Title
Title
We shortly outline the principal results concerning the reparametrization-invariant world-volume Lagrangian formulation of {\em lightlike} brane dynamics and its impact as a source for gravity and (nonlinear) electromagnetism in black hole and wormhole physics.
BJP-LL-branes.tex
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\title{Lightlike Membranes in Black Hole and  Wormhole Physics, and Cosmology
\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}{Lightlike 
Membranes in Black Hole $\&$ Wormhole Physics $\&$ Cosmology}

\received{}

\begin{Abstract}
We shortly outline the principal results concerning the reparametrization-invariant
world-volume Lagrangian formulation of {\em lightlike} brane dynamics and
its impact as a source for gravity and (nonlinear) electromagnetism in black hole 
and wormhole physics.
\end{Abstract}

\PACS{11.25.-w,04.70.Bw,04.50.-h}
%%%%%%%%%%%%%%%%%%%%%%
% 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}

Extended objects (strings and p-branes) are of primary importance for the
construction of self-consistent unified modern theory of fundamental forces in Nature
\ct{polchinski}.
In a series of recent papers of ours \ct{LL-main-1,Reg-BH,BR-kink} 
we have proposed for the first time 
in the literature a systematic world-volume Lagrangian description and studied in 
detail the physical properties of a new class of brane theories called 
{\em lightlike branes} (\textsl{LL-branes}), which are qualitatively distinct 
from the standard Nambu-Goto
type brane models which describe intrinsically {\em massive} world-volume modes.

As it is well known, \textsl{LL-branes} (also called \textsl{null-branes}) 
are of substantial interest in general relativity as they describe impulsive 
lightlike signals arising in various violent astrophysical events, \textsl{e.g.}, 
final explosion in cataclysmic processes such as supernovae and collision of 
neutron stars. \textsl{LL-branes} also play important
role in the description of various other physically important cosmological and 
astrophysical phenomena such as the ``membrane paradigm'' of black hole physics
and the thin-wall approach to domain walls coupled to gravity. For a detailed 
account, see \ct{barrabes-hogan}. More recently they became significant also in the
context of modern non-perturbative string theory \ct{nonperturb-string}.

Here we will shortly describe some of our principal results concerning the physics of
\textsl{LL-branes} and their implications in black hole and wormhole physics, 
and cosmology:

(a) Horizon ``straddling'' effect: the dynamics of \textsl{LL-branes} requires the 
bulk space-time geometry to possess one or more horizons, for instance, 
to be of black hole type, and it dictates that \textsl{LL-branes} automatically occupy 
(one of these) horizon(s).

(b) \textsl{LL-branes} are natural candidates for matter and charged sources of
``thin-shell'' traversable wormholes of various types (one- or multi-``throat''
``tube-like'', rotating etc.) \ct{LL-main-1}.

(c) \textsl{LL-branes} naturally produce regular black holes, i.e., black holes 
free of ``inside'' (below the inner horizon) physical space-time singularities
\ct{Reg-BH}.

(d) \textsl{LL-branes} trigger spontaneous compactification of space-time, 
as well as compactification/decompactification transitions \ct{BR-kink}.

(e) \textsl{LL-branes} are consistent matter sources for lightlike braneworlds 
\ct{Varna-11}.
% (the latter provide a natural explanation of the undetectability of (large) 
% extra dimensions from the point of view of an observer confined to the 
% lightlike brane ``universe'').

(f) \textsl{LL-branes} produce new wormhole ``universes'' exhibiting 
{\em charge-hiding} and {\em charge-confining} effects \ct{hide-confine}, physically analogous 
to the quark confinement mechanism in quantum chromodynamics.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Gravity and Nonlinear Gauge Fields Coupled to LL-Brane Sources}

