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\title{Brane World Dynamics and Conformal Bulk Fields}

\author{Rui Neves\footnote{E-mail: \tt rneves@ualg.pt}\hspace{0.2cm}
  and
Cenalo Vaz\footnote{E-mail: \tt cvaz@ualg.pt}\\
{\small \em Faculdade de Ci\^encias e Tecnologia,
Universidade do Algarve}\\
{\small \em Campus de Gambelas, 8000-117 Faro, Portugal}
}

\maketitle

\begin{abstract}
We investigate the dynamics of a spherically symmetric 3-brane world in the Randall-Sundrum
scenario by taking the point of view of a 5-dimensional observer. To analyze the
Einstein equations in the presence of matter fields in the bulk, we employ a global conformal
transformation whose factor characterizes the $Z_2$ symmetric warp. For a certain kind of conformal
stress-energy tensor and if the warp factor depends only on the fifth dimension, we find a
new set of exact dynamical collapse solutions which localize gravity in the vicinity of the brane.
Geometries inducing the brane world dynamics of inhomogeneous dust and generalized dark radiation
are shown to belong to this set. The conditions for singular or globally regular behavior and
the static marginally bound limits are discussed for these examples. Also explicitly demonstrated
is the complete consistency with the effective point of view of a 4-dimensional observer who
is confined to the brane and makes the same assumptions about the bulk degrees of freedom.
\vspace{0.5cm}

\noindent PACS numbers: 04.50.+h, 04.70.-s, 98.80.-k, 11.25.Mj
\end{abstract}

\section{Introduction}

Matter fields may be confined to a 3-brane world embedded in a higher dimensional space if gravity
propagates away from the brane in the extra dimensions \cite{EBWM,CED}. Remarkably, in this context
it is possible to solve the hierarchy problem following two alternative paths which admit a fundamental
Planck scale in the TeV range. In one approach the extra dimensions need to be compactified to a large
finite volume \cite{CLED}, in the other the observable brane must have a negative tension and the extra
dimensions need to be strongly warped \cite{RS1}-\cite{TM}. In the latter scenario it is even possible to
consider infinite and non-compact extra dimensions. Of the two branes, the visible one has then a positive
tension and the warp localizes gravity in its vicinity allowing the recovery of 4-dimensional Einstein
gravity at low energy scales \cite{RS2}-\cite{GiR}. However, the hierarchy problem is solved on the brane
with negative tension. An alternative setup involving only positive tension branes was proposed in \cite{LR}
and shown to solve the hierarchy problem on the brane on which gravity is not localized. Although gravity is
not localized on this brane, its low-energy behavior is indistinguishable from that of localized gravity.

The scenario with an infinite and non-compact extra dimension first
proposed by of Randall and Sundrum (RS) \cite{RS2}-\cite{LR} is to us
of special interest. In this framework
the observable universe is a boundary hypersurface of an infinitely extended $Z_2$ symmetric 5-dimensional
anti-de Sitter (AdS) space. The classical dynamics is defined by the Einstein equations with a negative bulk
cosmological constant $\Lambda_B$, a Dirac delta source representing the brane and a stress-energy tensor
describing other bulk field modes which may exist in the whole AdS space.

A set of vaccum solutions is given by the metric \cite{CHR}
\beq
d{\tilde{s}_5^2}={l\over{|z-{z_0}|+{z_0}}}\left(d{z^2}+d{s_4^2}\right),
\eeq
where $d{s_4^2}$ is a 4-dimensional line element generated by a Ricci flat metric. The parameter $l$ is the
AdS radius and is written as $l=1/\sqrt{-{\Lambda_B}{\kappa_5^2}/6}$ where ${\kappa_5^2}=8\pi/{M_5^3}$ with
$M_5$ the fundamental 5-dimensional Planck mass. The brane is located at $z={z_0}$ and is fine-tuned to have
zero cosmological constant, giving ${\Lambda_B}=-{\kappa_5^2}{\lambda^2}/6$ or $l=6/({\kappa_5^2}\lambda)$
where $\lambda$ denotes the positive brane tension.

The original RS solution \cite{RS2} belongs to this class and is obtained when the 4-dimensional subspace
has a Minkowski metric. Another example is the black string solution \cite{CHR} which induces the Schwarzschild
metric on the brane. However, the Kretschmann scalar diverges both at the AdS horizon and at the black string
singularity \cite{CHR} leading to a violation of the cosmic censorship conjecture \cite{CCC}. This solution also
appears to be unstable \cite{GLinst} near the AdS horizon and it may decay to a black cylinder localized near the brane which
is free from naked singularities \cite{CHR}. However, despite considerable effort, this still remains a
conjecture. Indeed, while exact solutions interpreted as static black holes localized on a brane have been
found for a 2-brane embedded in a 4-dimensional AdS space \cite{EHM}, a static black hole localized on a 3-brane
is yet to be discovered \cite{SS}-\cite{KOT}. The difficulty in finding a static solution has led to an
interesting conjecture \cite{EFK} which attempts to relate black hole solutions localized on the brane in an
AdS${}_{D+1}$ braneworld, which are found with the brane boundary conditions,  with quantum black holes in $D$
dimensions rather than classical ones.

Many other 5-dimensional solutions have been determined within the RS brane world scenario. These include the
extensions of the RS geometry to thick branes \cite{CEHS} and non-fine-tuned branes \cite{KR}, the 5-dimensional
metrics describing Friedmann-Robertson-Walker (FRW) cosmologies
\cite{5DBWCos,CosDR} and also other solutions envolving
the effect of scalar fields in the bulk \cite{SBSF}-\cite{CGRT}. The aplication of the covariant Gauss-Codazzi (GC)
approach \cite{CGC}-\cite{RM} describing the perspective of a
4-dimensional
observer restricted to the brane, has
permitted the analysis of even more brane world physics
\cite{DMPR}-\cite{RM4dp}. In addition, several connections
have been established between the AdS/CFT correspondence \cite{CFT}
and the RS brane world scenario \cite{GKR,EFK,RSCFT}.

In spite of the numerous 5-dimensional exact solutions already found in the RS scenario there are still many
effective 4-dimensional metrics which have not been shown to be associated with exact bulk spacetimes. This happens
for instance with the inhomogeneous dust and dark radiation dynamics on the brane. While the latter has been
deduced within the RS scenario using the covariant GC formulation \cite{RC}, the former has not yet been shown to
have a brane world description. It is thus the purpose of the present work to search for new exact 5-dimensional
solutions which might reproduce such effective 4-dimensional brane world dynamics. To attain this objective we
consider the generally inhomogeneous dynamics of a spherically symmetric RS 3-brane when field modes other than
the Einstein-Hilbert gravity are present in the whole AdS space. Such bulk fields and their influence on the
brane world dynamics have been extensively discussed within the RS scenario for example to stabilize the extra
dimensions \cite{GW,TM,SBSF,KKOP} and to analyze the possibility of generating a black hole localized on a
3-brane world \cite{LP,KT}.

We begin in section 2 with an analysis of the Einstein field equations for the most general 5-dimensional metric
consistent with the $Z_2$ symmetry and with spherical symmetry on the brane. We organize the Einstein equations
using a global conformal transformation whose factor characterizes the warp of the fifth dimension. Assuming that
the warp factor depends only on the fifth dimension, we show that the 4-dimensional dynamics induced on the brane
by the bulk fields may be decoupled from the fifth dimension only if the bulk fields are defined by a stress-energy
tensor with conformal weight $s=-4$. The induced matter dynamics on the brane is then defined by standard 4-dimensional
Einstein equations and in general it generates a non-zero pressure along the fifth dimension. Assuming further that
the bulk stress-energy tensor is diagonal we also show that the 5-dimensional pressure and the density and pressures
of the induced matter on the brane must satisfy a precise equation of state so as to be consistent with the
Einstein dynamics. Thus we obtain a new set of exact dynamical solutions for which gravity is localized near the
brane by the conformal warp factor.

