Killing Reduction of 5-Dimensional Spacetimes
Jianbing Shao
and Wei Zhou
Department of Physics
Beijing 100875
CHINA
In a 5-dimensional spacetime ($M,g_{ab}$) with a Killing vector field $\xi ^a$ which is either everywhere timelike or everywhere spacelike, the collection of all trajectories of $\xi ^a$ gives a 4-dimensional space $S$. The reduction of ($M,g_{ab}$) is studied in the geometric language, which is a generalization of Geroch's method for the reduction of 4-dimensional spacetime. A 4-dimensional gravity coupled to a vector field and a scalar field on $S$ is obtained by the reduction of vacuum Einstein's equations on $M$ , which gives also an alternative description of the 5-dimensional Kaluza-Klein theory. Besides the symmetry-reduced action from the Hilbert action on $M$, an alternative action of the fields on $S$ is also obtained, the variations of which lead to the same fields equations as those reduced from the vacuum Einstein equation on $M$.
gr-qc0307061.tex

\documentclass[preprint,showpacs,preprintnumbers,amsmath,amssymb]{article}

\usepackage{graphicx}
\usepackage{dcolumn}
\usepackage{bm}

\begin{document}

\title{Killing Reduction of 5-Dimensional Spacetimes}

\author{Xuejun Yang\footnote{yang\_xue\_jun@sohu.com},
Yongge Ma\footnote{yonggema@yahoo.com}, Jianbing Shao, and Wei Zhou\\
\small Department of Physics, Beijing Normal University, Beijing
100875, CHINA}

\date{\today}

\maketitle

\begin{abstract}
In a 5-dimensional spacetime ($M,g_{ab}$) with a Killing vector
field $\xi ^a$ which is either everywhere timelike or everywhere
spacelike, the collection of all trajectories of $\xi ^a$ gives a
4-dimensional space $S$. The reduction of ($M,g_{ab}$) is studied
in the geometric language, which is a generalization of Geroch's
method for the reduction of 4-dimensional spacetime. A
4-dimensional gravity coupled to a vector field and a scalar field
on $S$ is obtained by the reduction of vacuum Einstein's equations
on $M$ , which gives also an alternative description of the
5-dimensional Kaluza-Klein theory. Besides the symmetry-reduced
action from the Hilbert action on $M$, an alternative action of
the fields on $S$ is also obtained, the variations of which lead
to the same fields equations as those reduced from the vacuum
Einstein equation on $M$.

\end{abstract}

{PACS number(s): 04.50.+h, 04.20.Fy}

\section{Introduction}

   Spacetime reduction is very important in any high dimensional
theory of physics such as Kaluza-Klein
theory\cite{Duff}-\cite{Darabi}, high dimensional theory of
gravity\cite{Overduin} \cite{Weinberg} and string
theory\cite{Polchinski}-\cite{Yega}. The dimensional reduction can
make a high dimensional theory contact with the 4-dimensional
sensational world. It is also a useful approach to study
spacetimes with symmetries. The original Kaluza-Klein theory
unifies electromagnetic and gravitational interactions in four
dimensions by a 5-dimensional spacetime. Since the theory was
first proposed by Kaluza\cite{Kaluza}, it has been studied by a
large number of authors\cite{Duff}-\cite{Darabi} and also extended
to higher dimensions in order to give rise to a unification of
gravity with non-Abelian gauge theories\cite{Appelquist}. Along
with the successful description of the electroweak and strong
interactions as gauge theories, the K-K approach may serve as a
framework for studying their unification with gravity. Also using
the idea of supergravity in K-K theory, a geometric description of
both gauge fields and spinor matter is possible.

In ref.\cite{Geroch}, Geroch introduced a Killing reduction
formalism of 4-dimensional spacetime. Let ($\mathcal{M},g_{ab}$)
be a 4-dimensional spacetime with Killing vector field $\xi^a$
which is either everywhere timelike or everywhere spacelike. The
collection of all trajectories of $\xi^{a}$ gives a 3-dimensional
space $\Sigma$.
 The discussion of Geroch shows that there is a one-to-one
correspondence between tensor fields and tensor operations on
$\Sigma$ and the certain tensor fields and tensor operations on
$\mathcal{M}$. The differential geometry of $\Sigma$ will, in this
sense, be mirrored in $\mathcal{M}$. Geroch gives the relations of
geometric properties between $\Sigma$ and $\mathcal{M}$ and then
obtains the field equations on $\Sigma$, which describes a
3-dimensional gravity coupled to two massless scalar fields and is
equivalent to the vacuum Einstein equations on $\mathcal{M}$.