In Refs.\ct{LL-main-1,Reg-BH,BR-kink,Varna-11,hide-confine} we proposed and 
extensively studied a manifestly reparametrization
invariant world-volume Lagrangian action of \textsl{LL-branes}:
\br
S_{\rm LL}\lb q\rb  = - \h \int d^{p+1}\s\, T b_0^{\frac{p-1}{2}}\sqrt{-\g}
\llb \g^{ab} {\bar g}_{ab} - b_0 (p-1)\rrb \; ,
\lab{LL-action+EM} \\
{\bar g}_{ab} \equiv % \pa_a X^\m G_{\m\n} \pa_b X^\n 
g_{ab} - \frac{1}{T^2} (\pa_a u + q\cA_a)(\pa_b u  + q\cA_b) 
\quad , \quad \cA_a \equiv \pa_a X^\m A_\m \; .
\lab{ind-metric-ext-A}
\er
Here and below the following notations are used:
\begin{itemize}
\item
$\g_{ab}$ is the {\em intrinsic} world-volume Riemannian metric;
$g_{ab}=\pa_a X^{\m} G_{\m\n}(X) \pa_b X^{\n}$ is the {\em induced} metric on the 
world-volume, which becomes {\em singular} on-shell (manifestation of the lightlike 
nature); $b_0$ is world-volume ``cosmological constant'';
\item
$X^\m (\s)$ are the $p$-brane embedding coordinates in the
$D$-dimensional bulk space-time with Riemannian metric
$G_{\m\n}(x)$ ($\m,\n = 0,1,\ldots ,D-1$); 
$(\s)\equiv \(\s^0 \equiv \t,\s^i\)$ with $i=1,\ldots ,p$;
$\pa_a \equiv \partder{}{\s^a}$.
\item
$u$ is auxiliary world-volume scalar field defining the lightlike direction
of the induced metric;
\item
$T$ is {\em dynamical (variable)} brane tension;
\item
$q$ -- the coupling to bulk spacetime gauge field $\cA_\m$ is
\textsl{LL-brane} surface charge density.
\end{itemize}

The on-shell singularity, \textsl{i.e.}, the lightlike property of the induced metric
$g_{ab}$, directly follows from the equations of motion resulting from
\rf{LL-action+EM}:
\be
g_{ab} \({\bar g}^{bc}(\pa_c u  + q\cA_c)\) = 0 \; .
\lab{on-shell-singular-A}
\ee

Now, the full action of gravity and (nonlinear) gauge fields interacting
self-consistently with \textsl{LL-branes} reads (we specialize to $D=4$
space-time dimensions and use units with the Newton constant $G_N=1$):
\be
S = \int d^4 x \sqrt{-G} \Bigl\lb \frac{R(G) - 2\L_0}{16\pi} + L(F^2)\Bigr\rb 
+ \sum_{k=1}^N S_{\mathrm{LL}}\lb q^{(k)}\rb \; ,
% \;\; ,\;\; L(F^2) = - \frac{1}{4} F^2 - \frac{f}{2} \sqrt{- F^2} \; ,
\lab{gravity+GG+LL}
\ee
where the superscript $(k)$ indicates the $k$-th \textsl{LL-brane}. Here 
$R(G)= G^{\m\n}R_{\m\n}$ and $R_{\m\n}$ denote the Riemannian scalar curvature 
and the Ricci tensor of the bulk space-time geometry. $L(F^2)$ is the Lagrangian 
of a remarkable non-standard nonlinear electrodynamics containing 
\textsl{square root} of ordinary Maxwell Lagrangian \ct{GG}:
\be
L(F^2) = - \frac{1}{4} F^2 - \frac{f_0}{2} \sqrt{- F^2} \quad ,\quad
F^2 \equiv F_{\m\k} F_{\n\l} G^{\m\n} G^{\k\l} \; .
\lab{GG}
\ee
This is an explicit realization of `t Hooft's proposal (in flat space-time) for 
\textsl{infrared charge confinement} \ct{thooft} (see also next talk \ct{BJP-svet} at this congress).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Charge-Confinement via ``Tube-Like'' Wormhole}

The general scheme to construct ``lightlike thin-shell'' wormholes of 
static ``spherically-symmetric'' type (in Eddington-Finkelstein coordinates
$dt=dv-\frac{d\eta}{A(\eta)}$ and ``radial''-like coordinate
$\eta \in (-\infty,+\infty)$):
\br
ds^2 = - A(\eta) dv^2 + 2dv d\eta + C(\eta) h_{ij}(\th) d\th^i d\th^j \quad ,\quad
F_{v\eta} = F_{v\eta} (\eta)\; , %\; \mathrm{rest}=0 \; .
\lab{static-spherical-EF} \\
-\infty < \eta < \infty \quad, \;\; A(\eta^{(k)}_0) = 0 \;\; 
\mathrm{for} \;\; \eta^{(1)}_0 <\ldots<\eta^{(N)}_0 
\lab{common-horizons}
\er
is as follows (cf. Section 5 in Ref.\ct{hide-confine}):

