On the stability of black holes with nonlinear electromagnetic fields
Nora Bret\'on
Santiago Esteban Perez Bergliaffa
The stability of three static and spherically symmetric black hole solutions with nonlinear electromagnetism as a source is investigated in three different ways. We show that the specific heat of all the solutions displays an infinite discontinuity with a change of sign , but the turning point method indicates that the solutions are thermodinamically stable (much in the same way as is the case of the Reissner-Nordstrom geometry). We also show that the black holes analyzed here are dynamically stable, thus suggesting that there may be a relation between thermodynamical and dynamical stability for nonvacuum black holes.
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\begin{document}

\title{On the stability of black holes with nonlinear electromagnetic fields}

\author{Nora Bret\'on}
\affiliation{Departamento de F\'{\i}sica, Centro de Investigaci\'on y de Estudios Avanzados del I. P. N., Apdo. 14-740, D.F., M\'exico.}

\author{Santiago Esteban Perez Bergliaffa}
\affiliation{Departamento de F\'{\i}sica Te\'orica, Instituto de F\'{\i}sica, Universidade do Estado do Rio de Janeiro, 
Brasil.}

\begin{abstract}
The stability of three static and spherically symmetric black hole solutions with nonlinear electromagnetism as a source is investigated in three different
ways. We show that the specific heat of all the solutions displays an infinite discontinuity with a change of sign , but the turning point method indicates that the solutions are thermodinamically stable (much in the same way as is the case of the Reissner-Nordstrom geometry). We also show that the black holes analyzed here 
are dynamically stable, thus suggesting that there may be a 
relation between thermodynamical and dynamical stability for 
nonvacuum black holes. 


\end{abstract}

\pacs{04.70.Bw, 04.20.Jb, 04.40.Nr, 11.10.Lm}

\maketitle


\section{Introduction}


The stability of black hole solutions can be analyzed from 
different points of view. First, since black holes have been considered as thermodynamical systems
after the papers by Bekenstein (who postulated the relation between the area of the horizon and the entropy) \citep{Bekenstein1973}, and Hawking (who showed that they posses a nonzero temperature, due to Hawking radiation) \citep{Hawking1974}, one can wonder about their thermodynamical stability. 
In this regard, Davies \citep{Davies1977} demonstrated that the specific heat (at constant charge or angular momentum) 
of the Kerr-Newman black hole presents an infinite discontinuity along which it changes sign
(while the Gibbs free energy and its first derivative are continuous), and associated this discontinuity to an instability stating that it would lead to a second-order phase transition. The thermodynamical stability of black holes can also be studied using the so-called Poincar\`e (or turning point) method \footnote{See \citep{Poincare1885} for the original version, and 
\citep{Sorkin1982} and \citep{Schiffrin2013}
for updates.}, which asserts that 
changes in stability of a series of equilibrium states can only occur 
when there is a vertical tangent in the plot of conjugate pairs of variables 
(such as mass and the inverse temperature, in the case of a black hole), or when there is a bifurcation.
% \footnote{A bifurcation point is the point at the intersection of two sequences of 
%equilibrium states.}.
Using this method, isolated black hole solutions in General Relativity have been shown to be 
thermodynamically
stable in 
\citep{Katz1993, Kaburaki1993}, a result that is at odds with 
the findings of Davies
\footnote{For different points of view about this discrepancy, see
\citep{Sokolowski1980,Sorkin1982, Pavon1988, Pavon1991, Katz1993, Kaburaki1993,Lousto1993, Arcioni2004, Ablu2010,Parentani1995, Okamoto1995}}.
%), partly due to 
%fact that the domains of stability of different ensembles related by a Legendre %transformation are different \textbf{AMPLIAR}. Hence, 
%heat capacities and other thermodynamic
%coefficients may not be good indicators 
%of instability (see for instance \citep{Parentani1995, Okamoto1995}).
%
%the  
%lack of a unique microscopic model, which should be furnished by a quantum %theory of gravitation. 

Second, several analyses of the stability of black holes from a dynamical point of view have yield the result that black hole solutions of General Relativity in four dimensions are
stable at the linear level.
% by perturbing the corresponding equations of motion. 
In the case of Schwarzschild's solution, the stability was proven by Regge and Wheeler \citep{Regge1957}, while that of the Reissner-Nordstrom (RN) geometry was shown in \citep{Moncrief1974a, Moncrief1974b}. The proof of the stability of Kerr's solution
was given in \citep{Whiting1989}. 
%These results are in line with those obtained with the turning point method. 

The relation between thermodynamical and dynamical stability was recently discussed in \citep{Hollands2012} (see also \citep{Schiffrin2013,Green2013}). In particular, it was shown there that 
for vacuum black holes in General Relativity
dynamical stability is equivalent to thermodynamic stability,
for perturbations with $\delta M = \delta J = \delta A = 0$. 
Hence, a turning point implies in this case a dynamic instability. 
We would like to take here one step further in the understanding of the relation between these different types of stability by 
exploring in detail some exact solutions representing nonvacuum 
%charged 
black holes in General Relativity.
%, both from the dynamical and the thermodynamical point of view. 
In particular, 
we will focus on
static and spherically symmetric charged black holes with nonlinear electromagnetism as a source. 
%Nonlinear electromagnetism (NLEM) was first considered by Born and Infeld %\citep{bi}, in an attempt to cure at the classical level the singularity of the %electric field of a point charge. After QED was shown to be more effective in %this regard, NLEM was left aside for some decades, but is making a comeback %lately since it arises naturally in
%open superstring theory and the theory of D-branes \citep{fra}, and it also 
%manifests itself in some astrophysical phenomena (such as the magnetars \citep{magnetars}). 
%NLEM has received considerable attention recently in several areas REF.
These solutions, which  
generalize the RN geometry,
%of the Einstein-Maxwell field equations 
have received considerable attention recently. The Born-Infeld-Einstein
static and spherically symmetric (SSS) spacetime was presented in \citep{Garcia1984, Breton2003}, 
and its thermodynamical properties analyzed in \citep{Chemissany2008, Gunasekaran2012}. A regular black hole geometry in the presence of a nonlinear electromagnetic field
which reduces to that of Maxwell in the weak-field limit
was obtained in \citep{Ayon1998} using the dual formalism introduced in 
\citep{Salazar1987}, and further discussed in \citep{Baldovin2000} and \citep{Bronnikov2000}. The SSS black hole with the Euler-Heisenberg effective
Lagrangian of quantum electrodynamics as a source was examined in \citep{Yajima2000}, and
the same type of solutions with Lagrangian densities that are
powers of Maxwell's Lagrangian were analyzed in \citep{Hassaine2008}. 
Also worth mentioning are the general analysis of \citep{Diaz2009, Diaz2010}, the examination of the thermodynamics of black holes 
with an arbitrary nonlinear Lagrangian for the electromagnetic field 
in \citep{Rasheed1997} and of the Smarr formula in \citep{Breton2004},
and the enhanced no-hair conjecture presented in \citep{Garcia2012}. 
To close this incomplete list, we would like to point to the references
\citep{Moreno2002} and \citep{Breton2005}, where 
the dynamical stability of SSS black holes with a nonlinear electromagnetic source was examined, and 
\citep{Flachi2012}, where a study of the quasinormal modes of neutral
and charged scalar field perturbations on regular nonlinear electromagnetic black hole backgrounds was presented.

Our main goal is to compare the results of the three abovementioned ways of determining the stability in the case of SSS black holes with NLEM as a source. We shall consider both singular and regular solutions. In particular, we shall see that in all cases there is a discontinuity in the specific heat of the kind present in the RN solution, but no sign of instability according to the Poincar\`e method.
%, as well as with the dynamical analysis of \citep{Moreno2002}
%and \citep{Breton2005}.
As a byproduct, we will exhibit several expressions 
valid for any static and spherically symmetric charged black hole, and a study of the position of the horizons in the 
exact solutions under scrutiny. 
%W
%e shall also highlight some particular features of
%the thermodynamics of the regular solutions. 