In sections 3 and 4 we analyze two examples which belong to this new set of exact 5-dimensional solutions. Both
examples permit the use of synchronous coordinates. We consider the dynamics of induced pressureless dust and
generalized dark radiation in the presence of a non-zero brane cosmological constant also generated by the bulk
fields. We show that both 5-dimensional metrics induce on the brane dynamical and inhomogeneous LeM\^aitre-Tolman-Bondi
geometries. We also determine the static marginally bound limits and discuss the conditions for singular or globally
regular behavior.

As a consistency check, we consider the point of view of an observer confined to the brane in section 5 . Using the
effective GC formulation we show that if such an observer uses the same description of the field variables, then it
indeed leads to identical brane world dynamics. Explicit proofs are provided for dust and dark radiation.
We conclude in section 6.

\section{5-Dimensional Einstein Equations and Conformal
Transformations}

In a 5-dimensional RS manifold mapped by a set of comoving coordinates $(t,r,\theta,\phi,z)$ the most general
dynamical metric consistent with the $Z_2$ symmetry in $z$ and with 4-dimensional spherical symmetry on the brane
may be written using a global conformal transformation \cite{WaldBD} defined by a $Z_2$ symmetric warp function
$\Omega=\Omega(t,r,z)$,
\beq
{\tilde{g}_{\mu\nu}}={\Omega^2}{g_{\mu\nu}}.\label{gm4}
\eeq
The corresponding 5-dimensional line elements are related by
\beq
d{\tilde{s}_5^2}={\Omega^2}d{s_5^2},
\eeq
where
\beq
d{s_5^2}=d{z^2}+d{s_4^2}\label{gm2}
\eeq
has a 4-dimensional line element which depends on three $Z_2$ symmetric functions $A=A(t,r,z)$, $B=B(t,r,z)$ and
$R=R(t,r,z)$ as follows
\beq
d{s_4^2}=-{e^{2A}}d{t^2}+{e^{2B}}d{r^2}+{R^2}d{\Omega_2^2}.\label{gm3}
\eeq
$R(t,r,z)$ represents the physical radius of the 2-spheres.

When bulk field modes other than the Einstein-Hilbert gravity are present in the 5-dimensional space the dynamical
RS action corresponding to $\tilde{g}_{\mu\nu}$ is given by
\beq
\tilde{\mathcal{S}}=\int{d^4}xdz\sqrt{-\tilde{g}}
\left[{\tilde{R}\over{2{\kappa_5^2}}}-
{\Lambda_B}-{\lambda\over{\sqrt{\tilde{g}_{55}}}}\delta\left(z-{z_0}\right)+
{\tilde{\mathcal{L}}_B}\right].\label{5Dact}
\eeq
The brane is assumed to be located at $z={z_0}$ and is a fixed point of the $Z_2$ symmetry of the manifold. The
contribution of the bulk fields is defined by the lagrangian $\tilde{\mathcal{L}}_B$. A Noether variation on the
action (\ref{5Dact}) gives the classical Einstein field equations,
\beq
{\tilde{G}_\mu^\nu}=-{\kappa_5^2}\left[{\Lambda_B}{\delta_\mu^\nu}+
\lambda\delta
\left(z-{z_0}\right){\tilde{\gamma}_\mu^\nu}-{\tilde{T}_\mu^\nu}\right],
\label{5DEeq}
\eeq
where the induced metric on the brane is
\beq
{\tilde{\gamma}_\mu^\nu}={1\over{\sqrt{\tilde{g}_{55}}}}\left(
{\delta_\mu^\nu}-{\delta_5^\nu}{\delta_\mu^5}\right)
\eeq
and the stress-energy tensor associated with the bulk fields is defined by
\beq
{\tilde{T}_\mu^\nu}={\tilde{\mathcal{L}}_B}{\delta_\mu^\nu}-
2{{\delta{\tilde{\mathcal{L}}_B}}\over{\delta{\tilde{g}^{\mu\alpha}}}}
{\tilde{g}^{\alpha\nu}}
\eeq
and is conserved in the bulk,
\beq
{\tilde{\nabla}_\mu}{\tilde{T}_\nu^\mu}=0.\label{5Dceq}
\eeq

The Einstein field equations (\ref{5DEeq}) are extremely complex when the metric $\tilde{g}_{\mu\nu}$ is
considered in its full generality. To be able to solve them we need to introduce simplifying assumptions
about the field variables involved in the problem. Let us first assume that under the conformal transformation
(\ref{gm4}) the bulk stress-energy tensor has invariant conformal weight $s+2$ \cite{WaldBD},
\beq
{\tilde{T}_\mu^\nu}={\Omega^{s+2}}{T_\mu^\nu}.\label{ctst}
\eeq
Then substituting Eq. (\ref{ctst}) and using the known transformation properties of the other tensors
\cite{WaldBD} we re-write Eq. (\ref{5DEeq}) as
\bea
{G_\mu^\nu}&=&-6{\Omega^{-2}}\left({\nabla_\mu}\Omega\right){g^{\nu\rho}}
{\nabla_\rho}\Omega+
3{\Omega^{-1}}{g^{\nu\rho}}{\nabla_\rho}{\nabla_\mu}\Omega
-3{\Omega^{-1}}{\delta_\mu^\nu}{g^{\rho\sigma}}{\nabla_\rho}{\nabla_\sigma}
\Omega\nn\\
&&-{\kappa_5^2}
{\Omega^2}\left[{\Lambda_B}{\delta_\mu^\nu}+\lambda{\Omega^{-1}}
\delta(z-{z_0}){\gamma_\mu^\nu}-{\Omega^{s+2}}{T_\mu^\nu}\right].
\label{t5DEeq}
\eea
Similarly, Eq. (\ref{5Dceq}) also transforms under the conformal transformation. We have
\beq
{\nabla_\mu}{T_\nu^\mu}+{\Omega^{-1}}\left[(s+7){T_\nu^\mu}{\partial_\mu}
\Omega-T{\partial_\nu}\Omega\right]=0,\label{t5Dceq}
\eeq
where $T={T_\mu^\mu}$ is the trace of the bulk stress-energy tensor. If $\tilde{T}_\mu^\nu$ has conformal weight
$s=-4$, then it transforms as the conformally invariant electromagnetic free energy $\tilde{F}_\mu^\nu$ in 4
dimensions and it is possible (though not necessary) to separate Eq. (\ref{t5DEeq}) as follows
\beq
{G_\mu^\nu}={\kappa_5^2}{T_\mu^\nu},\label{r5DEeq}
\eeq
\bea
&6{\Omega^{-2}}{\nabla_\mu}\Omega{\nabla_\rho}
\Omega{g^{\rho\nu}}-
3{\Omega^{-1}}{\nabla_\mu}{\nabla_\rho}\Omega{g^{\rho\nu}}+3{\Omega^{-1}}
{\nabla_\rho}{\nabla_\sigma}\Omega{g^{\rho\sigma}}{\delta_\mu^\nu}=\nn\\
&-{\kappa_5^2}
{\Omega^2}\left[{\Lambda_B}{\delta_\mu^\nu}+\lambda{\Omega^{-1}}
\delta(z-{z_0}){\gamma_\mu^\nu}\right].\label{5DEeqwf}
\eea
In Eq. (\ref{r5DEeq}) we find the 5-dimensional Einstein equations with fields present in the bulk when the
brane and the bulk cosmological constant are absent. It does not depend on the conformal warp factor which is
dynamically defined by Eq. (\ref{5DEeqwf}) and is then the only effect reflecting the existence of the brane
or of the bulk cosmological constant in this setting. We emphasize that this is only possible for the special
class of bulk fields which have a stress-energy tensor with conformal weight $s=-4$. The decomposition
of (\ref{5DEeq}) according to (\ref{r5DEeq}) and (\ref{5DEeqwf}) implies a separation of Eq. (\ref{t5Dceq})
because of the Bianchi identity. We must have
\beq
{\nabla_\mu}{T_\nu^\mu}=0,\label{r5Dceq}
\eeq
\beq
3{T_\nu^\mu}{\partial_\mu}\Omega-T {\partial_\nu}\Omega=0.
\label{5Dceqwf}
\eeq
Note that now $T_\mu^\nu$ is a conserved tensor field which must satisfy the additional warp constraint equations
(\ref{5Dceqwf}). Since $s\not=-7$ it does not need to be traceless.