(1) Take ``vacuum'' solutions of Einstein and (nonlinear) Maxwell equations
resulting from \rf{gravity+GG+LL} (\textsl{i.e.}, without the delta-function 
\textsl{LL-brane} contributions) in each space-time region (separate ``universe'') 
given by $\bigl(-\infty\! <\!\eta\!<\!\eta^{(1)}_0\bigr),\ldots,$
$\bigl(\eta^{(N)}_0 \!<\!\eta \!<\!\infty\bigr)$ with common horizon(s) at 
$\eta=\eta^{(k)}_0$ ($k=1,\ldots ,N$).

(2) Each $k$-th \textsl{LL-brane} automatically locates itself on the
horizon at $\eta=\eta^{(k)}_0$ -- intrinsic property of \textsl{LL-brane} dynamics 
defined by the action \rf{LL-action+EM}. % (see \ct{}).

(3) Match discontinuities of the derivatives of the metric and
the gauge field strength  across each horizon at $\eta=\eta^{(k)}_0$ using the 
explicit expressions for the \textsl{LL-brane} stress-energy tensor and charge 
current density systematically derived from the action \rf{gravity+GG+LL} with
\rf{LL-action+EM}.

Let us now consider the gravity/nonlinear-gauge-field system coupled to two
oppositely charged \textsl{LL-branes}, \textsl{i.e.}, $N=2$ and $q_1 = -
q_2 \equiv q$ in \rf{gravity+GG+LL}. We obtain a particularly interesting
``two-throat'' wormhole-type solution exhibiting a QCD-like charge
confinement effect. The total space-time manifold consists of three
``universes'' with different geometry glued together at their common
horizons occupied by the two oppositely charged \textsl{LL-branes}: 

(i) ``Left-most'' non-compact ``universe'' comprising the exterior
region of a new kind of {\em non-standard} Schwarzschild-de-Sitter-type black hole, 
with additional {\em constant vacuum radial electric field} $\vec{E}_{\rm vac}$,
beyond the Schwarzschild-type horizon $r_0$ for the ``radial-like'' 
$\eta$-coordinate interval $-\infty < \eta < -\eta_0 
\equiv - \Bigl\lb 4\pi\(\sqrt{2}f_0|\vec{E}| - \vec{E}^2\) + \L_0 \Bigr\rb^{-\h}$,
where (using notations as in \rf{static-spherical-EF}):
\br
A(\eta) = 1 - \frac{2m}{r_0 - \eta_0 - \eta} 
% - \frac{\L_0 + 2\pi f_0^2}{3} (r_0 - \eta_0 - \eta)^2 \; ,
- \frac{\L_{\rm eff}}{3} (r_0 - \eta_0 - \eta)^2 \; ,
\lab{SdS-left-most-1}\\
C(\eta) = (r_0 - \eta_0 - \eta)^2 \;\; ,\;\;
|F_{v\eta}(\eta)| \equiv |\vec{E}_{\rm vac}| = \frac{f_0}{\sqrt{2}} <|\vec{E}| \; .
\lab{SdS-left-most-2}
\er
Here $\vec{E}$ is the constant electric field in the ``middle'' ``tube-like''
``universe'' (ii) (Eq.\rf{LCBR-middle-2} below);
$\L_{\rm eff}\equiv \L_0 + 2\pi f_0^2$ in \rf{SdS-left-most-1}
is {\em dynamically generated/shifted} cosmological constant, which is non-vanishing
even in the absence of the ``bare'' cosmological constant $\L_0$. Let us stress
that {\em constant vacuum radial electric fields} such as in \rf{SdS-left-most-2}
{\em do not} exist as solutions of ordinary Maxwell electrodynamics on
generic non-compact space-times -- the former are due exclusively to the nonlinear
``square-root'' term in \rf{GG}.