In this paper we extend Geroch's approach to a 5-dimensional
spacetime ($M,g_{ab}$) with a Killing vector field in order to
study its reduction. We first obtain a series of equations on $S$,
which is also the collection of all trajectories of $\xi^a$,
parallel to Geroch's. The results show that 5-dimensional gravity
with a Killing vector is equivalent to 4-dimensional gravity
$h_{ab}$ coupled to a vector field $A_a$ and a scalar field
$\lambda$, and hence it is consistent with the conclusion of the
Kaluza-Klein theory. We then study the reduction from the
viewpoint of variation principle. It turns out that the
symmetry-reduced action from the Hilbert action on $M$ would give
the correct reduced field equations only if one used $B_a$, which
has a trivial relation with the 5-metric components, rather than
$A_a$ as one of the arguments. Finally, we propose another
4-dimensional action on $S$. Its variations with respect to
$h^{ab}$, $\lambda$ and $A_a$ can give the same reduced field
equations.

\section{Symmetric Reduction of 5-Dimensional Spacetime}
 Let ($M,g_{ab}$) be a n-dimensional spacetime with a Killing vector field $\xi
^a$, which is either everywhere timelike or everywhere spacelike.
Let $S$ denote the collection of all trajectories of $\xi^a$. A
map $\psi$ from $M$ to $S$ can be defined as follows: For each
point p of $M$, $\psi(p)$ is the trajectory of $\xi^a$ passing
through p. Assume $S$ is given the structure of a differentiable
(n-1)-manifold such that $\psi$ is a smooth mapping. It is natural
to regard $S$ as a quotient space of $M$. The proof of Geroch
about the following conclusion is independent of the dimension of
$M$ \cite{Tavanfar}: There is a one-to-one correspondence between
tensor fields $\tilde{T}_{a...c}^{b...d}$ on $S$ and tensor fields
$T_{a...c}^{b...d}$ on $M$ which satisfy

\begin{equation}
\begin{array}{l}
\xi^a T^{b\cdots d}_{a\cdots c}=0,\ \cdots \ ,\xi_d T^{b\cdots
d}_{a\cdots
  c}=0 ,\\
{\mathcal{L}}_\xi T^{b\cdots d}_{a\cdots c}=0 . \label{Li D}
\end{array}
\end{equation}
The entire tensor field algebra on $S$ is completely and uniquely
mirrored by tensor field on $M$ subject to (\ref{Li D}). Thus, we
shall speak of tensor fields being on $S$ merely as a shorthand
way of saying that the fields on $M$ satisfy (\ref{Li D}).

The metric, inverse metric and the Kronecker delta on $S$ are
defined as
\begin{equation}
h_{ab}=g_{ab}-\lambda ^{-1}\xi _a\xi_b, \label{h_{ab}}
\end{equation}

\begin{equation}
h^{ab}=g^{ab}-\lambda ^{-1}\xi ^a\xi ^b,
 \label{h^{ab}}
\end{equation}

\begin{equation}
h_a^b=\delta _a^b-\lambda ^{-1}\xi _a\xi ^b,
 \label{h_a^b}
\end{equation}
where $\lambda \equiv\xi ^a\xi _a$. (\ref{h_a^b}) can also be
interpreted as the projection operator onto $S$. The covariant
derivative on $S$ is defined by
\begin{equation}
D_eT_{a\cdots c}^{b\cdots d}=h_e^ph_a^m\cdots h_c^nh_r^b\cdots
h_s^d\nabla _pT_{m\cdots n}^{r\cdots s},\label{D T}
\end{equation}
where $\nabla _p$ is the covariant derivative associated with the
metric $ g_{ab}$ on $M$ and $T_{a\cdots c}^{b\cdots d}$ is any
tensor field on $S$. Note that $D_e$ satisfies all the conditions
of a derivative operator and $D_c h_{ab}=0$.

