_{\rm eff}(\phi_0)$ is a {\em dynamically generated} cosmological constant:
\be
\L_{\rm eff}(\phi_0) = \frac{\L_0 +8\pi V(\phi_0)+2\pi e^2 f^2_0}{
1+8\a\(\L_0 +8\pi V(\phi_0)+2\pi e^2 f^2_0\)} \; .
\lab{h-CC-eff}
\ee
The black hole's static radial electric field contains
apart from the Coulomb term an {\em additional constant ``vacuum'' piece}:
% responsible for the confinement property:
\br
|F_{0r}| = |\vec{E}_{\rm vac}| -
\frac{Q}{\sqrt{4\pi}\, r^2} \Bigl(\frac{1}{e^2} + 
\frac{16\pi\a f_0^2}{1 + 8\a\(8\pi V(\phi_0) + \L_0 \)}\Bigr)^{-\h}
% \frac{1}{\sqrt{2}}f_{\rm eff}(\phi_0) e^2_{\rm eff}(\phi_0) 
% + \frac{Q}{\sqrt{4\pi}\, r^2} e_{\rm eff}(\phi_0) \; ,
\lab{cornell-sol-2} \\
|\vec{E}_{\rm vac}| \equiv 
% \frac{1}{\sqrt{2}}f_{\rm eff}(\phi_0) e^2_{\rm eff}(\phi_0) 
\Bigl(\frac{1}{e^2} + 
\frac{16\pi\a f_0^2}{1 + 8\a\(8\pi V(\phi_0) + \L_0 \)}\Bigr)^{-1}
\frac{f_0/\sqrt{2}}{1+8\a\(8\pi V(\phi_0)+\L_0\)} \; .
\lab{vacuum-radial}
\er
Let us emphasize again that constant vacuum radial electric fields do not exist as
solutions of ordinary Maxwell electrodynamics.

In the special case $\L_{\rm eff}(\phi_0)=0$ we obtain a non-standard
Reissner-Nordstr{\"o}m-type black hole with a ``hedgehog'' 
{\em non-flat-spacetime} asymptotics (cf. Refs.\ct{hedgehog}:
$A(r) \to 1 -\sqrt{8\pi}|Q|f_{\rm eff}(\phi_0) e_{\rm eff}(\phi_0) \neq 1$
for $ r\to\infty$.
% \be
% A(r) \to 1 -\sqrt{8\pi}|Q|f_{\rm eff}(\phi_0) e_{\rm eff}(\phi_0) \neq 1 
% \;\; {\rm for} \; r\to\infty \; .
% \lab{hedgehog}
% \ee

Apart from non-standard black hole solutions we also obtain ``tube-like''
space-time solutions of the Einstein-frame gravity Eqs.\rf{einstein-h-eqs},
which are of Levi-Civitta-Bertotti-Robinson type (cf. Refs.\ct{LC-BR}) with 
geometries $AdS_2 \times S^2$, $Rind_2 \times S^2$ and $dS_2 \times S^2$,
where $AdS_2$, $Rind_2$ and $dS_2$ are two-dimensional anti-de Sitter,
Rindler and de Sitter space, respectively. The corresponding metric is of the form:
\be
ds^2 = - A(\eta) dt^2 + \frac{d\eta^2}{A(\eta)} 
+ r_0^2 \bigl(d\th^2 + \sin^2 \th d\vp^2\bigr) \;\; ,\;\;  %\quad ,\quad
-\infty < \eta <\infty \; ,
\lab{gen-BR-metric-2}
\ee
carrying constant vacuum ``radial'' electric field $|F_{0\eta}|=|\vec{E}_{\rm vac}|$.
The radius of the spherical factor $S^2$ is given by 
(using short-hand $\L(\phi_0)\equiv 8\pi V(\phi_0) + \L_0$):
\be
\frac{1}{r_0^2} = \frac{4\pi}{1+8\a\L(\phi_0)}\Bigl\lb
\Bigl(1+8\a\(\L(\phi_0)+2\pi f_0^2\)\Bigr) \vec{E}_{\rm vac}^2 +
\frac{1}{4\pi}\L(\phi_0)\Bigr\rb \; ,
\lab{r0-eq}
\ee
and the metric coefficient in \rf{gen-BR-metric-2} reads:
$A(\eta) = 4\pi K(\vec{E}_{\rm vac}) \eta^2 \;,\; K(\vec{E}_{\rm vac}) >0$ for $AdS_2$;
$A(\eta) = \pm \eta$ for $\eta \in (0,+\infty)$ or  $\eta \in (-\infty,0)$
for $Rind_2$;
$A(\eta) = 1 - 4\pi |K(\vec{E}_{\rm vac})|\,\eta^2\; ,\;  K(\vec{E}_{\rm vac}) <0$ 
for $dS_2$, using notation: \\
$K(\vec{E}) \equiv \Bigl(1+8\a\(\L(\phi_0)+2\pi f_0^2\)\Bigr) \vec{E}^2 
- \sqrt{2}f_0 |\vec{E}| - \frac{1}{4\pi}\L(\phi_0)$.
