\section{Introduction.}
Gravity in 2+1 dimensions \cite{DJH} has provided a good theoretical
laboratory
both at the classical and quantum level. On the classical side most
attention has been devoted up to now to stationary solutions in
presence of point-like or string-like sources \cite{DJH} \cite{grilee}.
Many interesting
features have emerged. In particular the discovery by Gott \cite{gott}
of systems
of point spinless particles in which closed time-like curves (CTC) are
present has revived the problematics of causal consistency in general
relativity.

In ref.\cite{MS1,MS1b,MS2,MS3} it was shown that the adoption
of gauges of a radial
type \cite{modtol} allows to write down the general
solution of Einstein equations
by means of simple quadratures in term of the sources, given by the
energy momentum tensor. On the other hand gauges of pure radial
nature, even though useful to discuss certain special problems, present
the disadvantage of singling out a special event in space time. So
the Fermi-Walker \cite{ferwal} gauge while retaining all interesting
features of the radial gauges shows the advantage to be applicable both
to stationary and time dependent situations. It was also shown in
ref.\cite{MS1b,MS2} that if the problem is stationary (presence
of a time-like Killing vector) or if it possesses axial symmetry
(presence of a Killing vector with closed integral curves that are
space like at space infinity) it is possible to give a complete
discussion of the
support property of the energy momentum tensor. In addition the
physical nature of the Fermi-Walker gauge allows to exploit the energy
condition; this possibility turns out very important to
discuss the occurrence of closed time-like curves \cite{MS2,CFGO}.
With regard to the problem of causality in ref.\cite{CFGO} it was
shown that Gott like
configurations cannot occur in an open universe of spinless point
particles with total time-like momentum; this is an important general
result even though it does
not exclude the occurrence of CTC's.~ 't Hooft
\cite{tH1,tH2} for a system of point
spinless particles in a closed universe gave a construction of a
complete set of Cauchy
surfaces proving that in the evolution of such Cauchy surfaces no CTC
can occur.
Along the lines of the present paper a  result was obtained in
ref.\cite{MS2} for open stationary universes with axial symmetry,
proving that absence of CTC's at space infinity prevents, when
combined with the weak energy condition the occurrence of CTC's
anywhere.

Obviously given the simpler setting of 2+1 dimensional
gravity compared to $3+1$ gravity one should like to have
general simple statements regarding the
problem of CTC's for any universe whose matter satisfy either the weak
or the dominant energy condition but up to now such a general
statement is missing.

The present paper is organized as follows: in Sect.II we write down
 the defining equations of the Fermi-Walker gauge in the first order
formalism that  will be adopted in the sequel of the paper. In
Sect.III we solve the conservation and symmetry
equations for the  energy momentum tensor in the Fermi-Walker
gauge. In Sect.IV we turn to the time dependent problem with axial
symmetry. We solve explicitly the symmetry constraints and write down
the support equations for the energy-momentum tensor in terms of the
Lorentz and Poincar\'e holonomies and show how such conditions can be
explicitly implemented. For the time dependent problem we have no
assurance that the Fermi-Walker coordinate system gives a complete
description of space-time. On the other hand in the general stationary
situation, that is dealt with in Sect.V the completeness of the
Geroch projection assures the completeness (or even overcompleteness)
of the Fermi-Walker coordinate system. That provides a good working
ground for the stationary problem and explicit quadrature formulas are
given and the invariants of the metric are worked out in terms of Lorentz
and Poincar\'e holonomies.
In Sect. VI we turn to the problem of CTC's in the stationary
case. The absence of CTC's is proved for universes which are conical at
space infinity under the hypothesis that the WEC holds and that the
determinant of the dreibein $\det(e)$ in our gauge never vanishes.
The non vanishing of $\det(e)$ can be proved in case of axial
symmetry in the sense that the vanishing of $\det(e)$ implies
either the compactification of the whole space-time or the space
closure of the universe. In this context we are able to extend the
proof of the absence of CTC's for open universes with axial symmetry
also to closed universes with axial symmetry. On the other hand in
absence of axial symmetry, up to now we have no way to dispose of the
hypothesis $\det(e)\neq 0$.

\section{The Fermi-Walker gauge}
The Fermi-Walker (FW)\cite{ferwal}
gauge is considered the natural system of coordinates
for an accelerated observer: it connects in a simple way
 physical observables like
acceleration and rotation to geometrical invariant objects like geodesic
distances. These coordinates are usually defined in terms of their explicit
geometrical construction and the fields are given by a perturbative expansion
around the observer' s worldline\cite{Synge,MTW}.
This feature makes it difficult to handle with
them in practical computations.\\
In this section we follow a different approach to the FW gauge, which allows
us  to recover all their well-known properties and to point out some new ones.
Our starting point is the first order formalism.
Here FW coordinates are defined by \cite{MS1}
\begin{equation}
\label{form1}
\sum_i\xi^i\Gamma^a_{bi}=0
\end{equation}
and
\begin{equation}
\label{form2}
\sum_i\xi^i e^a_i=\sum_i\xi^i\delta^a_i,
\end{equation}
where the sums run only over space indices. (In the following the indices
$i,\ j,\ k,\ l \ {\rm and} \ m$ run over space indices, the quantities in a
generic gauge are denoted by a hat and the quantities without hat are
computed in the FW gauge).\\

First of all we shall discuss the possibility of recovering the usual approach
\cite{Synge,MTW}
as a consequence of  the conditions (\ref{form1}) and (\ref{form2})
by exploiting the geometrical content of them.
We know that the Christoffel symbol are given by
\begin{equation}
\label{Cris1}
\Gamma^\lambda_{\mu\nu}=e^\lambda_a \Gamma^a_{~b\nu}e^b_\mu+
e^\lambda_a\partial_\nu e^a_\mu.
\end{equation}
 From eqs. (\ref{form1}) and (\ref{form2}) we obtain
\begin{equation}
\label{xixi}
\xi^i\xi^j \Gamma^\lambda_{ij}(\xi)=\xi^i \xi^j e^\lambda_a\partial_i e^a_j=
e^\lambda_a(\xi^i \partial_i (e^a_j \xi^j)- e^a_j \xi^j)=0.
\end{equation}
Let us call now $x^\mu(\xi)$ the transformation of coordinates that connects
a generic system $\{x^\mu\}$ to the FW one $\{\xi^\mu\}$; the connections
 are related by the following equation
\begin{equation}
\Gamma^\lambda_{\mu\nu}(\xi)=
\frac{\partial \xi^\lambda}{\partial x^\alpha}\left (
\hat \Gamma^\alpha_{\rho\sigma}(x(\xi))~
 \frac{\partial x^\rho}{\partial \xi^\mu}
\frac{\partial x^\sigma}{\partial \xi^\nu} +\frac{\partial^2 x^\alpha}
{\partial \xi^\mu\partial \xi^\nu}\right ).
\end{equation}
Using equation (\ref{xixi}) we obtain
\begin{equation}
\label{x(xi)}
\xi^i\xi^j\frac{\partial^2 x^\alpha}{\partial \xi^i \partial \xi^j}+
\hat \Gamma^\alpha_{\rho\sigma}(x(\xi))~
 \xi^i \frac{\partial x^\rho}{\partial \xi^i}
 \xi^j\frac{\partial x^\sigma}{\partial \xi^j}=0.
\end{equation}
 From eq. (\ref{form2}) we have \footnote{The information given by eq.
(\ref{Eqq}) is partially contained
in eq. (\ref{x(xi)}). In fact  eq. (\ref{x(xi)}) implies that
$\xi^j\frac{\partial x^\mu}{\partial \xi^j} \hat g_{\mu\nu}(x(\xi))
\xi^i\frac{\partial x^\nu}{\partial \xi^i}$ is an homogeneous function of
degree $2$ in the variables $\xi^i$.}
\begin{equation}
\label{Eqq}
\xi^j\frac{\partial x^\mu}{\partial \xi^j} \hat g_{\mu\nu}(x(\xi))
\xi^i\frac{\partial x^\nu}{\partial \xi^i}=\xi^i g_{ij} (\xi)\xi^j=
\xi^i e^a_i(\xi) \eta_{ab} e^b_j (\xi)\xi^j=-\sum_i \xi^i\xi^i.
\end{equation}
These  two differential equations  define $x^\mu(\xi)$. The exact meaning of
this statement is discussed in appendix A. Obviously it is
impossible to write down the general solution, but the most interesting
properties can be derived without solving eqs. (\ref{x(xi)}) and (\ref{Eqq}).
In fact let us define $x^\mu (\lambda)$ by
\begin{equation}
x^\mu (\lambda)\equiv x^\mu(\xi^0,\lambda \xi^i)
\end{equation}
where $\xi^0$ and $\xi^i$
are kept constant, then from eq. (\ref{x(xi)}) we
obtain
\begin{equation}
\frac{d^2 x^\alpha}{d\lambda^2}+\hat \Gamma^\alpha_{\rho\sigma}(x)
\frac{d x^\rho }{d\lambda}\frac{d x^\sigma }{d\lambda}=0,
\end{equation}
that is $x^\alpha(\lambda)$ are geodesics for each value of
$\xi^0$ and $\xi^i$ and they all start from the curve $s^\alpha(\xi^0)=
x^\alpha(\xi^0,{\bf 0})$. In addition we can also show that these geodesics
are orthogonal to the line $s^\alpha(\xi^0)$.  This result, as we shall see,
is a trivial consequence of the following property of the FW $n$-bein.

