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\begin{flushright}{UFIFT-HEP-02-37\\}
\end{flushright}
\begin{flushright}{BRX-TH-512\\}
\end{flushright}




\vspace{2.5cm}
\begin{center}
{\LARGE \bf {Kaluza-Klein Monopole in AdS Spacetime}}\\
\vspace{1cm}
{\large Vakif K. Onemli$^{a,}$\footnote{e-mail:~ 
onemli@phys.ufl.edu},
\hskip 0.3 cm 
Bayram Tekin$^{b,}$\footnote{e-mail:~
tekin@brandeis.edu} }\\
\vspace{.5cm}
$^a${\it Physics Department, University of Florida, Gainesville, 
FL 32611, USA}\\  
\vspace{.5cm}
$^b${\it {Department of Physics, Brandeis University , Waltham, MA 02454,
USA}}\\

\end{center}


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\begin{abstract}
\baselineskip=18pt
We construct analogs of the flat space Kaluza-Klein monopoles 
in locally Anti-de Sitter (AdS) spaces for $D \ge 5+1$.  We show 
that unlike the flat space KK monopole, 
there are no five dimensional static KK monopoles in AdS
space. Thus, one needs at least two extra 
dimensions, one of which is compact, to get a static KK monopole 
for cosmological backgrounds.
\end{abstract}
\vfill

Keywords: ~ Kaluza-Klein Monopole, Solitons, AdS

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\section{Introduction}

Gravitational solitons are topologically stable, everywhere 
smooth, particle-like solutions of pure gravity theories. Not 
all dimensions accommodate these spacetimes. For example,  
$D=4$ General Relativity without 
a cosmological constant does not have solitons and thus, 
globally flat (Lorentzian) $R^4$ 
is the unique singularity free solution. This no-go result 
in four dimensions persists even if a cosmological constant ($\Lambda$) 
is added to Einstein's theory. As shown in \cite{boucher}, 
if the soliton is required to be asymptotically AdS ($\Lambda <0$), 
then globally AdS spacetime is the unique static non-singular solution. 
For de-Sitter ($\Lambda >0$) case , the cosmological horizon 
complicates a rigorous proof, but  
a similar no-soliton result is expected to hold \cite{boucher}.  
 
Non-trivial soliton solutions 
can be found if either one
or both of the following requirements are relaxed:\\
a) the spacetime is four dimensional,\\  
b) the solutions  are asymptotically flat or asymptotically AdS.\\  
In fact, adding an extra compact spatial dimension leads to
many solitons one of which is the well-known  Kaluza-Klein (KK) monopole
of Sorkin, Gross and Perry \cite{sorkin, gross}. 
This is a $4+1$ dimensional, static, 
Ricci-flat space-time that looks exactly like a Dirac monopole  
to a $3+1$ dimensional `effective' observer who cannot see 
the compact extra dimension. Magnetic field of the 
KK monopole spreads radially in the 3 dimensional 
space. Once the size of the extra dimension is fixed, to get 
the proper $3+1$ dimensional gravity plus Maxwell's Electromagnetic  
theory, monopole's 
{\it{inertial mass}} is determined to be  about three times the Planck 
mass ( $6 \times 10^{-5}$ g ) \cite{gross}. In spite of its finite
mass, the monopole does not exert 
any gravitational force on massive neutral test particles. It only interacts 
with moving charged particles, as a monopole should do.
Also, in asymptotically {\it{locally}} AdS spacetimes (as opposed 
to condition (b) ),  
a non-trivial solution (the AdS soliton) was 
found  more recently by Horowitz and Myers \cite{horowitz}. 
The AdS soliton, which exists even in four dimensions, is conjectured to 
have the minimum (negative) energy among all the asymptotically locally 
AdS spacetimes. 
The uniqueness of the AdS soliton as the lowest 
energy configuration provided support for a new 'positive energy 
conjecture' in gravity \cite{galloway}.  

Although the Ricci flat KK monopoles are well studied 
both in pure gravity theories and in string/M-theory \cite{townsend},
analogous solutions in cosmological backgrounds have not been
constructed before. In this paper, we show that in $D=4+1$ cosmological
Einstein theory, no such solution exists. However, for
$D \ge 5+1$, we find static KK monopole-like solitons with 
asymptotically locally AdS geometries. Given the recent interest 
in the AdS spacetime both in the context of AdS/CFT duality and the brane
world
scenarios, it seems proper to study monopoles in this background. 

