\section{Introduction}

In \cite{salg1} was shown that the standard $5$-dimensional General
Relativity (GR), without cosmological constant, can be obtained from
Chern-Simons gravity theory for a certain Lie algebra $\mathcal{B}$ \cit
{salg2}, \cite{salg3}. The Chern-Simons Lagrangian for the $\mathcal{B}$
algebra is given by

\begin{equation}
L_{\mathrm{ChS}}^{(5)}=\alpha _{1}l^{2}\varepsilon
_{abcde}R^{ab}R^{cd}e^{e}+\alpha _{3}\varepsilon _{abcde}\left( \frac{2}{3
R^{ab}e^{c}e^{d}e^{e}+2l^{2}k^{ab}R^{cd}T^{\text{ }e}+l^{2}R^{ab}R^{cd}h^{e
\right) ,  \label{26}
\end{equation
where $\alpha _{1}$ and $\alpha _{3}$ are arbitrary constants and $l$ can be
interpreted as a coupling constant that characterizes different regimes
within the theory.

The field content induced by the $\mathcal{B}$ algebra includes the vielbein 
$e^{a}$, the spin connection $\omega ^{ab}$ and two extra bosonic fields 
h^{a}$ and $k^{ab}$. From (\ref{26}) we can see that it is possible to
recover the $5$-dimensional Einstein gravity theory from a Chern-Simons
gravity theory in the limit where the coupling constant $l$ equals to zero
while keeping the effective Newton's constant fixed.

In the presence of matter described by the langragian 
L_{M}=L_{M}(e^{a},h^{a},\omega ^{ab})$, in Ref. \cite{salga1} was found that
the field equations admit black hole-type solutions for a spherically
symmetric metric.

On the other hand, in Ref. \cite{bra1}, \cite{bra2} was found that if one
replaces a Randall-Sundrum type metric \cite{randall} \cite{randall1} in the
Einstein-Chern-Simons (EChS) gravity Lagrangian (\ref{26}), the following
action can be obtained

\begin{eqnarray}
\tilde{S}[\tilde{e},\tilde{h}] &=&\int_{\Sigma _{4}}\tilde{\varepsilon
_{mnpq}\left( -\frac{1}{2}\tilde{R}^{mn}\tilde{e}^{p}\tilde{e}^{q}\right. + 
\notag \\
&&\left. +K\tilde{R}^{mn}\tilde{e}^{p}\tilde{h}^{q}-\frac{K}{4r_{c}^{2}
\tilde{e}^{m}\tilde{e}^{n}\tilde{e}^{p}\tilde{h}^{q}\right) ,  \label{s3}
\end{eqnarray
which is a gravity action with a cosmological constant for a $4$-dimensional
brane embedded in the $5$-dimensional spacetime of the so-called EChS
gravity.\textbf{\ }$\tilde{\varepsilon}_{mnpq}$\textbf{, }$\tilde{e}^{m}$, 
\tilde{R}^{mn}$\textbf{\ }and\textbf{\ }$\tilde{h}^{m}$\textbf{\ }represent,
respectively, the\textbf{\ }$4$\textbf{-}dimensional versions of the
Levi-Civita symbol,\textbf{\ }the vielbein, the curvature form, a\textbf{\ 
matter field and $r_{c}$\textbf{\ }is the so-called compactification radius. 
$K$ is a constant related to $\alpha _{3}$, $l$ and $r_{c}$. It is of
interest to note that the field $h^{a}$ gives rise to a form of positive
cosmological constant which appears as a consequence of modifications of the
Poincar\'{e} symmetries, carried out through the expansion procedure.\textbf
\ }The corresponding version of Eq. (\ref{s3}) in tensor language Ref. \cit
{bra1}, \cite{bra2} is given by

\begin{equation}
\tilde{S}[\tilde{g},\tilde{h}]=\int d^{4}\tilde{x}\sqrt{-\tilde{g}}\left[ 
\tilde{R}+2K\left( \tilde{R}\tilde{h}-2\tilde{R}_{\text{ }\nu }^{\mu }\tilde
h}_{\text{ }\mu }^{\nu }\right) -\frac{3K}{2r_{c}^{2}}\tilde{h}\right] ,
\label{31t'}
\end{equation
where $K=4\pi \alpha _{3}l^{2}/r_{c}$, so that when $l\longrightarrow 0$
then $K\longrightarrow 0$ and the actions (\ref{s3}) and (\ref{31t'})
becomes the $4$-dimensional Einstein-Hilbert action. \ 