\section{Background}
\label{backg}
Several equations that follow from the assumed symmetries of the problem will be deduced in this section. 
The action
for the gravitational and the electromagnetic field is
given by (the notation used in this paper agrees with that in \citep{Bronnikov2000})
%according to Bronnikov's paper \citep{Bronn},
\begin{equation}
S= \frac{1}{16 \pi} \int{d^4x \sqrt{-g}\;[R-{\cal L}(F)]},
\label{action}
\end{equation}
where $R$ is the scalar curvature, $F_{\mu \nu} \equiv \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the electromagnetic field tensor, and ${\cal L}$ is an arbitrary function of $F \equiv F^{\mu \nu} F_{\mu \nu}$. 
%We shall require that ${\cal L}$ reduce to Maxwell's theory for small values of %$F$.
%\section{General features}
The equations of motion that follow from Eqn.(\ref{action}) are
\begin{equation}
G_{\mu \nu}=-T_{\mu \nu}
\label{einstein}
\end{equation}
\begin{equation}
\nabla_{\mu}({\cal L}_{F}F^{\mu \nu})=0,\;\;\;\; \quad \nabla_{\mu}(\ast F^{\mu \nu})=0,
\label{maxwell}
\end{equation}
where the subindex $F$ means derivative w.r.t. $F$, 
%the dual is defined by $\ast F_{\mu \nu} = %\eta_{\mu\nu\alpha\beta}F^{\alpha\beta}$, 
$\ast F_{\mu \nu}$ is the dual of $F_{\mu \nu}$,
and $T_{\mu \nu}= -2{\cal L}_{F}F_{\mu \alpha}F_{\nu}^{\cdot \alpha}+ \frac{1}{2}g_{\mu \nu} {\cal L} $. From the \textit{Ansatz}
$$
ds^2 = e^{2\gamma (r)}dt^2 - e^{2\lambda (r)}dr^2 - r^2d\Omega^2,
$$
and Eqns.(\ref{einstein}), it follows that $\gamma = - \lambda$. Hence we adopt the notation
\begin{equation}
ds^2=\left(1- \frac{2{\cal M}(r)}{r}\right)dt^2-\left(1- \frac{2{\cal M}(r)}{r}\right)^{-1}dr^2-r^2 d\Omega^2.
\label{metric}
\end{equation}
%where $d\Omega^2=d\theta^2+ \sin^2{\theta} d\phi^2$. 
%From
%equations ???, the following relations can be obtained,
Einstein's equations also yield
\begin{equation}
{\cal M}(r) =  k +\frac{1}{2} \int_{r_h}^{r}{T^{0}_{0}(r) r^2dr}.
\label{m}
\end{equation}
where $k$ is an integration constant.
The electromagnetic tensor compatible with spherical symmetry has two nonzero components ($F_{01}=-F_{10}$ and $F_{23}=-F_{32}$), corresponding to the radial electric and magnetic fields. In each case, it follows from Eqns.(\ref{maxwell}) that
\begin{equation}
r^2{\cal L}_{F}F^{01}=Q_e, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; F_{23}=Q_m \sin \theta,
\end{equation}
where $Q_e$ and $Q_m$ are the electric and magnetic charges, respectively.
With the definitions 
%(as in \citep{Bronn}),
%\begin{equation}
%%
$$f_e \equiv 2F_{01}F^{10}=2Q_e^2{\cal L}_{F}^{-2}r^{-4} \ge 0, \;\;\;\;\;\;\quad f_m=2F_{23}F^{23}=2Q_m^2 r^{-4} \ge 0,
$$
%%\end{equation}
the energy-momentum tensor can be written as
\begin{eqnarray}
T^\mu_\nu
%&=& -2L_{F}F_{\mu \alpha}F_{\nu}^{\cdot \alpha}+ \frac{1}{2}\delta_{\mu \nu} L \nonumber\\
&=& \frac{1}{2}\; {\rm diag} \left({\cal L}+2f_e{\cal L}_{F},{\cal L}+2f_e{\cal L}_{F}, {\cal L}-2f_m{\cal L}_{F}, {\cal L}-2f_m
{\cal L}_{F}\right).
\label{EE}
\end{eqnarray}
It follows that $T^0_{\;0} =  T^1_{\;1}$, 
%(which is a particular instance of the general result found in %\citep{Achucarro1995} and \citep{Nunez1996}, valid at the horizon),
and $T^2_{\;2} = T^3_{\;3}$.
The conservation of the energy-momentum tensor, $\nabla_{\mu}T^{\mu \nu}=0$,
yields
\begin{equation}
2T^{1}_{1}+r(T^{\;\prime\;1}_{1})-2 T^{2}_{2}=0,
\label{ec}
\end{equation} 
%$$
%2T^{0}_{0}+r(T^{0}_{0})'-2 T^{2}_{2}=0,
%$$
where a prime denotes the derivative wrt $r$. 
Combining the previous equation with $R=T = 2T^0_0 + 2 T^2_2$ we find that 
\begin{equation}
R = r(T^{\prime\;0}_{0})+ 4 T^0_0.
\end{equation}
These expressions will be used in the forthcoming sections, as well as  
in the calculation of the specific heat for an SSS black hole with a nonlinear electromagnetic field as a source, which we present next.
%\section{Specific heat $C_Q$}
The thermal capacities of 
%a given thermoyinamical system are 
%defined by
%\begin{equation}
%C_X = T\left(\frac {dS}{dT} \right)_X,
%\end{equation}
%where $X$ is a set of parameters that are being held constant. 
%In the case of 
a black hole
%, using $S=A/4$ and $T=\kappa/(2\pi)$, where $\kappa$ is the surface gravity, $C_X$ can be written as
are given by 
$$
C_X = \frac{\kappa}{4}\left(\frac {dA}{d\kappa} \right)_X,
$$
where $X$ is a set of parameters that are being held constant, $\kappa$ is the surface gravity, and $A$ the area, and all the functions are evaluated 
at the horizon radius $r_h$. 
The usual way to calculate $C_X$ involves the use of
the Smarr formula
% and the definition of the temperature 
\citep{Davies1977}.
Since in the case of black holes with a nonlinear electromagnetic source
this formula is no longer valid \citep{Rasheed1997, Breton2004}, we shall follow a different route.  
For SSS black holes, the surface gravity $\kappa$ is given by  
$\kappa = \frac 1 2\left.\frac{d{g_{00}}}{dr}\right|_{r_h}$. Taking into account that  
$g_{00}= 1-2{\cal M}(r)/r$, and ${\cal M}(r)$ is given by Eqn.(\ref{m}),
%
%From Eqns.(\ref{metric}) and (\ref{m}), 
it follows that 
\begin{equation}
\kappa = \frac{1}{2r_h}\left[1-r_h^2T^0_{\;0}(r_h)\right],
\label{kappa}
\end{equation}
an expression which agrees with
the one obtained in \citep{Visser1992}.
The
specific heat at constant charge, defined by 
$C_Q = \frac{\kappa}{4}\left(\frac {dA}{d\kappa} \right)_Q
$
can be calculated 
using
\begin{equation}
C_Q = \frac{\kappa}{4}\left(\frac {dA}{dM}\frac {dM}{d\kappa} \right)_Q,
\end{equation}
with $\kappa = \kappa (r_h)$ given by Eqn.(\ref{kappa}), $A=4\pi r_h^2$,
and  
$r_h=r_h(M,Q)$, and $M$ and $Q$ are the total mass and charge
of the black hole \footnote{The calculation of $C_\Phi$, and of the analogous of the thermal expansion and the isotermal compressibility involve Legendre transformations, and the use of the Smarr formula, which is not available in this context.}.
A straightforward calculation (which is independent of the value of the integration constant $k$ in Eqn.(\ref{m}))
yields
\begin{equation}
C_Q=-2 \pi r_{h}^{2}\; \frac{1-r_{h}^{2} T^{0}_{\;0}(r_h)}{1+r_{h}^{2}T^{0}_{\;0}(r_h)+r_{h}^{3} T^{\prime\; 0}_{\;0}(r_h)},
\label{cq}
\end{equation}
where a prime denotes the derivative with respect to the radial coordinate.
%the EOM for the electromagnetic field have not been used to obtain this expression, so it 
This expression is actually valid for any charged SSS black hole, 
% with a given energy-momentum tensor as source. It 
and it reduces to the well-known formula
for the case
of the RN black hole, given by
%\begin{equation}
%C_{\rm{Sch}}=-2 \pi r_0^2 ,
%\end{equation}
\begin{equation}
C^{\rm{(RN)}}_Q=-2 \pi r_{h+}^2 \frac{r_{h+}^2-Q^2}{r_{h+}^2-3Q^2},
\end{equation}
%Este es f\'acil de graficar $C_q$ en funci\'on de $R_{+}$ y sabemos que para cargas chicas aprox.  $R_{+}$ es proporcional a $M$,
%$R_{+}=M+ \sqrt{M^2-q^2}$. Ver gr\'afico 8.4 en T. Ortin, p. 241.
where $r_{h+}$ is the external horizon of the RN solution.
The correct expression for the Schwarzschild solution follows from the latter when $Q=0$.

Defining $D = -(1+r_{h}^{2}T^{0}_{\;0}(r_h)+r_{h}^{3} T^{\prime\; 0}_{\;0}(r_h))$, 
and taking into account that \citep{Bekenstein1998}
\beq
0<T_0^0(r_h) \le 1/r_h^2,
\label{desbek}
\eeq
it follows that
the sign of $C_Q$ 
is given by that of $D$, and any possible divergence in $C_Q$ must arise
from a zero of $D$. In fact, using Eqns.(\ref{ec}) and (\ref{desbek}), it follows that
a necessary condition for $D=0$ is  
$$
\frac{1}{2r_h^2} < \left.(T^0_0-T^2_2)\right|_{r_h}\leq \frac{1}{r_h^2}.
$$
For completeness, we also give the expression of 
$C_Q$ in terms of ${\cal M}(r)$ and its derivatives evaluated at $r_h$, 
\begin{equation}
C_Q=-2 \pi r_h^2 \frac{1-2{\cal M}'(r_h)}{1+2{\cal M}'(r_h)+2r_h{\cal M}''(r_h)}.
\end{equation}
%The plots show a generic behaviour that for small charges the black hole %behaves Schwarzschild-like
%and increasing the charge the black hole presents the divergence like a %Reissner-Nordstrom black hole.
%The specific charge can be found from the zeroes in the denominator of the $C_q$ expression,
We shall present next the plots of $C_Q$ for several analytic SSS black hole solutions corresponding to different NLEM theories, along with an analysis of the position of the external horizon in each case. 

\subsection{Born-Infeld black hole}

\noindent The properties of this singular solution have been discussed in several articles (see for instance \citep{Garcia1984, Chemissany2008,Breton2002,Breton2003}). The Born-Infeld Lagrangian is given by
\begin{equation}
{\cal L}=4b^2\left(-1+\sqrt{1+\frac{F}{2b^2}}\right).
\end{equation}  
%Using 
%\begin{equation}
%T^\mu_\nu = 2{\cal L}_FF^\mu_\alpha F^\alpha_\nu + \frac{{\cal L}}{2},
%\end{equation}
%it follows that
%\begin{equation}
%T^0_0 =
%\end{equation}
With the metric element given by Eq.(\ref{metric}), 
%and using the notation
%$A(r) = 1 - 2{\cal M}(r)/r$, 
the black hole geometry is determined by the function \citep{Garcia1984}
\begin{equation}
g_{00}(R)=1-\frac{2m}{R}+\frac 2 3 R^2 \left(1-\sqrt{1+\frac{q^2}{R^4}}\right)+\frac 4 3 
\frac{q^2}{R}\int_{R}^{\infty}\frac{dz}{\sqrt{z^4+q^2}},
\label{abi}
\end{equation}
where $R\equiv br$, $q\equiv bQ$, and $m\equiv Mb$, and $b$ has units of $\rm [length]^{-1}$.