The outline of the paper is as follows: In Sec. 2, we review the KK
monopole and show that there is no static KK monopole solution in $5D$
cosmological backgrounds.
In Sec. 3, we construct a KK monopole in $D=6$ asymptotically locally AdS
background; find another new soliton in this dimension and interpret 
it as a brane-world. We conclude with Sec 4.  

\section{KK monopole and the no-go result in $5D$ AdS}
 
First let us review the essence of the Ricci-flat KK monopole 
construction \cite{sorkin, gross}, and try to formulate a recipe for the
AdS case. Our conventions are $(-,+,+, ... +)$,
$\left[\nabla_M, \nabla_N\right]V_L = R_{M N L}\,^S V_S $,\,\,
$ R_{M N} \equiv R_{M L N}\,^L$. The flat KK monopole was obtained by  
trivially adding a time direction to the four dimensional Taub-NUT (TN) 
space: 
\be
ds^2= -dt^2  + ds^2_{\mbox{{\scriptsize TN}}},
\ee
where
\be
ds^2_{\mbox{{\scriptsize TN}}}
=V^{-1}(r)\,
\left(dx^5 + \vec{A}\cdot d\vec{r}\right)^2 +
V(r)d\vec{r}\cdot d \vec{r}. 
\label{metric1}
\ee
TN metric is a gravitational instanton: a 
Euclidean, Ricci-flat metric 
that solves the $D=4$ self-dual Einstein's equations which reduce 
to a first order BPS-like equation : 
$\vec{\nabla} \times \vec{A}=\pm \vec{\nabla} V(r)$. 
Since the time direction is flat,
the $5D$ metric is also Ricci flat. The metric has a
time-like Killing vector, along with $g_{0i}=0 $.  BPS equation 
implies that $V(r)$ is an harmonic equation on $R^3$. One 
monopole solution with a $\delta$-function
singularity is:   
\be
V = 1 + {4 M\over r}\;.   
\ee
Thus, the vector potential $A$ and the magnetic field $\vec{B}(r)$ are:
\be
\vec{A}=
\frac{4M\left(1-\cos{\theta}\right)}{r\sin{\theta}}\hat\phi\; , \hskip 1
cm 
\vec{B}(r) = {4M \hat{r}\over r^2}\; .
\label{potential}
\ee
The metric ({\ref{metric1}) becomes regular everywhere
if  $x^5$ is compactified on a circle with radius $8M$.
This solution cannot decay to the trivial vacuum (zero charge, $M=0$
case)
because of its non-trivial topological structure
with 1 unit of Euler character $(\chi_E)$.
Multi-monopole solutions with equal charges (both
in sign and in magnitude ) located at various points  can also be
obtained,
since there are no forces between these BPS objects \cite{gibbons}.
In \cite{onemli}, it was shown that ``KK vortices'', with magnetic fields
trapped effectively in $2D$, can be constructed by summing up infinitely
many
KK monopoles on a line. Also, by adding 5 or 6 more spatial flat
directions
to the metric, one gets a $D5$ or $D6$ brane which is a BPS solution to
string or M theory, respectively \cite{townsend}.

How can one generalize the above construction to 
the spaces with cosmological constant?
First, there is the issue of the sign of the cosmological 
constant. Since the spatial hypersurfaces of de-Sitter space (dS) 
are compact, 
one should not expect a single monopole solution in dS. Rather 
magnetic dipole-like  totally neutral 
solutions might exist. [ Related is the issue of the
non-existence of a global time-like Killing vector in the relevant
solutions in dS.]   
On the other hand, in AdS which has open hypersurfaces, and hence a 
boundary to define flux,  
one expects monopole solutions. However,
this expectation will turn out to be wrong in $D=5$. Unlike the 
flat space case, for negative 
cosmological backgrounds, $D=6$ is the minimal number of dimensions
where a monopole-like solution exists for pure gravity equations. 