An interesting observation that if we consider a maximally symmetric
spacetime, then the equation 13.4.6 of Ref. \cite{weinberg} allows us to
write the field\textbf{\ }$\tilde{h}_{\mu \nu }$\textbf{\ }as\textbf{\ }

\begin{equation}
\tilde{h}_{\mu \nu }=\frac{1}{4}\tilde{F}(\tilde{\phi})\tilde{g}_{\mu \nu },
\label{66t}
\end{equation
where $\tilde{F}$ is an arbitrary function of an $4$-scalar field $\tilde
\phi}=$ $\tilde{\phi}(\tilde{x})$.\ This means

\begin{equation}
\tilde{R}_{\text{ }\nu }^{\mu }\tilde{h}_{\text{ }\mu }^{\nu }=\frac{1}{4
\tilde{F}(\tilde{\phi})\tilde{R}\text{ \ \ },\text{\ \ }\tilde{h}=\tilde{h
_{\mu \nu }\tilde{g}_{\text{ }}^{\mu \nu }=\tilde{F}(\tilde{\phi}),
\end{equation
so that the action (\ref{31t'}) takes the form 
\begin{equation}
\tilde{S}[\tilde{g},\tilde{\phi}]=\int d^{4}\tilde{x}\sqrt{-\tilde{g}}\left[ 
\tilde{R}+K\tilde{R}\tilde{F}(\tilde{\phi})-\frac{3K}{2r_{c}^{2}}\tilde{F}
\tilde{\phi})\right] ,  \label{32t}
\end{equation
which corresponds to an action for the $4$-dimensional gravity coupled
non-minimally to the scalar field. Note that this action has the form 
\tilde{S}=\tilde{S}_{g}+\tilde{S}_{g\phi }+\tilde{S}_{\phi },$ where, 
\tilde{S}_{g}$ is a pure gravitational action term, $\tilde{S}_{g\phi }$ is
a non-minimal interaction term between gravity and the scalar field, and 
\tilde{S}_{\phi }$ represents the action for a kind of scalar field
potential. In order to write down the action in the usual way, we define the
constant $\varepsilon $ and the potential $V(\tilde{\phi})$ as (removing the
symbols $\sim $ in (\ref{32t})) 
\begin{equation}
\varepsilon =\frac{4\kappa r_{c}^{2}}{3}\text{ \ \ },\text{\ \ }V(\phi )
\frac{3K}{4\kappa r_{c}^{2}}F(\phi ),  \label{100t}
\end{equation
where $\kappa $ is a constant. This permits to rewrite the action for a $4
-dimensional brane non-minimally coupled to a scalar field, immersed in a $5
-dimensional space-time a
\begin{equation}
S[g,\varphi ]=\int d^{4}x\sqrt{-g}\left[ R+\varepsilon RV(\phi )-2\kappa
V(\phi )\right] .  \label{33t}
\end{equation
Note that this action was obtained from the Lagrangian (\ref{26}) that does
not consider the presence of a cosmological constant.

The corresponding field equations describing the behavior of the $4
-dimensional brane in the presence of the scalar field $\phi $ are given by 
\cite{bra1}, \cite{bra2}

\begin{equation}
G_{\mu \nu }\left( 1+\varepsilon V\right) +\varepsilon H_{\mu \nu }=-\kappa
g_{\mu \nu }V,  \label{z1}
\end{equation
\begin{equation}
\frac{\partial V}{\partial \phi }\left( 1-\frac{\varepsilon R}{2\kappa 
\right) =0,  \label{z2}
\end{equation
wher
\begin{equation}
H_{\mu \nu }=g_{\mu \nu }\nabla ^{\lambda }\nabla _{\lambda }V-\nabla _{\mu
}\nabla _{\nu }V.  \label{z3}
\end{equation}

In this article we\ study black hole solutions of the $4$-dimensional
gravitational field equations (\ref{z1}),(\ref{z2}),(\ref{z3}). In Section
II we find that field equations (\ref{z1}),(\ref{z2}) and (\ref{z3}), for a
spherically symmetric metric, lead to a spacetime with a cosmological
constant inversely proportional to the square of the compactification radius
and to a solution dependent on an arbitrary constant $C$. In Section III we
discuss the case when this constant is negative, in particular, when such
negative constant is equal to the Schwarzschid radius. This solution is
reminiscent of a Schwarzschid-de Sitter (SdS) black hole. A procedure used
in the context of general relativity \cite{bonda} is generalized in Section
IV in order to apply it to Chern-Simons gravity. This method is based on the
symmetries of parity $r\longrightarrow -\tilde{r}$, time reversal 
t\longrightarrow -\tilde{t}$ and mass reversal $M\longrightarrow -\tilde{M}$
called $PTM$ symmetry. This symmetry: $\left( i\right) $ leaves the metric
invariant in form, $\left( ii\right) $ relates the different regions of the
extended solution of the Einstein field equations written in terms of the
Kruskal-Szekeres coordinates \cite{kr}, \cite{sz} and $\left( iii\right) $
reverse the signs of the Kruskal-Szekeres coordinates. In Ref. \cite{bonda}
the Schwarzschid case was studied and in Refs. \cite{chan}, \cite{fro} the
Reissner Nordstr\"{o}m's solution was considered. In this article, the
solutions of the field equations obtained from a four-dimensional
gravitational action, that was obtained by compactification of the so-called
EChS gravity in five dimensions, will be analyzed. Finally, in Section V we
present our conclusions. We will use $G=c=1$ units.