We now consider the special case where n=5. It can be shown that
$\varepsilon _{abcd}\equiv |\lambda|^{-\frac{1}{2} }\varepsilon
_{abcde}\xi ^e$ is the volume element associated with the metric
$h_{ab}$ on $S$, i.e.,

\begin{equation}
D_f\varepsilon_{abcd}=0, \label{silon}
\end{equation}
here $\varepsilon _{abcde}$ is the volume element associated with
the metric $ g_{ab}$ on $M$, i.e.,
 $\nabla _f\varepsilon
_{abcde}=0.$ We define the twist 2-form $\omega_{ab}$ of $\xi ^a$
by
\begin{equation}
\omega_{ab}:=\varepsilon _{abcde}\xi^c\nabla ^d\xi^e.
\label{omega}
\end{equation}
Clearly we have $\omega_{ab}=\omega_{[ab]}$ and $\omega_{ab}\xi
^a=0$. It is also easy to see that
\begin{equation}
\mathcal{L} _\xi \lambda =0,\hspace{1cm} \mathcal{L} _\xi
\omega_{ab}=0. \label{Li lambda}
\end{equation}
Hence, $\lambda $ and $\omega_{ab}$ are fields on $S$.

Lengthy but straightforward calculations lead to the following
results.

1) The Riemann tensor $R_{abcd}$ of ($S,h_{ab}$) is related to the
Riemann tensor $ \mathcal{R}_{abcd}$ of ($M,g_{ab}$) by
\begin{equation}
R_{abcd}=h_{[a}^ph_{b]}^qh_{[c}^rh_{d]}^s[\mathcal{R}_{pqrs}+2\lambda
^{-1}(\nabla _p\xi _q)(\nabla _r\xi_s)+2\lambda ^{-1}(\nabla
_p\xi_r)(\nabla _q\xi_s)]; \label{R abcd}
\end{equation}

2)The derivative of the Killing vector reads
\begin{equation}
\nabla _a\xi _b=-\frac {1}{4}\lambda ^{-1}\varepsilon _{abcde}\xi
^c\omega^{de}+\lambda ^{-1}\xi _{[b}D_{a]}\lambda ;\label{nablaxi}
\end{equation}

3)The second derivative reads
\begin{equation}
\nabla _a\nabla _b\xi _c=\mathcal{R}_{dabc}\xi ^d;
\label{nablanablaxi}
\end{equation}

4) Contracting (\ref{R abcd}) and using (\ref{nablaxi}) and
(\ref{nablanablaxi}), the relation between the Ricci tensor
$R_{ab}$ of ($S,h_{ab}$) and the Ricci tensor $\mathcal{R}_{ab}$
of ($M,g_{ab}$) can be obtained as
\begin{equation}
R_{ab}=\frac{1}{2} \lambda ^{-2}(\omega_a^m\omega_{bm}-\frac{1}{2}
h_{ab}\omega_{mn}\omega^{mn})+\frac{1}{2}\lambda
^{-1}D_aD_b\lambda -\frac{1}{4}\lambda ^{-2}(D_a\lambda)
D_b\lambda +h_a^mh_b^n\mathcal{R}_{mn} ;\label{R_{ab}}
\end{equation}

5) Taking the curl and divergence of (\ref{omega}) and using
(\ref{nablaxi}) and (\ref{nablanablaxi}), one can get
\begin{equation}
D_{[a}\omega_{bc]}=\frac{2}{3}\varepsilon _{abcmn}\xi
^m\mathcal{R}_p^{n}\xi ^p, \label{Domega}
\end{equation}
and
\begin{equation}
D^a\omega_{ab}=\frac{3}{2}\lambda ^{-1}\omega_{mb}D^m\lambda;
\label{Daaomega}
\end{equation}

6) Using (\ref{nablaxi}) and (\ref{nablanablaxi}) again, we have
\begin{equation}
D^2\lambda =\frac 12\lambda ^{-1}(D^a\lambda) D_a\lambda -\frac
12\lambda ^{-1}\omega^{ab}\omega_{ab}-2\mathcal{R}_{mn}\xi ^m\xi
^n, \label{D^2lambda}
\end{equation}
where $D^2\equiv D^aD_a$;

7) Contracting (\ref{R_{ab}}) by $h^{ab}$ and using
(\ref{D^2lambda}), we obtain
\begin{equation}
R=-\frac 14\lambda ^{-2}\omega_{mn}\omega^{mn}+\lambda
^{-1}D^2\lambda -\frac 12\lambda ^{-2}(D^a\lambda) D_a\lambda
+\mathcal{R}, \label{R}
\end{equation}
here, $R$ and $\mathcal{R}$ are the scalar curvature of
($S,h_{ab}$) and ($M,g_{ab}$) respectively.