Now let us adapt the essential ingredients of the above 
Ricci flat construction to AdS.
Needless to
say that in this case, 
the time direction cannot simply be added to a four dimensional gravitational 
instanton such as the Taub-NUT or the AdS-Taub-NUT. 
It has to be a warp 
product metric. The first thing we require is that
the magnetic field seen by an effective $3+1$ observer should now be
that of a $3D$ abelian ''hyperbolic monopole''
\cite{atiyah,chakrabarti} (instead of an $R^3$ monopole):
\be
\vec{B} = {4 M |\Lambda| \hat{r}\over
\sinh^2{(r \sqrt{|\Lambda|})}}\;,
\label{magnetic}
\ee
where $\Lambda$ is the (negative) cosmological constant of $AdS_3$. 
We also require that in the limit of vanishing cosmological
constant, the AdS monopole reduces to the flat KK monopole, since 
for $\Lambda \rightarrow 0$ we do not expect any discontinuity.

For $\Lambda := -L^2 < 0 $, Einstein spaces are solutions 
to the equations of motion with
\be
R_{MN} = - 2 L^2 g_{MN}\; .   
\ee
We first need to 
find the zero charge monopole (or the proper background ). 
The usual maximally symmetric 
$AdS_{4+1}$ spacetime is:
\be
ds^2 = -\cosh^2(L r/ \sqrt{2}) dt^2  + dr^2 + (2 / L^2)
\sinh^2 ( Lr/ \sqrt{2})d\Omega_3\; , 
\label{maxads}
\ee 
where the metric on $S^3$ can be (up to gauge invariance ) taken 
as the usual $S^2$ foliation form  $d\Omega_3 = 
d\psi^2 + \sin^2 \psi\,\left(d\theta^2 + \sin^2\theta \,d\phi^2\right)$ or
the
Clifford torus ($S^1\times S^1$) foliation  
$d\Omega_3 = d\psi^2 + \cos^2\psi\, d\theta^2 + \sin^2\psi \,d\phi^2$. But, 
it is clear that the metric (\ref{maxads}) 
cannot be deformed to have an effective
$3+1$ dimensional space whose spatial part is $AdS_3$ that supports the
hyperbolic monopole with the magnetic field given by (\ref{magnetic}).
In fact, the proper background  
should be of the $AdS_2\times AdS_3$ form
with the line element:
\be
ds^2 = - {1\over 2} \exp{(-2\sqrt{2}L x^5)} dt^2 + dx_5^2 + 
dr^2 + {1\over L^2} \sinh^2(L r) d\Omega_2\; ,             
\label{ads}
\ee 
where  $ d\Omega_2= d\theta^2 + \sin^2\theta d\phi^2 $. 
The sought-after monopole should yield (\ref{ads}) 
in the limit of zero charge. 
The total magnetic flux $\Phi$  of the hyperbolic monopole
(\ref{magnetic}) 
on a large sphere (for fixed $t, x^5$) is just like the flat space
case: $\Phi=16\pi M$.
With the definition, $ B :=  F = dA$,  
the gauge potential retains its flat space form: 
$A = 4M\left(1-\cos{ \theta }\right) d\phi$. 