\section{Field equations for a spherically symmetric me\-tric}

In $4$-dimensions a static and spherically symmetric metric can be written
in the form 
\begin{equation}
ds^{2}=-e^{\alpha (r)}dt^{2}+e^{\beta (r)}dr^{2}+r^{2}\left( d\theta
^{2}+\sin ^{2}\theta d\varphi ^{2}\right) ,
\end{equation
where $r,\theta $ and $\varphi $ are the usual spherical polar coordinates.
So, according to (\ref{z1}) and (\ref{z2}), the following field equations
are obtained (see Appendix)

\begin{eqnarray}
\left[ \frac{e^{\alpha }}{r^{2}}+e^{\alpha -\beta }\left( \frac{\beta
^{\prime }}{r}-\frac{1}{r^{2}}\right) \right] \left( 1+\varepsilon V\right)
+\varepsilon H_{00} &=&\kappa e^{\alpha }V,  \label{uno} \\
\left[ \frac{\alpha ^{\prime }}{r}+\frac{1}{r^{2}}-\frac{e^{\beta }}{r^{2}
\right] \left( 1+\varepsilon V\right) +\varepsilon H_{11} &=&-\kappa
e^{\beta }V,  \label{dos} \\
\left[ 1-\frac{r^{2}}{2}R-\frac{e^{-\beta }}{2}\left[ \left( \alpha ^{\prime
}-\beta ^{\prime }\right) r+2\right] \right] \left( 1+\varepsilon V\right)
+\varepsilon H_{22} &=&-\kappa r^{2}V,  \label{tres}
\end{eqnarray}

\begin{eqnarray}
\left[ 1-\frac{r^{2}}{2}R-\frac{e^{-\beta }}{2}\left[ \left( \alpha ^{\prime
}-\beta ^{\prime }\right) r+2\right] \right] \left( 1+\varepsilon V\right)
\sin ^{2}\theta +\varepsilon H_{33} &=&-\kappa r^{2}V\sin ^{2}\theta ,
\label{cuatr} \\
\frac{\partial V}{\partial \phi }\left( 1-\frac{\varepsilon R}{2\kappa 
\right) &=&0,  \label{cinco} \\
g_{\mu \nu }\nabla ^{\lambda }\nabla _{\lambda }V-\nabla _{\mu }\nabla _{\nu
}V &=&H_{\mu \nu },  \label{seis}
\end{eqnarray
where the "prime" symbol denotes derivative respect to $r$. We have seen
that when $l\longrightarrow 0$, i. e., $V\rightarrow 0$, the equations (\re
{z1}),(\ref{z2}),(\ref{z3}) lead to Einstein's equations. This means that
the equations (\ref{uno}-\ref{seis}) must lead, in this limit, to the
corresponding Einstein field equations for a spherically symmetric metric.
For this limit to be fulfilled, it must be satisfied that $\alpha (r)=-\beta
(r)$. On the other hand, the spherical symmetry condition implies that 
V=V(r)$ and $F=F(r)$, so that $\partial V/\partial \phi \rightarrow \partial
V/\partial r.$

By imposing these conditions on the equations (\ref{uno}-\ref{seis}) we
obtain

\begin{eqnarray}
\left[ \frac{e^{\alpha }}{r^{2}}-e^{2\alpha }\left( \frac{\alpha ^{\prime }}
r}+\frac{1}{r^{2}}\right) \right] \left( 1+\varepsilon V\right) +\varepsilon
H_{00} &=&\kappa e^{\alpha }V,  \label{7} \\
\left[ \frac{\alpha ^{\prime }}{r}+\frac{1}{r^{2}}-\frac{e^{-\alpha }}{r^{2}
\right] \left( 1+\varepsilon V\right) +\varepsilon H_{11} &=&-\kappa
e^{-\alpha }V,  \label{8} \\
\left[ 1-\frac{r^{2}}{2}R-e^{\alpha }\left( r\alpha ^{\prime }+1\right)
\right] \left( 1+\varepsilon V\right) +\varepsilon H_{22} &=&-\kappa r^{2}V,
\label{9} \\
\left[ 1-\frac{r^{2}}{2}R-e^{\alpha }\left( r\alpha ^{\prime }+1\right)
\right] \left( 1+\varepsilon V\right) \sin ^{2}\theta +\varepsilon H_{33}
&=&-\kappa r^{2}V\sin ^{2}\theta ,  \label{10}
\end{eqnarray
\begin{equation}
\frac{\partial V}{\partial r}\left( 1-\frac{\varepsilon R}{2\kappa }\right)
=0,  \label{12}
\end{equation
\begin{eqnarray}
H_{00} &=&-e^{2\alpha }\left[ \frac{V^{\prime }\alpha ^{\prime }}{2
+V^{\prime \prime }+\frac{2}{r}V^{\prime }\right] , \\
\text{ \ }H_{11} &=&\frac{V^{\prime }\alpha ^{\prime }}{2}+\frac{2}{r
V^{\prime }=-e^{-2\alpha }\left( H_{00}-V^{\prime \prime }\right) ,
\label{13'} \\
\text{ }H_{22} &=&r^{2}e^{\alpha }\left[ V^{\prime \prime }+\left( \alpha
^{\prime }+\frac{1}{r}\right) V^{\prime }\right] , \\
H_{33} &=&H_{22}\sin ^{2}\theta .  \label{14}
\end{eqnarray}