For regular fields
(i.e. continuous with bounded derivatives),
\begin{equation}
e^a_i (\xi^0,{\bf 0})=\delta^a_i,\ \ \ e^a_0 (\xi^0,{\bf 0})=\phi(\xi^0)
\delta^a_0\ \ \ {\rm and}\ \ \ \Gamma^a_{~bi} (\xi^0,{\bf 0})= 0,
\end{equation}
if we assume that our $n$-bein is orthonormal, i.e.
$(e^a,e^b)=\eta^{ab}$.\footnote{The presence of this arbitrary function
$\phi(\xi^0)$ is related to the residual gauge invariance. In particular
we can set $\phi(\xi^0)=1$ if
we choose  to parametrize the observer' s world-line with its proper
time.}

In fact taking the derivative of
(\ref{form2}) with respect to $\xi^j$, we have
\begin{equation}
e^a_j (\xi)=\delta^a_j+\xi^i\partial_j e^a_i(\xi).
\end{equation}
For $\xi^j=0$ we have $e^a_i (\xi^0,{\bf 0})=\delta^a_i$.
The fact that the $n$-bein is orthonormal then fixes $e^a_0 (\xi^0,{\bf 0})=
\phi(\xi^0)\delta^a_0$.
The equivalent statement for $\Gamma^a_{~bi}(\xi)$ is reached
by using the same procedure.\\

\noindent Coming back to the proof that the geodesics start orthogonal to the
line $s^\alpha(\xi^0)=x^\alpha(\xi^0,{\bf 0})$, from the previous
result we have that
\begin{equation}
g_{ij}(\xi^0,{\bf 0})=-\delta_{ij}\  \ \ {\rm and}\ \ \
g_{0\mu}(\xi^0,{\bf 0})=\phi(\xi^0)^2 \eta_{0\mu}.
\end{equation}
Thus the following equalities hold
\begin{equation}
\left(\frac{dx}{d\lambda},\frac{ds}{d\xi^0}\right)\biggr |_{\xi^i=0}=
\hat g_{\mu\nu} (x)\xi^i \partial_i x^\mu (\xi^0,\lambda \xi^i)\partial_0
x^\nu(\xi)\biggr |_{\xi^i=0}=g_{i0}(\xi^0,{\bf 0}) \xi^i=0,
\end{equation}
where we have used the rule of transformation of the metric under change of
coordinates. This completes the proof of our statement.

It is interesting that the converse property is true as well. In particular
if we consider the family of geodesics that start orthogonal to the worldline
$s^\alpha (\xi^0)=x^\mu(\xi^0,{\bf 0})$,
they define a function  $x^\mu (\xi^0,\lambda \xi^i)$
that satisfies eqs. (\ref{x(xi)}) and (\ref{Eqq}).
The proof is a straightforward application of the theory of
differential equations.

Another property, which  we are going to use in the following, is that the
FW $n$-bein $e^a(\xi)$ is parallel transported fields along the
geodesics $x^\mu(\lambda)$. In
fact  let us consider the geodesics  $x^\mu(\lambda)$ defined before,
in the FW coordinates they have the simple form $\xi^0={\rm constant}$
and $\xi^i=\lambda v^i$ with $v^i$ constant vector. Then we obtain
\begin{equation}
\frac{D e^a}{d\lambda}=v^i\Gamma^a_{~b i}(\xi(\lambda)) e^b=0,
\end{equation}
where we have used eq. (\ref{form2}). Thus the covariant derivative of
the $n$-bein is zero along this curves. This  means that they are parallel
transported.  This property defines completely the $n$-bein
up to their redefinition  along the line $s^\alpha(\tau)$.

A geometrical construction of this coordinates is now a straightforward
consequence of the previous properties. In fact given a worldline
$s^\alpha(\tau)$, we fix a basis  $E_a(s)$ along the curve with
the following properties\footnote{The fact that $E_0(s)$ must be parallel
to $\frac{ds}{d\tau}$ is a consequence of the previous property.}
\begin{equation}
(E_a(s),E_b(s))=\eta_{ab}\ \ \ \ {\rm  and}\ \ \ \
E_0(s) {\rm \   parallel\  to \ \ }{\frac{ds}{d\tau}}.
\end{equation}
Then the FW coordinates of a generic point with respect to
this curve can be constructed in this way.
Let us consider the geodesic that
starts orthogonal to $s^\alpha(\tau)$ and reaches $P_0$ i.e.
\begin{equation}
x^\alpha(0)=s^\alpha (\tau), \ \ \ x^\alpha(1)=x(P_0 ),\ \ \
(\frac{dx}{d\lambda},\frac{ds}{dt})\vert_{\xi^i=0}=0.
\end{equation}
Using what we have shown before, this geodesics must have the form
\begin{equation}
x^\mu(\lambda)=x^\mu (\xi^0,\lambda\xi^i) ,
\end{equation}
where $x^\mu(\xi)$ is the transformation of coordinates that connects the
coordinates $\{ x^\mu\} $ with the FW coordinates $\{ \xi^\mu\}$.
Then the new coordinates are simply obtained by
\begin{equation}
\xi^0=\tau \ \ \ \ {\rm and}\ \ \ \
\xi^i=
\frac{dx^\mu}{d\lambda}\biggr |_{\lambda=0} E^i_\mu(s(\tau)).
\end{equation}
One can recognize in this construction the usual one. However
 in our case  we have to specify how to construct the
field $e^a_\mu(\xi)$ at each point of the space-time.
The new feature arises from the fact
that we are in
the first order formalism and beyond the diffeomorphisms we have the
local Lorentz invariance. \\
The construction of the FW coordinates associates to each point
$P_0$ a geodesics starting from the wordline $s^\alpha(\tau)$ and  the fields
$e^a_\mu(\xi)$ at the point $P_0$, using  a property previously derived,
are the parallel transported ones along this geodesic of the $n$-bein
$E_a(s)$  that we have at the intersection point of the two curves.
This completes the geometrical construction of the coordinate
system and of the fields.