Thus, the basic equations for a 5-dimensional spacetime with a
Killing vector field are a set of differential equations on three
variables, the metric $h^{ab}$, the norm $\lambda $ and the twist
2-form $\omega_{ab}$ of the Killing vector. They
 are formulated as (\ref{R_{ab}}), (\ref{Domega}),
(\ref{Daaomega}) and (\ref{D^2lambda}).

 When the spacetime ($M,g_{ab}$) is source free ($\mathcal{R}_{ab}=0$),
(\ref{Domega}) implies $(d\omega)_{abc}$ on $S$ is zero. Hence at
least locally there is a 1-form $A_a$ on $S$ such that
$\omega_{ab}=(dA)_{ab} \equiv 2D_{[a}A_{b]}$. Then the equations
(\ref{R_{ab}})-(\ref{D^2lambda}) become

\begin{equation}
\begin{array}{lll}
R_{ab}&=&\frac 12\lambda ^{-2}[(dA)_a^m(dA)_{bm}-\frac
12h_{ab}(dA)_{mn}(dA)^{mn}]\\
 &&+\frac 12\lambda
^{-1}D_aD_b\lambda -\frac 14\lambda ^{-2}(D_a\lambda) D_b\lambda;
 \label{R_{ab}1}
\end{array}
\end{equation}

\begin{equation}
D^a(dA)_{ab}=\frac 32\lambda ^{-1}(dA)_{mb}D^m\lambda;
\label{D(ab)}
\end{equation}

\begin{equation}
D^2\lambda =\frac 12\lambda ^{-1}(D^a\lambda) D_a\lambda -\frac
12\lambda ^{-1}(dA)_{ab}(dA)^{ab}. \label{D^2lambda1}
\end{equation}
Equations (\ref{R_{ab}1})-(\ref{D^2lambda1}) describe a
4-dimensional gravity coupled to a vector field and a scalar
field, which are equivalent to 5-dimensional vacuum Einstein's
equation with a Killing symmetry. When the Killing vector field is
spacelike, it coincides with the conclusion of the 5-dimensional
Kaluza-Klein theory. Thus, by reducing Einstein's equation on $M$
by an extension of  Geroch's method, we can provide an alternative
description of the Kaluza-Klein theory in 4+1 dimensions.

\section{Symmetry-Reduced Hilbert Action on $S$ }

For practical calculations, it is convenient to take a coordinate
system adapted to the congruence:

\begin{equation}
(\frac \partial {\partial x^5})^a=\xi ^a .\label{x5}
\end{equation}
From (\ref{h_{ab}}), we have

\begin{equation}
g_{ab}=h_{ab}+\lambda ^{-1}\xi _a\xi _b. \label{g_{ab}h_{ab}}
\end{equation}
The components of (\ref{g_{ab}h_{ab}}) are