In order to use Eqn.(\ref{cq}) to calculate $C_q$, the radius of the 
(external) horizon is needed. This is the value $R_h$ such that 
$g_{00}(R_h)=0$. The plots in Fig.\ref{fig13}
show the values of $R_h$ for different values of the parameters. 
\begin{figure}
\centering
\mbox{\subfigure{\includegraphics[width=2.3in]{horizonsbi1.eps}
\quad
\subfigure{\includegraphics[width=2.3in]{horizonsbi2.eps}}}}
\caption{The mass of the Born-Infeld black hole as a function of $R_h$ for different values of the charge.}
\label{fig13}
\end{figure}
Depending on the value of the mass, for
$q < q_{\rm min} = 0.5$, there may be one or no horizon, 
while for $q\geq q_{\rm min}$ there may be two, one, or no horizons
\footnote{The extremal case, in which the two horizons coalesce into one, is given by 
$q^2=\frac{1}{4}+R_h^2$ \citep{Chemissany2008}.}. 
The plots also show that for $R_h>>q$, $R_h \approx 2m$, as can be seen from Eq.(\ref{abi}).

The expression for $C_q$ 
for the electrically charged Born-Infeld black hole 
%\footnote{The magnetic case...} 
can be computed from Eqn.(\ref{cq}). The 
result is
\begin{equation}
C_{q}=-2 \pi R_{h}^2 \frac{(1+2R_{h}^2)\sqrt{R_{h}^4+q^2}-2(R_{h}^4+q^2)}{(1-2R_{h}^2)\sqrt{R_{h}^4+q^2}+2(R_{h}^4-q^2)}.
\label{ecqbi}
\end{equation}
Fig.\ref{cqbi} shows the plots of $C_q$ as a function of $m$ for two different values of $q$. 
They show a divergence that resembles that of the RN black hole, and reduce to the Schwarzschild's case in the limit $R_h>>q$, as can be checked from Eqn.(\ref{ecqbi}).
\begin{figure}
\centering
\mbox{\subfigure{
\quad
\subfigure{\includegraphics[width=2.8in]{CqBI3.eps}
\quad
\subfigure{\includegraphics[width=2.8in]{CqBI4.eps}}}}}
\caption{Plot of $C_q$ as a function of $m$ with $q=0.6$ (left) and $q=1$ 
(right) for the Born-Infeld black-hole.}
\label{cqbi}
\end{figure}
%\begin{figure}[h]
%       \centering  % figura centralizada
%       \includegraphics[width=0.8\textwidth]{BICqdot.eps}
%       \caption{Plot of $C_q$ as a function of $m$ with $q=0.6$ (left) and %$q=1$ 
%(right) for the Born-Infeld black-hole.}
%       \label{cqbi}
%\end{figure}

\subsection{Nonsingular solutions}

In this section, we shall consider two solutions that are different from the one analyzed in the previous section at least in two respects: (a) they are non-singular, and 
(b) the parameter which measures the gravitational mass 
of the point sources of the electromagnetic field (namely, the integration constant $k$ in Eq.(\ref{m})) is zero, in spite of the fact that these geometries approach asymptotically a RN-like solution. This latter fact is due to the emergence of 
an effective electromagnetic mass $M_{EM}$, defined  in  terms  of  the  asymptotic  behavior  of  the related gravitation field (see for instance
\citep{Pellicer1969}) in such a way that, as we shall see below, $M_{EM}\propto Q^{3/2}$. 

\subsubsection{Bronnikov solution}

This is a static, spherically symmetric and nonsingular 
magnetic 
black hole solution derived in \citep{Bronnikov2000}, for a theory with Lagrangian given by
%${\cal L}(F)$ and its derivative are
\begin{eqnarray}
{\cal L}(F)&=& F {\rm sech}^2 \left[{a\left({\frac{F}{2}}\right)^{1/4}}\right], %\\
%{\cal L}_F(F)&=& {\rm sech}^2\left[{a \left({\frac{F}{2}}\right)^{1/4}}\right]
%\left[1- {\frac{a}{2} \left({\frac{F}{2}}\right)^{1/4}} %\tanh\left({a\left({\frac{F}{2}}\right)^{1/4}}\right)\right], 
\end{eqnarray}
where $a$ is a parameter \footnote{This Lagrangian was inspired in the one presented in \citep{Ayon1999}.}.
%line element
%is given by
%\begin{equation}
%ds^2 = A(r)dt^2-A(r)^{-1}dr^2-r^2 d \Omega^2,
%\label{metric2}
%\end{equation}
The
relevant metric function is given by
\begin{equation}
g_{00}(r) =  1- \frac{Q_m}{a^2}\left({\frac{a \sqrt{Q_m}}{r}}\right) \left[{1- \tanh{\left({\frac{a \sqrt{Q_m}}{r}}\right)}}\right],
\label{g00bron}
\end{equation}
%with $g=q_m$ the magnetic charge. 
%It is a regular solution, since $A(r=0)=1$. 
and the effective electromagnetic mass for this geometry is
\begin{equation}
M_{EM}= M(r \to \infty)=\frac{Q_m^{3/2}}{2a}.
\end{equation}
%
%Therefore $4\xi^2=g/a^2$, and
%
The solution behaves as a black hole or as a soliton depending on the value of the parameter $\xi=\frac{M_{EM}}{Q_m}$.
If $\xi=\xi_0\approx 0.96$, then $g_{00}(r)=0$ has only one (double) root, in which case the black hole would be extreme.
If $\xi > \xi_0$, $g_{00}(r)=0$ has two different real roots, hence the black hole has two horizons. Otherwise the solution represents a soliton ($g_{00}(r)>0$ for all values of the radial coordinate). 
The position of the zeros of Eqn.(\ref{g00bron}) is given by \footnote{As shown in \citep{Matyjasek2000}, the location of the horizons can be expressed in terms of the Lambert function.} 
\beq
1-\tanh \left(\frac{1}{2\xi^2R_h}\right) = \frac{R_h}{2},
\label{zerosb}
\eeq
where $R_h\equiv r_h/M$.
The radius of the external horizon in terms of $\xi$ is shown in Fig.\ref{horbron}. It tends to $2M$ for large values of $\xi$.
\begin{figure}
%\centering
%\mbox{\subfigure{
\includegraphics[width=3in]{horbron.eps}
%\quad
%\subfigure{\includegraphics[width=2in]{horbron1.eps}
%\quad
%\subfigure{\includegraphics[width=2in]{horbron3.eps}}}}
\caption{Size of the external horizon $R_h$ of Bronnikov's solution in terms of $\xi$, determined by Eqn.(\ref{zerosb})
.}
\label{horbron}
\end{figure}
The expression for the specific heat follows from Eqn.(\ref{cq}):  
\begin{equation}
\frac{C_{Q}}{M^2}=-4\pi R_h^3\xi^2\frac{4R_h\xi^2+R_h-4}{8R_h^2\xi^4-(4-R_h)(6R_h\xi^2+R_h-2)},
\label{cqbron}
\end{equation}
and is plotted as a function of $\xi$ in 
Fig.\ref{cqbr}.  
\begin{figure}[h]
       \centering  % figura centralizada
       \includegraphics[width=3in]{cqbron.eps}
       \caption{Plot of $C_q$ as a function of $\xi$ for the black hole represented by Eqn.(\ref{g00bron}).}
       \label{cqbr}
\end{figure}
The plot displays a divergence similar to that of the RN solution, 
and we have verified that it tends to $C_Q^{\rm{(Schw)}}$ for large $\xi$, 
as can be seen from Eqn.(\ref{cqbron}).

\subsection{Dymnikova solution}

The regular, static and spherically symmetric geometry studied in \citep{Dymnikova2004} has a nonzero electric field 
described by the Lagrangian
\begin{equation}
{\cal L} = \frac{F}{(1+\alpha\sqrt F)^2},
\end{equation}
where $F=-2Q_e^2r^8/(r^2+r_e^2)^6$, $Q_e$ is the electric charge, 
$\alpha = r_e^2/(\sqrt{2}Q_e)$, and $r_e = \pi Q_e^2/(8M)$ is proportional to the classical electromagnetic radius. The metric is determined   by
\begin{equation}
g_{00}(r) = 1-\frac{4M}{\pi r}\left[\arctan\left(\frac{r}{r_e}\right)-\frac{rr_e}{r^2+r_e^2}\right],
\label{metdymn}
\end{equation}
and it goes to a RN-like metric at infinity, as in the previous case. 
The parameter $\beta$ given by
\beq
\beta = \frac{8}{\pi^2}\left(\frac{2M}{Q_e}\right)^2
\eeq
discriminates between a 
regular electrically charged black hole and
a self-gravitating particle-like structure. 
A single-horizon black hole is described by 
$\beta = \beta_0 \approx 2.82$, while the solution 
with two horizons has $\beta > \beta_0$ \footnote{This solution evades the no-go theorem presented in \citep{Bronnikov2000} because the Lagrangian does not go
to the Maxwell's limit at the center,
see \citep{Dymnikova2004}.}.
The position of the horizon(s) is given by 
\begin{equation}
\arctan (R_h\beta) - \frac{R_h\beta}{\beta^2R_h^2+1} = R_h,
\end{equation}
where $R_h \equiv r_h/\mu$, and $\mu \equiv 4M/\pi$.
The variation of the radius of the external horizon 
with $\beta$ is plotted in 
Fig.\ref{hordymn}. The plot shows that the solution tends to the Scharzschild 
black hole for large $\beta$.
\begin{figure}
\centering
%\mbox{\subfigure{
\includegraphics[width=3in]{hordymn.eps}
%\quad
%\subfigure{\includegraphics[width=2in]{horbron1.eps}
%\quad
%\subfigure{\includegraphics[width=2in]{hordymn1.eps}}}}
\caption{Plot of the external horizon as a function of $\beta$ for the solution given in Eqn.(\ref{metdymn}).}
\label{hordymn}
\end{figure}
The expression for the specific heat is in this case
%\begin{eqnarray}
%C_{q}&=&-2 \pi r_{0}^2 %\;\frac{(r_{0}^2+r_{e}^2)^3-q^2r_{0}^2(r_{0}^2+r_{e}^2)}{(r_{0}^2+r_{e}^2)^3+q^2r_{0}^2
%(r_{e}^2-3r_{0}^2)},
%\end{eqnarray}
\begin{equation}
\frac{C_{Q}}{\mu^2} = -2\pi R_h^2\;(\beta^2R_h^2+1)\;\frac{(1+R_h^2\beta^2)^2-2\beta^3R_h^2}{(\beta^2R_h^2+1)^3+2\beta^3R_h^2(1-3\beta^2R_h^2)},
\label{cqdymn}
\end{equation}
%\begin{figure}[h]
%       \centering  % figura centralizada
%       \includegraphics[width=0.4\textwidth]{hordymn.eps}
%       \caption{Position of the external horizon as a function of $\beta$ in %the solution ....}
%       \label{hd}
%\end{figure}
and is plotted as a function of $\beta$ in Fig.\ref{cqd}.
\begin{figure}[h]
       \centering  % figura centralizada
       \includegraphics[width=3in]{cqdymn.eps}
       \caption{Plot of the heat capacity at constant $Q$ as a function of $\beta$ for the black hole described by Eqn.(\ref{metdymn}).}
       \label{cqd}
\end{figure}
The RN-like divergence is again present. We have verified that the plot of $C_Q$ tends to $C_Q^{\rm{(Schw)}}$ for large $\beta$, in accordance with Eqn.(\ref{cqdymn}).