Bearing these constraints in mind, we 
write down the generic, static 
ansatz which might solve the equations of motion: 
\be
ds^2 = -{1\over 2}a^2(r, x^5)\exp{(-2\sqrt{2}Lx^5)} dt^2 + 
b^2(r,x^5)\left[dx^5 +4M\left(1-\cos{\theta}\right) d\phi
\right]^2\nonumber\\  
+v^2(r,x^5)\left[dr^2 + (1/ L^2) \sinh^2(L r)d\Omega_2\right]\; .      
\label{metric2}
\ee
For zero magnetic charge case $(M=0 )$, we require $a(r,x^5) = b(r,x^5)
= v(r,x^5) = 1 $. And for $L =0$ the solution should yield: 
\be
a^2(r,x^5) = 2  \hskip 1 cm  b^{-1}(r,x^5) = 
v(r,x^5) = 1 + {4M\over r} \; .
\ee
We claim that there are no non-trivial monopole-like solutions 
of the form (\ref{metric2}) with the desired properties laid out above.
The easiest way to show this, is 
to work with directly the equations of motion. We only need to 
look at the simplest components of the Ricci tensor, the ones that are
supposed to vanish. 
Let us start with $R_{\theta x^5}$:
\be
R_{\theta x^5}=
\frac{
8M^2L^2 b^2(r,x^5)}{v^2(r,x^5)\sinh^2{(Lr)}}
\,\frac{\left(\cos{\theta} 
-1\right)}{\sin{\theta}}
\left[\sqrt{2}L-3
\frac{b'(r, x^5)}{b(r, x^5)}-\frac{a'(r, x^5)}{a(r, 
x^5)}+\frac{v'(r, x^5)}{v(r, x^5)} \right] \; ,
\ee
where $'$ denotes $\partial_{x^5}$. For non-vanishing $M$ and $L$, 
setting $R_{\theta x^5} =0$, we find:
\be
b(r,x^5)= \left[{{v(r,x^5)}\over
{a(r,x^5)}}\right]^{1/3}f(r)\exp{(\sqrt{2}L x^5/3)}\; ,
\label{cond1}
\ee 
where $f(r)$ is an arbitrary function. Plugging (\ref{cond1})
into $R_{\theta\phi}$ yields: 
\be
R_{\theta \phi} = \frac{4M}{3} {{\left(\cos{\theta}-1\right)^2} \over 
{\sin \theta}} 
\left[\frac{a'(r,x^5)}{a(r,x^5)} +
2\frac{v'(r,x^5)}{v(r,x^5)}-\sqrt{2}L \right]\; .
\label{cond2}
\ee
Then, the equation $R_{\theta \phi} = 0$ is solved by:
\be 
v^2(r,x^5)={g(r)\over {a(r,x^5)}}{\exp{(\sqrt{2}Lx^5)}}\; ,
\label{cond3}
\ee
where again $g(r)$ is arbitrary. Using  (\ref{cond1}, \ref{cond3}), 
$R_{rx^5}$ is simplified as: 
\be
R_{r x^5}=\frac{3}{2}\frac{\partial_r a(r,x^5)}{a^2(r,x^5)}\left[\sqrt{2}L
a(r,x^5)
-a'(r,x^5)\right]\; . 
\ee
For the metric to be Einstein we set $R_{r x^5}=0$, 
leading to:  
\be
a(r,x^5)=h(r)\exp{(\sqrt{2}Lx^5)}\; . 
\ee
Hence, collecting everything, one can write the final metric as:
\be
ds^2=-\frac{1}{2}h^2(r)\,dt^2+\frac{g(r)}{h(r)}\left[ dr^2 + (1/L^2)
\sinh^2(L r)d\Omega_2\right ]\nonumber\\
+\frac{f^2(r)g^{1/3}(r)}{h(r)}\left[ dx^5
+4M\left(1-\cos{ \theta}\right) d\phi\right]^2\; .
\label{metric3}
\ee
Observe that $x^5$ dependence is completely dropped out. Therefore,  
it is clear that $M=0$ does not yield the background metric
(\ref{ads}). Thus, there is no $5 D$ KK monopole in AdS space. 

\section{ 6D AdS monopoles}

In the previous section, we have seen that we cannot embed the 
$3D$ hyperbolic monopole in a $5D$ cosmological space. The main problem 
is that the time direction cannot simply be added to the spatial 
part. On the other hand, as we show here, in $D \ge 5+1$ 
cosmological spacetimes, one can construct solitons which 
resemble the flat space KK monopoles. 

For monopole solutions, the need for at least two extra 
dimensions in the cosmological case should not 
be surprising. In fact, the only other non-trivial 
soliton we know for $\Lambda<0$ -the AdS soliton  
has to be at least 6-dimensional if 3-dimensional 
spherical symmetry is imposed \cite{horowitz}. 
Indeed, the KK monopole we are searching for, 
is also spherically symmetric. [ Note that in the metric,
one breaks the spherical symmetry by choosing a gauge 
for the gauge potential $A$. But, physically, the spacetime
is spherically symmetric. ]


