Conditional escape of gravitons from the brane
et al.
In this paper we consider a cosmological Friedmann Robertson Walker brane world embedded in a 5-dimensional anti-de Sitter Schwarzschild bulk. We show, using potential diagrams, that for an anti-de Sitter bulk the null geodesics never return to the brane. Null geodesics do however return for $k=+1$ when we include the Schwarzschild like mass and the condition of return is obtained from the corresponding effective potential. We next obtain the condition for a gravitational signal to be emitted from a FRW brane with isotropically distributed matter. We use these results to investigate the conditions under which shortcuts through the bulk are possible.
conditional_escape2.tex
\documentclass[10pt]{iopart}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%version_2
%25_April_2005
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\usepackage{mathrsfs}
\usepackage{iopams}
\usepackage{cite}
\usepackage{epsfig}
\usepackage{graphicx}

\newcommand{\reff}[1]{(\ref{#1})}
\def\udot{\dot{u}}
\def\3nab{\tilde{\nabla}}
\def\lgl{\langle}
\def\rgl{\rangle}
\def\la{\langle}
\def\ra{\rangle}
\def\c{\mbox{curl}}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\ba{\begin{eqnarray}}
\def\ea{\end{eqnarray}}
\newcommand{\bra}[1]{\left(#1\right)}
\newcommand{\bras}[1]{\left[#1\right]}
\newcommand{\brac}[1]{\left\{#1\right\}}
\newcommand{\bfe}{\boldsymbol{e}}
\newcommand{\sfr}[2]{{\textstyle\frac{#1}{#2}}}
\newcommand{\ffr}[2]{{\frac{#1}{#2}}}
\newcommand{\hs}{\;-\;}
\jl{6}

\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{Conditional escape of gravitons from the brane}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\author[J.A. Leach {\it et al.}]{Jannie A. Leach \dag \footnote{Present address: Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch,
7701, South Africa}\ and William M. Lesame \dag} \vspace{3mm}

\address{\dag \ Department of Mathematical Sciences, University of South Africa, Pretoria 0003, South Africa}

\date{\today}

\eads{leachja@nassp.uct.ac.za and lesamwm@unisa.ac.za}

\begin{abstract}
In this paper we consider a cosmological Friedmann Robertson
Walker brane world embedded in a 5-dimensional anti-de Sitter
Schwarzschild bulk. We show, using potential diagrams,
that for an anti-de Sitter bulk the null geodesics never return to
the brane. Null geodesics do however return for $k=+1$ when we include
the Schwarzschild like mass and the condition of return is
obtained from the corresponding effective potential. We next
obtain the condition for a gravitational signal to be emitted from a FRW brane with isotropically
distributed matter. We use these results to investigate the conditions under which shortcuts through the bulk are possible.
\end{abstract}

\pacs{98.80.Cq, 04.50.+h}

%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%

Causality plays an important role in the brane-world
scenario. Null geodesics passing through the extra
dimension may connect points in the lower dimension which are
causally disconnected in terms of the null geodesics confined to
the brane \cite{Chung}. Thus instead of inflation, such null
geodesics could be used to solve the cosmological horizon problem.
The condition for the existence of such ``causality violating" null
geodesics, which pass through the $AdS_5$ bulk, is merely the
deviation from a pure tension-like stress energy tensor, i.e.
provided that the energy density $\rho_b$ and the isotropic
pressure $p_b$ of the brane satisfies \cite{Ishi01}
\[
\rho_{b}+p_{b}>0,
\]
then the extrinsic curvature bends the brane concave towards the
bulk, allowing the existence of such null geodesics.

Initially most authors considered these ``shortcuts'' through the
bulk for a static cosmological brane (brane based approach). They
made use of the most general metric (see \cite{BinLang,BinDef}) of the form
\begin{equation*}
ds^{2}=-n^{2}\left( t ,y\right) dt ^{2}+a^{2}\left( t ,y\right)
\gamma _{\mu \nu }dx^{\mu }dx^{\nu }+b^{2}\left( t ,y\right)
dy^{2}.
\end{equation*}
Null bulk geodesic equations of motion has been computed for static and
cosmological single brane models (see \cite{Youm01,Davis,Abdalla02}). It was found that
although there are effective acausality in some cases, there is no \textit{a priori} solution to the horizon
problem.

A more realistic scenario was found by considering a dynamical
brane (bulk based approach). It has been shown that the
motion of a domain wall in the bulk can be interpreted as cosmological
expansion or contraction on the wall (see \cite{Kraus}). A generalization of Birkhoff's
theorem can be used to show that a fixed wall embedded in a non-static spacetime
is strictly equivalent to a moving wall embedded in a
static spacetime and so establishing in full generality
the equivalence between the two approaches (see \cite{Bowcock}). Solutions have also been
found for higher dimensional dynamic domain walls that couples to bulk matter \cite{Chamblin}.

Causality can now be investigated in a different way. The
trajectories of the null geodesics and the brane's equations of
motion in the bulk are first obtained. These are then plotted on
the same set of axes, with both curves having the same initial
conditions. If the curves intersect again, then it can be said
that two points are causally connected through the bulk. The
brane's equations of motion for a FRW brane in a $AdS_5$ bulk
(only considering the tension of the brane and not the matter
inside) was obtained in \cite{Kraus}. Shortcuts through the bulk
could thus be investigated from the dynamical or bulk based
approach (see e.g.\cite{Caldwell,Abdalla05}). A FRW brane embedded
in a $AdS_5$ was used and it was found that although the distance
covered by a graviton passing through the bulk is longer than the
corresponding distance on the brane, it is not enough to solve the
horizon problem \cite{Caldwell}. Shortcuts have also been
investigated in various other models, such as; six dimensional
brane worlds \cite{Abdalla02b}, $(D-2)$-branes (domain walls)
\cite{Abdalla03b} and for nonlinear dynamical universes
\cite{Melgar04}.

In non of the references above (and no reference to our knowledge),
has the condition of escape been determined for a graviton to
actually leave the brane. So far most authors have assumed that
gravitons escape freely into the bulk. If this was the case, why
then don't all the gravitons escape into the bulk? This is the
main question which we will try and answer in this paper. We will
attempt to firstly determine whether null geodesics return to the
brane after they have escaped and secondly the condition for a
graviton to escape from the brane.