\section{Stability according to the Poincar\`e method}

We shall analyze next the stability of the black hole configurations
of the previous sections following the Poincar\`e method.
Let us start by giving a short summary of it, following the presentation in  \citep{Arcioni2004}. 
%As discussed in the Introduction, 
%local thermodynamical stability does not imply dynamical stability in the case %of
%black holes, since these are nonextensive systems {\textcolor{red} {CUIDADO}}. 
The Poincar\`e method of stability 
%(aka the turning point method) 
succeeds in showing the existence of unstable modes from the properties of the equilibrium sequence alone, without the need of analyzing the 
Hessian of the system. 
%We shall now briefly review it mostly following the presentation in \citep{Arcioni2004}. 
Let ${\cal M}$ be the configuration space of the system under scrutiny, and $X$ a point in ${\cal M}$. The set of independent thermodynamical variables that specify the ensemble shall be denoted by $\{\mu^i\}$, and $S$ will be the corresponding Massieu function. 
The equilibrium states are defined as the points in ${\cal M}$ that are extrema of $S$ under displacements $dX$ for which $d\mu^i = 0$. At any such point, the conjugate thermodynamical variables ${\beta_i}$ are defined by 
\begin{equation}
dS = \beta_id\mu^i
\label{ds}
\end{equation}
for all $dX$. The set of equilibrium points form a submanifold in ${\cal M}$, referred to as ${\cal M}_{eq}$, each point of which can be labelled by the corresponding set of values $\{\mu^i\}$. The maximum entropy postulate states that unconstrained locally stable equilibrium points take place at points in ${\cal M}_{eq}$ in which $S$ is a local maximum with respect to arbitrary variations $dX$ in ${\cal M}$ that preserve $d\mu^i=0$. The fundamental relation $S_{eq} = S(\mu^i)$ can be obtained from Eq.(\ref{ds}), and then 
$$
\beta_i(\mu^i)=\frac{\partial S_{eq}}{\partial \mu^i}
$$
are the equations of state, which give the value of the conjugate variables at equilibrium. 

The starting point of the Poincar\`e method is the fact that the maximum entropy  postulate refers to the behaviour of 
$S$ along off-equilibrium states, not about its variation on ${\cal M}_{eq}$. In other words, the function $S_{eq}$ is not related to local stability without assuming additivity. To inquiry about local stability, an extended Massieu function $\hat S$ is needed, such that it gives the behaviour of $S$ near the equilibrium states. Although $\hat S$ is generally unknown, information about it (and hence about stability) can be obtained with the Poincar\`e method by using only the equilibrium equations of state. We shall give here the recipe that is to be used to identify changes in stability (for the proof of the method see \citep{Katz1978, Parentani1995, Sorkin1981, Sorkin1982}):
a change of stability happens when the plot of a conjugacy diagram $\beta_a(\mu^a)$ (for fixed $a$) along the equilibrium series has a vertical tangent (thus showing a turning point). In other words, a turning point necessarily implies instability, without any other assumptions 
\footnote{Actually, there is another instance that implies a change of stability, namely the existence of a bifurcation point (\textit{i.e.} the crossing point of two sequences of equilibria).}. 

Next we shall apply the turning point method
to the
three static and spherically symmetric black hole solutions with a nonlinear electromagnetic field as a source presented above. 
%Notice that singular and regular solutions
%need to be treated separately, since in the latter case, the charge and the %mass are not independent parameters (see below).
%
%\subsection{Singular solutions}
%
%In the case of singular charged black holes with a NLEM source, 
In the solutions under scrutiny there are two pairs of conjugate parameters that are of interest regarding stability. These are $(M,\beta_M)$ and $(Q,\beta_Q)$. From the first law of black hole mechanics, 
\begin{equation}
dA = \frac{8\pi}{\kappa_h}dM-8\pi \frac{\Phi_h}{\kappa_h} dQ,
\end{equation} 
where 
the electric potential $\Phi$ at the horizon
is given by
$$
\Phi_h = Q\int_{r_h}^{\infty} \frac{dr}{r^2{\cal L}_F},
$$
we see that $\beta_M = \frac{8\pi}{\kappa_h}$ and $\beta_Q = -8\pi \frac{\Phi_h}{\kappa_h}$. 
It follows from these expressions that the corresponding plots should display divergencies in the case of an extremal black hole
(for which $T=0$ \footnote{See however the discussion in \citep{Liberati2000}.}). We shall see below that this is the case.  

We shall try first to keep
the stability analysis general. Hence, 
no commitment to a specific form of ${\cal L}$ will be adopted. In the case of $(M,\beta_M)$, although we do not have the fundamental relation, the symmetries of the solution allow the calculation of $\frac{\partial M}{\partial \beta_M}$ and of the second derivative for an arbitrary Lagrangian, much in the same way as the $C_Q$ was calculated. 
The results are:  
\begin{equation}
\frac{\partial M}{\partial \beta_M}=\frac{r_h^3}{4\pi}\frac{\kappa_h^3}{1+r_h^2T^0_0(r_h)+r_0^3T^{\prime 0}_0(r_h)},
\label{stab}
\end{equation}
\begin{equation}
\frac{\partial^2 M}{\partial \beta_M^2}  =  \frac{r_h^3\kappa_h^4}{32\pi^2}\;
\frac{
\left[-4r_h^2T^0_0(r_h)(2+r_h^2T^0_0(r_h))
-r_h^3T^{\prime 0}_0(r_h)(7+5r_h^2T^0_0(r_h)+3r_{h}^{3}T^{\prime 0}_{0}(r_h))+r_{h}^{4}
T^{\prime\prime 0}_{0}(r_h)
(r_{h}^{2}T^{0}_{0}(r_h)-1)\right]}{(1+r_h^2T^0_0(r_h)+r_h^3T^{\prime 0}_0(r_h))^3}.
\label{stab2}
\end{equation}
From these expressions we see that the plot of $M$ against $\beta_M$  
cannot have extrema for $\kappa_h \neq 0$. Hence, there are no turning points in the 
($\beta_M, M)$ plane (we shall see below that the plots for the Born-Infeld black hole confirm this assertion). It is also seen that $\frac{\partial\beta_M}{\partial M}$ diverges for the extremal case. Note that these are general statements, in the sense that they are valid for any charged black hole with
the assumed symmetries. 

We can proceed in a similar way with the pair of conjugate variables $(Q,\beta_Q)$. 
The charge of the black hole is given by \citep{Rasheed1997}
\beq
Q = -\frac{1}{8\pi}\varoint dS_{\mu\nu}G^{\mu\nu},
\eeq
where 
\beq
G^{\mu\nu} \equiv -\frac{1}{2} \frac{\partial{\cal L}}{\partial F_{\mu\nu}}.
\eeq
The integral may be calculated over any closed 2-surface enclosing the charge
and is independent of the particular surface chosen. 
In the case of an electric charge, and choosing the horizon as the surface, we get
\beq 
Q = r_h\left.({\cal L}_{F}F^{tr})\right|_h,
\eeq 
which follows also from Eqn.(\ref{maxwell}). Since the calculation of $\beta_Q = -8\pi \frac{\Phi_h}{\kappa_h}$ is carried out at fixed $M$, the rhs is a function of $r_h$. It follows that
\beq
\frac{\partial Q}{\partial \beta_Q} = \frac{\partial Q}{\partial r_h}\frac{\partial r_h}{\partial \beta_Q}.
\eeq
Hence,
\beq
\frac{\partial Q}{\partial \beta_Q} = \frac{\kappa_h^2}{8\pi}\;\frac{
\frac{\partial Q}{\partial r_h}}
{\Phi_h\frac{\partial\kappa}{\partial r_h}-\kappa_h 
\frac{\partial \Phi_h}{\partial r_h}}.
\eeq
To have a turning point, it is necessary that $\frac{\partial Q}{\partial r_h} = 0$. In the case of the R-N black hole, $Q = \sqrt{M^2-(r_h-M)^2}$, and the necessary condition is satisfied only in the extremal case, $r_h = M$. To examine other black hole solutions, the equation $\frac{\partial Q}{\partial r_h} = 0$ should be solved numerically, and the sign of the second derivative should be calculated at the zero of the first derivative. In the next sections we will take an alternative road, which consists in directly plotting $\beta_Q$ 
%as a function of $Q$ for different values of $M$, 
for the black hole solutions presented in the previous sections. 
As in the previous pair of conjugate variables, $\frac{\partial\beta_Q}{\partial Q}$ diverges for the extremal case. 