Let us now consider Eq. (\ref{r5DEeq}). Expanding the Einstein tensor in terms of the metric functions $A$, $B$
and $R$ reveals that its only non-zero off-diagonal elements are $G_t^r$, $G_t^z$ and $G_r^z$. Assuming that
$A=A(t,r)$, $B=B(t,r)$ and $R=R(t,r)$ sets $G_t^z$, $G_r^z$ and the corresponding stress-energy tensor components
to zero. Hence, we have
\beq
{T_a^z}=0,\label{eqst1}
\eeq
where the latin index represents the 4-dimensional brane coordinates $t$, $r$, $\theta$ and $\phi$.
If in addition $\Omega=\Omega(z)$ then Eq. (\ref{5DEeqwf}) turns out to be independent of the metric functions
$A$, $B$, $R$ and reads
\[
6{\Omega^{-2}}{{({\partial_z}\Omega)}^2}=
-{\kappa_5^2}{\Omega^2}{\Lambda_B},
\]
\beq
3{\Omega^{-1}}{\partial_z^2}\Omega=-{\kappa_5^2}{\Omega^2}
\left[{\Lambda_B}+\lambda{\Omega^{-1}}\delta(z-{z_0})\right].\label{rswf}
\eeq
For definiteness, take as solution of Eq. (\ref{rswf}) the RS conformal warp factor
\beq
\Omega={\Omega_{\mbox{\tiny RS}}}\equiv {l\over{|z-{z_0}|+{z_0}}}\label{RSwf1}
\eeq
where $l=\sqrt{-6/({\Lambda_B}{\kappa_5^2})}$ and
\beq
{\Lambda_B}+{{{\kappa_5^2}{\lambda^2}}\over{6}}=0.\label{RSwf2}
\eeq
Of course, other solutions with warp factors which depend only on the 5-dimensional coordinate $z$ such as those
corresponding to non-fine-tuned branes \cite{KR} or thick branes \cite{CEHS} may also be considered (see
\cite{KT}).

Consequently, Eq. (\ref{5Dceqwf}) constrains $T_\mu^\nu$ to satisfy the equation of state
\beq
2{T_z^z}={T_c^c}.\label{eqst2}
\eeq
Assume further that $T_r^t$ is equal to zero. This condition  leads to the diagonal form
\beq
{T_\mu^\nu}=diag\left(-\rho,{p_r},{p_T},{p_T},{p_z}\right),\label{bmten}
\eeq
where $\rho$, $p_r$, $p_T$ and $p_z$ denote the bulk matter density and pressures. Then Eq. (\ref{eqst2}) is
re-written as
\beq
\rho-{p_r}-2{p_T}+2{p_z}=0.\label{eqst3}
\eeq
Because the metric functions $A$, $B$ and $R$ do not depend on $z$ Eq. (\ref{eqst1}) applied to Eq. (\ref{r5Dceq})
leads to ${\partial_z}{p_z}=0$ and to
\beq
{\nabla_a}{T_b^a}=0,\label{4Dceq}
\eeq
So $\rho$, $p_r$ and $p_T$ must also be independent of $z$.

Since all the off-diagonal components are zero Eq. (\ref{r5DEeq}) also admits a similar dimensional reduction.
The 4-dimensional Einstein equations are given by
\beq
{G_a^b}={\kappa_5^2}{T_a^b}\label{4DEeq}
\eeq
but now the 5-dimensional component is in general non-zero because of the existing pressure $p_z$,
\beq
{G_z^z}={\kappa_5^2}{p_z}.\label{5DEeqz}
\eeq
Thus, the conformal bulk fields induce matter on the brane which has a generally inhomogeneous gravitational
dynamics defined by Eqs. (\ref{4Dceq}) and (\ref{4DEeq}). Gravity is always localized in the vicinity of the brane
due to the RS warp factor (\ref{RSwf1}) but the bulk fields are not because the conformal weight of the stress-energy
tensor is $s=-4$. However, the induced matter is dynamically trapped on the brane by the localized gravitational
field. There is no energy flux in the bulk, but the matter dynamics on the brane generates a pressure $p_z$ along
the fifth dimension which must consistently be given by Eqs. (\ref{eqst3}) and (\ref{5DEeqz}). A priori, there
are no other constraints on the type of diagonal matter generated on the brane by this special kind of conformal
bulk fields.

\section{Dust Dynamics on the Brane}

The simplest kind of matter which may be induced on the brane by the conformal bulk fields is 4-dimensional
pressureless dust. This system is characterized by the equation of state ${p_r}={p_T}=0$ and according to
Eq. (\ref{eqst3}) its density $\rho={\rho_{\mbox{\tiny D}}}$ must generate a pressure $p_z$ along the fifth
dimension which satisfies ${p_z}=-{\rho_{\mbox{\tiny D}}}/2$. If the effect of a brane cosmological constant
$\Lambda$ is added the density $\rho$ and the pressures $p_r$, $p_T$ should be given by
\beq
\rho={\rho_{\mbox{\tiny D}}}+{\Lambda\over{\kappa_5^2}},\quad {p_r}={p_T}=
-{\Lambda\over{\kappa_5^2}}.\label{dustes1}
\eeq
Then
\beq
{p_z}=-{1\over{2}}\left({\rho_{\mbox{\tiny
        D}}}+4{\Lambda\over{\kappa_5^2}}
\right).\label{dustes2}
\eeq
Since the warp factor has already been chosen to be the RS solution (\ref{RSwf1}) in order to find the 5-dimensional
metric describing the localized brane world dynamics of dust in the presence of a cosmological constant we just
have to solve Eqs. (\ref{4Dceq}), (\ref{4DEeq}) and (\ref{5DEeqz})
under conditions (\ref{dustes1}) and (\ref{dustes2}).

Using Eq. (\ref{dustes1}) we write Eq. (\ref{4Dceq}) \cite{TPS} as follows
\beq
\dot{B}{\rho_{\mbox{\tiny D}}}=-\dot{\rho_{\mbox{\tiny
      D}}}-2{\dot{R}\over{R}}
{\rho_{\mbox{\tiny D}}},\label{dceq1}
\eeq
\beq
A'{\rho_{\mbox{\tiny D}}}=0,\label{dceq2}
\eeq
where the dot and the prime denote, respectively, partial
differentiation with respect to $t$ and $r$. Because of Eq. (\ref{dceq2})
we have to take the synchronous frame where $A=0$ to avoid setting
$\rho_{\mbox{\tiny D}}$ to zero. Then the off-diagonal equation is given by
\beq
{G_r^t}={2\over{R}}\left(\dot{R}'-\dot{B}R'\right)=0
\eeq
and has the solution
\beq
{e^B}={R'\over{H}},\label{eB}
\eeq
where $H=H(r)$ is an arbitrary positive function of $r$. Introducing
Eq. (\ref{eB}) in Eq. (\ref{dceq1}) we get the required dust density \cite{PSJ},
\beq
{\rho_{\mbox{\tiny D}}}={{2{G_N}M'}\over{{\kappa_5^2}{R^2}R'}},\label{dustd}
\eeq
where $G_N$ is Newton's gravitational constant and $M=M(r)$ is an arbitrary positive function of $r$ which represents the
dust mass inside a shell labelled by $r$. Note that ${\rho_{\mbox{\tiny D}}}>0$ is equivalent to $M'/R'>0$. This
implies that the weak, strong and dominant energy conditions
\cite{WaldBD} are satisfied in 4 and 5 dimensions.