From (\ref{13'}), (\ref{7}) and (\ref{8}) it direct to see that $V^{\prime
\prime }=0$, which implies that $V(r)\sim r$.\ From (\ref{12}) it is direct
to see that when $\partial V/\partial r\neq 0$\ we hav
\begin{equation}
R=\frac{3}{2r_{c}^{2}},  \label{20}
\end{equation
which indicates the presence of a cosmological constant given b
\begin{equation}
\Lambda =\frac{3}{8r_{c}^{2}}.  \label{20'}
\end{equation
This result allow us to conjecture that the $4$-dimensional cosmological
constant could have its origin in the extra dimension. In other words,\ the
cosmological constant could have its origin in the projection of the extra
dimension on the brane.

The equations (\ref{z1}),(\ref{z2}) can be written in the form

\begin{eqnarray}
G_{\mu \nu } &=&R_{\mu \nu }-\frac{1}{2}Rg_{\mu \nu }=\kappa T_{\mu \nu }; \\
T_{\mu \nu } &=&-\frac{1}{1+\varepsilon V}\left( \frac{2}{R}H_{\mu \nu
}+g_{\mu \nu }V\right) ,  \label{z2'}
\end{eqnarray
such that, in absence of sources $G_{\mu \nu }=0$ and then $R=0$. If we
include cosmological constant we obtain $R=4\Lambda $. In the present case 
R=3/2r_{c}^{2}$, i.e., $\Lambda =3/8r_{c}^{2}$ and we have a source given b
\textbf{\ }$T_{\mu \nu }$ \textbf{(}$V\neq 0$\textbf{)}.

Introducing these results into (\ref{7}) and (\ref{9}) and carrying out the
change of variables $\alpha (r)=\ln F(r)$, we can write

\begin{equation}
e^{\alpha (r)}=F(r)=1+\frac{C}{r}-\frac{\Lambda }{3}r^{2},  \label{17}
\end{equation
where $C$ is a constant of integration. So, the line element

\begin{equation}
ds^{2}=-\left( 1+\frac{C}{r}-\frac{\Lambda }{3}r^{2}\right) dt^{2}+\left( 1
\frac{C}{r}-\frac{\Lambda }{3}r^{2}\right) ^{-1}dr^{2}+r^{2}d\Omega ^{2},
\label{18}
\end{equation
is a $sgn(C)$-dependent solution. This means that the $F(r)$ function is
subject to the discution of $C$, that is, its sign and size. As we will see
in the next Section, if $C=0$ we obtain the usual solution that leads to a
black hole whose even horizon is a cosmological one. For $C<0$\textbf{\ }we
obtain a SdS black hole if we consider $C=-r_{s}$, where $r_{s}$ is the
Schwarzschild radius.

The case $C>0$, discussed in Section IV, \ leads to a white hole solution
which is obtained applying to the solution with $C<0$, the discrete
coordinate transformation $PT$ accompanied by the transformation 
C\longrightarrow -\tilde{C}$, which can be interpreted as the mass reversal.

\section{$\mathbf{C<0}$ case. \textbf{SdS black hole solution}}

From (\ref{18}), we writ
\begin{equation}
ds^{2}=-F(r)dt^{2}+\frac{dr^{2}}{F(r)}+r^{2}d\Omega ^{2},  \label{38}
\end{equation
where $F(r)$ is given by (\ref{17}). If $C=0$, then the function $F(r)$
takes the for
\begin{equation}
F(r)=1-\frac{\Lambda }{3}r^{2},  \label{39}
\end{equation
and we have a divergence at $r_{0}=\sqrt{3/\Lambda }$, case that can be seen
as the limit at large $r$ ($dS$ space limit) of the SdS black hole.