\subsubsection{Born-Infeld black hole}

We have analitically shown above that there are no turning points in the $(M,\beta_M)$ plane
for any ${\cal L}(F)$. The plots in Fig.\ref{qfija} for the Born-Infeld solution 
are in agreement with this assertion, and display the divergence mentioned above for the extremal black hole.
\begin{figure}
\centering
\mbox{\subfigure{\includegraphics[width=2in]{biq02.eps}
\quad
\subfigure{\includegraphics[width=2in]{biq6.eps}
\quad
\subfigure{\includegraphics[width=2in]{biq12.eps}}}}}
\caption{Variation of $\beta_m$ with $m$ at fixed $q=0.2,3,10$, from left to right for the Born-Infeld black hole. The plots do not exhibit turning points.}
\label{qfija}
\end{figure}
For the plot of $\beta_Q = -8\pi \frac{\Phi_h}{\kappa_h}$ as a function of $Q$, the electric potential and the surface gravity are needed. They are given by
$$
\Phi_h = q\int_{R_h}^{\infty} \frac{dx}{\sqrt{q^2+x^4}},
$$
\begin{equation}
\frac{\kappa_h}{b}=\frac{1}{ 2 R_h}\left[1+2R_h^2\left(1-
\sqrt{1+\frac{q^2}{R_h^4}}\right)\right]
\label{kappa2}
\end{equation}
Fig.{\ref{mfija} shows that there are no turning points in the 
$\beta_Q$ vs $Q$ plot. The divergence corresponds to the extreme case, given by 
$q^2=\frac{1}{4}+R_h^2$ which, from Eq.(\ref{kappa})
is also where $\kappa_h = 0$.

\begin{figure}
\centering
\mbox{\subfigure{\includegraphics[width=2in]{bim1.eps}
\quad
\subfigure{\includegraphics[width=2in]{bim5.eps}
\quad
\subfigure{\includegraphics[width=2in]{bim10.eps}}}}}
\caption{The plots show the variation of $\beta_q$ with the charge, for $m=1,5,10$, from left to right.}
\label{mfija}
\end{figure}
Hence, it follows from the Poincar\`e method that no changes of stability are possible in this solution. 
Since this geometry reduces in the case $Q = 0$ to Schwarschild's solution (which is stable \citep{Vishveshwara1970}), 
our result indicates that the solution is stable
(if no bifurcations are present). 
%We shall see below that this is consistent with the dynamical stability
%showed in \citep{Moreno2002}.
%
%we conclude that this solution is stable with respect to the perturbations
%encompassed by the turning point method.

%\subsection{Nonsingular solutions}%
%
%In this type of black holes, there is a relation between the mass and the %charge... Hence the first law must be written as $
%dA = \beta_Q dQ,
%$
%where 
%$$
%\beta_Q = %\frac{8\pi}{\kappa_0}\left(\left.\frac{dM}{dQ}\right|_0-\Phi_0\right). 
% $$
We shall present next the plots of $\beta_Q$ as a function of $Q$ for the two nonsingular solutions mentioned above. 

\subsubsection{Bronnikov solution}

As discussed in the begining of this section, 
we need only to examine the behaviour in terms of $Q$
of the function $\beta_Q$. The necessary quantities 
for this calculation are the magnetic potential and the surface gravity, given by
\beq
\psi_h = \frac{1}{\xi}\left\{\frac{1}{4R_h}{\rm sech}^2\left(\frac{1}{2R_h\xi^2}\right)
+\frac 3 2 \xi^2 \tanh\left(\frac{1}{2R_h\xi^2}\right)\right\},
\eeq
\beq
M\kappa_h = \frac{1}{2R_h}\left\{1-\frac{1}{\xi^2 R_h^2}\cosh^{-2}\left(\frac{1}{2R_h\xi^2}
\right)\right\}.
\eeq
The plot of $\beta_Q$ as a function of $Q$ shows no turning points, and the divergence corresponding to the extremal case. Since 
this solution reduces to that of Schwarzschild for large values of $\xi$,
it follows that it is stable (if no 
bifurcations are present).
\begin{figure}[h]
       \centering  % figura centralizada
       \includegraphics[width=0.5\textwidth]{Beta_qBronn4.eps}
       \caption{Plot of the conjugate parameters for the Bronnikov solution.}
       \label{pb}
\end{figure}


\subsubsection{Dymnikova solution}
The relevant quantities for the examination of the function $\beta_Q$ are the electric potential and the surface gravity, respectively given by
\beq
\phi_h = \frac{3\sqrt {2\beta}}{8}\left\{\frac\pi 2 -R_h+\frac{2}{3}\beta^3 \frac{R_h^3}{(\beta^2 R_h^2+1)^2}\right\},
\eeq
\beq
\mu\kappa_h = \frac{1}{2R_h}\left(1-2\beta^3\frac{R_h^2}{(\beta^2 R_h^2+1)^2}\right).
\eeq
The dependence of $\beta_Q$ with $Q$ (displaying the divergence for the extremal black hole) is shown in 
Fig.\ref{mfijadymni}.
%\begin{equation}
%{\cal L} = \frac{F}{(1+\alpha\sqrt F)^2}
%\end{equation}
%where $\alpha = r_0^2/(\sqrt{2}g)$, $r_0 = \pi g^2/(8m)$, $F=2g^2/r^4$, and $g$ %is the magnetic charge. The corresponding mass function
%is given by
%\begin{equation}
%M(r) = %\frac{2m}{\pi}\left[\arctan\left(\frac{r}{r_0}\right)-\frac{rr_0}{r^2+r_0^2}\right]
%\end{equation}
\begin{figure}[h]
       \centering  % figura centralizada
       \includegraphics[width=0.5\textwidth]{Beta_qDimnik3.eps}
       \caption{The plot shows the variation of $\beta_Q$ with $Q$ for the Dymnikova solution.}
      \label{mfijadymni}
\end{figure}
%\begin{figure}
%\centering
%\mbox{\subfigure{\includegraphics[width=2in]{D0_6.eps}
%\quad
%\subfigure{\includegraphics[width=2in]{D1.eps}
%\quad
%\subfigure{\includegraphics[width=2in]{D2.eps}}}}}
%\caption{The plots show the variation of $\beta_q$ with $q$ at fixed %$m=0.6,1,2$, from left to right for the Dymnikova solution.}
%\label{mfijadymni}
%\end{figure}
As in the previous case, 
the solution is stable (if no 
bifurcations are present), since it reduces to the Schwarzschild black hole for 
large $\beta$.



\section{Comparison of the different methods} 

In this section we compare the results obtained via thermodynamics with 
those that follow from the dynamics of gravitational perturbations
\footnote{Let us remark that the stability of the solutions examined here was inspected in \citep{Breton2005}using the 
Ashtekar-Corichi-Sudarsky (ACS) conjecture. 
In the present work, the situation is different from the one considered in 
\citep{Breton2005} in the boundary conditions. 
The range we considered for the dynamical stability test is the domain of outer communication (DOC), $r \ge r_{h}$, whereas the solitonic solution is not considered at all. In other words, 
we are analyzing completely isolated black holes, otherwise the microcanonical ensemble would not be appropriate for  the thermodynamical analysis, and therefore, the results in \cite{Breton2005} do not correspond to the situation analyzed in this paper.
}.
We have shown that in the three cases examined there is a divergence in the plots of the specific heat at constant charge, as in the RN solution. 
While this divergence has been taken as an indication of thermodynamical instability, our results obtained using the Poincar\`e method showed that
the 
Born-Infeld, Bronnikov, and Dymnikova black holes
are stable, assuming in all cases that no bifurcations are present.

The results obtained via thermodynamics can be compared to those coming 
from a dynamical analysis. 
The dynamical stability of the Born-Infeld black hole under gravitational perturbations has been established in \citep{Fernando2004}.
While we are not aware of any calculation about the dynamical stability
of the two the regular solutions considered here, 
a general analysis of the dynamical stability of SSS black holes with NLEM as a source has been 
presented in \citep{Moreno2002}. As a result, sufficient conditions for dynamical stability were given. 
In the magnetic case they establish that, if 
${\cal L}>0$, ${\cal L}_y>0$, ${\cal L}_{yy}>0$, and $3{\cal L}_y>yg_{00}{\cal L}_{yy}$, the solution is stable, where $y=\sqrt{Q_mF/2}$. All these conditions are satisfied by Bronnikov's solution in the allowed range of parameters, hence it is dynamically stable.

For the electric case, it is convenient to work in the so-called P frame
of nonlinear electrodynamics (see for instance \citep{Salazar1987}). In this case, the sufficient
stability conditions given in \citep{Moreno2002} read
${\cal H}<0$, ${\cal H}_x<0$, ${\cal H}_{xx}<0$, and $3{\cal H}_x\leq x g_{00}{\cal H}_{xx}$, where $x = \sqrt{-2Q_e^2P}$. For Dymnikova's solution, 
$$
{\cal H} = \frac{P}{(1+\alpha\sqrt{-P})^2},$$ and it follows that the solution is dynamically stable when the allowed range of parameters is considered. 

\section{Conclusions}

%In order to 
%analyze the thermodynamical stability of three static and spherically symmetric black holes with a nonlinear electromagnetic field as source, 
We derived an expression for the specific heat at constant charge valid for any charged SSS black hole. The $C_Q$ displays, in the three cases examined, a divergence reminiscent of that of the RN solution. 
As a byproduct, 
plots of the radius of the horizon(s) with the relevant parameters of the three solutions considered were obtained. \\
We also studied the thermodynamical stability using the Poincar\`e method. 
Our results for the diagrams of the conjugate variables show that no instability
is present, as is also the case in the RN solution. 
In particular, we have shown that no turning point is possible in the $(\beta_M, M)$ plane for any charged SSS black hole. 

We have also compared our results with those coming from 
studies of dynamical stability. 
Both the Born-Infeld black hole
and the regular solutions are dynamically stable. The absence of turning points for these solutions
might be a hint to the fact that for charged SSS black holes, 
dynamical stability may be equivalent to 
thermodynamical stability \footnote{The same could be said of the RN solution.}, as was shown in \citep{Hollands2012} for vacuum black holes
in General Relativity.  This issue deserves further investigation. 