Next, consider the trace equation
\beq
-{G_t^t}+{G_r^r}+2{G_\theta^\theta}=-2{{\ddot{R}'}\over{R'}}-
4{\ddot{R}\over{R}}={{2{G_N}M'}\over{{R^2}R'}}-2\Lambda.
\eeq
Integrating twice we find
\beq
{\dot{R}^2}={{2{G_N}M}\over{R}}+{\Lambda\over{3}}{R^2}+f,
\eeq
where $f\equiv f(r)$ is an arbitrary function of $r$ to be interpreted as the energy inside a shell labelled by
$r$. Imposing on the initial hypersurface $t=0$ the condition $R(0,r)=r$ we obtain
\beq
\pm t+\psi=\int{{dR}\over{\sqrt{{{2{G_N}M}\over{R}}
+{\Lambda\over{3}}{R^2}+f}}},\label{dust2}
\eeq
where the signs $+$ or $-$ refer to expansion or collapse and $\psi=\psi(r)$ is given by the evaluation at $t=0$
of the integral in the r.h.s. Applying the radial equation
\beq
{G_r^r}=-2{\ddot{R}\over{R}}+{{H^2}\over{R^2}}-{1\over{R^2}}-
{{\dot{R}^2}\over{R^2}}=-\Lambda
\eeq
we obtain $H=\sqrt{1+f}$ and this restricts $f$ to satisfy $f>-1$. Then it is easy to see that Eq. (\ref{5DEeqz}),
\beq
{G_z^z}=-{{\ddot{R}'}\over{R'}}-2{\ddot{R}\over{R}}+
{{f-{\dot{R}^2}}\over{R^2}}
+{{(f-{\dot{R}^2})'}\over{RR'}}=-{{{G_N}M'}\over{{R^2}R'}}-2\Lambda,
\eeq
is an identity for all $M$ and $f$.

We have thus obtained the 5-dimensional dust solutions
\beq
d{\tilde{s}_5^2}={\Omega_{\mbox{\tiny RS}}^2}\left(d{z^2}+d{s_4^2}
\right),\label{vacs1}
\eeq
where the 4-dimensional metric has the LeMa\^{\i}tre-Tolman-Bondi (LTB) form
\beq
d{s_4^2}=-d{t^2}+{{{R'}^2}\over{1+f}}d{r^2}+{R^2}d{\Omega_2^2},\label{vacs2}
\eeq
with the physical radius satisfying Eq. (\ref{dust2}).

The marginally bound models (corresponding to $f=0$) with constant mass function, $M(r) = \mathrm{const.}$, describe
static solutions. Indeed, using the standard transformation from the LTB coordinates $(t,r)$ to the curvature
coordinates $(T,R)$,
\beq
T=t+\int dR {{\sqrt{{\Lambda\over{3}}{R^4}+2{G_N}M R}}\over{
{\Lambda\over{3}}{R^3}-R+2{G_N}M}},
\eeq
we obtain an AdS/dS-Schwarzschild black string solution given by Eq. (\ref{vacs1}) where
\beq
d{s_4^2}=-\left(1-{{2{G_N}M}\over{R}}-{\Lambda\over{3}}{R^2}\right)d{T^2}+
{{\left(1-{{2{G_N}M}\over{R}}-{\Lambda\over{3}}{R^2}\right)}^{-1}}d{R^2}+
{R^2}d{\Omega_2^2}.
\eeq
In the vaccum we obtain the original RS static solution \cite{RS2} and not the Schwarzschild black string
\cite{CHR}. In agreement with \cite{KT} we have not found any more static vaccum solutions.

For $f\not=0$ the solutions are dynamical and inhomogeneous. An
analysis of the potential $V={\Lambda\over{3}}{R^3}
+fR+2{G_N}M$ uncovers a rich set of singular and globally regular solutions \cite{DJCJ}.

\section{Generalized Dark Radiation Dynamics on the Brane}

Let us now consider the possibility of generating on the brane the localized gravitational interaction between
a generalized form of inhomogeneous dark radiation and a cosmological constant. This system is defined by
conformal bulk fields with the equations of state
\beq
\rho+{p_r}=0,\quad{p_T}+\eta\rho+{\Lambda\over{\kappa_5^2}}
\left(1-\eta\right)=0,\label{gdres1}
\eeq
where $\eta$ is the parameter characterizing the dark radiation model and $\rho$ is given by
\beq
\rho={\rho_{\mbox{\tiny DR}}}+{\Lambda\over{\kappa_5^2}}.\label{gdres2}
\eeq
Applying Eq. (\ref{eqst3}) we find
\beq
{p_z}=-\left(1+\eta\right)\rho-{\Lambda\over{\kappa_5^2}}\left(1-\eta\right).
\label{gdres3}
\eeq
Thus only for standard dark radiation \cite{CosDR,RC} with $\eta=-1$ and the four
dimensional cosmological constant $\Lambda=0$,
is the fifth dimensional pressure $p_z$ equal to zero. Furthermore, the trace of the stress-energy tensor is
\beq
{T_\mu^\mu}={T_a^a}+{p_z},
\eeq
and ${T_a^a}=2{p_z}$, so its trace is only zero for $\eta=-1$ and
$\Lambda=0$. This implies that only the standard form
of dark radiation may be associated with the traceless projected Weyl tensor and so with the 4-dimensional brane
world vaccum in the effective Gauss-Codazzi approach \cite{RC}.

After decoupling the RS warp factor the determination of the 5-dimensional metric for the dark radiation system
requires the solution of Eqs. (\ref{4Dceq}), (\ref{4DEeq}) and (\ref{5DEeqz}) under conditions
(\ref{gdres1})-(\ref{gdres3}). Let us start by introducing Eqs. (\ref{gdres1}) and (\ref{gdres2}) in Eq. (\ref{4Dceq}).
Since the contribution of the cosmological constant cancels out we
find the following dark radiation conservation equations
\beq
\dot{{\rho_{\mbox{\tiny DR}}}}+2(1-\eta){\dot{R}\over{R}}{\rho_{\mbox{\tiny
      DR}}}=0=
{\rho_{\mbox{\tiny DR}}}'+2(1-\eta){{R'}\over{R}}{\rho_{\mbox{\tiny
      DR}}}.\label{drceq1}
\eeq
Note that for this generalized dark radiation system we may also safely take the synchronous frame ($A=0$).
Because of the equation of state $\rho+{p_r}=0$ the dark radiation has a density defined independently of $A$
as a consequence of which, despite the existing pressures, it admits a synchronous solution. Note as well
that this is independent of the relation between $\rho$ and $p_T$. As a consequence the intrincate general
equations \cite{TPS} simplify to give Eq. (\ref{drceq1}). The corresponding inhomogeneous density solution
is given by

\beq
{\rho_{\mbox{\tiny
      DR}}}={{Q_\eta}\over{\kappa_5^2}}{R^{2\eta-2}}
\label{gdrden},
\eeq
where the constant $Q_\eta$ is the dark radiation tidal charge. If ${Q_\eta}>0$ then the dark radiation density
is positive. As a consequence the weak, the dominant and the strong
energy conditions in 4 dimensions imply,
respectively, $\eta\leq 1$, $|\eta|\leq 1$ and $\eta\leq 0$. If these conditions are imposed in 5 dimensions then
we find that they lead, respectively, to $\eta\leq 0$, $-2\leq\eta\leq 0$ and $\eta\leq -1/3$. Of course if
${Q_\eta}<0$ then the dark radiation density is negative and all the energy conditions are violated.