(ii) ``Middle'' ``tube-like'' ``universe'' of
Levi-Civita-Bertotti-Robinson type \ct{LC-BR} with geometry $dS_2 \times S^2$ 
($dS_2$ denotes two-dimensional de Sitter space; $S^2$ -- sphere with
constant radius $r_0$), comprising the finite extent 
(w.r.t. $\eta$-coordinate) region between the two horizons of $dS_2$ at 
$\eta = \pm \eta_0$ occupied by the two \textsl{LL-branes} with charges $\pm q$:
\be
-\eta_0 < \eta < \eta_0 
\equiv \Bigl\lb 4\pi\(\sqrt{2}f_0|\vec{E}| - \vec{E}^2\) + \L_0 \Bigr\rb^{-\h} \; ,
\lab{middle-interval}
\ee
where the metric coefficients and electric field are:
\br
A(\eta) = 
1 - \Bigl\lb 4\pi\(\sqrt{2}f_0|\vec{E}| - \vec{E}^2\) + \L_0 \Bigr\rb\,\eta^2
\;\; ,\;\; A(\pm \eta_0) = 0 \; ,
\lab{LCBR-middle-1} \\
C(\eta) = r_0^2 = \frac{1}{4\pi \vec{E}^2 + \L_0} \;\; ,\;\;
% |F_{v\eta}|\equiv 
|\vec{E}| = |q| + \frac{f_0}{\sqrt{2}} = {\rm const} \; .
% < \frac{f_0}{\sqrt{2}}\Bigl( 1 + \sqrt{1 + \frac{\L_0}{2\pi f_0^2}}\Bigr) \; ,
\lab{LCBR-middle-2}
\er
% with $\L_0 > - 2\pi f_0^2$, in particular, for $\L_0 = 0$.

(iii) ``Right-most'' non-compact ``universe'' of the same type as (i) above
for the ``radial-like'' $\eta$-coordinate interval ~$\eta_0 < \eta < \infty$ 
($\eta_0$ as in \rf{middle-interval}). Its metric is given by 
Eq.\rf{SdS-left-most-1} upon changing $-\eta \to \eta$ and the elecric
field is the same as in \rf{SdS-left-most-2}.

The equations for the electric field (second relations in \rf{SdS-left-most-2} and 
\rf{LCBR-middle-2}) have profound consequences:

\begin{itemize}
\item
The ``left-most'' and ``right-most'' non-compact ``universes'' are
two identical copies of the {\em electrically neutral} exterior region of
Schwarzschild-de-Sitter black hole beyond the Schwarzschild horizon. They both
carry a constant vacuum radial electric field with magnitude 
$|\vec{E}|=\frac{f_0}{\sqrt{2}}$ pointing inbound/outbound w.r.t. pertinent
% towards the horizon in one of these ``universes'' and pointing outbound w.r.t. 
% the horizon in the second ``universe''. 
horizon. The corresponding electric displacement field 
$\vec{D}=\Bigl( 1 - \frac{f_0}{\sqrt{2}|\vec{E}|}\Bigr)\,\vec{E}=0$, 
so there is {\em no} electric flux there.
\item
The whole electric flux produced by the two charged \textsl{LL-branes} with 
opposite charges $\pm q$ at the boundaries of the above non-compact ``universes''
is {\em confined} within the finite-extent ``tube-like'' middle ``universe'' of 
Levi-Civitta-Robinson-Bertotti type
with geometry $dS_2 \times S^2$, where the constant electric field is
$|\vec{E}|=\frac{f_0}{\sqrt{2}} + |q|$ with associated non-zero electric 
displacement field $|\vec{D}|= |q|$ . This is {\em QCD-like confinement}.
\end{itemize}