%%%%%%%%%%%%%%%%%%%%%%
\section{Null geodesic equations}
%%%%%%%%%%%%%%%%%%%%%%

We start with the 5-dimensional anti-de Sitter Schwarzschild
metric
\begin{equation}
ds^{2}=-f_{k}(R)dT^{2}+f_{k}(R)^{-1}dR^{2}+R^{2}d\Sigma _{k}^{2}
\label{AdSS5 metric general}
\end{equation}
where $T$ is the global bulk time, $R$ is the bulk dimension and $d\Sigma
_{k}$ is the metric for a maximally symmetric three-dimensional \ spaces ($%
k=0$ for a flat space, $k=1$ for a three sphere and $k=-1$ for a
hyperbolic three space). The function $f_{k}(R)$ is given by
\begin{equation}
f_{k}(R)=k+\frac{R^{2}}{\ell^{2}}-\frac{\mu}{R^2}  \label{function
fk}
\end{equation}
where $\ell\equiv \sqrt{\frac{-6}{\Lambda}}$ is the anti-de -Sitter
radius of curvature, $\Lambda$ is the negative bulk cosmological
constant (i.e. $\Lambda<0$) and $\mu$ is the five dimensional
Schwarzschild-like mass. Since the third term in the equation above scales like radiation, $\mu$
is sometimes called the dark radiation or Weyl parameter. The solutions are anti-de Sitter when
$\mu=0$ and when $\mu\neq 0$ we have a bulk black hole with a
horizon at
\begin{equation}
R_h^2=\frac{\ell^2}{2}\left(-k+\sqrt{k^2+\frac{4\mu}{\ell^2}}\right).
\label{horizon}
\end{equation}

A spherical coordinate system $(r,\theta ,\phi )$ can now be introduced in the brane which is
centered at a point A, so that any path can be described by a
radial geodesic \cite{Caldwell}. Then due to the radial geodesics we may ignore
the angular variables $\theta $ and $\phi $, so that we are left
with a three dimensional problem with the resulting metric being
given by
\begin{equation}
ds^{2}=-f_{k}(R)dT^{2}+f_{k}(R)^{-1}dR^{2}+R^{2}dr^{2}.
\label{3d adss5 metric}
\end{equation}
The geodesics may be easily obtained by using the symmetries to
obtain the Killing vectors, namely $\left(\frac{\partial}{\partial
T}\right)^a$ and $\left(\frac{\partial}{\partial r}\right)^a$ (see
\cite{Caldwell} and \cite{Misner}). Then the vector tangent to
the geodesics, $p^a$, is given by
\begin{equation}
p_T=-f_{k}(R)\frac{dT}{d\lambda }=-E\quad (a\;constant) \label{pT}
\end{equation}
and
\begin{equation}
p_r=R^{2}\frac{dr}{d\lambda }=P\quad (a\;constant) \label{pr}
\end{equation}
where $\lambda$ is an affine parameter that relates each of the
three parameters $(R,T,r)$ to one another. Both $E$ and $P$ are
positive constants which are similarly defined as the ``energy at
infinity" $E$ and ``angular momentum" $L$ in standard general
relativity (see \cite{Misner}).

For a null geodesic we furthermore have
\begin{equation}
\left( \frac{dR}{d\lambda }\right)
^{2}=E^{2}-P^{2}\frac{f_{k}(R)}{R^{2}} \label{dR/dlam}
\end{equation}
Equations (\ref{pT}), (\ref{pr}) and (\ref{dR/dlam}) may be rewritten in the standard GR form of \cite{Misner}
 by setting $\tilde{\lambda}=P\lambda$ so that the
full set of geodesic equations are
\begin{eqnarray}
\frac{dT}{d\tilde{\lambda}}&=& \frac{E}{P f_k(R)}, \label{geodesic
T} \\
\frac{dr}{d\tilde{\lambda}}&=& \frac{1}{R^{2}}
\label{geodesic r}, \\
\left(\frac{dR}{d\tilde{\lambda}}\right) ^2&=&
\frac{E^2}{P^2}-\frac{f_k(R)}{R^{2}}. \label{geodesic R null}
\end{eqnarray}


%%%%%%%%%%%%%%%%%%%%%%%%
\section{Brane motion}
%%%%%%%%%%%%%%%%%%%%%%%%

We start by introducing the proper time $t$ on the brane, which is
defined by \cite{Caldwell} as
\begin{equation}
dt^2=f(R_b)dT^2-f(R_b)^{-1}dR_b^2 \label{dT/dt 2}
\end{equation}
so that
\begin{equation}
\frac{dT}{dt}=\frac{\sqrt{f(R_b)+\dot{R}_b^2}}{f(R_b)}
\label{dT/dt}
\end{equation}
where $\dot{R}_b=\frac{dR_b}{dt}$. $R_b (t)\equiv a(t)$ is the
usual cosmological scale factor in the brane-world which can be
obtained from the modified Friedmann equation which is determined
from the junction conditions.

The Israel junction conditions \cite{Israel} relate the
discontinuity in the second fundamental form (extrinsic curvature)
through the brane and the energy momentum tensor in the brane. If
$K^+_{ab}$ and $K^-_{ab}$ are the jump in extrinsic curvature on
the two sides of the brane, then the junction conditions for a $Z_2$ symmetric brane is \cite{Ishi01}
\begin{equation}
K^+_{ij}=-K^-_{ij}=-\frac{1}{2}\kappa^2_{5}\left(S_{ij}-\frac{1}{3}h_{ij}
S\right). \label{junction cond brane adss}
\end{equation}
The extrinsic curvature of the hypersurface can be computed using
the unit norm vector (see e.g. \cite{Abdalla05,Brax03a,Brax03}).
 If we assume an isotropic distribution of matter on the brane,
then the energy momentum tensor $S_{ij}$ in the brane is given by
\begin{equation}
S_{ij}=\rho_b u_i u_j +p_b\left(h_{ij}+u_i u_j\right),\label{Sij
isotropic}
\end{equation}
where $\rho_b$ is the energy density and $p_b$ is the isotropic
pressure of the brane.