Substituting Eq. (\ref{gdrden}) in the Einstein trace equation we obtain
\beq
-{G_t^t}+{G_r^r}+2{G_\theta^\theta}=-2{{\ddot{R}'}\over{R'}}
-4{\ddot{R}\over{R}}=-2\eta{Q_\eta}{R^{2\eta-2}}-2\Lambda.
\eeq
After two integrations we find
\beq
{\dot{R}^2}={{Q_\eta}\over{2\eta+1}}{R^{2\eta}}+{\Lambda\over{3}}{R^2}
+f,\label{gdr1}
\eeq
where $f\equiv f(r)$ is the arbitrary function of $r$ interpreted as the energy inside a shell labelled by $r$.
Imposing the condition $R(0,r)=r$ on the initial hypersurface $t=0$ and integrating Eq. (\ref{gdr1}) we get
\beq
\pm t +\psi=\int{{dR}\over{\sqrt{{\Lambda\over{3}}{R^2}+f
+{{Q_\eta}\over{2\eta+1}}{R^{2\eta}}}}},\label{gdr2}
\eeq
Note that we have assumed $\eta\not=-1/2$. For $\eta=-1/2$ we obtain
\beq
{\dot{R}^2}={{Q_{\mbox{\tiny -1/2}}}\over{R}}\left(1+\ln R\right)
+{\Lambda\over{3}}{R^2}+f,\label{gdr11}
\eeq
and then
\beq
\pm t +\psi=\int{{dR}\over{\sqrt{{\Lambda\over{3}}{R^2}+f
+{{Q_{\mbox{\tiny -1/2}}}\over{R}}\left(1+\ln R\right)}}}.\label{gdr22}
\eeq
Note as well that if the evolution is to be dominated by the cosmological constant as $R$ goes to infinity
then $\eta$ should satisfy $\eta<1$. This is true when ${Q_\eta}>0$ and any one of the energy conditions is
satisfied. Applying the radial equation
\beq
{G_r^r}=-2{\ddot{R}\over{R}}+{{H^2}\over{R^2}}-{1\over{R^2}}-
{{\dot{R}^2}\over{R^2}}=-{Q_\eta}{R^{2\eta-2}}-\Lambda
\eeq
we again obtain $H=\sqrt{1+f}$ with $f>-1$. Then Eq. (\ref{5DEeqz}),
\beq
{G_z^z}=-{{\ddot{R}'}\over{R'}}-2{\ddot{R}\over{R}}
+{{f-{\dot{R}^2}}\over{R^2}}
+{{(f-{\dot{R}^2})'}\over{RR'}}=-\left(1+\eta\right)
{Q_\eta}{R^{2\eta-2}}-2\Lambda = \kappa_5^2 p_z,
\eeq
is an identity for all functions $R$ and $f$. Thus we conclude that the metric has the RS-LTB form (\ref{vacs1})
and (\ref{vacs2}) with the physical radius given by Eq. (\ref{gdr2}) for $\eta\not=-1/2$ and by Eq. (\ref{gdr22})
for $\eta=-1/2$.

\subsection{Static Limits}

Of the dynamical dark radiation models the marginally bound correspond to $f=0$ and are actually static solutions.
Consider first $\eta\not=-1/2$. Transforming from the LTB coordinates $(t,r)$ to the curvature coordinates $(T,R)$
defined by
\beq
T=t+\int dR
{{\sqrt{{\Lambda\over{3}}{R^2}+{{Q_\eta}\over{2\eta+1}}
{R^{2\eta}}}}\over{
{\Lambda\over{3}}{R^2}-1+{{Q_\eta}\over{2\eta+1}}{R^{2\eta}}}}
\eeq
we find new black string solutions given by Eq. (\ref{vacs1}) with
\beq
d{s_4^2}=-\left(1-{{Q_\eta}\over{2\eta+1}}{R^{2\eta}}
-{\Lambda\over{3}}{R^2}\right)d{T^2}+
{{\left(1-{{Q_\eta}\over{2\eta+1}}{R^{2\eta}}
-{\Lambda\over{3}}{R^2}\right)}^{-1}}d{R^2}+
{R^2}d{\Omega_2^2}.\label{RNBH}
\eeq
The 4-dimensional solution (\ref{RNBH}) for $\eta=-1$ is the inhomogeneous static exterior of a collapsing sphere of
homogeneous standard dark radiation \cite{BGM,GD,RC}. When $\Lambda=0$ it corresponds to the zero mass limit of the
tidal Reissner-Nordstr\"om black hole on the brane \cite{DMPR}. The horizons covering the physical singularity at
${R_s}=0$ are defined by the transcendental equation
\beq
1-{{Q_\eta}\over{2\eta+1}}{R^{2\eta}}
-{\Lambda\over{3}}{R^2}=0.
\eeq
For $\Lambda=0$ and if the other parameters allowed it there may be an horizon located at
\beq
{R_h}={{\left({{2\eta+1}\over{Q_\eta}}\right)}^{1\over{2\eta}}}.
\eeq
If $\Lambda\not=0$ then in general it is not possible to obtain the exact location of the horizons. The two single
exceptions are the models corresponding to $\eta=-1$ and $\eta=1/2$. For $\eta=-1$ we find the standard dark radiation
horizons \cite{BGM,RC}. For $\eta=1/2$ the horizons are given by
\beq
{R_h}={{3{Q_{\mbox{\tiny
          1/2}}}}\over{4\Lambda}}\left(-1\pm\sqrt{1+{{16\Lambda}
\over{9{Q_{\mbox{\tiny 1/2}}^2}}}}\right).
\eeq
If $\Lambda<0$ and ${Q_{\mbox{\tiny 1/2}}}>0$ then we have an inner horizon $R_h^-$ and an outer horizon $R_h^+$. The
two horizons merge for ${Q_{\mbox{\tiny 1/2}}}=4\sqrt{-\Lambda}/3$ and for ${Q_{\mbox{\tiny 1/2}}}<4\sqrt{-\Lambda}/3$
the singularity at ${R_s}=0$ becomes naked. If $\Lambda>0$ and ${Q_{\mbox{\tiny 1/2}}}>0$ there is a single horizon
at $R={R_h^+}$ and for $\Lambda>0$ and ${Q_{\mbox{\tiny 1/2}}}<0$ the horizon is at $R_h^-$. Note that for $\eta=-1$
the dark radiation with a positive tidal charge satisfies all energy conditions. For $\eta=1/2$ this is no longer
true. Indeed, the strong condition is violated in 4 dimensions and in 5 dimensions none of the energy conditions holds.

For $\eta=-1/2$ we proceed analogously to find a black string given by Eq. (\ref{vacs1}) and
\bea
d{s_4^2}&=&-\left[1-{{Q_{\mbox{\tiny -1/2}}}\over{R}}\left(1+\ln R\right)
-{\Lambda\over{3}}{R^2}\right]d{T^2}+
{{\left[1-{{Q_{\mbox{\tiny -1/2}}}\over{R}}\left(1+\ln R\right)
-{\Lambda\over{3}}{R^2}\right]}^{-1}}d{R^2}\nn\\
&+&{R^2}d{\Omega_2^2}.
\eea

\subsection{Exact Dynamical Solutions}
Let us now consider the non-marginally bound models corresponding to $f\not=0$. These are the ones which actually
lead to dynamical and inhomogeneous evolutions. In general it is not possible to determine the exact solutions of
Eqs. (\ref{gdr2}) and (\ref{gdr22}). The only exceptions are $\eta=-1$ and $\eta=1/2$. If $\eta=-1$ we have
standard dark radiation and the solutions have already been determined in \cite{RC}. They are inhomogeneous
cosmologies characterized by $\Lambda$, ${Q_{\mbox{\tiny -1}}}\equiv Q$ and $f$. The corresponding rich structure
of physical singularities and regular rebounces was also identified in \cite{RC}. However, note that for
the 5-dimensional solutions defined by Eqs. (\ref{vacs1}), (\ref{vacs2}) and (\ref{gdr2}) the gravitational field
in always localized near the brane and the inhomogeneous dark radiation dynamics has been induced on the brane by
the bulk fields. They are not a pure vaccum phenomena even if $\eta=-1$ and for such a vaccum gravity is not always
confined to the vicinity of the brane \cite{RC}.