Let us consider now the conditions under which the field equations admit
black hole-type solutions. The solution (\ref{38}) with $F(r)$ given by (\re
{17}) shows a singular behaviour at $F(r_{0})=0$, i.e., 
\begin{eqnarray}
r_{0}^{3}+\left( -8r_{c}^{2}\right) r_{0}+\left( -8r_{c}^{2}C\right) &=&0
\text{ }  \label{39'} \\
i.e.,\text{ }C &=&\frac{1}{8r_{c}^{2}}\left( r_{0}-2\sqrt{2}r_{c}\right)
\left( r_{0}+2\sqrt{2}r_{c}\right) r_{0}.  \label{39''}
\end{eqnarray
From (\ref{39'}) we see that the discriminant is 
\begin{equation}
\Delta =-64r_{c}^{4}\left( -32r_{c}^{2}+27\tilde{C}^{2}\right) >0,
\label{40'}
\end{equation
so that $-32r_{c}^{2}+27C^{2}<0$. This means that $C<0$ as long as $r_{0}<
\sqrt{2}r_{c}$ with 
\begin{equation}
C=-\Sigma ^{2}<-\sqrt{\frac{2}{3}}\left( \frac{4}{3}r_{c}\right) .
\label{90'}
\end{equation
Hence (\ref{18}) can be written in the form\ \ \ 

\begin{equation}
ds^{2}=-\left( 1-\frac{\Sigma ^{2}}{r}-\frac{\Lambda }{3}r^{2}\right)
dt^{2}+\left( 1-\frac{\Sigma ^{2}}{r}-\frac{\Lambda }{3}r^{2}\right)
^{-1}dr^{2}+r^{2}d\Omega ^{2}.  \label{18'}
\end{equation}
\ 

On another hand, from (\ref{39'}) and (\ref{40'}) we have three real roots 
\begin{equation}
r_{0k}\left( C\right) =-\frac{4\sqrt{2}}{\sqrt{3}}r_{c}\cos \left[ \frac{1}{
}\arccos \left( \frac{3\sqrt{3}}{4\sqrt{2}}\frac{\Sigma ^{2}}{r_{c}}\right) 
\frac{2}{3}\left( k-1\right) \pi \right] ,  \label{92}
\end{equation
where $k=0,1,2$. The solution for $k=1$can be withdrawn as non-physical.
Then, the equation $F\left( r_{0}\right) =0$ tells us that the physical
horizons are $r=r_{02}$ and $r=r_{00}$, where $r_{02}\leq r_{00}$. \ So,
according to (\ref{90'}) and with $\Sigma ^{2}=r_{s}$ 
\begin{equation}
r_{s}>\sqrt{\frac{2}{3}}\left( \frac{4}{3}r_{c}\right) ,  \label{93}
\end{equation
we identify $r_{02}$ as the black hole event horizon and $r_{00}$ as the
cosmological horizon.

The solution (\ref{18'}) coincides with the SdS black hole solution when\ 
C=-\Sigma ^{2}=-r_{s}$, where $r_{s}=2M$ is the Schwarzschild radius.

Defining $x=4\sqrt{2}r_{c}/\sqrt{3}r_{s}$ and then taking the limit when 
r_{c}\longrightarrow \infty $, we have 
\begin{equation}
\lim_{r_{c}\longrightarrow \infty }r_{00}=r_{s}\lim_{x\longrightarrow \infty
}\left( x\cos \frac{1}{3}\left[ \arccos \left( \frac{3}{x}\right) -\frac
2\pi }{3}\right] \right) =r_{s}=\lim_{r_{c}\longrightarrow \infty }r_{02}.
\end{equation
This shows that the solution (\ref{18'}) with $\Sigma ^{2}=r_{s}$ is the SdS
black hole solution, where $r_{s}=2M$, where $M$ is the black hole mass.

Now, from (\ref{17}) and (\ref{18'}) we can see that $F\left( r_{0}\right)
=0 $ leads in this case to 
\begin{equation*}
\Lambda r_{0}^{3}=3\left( r_{0}-r_{s}\right) ,
\end{equation*
and from here it is straightforward to see that $r_{c}\rightarrow \infty
\longleftrightarrow $ $\Lambda \rightarrow 0$, then $r_{0}\rightarrow r_{s}
. Similarly, when $r_{s}<<r_{0}$ we have to $r_{0}\rightarrow \sqrt
3/\Lambda }$, i.e., "we live" in a $r_{0}$-range given by

\begin{equation}
r_{s}<r_{0}<2\sqrt{2}r_{c},
\end{equation
i.e., we live in an inflationary $\left( \Lambda \right) $ universe whose
"center" is the singularity of the black hole, singularity that we do not
see given the event horizon. As we know, in GR can have black holes with
event horizon and also with cosmological horizon being the SdS black hole
the simplest solution wich has both \cite{footnote1}.