However there are some  issues that we have to clarify and that are related
to problem of the residual gauge  invariance.  In this construction we make
three arbitrary choices
\begin{itemize}
\item the world-line $s^\alpha(\tau)$ and its parameter $\tau$
\item the affine parameter $\lambda$ for our geodesics $x(\lambda)$
\item the choice of the basis $E_a (s(\tau))$ along the line $s^\alpha(\tau)$
\end{itemize}
The invariance  related to this three points is discussed in \cite{Sem}.
Here we want to make only a few remarks.\\
Eqs. (\ref{form1}) and (\ref{form2}) do not contain any information about
the observer, and in particular its worldline; they express general geometrical
 properties of the FW system which are valid whatever the observer is.
Any information about the observer is an extra degree of freedom that we are
free to fix. From a technical point of view, we can say that we have to
specify  the initial condition
\begin{equation}
x^\mu(\xi,{\bf 0})=s^\mu(\tau)\bigr |_{\tau=\xi^0}
\end{equation}
if we want to solve eqs. (\ref{x(xi)}) and (\ref{Eqq}).\\
Then the possibility of choosing a different affine parameter $\lambda^\prime=
a~\lambda+b$ for our geodesics is only a matter of convenience. It gives
a global rescaling of our coordinates. The choice $x^\mu(0)=s^\mu (\tau)$ and
$x^\mu(1)=x^\mu(P)$ is the usual one.\\
Having a different basis  $E_{a}(s(\tau))$ along the observer' s world
line means that
we can choose the angular velocity for our observer. In other words the
transformation
\begin{eqnarray}
&&\xi^0=\bar\xi^0\ \ \ \ {\rm and}\ \ \ \ \xi^i=\Omega^i_j(\bar\xi^0)\bar \xi^j
\nonumber\\
&& e^0=\bar e^0\ \ \ \ {\rm and }\ \ \ \ e^i=\Omega^i_j(\bar\xi^0)\bar e^j
\end{eqnarray}
where $\Omega$ is a euclidean
rotation in $(n-1)$ dimensional space, preserves the FW structure of
our fields.
It is easy to see that this residual
gauge transformation allows to choose the space-space components of
$\Gamma_0 (\xi^0,{\bf 0})$ vanishing\footnote{We have at our disposal
$(n-2)(n-1)/2$ degrees of freedom in
$\dot\Omega$ which can be use to put to zero the space-space components
of $\Gamma_0({\bf 0},t)$. This is obtained by solving by means of the
standard time ordered integral the equation
$\dot\Omega=-\Gamma~\Omega$ in the $(n-1)\times (n-1)$ space components.}.
We have only the spatial rotation and not a Lorentz transformation because we
are obliged to keep $E_0(s)$ parallel to the observer' s worldline.\\

At the beginning of this section we have recalled that the FW field
are given, in the usual approach, by a perturbative expansion around the
observer' s worldline. The situations changes in this first order formulation.
In fact one can express the fields in term of quadratures of the Riemann and
torsion two forms. In particular one finds \cite{MS1} that
\begin{equation}
\label{A1}
\Gamma^a_{bi}(\xi)=\xi^j\int^1_0 R^a_{bji}(\xi^0,\lambda \pmb{$\xi$})\lambda d\lambda.
\end{equation}
\begin{equation}
\label{A2}
\Gamma^a_{b0}({\xi})=\Gamma^a_{b0}(\xi^0,{\bf 0})+\xi^i\int^1_0 R^a_{bi0}(\xi^0,
\lambda {\pmb{$\xi$}})
d\lambda,
\end{equation}
\begin{equation}
\label{A9}
e^a_0({\bf \xi})=\delta^a_0+\xi^i\int^1_0\Gamma^a_{i0}(\xi^0,\lambda{\pmb{$\xi$}}) d\lambda+
\xi^j\int^1_0 S^a_{j0}(\xi^0,\lambda\pmb{$\xi$})d\lambda.
\end{equation}

\begin{equation}
\label{A10}
e^a_i(\xi)=\delta^a_i+\xi^j\int^1_0\Gamma^a_{ji}(\xi^0,\lambda {\pmb{$\xi$}})\lambda d\lambda
+\xi^j\int^1_0 S^a_{ji}(\xi^0,\lambda\pmb{$\xi$}) \lambda d\lambda
\end{equation}
being $R^a_{bji}$ the curvature and $S^a_{i0}$ the torsion. These formulae
contain the additional hypothesis that $\xi^0$ is identified with the proper
time of the observer. (See footnote pag. 5). The arbitrary functions
$\Gamma^a_{~b0}(\xi^0,{\bf 0})$ that appear in the expression are obviously
related to the residual gauge invariance discussed before.
$\Gamma^0_{~b0}(\xi^0,{\bf 0})$  is the observer' s acceleration and
$\Gamma^i_{~j0}(\xi^0,{\bf 0})$ his angular velocity with respect to
gyroscopic directions \cite{HawE}.

\section{CONSTRAINT EQUATIONS FOR THE ENERGY MOMENTUM TENSOR}

The above given treatment is valid in any dimension; in $2+1$
dimensions a simplifying feature intervenes because the Riemann
tensor, being a linear function of the Ricci tensor, can be written
directly in terms of the energy-momentum tensor.
\begin{equation}
\label{einstein}
\varepsilon_{abc} R^{ab}= -{2\kappa} T_c,
\end{equation}
\begin{equation}
R^{ab}=-{\kappa}\varepsilon^{abc} T_c=-{\kappa\over 2}
\varepsilon^{abc}~\varepsilon_{\rho\mu\nu}\tau^{~\rho}_c dx^\mu\wedge dx^\nu,
\end{equation}
where $\kappa=8\pi G$ and $T_c$ is the energy momentum two form and
$R^{ab}$ is the curvature two form.

Thus also in the time
dependent case eqs.(\ref{A1},\ref{A2},\ref{A9},\ref{A10}) provide a
solution by quadrature of Einstein's
equations. Nevertheless one has to keep in mind that the solving
formulas are true only in the Fermi-Walker gauge, in which the energy
momentum tensor is not an arbitrary function of the coordinates but is
subject to the covariant conservation law and symmetry property,
that are summarized by the equations
\begin{equation}
{\cal D} T^a=0,
\end{equation}
and by
\begin{equation}
\label{symmetry}
\varepsilon_{abc} T^b\wedge e^c =0.
\end{equation}
It will be useful as done in ref.\cite{MS2} to introduce the
cotangent vectors
$\displaystyle{T_{\mu}=\frac{\partial \xi^0}{\partial \xi^
\mu}}$,
$\displaystyle{P_{\mu}=\frac{\partial \rho}{\partial \xi^
\mu}}$ and
$\displaystyle{\Theta_{\mu}=\rho\frac{\partial \theta}{\partial \xi^
\mu}}$ where $\rho$ and  $\theta$ are the polar variables in the
$(\xi^1,\xi^2)$ plane. In addition  we notice that in $2+1$ dimensions
the most general form of a connection satisfying eq.(\ref{form1}) is
\begin{equation}
\label{connection}
\Gamma^{ab}_\mu(\xi)=\varepsilon^{abc}\varepsilon_{\mu\rho\nu}P^\rho
A^\nu_c(\xi).
\end{equation}
where in eq.(\ref{connection}) obviously the component of $A$ along
$P^\rho$ is irrelevant.