\begin{equation}
g_{\mu\nu} =h_{\mu\nu}+\lambda ^{-1}\xi _{\mu}\xi
_{\nu},\hspace{1cm} \mu, \nu =1,...,5, \label{g_{munu}}
\end{equation}
Particularly one has
\begin{equation}
g_{\mu 5}=g_{ab}(\frac \partial {\partial x^{\mu}})^a\xi ^b=(\frac
\partial
{\partial x^{\mu}})^a\xi _a=\xi _{\mu}\label{g_{mu5}}
\end{equation}
and
\begin{equation}
g_{55}=g_{ab}\xi ^a\xi ^b=(\frac \partial {\partial x^5})^a\xi
_a=\xi _5=\lambda. \label{g_{55}}
\end{equation}
Let
\begin{equation}
B_a^{\prime }=\lambda ^{-1}\xi _a-(dx^5)_a, \label{Bprime}
\end{equation}
then $\xi ^aB_a^{\prime }=0$. One can also prove that $\mathcal{L}
_\xi B_a^{\prime }=0$, hence $B_a^{\prime }$ is a 1-form on $S$.
Note that $B_a^{\prime }$ is dependent of the coordinate system
chosen. Using (\ref{nablaxi}) and (\ref{Bprime}), it can be shown
that
\begin{equation}
(dB^{\prime })_{ba}\equiv 2D_{[b}B_{a]}^{\prime }=\frac 12|
\lambda |^{-\frac 32}\varepsilon _{abcd}\omega^{cd}\equiv
\bar{\omega}_{ba}. \label{dBprime}
\end{equation}
Eq. (\ref{dBprime}) means $(d\bar{\omega})_{cba}=0$, hence there
is at least locally a one-form $B_a$ on $S$ such that
$\bar{\omega}_{ba}=(dB)_{ba}$. This definition of $B_a$ is
independent of any coordinate system and hence purely geometric.
Thus, we obtain
\begin{equation}
\begin{array}{lll}
(dB)_{ba}&=&\frac 12|\lambda|^{-\frac 32}\varepsilon
_{abcd}\omega^{cd},\\
\hspace{0.5cm}\omega_{ab}&=&\pm \frac 12|\lambda|^{\frac
32}\varepsilon _{abcd}(dB)^{cd}.
\end{array}
\label{dB}
\end{equation}
Hereafter, when $\lambda >0$ or $\lambda <0$, the sign ''$\pm $''
means ''+'' or ''-'' respectively. The Hilbert action on $M$ reads
\begin{equation}
S[g^{ab}]=\int_M\sqrt{-g}\mathcal{R} .\label{Sg}
\end{equation}
Since the principle of symmetric criticality is valid in our one
Killing vector model \cite{Fels}, one expects that the reduced
field Eqs. (\ref{R_{ab}1})-(\ref{D^2lambda1}) could be obtained by
the variation of the action from the symmetric reduction of
(\ref{Sg}). Using (\ref{g_{munu}}), we obtain
\begin{equation}
g=\lambda h, \label{gh}
\end{equation}
where $g$ and $h$ are respectively the determinants of components
$g_{\mu\nu} (\mu, \nu=1,2,...5)$ and $h_{\mu\nu} (\mu,
\nu=1...4)$. Using (\ref{R}) and (\ref{gh}), the action (\ref{Sg})
on $M$ is reduced to the following action on $S$ up to a boundary
term

\begin{equation}
S[h^{ab},\lambda ,\omega_{ab}]=\int_S|\lambda |^{\frac
12}\sqrt{|h|}[ R +\frac 14\lambda ^{-2}w_{mn}w^{mn}].
\label{Shomega}
\end{equation}
The variation of action (\ref{Shomega}) with respect to ($h^{ab}$,
$\lambda$, $\omega_{mn}$) can not give the correct reduced field
equations. In order to get those equations, one has to use the
5-metric components rather than $\omega_{mn}$ as the arguments of
the reduced action. This subtlety exists also in the symmetric
reduction of 4-dimensional spacetimes\cite{HeMaYang}.

From (\ref{dB}), we have
\begin{equation}
\omega^{mn}\omega_{mn}=-\lambda ^3(dB)_{ab}(dB)^{ab}.
\label{omega^{mn}omega_{mn}}
\end{equation}
Substituting (\ref{omega^{mn}omega_{mn}}) into (\ref{Shomega}), we
obtain the action on $S$ in terms of basic variables $h^{ab}$,
$\lambda $ and $B_a$ as
\begin{equation}
S[h^{ab},\lambda ,B_a]=\int_S|\lambda|^{\frac 12}\sqrt{|
h|}[R-\frac 14\lambda (dB)_{ab}(dB)^{ab}], \label{ShB}
\end{equation}
which has the same form as the reduced action in 5-dimensional K-K
theory. The variations of this action with respect to ($h^{ab}$,
$\lambda$, $B_c$) will give the correct reduced fields equations
on $S$.
 Especially, the variation of action (\ref{ShB})
with respect to $B_a$ gives
\begin{equation}
D^c(dB)_{ac}=-\frac 32\lambda ^{-1}(D^c\lambda) (dB)_{ac}.
\label{DdB}
\end{equation}
Substituting (\ref{dB}) into (\ref{DdB}), we get
\begin{equation}
D_{[a}\omega_{bc]}=0 \label{Dabcomega}.
\end{equation}
Eq. (\ref{Dabcomega}) means that there is at least locally a
1-form $A_a$ on $S$ such that
\begin{equation}
\omega_{ab}=(dA)_{ab}. \label{omega1}
\end{equation}
Substituting $A_a$ for $B_a$ in Eq. (\ref{DdB}) through Eqs.
(\ref{omega1}) and (\ref{dB}), we obtain Eq. (\ref{D(ab)}). Eqs.
(\ref{R_{ab}1}) and (\ref{D^2lambda1}) can also be obtained by
substituting $A_a$ for $B_a$ after the corresponding variations.