In order to have a rough idea of the present scales, it is good to remember
that according to the current observational data $\Lambda \sim 10^{-52}\left[
m^{-2}\right] $ and $a_{0}\sim 10^{26}\left[ m\right] $ ($a_{0}$ is the
causal "size" of the observable universe). In our case, from the current
value for $\Lambda =3/8r_{c}^{2}$ we obtain $r_{c}\sim 10^{25}\left[ m\right]
$, i.e., $r_{c}\sim a_{0}/10$. Thus, we could say that for $z\gtrsim 9$ our
universe remained inside $r_{c}$. What implications could it have this fact
if we are thinking \ as feasible the existence of one extra dimension? It is
an open question \cite{footnote2}.

We finish this Section highlighting the relation between $\Lambda $ and 
r_{c}$. This situation is interesting in itself. As we already said, the
cosmological constant could have its origin in the projection of the extra
dimension over our $4$-spacetime. Seen in reverse, the observational value
of $\Lambda $ could give us information on $r_{c}$ accepting that our $4
-reality is a sort of "shadow" of a $5$-reality.

\section{$C>0$ case. White hole solution}

This case leads to a solution that can be understood as a white hole. To
prove this we will generalize to the case of a Chern-Simons theory, a
procedure introduced in the context of GR \cite{bonda} to construct a white
hole solution from a black hole solution.

From (\ref{38}) we see that exist a singular behaviour at $F(r_{0})=0$,
i.e., at $r_{0}^{3}=8\left( r_{0}+C\right) r_{c}^{2},$ such that 
\begin{equation}
C=\frac{1}{8r_{c}^{2}}\left( r_{0}-2\sqrt{2}r_{c}\right) \left( r_{0}+2\sqrt
2}r_{c}\right) r_{0},  \label{35'}
\end{equation
and this means that $C>0$ as long as $r_{0}>2\sqrt{2}r_{c}$. \ The equation 
F(r_{0})=0$ has only a real and positive root and its discriminant\ tells us
that 
\begin{equation}
C>\frac{4}{3}\sqrt{\frac{2}{3}}r_{c}=\left( \frac{2}{3\sqrt{3}}\right) 
\sqrt{2}r_{c},  \notag
\end{equation
and the solution for $r_{0}$ is given by

\begin{eqnarray}
r_{0} &=&\frac{1}{3}\left[ \left( 108C\pm 12\sqrt{3}C\sqrt
27-32r_{c}^{2}/C^{2}}\right) r_{c}^{2}\right] ^{1/3}  \notag \\
&&+\frac{8r_{c}^{2}}{\left[ \left( 108C\pm 12\sqrt{3}C\sqrt
27-32r_{c}^{2}/C^{2}}\right) r_{c}^{2}\right] ^{1/3}}.  \label{69''}
\end{eqnarray}

The solution (\ref{38}) appears to have a singularity at $r=0$ and $r=r_{0}.$
The singularity at $r=r_{0}$ divides the coordinates in two disconnected
patches. The exterior solution with $r>r_{0}$ is the one that is related to
the gravitational fields of massive objects. The interior solution with 
0\leq r\leq r_{0}$, which contains the singularity at $r=0$, is completely
separated from the outer patch by the singularity at $r=r_{0}$.

The singularity at $r=r_{0}$ is a coordinate singularity. This singularity
arises from a bad choice of coordinates or coordinate conditions. A easiest
way to see that at $r=r_{0}$ is to consider the metric in the
Eddington-Finkelstein (EF) coordinates. For the ingoing EF coordinates 
\left( v,r,t\right) $, which describe the line element in the $I$-$II$\
regions\textbf{\ (}the form of this regions as well as the regions $III
\textbf{\ }and\textbf{\ }$IV$ are similar to that of the case of GR which
are shown in Ref.\cite{chan}), we hav
\begin{equation}
ds^{2}=-F(r)dt^{2}+2dvdr+r^{2}d\Omega ^{2},  \label{68'}
\end{equation
which is regular at future horizon and the past horizon is at $v=t+r^{\ast
}=\infty $. \ 

Correspondingly, the outgoing EF coordinates $\left( u,r,t\right) $ which
describe the metric in the $III$-$IV$\ regions has the following form

\begin{equation}
ds^{2}=-F(r)dt^{2}-2dudr+r^{2}d\Omega ^{2},  \label{69'}
\end{equation
which is regular at past horizon and the future horizon is at $u\equiv
t-r^{\ast }$, which can be interpreted as a new time coordinate where 
r^{\ast }$ is given b
\begin{eqnarray}
r^{\ast } &=&\beta \ln \left( \frac{r-r_{0}}{2r_{c}\sqrt{\left(
r/2r_{c}\right) ^{2}+r_{0}r/4r_{c}^{2}+2C/r_{0}}}\right)  \notag \\
&&-\frac{8r_{c}\left( 3C+r_{0}\right) }{\left( 2r_{0}+3C\right) \sqrt
8\left( 3C-r_{0}\right) /r_{0}}}\arctan \left( \frac{2r+r_{0}}{r_{c}\sqrt
8\left( 3C-r_{0}\right) /r_{0}}}\right) ,  \label{70}
\end{eqnarray
and 
\begin{equation}
\beta =-\frac{r_{0}^{2}}{2r_{0}+3C}.  \label{71}
\end{equation}