Writing $A^\nu_c$ in the form \cite{MS2}
\begin{equation}
\label{A}
A^\rho_c(\xi)= T_c \left [ \Theta^\rho\beta_1+T^\rho\frac{(\beta_2-1)}{\rho}
\right ]+\Theta_c \left [\Theta^\rho\alpha_1 +T^\rho\frac{\alpha_2}{\rho}
\right ]+P_c \left [\Theta^\rho \gamma_1+T^\rho\frac{\gamma_2}{\rho}
\right ],
\end{equation}
we have for the components $\tau^\rho_c$ of the energy momentum two
form
\begin{eqnarray}
\label{taumisto}
&&\tau^{\rho}_c=-\frac{1}{\kappa}\biggl \{ T_c \biggl ( T^\rho
\frac{\beta^\prime_2}{\rho}+\Theta^\rho \beta^\prime_1\biggr )+
\Theta_c
\biggl ( T^\rho
\frac{\alpha^\prime_2}{\rho}+\Theta^\rho\alpha^\prime_1
\biggr )+
 P_c\biggl (T^\rho \frac{\gamma^\prime_2}{\rho}+
\Theta^\rho
\gamma^\prime_1\biggr ) +\\
&&\frac{1}{\rho}P^\rho
\biggl [ T_c\biggl (\alpha_1\gamma_2-\alpha_2\gamma_1-
\frac{\partial\beta_1}{\partial \theta}+ {\partial \beta_2\over \partial t}\biggr )+
\Theta_c \biggl (\beta_1\gamma_2-\beta_2\gamma_1-
\frac{\partial\alpha_1}{\partial \theta}+{\partial \alpha_2\over \partial t}\biggr )
+\\
&&P_c \biggl (\alpha_1\beta_2
-\alpha_2\beta_1-
\frac{\partial\gamma_1}{\partial \theta}+ {\partial \gamma_2\over \partial t}
\biggr )\biggr ]\biggr \}.\nonumber
\end{eqnarray}
The dreibein and the metric are now given by
\begin{equation}
\label{dreibein}
e^a_\mu(\xi)= -T^a (T_\mu A_1+\frac{1}{\rho} \Theta_\mu A_2)-
\Theta^a (T_\mu B_1+\frac{1}{\rho} \Theta_\mu B_2)-P^a P_\mu
\end{equation}
and
\begin{equation}
ds^2= (A_1^2-B_1^2) dt^2 + 2 (A_1 A_2-B_1 B_2)dtd\theta +
(A_2^2-B^2_2) d\theta^2-d\rho^2
\end{equation}
where $A_i$ and $B_i$ are defined by
\begin{eqnarray}
&&A_1(\xi)=\rho \int^1_0 \alpha_1(\lambda \pmb{$\xi$},t) d\lambda-1\ \ {\  ,\  }
\ \ \ B_1(\xi)=\rho \int^1_0 \beta_1(\lambda \pmb{$\xi$},t) d\lambda \nonumber,   \\
&&A_2(\xi)=\rho \int^1_0 \alpha_2(\lambda \pmb{$\xi$},t) d\lambda\phantom{-1}
\ \ {\rm and}
\ \ \ B_2(\xi)=\rho \int^1_0 \beta_2(\lambda \pmb{$\xi$},t) d\lambda.
\end{eqnarray}
The above equation (\ref{taumisto}) is obtained by substituting
eq.(\ref{A})
into eq.(\ref{einstein}) and as
such the resulting energy-momentum tensor is covariantly conserved.
On the other hand the imposition of the symmetry constraint
eq.(\ref{symmetry}) gives
\begin{mathletters}
\begin{eqnarray}
\label{22a}
&&A_1\alpha_2^\prime -A_2 \alpha_1^\prime
+B_2\beta_1^\prime
 -B_1 \beta_2^\prime=0\\
\label{22b}
&&\alpha_2 \gamma_1-\alpha_1 \gamma_2+A_2\gamma_1^\prime -A_1
\gamma_2^\prime
+\frac{\partial\beta_1}{\partial\theta}-{\partial\beta_2\over \partial t}=0\\
\label{22c}
&&\beta_2 \gamma_1-\beta_1 \gamma_2+B_2\gamma_1^\prime -B_1
\gamma_2^\prime +\frac{\partial\alpha_1}{\partial\theta}-
{\partial\alpha_2\over \partial t}=0.
\end{eqnarray}
\end{mathletters}
These relations can be integrated with respect to $\rho$ and taking into
account the regularity conditions of $\alpha_i$, $\beta_i$ and $\gamma_i$
at the origin we reach
\begin{mathletters}
\label{symmetryeq}
\begin{eqnarray}
\label{symmetry1}
&&A_1\alpha_2-A_2 \alpha_1+B_2
\beta_1-B_1\beta_2=0\\
\label{symmetry2}
&&A_2\gamma_1 -A_1
\gamma_2 +\frac{\partial B_1}{\partial\theta}-
{\partial B_2\over \partial t}=0\\
\label{symmetry3}
&&B_2\gamma_1-B_1
\gamma_2 +\frac{\partial A_1}{\partial\theta}-{\partial A_2\over
\partial t}=0.
\end{eqnarray}
\end{mathletters}
\section{Time dependent case with axial symmetry}
In presence of axial symmetry eq.(28) simplify to

\begin{mathletters}
\begin{eqnarray}
\label{23a}
&&A_1\alpha_2^\prime -A_2 \alpha_1^\prime
+B_2\beta_1^\prime
 -B_1 \beta_2^\prime=0\\
\label{23b}
&&\alpha_2 \gamma_1-\alpha_1 \gamma_2+A_2\gamma_1^\prime -A_1
\gamma_2^\prime-{\partial\beta_2\over \partial t}=0\\
\label{23c}
&&\beta_2 \gamma_1-\beta_1 \gamma_2+B_2\gamma_1^\prime -B_1
\gamma_2^\prime-{\partial\alpha_2\over \partial t}=0.
\end{eqnarray}
\end{mathletters}

Eq.(\ref{taumisto}) implies that
outside the sources the following quantities are constant both with
respect to $\rho$ and $t$
\begin{equation}
\label{qua2}
\beta_2^2-\alpha_2^2-\gamma_2^2= {\rm const}.
\end{equation}

\begin{equation}
\label{qua3}
\alpha_2 B_2-\beta_2 A_2= {\rm const}.
\end{equation}
We shall relate such conserved quantities with the energy
and the angular momentum of the system.

The procedure for proving the independence both on $\rho$ and $t$ of
eq.(\ref{qua2}) and eq.(\ref{qua3}) is completely similar to the one
used to prove the
constancy in $\rho$ and $\theta$ of $\beta_1^2-\alpha_1^2-\gamma_1^2$
and $\alpha_1 B_1-\beta_1 A_1$ in the stationary case for which we refer to
\cite{MS2}.

We come now to the computation of the Lorentz holonomy of the connection
(\ref{connection}) along a circle at constant time $t$ and which
encompasses all matter.
It is of the form $W_L=e^{iJ_a\Delta^a}$
where $J_a$ are the generators of the Lorentz group
in the fundamental representation with commutation relations
$[J_a,J_b] = i \varepsilon_{ab}^{~~c} J_c$, and it is obtained by
substituting eq.(\ref{connection}) and (\ref{A}) into
\begin{equation}
{\rm Pexp}\left [-i \oint J_a \Gamma^a_i ~dx^i\right ]
 \equiv e^{iJ_a \Delta^a}.
\end{equation}
We have
\begin{equation}
J_a\Gamma^a_1 dx^1 + J_a\Gamma^a_2 dx^2 = [-J_0 (1-\beta_2) - (J_2
\cos\theta-J_1 \sin\theta)\alpha_2 - (J_1 \cos\theta + J_2
\sin\theta)\gamma_2] d\theta
\end{equation}
from which using ref.\cite{VolDol} we have
\begin{equation}
{\rm Pexp}(-i \int J_a \Gamma^a_i dx^i) = \exp(i J_0\theta)
{\rm Pexp}(-i \int J_a V^a d\theta)
\end{equation}
with $V^a=(\beta_2,-\gamma_2,-\alpha_2)$ which for $\theta=2\pi$ and $V$
time-like becomes
\begin{equation}
\exp(2\pi i J_a\hat V^a (1-V))
\end{equation}
with $V=\sqrt{\beta_2^2-\alpha_2^2-\gamma_2^2}$. Then the angular
deficit and the mass are given by
$2\pi\left (1-\sqrt{\beta_2^2-\alpha_2^2-\gamma_2^2}\right )= 8 \pi G M$.