To construct a $D= 6$ soliton with a $3D$ magnetic field 
of the form (\ref{magnetic}), we take the following 
$AdS_2 \times AdS_4$ background:
\be
ds^2 = -\exp{(- 2 L x^6)} dt^2 + dx_6^2
+\frac{3}{2}\left[ dr^2 + (1/L^2) \sinh^2(r L ) d\Omega_2
+ \cosh^2(r L) dx_5^2\right]\; ,  
\ee
and deform it to:
\be
ds^2 = -\exp{(- 2 L x^6)} dt^2 + dx_6^2 +\frac{3}{2}H^2(r)
\Bigg\{V(r) \left( dr^2 + (1/L^2) \sinh^2(r L) d\Omega_2\right)
\nonumber\\
+{\cosh^2(r L)\over V(r)}\left[ dx^5 + 
4M\left( 1- \cos \theta \right)d\phi\right]^2 \Bigg\}\; .
\ee
This metric is the most general one that meets 
our requirements of smoothness for various limits. The non-trivial 
$4D$ part of the metric , with $S^3$ foliations instead of $AdS_3$,  
was studied before by Pedersen \cite{pedersen}. 
Staticity, ``spherical symmetry'' and diffeomorphism invariance 
leave two independent functions in the metric, best parametrized
as above. For various limits, we should get: 
\be
M=0  \rightarrow  H(r)= V(r) = 1 \hskip 1 cm  \Lambda =0 \rightarrow 
H(r)=1 ,\hskip 0.5 cm  V(r) = 1 +{4 M\over r}
\ee
Equations of motion can be simplified by observing that for non-zero 
$\Lambda$, $r \rightarrow  \sinh(r L)/L$ in the magnetic field.  
Therefore, one can take the ansatz:
\be
V(r) = 1 +{4 M L\over \sinh(r L)}\; .
\ee
Then the remaining function $H(r)$ 
can be determined from the equations of motion:
\be
H(r) =  {1\over {1 - 4 M L \sinh(r L)}}\; .
\ee
Thus, the $D=6$ AdS KK monopole metric is:
\be
ds^2&=&-\exp{(- 2 L x^6)} dt^2 + dx_6^2\nonumber \\ 
&&+{{3/2}\over ( 1- 4M L \sinh(r L) )^2} 
\Bigg\{\left[ 1+ {{4M L\over \sinh(r L)} }\right]
\left( dr^2 + (1/L^2)\sinh^2(r L)d\Omega_2^2\right) \nonumber \\ 
&&+\cosh^2(r L)
{\left[ 1+ {4M L \over \sinh(r L)}\right]}^{-1} \left[ dx^5 + 
4M\left(1- \cos \theta \right)d\phi\right]^2 \Bigg\}
\label{fullmetric}\; .
\ee
The monopole is located at  $r= 0$, where the $4D$ part of the metric 
reduces to the singular flat Taub-NUT space. Again, to get rid of the 
singularity at the origin, $x^5$ should be compactified on a 
circle with radius $8M$.
Because of its direct product structure, 
there are two disconnected boundaries of the above metric: One at 
$x^6 = - \infty$ and the other at  $L r = \sinh^{-1}( 1/4ML)$. [ 
$x^6 = + \infty$ is a Killing horizon.] The first boundary 
is the usual-trivial boundary of $AdS_2$.
In $r$-coordinates, the latter boundary is at a finite distance but
can be moved to 'infinity' by  
transforming the metric into the new coordinates: $\tilde{r} := \int dr
H(r)\sqrt{V(r)}$. For  $r \in [0, (1/L)\sinh^{-1}( 1/4ML)]$, 
$\tilde{r} \in [0, \infty]$.
Only ``light'' can travel to the boundary at a finite time.
Strictly speaking for $4D$ part alone, 
there is of course no light-like geodesic. But to understand 
the boundary let us set $\phi = \theta = \mbox{constant}$ 
and set $ds^2 =0$ to find the Euclidean time: 
$ ix^5 = \int_\epsilon^{(1/L)\sinh^{-1}( 1/4ML)}
dr \cosh( rL ) V(r) = \mbox{finite}$,
where $\epsilon > 0$. [ Note that if the light starts its journey 
at $r=0$, it would take an infinite time to reach the boundary, since 
$r= 0$ is a Killing horizon.] Thus, the metric (\ref{fullmetric}) 
describes a non-singular 
geodesically complete space. Another way to see the smoothness of the  
metric 
is to check whether curvature invariants are regular or not. 
It suffices to look at $4D$ part of the metric. 
Among the real invariants \cite{Carminati, Zakhary}, apart 
from the obvious ones like the Ricci scalar, $R_{\mu \nu}R^{\mu \nu}$ 
{\it{etc.}},  the only non-vanishing invariant are the Weyl scalars:
\be
\frac{1}{8}C_{\mu \nu \sigma \rho }
C^{\mu \nu \sigma \rho}\!\!\!\!&=&\!\!\!\!\frac{1}{8}
C^{*}_{\mu \nu \sigma \rho }C^{\mu \nu \sigma \rho}=
\frac{64
M^2\, L^6}{3}\left(\frac{1-4ML\sinh{(Lr)}}{4ML+\sinh{(Lr)}}\right)^6\; ,
\nonumber\\
-\frac{1}{16}C_{\mu \nu }\,^{\sigma \rho }C_{\sigma \rho }\,
^{\lambda \eta }C_{\lambda \eta }\,^{\mu \nu}\!\!\!\!&=&\!\!\!\!
-\frac{1}{16}C^{*}_{\mu \nu}\,^{\sigma \rho }C_{\sigma \rho }\,
^{\tau \eta }C_{\tau \eta }\,^{\mu \nu}=\frac{512\,
M^3\,L^9}{9}\left(\frac{-1+4ML\sinh{(Lr)}}{4ML+\sinh{(Lr)}}\right)^9
\ee
where $C_{\mu\nu\sigma\rho}$ is the the Weyl tensor  and its dual is :
$C^*_{\mu\nu\sigma\rho} := \frac{1}{2}
\epsilon_{\mu\nu\lambda\eta}C^{\lambda\eta}\,_{\sigma\rho}$. None of the
invariants are
singular. 