% \be
% K(\vec{E}) \equiv \Bigl(1+8\a\(\L(\phi_0)+2\pi f_0^2\)\Bigr) \vec{E}^2 
% - \sqrt{2}f_0 |\vec{E}| - \frac{1}{4\pi}\L(\phi_0) \; .
% \lab{K-def}
% \ee

To conclude, let us particularly stress that all physically relevant
features described above are exclusively due to the {\em combined} effect of
both square-root nonlinearity $\sqrt{-F^2}$ in the gauge field Lagrangian as well 
as the $R^2$ term in the gravity action.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% \section{Conclusions}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgments}
We gratefully acknowledge support of our collaboration through the academic exchange 
agreement between the Ben-Gurion University and the Bulgarian Academy of Sciences.
S.P. has received partial support from COST action MP-1210.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}
%%
\bib{thooft}
G. `t Hooft (2003) {\em Nucl. Phys. B (Proc. Suppl.)} {\bf 121} 333-340;
% ~(\textsl{arxiv:hep-th/0208054}); %%\\
(2007) {\em Progr. Theor. Phys. Suppl.} {\bf 167} 144-154.
%%
\bib{GG-1}
E. Guendelman (2004) {\em Int. J. Mod. Phys.} {\bf A19} 3255. 
% ~(\textsl{arxiv:hep-th/0306162}); \\
\bib{GG-2}
P. Gaete and E. Guendelman (2006) {\em Phys. Lett.} {\bf B640} 201-204.
%~(\textsl{arxiv:hep-th/0607113});\\
\bib{GG-3}
P. Gaete, E. Guendelman and E. Spalluci (2007) {\em Phys. Lett.} {\bf B649} 217; 
% ~(\textsl{arxiv:hep-th/0702067}); \\
E. Guendelman (2007) {\em Mod. Phys. Lett.} {\bf A22} 1209-15; 
% ~(\textsl{arxiv:hep-th/0703139});
E. Guendelman (2010) {\em Int. J. Mod. Phys.} {\bf A25} 4195-4220; 
% ~(\textsl{arxiv:1005.1421}); \\
I. Korover and E. Guendelman (2009) {\em Int. J. Mod. Phys.} {\bf A24} 1443-56.
%%
\bib{our-main-1}
E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva (2011) {\em Phys. Lett.}
{\bf 704} 230-233, erratum {\bf 705} 545; (2011) {\em Int. J. Mod. Phys.} {\bf A26}
5211-39; (2013) in {\em Seventh Mathematical Physics Meeting}, B. Dragovic and
Z. Rakic (eds.), Belgrade Inst. Phys. Press.
%%
\bib{our-main-2} % f(R)-maxwell
E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva (2013) {\em Phys. Lett.}
{\bf B718} 1099-1104.
%%
\bib{f(R)-grav}
S. Capozziello, M. De Laurentis, V. Faraoni, \textsl{The Open Astronomical
Journal} {\bf 3} (2010) 49-72; % ~(\textsl{arXiv:0909.4672}[gr-qc]); %\\
A. De Felice, S. Tsujikawa, \textsl{Living Rev. Rel.} {\bf 13} (2010) 3-163.
% ~(\textsl{arXiv:1002.4928}[gr-qc]).
%%
\bib{starobinsky}
A. Starobinsky, \PLB{91}{1980}{99-102}.
%%
\bib{LC-BR}
% \bib{LC-BR-1}
T. Levi-Civita (1917) % The Physical Reality of Some Normal Spaces of Bianchi. 
{\sl Rendiconti della Reale Accademia dei Lincei} {\bf 26} 519-531;
% \bib{LC-BR-2}
R. Bertotti (1959) % Uniform Electromagnetic Field in the Theory of General Relativity. 
{\em Phys. Rev.} {\bf 116} 1331-33;
% \bib{LC-BR-3}
I. Robinson (1959) % A Solution of the Maxwell-Einstein Equations. 
\textsl{Bull. Akad. Pol.} {\bf 7} 351-352.