Coming now to the Poincar\'e holonomy we have, with $P_a$ the
translation generators of $ISO(2,1)$ in the fundamental representation
\begin{eqnarray}
&& W_P=e^{iJ_a \Delta^a + i P_a \Xi^a}=
e^{2\pi i J_a \hat V^a } e^{-2\pi i(J_0\beta_2 -J_1\gamma_2 - J_2\alpha_2 +P_2
B_2-P_0 A_2)} = \\
&& e^{2\pi i J_a \hat V^a} e^{-2\pi i
(J_0 \beta_2 -J_1 \gamma_2 - J_2\alpha_2)}
e^{-2\pi i(P_2
\tilde B_2-P_0 \tilde A_2)}
\end{eqnarray}
with $\tilde B_2 \alpha_2 - \tilde A_2 \beta_2 = B_2 \alpha_2 - A_2
\beta_2 $.
Such a quantity, as discussed in \cite{MS3} \cite{AcTo} \cite{Witt},
is an invariant under gauge transformations and in particular  under
deformations of the loop which do not intersect matter. Then we can
rewrite
the holonomy in the form
\begin{equation}
W_P= \exp( 2\pi i J_a\hat V^a (1-V)) \exp(-2\pi i(P_2
\tilde B_2-P_0 \tilde A_2))
\end{equation}
and the angular momentum is related to the written invariant according
to the general formula \cite{MS3}
\begin{equation}
{\cal J} = {\Delta_a\Xi^a \over 8\pi G \sqrt{\Delta_a\Delta^a}}=
{\alpha_2 B_2 -\beta_2 A_2\over 4 G V}.
\end{equation}
Choosing $\alpha_2,\beta_2,\gamma_2$ satisfying
eq.(\ref{qua2},\ref{qua3}) as input functions specifying an
energy-momentum tensor with bounded support it is possible to write
down using a procedure completely similar to that used in the
stationary case \cite{MS2}, quadrature formulas which express the
metric in term of the $\alpha_2,\beta_2,\gamma_2$.

\section{STATIONARY CASE}
The stationary problem in the Fermi-Walker gauge has been introduced
in \cite{MS1}\cite{MS2}. A major difficulty of the applicability of
the Fermi-Walker
coordinates is whether they give a complete description of the space
time manifold. For the time dependent case we know that in general
this is not the case \cite{MiThWe}. On the other hand in the
stationary case there exists a powerful projection technique due to
Geroch that allows to infer from the completeness of such a
projection the completeness of the Fermi-Walker coordinate system.

Given a Killing vector field, that in our case will be assumed to be
everywhere time-like and never vanishing, one defines the Geroch
projection $S$ \cite{geroch} of the $2+1$ manifold $M$ as the quotient
space of $M$ by the motion generated by the Killing vector field; in
different words $S$ is the collection of all trajectories in $M$ which
are everywhere tangent to the Killing vector field. We shall assume
following \cite{geroch} that $S$ has the structure of a differentiable
manifold; then Geroch \cite{geroch} shows that $S$ is endowed of a
metric structure with metric tensor given by
\begin{equation}
h_{ab}=g_{ab}- {K_a K_b\over K^2}
\end{equation}
which, being the Killing vector $K$ time-like, is negative definite
(space-like metric).

If such a manifold is metrically complete, (if it is not we can
consider its metric completion)  then we know from the
theorem of Hopf-Rinew-De Rahm \cite{spivak}, that it is also geodesically
complete. This means that given two points is $S$ there exists always
at least one geodesic connecting them. Given an event in $M$ we shall
consider the geodesic in $S$ which connects its projection with the
projection of the world line of the stationary observer.
We want now to relate such geodesics on the Geroch projection $S$ to
the geodesics in the $2+1$ dimensional manifold $M$, that where used
to define the Fermi-Walker coordinate system. To this end let us
consider the integral curve starting from our event, of the vector
field on $M$ which is orthogonal to the Killing vector $K$ and
possesses as Geroch projection the tangent vector to the geodesic in
$S$. Then we have
\begin{equation}
v^\alpha K_\alpha=0,~~~~~~~~v^\mu h^\alpha_{\beta}\nabla_\mu v^\beta=0.
\end{equation}
But then, keeping in mind that
\begin{equation}
h^\alpha_\beta=\delta^\alpha_\beta - K^\alpha K_\beta/K^2
\end{equation}
and that
\begin{equation}
v^\mu K_\alpha\nabla_\mu v^\alpha=-v^\mu v^\alpha \nabla_\mu K_\alpha=
v^\mu v^\alpha \nabla_\alpha K^\mu=0
\end{equation}
we have $v^\mu\nabla_\mu v^\alpha=0$ i.e. the considered curve is a
geodesic in
$M$. Such geodesic will meet the line $O$ at a certain  time $t$ and
will be orthogonal to it.
Thus we have reached the conclusion that the Fermi-Walker coordinate
system constructed in \cite{MS1} is complete, and possibly
overcomplete.

In ref.\cite{MS2} we proved algebraically, starting from eq.(28) and
eq.(\ref{taumisto})  for the general stationary problem, that
outside the sources we have two invariants i.e. the expressions
\begin{equation}
\beta_1^2-\alpha_1^2-\gamma_1^2 = C_1
\end{equation}
and
\begin{equation}
\alpha_1 B_1 - \beta_1 A_1 = C_2
\end{equation}
become independent of $\rho$ and $\theta$, outside the sources. We
shall give here $C_1$ and $C_2$ an interpretation in term of Lorentz
and Poincar\'e holonomies.

Let us consider a Wilson loop that has two branches $AB$ and $CD$
parallel to the time Killing vector field  and of unit length in the
Killing time and connect them by two
arcs $BC$ and $DA$ each of which develops at constant time. We have
\begin{equation}
W= W_{AD}W_{DC}W_{CB}W_{BA}=1
\end{equation}
if the whole Wilson loop is taken outside the sources. Then as the two
Wilson arcs $W_{CB}$ and $W_{DA}$, owing to the stationary nature of the
problem are equal $W_{CB}=W_{DA}=U$, we have
\begin{equation}
W_{BA}= U^{-1} W_{CD} U.
\end{equation}
In words, two Wilson lines parallel to time and translated in $\rho$
and $\theta$ without intersecting the sources are related by a
similitude transformation. Let us compute now $W_{BA}$.
We have
\begin{equation}
J_a\Gamma^a_0= J_0 \beta_1-\tilde J_1\gamma_1 -\tilde J_2\alpha_1
\end{equation}
and
\begin{equation}
P_a e^a_0= -P_0 A_1+\tilde P_2 B_2
\end{equation}
where $\tilde J_1= \cos\theta J_1+ \sin \theta J_2$, $\tilde J_2=
\cos\theta J_2 - \sin \theta J_1$, $\tilde P_1= \cos\theta P_1+
\sin \theta P_2$  and $\tilde P_2= \cos\theta P_2 - \sin \theta P_1$.
As $J_0, \tilde J_1  , \tilde J_2 , P_0, \tilde P_1  , \tilde P_2$
satisfy the same commutation relations as  $J_0, J_1, J_2, P_0, P_1,
P_2$ we have the
two invariants $C_1$ and $C_2$ given by the combinations
\begin{equation}
\Delta^a \Delta_a=2\pi (\beta_1^2-\alpha_1^2-\gamma_1^2)~~~~~~~~
{\rm and}~~~~~~~~ \Delta^a \Xi_a=2\pi (A_1\beta_1-B_1\alpha_1).
\end{equation}
In appendix A the two written invariants are expressed in terms the norm
of the vorticity of the Killing vector and the projection of the curl
of the Killing vector along the Killing vector itself.