The coordinates used in (\ref{68'}) and (\ref{69'}) have the advantage that
they describe the neighbourhood of the surface $r=r_{0}$ in a satisfactory
way. However, the metrics (\ref{68'}) and (\ref{69'}) has still a deficienc
\textbf{,} analogue to that appears in the Schwarzschild solution of GR.
This deficiency is avoided in the Kruskal coordinates, which describe a
geodesically complete space. Defining 
\begin{equation}
U=-\exp \left( -\frac{u}{2\beta }\right) \text{ \ \ },\text{ \ }V=\exp
\left( \frac{v}{2\beta }\right) ,\text{ \ }  \label{73'}
\end{equation
we have that the Kruskal coordinates are given by

\begin{equation}
T=\frac{1}{2}(U+V)\text{ \ \ },\text{ \ }X=\frac{1}{2}(U-V),  \label{83}
\end{equation
with which the line element takes the form 
\begin{equation}
ds^{2}=-F_{C}(r)\left( dT^{2}-dX^{2}\right) +r^{2}d\Omega ^{2},  \label{85}
\end{equation
wher
\begin{equation}
F_{C\text{ }>0}(r)=\frac{32r_{0}r_{c}^{2}}{3r_{0}^{2}-8r_{c}^{2}}F(r)\exp
\left( \frac{3r_{0}^{2}-8r_{c}^{2}}{8r_{c}^{2}}\frac{r^{\ast }}{r_{0}
\right) ,\quad r>0,  \label{79}
\end{equation}

Following the usual procedure we construct a spacetime diagram in the 
\left( T,X\right) $ plane (with $\theta ,\varphi $ suppressed) known as the
Kruskal diagram, which represents the extended spacetime corresponding to
the metric (\ref{85}), whose form is similar to that of the case of GR (see
Ref. \cite{chan} where the aforementioned regions $I,II,III,IV$\ are
shown).\ 

Following the reference \cite{bonda}, the next step is to continue the
metric in the region of the additive inverse of $C$ carrying out the
substitution $C\longrightarrow -\tilde{C}$, with $\tilde{C}>0$, which must
be complemented with the transformation $t\longrightarrow -\tilde{t}$, 
r\longrightarrow -\tilde{r}$, $r_{c}\longrightarrow -\tilde{r}_{c}$, because
it is necessary to preserve the dynamics. Under this transformation we see
that (\ref{69''}) transforms as

\begin{eqnarray}
r_{0} &\longrightarrow &-\tilde{r}_{0}=\frac{1}{3}\left[ -\left( 108\tilde{C
\pm 12\sqrt{3}\tilde{C}\sqrt{27-32\tilde{r}_{c}^{2}/\tilde{C}^{2}}\right)
r_{c}^{2}\right] ^{1/3}  \notag \\
&&+\frac{8\tilde{r}_{c}^{2}}{\left[ -\left( 108\tilde{C}\pm 12\sqrt{3}\tilde
C}\sqrt{27-32\tilde{r}_{c}^{2}/\tilde{C}^{2}}\right) r_{c}^{2}\right] ^{1/3}
.  \label{86}
\end{eqnarray
This means that $C\longrightarrow -\tilde{C}$ implies $r_{0}\longrightarrow 
\tilde{r}_{0}$. On the other hand under this transformation it is
straightforward to see that $\tilde{\beta}=-\beta $ and that the EF
coordinate $v=t+r^{\ast }$ takes the form

\begin{eqnarray}
\tilde{v} &=&-\tilde{t}-\left\{ \tilde{\beta}\ln \left( \frac{\left( \tilde{
}-\tilde{r}_{0}\right) }{2\tilde{r}_{c}\sqrt{\left( \tilde{r}/2\tilde{r
_{c}\right) ^{2}+\tilde{r}_{0}\tilde{r}/4\tilde{r}_{c}^{2}+2\tilde{C}/\tilde
r}_{0}}}\right) \right.  \notag \\
&&\left. -\frac{8\tilde{r}_{c}\left( 3\tilde{C}+\tilde{r}_{0}\right) }
\left( 2\tilde{r}_{0}+3\tilde{C}\right) \sqrt{8\left( 3\tilde{C}-\tilde{r
_{0}\right) /\tilde{r}_{0}}}\arctan \left( \frac{2\tilde{r}+\tilde{r}_{0}}
\tilde{r}_{c}\sqrt{8\left( 3\tilde{C}-\tilde{r}_{0}\right) /\tilde{r}_{0}}
\right) \right\} -2i\pi \tilde{\beta},  \notag \\
\tilde{v} &=&-\tilde{t}-\tilde{r}^{\ast }-2i\pi \tilde{\beta}=-\nu -2i\pi 
\tilde{\beta},  \label{87}
\end{eqnarray
where $\tilde{r}^{\ast }$ is given by the expression in parentheses brace.