From the junction conditions above one obtains two sets of equation governing the evolution of the brane (see \cite{Kraus,Bowcock,BinLang}):
\begin{equation}
\frac{d\rho_b}{dt}=-3\frac{\dot{R}_b}{R_b}\left(\rho_b+p_b\right).
\label{jc isomat 2}
\end{equation}
and
\begin{equation}
\frac{\dot{R}^2_b}{R_b^2}+\frac{f(R_b)}{R_b^2}=\frac{\kappa^4_{5}
\rho_b^2}{36}. \label{jc isomat 1}
\end{equation}

We require some equation of state that relates the matter
density $\rho$ and pressure $p$ on the brane. We follow convention
by setting
\begin{equation*}
\rho_b=\rho +\sigma, \ \ \ \ p_b=p-\sigma,
\end{equation*}
where $\sigma$ is tension of the wall. The equation of state then
may be given by
\begin{equation*}
p=\left(\gamma -1\right)\rho.
\end{equation*}
Substitution then yields the conservation equations as
\begin{equation}
\frac{d\rho}{dt}=-3\frac{\dot{R}_b}{R_b}\left(\rho+p\right),
\label{cons eqn brane}
\end{equation}
with the solution
\begin{equation*}
\rho=\rho_0\left(\frac{R_0}{R_b}\right)^{3\gamma} \label{sol cons
eqn}
\end{equation*}
where the subscript $0$ means any given time. Similarly we may
obtain the modified Friedmann equation of \cite{BinDef} in the form
\begin{equation}
\dot{R}_b^2+\left(1-\left(\frac{\rho+\sigma}{\sigma_c}\right)^2\right)\frac{R_b^2}{\ell^2}-\frac{\mu}{R_b^2}=-k,
\label{matter FRW eqn}
\end{equation}
where $\sigma_c\equiv3/(4\pi G\ell^2)=6/(\kappa^2_{5} \ell)$ is
the critical tension for fine tuning. The cosmological constant in
the brane then has the form
\begin{equation*}
\frac{\Lambda_4}{3}=\frac{\kappa^4_{5}\sigma^2}{36}-\frac{1}{\ell^2}=\frac{1}{\ell^2}\left(\frac{\sigma^2}{\sigma_c^2}-1\right).
\end{equation*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Graviton emission and capture}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Potential diagrams}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In the case of a domain wall, there is no matter on the wall so we do not expect the escaping gravitons to be attracted back to the brane. A more realistic scenario would be to consider a brane with isotropically distributed matter. Analytically computing trajectories of the moving brane with matter is difficult so we make use of potential diagrams to study whether gravitational signals will return to the brane. In classical GR, potential diagrams have been found useful for
qualitative analysis of orbits of particles and photons in Schwarzschild geometry. Potential diagrams has been used before to investigate Lorentz violations in $AdSS_5$ bulk background \cite{Csaki}. Here we extend that analysis to brane motion and graviton motion in $AdS_5$ and $AdSS_5$.

We start by rewriting equation (\ref{geodesic R null}) as
\begin{equation}
\left(\frac{dR}{d\tilde{\lambda}}\right) ^2+B^2(R)=b^2,
\label{geodesic R potential}
\end{equation}
where
\begin{equation}
B^2(R)=\frac{f_k(R)}{R^{2}} \label{effective poten}
\end{equation}
defines a kind of ``effective potential of a graviton", and
\begin{equation}
b^2=\frac{E^2}{P^2} \label{impact param}
\end{equation}
defines the ``impact parameter".

The potential impact parameter $B(R)$ has the following useful
interpretation: For a graviton to reach the point $R$, it must have
a impact parameter $b$ greater than or equal to $B(R)$, i.e.
\begin{equation}
b\geq B(R). \label{cond. acces}
\end{equation}
This is referred to as the \textit{condition of accessibility} \cite{Misner}.
\vspace{5mm}

\noindent \underline{$AdS_5$ ($\mu=0$)}: The effective potential (\ref{effective poten}) for
the various values of $k$ are
\begin{eqnarray}
B^2(R)&=& \frac{1}{\ell^{2}}+\frac{1}{R^{2}}, \ \ \ \ k=+1\\
B^2(R)&=&\frac{1}{\ell^{2}}, \ \ \ \ \ \ \ \ \ \ \ \ k=0 \\
B^2(R)&=&\frac{1}{\ell^{2}}-\frac{1}{R^{2}}, \ \ \ \ k=-1
\end{eqnarray}
and are plotted in Figure 1. Clearly none of the potential curves
have turning points, i.e if a graviton escape from the brane
(domain wall or brane with matter) into an anti-de
Sitter bulk, it will not return to the brane. For $k=-1$ we have a
horizon at $R=\ell$.

\begin{figure}[tbp]
\begin{center}
\epsfig{file=potAds.eps, scale=0.9}
\caption{Potential diagram for $AdS_5$ ($\mu=0$) bulk geometry.}
\end{center} \label{Poten Diagram mu=0}
\end{figure}


\noindent \underline{$AdSS_5$ ($\mu\neq0$)}: The effective potential (\ref{effective
poten}) for the various values of $k$ are
\begin{eqnarray}
B^2(R)&=& \frac{1}{\ell^{2}}+\frac{1}{R^{2}}-\frac{\mu}{R^4}, \ \ \ \ k=+1\\
B^2(R)&=&\frac{1}{\ell^{2}}-\frac{\mu}{R^4}, \ \ \ \ \ \ \ \ \ \ \ \ k=0 \\
B^2(R)&=&\frac{1}{\ell^{2}}-\frac{1}{R^{2}}-\frac{\mu}{R^4}, \ \ \
\ k=-1
\end{eqnarray}
The case where $k=+1$ is plotted in Figure 2. Both the
$k=0$ and $k=-1$ cases look similar to $k=-1$ when $\mu=0$ (see Figure 1), i.e they have
no turning points. The horizons for all $k$'s are given by
(\ref{horizon}).

\begin{figure}[tbp]
\begin{center}
\epsfig{file=potsch1.eps, scale=0.9}
\caption{Potential diagram for $\mu\neq 0$ and $k=+1$.}
\end{center} \label{Poten Diagram black k=+1}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Condition of return}
%%%%%%%%%%%%%%%%%%%%%%%%%

The turning point on Figure 2, is at
\begin{equation}
R_t=\left(2\mu\right)^\frac{1}{2} \label{Rt}
\end{equation}
which implies a maximum effective potential of
\begin{equation*}
B^2(R_t)=\frac{1}{\ell^{2}}+\frac{1}{4\mu}.
\end{equation*}

Combining this with the condition of accessibility (\ref{cond.
acces}), we can summarize the qualitative features of the
trajectories of gravitons emitted in the $AdSS_5$ bulk with $k=+1$ (see \cite{Misner}) as follows:

When a graviton is emitted near the horizon $R_h$ (i.e. at some point $R_{em}$, $R_h<R_{em}<R_t$), it will:
\begin{enumerate}
\item escape to infinity when
\begin{equation}
b>\sqrt{\frac{1}{\ell^2}+\frac{1}{4\mu}};
\end{equation}
\item or return when
\begin{equation}
0<b<\sqrt{\frac{1}{\ell^2}+\frac{1}{4\mu}}. \label{cond.ret}
\end{equation}
\end{enumerate}

Equation (\ref{cond.ret}) in effect gives us the \textit{condition of return}
for a graviton emitted near the bulk horizon $R_h$. The lower limit arise from the fact that both
$E$ and $P$ are positive constants, and hence $b>0$. Clearly we can only expect returning
gravitons at early times when $R_{em}<R_t$ when the effective potential is increasing.
When a graviton is emitted at $R_{em}>R_t$ it will escape to infinity (since $B(R)$ is decreasing).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Conditions of escape}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For a gravitational signal to escape from the brane, it must move
faster than the brane with respect to an observer in the bulk.
Thus the relative velocity of the gravitational signal $v_g$ must
be greater than the relative velocity of the brane $v_b$ at the point of emission $R_{em}$. Therefore, we now seek to obtain the relative velocities of the
gravitational signal and the brane.