\section{The problem of closed time-like curves in the stationary
case}

The problem of CTC's in the stationary case for open universes in
presence of  axial
symmetry has been dealt with in ref.\cite{MS2}.  The result is
simple to state:
If the matter sources satisfy the weak energy condition (WEC) , the
universe in open
and there are no CTC at space infinity, in presence of axial symmetry
there are no CTC
at all. We recall in addition that explicit examples show that the
hypothesis of absence of CTC's at space infinity is a necessary one.

Here we shall give a simplified treatment that extends the result to
closed universes
always with axial symmetry, and that  under a certain assumption about
the behavior
of the determinant of the dreibein in the Fermi-Walker gauge, extends also to
stationary universes in absence of axial symmetry.

First we notice that if for $\rho >0$, $A_2^2-B_2^2\equiv
g_{\theta\theta} <0$ there
cannot be CTC's.
In fact given the closed curve $\xi(\lambda)$ let us consider a point
where $\displaystyle{{\partial \xi^0\over \partial\lambda}=0}$. There we have
\begin{equation}
ds^2=(A_2^2-B_2^2) d\theta^2-d\rho^2
\end{equation}
and if $ A_2^2-B_2^2 \leq 0$ it cannot be an element of a CTC.
Then
also for the non axially symmetric stationary problem, it is enough
to prove that for
$\rho >0$, $ A_2^2-B_2^2 \leq 0$. Without committing ourselves to the axially
symmetric case we shall start proving the following lemma.

Lemma: If the WEC holds, $\det(e)>0$ for $\rho>0$ and $g_{00}\equiv
A_1^2-B_1^2 >0$ (and
thus never vanishes), and the space is conical at infinity then
there are no CTC at all.

We notice that $g_{00}\equiv A_1^2-B_1^2 >0$ express the requirement of the
existence of a non singular time-like Killing vector field.

Proof: $ A_2^2-B_2^2$ can vanish either for $A_2 = B_2$  (zero of type
$+$)  or
for $A_2 = -B_2$  (zero of type $-$). If $ A_2^2-B_2^2$, which is zero at the
origin and due to the behavior of $A_2$ and $B_2$ is negative in a
neighborhood of the  origin, changes sign for a certain $\theta$ and
$\rho$ it has to revert, as $\rho$ increases, to the negative
sign according to the results of Appendix C.
Let us consider the first zero
of $A_2^2- B_2^2$ after which it becomes again negative and suppose
this zero to be of type $+$,
i.e. $A_2=B_2$ and let call it $\rho_+$. Then in $\rho_+$ we must have for
$g_{\theta\theta}$ a non positive derivative i.e.
\begin{equation}
A_2(\alpha_2-\beta_2)\leq 0
\end{equation}
with $A_2\neq 0$ because we cannot have  $A_2=B_2=0$ otherwise $\det(e)=0$.
We also have
\begin{equation}
\det(e)=A_2(B_1-A_1)>0
\end{equation}
which gives as a consequence
\begin{equation}
(\alpha_2-\beta_2)(B_1-A_1)\leq 0.
\end{equation}
Then defined
\begin{equation}
\label{WECdelta}
E^{(\pm)} (\rho)\equiv
(B_2\pm A_2)(\alpha_1\pm\beta_1)-
(\alpha_2\pm\beta_2)(B_1\pm A_1)
\end{equation}
in $\rho_+$ we have
\begin{equation}
E^{(-)}(\rho_+)= -(\alpha_2-\beta_2)(B_1-A_1)\geq 0.
\end{equation}
We recall however \cite{MS2} that as a direct consequence of the WEC,
$E^{(\pm)}$ are non increasing functions of $\rho$. This fact implies
$E^{(-)}(\rho <\rho_+)\geq 0$.
Now we consider the following identity
\begin{equation}
{d\over d\rho}({B_2-A_2\over B_1-A_1}) = { E^{(-)}(\rho)\over (B_1-A_1)^2}
\geq 0
\end{equation}
for $\rho <\rho_+$. We recall in addition that $ B_1-A_1>0$ because it cannot
vanish and at the origin equals $1$. But this implies that $A_2-B_2$
is identically
$0$ from the origin to $\rho_+$ which contradicts the fact that in a
neighborhood of
the origin the same quantity has to be negative. Then the above described zero
at $\rho_+$ cannot exist. Similarly one reasons for a zero of type $-$
and we reach the
conclusion that $A_2^2-B_2^2$ has to be always negative except at the
origin where has the value $0$.

This lemma is already sufficient to exclude CTC for open universes whenever
$\det(e)$ never vanishes for $\rho>0$. The non vanishing of $\det(e)$
for $\rho >0$ is a rather strong requirement in
absence of axial symmetry. On the other hand it was proved in
ref.\cite{MS2}  that in
presence of axial symmetry the vanishing of $\det(e)$ leads either to
the closure of the universe or to the compactification of the
$2+1$ dimensional manifold (see Appendix D).
Referring now to the case of axial symmetry we extend the result on the
absence of CTC's to closed universes. We shall prove in what
follows that the
WEC plus axial symmetry implies the absence of CTC's in any closed
stationary universe.

If the universe closes ( with the topology of a sphere due to the
axial symmetry ) then for a certain
$\rho_0$ we must have $\det(e)(\rho_0)=0$ as it is imposed by the
vanishing of the component
$\gamma_{\theta\theta}$ of the space metric and in addition (see
Appendix D)  in $\rho_0$ $A_2^2-B_2^2=0$.
In $\rho_0$ we
must have necessarily $A_2=B_2=0$
otherwise substituting into $\det(e)=0$ we would get either $B_1-A_1=0$ or
$B_1+A_1=0$ which would make the time-like Killing vector field
singular at that point.
We notice furthermore that in $\rho_0$, $(A_1^2-B_1^2)
(\alpha_2^2-\beta_2^2) <0$ as can be
seen form the symmetry equation (\ref{symmetry1}) at $\rho_0$ written
in the form
\begin{equation}
(\alpha_2+\beta_2)(A_1-B_1)=-(\alpha_2-\beta_2)(A_1+B_1)
\end{equation}
and being $A_1^2-B_1^2>0$ we have $\alpha_2^2-\beta_2^2 <0$. This
means that in the neighborhood of $\rho_0$ there cannot be CTC's and
thus we are under the same hypothesis of the proof for open axially
symmetric universes.

\section{Conclusions}
The Fermi-Walker gauge in 2+1 dimensional gravity has
been successful both in dealing with extended sources and time
dependent problems. In this gauge it is possible to write down general
resolvent formulas that contain only quadratures and express the
metric in term of the source of the gravitational field i.e. the
energy-momentum tensor. When a Killing vector exists (axially symmetric
problem  or stationary problem ) it is also possible to treat
explicitly the support of the energy-momentum tensor. The compactness
condition on the sources can be expressed algebraically in terms of
the Poincar\'e holonomies which in the stationary case can be related
to the vorticity of the Killing vector.
We have proven here that for the stationary problem the completeness
of the Geroch projection implies the completeness of the Fermi-Walker
coordinate system. In this context we gave an extension of the theorem
on the absence of CTC \cite{MS2} to the case of closed universes, with
axial symmetry.  In addition whenever the determinant of the dreibein
in the Fermi-Walker system does not vanish, the proof extends also to
the stationary case in absence of axial symmetry.