The dynamics for $\eta=1/2$ is actually similar to that of standard
dark radiation \cite{RC}. Indeed, the solutions may also
be organized by $\Lambda$ and by the functions $Y$ and $\beta$ now defined as
\beq
Y=R+{{3{Q_{\mbox{\tiny 1/2}}}}\over{4\Lambda}},\quad
\beta={3\over{\Lambda}}\left({{3{Q_{\mbox{\tiny
            1/2}}^2}}\over{16\Lambda}}
-f\right).
\eeq
To ilustrate consider $\Lambda>0$ \cite{EXP} and allow $Q_{\mbox{\tiny
    1/2}}$ to be a real parameter as $Q$ \cite{qexp}. If $\beta>0$ then $-1<f<3{Q_{\mbox{\tiny 1/2}}^2}/(16\Lambda)$ and so the solutions
are
\beq
\left|R+{{3{Q_{\mbox{\tiny 1/2}}}}\over{4\Lambda}}\right|
=\sqrt{\beta}\cosh\left[\pm
\sqrt{{\Lambda\over{3}}}t+{\cosh^{-1}}\left(
{{\left|r+{{3{Q_{\mbox{\tiny 1/2}}}}\over{4\Lambda}}\right|}
\over{\sqrt{\beta}}}\right)\right].\label{drsol1}
\eeq
If $\beta<0$ then $f>3{Q_{\mbox{\tiny 1/2}}^2}/(16\Lambda)$ and we obtain
\beq
R+{{3{Q_{\mbox{\tiny 1/2}}}}\over{4\Lambda}}=\sqrt{-\beta}\sinh\left[\pm
\sqrt{{\Lambda\over{3}}}t+{\sinh^{-1}}
\left({{r+{{3{Q_{\mbox{\tiny 1/2}}}}\over{4\Lambda}}}
\over{\sqrt{-\beta}}}\right)\right].\label{drsol2}
\eeq
If $\beta=0$ then $f=3{Q_{\mbox{\tiny 1/2}}^2}/(16\Lambda)$ and we get
an homogeneous solution
\beq
\left|R+{{3{Q_{\mbox{\tiny 1/2}}}}\over{4\Lambda}}\right|=
\left|r+{{3{Q_{\mbox{\tiny 1/2}}}}\over{4\Lambda}}
\right|\exp\left(\pm \sqrt{{\Lambda\over{3}}}t\right).\label{drhsol}
\eeq
Clearly, solutions (\ref{drsol1}) and
(\ref{drsol2}) are intrinsically dependent on $r$ and so correspond to
inhomogeneous cosmologies which cannot be reduced to the standard
homogeneous dS or Robertson-Walker spaces \cite{HE} by any coordinate
transformation.

\subsection{Singularities and Regular Rebounces}

To analyze the space of solutions of the dark radiation models let us first take $\eta\not=-1/2$. Then we
consider Eq. (\ref{gdr1}) written as
\beq
{R^\sigma}{\dot{R}^2}=V,
\eeq
where the potential $V$ is
\beq
V=V(R,r)={{Q_\eta}\over{2\eta+1}}{R^{2\eta+\sigma}}
+{\Lambda\over{3}}{R^{2+\sigma}}
+f{R^\sigma},
\eeq
and the parameter $\sigma\geq 0$ is only non-zero when $\eta$ is negative. For example if $\eta=-1$ then
$\sigma=-2\eta=2$. Again in general it is not possible to study this potential exactly. The same is true for
$\eta=-1/2$ which involves a logarithm of $R$. Only for $\eta=-1$ and $\eta=1/2$ is such exact analysis
possible. For $\eta=-1$ a rich structure of singularities and regular rebounces was found and discussed in
\cite{RC}. A similar structure of solutions may now be shown to exist for $\eta=1/2$. To ilustrate consider
$\Lambda>0$. The potential is written as
\beq
V={\Lambda\over{3}}{R^2}+{{Q_{\mbox{\tiny 1/2}}}\over{2}}R+f=
{{\Lambda}\over{3}}\left({Y^2}-\beta\right).
\eeq
Then as for $\eta=-1$ there are at most two regular rebounce epochs and a phase of continuous accelerated
expansion to infinity.

For $\beta<0$ it is clear that $V>0$ for all values of $R\geq 0$. It satisfies $V(0,r)=f$ with
$f>3{Q_{\mbox{\tiny 1/2}}^2}/(16\Lambda)$ and it grows to infinity with $R$ as $\Lambda{R^2}$. The dark
radiation shells may either expand forever or collapse to the singularity after a proper time $t={t_s}(r)$
given by
\beq
{t_s}(r)=\sqrt{{3\over{\Lambda}}}\left[{\sinh^{-1}}
\left({{r+{{3{Q_{\mbox{\tiny 1/2}}}}\over{4\Lambda}}}
\over{\sqrt{-\beta}}}\right)-{\sinh^{-1}}\left({{3{Q_{\mbox{\tiny 1/2}}}}
\over{4\Lambda\sqrt{-\beta}}}\right)\right].
\eeq
For $\beta>0$ and independently of $Q_{\mbox{\tiny 1/2}}$ the configuration $-1<f<0$ leads to globally regular
solutions with a single rebounce epoch at $R={R_*}$ where
\beq
{R_*}=-{{3{Q_{\mbox{\tiny 1/2}}}}\over{4\Lambda}}+\sqrt{\beta}.
\eeq
Indeed, this is the minimum radius a collapsing dark radiation shell can have and it is reached after the time
$t={t_*}(r)$ where
\beq
{t_*}(r)=\sqrt{{3\over{\Lambda}}}{\cosh^{-1}}\left(
{{r+{{3{Q_{\mbox{\tiny 1/2}}}}\over{4\Lambda}}}
\over{\sqrt{\beta}}}\right).
\eeq
At this point the shells reverse their motion and expand continuously with ever increasing speed.

If $\beta>0$ and $0<f<3{Q_{\mbox{\tiny 1/2}}^2}/(16\Lambda)$ then for ${Q_{\mbox{\tiny 1/2}}}>0$ there are no
rebounce points in the allowed dynamical region $R\geq 0$. The time to reach the singularity is now given by
\beq
{t_s}(r)=\sqrt{{3\over{\Lambda}}}\left[{\cosh^{-1}}
\left({{r+{{3{Q_{\mbox{\tiny 1/2}}}}\over{4\Lambda}}}
\over{\sqrt{\beta}}}\right)-{\cosh^{-1}}\left({{3{Q_{\mbox{\tiny 1/2}}}}
\over{4\Lambda\sqrt{\beta}}}\right)\right].
\eeq
On the other hand for ${Q_{\mbox{\tiny 1/2}}}<0$ there are two rebounce epochs at $R=R_{*\pm}$ with
\beq
{R_{*\pm}}=-{{3{Q_{\mbox{\tiny 1/2}}}}\over{4\Lambda}}\pm\sqrt{\beta}.
\eeq
Since $V(0,r)=f>0$ a singularity also forms at ${R_s}=0$. The phase space of allowed dynamics is divided in two
disconnected regions separated by the forbidden interval ${R_{*-}}<R<{R_{*+}}$ where the potential is negative.
For $0\leq R\leq{R_{*-}}$ the dark radiation shells may expand to a maximum radius $R={R_{*-}}$ in the time
${t_{*-}}={t_*}$ where
\beq
{t_*}(r)=\sqrt{{3\over{\Lambda}}}{\cosh^{-1}}\left(
{{\left|r+{{3{Q_{\mbox{\tiny 1/2}}}}\over{4\Lambda}}\right|}
\over{\sqrt{\beta}}}\right).
\eeq
At this rebounce epoch the shells start to fall towards the singularity which is reached after the time
\beq
{t_s}(r)=\sqrt{{3\over{\Lambda}}}{\cosh^{-1}}
\left({{|3{Q_{\mbox{\tiny 1/2}}}|}\over{4\Lambda\sqrt{\beta}}}\right).
\eeq
If $R\geq{R_{*+}}$ then there is a collapsing phase to the minimum radius $R={R_{*+}}$ taking the time
${t_{*+}}={t_*}$ followed by reversal and subsequent accelerated continuous expansion. The singularity at ${R_s}=0$
does not form and so the solutions are globally regular.