\section{An Alternative Action on $S$ }

If one wants to take $h^{ab}$, $\lambda $ and $A_a$ as basic
variables, action (\ref{Shomega}) fails to be the right action.
Another action is thus needed. When the Killing field $\xi ^a$ is
hypersurface orthogonal, from Eqs. (\ref{g_{mu5}}),
(\ref{g_{55}}), (\ref{Bprime}), (\ref{dBprime}) and (\ref{R}), we
have

\begin{equation}
\omega_{ab}=0 , \label{omega2}
\end{equation}

\begin{equation}
\mathcal{R}=R-\lambda ^{-1}h^{ab}D_aD_b\lambda +\frac 12\lambda
^{-2}h^{ab}(D_a\lambda) D_b\lambda .\label{tildeR5}
\end{equation}
In this case, up to a boundary term the Hilbert action (\ref{Sg})
on $M$ is reduced on $S$ as \cite{Fels}

\begin{equation}
S[h^{ab},\lambda ]=\int_S|\lambda|^{\frac 12}\sqrt{|h|}R.
\label{Shlambda}
\end{equation}
To obtain a regular form of action on $S$, we conformally
transform $h^{ab}$ as
\begin{equation}
\tilde{h}_{ab}=\Omega ^{-2}h_{ab}. \label{tildeh}
\end{equation}
Let $\tilde{D}_a$ be the covariant derivative operator determined
by metric $\tilde{h}_{ab}$, i.e.,
\begin{equation}
\tilde{D}_a\tilde{h}_{bc}=0. \label{tildeDtildeh}
\end{equation}
The relation of $\tilde{D}_a$ and $D_a$ is \cite{Wald}
\begin{equation}
\begin{array}{lll}
D_a v_b =\tilde{D}_a v_b-C_{ab}^c
v_c,\hspace{1cm}\forall\hspace{0.3cm} v_a \in \mathcal{F} _S(0,1),
\end{array}
\label{Dv}
\end{equation}
where
\begin{equation}
\begin{array}{lll}
C_{ab}^c &=&\frac
12h^{cd}(\tilde{D}_ah_{bd}+\tilde{D}_bh_{ad}-\tilde{D}_dh_{ab})
\\
&=&\tilde{h}_b^c\tilde{D}_a\ln \Omega +\tilde{h}_a^c\tilde{D}_b\ln
\Omega -\tilde{h} _{ab}\tilde{h}^{cd}\tilde{D}_d\ln \Omega.
\end{array}
\label{C_{ab}^c}
\end{equation}
Let
\begin{equation}
\Omega =|\lambda|^{-\frac14}. \label{Omega1}
\end{equation}
Ignoring the boundary term (\ref{Shlambda}) becomes
\begin{equation}
S[\tilde{h}^{ab},\lambda
]=\int_S\sqrt{|\tilde{h}|}[\tilde{R}-\frac 38\lambda ^{-2}
\tilde{h}^{ab}(\tilde{D}_a\lambda) \tilde{D}_b\lambda ].
\label{Stildehlambda1}
\end{equation}
Let $\Lambda =\sqrt{6}\ln \Omega $, then (\ref{Stildehlambda1})
becomes
\begin{equation}
S[\tilde{h}^{ab},\Lambda
]=\int_S\sqrt{|\tilde{h}|}[\tilde{R}-\tilde{h}^{ab}(\tilde{D}
_a\Lambda) \tilde{D}_b\Lambda ].\label{StildehLambda}
\end{equation}
This is the action of a 4-gravity $\tilde{h}^{ab}$ coupled to a
massless scalar field. Thus in the source-free case, a
5-dimensional spacetime with a hypersurface orthogonal Killing
vector field which is either everywhere timelike or everywhere
spacelike is "comformally" equivalent to 4-dimensional gravity
coupled to a massless Klein-Gordon field.