Regarding the nonsingular solutions, it must be noted that their chararacter seems to change through the addition of gravitational mass, since the coefficient $k$ in Eqn.(\ref{m}) becomes different from zero. As long as the term
$2M_{{\rm grav}}/r$ in this equation remains small, the radius of the horizons (and hence the thermodynamics) will not be very different from the results obtained here. The opposite case, as well as the change in the solution, warrants a closer examination, that will be taken elsewhere.  

%To close this section, we would like to comment that as a result of the addition of gravitational mass through processes with $Q=$ constant, the two nonsingular black holes analyzed above will develop a nonzero ``purely gravitational'' mass
%$M_{\rm{grav}}$. Hence, the results plotted in Figs. (\ref{cqbr}) and (\ref{cqd}) are valid as long as the term $2M_{\rm{grav}}/r$ in Eqn.(\ref{m}) is small. 

%It must be emphasized that the results plotted in Figs.(\ref{pb}) and (\ref{mfijadymni}) are valid as long as the term $2M_{grav}/r$ in Eqn.(\ref{m}) is small. 
%Extensions of this work to SSS black holes with (A)dS behaviour at infinity %(which are interesting
%in view of a possible phase transition \citep{Chamblin1999}) are in progress, %as well 
%as to the case in which the Lagrangian is a function of the invariant $F$ and
%$G=F\ast F$, with the possibility of duality invariance \citep{Gibbons1995}. 


%**********************************************************************
\begin{acknowledgments}
N. B. acknowledges partial support from CONACyT, project 166581.
SEPB would like to acknowledge support from FAPERJ, CLAF, UERJ, and CNPQ. 

\end{acknowledgments}

%\bibliographystyle{apsrev4-1} %archivo.bst
\bibliographystyle{plain} %archivo.bst
\bibliography{bibliography} %archivo.bib

\end{document}

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\bibitem{Quevedo}
D. N\'u\~nez, H. Quevedo and D. Sudarsky, 
Phys. Rev. lett. \textbf{76} 571 (1996).

\bibitem{Mayo}
A. Mayo and J. D. Bekenstein,
Phys. Rev. D \textbf{54} 5059 (1996).

\bibitem{Rasheed}
Rasheed, D. A., \texttt{hep-th/9702087}.

\bibitem{Heusler}
Heusler, M., Class. Quant. Grav.
\textbf{12}, 779 (1995).

\bibitem{Komar}
A. Komar, Phys. Rev. \textbf{127}1411 (1962).

\bibitem{vol_element}
R. M. Wald, \textit{General Relativity}, The Univ. of Chicago Press (1984)
Appendix B. See also: T. Ort\'{\i}n, \textit{Gravity and Strings}, Cambridge Univ. Press (2004), Sects. 1.6 and 1.7.

\bibitem{mio} Santiago Esteban Perez Bergliaffa, Yves Eduardo Chifarelli de Oliveira Nunes,
Phys.Rev. D84 (2011) 084006, \texttt{1107.5727 [gr-qc]}.

\bibitem{visserdirty} 
Matt Visser, Phys.Rev. D46 (1992) 2445-2451,
\texttt{hep-th/9203057}.

\bibitem{ragi} G.W. Gibbons, D.A. Rasheed, 
Nucl.Phys. B454 (1995) 185-206, \texttt{hep-th/9506035}.

\bibitem{cha} See for instance 
A. Chamblin, R. Emparan, C. V. Johnson, R. C. Myers,
Phys.Rev. D60 (1999) 064018,
\texttt{hep-th/9902170}.

\bibitem{arci}  
G. Arcioni, E. Lozano-Tellechea, Phys.Rev. D72 (2005) 104021, \texttt{hep-th/0412118}.

\bibitem{novelloetal} See for instance
M. Novello, V.A. De Lorenci, J.M. Salim, Renato Klippert, Phys.Rev. D61 (2000) 045001, \texttt{gr-qc/9911085}.


\bibitem{dymni} I. Dymnikova, Class. Quant. Grav. 21 4417 (2004).

\bibitem{poin} H. Poincar\`e, Acta Math. 7, 259 (1885).

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817 (1979).

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Gravity 12, 443 (1995).

\bibitem{sorkin} R. D. Sorkin, Astrophys. J. 249, 254 (1981), Astrophys. J. 257, 847 (1982).

\bibitem{ss} See for instance C. V. Vishveshwara, Phys. Rev. D \textbf{1}, 2870�2879 (1970).


\end{thebibliography}

\end{document}


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\newpage

\section{Inequalities}

In what follows we shall use the general result \citep{Mayo}
\begin{equation}
T^{0}_{0}(r_0) \le \frac{1}{r_0^2}
\end{equation}
to obtain limits on the parameters of the black hole, and also necessary conditions for the existence of a black hole with the assumed symmetries.
We shall also assume that $T^{0}_{0}>0$, hence 
\begin{equation}
0 \le T^{0}_{0}(r_0)\le \frac{1}{r_0^2},
\label{in0}
\end{equation}
or
\begin{equation}
0\le {\cal L}_F(r_0) f_e(r_0)+\frac{{\cal L}(r_0)}{2}\le \frac{1}{r_0^2}.
\label{in1}
\end{equation}
To see how this inequality leads to the abovementioned restrictions, 
we shall consider several cases (notice that all the quantities are evaluated at the horizon): 
\begin{itemize}
\item ${\cal L}_F>0$ and ${\cal L}>0$. It follows from (\ref{in1}) that 
$$ 
0\le \frac{q_e^2}{r_0^4}\le \frac{{\cal L}_F}{2}\left(\frac{1}{r_0^2}-\frac{{\cal L}}{2}\right),
$$
which yields an upper limit for the electric charge of the black hole in the case $\frac{1}{r_0^2}-\frac{{\cal L}}{2} > 0 $. If $\frac{1}{r_0^2}-\frac{{\cal L}}{2} <0$, there is no black hole. 

\item ${\cal L}_F>0$ and ${\cal L}<0$. It follows from inequality (\ref{in1}) that
there is both a lower limit for $q_e$ (given by $\frac{-{\cal L}{\cal L}_F}{4}$), and an upper limit,
given by $\frac{{\cal L}_F}{2}\left(\frac{1}{r_0^2}-\frac{{\cal L}}{2}\right)$.

\item The case ${\cal L}_F < 0$ and ${\cal L} < 0$ is not allowed since it yields $T^0_{\;0}<0$.

\item ${\cal L}_F < 0$ and ${\cal L} > 0$ yield an upper limit for the electric charge, given by
$\frac{-{\cal L}{\cal L}_F}{4}$, and a lower limit, given by 
$\frac{{\cal L}_F}{2}\left(\frac{1}{r_0^2}-\frac{{\cal L}}{2}\right)$, or zero.
\end{itemize}

%Since $
%R_0=T_0= 2T^{0}_{0} + 2T^{2}_{2}$,
%we obtain
%and considering the convention for the Einstein Equations as 
%$T_{\mu \nu}=-G_{\mu \nu}$, it turns out


%\begin{equation}
%R_0=T_0= 2T^{0}_{0} + 2T^{2}_{2} \le \frac{4}{r_0^2},
%\end{equation}
%that can be written as

%\begin{equation}
%T=\left[ 2{\cal L}+2(f_e-f_m){\cal L}_{F}\right]_{0} \le \frac{4}{r_0^2},
%\end{equation}
%or
%\begin{equation}
%\left[ {\cal L}+(f_e-f_m)l_{F}\right]_{0} \le \frac{2}{r_0^2},
%\end{equation}
%From the last equation we can obtain two inequalities that should be fulfilled by magnetic or electric charges if a %magnetic or electric black hole does exist,

%i) 
%\begin{equation}
%q_e=0, \quad q_m^2 \ge - \frac{r_0^2}{2 l_{F}(r_0)}\left({{2}-{r_0^2}l(r_0) }\right), \quad (l_{F} >0),
%\end{equation}

%ii)
%\begin{equation} 
%q_m=0, \quad q_e^2 \le - \frac{r_0^2}{2}l_{F}(r_0)\left({{2}-{r_0^2}l(r_0) }\right), \quad (l_{F} >0),
%\end{equation}

For the Schwarzschild and RN black holes the previous inequalities are trivially satisfied.

%Further constraints can be obtained combining Eqns.(\ref{in0}) and (\ref{t22}).
%It is easy to show that $T^2_{\;2}(r_0)$ is negative (as in RN solution)
%for $Q^2=q_e^2/R_0^2\ge 1/2$, and also for $r_0/R_0<(1-Q^2)/(1-2Q^2)$ (with %$Q^2<1/2$). It follows that 
%\begin{equation}
%\frac{r_0}{R_0}\frac{1-2Q^2}{1-Q^2}-1 \le T^2_{\;2}(r_0) < 0.
%\end{equation}
%Defining $-\alpha^2=\frac{r_0}{R_0}\frac{1-2Q^2}{1-Q^2}-1$, the different %possibilities (with all quantities evaluated at the horizon) are:
%\begin{itemize}
%
%\item ${\cal L}_F>0\Rightarrow \frac{{\cal L}}{4{\cal L}_F}\le %\frac{q_m^2}{r0^4} \le \frac{1}{2{\cal L}_f}\left(\frac{{\cal %L}}{2}+\frac{\alpha^2}{r_0^2}\right).$ If $\left(\frac{{\cal %L}}{2}+\frac{\alpha^2}{r_0^2}\right)<0$, there is no black hole.
%
%\item ${\cal L}_F<0\Rightarrow \frac{1}{2{\cal L}_f}\left(\frac{{\cal %L}}{2}+\frac{\alpha^2}{r_0^2}\right)\le \frac{q_m^2}{r0^4} \le 
%\frac{{\cal L}}{4{\cal L}_F}.$ This gives an upper limit for $q_m$,
%the lower limit being 0.
%
%\item $q_m=0\rightarrow \frac{\-\alpha^2}{r_0^2}\le \frac{{\cal L}}{2}\le 0$.
%Hence if ${\cal L}$ is positive, there is no black hole.
%
%
%\end{itemize}

Yet another constraint comes from the tidal forces at horizon. As discussed in \citep{mio},
for the radial component to be positive at the horizon, the condition 
$$
2M'(r_0)-r_0M''(r_0)<1
$$
must be fulfilled. This condition is equivalent to 
\begin{equation}
T^{0}_{0} -T^{2}_{2} \le \frac{1}{r_0^2},
\end{equation} 
If we also assume that ${\cal L}_F>0$ (one of the sufficient conditions for $T^0_{\;0}$ to be positive), 
it follows that $T^{0}_{0} -T^{2}_{2} \ge 0$. Hence,
\begin{equation}
0<T^{0}_{0} -T^{2}_{2} \le \frac{1}{r_0^2}.
\label{tf}
\end{equation}
The consequences of this inequality are considered next for the electric and the magnetic case:
\begin{itemize} 
\item $q_e=0\rightarrow \quad \frac{q_m^2}{r_0^2} <  \frac{1}{2 {\cal L}_{F}(r_0)}.$

\item   
$q_m=0 \rightarrow  \quad \frac{q_e^2}{r_0^2} < \frac{ {\cal L}_{F}(r_0)}{2}.$
\end{itemize}

These conditions furnish a maximum value for the charge of the black hole, and they can also be
taken as yelding a minimum value for $r_0$ (hence for the entropy of the black hole).