\typeout{inizio}


\section{Introduction}

The recent efforts that are being invested in the study of
2-dimensional quantum gravity ---in particular dilatonic
gravity, with or without matter \cite{14}-\cite{8}--- have a
variety of motivations: from the elementary fact that it is much easier
to study quantum gravity (QG) in two rather than in four dimensions,
to the much more fundamental reason, that such kind of theories
appear  naturally as string-inspired models. However,
first indications that some of the hard problems of 4D QG ---as
those associated with
the Hawking radiation and black-hole evaporation \cite{14,17}
(see \cite{19} for a review)---
might be much easier to understand in frames of 2-dimensional
dilatonic gravity, appear now not to be quite correct. The
impression nowadays is that 2-dimensional dilatonic gravity
does not look any more as a very `simple' toy model. More
effort should be invested in the study of dilatonic gravity
in exactly (or near) two dimensions, having always in mind
its subsequent generalization to higher dimensions.

As is well accepted, 2-dimensional Einstein gravity is no more of
ultimate interest, since this action represents a topological
invariant in two dimensions. However, it is quite an old idea
\cite{5} to try to study Einstein gravity in
$2+\epsilon$ dimensions, what might help to remedy this shortcoming.
The gravitational coupling constant in such a theory ---near
two dimensions--- shows an asymptotically free behavior \cite{5},
what can actually be very exciting in attempts to solve the
problem
of non-renormalizability of 4-dimensional Einstein gravity
\cite{11}.
Unfortunately, it was shown that Einstein gravity in
$2+\epsilon$  dimensions is on its turn a non-renormalizable
theory
\cite{20} (in other words, it has the oversubtraction problem
\cite{21}).

Furthermore, dynamical triangulations in more than two
dimensions (see, for example, \cite{22,28}) have clearly shown the
existence of a phase transition to a strong-coupling phase of
similar nature as 2-dimensional quantum gravity \cite{23}.
Because
of this,  of considerable interest is still to try to
construct
a consistent theory of QG  in  $2+\epsilon$  dimensions (having in
mind
a subsequent continuation of $\epsilon$ to $\epsilon =1$ or
              $\epsilon =2$).

It has been suggested recently \cite{6}, that a thing to do would be
to study dilatonic gravity near two dimensions (similarly,
it
was the idea in ref. \cite{13} to consider Einstein gravity with
a
conformal scalar field near two dimensions).
Notice, however, that unlike
Einstein gravity, dilatonic gravity possesses a smooth $\epsilon
\rightarrow 0$ limit. This difference manifests itself in the
fact
that dilatonic gravity ---which after a proper field definition
can be presented as an Einsteinian theory with a scalar field--- is
{\it not} equivalent to Einstein gravity in {\it exactly}
 two dimensions. Moreover, dilatonic gravity is renormalizable
and,
hence, it does not suffer from the oversubtraction problem. Even
more
\cite{6}, the gravitational coupling constant in this theory also has an
ultraviolet stable fixed point (asymptotic freedom) for $\epsilon
>0$
and $n<24$ ($n$ is the number of scalars or matter central
charge).
Thus, the matter central charge in dilatonic gravity is bounded,
as
it also happens in Einstein's theory.

In the present paper we consider dilatonic-Maxwell gravity with
scalar
matter in $2+\epsilon$ dimensions. This theory, which includes a
dilatonic potential and a Maxwel term with an arbitrary
dilatonic-vector
coupling function, may be considered as a toy model for
unification of
gravity with matter (scalar and vector fields). We will show that
in the
ultraviolet stable fixed point the gravitational coupling
constant has
the value $G^* = 3\epsilon /[2(30-n)]$ and, hence, owing to the
contribution of the vectors, the matter central charge of our
universe
in such a model can be naturally increased from $0<n<24$ (for
pure
dilatonic gravity) to $0<n<30$ (for dilatonic-Maxwell gravity). With
this,
we will show explicitly that it is possible to have an even wider
window for the matter central charge, provided we consider
non-abelian
gauge fields. We will also present an evaluation of the corresponding,
generalized beta functions, and their fixed-point solutions will
be
found. At the fixed point the theory can be cast under the form
of
a standard, string-inspired model with a dilatonic potential of
Liouville type.

The paper is organized as follows. In the next section we discuss
the classical action of dilatonic-Maxwell gravity with matter and
we derive the corresponding one-loop effective action. Section 3
is devoted to the study of the renormalization of the model and,
in particular, to the derivation of the beta functions (to first
order in the gravitational coupling constant $G$). The
fixed-point
solutions of the renormalization group (RG) equations together
with
a detailed analysis of their stability are presented in section
4.
The form of the action at the fixed points is discussed. In
section
5 we investigate the issue of gauge dependence of the position of the
fixed point for the dilatonic coupling function. Finally, in
section 6 we present the conclusions of our investigation and an
outlook.
\bigskip

\section{Dilatonic-Maxwell gravity with matter in $2+\epsilon$
dimensions}

In this section we are going to consider a theory of dilatonic
gravity
interacting with scalars and vectors via dilatonic couplings in
$2+\epsilon$ dimensions. This theory can be considered as a toy
model
for a theory of unified gravity with matter (since fermions can
be
included without any problem). The action is
 \begin{equation}
S = \int d^d x\,  \sqrt{-g} \biggl[{1 \over 2} Z(\phi) g^{\mu\nu}
\partial_\mu \phi\partial_\nu \phi + {\mu^\epsilon \over 16\pi
G} R C(\phi )+ V(\phi)
- {1 \over 2} f(\phi) g^{\mu\nu} \partial_\mu \chi_i \partial_\nu
\chi^i+ {1 \over 4} f_1(\phi) F_{\mu\nu}^2 \biggr], \label{1}
\end{equation}
where $g_{\mu\nu}$ is the ($2+\epsilon$)-dimensional metric, $R$
the corresponding curvature, $\chi_i$ are scalars ($i=1,2,
\ldots,
n$), $F_{\mu\nu} = \nabla_\mu A_\nu - \nabla_\nu A_\mu$, with
$A_\mu$ a vector, and where the smooth functions $Z(\phi)$,
$C(\phi)$, $V(\phi)$, $f(\phi)$ and $f_1(\phi)$ describe the
dilatonic interactions. Notice that $V(\phi)$ is a dimensional
function and it is therefore convenient to redefine $V
\rightarrow m^2V$, where $m$ is some parameter with dimensions of
mass. Notice that in the absence of an electromagnetic sector
the action (\ref{1}) can be easily presented under the form
of a non-linear sigma model \cite{24}. Then the $\epsilon
\rightarrow 0$  limit of the theory (\ref{1}) can be
discussed in a string effective action manner \cite{25}.

One can easily prove that the above theory (\ref{1}) is
renormalizable in a generalized sense. The one-loop
counterterms corresponding to (\ref{1}) in two dimensions have
been calculated in refs. \cite{1,2}. Our purpose here will be  to
study in some detail the renormalization structure of (\ref{1})
in $2+\epsilon$ dimensions and in particular the
corresponding renormalization
group equations. The very remarkable point that gives sense to
this study is the fact that the theory (\ref{1}) (unlike Einstein
gravity \cite{5}) has a smooth limit for $\epsilon \rightarrow
0$. This property allows for the possibility to study the behavior
of (\ref{1}) in $2+\epsilon$ dimensions by simply using the
counterterms  calculated already in 2 dimensions ---in close
analogy with quantum field theory in frames of the
$\epsilon$-expansion technique (for a review see \cite{3}
and \cite{4}).