%%
\bib{BJP-LL-branes}
E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva (2013)
\textsl{Lightlike Membranes in Black Hole and Wormhole Physics, and Cosmology}, 
this proceedings.
%%
\bib{damour-polyakov}
T. Damour and A. Polyakov (1994) {\em Nucl. Phys.} {\bf B423} 532-558.
% ~(\textsl{arXiv:hep-th/9401069}).
%%
\bib{hedgehog}
M. Barriola and A Vilenkin (1989) {\em Phys. Rev. Lett.} {\bf 63} 341; % \\
%%
E. Guendelman and A. Rabinowitz (1991) {\em Phys. Rev.} {\bf D44} 3152.
%%
\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}

1310.5337.extracted_bib
\begin{thebibliography}{99}
%%
\bib{thooft}
G. `t Hooft (2003) {\em Nucl. Phys. B (Proc. Suppl.)} {\bf 121} 333-340;
% ~(\textsl{arxiv:hep-th/0208054}); %%\\
(2007) {\em Progr. Theor. Phys. Suppl.} {\bf 167} 144-154.
%%
\bib{GG-1}
E. Guendelman (2004) {\em Int. J. Mod. Phys.} {\bf A19} 3255. 
% ~(\textsl{arxiv:hep-th/0306162}); \\
\bib{GG-2}
P. Gaete and E. Guendelman (2006) {\em Phys. Lett.} {\bf B640} 201-204.
%~(\textsl{arxiv:hep-th/0607113});\\
\bib{GG-3}
P. Gaete, E. Guendelman and E. Spalluci (2007) {\em Phys. Lett.} {\bf B649} 217; 
% ~(\textsl{arxiv:hep-th/0702067}); \\
E. Guendelman (2007) {\em Mod. Phys. Lett.} {\bf A22} 1209-15; 
% ~(\textsl{arxiv:hep-th/0703139});
E. Guendelman (2010) {\em Int. J. Mod. Phys.} {\bf A25} 4195-4220; 
% ~(\textsl{arxiv:1005.1421}); \\
I. Korover and E. Guendelman (2009) {\em Int. J. Mod. Phys.} {\bf A24} 1443-56.
%%
\bib{our-main-1}
E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva (2011) {\em Phys. Lett.}
{\bf 704} 230-233, erratum {\bf 705} 545; (2011) {\em Int. J. Mod. Phys.} {\bf A26}
5211-39; (2013) in {\em Seventh Mathematical Physics Meeting}, B. Dragovic and
Z. Rakic (eds.), Belgrade Inst. Phys. Press.
%%
\bib{our-main-2} % f(R)-maxwell
E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva (2013) {\em Phys. Lett.}
{\bf B718} 1099-1104.
%%
\bib{f(R)-grav}
S. Capozziello, M. De Laurentis, V. Faraoni, \textsl{The Open Astronomical
Journal} {\bf 3} (2010) 49-72; % ~(\textsl{arXiv:0909.4672}[gr-qc]); %\\
A. De Felice, S. Tsujikawa, \textsl{Living Rev. Rel.} {\bf 13} (2010) 3-163.
% ~(\textsl{arXiv:1002.4928}[gr-qc]).
%%
\bib{starobinsky}
A. Starobinsky, \PLB{91}{1980}{99-102}.
%%
\bib{LC-BR}
% \bib{LC-BR-1}
T. Levi-Civita (1917) % The Physical Reality of Some Normal Spaces of Bianchi. 
{\sl Rendiconti della Reale Accademia dei Lincei} {\bf 26} 519-531;
% \bib{LC-BR-2}
R. Bertotti (1959) % Uniform Electromagnetic Field in the Theory of General Relativity. 
{\em Phys. Rev.} {\bf 116} 1331-33;
% \bib{LC-BR-3}
I. Robinson (1959) % A Solution of the Maxwell-Einstein Equations. 
\textsl{Bull. Akad. Pol.} {\bf 7} 351-352.
%%
\bib{BJP-LL-branes}
E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva (2013)
\textsl{Lightlike Membranes in Black Hole and Wormhole Physics, and Cosmology}, 
this proceedings.
%%
\bib{damour-polyakov}
T. Damour and A. Polyakov (1994) {\em Nucl. Phys.} {\bf B423} 532-558.
% ~(\textsl{arXiv:hep-th/9401069}).
%%
\bib{hedgehog}
M. Barriola and A Vilenkin (1989) {\em Phys. Rev. Lett.} {\bf 63} 341; % \\
%%
E. Guendelman and A. Rabinowitz (1991) {\em Phys. Rev.} {\bf D44} 3152.
%%
\end{thebibliography}