We first obtain the relative velocity of the gravitational signal
by dividing equation (\ref{geodesic T}) into equation
(\ref{geodesic R null}), and then taking the square root to find
\begin{equation}
v_g=\frac{dR_g}{dT}=f_k(R_g)\sqrt{1-\frac{P^2 f_k(R_g)}{E^2 R_g^2}}=f_k(R_g)\sqrt{1-\frac{B^2(R_g)}{b^2}}.
\label{vg}
\end{equation}
We see that this equation will hold if the condition of accessibility (\ref{cond. acces}) is satisfied
(where we made use of (\ref{effective poten}) and (\ref{impact param})). The velocity will be zero when
$b=B(R_g)$, i.e. we will have a turning point for a gravitational signal.

Next we find the relative velocity of the brane by substituting
equation (\ref{dT/dt 2}) for $dt$ into (\ref{jc isomat 1}),
\begin{equation}
dR_b^2=\left[\left(\frac{\rho+\sigma}{\sigma_c}\right)^2\frac{R_b^2}{\ell^2}-f_k(R_b)\right]
\left[f_k(R_b)dT^2-f_k^{-1}(R_b)dR_b^2\right],
\end{equation}
which on simplification yields
\begin{equation}
v_b=\frac{dR_b}{dT}=f_k(R_b)\sqrt{1-\frac{\ell^2f_k(R_b)}{\left(\frac{\rho+\sigma}{\sigma_c}\right)^2
R_b^2}}. \label{vb}
\end{equation}
The equation for the relative velocity of the brane will only hold if
\begin{equation}
\hat{\beta}=\frac{1}{\ell}\left(\frac{\rho+\sigma}{\sigma_c}\right)\geq B(R_b).
\end{equation}
In the limit of $R\rightarrow 0$ this imply that $\mu\geq 0$ and in the limit of $R\rightarrow \infty$
we must have $\sigma\geq \sigma_c$. We are therefore only considering positive dark radiation.

The ratio of the two relative velocities is then given by
\begin{equation}
\frac{v_g}{v_b}=\frac{f_k(R_g)}{f_k(R_b)}\cdot\sqrt{\frac{1-\frac{P^2 f_k(R_g)}{E^2
R_g^2}}{1-\frac{\ell^2f_k(R_b)}{\left(\frac{\rho+\sigma}{\sigma_c}\right)^2
R_b^2}}}
\end{equation}

We seek to find the conditions of escape, and therefore we only
consider the position of the brane and the energy density at the
time of emission, i.e. $R_b=R_g=R_{em}$ and $\rho=\rho_{em}$ respectively.
The graviton will have the same position as the brane at the bulk time of emission. A gravitational
signal can be emitted if and only if $v_g>v_b$, that is when
\begin{equation*}
1-\frac{P^2 f_k(R_{em})}{E^2
R_{em}^2}>1-\frac{\ell^2f_k(R_{em})}{\left(\frac{\rho_{em}+\sigma}{\sigma_c}\right)^2
R_{em}^2},
\end{equation*}
which after simplification reduces to
\begin{equation}
\frac{E}{P}>\frac{1}{\ell}\left(\frac{\rho_{em}+\sigma}{\sigma_c}\right) \label{cond escape}
\end{equation}
or
\begin{equation}
b>\hat{\beta}_{em}.
\end{equation}
This condition can be considered as the \textit{condition of escape}, that is the condition
which needs to be satisfied for a graviton to escape from the brane.

When $v_g=v_b$ we have the condition
\begin{equation}
\frac{E}{P}=\frac{1}{\ell}\left(\frac{\rho_{em}+\sigma}{\sigma_c}\right)
\end{equation}
which implies that the graviton is stuck on the brane and
therefore appears to be moving with the brane at the same relative
velocity. In the case when $v_g<v_b$,
the gravitational signal is lagging the brane. This is clearly not possible.

%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary and discussion}
%%%%%%%%%%%%%%%%%%%%%%%%

The only case which showed the possibility of emitting a graviton
with it returning to the brane, is $k=+1$ when $\mu\neq 0$. The
potential diagram for this case has a classical potential form
(see \cite{Misner} and Figure 2) which meant that if a graviton is emitted from the brane at $R_{em}$
(where $R_h<R_{em}<R_t$) it would return if the condition (\ref{cond.ret}) is satisfied. Comparing the
velocities of the graviton and the brane we were able to determine that a graviton will be able to escape
from the brane if the condition (\ref{cond escape}) is satisfied.

The condition for a graviton to escape from the brane at some early time (when $R_h<R_{em}<R_t$) and return,
is then given by
\begin{equation}
\frac{1}{\ell}\left(\frac{\rho_{em}+\sigma}{\sigma_c}\right)<b<\sqrt{\frac{1}{\ell^2}+\frac{1}{4\mu}},
\end{equation}
where $\mu>0$ and $\sigma\geq\sigma_c$. This condition can also be expressed in terms of the 5-dimensional
 fundamental mass scale $M_5$, by using the
relationship $\kappa_5^2=8\pi G\ell=M_5^{-3}$, namely
\begin{equation}
\frac{\rho_{em}+\sigma}{6M^3_5}<b<\sqrt{\frac{1}{\ell^2}+\frac{1}{4\mu}}.
\end{equation}

Since the impact parameter has no physical significance, we rather use the conditions above to find the
relationship between parameters that do have physical significance. We find that the conditions of escape
and return impose the following restrictions on the dark radiation $\mu$ (Schwarzschild-like mass)
\begin{equation}
\mu<\frac{\ell^2}{4}\left[\left(\frac{\rho_{em}+\sigma}{\sigma_c}\right)^2-1\right]^{-1},
\end{equation}
for all $\rho_{em}+\sigma>\sigma_c$. This in turn tells us that we may only have returning gravitons before the
brane reaches the point $R_t$, where using (\ref{Rt}) we have the condition
\begin{equation}
R_t<\frac{\ell}{\sqrt{2}}\left[\left(\frac{\rho_{em}+\sigma}{\sigma_c}\right)^2-1\right]^{-\frac{1}{2}}.
\end{equation}

We can realistically only expect emitted gravitons to return to
the brane at very early times ($\rho_{em}+\sigma>>\sigma_c$) when
$R_{em}\rightarrow R_h$ and thus $B(R_{em})\rightarrow 0$. At late
times ($\rho \rightarrow 0$) we are essentially looking at a
domain wall moving in an $AdS_5$ space where gravitons do not
return to the brane (see appendix). In the limit
$\rho_{em}+\sigma>>\sigma_c$ the restrictions above reduce to
\begin{equation}
\mu<\frac{1}{(\rho_{em}+\sigma)^2}
\end{equation}
and hence
\begin{equation}
R_t<\frac{1}{\rho_{em}+\sigma}.
\end{equation}
The upper limit for the dark radiation $\mu$ obtained here still require
further investigation, especially in the light of the nucleosynthesis limits \cite{Ichiki}.