%An additional inequality should be fulfilled related to the gravity surface

%\begin{equation}
%\kappa= \frac{1}{2} \lim_{r \to r_0} g_{00,r}= \frac{1}{2r_0} (1-T^{0}_{0}r_0^2),
%\end{equation}
%that in terms of the lagrangian gives,
% 
%\begin{equation} 
%- r_0^2 \le -\left({ \frac{2q_e^2}{r_0^2 l_{F}(r_0)} + \frac{lr_0^2}{2}}\right) < 0, \quad (l_{F} >0),
%\end{equation}

%Considering the sign of $l_F$, the information provided by the previous  
%inequalities can be summarized as follows:
%
%Electric black hole ($q_m=0$):
%
%\begin{equation} 
%0 \le l_{F}(r_0)f_e r_0^2 + \frac{lr_0^2}{2} \le 1,
%\end{equation}%
%
%i) Case $l_{F} >0 \Rightarrow$%
%
%\begin{equation} 
%- \frac{l r_0^2 l_F}{4} \le \frac{q_e^2}{r_0^2} \le \frac{l_{F}}{2}\left( 1-\frac{r_0^2 l}{2}\right),
%\end{equation}
% 
% 
%ii) Case $l_{F} < 0, l_F=- \left|l_F\right| \Rightarrow$%

%\begin{equation} 
%\frac{\left|l_{F}\right|}{2} \left(\frac{r_0^2 l}{2} -1 \right) \le \frac{q_e^2}{r_0^2} \le  \frac{l r_0^2 %\left|l_F\right|}{4},
%\end{equation} 
%
%the subcase with $l<0$ should be dismissed because it corresponds to a negative energy density ($\rho <0$),
%
%Magnetic black hole ($q_e=0$):
%
%\begin{equation} 
%0 \le -l_{F}(r_0)f_mr_0^2 + \frac{lr_0^2}{2} \le 1, \quad
%0 \le -l_{F}(r_0) \frac{2q_m}{r_0^2} + \frac{lr_0^2}{2} \le 1,
%\end{equation}
%
%i) Case $l_{F} >0 \Rightarrow$
%
%\begin{equation} 
%\frac{1}{2l_{F}}\left( \frac{r_0^2 l}{2} -1 \right) \le \frac{q_m^2}{r_0^2} \le
%\frac{l r_0^2}{ 4l_F}, 
%\end{equation}
%the subcase with $l<0$ should be dismissed because it corresponds to a negative energy density ($\rho <0$), 
% 
%
%ii) Case $l_{F} < 0, l_F=- \left|l_F\right| \Rightarrow$
%
%\begin{equation} 
%-\frac{l r_0^2}{ 4\left|l_F\right|} \le \frac{q_m^2}{r_0^2} \le 
%\frac{1}{2\left|l_{F}\right|} \left(1-\frac{r_0^2 l}{2}\right),
%\end{equation} 
% 
%again the subcase with $l<0$ should be dismissed because it corresponds to a negative energy density ($\rho <0$),

We shall see below how to use all the inequalities obtained in this section to limit tha parameters of a given theory for which a black hole with the given symmetries exist, or to discard the existence of such black holes in a given theory. 



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

\section{Example 1: The Bronnikov Regular Magnetic Black Hole}
  
In this section we apply the derived inequalities
to the regular magnetic black hole derived in \citep{Bronnikov2000}. The line element
is
\begin{eqnarray}
ds^2&=&A(r)dt^2-A(r)^{-1}dr^2-r^2 d \Omega^2,\\
A(r)&=& 1- \frac{2M(r)}{r}=1- \frac{g}{a^2}\left({\frac{a \sqrt{g}}{r}}\right) \left[{1- \tanh{\left({\frac{a \sqrt{g}}{r}}\right)}}\right],
\end{eqnarray}
where $g=q_m$ is the magnetic charge, $a$ is a parameter. It is a regular solution, since $A(r=0)=1$. The solution behaves as a black hole or as a soliton depending on the value of the parameter $\xi$, related to the ADM mass,

\begin{equation}
\xi=\frac{m}{g}, \quad m=M_{ADM}=M(r \to \infty)=\frac{g^{3/2}}{2a},
\end{equation}
%
%Therefore $4\xi^2=g/a^2$, and
%
If $\xi=\xi_0=0.96$, then $A(r)=0$ has only one (double) root, in which case the black hole would be extreme.
If $\xi > \xi_0$ $A(r)=0$ has two different real roots, hence the black hole has two horizons. Otherwise the solution represents a soliton ($A(r)>0, \forall r$ ).  

The corresponding Lagrangian ${\cal L}(F)$ and its derivative are
\begin{eqnarray}
{\cal L}(F)&=& F {\rm sech}^2 \left[{a\left({\frac{F}{2}}\right)^{1/4}}\right], \\
{\cal L}_F(F)&=& {\rm sech}^2\left[{a \left({\frac{F}{2}}\right)^{1/4}}\right]
\left[1- {\frac{a}{2} \left({\frac{F}{2}}\right)^{1/4}} \tanh\left({a\left({\frac{F}{2}}\right)^{1/4}}\right)\right], 
\end{eqnarray}
Substituting 
$F(r)= 2g^2/r^4$, and adopting the more convenient variable $x=a \sqrt{g} /r$ the previous expressions can be written as
\begin{eqnarray}
{\cal L}&=&\frac{2x^4}{a^4}{\rm sech}^2x,\\
{\cal L}_F&=& {\rm sech}^2x\left({1- \frac{x}{2}\tanh x}\right),
\end{eqnarray}  
It is easy to check that ${\cal L}>0, \forall x$ and ${\cal L}_F>0$ if $x\leq 2.065$ or $r\geq a \sqrt{g}/2.065$.
Substituting these values into the inequalities calculated in the previous section, we obtain,
\begin{itemize}

\item
$T_0^0 \in [0, 1/r_0^2] \Rightarrow$ 

\begin{eqnarray}
0 & \le & {\cal L}_F f_er_0^2 + \frac{{\cal L}r_0^2}{2} <1, \quad q_e=0, \nonumber\\
0 & \le &  \frac{{\cal L}r_0^2}{2} <1,  \nonumber\\
 &&\frac{2g^2}{r_0^4} {\rm sech}^2{x} \le \frac{2}{r_0^2},  \nonumber\\
4 \xi_0^2 & \le & 4 \xi^2= \frac{g}{a^2} \le \frac{\cosh^2{x}}{x^2},
\end{eqnarray}

\item 
Tidal forces condition implies that

\begin{equation}
4 \xi^2= \frac{g}{a^2} < \left[{2x^2\left({1- \frac{x}{2} \tanh{x}}\right) 
{\rm sech}^2{x}}\right]^{-1},
\end{equation}

that is a stronger condition than the one derived from the positiveness of energy density, as seen from Fig.\ref{bm}.
\begin{figure}[H]
       \centering  % figura centralizada
       \includegraphics[scale=1.2]{bm.eps}
       \caption{The first two inequalities are plotted for the Bronnikov magnetic black hole. The horizontal line stands for $4 \xi_0^2$, and the  allowed values for the charge of the black hole are above this line.
The dotted line corresponds to $T_0^0 \le 1/r_0^2$. The dot-dashed curve is the restriction imposed by tidal forces. The allowed values for the charge of the black hole must be below both curves. To the left of the continuous curve ($0<x \le 0.6392$) is the domain of outer communication (DOC) of the black hole with charge $4 \xi^2$.}
       \label{bm}
\end{figure}


%\item
%the surface gravity condition amounts to,
%\begin{equation}
%1- \frac{r_0^2 l}{2} \le 1, \quad  -\frac{r_0^2}{2}l \le 0,
%\end{equation}
%since $l>0$ the surface gravity condition is trivially fulfilled.
\end{itemize}
Recall that the inequalities holds at the horizon, $x_h=a \sqrt{g}/r_h$, and for the values corresponding to black hole, $\xi \geq \xi_0$.
In Figure \ref{bmcq}, we plot the specific heat $C_q$
for the magnetic (regular) black hole in Bronnikov's theory. 
\begin{figure}[h]
       \centering  % figura centralizada
       \includegraphics[scale=1.2]{bmcq.eps}
       \caption{Specific heat for the Bronnikov magnetic black hole. The dashed-dotted line corresponds to
$\xi=0.355$, in which case the associated behaviour is Schwarzschild-like. The continuous curve is for $\xi=0.75$, and the behaviour in this case is Reissner-Nordstrom-like. The dotted curve is for a charge $\xi=0.575$ and the behaviour is between Schwarzschild and RN-like.
}
       \label{bmcq}
\end{figure}

\subsection{Anexo: nueva parametrizaci\'on}

Del paper de Bronnikov:
$$
M(r) = m \left(1-\tanh \frac{a\sqrt{|q|}}{r}\right)
$$
con 
\begin{equation}
a = \frac{|q|^{3/2}}{2m}
\label{masa}
\end{equation}
La posici\'on del horizonte viene dada por
\begin{equation}
m \left(1-\tanh \frac{a\sqrt{|q|}}{r_0}\right) = \frac{r_0}{2},
\label{r0}
\end{equation}
y substituyendo $a$:
\begin{equation}
1-\tanh \frac{|q|^2}{2mr_0} = \frac{r_0}{2m}.
\label{r0}
\end{equation}
La soluci\'on tiene 4 par\'ametros $(m,a,q,r_0)$, pero s\'olo dos son independientes, debido a las relaciones (\ref{masa}) y (\ref{r0}). 