The {\em charge-confining} wormhole geometry is visualized on Fig.1.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \usepackage{graphicx}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
% \includegraphics{Confining-WH_3.eps}
\includegraphics[scale=0.7,angle=270,keepaspectratio=true]{Confining-WH_3}
\caption{Shape of $t=const$ and $\th=\frac{\pi}{2}$ slice of
charge-confining wormhole geometry. The whole electric flux is confined
within the middle cylindric tube.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \usepackage{epsfig,cite}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{figure}[b]
% \centering{\epsfig{file=Confining-WH_3.eps,width=70mm}}
% \caption{Shape of $t=const$ and $\th=\frac{\pi}{2}$ slice of
% charge-confining wormhole geometry. The whole electric flux is confined
% within the middle cylindric tube. \label{f1}}
% \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

To conclude let us emphasize that the existence of charge-confining
``thin-shell'' wormholes is entirely due to the combined effect of the exceptional 
properties of \textsl{LL-brane} dynamics and the ``square-root'' nonlinear electrodynamics.

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

% \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{polchinski}
J. Polchinski (1998) {\em ``String Theory''}, vols.1,2, Cambridge Univ. Press., 
Cambridge Mass.
%%
\bib{LL-main-1}
E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva (2005)
{\em Phys. Rev.} {\bf 72} 0806011. % ~(\textsl{arxiv:hep-th/0507193});
%%
% E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva, 
% \textsl{Fortschr. der Physik} {\bf 55} (2007) 579 ~(\textsl{hep-th/0612091});
% \textsl{Fortschr. der Phys.} {\bf 57} (2009) 566
% ~(\textsl{arxiv:0901.4443}[hep-th]); \\
%%
% E. Guendelman, A. Kaganovich, E. Nissimov, S. Pacheva,
(2009) {\em Phys. Lett.} {\bf 673} 288-292; % ~(\textsl{arxiv:0811.2882});
(2009) {\em Phys. Lett.} {\bf 681} 457-462; % ~(\textsl{arxiv:0904.3198}); 
(2010) {\em Int. J. Mod. Phys.} {\bf A25} 1405-28. % ~(\textsl{arxiv:0904.0401})
%%
\bib{Reg-BH}
E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva
(2010) {\em Int. J. Mod. Phys.} {\bf A25} 1571-96. % ~(\textsl{arxiv:0908.4195}).
%%
\bib{BR-kink}
E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva (2011) 
\textsl{Gen. Rel. Grav.} {\bf 43} 1487-1513.
% ~(\textsl{arxiv:1007.4893}).
%%
\bib{barrabes-hogan}
C. Barrab{\'e}s and P. Hogan (2003) {\em ``Singular Null Hypersurfaces in General
Relativity''}, World Scientific.
%%
\bib{nonperturb-string}
% \bib{kogan-01}
I. Kogan and N. Reis (2001) {\em Int. J. Mod. Phys.} {\bf A16} 4567; % \lb\textsl{hep-th/0107163}\rb.
% \bib{mateos-02}
D. Mateos and S. Ng (2002) {\em JHEP} {\bf 0208} 005; % \lb\textsl{hep-th/0205291}\rb.
% \bib{nonperturb-string-1}
J. Harvey, P. Kraus and F. Larsen (2001) {\em Phys. Rev.} {\bf D63} 026002;
% \lb\textsl{hep-th/0008064}\rb.
% \bib{nonperturb-string-2}
D. Mateos, T. Mateos and P.K. Townsend (2003) {\em JHEP} {\bf 0312} 017;
% \lb\textsl{hep-th/0309114}\rb.
% \bib{nonperturb-string-3}
A. Bredthauer, U. Lindstr{\"o}m, J. Persson and L. Wulff (2004) {\em JHEP} 
{\bf 0402} 051. % \lb\textsl{hep-th/0401159}\rb.
%%
\bib{Varna-11}
E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva (2013) in {\em Springer Proceedings 
in Mathematics and Statistics} {\bf 36} 215-230, ed. V. Dobrev, Springer.
%%
\bib{hide-confine}
E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva (2011) {\em Int. J. Mod. Phys.} 
{\bf A26} 5211-39. % ~(\textsl{arxiv:1109.0453}[hep-th]).
%%
\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{GG}
E. Guendelman (2004) {\em Int. J. Mod. Phys.} {\bf A19} 3255; 
% ~(\textsl{arxiv:hep-th/0306162}); \\
P. Gaete and E. Guendelman (2006) {\em Phys. Lett.} {\bf B640} 201-204;
%~(\textsl{arxiv:hep-th/0607113});\\
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{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{BJP-svet}
E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva (2013)
\textsl{Dynamical Couplings and Charge Confinement/Deconfinement from Gravity 
Coupled to Nonlinear Gauge Fields}, this proceedings.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{thebibliography}