We see that there exist a small window of opportunity, before the brane reaches $R_t$,
for gravitons to be emitted from the brane and then return to it. After the brane reaches $R_t$,
all gravitons emitted never return to the brane. An alternative might be to look at a bulk with
some sort of field (dilatonic etc.) which forces the path back to the brane when it has gone beyond $R_t$.
Another possibility is to have a black hole on the brane that will give us
the required curvature. Black holes on the brane have been
consider by a number of authors (see \cite{ChamHawk,Dadhich,Emparan,Chamblin01,Pravda})
most notably in the context of causality, by \cite{Chamblin01b}.

Another issue that needs investigation, is that of conservation of
energy for a brane that is radiating these graviton. Clearly the
form (\ref{jc isomat 2}) does not take into consideration the
gravitons that are emitted. Radiating braneworlds have been
considered in \cite{Langlois} and \cite{Leeper}, which might show
more promising results. The mechanisms that could produce
gravitons with sufficient energy to leave the brane has also as
yet not been explored properly.

In conclusion, there are two important aspect of escaping gravitational signals.
In \S 4 we clearly saw that the condition of return does not
depend on the type of brane that we choose, but rather on the type
of bulk (i.e. on its 5-D spacetime). We also discovered that
the condition of escape depends on the type of brane we choose and
is independent of the bulk space time. We note that no matter
which values of $\mu$ ($AdSS_5$) and charge $Q$ (Reissner-Nordstr\"{o}m) we choose,
the escape condition still stays the same. The condition of escape is also indepent of the curvature values $k$.

We have thus shown that there are certain conditions which first
needs to be satisfied before one may assume that a signal can
escape and then return to the brane. Two important conditions need
to be satisfied, firstly we need a condition under which a
graviton can escape from the brane (or be captured by the brane),
and secondly the conditions which allow it to return to the brane.
Furthermore,
 these two conditions together restrict gravitons to escaping and returning to
 the brane before the brane reaches a point $R_t$ in the bulk.
If all of these conditions are not satisfied one can not expect to have
causally connected points through the bulk.

%%%%%%%%%%%%%%%%%%%
\ack
%%%%%%%%%%%%%%%%%%%

This research was supported by the National Research Foundation (South Africa).
J.L. also acknowledges the postgraduate assistantship from the University of South Africa (UNISA).

\appendix

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Domain wall motion in $AdS_5$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we give the results for the special case where we
are considering a empty brane (domain wall) moving in an $AdS_5$
bulk.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Null geodesics solutions for $AdS_5$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

When the bulk is anti-de Sitter $AdS_5$, i.e $\mu=0$, one can
readily obtain the solutions of the null geodesic equations
(\ref{geodesic T})-(\ref{geodesic R null}). For $k=0$ the
solutions are
\begin{equation}
R_g=\frac{\mathcal{B}_g}{\mathcal{A}_{g}-T}, \ \ \ \ \ \alpha>0,
\label{k=0 solution R(T)}
\end{equation}
and
\begin{equation}
R_g=\frac{\mathcal{B}_g}{T-\mathcal{A}_{g}}, \ \ \ \ \ \alpha<0,
\label{k=0 solution R a<0}
\end{equation}
where $\alpha^2=\frac{E^2}{P^2}-\frac{1}{\ell^{2}}$ and
\begin{equation*}
\mathcal{A}_g=T_0\pm\frac{E\ell^{2}}{\alpha P R_0}, \ \ \ \
\mathcal{B}_g=\frac{E\ell^{2}}{\alpha P}.
\end{equation*}
The $+$ sign in $\mathcal{A}_g$ holds when $\alpha>0$, the $-$
sign when $\alpha<0$.

In the case where $k=-1$, the solutions are
\begin{equation}
R_g^2=\frac{E^2\ell^2}{\alpha^2P^2}\coth^2
\left(\frac{\mathcal{C}_g-T}{\ell}\right)-\frac{1}{\alpha^2}, \ \
\ \ \ R>\ell, \label{k=-1 sol R(T)}
\end{equation}
and
\begin{equation}
R_g^2=\frac{E^2\ell^2}{\alpha^2P^2}\tanh^2
\left(\frac{\mathcal{D}_g-T}{\ell}\right)-\frac{1}{\alpha^2}, \ \
\ \ \ R<\ell, \label{k=-1 sol R(T) R<l}
\end{equation}
where the initial conditions are
\begin{equation*}
\mathcal{C}_g=T_{0}+\ell\coth ^{-1}\frac{P\sqrt{\alpha
^{2}R_0^{2}+1}}{E\ell},
\end{equation*}
and
\begin{equation*}
\mathcal{D}_g=T_{0}+\ell\tanh ^{-1}\frac{P\sqrt{\alpha
^{2}R_0^{2}+1}}{E\ell}.
\end{equation*}

Similarly for $k=+1$ the solutions are
\begin{equation}
R_g^2=\frac{E^2\ell^2}{\alpha^2P^2}\tan^2\left(\frac{T-\mathcal{F}_g}{\ell}\right)+\frac{1}{\alpha^2},
\label{k=1 solution R(T)}
\end{equation}
where
\begin{equation*}
\mathcal{F}_g=T_0-\ell\tan ^{-1}\frac{P\sqrt{%
\alpha ^{2}R_0^{2}-1}}{E\ell}.
\end{equation*}

The solutions to the geodesic equation for the radial coordinate
$r$ in terms of the global bulk time $T$, for all $k=0,-1$ and
$+1$, are given by
\begin{equation}
r=r_0+\frac{P}{E\ell^2}\left( T-T_0\right). \label{solution r(T)}
\end{equation}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Trajectories for a moving domain wall}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