In general case, from Eqs. (\ref{R_{ab}1})-(\ref{D^2lambda1}) we
know that besides a scalar field $ \lambda $, the 4-gravity
couples also to a vector field $A_a$ on $S$. By carefully
observing the reduced field equations and the special action
(\ref{Stildehlambda1}), we suppose the following action being the
one we are looking for,
\begin{equation}
S[\tilde{h}^{ab},\lambda
,A_a]=\int_S\sqrt{|\tilde{h}|}[\tilde{R}-\frac 38\lambda
^{-2}\tilde{h}^{ab}(\tilde{D}_a\lambda) \tilde{D}_b\lambda -\frac
14|\lambda|^{-
\frac32}\tilde{h}^{ac}\tilde{h}^{bd}(dA)_{ab}(dA)_{cd}].
\label{ShA}
\end{equation}
Varying the action (\ref{ShA}) with respect to $\tilde{h}^{ab}$,
$\lambda $ and $A_a$ respectively, we get
\begin{equation}
\begin{array}{lll}
\tilde{R}_{ab}&=&\frac 38\lambda ^{-2}(\tilde{D}_a\lambda)
\tilde{D}_b\lambda -\frac 18|\lambda
|^{-\frac32}\tilde{h}_{ab}\tilde{h}^{ec}\tilde{h}
^{fd}(dA)_{ef}(dA)_{cd}\\
&&+\frac 12|\lambda|^{-\frac32}\tilde{h} ^{cd}(dA)_{ac}(dA)_{bd};
\end{array}
\label{tildeRab1}
\end{equation}

\begin{equation}
\tilde{h}^{ab}\tilde{D}_a\tilde{D}_b\lambda =\lambda
^{-1}\tilde{h}^{ab}(\tilde{D} _a\lambda) \tilde{D}_b\lambda +\frac
12|\lambda|^{-\frac12}\tilde{h}^{ac}\tilde{
h}^{bd}(dA)_{ab}(dA)_{cd}; \label{tildehDaDb}
\end{equation}

\begin{equation}
D^a(dA)_{ab}=\frac 32\lambda ^{-1}(dA)_{ab}D^a\lambda. \label{DdA}
\end{equation}
Making the conformal transformation (\ref{tildeh}) inversely, Eqs.
(\ref{tildeRab1}) and (\ref{tildehDaDb}) become

\begin{equation}
\begin{array}{lll}
R_{ab}&=&\frac 12\lambda ^{-2}[h^{cd}(dA)_{ac}(dA)_{bd}-\frac
12h_{ab}h^{ec}h^{fd}(dA)_{ef}(dA)_{cd}]\\
&&+\frac 12\lambda ^{-1}D_aD_b\lambda -\frac 14\lambda
^{-2}(D_a\lambda) D_b\lambda;
\end{array}
\label{Rab}
\end{equation}

\begin{equation}
D^2\lambda =\frac 12\lambda ^{-1}h^{ab}(D_a\lambda) D_b\lambda
-\frac 12\lambda ^{-1}h^{ac}h^{bd}(dA)_{ab}(dA)_{cd}.
\label{D^2lambda3}
\end{equation}
Eqs. (\ref{Rab}), (\ref{DdA}) and (\ref{D^2lambda3}) are exactly
the same as (\ref{R_{ab}1})-(\ref{D^2lambda1}). Therefore, the
action (\ref{ShA}) is just the one which gives also the right
reduced field equations.

In conclusion, we have studied the symmetric reduction of
5-dimensional spacetime in three hands. First, we obtain
4-dimensional gravity coupled to a vector field and a scalar field
on $S$ by a direct reduction of vacuum Einstein's equation on $M$.
Then, taking $h^{ab}$, $\lambda $ and $B_a$ as basic variables, we
get reduced field equations by varying the symmetry-reduced action
on $S$, which is obtained by the reduction of the Hilbert action
on $M$. Finally, we propose an alternative action on $S$, which
allows us to take $h^{ab}$, $\lambda $ and $A_a$ as basic
variables, and its variations gives also the right reduced field
equations. All discussions are geometrically presented. The scheme
might also be extended to higher-dimensional Kaluza-Klein theories
that attempt to unify gauge theories with gravity.

\section*{ Acknowledgments}

This work is  supported in part by NSFC (10205002), YSRF for ROCS,
SEM, and Young Teachers Foundation of BNU. X. Yang would also like
to acknowledge support from NSFC (10073002).