\end{document}

1310.5334.extracted_bib
\begin{thebibliography}{99}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bib{polchinski}
J. Polchinski (1998) {\em ``String Theory''}, vols.1,2, Cambridge Univ. Press., 
Cambridge Mass.
%%
\bib{LL-main-1}
E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva (2005)
{\em Phys. Rev.} {\bf 72} 0806011. % ~(\textsl{arxiv:hep-th/0507193});
%%
% E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva, 
% \textsl{Fortschr. der Physik} {\bf 55} (2007) 579 ~(\textsl{hep-th/0612091});
% \textsl{Fortschr. der Phys.} {\bf 57} (2009) 566
% ~(\textsl{arxiv:0901.4443}[hep-th]); \\
%%
% E. Guendelman, A. Kaganovich, E. Nissimov, S. Pacheva,
(2009) {\em Phys. Lett.} {\bf 673} 288-292; % ~(\textsl{arxiv:0811.2882});
(2009) {\em Phys. Lett.} {\bf 681} 457-462; % ~(\textsl{arxiv:0904.3198}); 
(2010) {\em Int. J. Mod. Phys.} {\bf A25} 1405-28. % ~(\textsl{arxiv:0904.0401})
%%
\bib{Reg-BH}
E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva
(2010) {\em Int. J. Mod. Phys.} {\bf A25} 1571-96. % ~(\textsl{arxiv:0908.4195}).
%%
\bib{BR-kink}
E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva (2011) 
\textsl{Gen. Rel. Grav.} {\bf 43} 1487-1513.
% ~(\textsl{arxiv:1007.4893}).
%%
\bib{barrabes-hogan}
C. Barrab{\'e}s and P. Hogan (2003) {\em ``Singular Null Hypersurfaces in General
Relativity''}, World Scientific.
%%
\bib{nonperturb-string}
% \bib{kogan-01}
I. Kogan and N. Reis (2001) {\em Int. J. Mod. Phys.} {\bf A16} 4567; % \lb\textsl{hep-th/0107163}\rb.
% \bib{mateos-02}
D. Mateos and S. Ng (2002) {\em JHEP} {\bf 0208} 005; % \lb\textsl{hep-th/0205291}\rb.
% \bib{nonperturb-string-1}
J. Harvey, P. Kraus and F. Larsen (2001) {\em Phys. Rev.} {\bf D63} 026002;
% \lb\textsl{hep-th/0008064}\rb.
% \bib{nonperturb-string-2}
D. Mateos, T. Mateos and P.K. Townsend (2003) {\em JHEP} {\bf 0312} 017;
% \lb\textsl{hep-th/0309114}\rb.
% \bib{nonperturb-string-3}
A. Bredthauer, U. Lindstr{\"o}m, J. Persson and L. Wulff (2004) {\em JHEP} 
{\bf 0402} 051. % \lb\textsl{hep-th/0401159}\rb.
%%
\bib{Varna-11}
E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva (2013) in {\em Springer Proceedings 
in Mathematics and Statistics} {\bf 36} 215-230, ed. V. Dobrev, Springer.
%%
\bib{hide-confine}
E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva (2011) {\em Int. J. Mod. Phys.} 
{\bf A26} 5211-39. % ~(\textsl{arxiv:1109.0453}[hep-th]).
%%
\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{GG}
E. Guendelman (2004) {\em Int. J. Mod. Phys.} {\bf A19} 3255; 
% ~(\textsl{arxiv:hep-th/0306162}); \\
P. Gaete and E. Guendelman (2006) {\em Phys. Lett.} {\bf B640} 201-204;
%~(\textsl{arxiv:hep-th/0607113});\\
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{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{BJP-svet}
E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva (2013)
\textsl{Dynamical Couplings and Charge Confinement/Deconfinement from Gravity 
Coupled to Nonlinear Gauge Fields}, this proceedings.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{thebibliography}