When one considers a domain wall (i.e. $\rho=0$) moving in an
anti-de Sitter bulk ($\mu=0$), the modified Friedmann equation
(\ref{matter FRW eqn}) takes the form \cite{Kraus}
\begin{equation}
\dot{R}_b^2+\left(1-\left(\frac{\sigma}{\sigma_c}\right)^2\right)\frac{R_b^2}{\ell^2}=-k.
\label{domain FRW eqn}
\end{equation}
The solutions of these, in terms of the proper time on the brane
$t$, are given in Kraus \cite{Kraus}. We are however interested in
the brane's trajectory in the bulk. We therefore transform Kraus'
equations by making use of (\ref{dT/dt}). The trajectories are
\begin{enumerate}

\item[]\underline{$k=0$, $\sigma=\sigma_c$:}  $R_b = constant$

\item[]\underline{$k=0$, $\sigma>\sigma_c$:}

\underline{solution 1} (expanding brane):
\begin{equation}
R_b=\frac{1}{H}\left(\frac{\sigma}{\sigma_c}\right)\frac{\ell}{\mathcal{A}_b-T},\
{\rm where} \ \ \mathcal{A}_b=T_0+\frac{\sigma}{\sigma_c}\frac{\ell}{R_0
H}, \label{Rb(T)2 s>sc}
\end{equation}
and where
$H=\frac{1}{\ell}\sqrt{\left|\left(\frac{\sigma}{\sigma_c}\right)^2-1\right|}$.

\underline{solution 2} (contracting brane):
\begin{equation}
R_b=\frac{1}{H}\left(\frac{\sigma}{\sigma_c}\right)\frac{\ell}{T-\mathcal{A}_b},\
{\rm where} \ \ \mathcal{A}_b=T_0-\frac{\sigma}{\sigma_c}\frac{\ell}{R_0
H}. \label{Rb(T)1 s>sc}
\end{equation}

\item[]\underline{$k=-1$, $\sigma=\sigma_c$:}
\begin{equation}
R_b^2=\ell^2-\left(\ell^2-R_0^2\right)e^{\frac{2(T-T_0)}{\ell}}, \
\ \ \ R<\ell, \label{k=-1 Rb(T) sol1}
\end{equation}
and
\begin{equation}
R_b^2=\left(R_0^2-\ell^2\right)e^{\frac{2(T-T_0)}{\ell}}+\ell^2, \
\ \ \  R>\ell. \label{k=-1 Rb(T) sol2}
\end{equation}

\item[]\underline{$k=-1$, $\sigma>\sigma_c$:}
\begin{equation}
R_b^2=\frac{1}{H^2}\left(\left(\frac{\sigma}{\sigma_c}\right)^2
\coth^2 \left(\frac{\mathcal{C}_b-T}{\ell}\right)-1\right), \ \ \
\ R>\ell, \label{k=-1 Rb(T) s>sc}
\end{equation}
and
\begin{equation}
R_b^2=\frac{1}{H^2}\left(\left(\frac{\sigma}{\sigma_c}\right)^2
\tanh^2 \left(\frac{\mathcal{D}_b-T}{\ell}\right)-1\right), \ \ \
\ R<\ell, \label{k=-1 Rb(T) s>sc R<l}
\end{equation}
where
\begin{equation*}
\mathcal{C}_b=T_0+\ell\coth^{-1}\left(\frac{\sqrt{H^2R_0^2+1}}{\left(\frac{\sigma}{\sigma_c}\right)}\right),
\end{equation*}
and
\begin{equation*}
\mathcal{D}_b=T_0+\ell\tanh^{-1}\left(\frac{\sqrt{H^2R_0^2+1}}{\left(\frac{\sigma}{\sigma_c}\right)}\right).
\end{equation*}

\item[]\underline{$k=-1$, $\sigma<\sigma_c$:}
\begin{equation}
R_b^2=\frac{1}{H^2}\left(1-\left(\frac{\sigma}{\sigma_c}\right)^2
\coth^2 (\frac{T-\mathcal{C}_{b2}}{\ell})\right),\ \ \ \ R<\ell,
\label{k=-1 Rb(T) s<sc}
\end{equation}
and
\begin{equation}
R_b^2=\frac{1}{H^2}\left(1-\left(\frac{\sigma}{\sigma_c}\right)^2
\tanh^2 (\frac{T-\mathcal{D}_{b2}}{\ell})\right),\ \ \ \ R>\ell,
\label{k=-1 Rb(T) s<sc2}
\end{equation}
where
\begin{equation*}
\mathcal{C}_{b2}=T_0-\ell\coth^{-1}\left(\frac{\sqrt{1-H^2R_0^2}}{\left(\frac{\sigma}{\sigma_c}\right)}\right),
\end{equation*}
and
\begin{equation*}
\mathcal{D}_{b2}=T_0-\ell\tanh^{-1}\left(\frac{\sqrt{1-H^2R_0^2}}{\left(\frac{\sigma}{\sigma_c}\right)}\right).
\end{equation*}

\item[]\underline{$k=+1$, $\sigma>\sigma_c$:}
\begin{equation}
R_b^2=\frac{1}{H^2}\left(\left(\frac{\sigma}{\sigma_c}\right)^2
\tan^2 (\frac{T-\mathcal{F}_b}{\ell})+1\right), \label{k=1 Rb(T)}
\end{equation}
where
\begin{equation*}
\mathcal{F}_b=T_0-\ell\tan^{-1}\left(\frac{\sqrt{H^2R_0^2-1}}{\left(\frac{\sigma}{\sigma_c}\right)}\right).
\end{equation*}

\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Analysis}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We now have the trajectories for both the empty brane (domain
wall) and an gravitational signal in $AdS_5$. We plot them on the
same set of axes, with the same initial conditions so as to see
the behaviour of the gravitational signal after it has left the
brane.

In order for us to compare the trajectories we set
\begin{equation}
\omega=\frac{E\ell}{P}
\end{equation}
in the null geodesic equations, and set
\begin{equation}
\beta=\frac{\sigma}{\sigma_c}
\end{equation}
in the brane's equations of motion. We assume that
$\ell\sim10^{-4}m$.