Finally, let us say a few words about  
the energy of the metric ({\ref{fullmetric}}).
The gravitational energy of the flat space KK monopole
was studied in \cite{sorkin2, soldate}. As often the case
for the gravitational systems with compact dimensions,
a useful energy definition is rather tricky because of the
complications in choosing a proper background metric with 
respect to which the energy should be defined.
The total gravitational energy of a given metric makes sense,
if it can be considered as a `particle'
or a 'perturbation' in a given background or vacuum. The 
background, by definition, solves
the equations of motion everywhere and it has zero energy.
The topology of the background metric ought to be the same as the
metric whose energy is measured.  
The flat space KK monopole is a perfectly smooth 
solution and there is no other one, with the same topology,
whose energy can be compared with the monopole.
Therefore KK monopole, considered  as {\it{the}} vacuum, 
has trivially zero energy.
We should note that in \cite{sorkin2,soldate}, the
chosen background metrics are {\it{not}} solutions everywhere. 
They are {\it{only}} asymptotic solutions.
Our metric ({\ref{fullmetric}}), being a smooth solution everywhere 
also has zero energy. This being a rather
boring result, one might, like
\cite{sorkin2,soldate}, relax the condition on the background and
take an asymptotic solution as a background. Here, we do not go into
details but sketch a heuristic argument that the AdS KK monopole's energy
expression is similar to the flat space one.
Following Abbott and Deser (AD) \cite{abbott}~\footnote{See also {\cite{dt}}},
let us choose $\bar{g}_{\mu \nu}$ as the background for which $V(r) =1$ 
in (\ref{fullmetric}),  and  $h_{\mu \nu}$
as the perturbation part, defined as  $ h_{\mu \nu} :=  g_{\mu \nu}
- \bar{g}_{\mu \nu}$. Then, $O(h)$ terms in the equations of motion:
is defined as the background conserved energy momentum tensor
\be
R_{\mu \nu}^L - {1\over 2} \bar{g}_{\mu \nu} R^L - 2\Lambda h_{\mu \nu}
:= T_{\mu \nu } (h),
\ee
where $L$ denotes linearization. Given ${\bar{\xi}^{\mu}}$ as a 
background Killing vector, an ordinarily conserved current 
can be constructed: $\bar{\nabla}_{\mu}( \sqrt{-\bar{g}}\, T^{\mu \nu }
\bar{\xi}_\nu ) = 
\partial_{\mu}( \sqrt{-\bar{g}}\, T^{\mu \nu }\bar{\xi}_\nu )$. 
Using this, we write the AD Killing energy, up to trivial
numerical factors, as
 E= $\int d^5 x \sqrt{-\bar{g}}\,T^{0 \nu}\bar{\xi}_\nu $. For
the metric (\ref{fullmetric}), $\bar{\xi}_\nu = (1, {\bf{0}})$
and so  $E =  \int d^5 x \sqrt{-\bar{g}}\, T^{0 0} $. Since
the AdS KK monopole is an Einstein space,
$R^L = 0 \rightarrow R_{\mu \nu}^L = 2 \Lambda h_{\mu \nu}$ and thus
$T^{00} = 0$ everywhere, except at $r=0$, where derivatives give
$\delta$-function singularities. Therefore, all the contributions to
the energy volume integral come from the origin. We already 
know that the non-trivial part of the metric $(\ref{fullmetric})$
reduces to the flat-space KK monopole metric around the origin. Thus,
the AdS KK monopole 
has the same energy expression as the flat one, which was
obtained to be $M$ in \cite{sorkin2, soldate}.