If $\beta=0$ then $f=3{Q_{\mbox{\tiny 1/2}}^2}/(16\Lambda)$. For ${Q_{\mbox{\tiny 1/2}}}<0$ there is one rebounce
point at
\beq
{R_*}=-{{3{Q_{\mbox{\tiny 1/2}}}}\over{4\Lambda}}.
\eeq
In this case $V(0,r)=f>0$ and then a singularity also forms at ${R_s}=0$. There is no forbidden region in phase
space but the point at $R_*$ turns out to be a regular fixed point which divides two distinct dynamical regions.
Indeed if a shell starts at $R={R_*}$ then it will not move for all times. If initially $R<{R_*}$ then either
the shell expands towards $R_*$ or it collapses to the singularity. The time to get to the singularity is finite,
\beq
{t_s}(r)={3\over{\Lambda}}\ln\left({{{|3{Q_{\mbox{\tiny 1/2}}}|}\over{4\Lambda}}
\over{\left|
r+{{3{Q_{\mbox{\tiny 1/2}}}}\over{4\Lambda}}\right|}}\right),
\eeq
but the time to expand to ${R_*}$ is infinite. If initially $R>{R_*}$ then the collapsing dark radiation shells
also take an infinite time to reach $R_*$. If ${Q_{\mbox{\tiny 1/2}}}>0$ there are no real rebounce epochs and the
collapsing dark radiation simply falls to the singularity at ${R_s}=0$. The colision proper time is
\beq
{t_s}(r)=-{3\over{\Lambda}}\ln\left({{{3{Q_{\mbox{\tiny 1/2}}}}\over{4\Lambda}}
\over{r+{{3{Q_{\mbox{\tiny 1/2}}}}\over{4\Lambda}}}}\right).
\eeq

\section{Gauss-Codazzi Equations and the Localization of Gravity on the Brane }

In the special conformal setting we have defined the RS warp solution (\ref{RSwf1}) has been factored out of the
5-dimensional problem. This is then reduced to the resolution of Eqs. (\ref{4Dceq}) and (\ref{4DEeq}) subjected to
the $p_z$ conditions (\ref{eqst3}) and (\ref{5DEeqz}). A set of effective 4-dimensional brane world geometries
are generated and two examples are the inhomogenous dust and dark radiation dynamics in the presence of a
brane cosmological constant. These effective 4-dimensional metrics should be deduced by an observer confined to
the brane which makes the same assumptions about the bulk degrees of freedom. Moreover, in this circunstances the
4-dimensional observer should also agree about the localization of gravity in the vicinity of the brane. Let us
now show that the 5-dimensional approach developed in this work is in this sense consistent with the effective
Gauss-Codazzi formulation \cite{CGC}-\cite{RM}.

Consider then the Gauss-Codazzi approach for the RS brane world scenario and assume for simplicity that
the matter degrees of freedom which exist on the brane are only originated by field modes present in the AdS bulk
space. Then the effective 4-dimensional Einstein equations are given by
\bea
{\mathcal{G}_\mu^\nu}&=&{{2{\kappa_5^2}}\over{3}}
\left[{\mathcal{T}_\alpha^\beta}
{q_\mu^\alpha}{q_\beta^\nu}+\left({\mathcal{T}_\alpha^\beta}{n^\alpha}
{n_\beta}-{1\over{4}}{\mathcal{T}_\alpha^\alpha}\right){q_\mu^\nu}
\right]+{\mathcal{K}_\alpha^\alpha}{\mathcal{K}_\mu^\nu}\nn\\
&-&{\mathcal{K}_\mu^\alpha}{\mathcal{K}_\alpha^\nu}-{1\over{2}}{q_\mu^\nu}
\left({\mathcal{K}^2}-{\mathcal{K}_\alpha^\beta}{\mathcal{K}_\beta^\alpha}
\right)
-{\mathcal{E}_\mu^\nu},\label{GCEeq1}
\eea
where
${\mathcal{G}_\mu^\nu}={G_\alpha^\beta}{q^\alpha_\mu}{q_\beta^\nu}$,
${n^\mu}={\delta_z^\mu}$ is the unit normal to the brane,
${q_\mu^\nu}={\delta_\mu^\nu}-{n_\mu}{n^\nu}$ is the tensor which
projects orthogonaly to $n^\mu$,
\beq
{\mathcal{T}_\mu^\nu}=-{\Lambda_B}{\delta_\mu^\nu}+{T_\mu^\nu}\label{GCst}
\eeq
is the stress-energy tensor,
\beq
{\mathcal{K}_\mu^\nu}={\lim_{z\to{z_0}+}}{K_\mu^\nu}=
-{{{\kappa_5^2}\lambda}\over{6}}{q_\mu^\nu}\label{GCec}
\eeq
is the extrinsic curvature and
\beq
{\mathcal{E}_\mu^\nu}={\lim_{z\to{z_0}+}}{C_{\rho\alpha\sigma\beta}}{n^\alpha}
{n^\beta}{q_\mu^\rho}{q^{\sigma\nu}}\label{weyldef}
\eeq
is the traceless projection of the 5-dimensional Weyl tensor.

Substituting Eqs. (\ref{GCst}) and (\ref{GCec}) in Eq. (\ref{GCEeq1}) we find
\beq
{\mathcal{G}_\mu^\nu}=-{{\kappa_5^2}\over{2}}\left({\Lambda_B}+
{{{\kappa_5^2}{\lambda^2}}\over{6}}\right){\delta_\mu^\nu}+
{{2{\kappa_5^2}}\over{3}}
\left[{T_\alpha^\beta}
{q_\mu^\alpha}{q_\beta^\nu}+\left({T_\alpha^\beta}{n^\alpha}{n_\beta}-
{1\over{4}}{T_\alpha^\alpha}\right){q_\mu^\nu}
\right]-{\mathcal{E}_\mu^\nu}.\label{GCEeq2}
\eeq
Applying the covariant derivative it is clear that in general the projected Weyl tensor is not conserved
because of the fields present in the bulk.

In the effective 4-dimensional point of view the metric $g_{\mu\nu}$ is defined by Eqs. (\ref{gm2}) and
(\ref{gm3}). Then we obtain ${\mathcal{E}_z^\mu}=0$. If in addition the RS identity (\ref{RSwf2}) is assumed
to be satisfied then Eq. (\ref{GCEeq2}) is written as
\beq
{G_a^b}={{2{\kappa_5^2}}\over{3}}
\left({T_a^b}+{1\over{4}}{T_z^z}{\delta_a^b}
\right)-{\mathcal{E}_a^b}.\label{GCEeq3}
\eeq
Moreover, if $T_a^b$ is conserved as in Eq. (\ref{4Dceq}) then the projected Weyl tensor must satisfy
\beq
{\nabla_a}{\mathcal{E}_b^a}={{\kappa_5^2}\over{6}}{\nabla_b}{T_z^z}.
\eeq
If it is verified that
\beq
{\mathcal{E}_a^b}={{\kappa_5^2}\over{3}}
\left(-{T_a^b}+{1\over{2}}{T_z^z}{\delta_a^b}
\right)\label{GCweyl1}
\eeq
then Eq. (\ref{GCEeq3}) becomes Eq. (\ref{4DEeq}) and so the 4-dimensional observer does find the same
induced matter dynamics on the brane.