In the same way it is found that the Eddington-Finkelstein coordinate 
u=-t+r^{\ast }$ transform as

\begin{equation}
u=-\tilde{t}+\tilde{r}^{\ast }+2i\pi \tilde{\beta}=-u+2i\pi \tilde{\beta}.
\label{88}
\end{equation}

Equations (\ref{87}) and (\ref{88}) allow finding the continuation of
equations (\ref{73'}) in the region of the additive inverse of $C,$ which
can be interpreted as the negative mass region. In fact, under the
transformation $C\longrightarrow -\tilde{C}$, $t\longrightarrow -\tilde{t}$, 
$r\longrightarrow -\tilde{r}$, $r_{c}\longrightarrow -\tilde{r}_{c}$, the
equations (\ref{73'}) transform as 
\begin{eqnarray*}
U &\longrightarrow &\tilde{U}=\exp \left( -\frac{\tilde{u}}{2\tilde{\beta}
\right) =-U \\
V &\longrightarrow &\tilde{V}=-\exp \left( \frac{\tilde{v}}{2\tilde{\beta}
\right) =-V.
\end{eqnarray*
It is straightforward to see that this transformation implies that the
Kruskal coordinates transform as

\begin{eqnarray*}
T &\longrightarrow &\tilde{T}=\frac{1}{2}(\tilde{U}+\tilde{V})=-T, \\
\tilde{X} &=&\frac{1}{2}(\tilde{V}-\tilde{U})=-X,
\end{eqnarray*
that is, the inversion of the Kruskal coordinates. As in the case of general
relativity, studied in \cite{bonda}, the spacetime with the variables $
\tilde{U},\tilde{V})$, found using the transformations $C\longrightarrow 
\tilde{C}$, $t\longrightarrow -\tilde{t}$, $r\longrightarrow -\tilde{r}$, 
r_{c}\longrightarrow -\tilde{r}_{c}$, can be understood as a space
corresponding to a spacetime of the a white hole. \ This means that region 
I $ with coordinates $(\tilde{U},\tilde{V})$ corresponds to region $III$
when we use coordinates $\left( U,V\right) $.

In the same way, region $IV$, which in the usual coordinates corresponds to
a white hole, is transformed into region $II$ which corresponds, in the new
coordinates, to a black hole. Obviously, region $II$, which in the usual
coordinates corresponds to a black hole, will be transformed into region $IV$
which, in the new coordinates, corresponds to a white hole.

The interesting thing about this result is that the solutions for $C<0$ and 
C>0$ are related through the transformation $C\longrightarrow -\tilde{C}$,
with $\tilde{C}>0$, $t\longrightarrow -\tilde{t}$, $r\longrightarrow -\tilde
r}$, $r_{c}\longrightarrow -\tilde{r}_{c}$.

\section{\textbf{Concluding Remarks }}

We have found black hole solutions from the $4$-dimensional gravitational
field equations corresponding to a four-dimensional gravity action, which
was obtained by compactifying the so-called five-dimensional EChS gravity
action using a mechanism developed in \cite{randall} and \cite{randall1}. We
have shown that such field equations lead to a spacetime with a cosmological
constant inversely proportional to the square of the compactification radius
and to a solution dependent on an arbitrary constant $C$ which gives rise to
black white holes solutions: ($i$) when $C$ is negative, we find a SdS black
hole, ($ii$) when $C=0$, we have a black hole whose event horizon is a
cosmological one and ($iii$)\ it is shown that the application of the $PTM$
transformation to the solution for $C<0$ leads to a solution for $C>0$
equivalent to an SdS white hole solution. The $PTM$ transformation means
changing $r\longrightarrow -\tilde{r},$ $t\longrightarrow -\tilde{t},$ 
r_{c}\longrightarrow \tilde{r}_{c}$ in addition to changing 
M\longrightarrow -\tilde{M}$. The change of sign in the mass can be
understood as a consequence of the requirement of the invariance of the
metric under the $PTM$ symmetry.

\textbf{Acknowledgements}

This work was supported in part by\textit{\ }FONDECYT Grant\textit{\ }No
\textit{\ }1180681 from the Government of Chile. One of the authors (CR) was
supported by ANID fellowship No. 2219610 from the Government of Chile and
from Universidad de Concepci\'{o}n, Chile.