In Figure A1 we start with the $k=0$ case for an expanding
universe (solution 1) with $\sigma>\sigma_c$. We set $\beta=2$ and
$R_0=1$ at $T_0=0$, in (\ref{Rb(T)2 s>sc}) for the brane equation
of motion. For the gravitational signals path (null geodesic), we
consider $\omega=4$ and $\omega=1.5$ in (\ref{k=0 solution R(T)})
with the same initial conditions.  In Figure A2 we plot the
results for a contracting universe (solution 2). The two cases for
$k=-1$ are plotted in Figure A3 and Figure A4 and the case when
$k=1$ is plotted in Figure A5 (all with $\sigma>\sigma_c$).
\begin{figure}[tbp]
\begin{center}
\epsfig{file=k0ag.eps, scale=0.9} \caption{Expanding brane: $R$
versus $T$ for $k=0$, $\sigma>\sigma_c$ and $\alpha>0$.}
\end{center} \label{R v T k=0 s=sc}
\end{figure}

\begin{figure}[tbp]
\begin{center}
\epsfig{file=k0as.eps, scale=0.9} \caption{Contracting brane: $R$
versus $T$ for $k=0$, $\sigma>\sigma_c$ and $\alpha<0$.}
\end{center} \label{R v T k=0 s>sc}
\end{figure}

\begin{figure}[tbp]
\begin{center}
\epsfig{file=kn1Rg.eps, scale=0.9} \caption{$R$ versus $T$ for
$k=-1$ and $\sigma>\sigma_c$ and $R>\ell$}
\end{center} \label{R v T k=-1 s=sc}
\end{figure}

\begin{figure}[tbp]
\begin{center}
\epsfig{file=kn1Rs.eps, scale=0.9} \caption{$R$ versus $T$ for
$k=-1$, $\sigma>\sigma_c$ and $R<\ell$.}
\end{center} \label{R v T k=-1 s>sc}
\end{figure}

\begin{figure}[tbp]
\begin{center}
\epsfig{file=k1sg.eps, scale=0.9} \caption{$R$ versus $T$ for
$k=1$ and $\sigma>\sigma_c$}
\end{center} \label{R v T k=1 s>sc}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{$\omega=\beta$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

When we set $\omega=\beta$ we find that the gravitational signal
follow the same path as the brane for all cases of $k$ when
$\sigma>\sigma_c$. We can interpreted this condition as the
graviton being confined to the brane and therefore having the same
coordinate in the bulk. We can restate this condition in terms of
the impact parameter
\begin{equation}
b=\frac{E}{P}=\frac{1}{\ell}\left(\frac{\sigma}{\sigma_c}\right)=\frac{\sigma}{6M_5^3}. \label{condition grav}
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{$\omega>\beta$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

When $\omega>\beta$ the geodesic equation is not the same as the
equation of motion. Thus a graviton can be emitted along the
extra-dimensional geodesic, i.e. can be emitted from the brane.
This can therefore be seen as the condition for a graviton to be
emitted from a domain wall with tension $\sigma$ into a $AdS_5$
space. We can express this condition in the form
\begin{equation}
b=\frac{E}{P}>\frac{1}{\ell}\left(\frac{\sigma}{\sigma_c}\right).
\label{condition emit}
\end{equation}
This condition amounts to the gravitational signal having a
greater velocity than the brane and may serve as a late time approximation of (\ref{cond escape}) since at late times the $AdSS_5$ becomes $AdS_5$ and the brane a domain wall. We can therefore see from Figure A5 that no gravitational signals return to the brane at late times.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{$\omega<\beta$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In the $k=0$ and $k=-1$ cases we see that the possibility of
$\omega<\beta$ also gives paths that are different to the brane's
path. However, in these cases the brane is moving faster than the
gravitational signal. Clearly these cases are not valid. The
initial conditions for the $k=1$, exclude the possibility of
$\omega<\beta$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{References}
%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{99}


\bibitem{Chung} Chung D and Freese K, \textit{Phys. Rev.} \textbf{D62}, 063513 (2000).

\bibitem{Ishi01} Ishihara H, \textit{Phys. Rev. Lett.} \textbf{86}, 381 (2001).

%\bibitem{Ishi02} Ishihara H, \textit{Phys. Rev.} \textbf{D66}, 023513 (2002).

\bibitem{BinLang}  Bin\'{e}truy P, Deffayet C and Langlois D, \textit{Nuc. Phys.}
\textbf{B565}, 269 (2000).

\bibitem{BinDef}  Bin\'{e}truy P, Deffayet C, Ellwanger U and Langlois D, \textit{Phys. Lett.}
\textbf{B477}, 285 (2000).

%\bibitem{Youm00} Youm D, \textit{Phys. Rev.} \textbf{D62}, 084002 (2000).

\bibitem{Youm01} Youm D, \textit{Mod. Phys. Let.} \textbf{A16}, 2371 (2001).

\bibitem{Davis} Davis A, Rhodes C and Vernon I, \textit{JHEP} \textbf{0111}, 015 (2001).

\bibitem{Abdalla02} Abdalla E, Cuadros-Melgar B, Feng S and
Wang B, \textit{Phys. Rev.} \textbf{D65}, 083512 (2002).

\bibitem{Kraus}  Kraus P, \textit{JHEP} \textbf{9912}, 011
(1999).

\bibitem{Bowcock}  Bowcock P, Charmousis C and Gregory R,
\textit{Class. Quant. Grav.} \textbf{17}, 4745 (2000).

\bibitem{Chamblin} Chamblin A and Reall H S, \textit{Nuc. Phys.}
\textbf{B562}, 133 (1999).

\bibitem{Caldwell}  Caldwell R and Langlois D, \textit{Phys. Lett. }%
\textbf{B511}, 129 (2001).

\bibitem{Abdalla05} Abdalla E, Casali A G and Cuadros-Melgar B,{\it Int. J .Theor.
Phys.} {\bf 43}, 801 (2004).

\bibitem{Abdalla02b} Abdalla E, Casali A G and Cuadros-Melgar B,
\textit{Nuc. Phys.} \textbf{B644}, 201 (2002).

\bibitem{Abdalla03b} Abdalla E and Cuadros-Melgar B, \textit{Phys. Rev.}
\textbf{D67}, 084012 (2003).

\bibitem{Melgar04} Cuadros-Melgar B, \textit{Class. Quant. Grav.} \textbf{21}, 2669 (2004) .

\bibitem{Misner}  Misner C W, Thorne K S and Wheeler J A, \textit{
Gravitation} (Freeman, San Francisco 1973).

\bibitem{Israel}  Israel W, \textit{Nuovo Cimento}
\textbf{B44}, 1 (1966), erratum: \textbf{B48}, 463 (1967).

\bibitem{Brax03a} Brax P, Langlois D and Rodr\'{i}guez-Martinez M,
\textit{Phys. Rev.} \textbf{D67}, 104022 (2003).

\bibitem{Brax03} Brax P and van de Bruck C, \textit{Class. Quant. Grav.} \textbf{20}, R201 (2003) [arXiv: hep-th/0303095].