This may be explicitly checked for the dust and dark radiation systems. First determine $\mathcal{E}_\mu^\nu$
using the alternative Eqs. (\ref{weyldef}) and (\ref{GCweyl1}). For dust both lead to the same result
\beq
{\mathcal{E}_t^t}={{\kappa_5^2}\over{4}}{\rho_{\mbox{\tiny D}}},\quad
{\mathcal{E}_r^r}={\mathcal{E}_\theta^\theta}={\mathcal{E}_\phi^\phi}=
-{{\kappa_5^2}\over{12}}{\rho_{\mbox{\tiny D}}}.\label{dustE}
\eeq
Thus Eq. (\ref{GCweyl1}) is indeed verified and then it is easy to see that Eq. (\ref{GCEeq3}) reduces to the
4-dimensional Einstein equations for dust and a cosmological constant
\beq
{G_t^t}=-{\kappa_5^2}{\rho_{\mbox{\tiny D}}}-\Lambda,\quad
{G_r^r}={G_\theta^\theta}={G_\phi^\phi}=
-\Lambda.
\eeq
For the dark radiation system we also conclude that both Eqs. (\ref{weyldef}) and (\ref{GCweyl1}) lead to the
same result
\beq
{\mathcal{E}_t^t}={\mathcal{E}_r^r}=
{{\kappa_5^2}\over{6}}(1-\eta){\rho_{\mbox{\tiny DR}}},\quad
{\mathcal{E}_\theta^\theta}={\mathcal{E}_\phi^\phi}=-{{\kappa_5^2}\over{6}}
(1-\eta){\rho_{\mbox{\tiny DR}}}\label{drE}
\eeq
so that Eq. (\ref{GCweyl1}) is satisfied. Then Eq. (\ref{GCEeq3}) reduces to the 4-dimensional Einstein equations
for dark radiation and a cosmological constant
\beq
{G_t^t}={G_r^r}=-{\kappa_5^2}{\rho_{\mbox{\tiny
      DR}}}
-\Lambda,\quad{G_\theta^\theta}={G_\phi^\phi}={\kappa_5^2}
{\rho_{\mbox{\tiny
      DR}}}
-\Lambda.
\eeq
In the 5-dimensional picture it is clear that whatever the effective 4-dimensional solution to be considered,
the conformal RS warp factor ensures that gravity is always localized in the vicinity of the brane. Let us now
show that the observer confined to the brane may reach the same conclusion. In the covariant Gauss-Codazzi
approach the tidal acceleration away from the brane is defined by
\cite{RM,RM4dp}
\beq
{a_t}=-{\lim_{z\to{z_0}+}}{R_{\mu\nu\alpha\beta}}{n^\nu}{u^\nu}
{n^\alpha}{n^\beta},
\eeq
where ${u^\mu}={\delta_t^\mu}$ is the extension off the brane of the 4-velocity field which satisfies ${u^\mu}{n_\mu}=0$
and ${u^\mu}{u_\mu}=-1$. For the gravitational field to be localized near the brane $a_t$ must be negative. Consider
the identity \cite{SMS}
\beq
{R_{\mu\nu\alpha\beta}}={C_{\mu\nu\alpha\beta}}+{2\over{3}}\left\{
{g_{\mu[\alpha}}{R_{\beta]\nu}}+{g_{\nu[\beta}}{R_{\alpha]\mu}}\right\}-
{1\over{6}}R{g_{\mu[\alpha}}{g_{\beta]\nu}},
\eeq
where the square brackets denote anti-symmetrization. Introducing the 5-dimensional Einstein equation
\beq
{G_\mu^\nu}={\kappa_5^2}{\mathcal{T}_\mu^\nu}
\eeq
we find for the tidal acceleration the following expression
\beq
{a_T}={{{\kappa_5^2}{\Lambda_B}}\over{6}}-{{\kappa_5^2}\over{3}}\left(
{T_\alpha^\beta}{u^\alpha}{u_\beta}-{T_z^z}+{{T_\alpha^\alpha}\over{2}}
\right)-
{\mathcal{E}_\alpha^\beta}{u^\alpha}{u_\beta}.
\eeq
If the contributions of $T_\mu^\nu$ and $\mathcal{E}_\mu^\nu$ cancel each other,
\beq
{\mathcal{E}_\alpha^\beta}{u^\alpha}{u_\beta}=-{{\kappa_5^2}\over{3}}\left(
{T_\alpha^\beta}{u^\alpha}{u_\beta}-{T_z^z}+{{T_\alpha^\alpha}\over{2}}\right)
,\label{GCweyl2}
\eeq
then the tidal acceleration is given by
\beq
{a_t}={{{\kappa_5^2}{\Lambda_B}}\over{6}},\label{GCta}
\eeq
which is indeed always negative in the RS 5-dimensional AdS space.

This may also be explicitly confirmed for the dust and dark radiation systems. For dust Eqs. (\ref{dustes1})
and (\ref{dustes2}) imply
\beq
{a_T}={{{\kappa_5^2}{\Lambda_B}}\over{6}}-{{\kappa_5^2}\over{4}}{\rho_D}-
{\mathcal{E}_\alpha^\beta}{u^\alpha}{u_\beta}.
\eeq
Then using Eq. (\ref{dustE}) it is easy to see that Eq. (\ref{GCweyl2}) is indeed verified. In the vaccum
there is no brane cosmological constant, the solutions are conformally flat and so we also obtain the same result.
Note, however, that using the effective Gauss-Codazzi approach it is possible to find a much richer non-conformally
flat vaccum dynamics which also admits a brane cosmological constant \cite{RC}. Then it turns out that gravity
is not always confined to the vicinity of the brane \cite{RC}. On the other hand for dark radiation generated on
the brane by the conformal bulk fields Eqs. (\ref{gdres1})-(\ref{gdres3}) imply
\beq
{a_T}={{{\kappa_5^2}{\Lambda_B}}\over{6}}-{{\kappa_5^2}\over{3}}(1-\eta)
{\rho_{\mbox{\tiny
      DR}}}-
{\mathcal{E}_\alpha^\beta}{u^\alpha}{u_\beta}.
\eeq
Once more Eq. (\ref{GCweyl2}) is verified as may be checked introducing Eq. (\ref{drE}). Then the gravitational
field is always bound to the vicinity of the brane.

\section{Conclusions}

In this work we have analyzed in some detail the dynamics of a spherically symmetric RS brane world from a
5-dimensional perspective. Allowing for bulk field modes other than the Einstein-Hilbert gravity, we have
applied a global conformal transformation to give a clarifying organization to the Einstein equations for the
most general 5-dimensional metric consistent with the $Z_2$ symmetry and with the spherical symmetry on the brane.
We have shown that if the bulk stress-energy tensor has conformal weight $s=-4$ and if only the conformal warp
factor depends on the fifth dimension, then matter dynamics is induced on the brane which is defined by standard
4-dimensional Einstein equations. We have seen that such dynamics has an intrincate influence on the curvature
near the brane and that it must in general produce a pressure along the fifth dimension. Imposing further that
the conformal factor is only a function of the 5-dimensional coordinate and that the bulk stress-energy tensor
is diagonal, we have shown that the 5-dimensional pressure is required to satisfy a well defined equation of
state with the density and pressures characterizing the induced matter on the brane.

With this analysis we have discovered a new class of exact dynamical solutions for which the conformal warp
factor localizes gravity in the vicinity of the brane. We have considered the two specific examples of synchronous
5-dimensional geometries which induce on the brane the non-conformally flat inhomogeneous gravitational dynamics
of dust and generalized dark radiation in the presence of a brane cosmological constant also generated by the bulk
field modes. For these examples we have discussed the static marginally bound limits and the conditions defining
the solutions as singular or as globally regular.

Finally, we have also analyzed the point of view of an observer confined to the brane to show that an identical
description of the variables in the problem consistently leads to the same localized brane world dynamics.
\vspace{1cm}

\centerline{\bf Acknowledgements}
\vspace{0.25cm}

We are grateful for financial support from the {\it Funda\c {c}\~ao para a Ci\^encia e a Tecnologia} (FCT) as well
as the {\it Fundo Social Europeu} (FSE) under the contracts SFRH/BPD/7182/2001 and POCTI/32694/FIS/2000 ({\it III
Quadro Comunit\'ario de Apoio}).

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\end{document}