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\bibitem{Giudice}G. F. Giudice, R. Rattazzi and J. D. Wells, Nucl. Phys. B{\bf 544}, 3(1999).

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\bibitem{Weinberg}S. Weinberg, Phys. Lett. B{\bf 125}, 265(1983).

\bibitem{Polchinski}J. Polchinski, String Theory I, II (Cambridge University
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\bibitem{Antoniadis} I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and
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\bibitem{Yega}H. J. de Vega and N. Sanchez, Phys. Lett. B{\bf 197}, 320(1987).

\bibitem{Kaluza}T. Kaluza, Sitzungsber, Preuss. Akad. Wiss. Phys. Mat. Klasse, 966(1921).

\bibitem{Appelquist}T. Appelquist, A. Chodos and P. G. O. Freund, Modern Kaluza-Klein
Theories, Addison-Wesley Publishing Company, Menlo Park
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\bibitem{Geroch}R. Geroch, Jour. Math. Phys. {\bf 12}, 918(1971).

\bibitem{Tavanfar}M. Nouri-Zonoz and A. R. Tavanfar, JHEP {\bf 0302}, 059
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\bibitem{Fels}M. E. Fels and C. G. Torre, Class. Quant. Grav. {\bf 19}, 641(2002).

\bibitem{HeMaYang} H. He, Y. Ma, and X. Yang, Actions for vacuum Einstein's
equation with a Killing symmetry, gr-qc/0212101, accepted for
publication in Int. J. Mod. Phys. D.

\bibitem{Wald}  R. M. Wald, General relativity, (The University of Chicago Press,
1984).

\end{thebibliography}

\end{document}

gr-qc0307061.extracted_bib
\begin{thebibliography}{99}

\bibitem{Duff} M. J. Duff and B. E. W. Nilsson, Phys. Rep. {\bf 130}, 1 (1986).

\bibitem{Dimopoulos} N. Arkani-Hamed, S. Dimopoulos and
          G. Dvali, Phys. Lett. B{\bf 429}, 263 (1998); Phys. Rev. D{\bf 59}, 086004 (1999).

\bibitem{Han} T. Han, J. D. Lykken and R. Zhang, Phys. Rev. D{\bf 59}, 105006 (1999).

\bibitem{Giudice}G. F. Giudice, R. Rattazzi and J. D. Wells, Nucl. Phys. B{\bf 544}, 3(1999).

\bibitem{Darabi}F. Darabi and P. S. Wesson, Phys. Lett. B{\bf 527}, 1(2002).

\bibitem{Overduin}J. M. Overduin and P. S. Wesson, Phys. Rep. {\bf 283}, 305(1997).

\bibitem{Weinberg}S. Weinberg, Phys. Lett. B{\bf 125}, 265(1983).

\bibitem{Polchinski}J. Polchinski, String Theory I, II (Cambridge University
Press, Cambridge, 1998).

\bibitem{Antoniadis} I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and
          G. Dvali, Phys. Lett. B{\bf 436}, 257 (1998).

\bibitem{Yega}H. J. de Vega and N. Sanchez, Phys. Lett. B{\bf 197}, 320(1987).

\bibitem{Kaluza}T. Kaluza, Sitzungsber, Preuss. Akad. Wiss. Phys. Mat. Klasse, 966(1921).

\bibitem{Appelquist}T. Appelquist, A. Chodos and P. G. O. Freund, Modern Kaluza-Klein
Theories, Addison-Wesley Publishing Company, Menlo Park
,California(1987).

\bibitem{Geroch}R. Geroch, Jour. Math. Phys. {\bf 12}, 918(1971).

\bibitem{Tavanfar}M. Nouri-Zonoz and A. R. Tavanfar, JHEP {\bf 0302}, 059
(2003).

\bibitem{Fels}M. E. Fels and C. G. Torre, Class. Quant. Grav. {\bf 19}, 641(2002).

\bibitem{HeMaYang} H. He, Y. Ma, and X. Yang, Actions for vacuum Einstein's
equation with a Killing symmetry, gr-qc/0212101, accepted for
publication in Int. J. Mod. Phys. D.

\bibitem{Wald}  R. M. Wald, General relativity, (The University of Chicago Press,
1984).

\end{thebibliography}