\bibitem{Csaki} Cs\'{a}ki C, Erlich J and Grojean C, \textit{Nuc. Phys.} \textbf{B604}, 312 (2001).

\bibitem{Ichiki} Ichiki K, Yahiro M, Kajino T, Orito M and Mathews G J, \textit{Phys. Rev.}
\textbf{D66}, 043521 (2002).

\bibitem{ChamHawk} Chamblin A, Hawking S W and
Reall H S, \textit{Phys. Rev.} \textbf{D61}, 065007 (2000).

\bibitem{Dadhich} Dadhich N, Maartens R, Papadopoulos P and Rezania V, \textit{Phys. Lett.}
\textbf{B487}, 1 (2000).

\bibitem{Emparan} Emparan R, Horowitz G T and Myers R C, \textit{JHEP} \textbf{0001}, 007 (2000).

\bibitem{Chamblin01} Chamblin A, Reall H.S, Shinkau H. and Shiromizu T., \textit{Phys. Rev.}
\textbf{D63}, 064015 (2001).

\bibitem{Pravda} Pravda V and Pravdova A, \textit{Class. Quant. Grav.} \textbf{18}, 1205 (2001).

\bibitem{Chamblin01b} Chamblin A, \textit{Class. Quant. Grav.} \textbf{18}, L17 (2001).

\bibitem{Langlois} Langlois D, Sorbo L and Rodr\'{i}guez-Mart\'{i}nez M, \textit{Phys. Rev.
Lett.} \textbf{89}, 171301 (2002).

\bibitem{Leeper} Leeper E, Maartens R and Sopuerta C F, \textit{Class. Quant. Grav.} \textbf{21}, 1125 (2004).
\end{thebibliography}

\end{document}

gr-qc0502109.extracted_bib
\begin{thebibliography}{99}


\bibitem{Chung} Chung D and Freese K, \textit{Phys. Rev.} \textbf{D62}, 063513 (2000).

\bibitem{Ishi01} Ishihara H, \textit{Phys. Rev. Lett.} \textbf{86}, 381 (2001).

%\bibitem{Ishi02} Ishihara H, \textit{Phys. Rev.} \textbf{D66}, 023513 (2002).

\bibitem{BinLang}  Bin\'{e}truy P, Deffayet C and Langlois D, \textit{Nuc. Phys.}
\textbf{B565}, 269 (2000).

\bibitem{BinDef}  Bin\'{e}truy P, Deffayet C, Ellwanger U and Langlois D, \textit{Phys. Lett.}
\textbf{B477}, 285 (2000).

%\bibitem{Youm00} Youm D, \textit{Phys. Rev.} \textbf{D62}, 084002 (2000).

\bibitem{Youm01} Youm D, \textit{Mod. Phys. Let.} \textbf{A16}, 2371 (2001).

\bibitem{Davis} Davis A, Rhodes C and Vernon I, \textit{JHEP} \textbf{0111}, 015 (2001).

\bibitem{Abdalla02} Abdalla E, Cuadros-Melgar B, Feng S and
Wang B, \textit{Phys. Rev.} \textbf{D65}, 083512 (2002).

\bibitem{Kraus}  Kraus P, \textit{JHEP} \textbf{9912}, 011
(1999).

\bibitem{Bowcock}  Bowcock P, Charmousis C and Gregory R,
\textit{Class. Quant. Grav.} \textbf{17}, 4745 (2000).

\bibitem{Chamblin} Chamblin A and Reall H S, \textit{Nuc. Phys.}
\textbf{B562}, 133 (1999).

\bibitem{Caldwell}  Caldwell R and Langlois D, \textit{Phys. Lett. }%
\textbf{B511}, 129 (2001).

\bibitem{Abdalla05} Abdalla E, Casali A G and Cuadros-Melgar B,{\it Int. J .Theor.
Phys.} {\bf 43}, 801 (2004).

\bibitem{Abdalla02b} Abdalla E, Casali A G and Cuadros-Melgar B,
\textit{Nuc. Phys.} \textbf{B644}, 201 (2002).

\bibitem{Abdalla03b} Abdalla E and Cuadros-Melgar B, \textit{Phys. Rev.}
\textbf{D67}, 084012 (2003).

\bibitem{Melgar04} Cuadros-Melgar B, \textit{Class. Quant. Grav.} \textbf{21}, 2669 (2004) .

\bibitem{Misner}  Misner C W, Thorne K S and Wheeler J A, \textit{
Gravitation} (Freeman, San Francisco 1973).

\bibitem{Israel}  Israel W, \textit{Nuovo Cimento}
\textbf{B44}, 1 (1966), erratum: \textbf{B48}, 463 (1967).

\bibitem{Brax03a} Brax P, Langlois D and Rodr\'{i}guez-Martinez M,
\textit{Phys. Rev.} \textbf{D67}, 104022 (2003).

\bibitem{Brax03} Brax P and van de Bruck C, \textit{Class. Quant. Grav.} \textbf{20}, R201 (2003) [arXiv: hep-th/0303095].

\bibitem{Csaki} Cs\'{a}ki C, Erlich J and Grojean C, \textit{Nuc. Phys.} \textbf{B604}, 312 (2001).

\bibitem{Ichiki} Ichiki K, Yahiro M, Kajino T, Orito M and Mathews G J, \textit{Phys. Rev.}
\textbf{D66}, 043521 (2002).

\bibitem{ChamHawk} Chamblin A, Hawking S W and
Reall H S, \textit{Phys. Rev.} \textbf{D61}, 065007 (2000).

\bibitem{Dadhich} Dadhich N, Maartens R, Papadopoulos P and Rezania V, \textit{Phys. Lett.}
\textbf{B487}, 1 (2000).

\bibitem{Emparan} Emparan R, Horowitz G T and Myers R C, \textit{JHEP} \textbf{0001}, 007 (2000).

\bibitem{Chamblin01} Chamblin A, Reall H.S, Shinkau H. and Shiromizu T., \textit{Phys. Rev.}
\textbf{D63}, 064015 (2001).

\bibitem{Pravda} Pravda V and Pravdova A, \textit{Class. Quant. Grav.} \textbf{18}, 1205 (2001).

\bibitem{Chamblin01b} Chamblin A, \textit{Class. Quant. Grav.} \textbf{18}, L17 (2001).

\bibitem{Langlois} Langlois D, Sorbo L and Rodr\'{i}guez-Mart\'{i}nez M, \textit{Phys. Rev.
Lett.} \textbf{89}, 171301 (2002).

\bibitem{Leeper} Leeper E, Maartens R and Sopuerta C F, \textit{Class. Quant. Grav.} \textbf{21}, 1125 (2004).
\end{thebibliography}