Habiamos definido
$$
x_0 = \frac{a\sqrt{|q|}}{r_0},
$$
Es mejor escribir esta como
$$
x_0 = \frac{q^2}{2mr_0} \equiv \alpha \beta,
$$
donde $\alpha = q/2m$ y $\beta = r_0/2m$. Estas nuevas variables son mas significativas fisicamente. En terminos de estas, la ec. para el horizonte es
$$
1-\tanh \frac{\alpha^2}{\beta} = \beta.
$$
De la desigualdad $0\leq T^0_0\leq 1/r_0^2$ tenemos que
\begin{equation}
0\leq {\rm sech}^2(x_0) (3-x_0\tanh x_0)\leq \frac{r_0^2}{q^2}
\label{db}
\end{equation}
que en t\'erminos de las nuevas variables queda
$$
0\leq {\rm sech}^2\left(\frac{\alpha^2}{\beta}\right) (3-\frac{\alpha^2}{\beta}\tanh\left(\frac{\alpha^2}{\beta}\right) ) \leq \frac{\alpha^2}{\beta}
$$
Pasando a la variable $\xi = 1/2\alpha$ usada por Bronnikov, 
y redefiniendo $\beta$ como $r_0/m$, tenemos
\begin{equation}
1-\tanh \frac{1}{2\xi^2\beta} = \frac{\beta}{2},
\label{r1}
\end{equation}
\begin{equation}
0\leq {\rm sech}^2\left(\frac{1}{2\xi^2\beta}\right) (3-\frac{1}{2\xi^2\beta}\tanh\left(\frac{1}{2\xi^2\beta}\right) ) \leq \frac{1}{\xi^2\beta^2}
\label{db1}
\end{equation}
En un caso en el que no conocemos la soluci\'on, 
no hay relaci\'on entre $\beta$ y $\xi$, y 
tendriamos a disposici\'on s\'olo la ec.
(\ref{db1}), que define dos superf\'icies en terminos de $\xi$ y $\beta$. 
En el caso de la sol. de Bronnikov, tenemos tb la relaci\'on (\ref{r1}) 
entre $\alpha$ y $\beta$. 
\begin{figure}[H]
       \centering  % figura centralizada
       \includegraphics[scale=1.2]{surfaces.jpg}
       \caption{La desigualdad (\ref{db1}) limita los valores de los par\'ametros a aquellos para los cuales la superficie amarilla est� abajo de la 
verde.}
.}
       \label{surf}
\end{figure}







\section{Example 2: Electric Bronnikov case}

Differently from the magnetic case, the electric bh solution in Bronnikov's theory has not been obtained. We shall use the inequalities derived above to 
discuss its features. 
The electric charge can be constrained by considering the expression for the Lagrangian in terms of the electric field $E$, given by
\begin{eqnarray}
F_{10}&=& \frac{q_e}{r^2 {\cal L}_F}=E, \quad \rightarrow r^2=\frac{q_e}{l_F E},
\nonumber\\
\left|{\frac{F}{2}}\right|^{1/4} &=& \sqrt{E}, \quad f_e=2E^2. \nonumber\\  
\end{eqnarray}
Bronnikov's Lagrangian and its first derivative in terms of the electric field are given by
\begin{eqnarray}
{\cal L}(E)&=& -2E^2 {\rm sech}^2{x},\nonumber\\
{\cal L}_F(E) &=& \left({{1- \frac{x}{2} \tanh{x}}}\right) {\rm sech}^2{x},  \quad x= a \sqrt{E},
\end{eqnarray}
The inequalities in this case take the form
\begin{eqnarray}
\frac{q_e}{a^2}&=& \frac{1- \frac{x}{2}\tanh{x}}{x^2(1-{x} \tanh{x})}, \nonumber\\
\frac{q_e}{a^2}&=& \frac{1}{2x^2},
\end{eqnarray}
Demanding tidal forces condition is stronger than the positiveness of energy density since
\begin{equation}
\frac{1}{2x^2} < \frac{1- \frac{x}{2}\tanh{x}}{x^2(1-{x} \tanh{x})}.
\end{equation}

\begin{figure}[H]
       \centering  % figura centralizada
       \includegraphics[scale=1.2]{be.eps}
       \caption{The dotted curve corresponds to the restriction imposed by the positiveness of the energy density: the black hole charge must between this curve and the positive horizontal axis. The dashed curve comes from the tidal forces requirement. Again the charge should be located below the curve.}
       \label{be}
\end{figure}

\section{Effective metric}

It is by now a well-known result that high-energy photons of a nonlinear electromagnetic theory do not follow geodesics of the background spacetime
but those of an effective metric given by \citep{novelloetal}
\begin{equation}
\tilde g^{\mu\nu} = {\cal L}_F g^{mu\nu} - 4 {\cal L}_{FF}F^\mu_\alpha F^{\alpha\nu}
\end{equation}
where $g^{\mu\nu}$ is the background metric. 
In the case of a black hole with a NLEM source, described by the metric 
in Eqn.(\ref{metric}), 
we obtain
$$
ds^2 = \frac{A}{\Phi}dt^2-\frac{dr^2}{A\Phi}-\frac{r^2}{{\cal L}_F}d\Omega^2
$$
in the case of an electric field, where $A=1-2M(r)/r$, and
$$
\Phi = {\cal L}_F \alpha - 4{\cal L}_{FF}\left(\frac{q_e}{{\cal L}_Fr^2}\right)^2.
$$
The radial EOM that follow from the geodesic equations 
can be written as 
$$
\dot r^2 = \Phi \left(E^2\Phi - \frac{A L^2}{{\cal L}_Fr^2}\right)
$$
We see that there are turning points where $\Phi =0$ or where
$$
E^2\Phi = 
\frac{A L^2 {\cal L}_F}{r^2}
$$
so we need to impose
that $\Phi \neq 0$ and 
$$
E^2\Phi \neq
\frac{A L^2 {\cal L}_F}{r^2}.
$$
The angular motion is governed by
$$
\dot\phi = -\frac{L{\cal L}_F}{r^2}
$$
so 
$$
{\cal L}_F \neq 0
$$

In the magnetic case we have
$$
ds^2 = \frac{B}{{\cal L}_F}dt^2-\frac{dr^2}{B{\cal L}_F}-\frac{r^2}{\Psi}d\Omega^2
$$
where
$$
\Psi = {\cal L}_F+4{\cal L}_{FF}\left(\frac{q_m}{r^2}\right)^2
$$
and $B(r) $ is the background magnetic bh solution.
For the corresponding EOM we get
$$
\dot r^2 = {\cal L}_F\left( E^2 {\cal L}_F - \frac{B L^2\Psi}{r^2}\right)
$$
so there may be turning points at ${\cal L}_F = 0 $
or 
$$
E^2{\cal L}_F = B \Psi\frac{L^2}{r^2}
$$
so we need to impose
${\cal L}_F \neq 0 $
$$
E^2{\cal L}_F \neq B \Psi\frac{L^2}{r^2}
$$

$$
\dot\phi = -\frac{L^2\Psi}{r^2}
$$
so 
$$
\Psi \neq 0
$$
$$
e^{-2\lambda} = 1 - \frac{r_0}{r}-\frac{1}{r}\int_{r_0}^r T^0_0 r^2dr
$$
$$
T^0_0-T^2_2 = {\cal L}_F(f_e+f_m) 
$$
has the sign of ${\cal L}_F$\\
For ${\cal L}_F > 0$
$$
T^0_0\geq T^2_2
$$
and we have 
$$
0\leq T^2_2 (r_0)\leq \frac{1}{r_0^2}
$$
if $T^2_2 (r_0) \geq 0 $. 



In Ref. \cite{Breton2005} the  Born-Infeld, Bronnikov and Dimnikova solutions were considered, to determine if the 
Ashtekar-Corichi-Sudarsky (ACS) conjecture \cite{Ashtekar2001} was valid for those nonlinear electromagnetic systems.
The ACS conjecture refers to hairy black holes and claims that the ADM and horizon black hole masses are related to the soliton mass through $M_{ADM}=M_{hor}+M_{sol}$. Moreover, according to the ACS conjecture, the unstability of the black hole is a consequence of the solitonic unstability.

Since the studied solutions (Born-Infeld, Bronnikov and Dimnikova) possess both solitonic and black hole counterparts, and that the masses relation is fulfilled by them, in \cite{Breton2005} it was tested if the ACS conjecture hold for those NLED systems. It turned out that the Bronnikov and Dimnikova solitonic solutions are unstable, then assuming the validity of the conjecture, the corresponding black holes should be unstable as well. The soliton BI solution being stable, does not allow to test the ACS conjecture.

In the present work, the situation is different concerning the boundary conditions. 

The range we considered for the dynamical stability test is the domain of outer communication (DOC), $r \ge r_{h}$, whereas the solitonic solution is not considered at all. In other words, 

we are analyzing completely isolated black holes, otherwise the microcanonical ensemble would not be appropriate for  the thermodynamical analysis, and therefore, the results in \cite{Breton2005} do not correspond to the situation analyzed in this paper.



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See \protect \citep {Poincare1885} for the original version, and \protect
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Let us remark that the stability of the solutions examined here was inspected
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The same could be said of the RN solution.

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The extremal case, in which the two horizons coalesce into one, is given by
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This Lagrangian was inspired in the one presented in \protect \citep
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As shown in \protect \citep {Matyjasek2000}, the location of the horizons can
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Actually, there is another instance that implies a change of stability, namely
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