{\bf{ A 6D KK Monopole Brane-World}}

Now that we have six dimensions at our disposal, we can construct more 
than one kind of soliton. Here, we relax our earlier condition that 
the low energy spacetime has a negative cosmological constant.   
For example, using the five dimensional flat Kaluza-Klein monopole,we 
can construct a six dimensional soliton:
\be
ds^2= \exp{( 2 k L x^6)}\left \{-dt^2+ ds^2_{\mbox{{\scriptsize TN}}}
\right \} +dx_6^2\; ,
\label{example1}
\ee
which satisfies $R_{MN}=-5 k^2L^2g_{MN}$.
Multi-monopoles with equal charges, located at 
different spatial points can be constructed as in the flat space case.  
$x^5$ direction  has to be compact for the solution to be smooth but 
the $x^6$ direction has infinite extent. Generalization to  $D$ 
dimensions is straightforward, since one can add more flat directions 
as long as they are multiplied with $\exp{(2k Lx^6)}$.

Let us choose $k=-1$ for definiteness. 
Then, $x^6 = -\infty$ is the boundary of the 
space and $x^6 = + \infty $ is the Killing horizon. In this form,
all dimensions (except the compact $x^5$ directions) are accessible to
low energy observers.  To localize gravity, and get an effective 
$3+1$ dimensional brane-world, 
we can cut and paste (\ref{example1}) into the the Randall-Sundrum 
$Z_2$ invariant form \cite{randall}: 
\be
ds^2=\exp{(- 2 L|x^6|)}\left \{-dt^2+
ds^2_{\mbox{{\scriptsize TN}}}
\right \} +dx_6^2\; .
\label{example2}
\ee
This metric is not differentiable at the 'origin', where
the brane ($D4$ Taub-NUT brane ) is located.  
Physically, $3+1$ dimensional 
observers see a flat-space KK monopole. The metric ({\ref{example2}) 
is a six dimensional brane world with gravity localized 
on a Ricci-flat brane.

\section{Conclusions}

We have shown that five dimensional cosmological 
Einstein gravity (with a Lorentzian 
signature and a negative cosmological constant ) does not have Kaluza-Klein 
monopole type static soliton solutions. On the other hand in $D \ge 6$, 
we have constructed analogs of the flat space KK monopoles which are 
asymptotically locally AdS space-times. It would be interesting 
to find out if these solutions preserve some supersymmetry and if
they can be embedded in string or M-theory. In the presence of 
anti-symmetric tensor fields, in addition to the usual translational 
collective degrees of freedom, the flat space KK monopole 
($D5$ or $D6$ brane ) can 
have dyonic deformations \cite{sen}. 
An extension of these ideas to the AdS KK monopole would also be worth 
studying.

In this paper, we have insisted that the solitons 
be static. If we relax this condition,
we can find  time-dependent $4+1 D$ or Euclidean $5D$ solutions.
One such (Euclidean ) example with Ricci-flat 4D slices would be:
\be
ds^2 =  dt^2 + \exp{(- 2tL)}\, ds^2_{\mbox{{\scriptsize TN}}}\; .
\ee
Another example, with negatively curved $4D$ slices is:
\be
ds^2 = - dt^2+{\sin^2(t L)\over ( 1- 4M L \sinh(r L) )^2}
\Bigg\{\left[ 1+ {{4M L\over \sinh(r L)} }\right]
\left( dr^2 + (1/L^2)\sinh^2(r L)d\Omega_2^2\right) \nonumber \\
+\cosh^2(r L)
{\left[ 1+ {4M L \over \sinh(r L)}\right]}^{-1} \left[ dx^5 +
4M\left(1- \cos \theta \right)d\phi\right]^2 \Bigg\}\; .
\ee

Finally, the KK monopole in AdS has a disconnected boundary. Thus, 
presumably, it is not suitable for the usual AdS/CFT applications, 
which require  a simply connected boundary \cite{yau}.  

{\bf{Acknowledgments}}

Authors would like to thank S. Deser, A. Lawrence, P. Ramond, 
P. Sikivie, R. P. Woodard and N. Wyllard for useful discussions, 
encouragement and support. The work of B.T. 
is supported by NSF grant PHY99-73935. V. K. O. is supported by 
DOE grant DE-FG02-97ER41029.       

\vskip 1cm

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\myend