Before we start working with action (\ref{1}) we will perform
some simplifications. First, motivated by the form
of the string effective action, we choose the dilatonic
couplings in (\ref{1}) as the exponents of convenient dilatonic
functions. Using the rigid rescaling of scalars and vectors:
\begin{equation}
\chi_i \longrightarrow a_1 \chi_i, \ \ \ \ \ \
A_\mu \longrightarrow a_2 A_\mu, \label{2} \end{equation}
with $a_1$ and $a_2$ constants, one can always normalize the
dilatonic couplings in such a manner that
\begin{equation}
f(0) =1, \ \ \ \ \ \ f_1(0)=1. \label{3} \end{equation}
Second, as has been done in refs. \cite{7,8}, we can also use a
local Weyl transformation of the metric of the form
\begin{equation}
g_{\mu\nu} \longrightarrow e^{-2\sigma (\phi)} \, g_{\mu\nu} ,
 \label{4} \end{equation}
in order to simplify the dilatonic sector of (\ref{1}). In
particular, one can use (\ref{4}) of a specific form such that
$Z(\phi)=0$. Finally, we can  perform a transformation of the
dilaton field in order to simplify the function $C(\phi)$, namely
to reduce it to the form $C(\phi)=e^{-2\phi}$. With all this taken
into account, the action (\ref{1}) can be written as follows
 \begin{equation}
S = \int d^d x\,  \sqrt{-g}\,  \biggl[ {\mu^\epsilon \over 16\pi
G}
R e^{-2\phi}
- {1 \over 2} g^{\mu\nu} \partial_\mu \chi_i \partial_\nu \chi^i
e^{-2\Phi (\phi)}+ \mu^\epsilon m^2 e^{-V(\phi )}+ {1 \over 4}
e^{-f_2(\phi)} F_{\mu\nu}^2 \biggr], \label{5}
\end{equation}
being $\mu$ a mass parameter and where we have choosen the
dilatonic couplings so that $\Phi (0) =f_2(0)=V(0) =0$. The
first two terms in (\ref{5}) correspond to the dilatonic gravity
action in ref. \cite{6}, which was considered in $2+\epsilon$
dimensions. With our choice of (\ref{5}) the zero modes of the
dilatonic couplings are fixed with the help of reference
operators in the gravitational sector \cite{6} and with the
renormalization (by constants) of the scalar, vector, and mass
$m^2$ in the matter sector.

We can now start the study of the divergences of the action
(\ref{5}). The study of the one-loop divergences of dilatonic
gravity in the covariant formalism, initiated in ref. \cite{9}, has been
continued
in refs. \cite{7,8,1,2,10,12}. It is by now well under control, and we
do not consider it necessary to repeat the details of such a
calculation for the case of the model (\ref{5}). Let us just
recall that it is based on the t'Hooft-Veltman prescription
\cite{11}. The gauge-fixing conditions that are most convenient
to use are the following. The covariant gauge-fixing action is
\begin{equation}
S_{gf} = -  {\mu^\epsilon \over 32\pi G} \int d^d x\,
\sqrt{-g}\,
 g_{\mu\nu} \chi^\mu \chi^\nu  e^{-2\phi}, \label{6}
\end{equation}
where
\begin{equation}
\chi^\mu =   \nabla_\nu \bar{h}^{\mu\nu} + 2 \nabla^\mu \varphi
\label{7}
\end{equation}
and $\bar{h}^{\mu\nu}$ is a traceless quantum gravitational field
and $\varphi$ a quantum scalar field, in the background field
method.
In the electromagnetic sector the gauge-fixing action is choosen
as
\begin{equation}
S_{L} = \int d^d x\,  \sqrt{-g}\,
 (\nabla_\mu Q^\mu)^2  e^{-f_2(\phi)}, \label{8}
\end{equation}
where $Q_\mu$ is a quantum vector field. Notice that for the
background fields we  use the same notations as for the classical
fields in (\ref{5}).

The calculation of the one-loop effective action corresponding to
(\ref{5}) in the gauges (\ref{6}) and (\ref{8}) can be performed
in close analogy with the one in refs. \cite{1,2} (which was
actually carried out for a more general theory in two
dimensions). Owing to the smooth behavior of (\ref{5}) for
$\epsilon \rightarrow 0$, the divergences of (\ref{5}) can be
also calculated in exactly two dimensions, what will also provide
the result for $2+\epsilon$ dimensions. After some algebra, we
get
\begin{eqnarray}
\Gamma_{div} &=& \frac{1}{4\pi \epsilon}  \int d^d x\,
\sqrt{-g}\,
\left\{ \frac{30-n}{6} R +m^2 e^{2\phi - V(\phi)} 16\pi G [2 +
V'(\phi)] -  g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi [8-
n\Phi'(\phi)^2] \right. \nonumber \\
&& + \left.  {4\pi G \over \mu^\epsilon} e^{2\phi-f_2(\phi)}
F_{\mu\nu}^2 [f_2'(\phi) -2 ] \right\},
\label{9}
\end{eqnarray}
where $\epsilon =d-2$.
A few remarks about the comparison of (\ref{9}) in some
particular cases with results that have appeared in the
literature are in order. First, in the absence of scalars and
vectors (pure dilatonic gravity), we obtain
\begin{equation}
\Gamma_{div} = \frac{1}{4\pi \epsilon}  \int d^d x\,  \sqrt{-g}\,
\left\{ 4R +m^2 e^{2\phi - V(\phi)} 16\pi G [2 + V'(\phi)] - 8
g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi \right\}.
\label{10}
\end{equation}
This expression coincides with the results of refs.
\cite{9,7,8,1,2,10,12} in the same covariant gauge. Moreover, in
ref. \cite{8}, $\Gamma_{div}$ eq. (\ref{10}) has been calculated
in a one-parameter dependent gauge, what gives an additional
check of (\ref{10}) at the value of the gauge parameter
corresponding to the gauge choice (\ref{6}). For dilatonic
gravity with a Maxwell sector, (\ref{9}) coincides with the
results in \cite{1,2,10}. Notice that the calculation of
$\Gamma_{div}$ eq. (\ref{9}) for the theory (\ref{5}) without
vectors and for $e^{-V(\phi)}\equiv 0$ has been done in ref.
\cite{6}, in the ($2+\epsilon$)-dimensional formalism but in a
slightly different gauge. When comparing the results, there is
here a difference with the coefficient of the $g^{\mu\nu}
\partial_\mu \phi \partial_\nu \phi$ term in (\ref{9}). On one
hand, we get a different contribution from the scalar dilaton
coupling, what may be due to the slightly different gauges. On
the other, instead of our coefficient 8 there appears in \cite{6}
a coefficient 4, what might be an indication of some mistake,
or maybe due to the use of the inconvenient background field
method
---since these authors also employ a gauge of harmonic type and do not
agree with the results of \cite{1,7,8,10} either. Later we will
discuss the gauge dependence of  $\Gamma_{div}$.
\bigskip

\section{Beta functions and renormalization group analysis}

We turn now to the study of the RG corresponding to our model in
$2+\epsilon$ dimensions. We will follow here the approach of ref.
\cite{6}, which is actually based on earlier considerations in
\cite{5}. The main idea of the whole approach is to use $G$ as
the coupling constant in perturbation theory and work at some
fixed power of $G$.
The counterterm which follows from (\ref{9}) can be written as
\begin{equation}
\Gamma_{count} =- \mu^\epsilon  \int d^d x\,  \sqrt{-g}\,
\left[
RA_1(\phi) + g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi
A_2(\phi) + m^2A_3(\phi) +  \frac{\mu^{-\epsilon}}{4}
F_{\mu\nu}^2 A_4(\phi) \right],
\label{11}
\end{equation}
where
\begin{eqnarray}
A_1(\phi) &=& \frac{30-n}{24\pi\epsilon} \equiv A_1, \nonumber \\
A_2(\phi) &=& \frac{1}{4\pi\epsilon} [n\Phi'(\phi)^2-8] , \nonumber
\\
A_3(\phi) &=& \frac{4G}{\epsilon}[2 + V'(\phi)]e^{2\phi -V(\phi)}
\equiv e^{-V(\phi)} \widetilde{A}_3 (\phi), \nonumber \\
A_4(\phi) &=& \frac{4G}{\epsilon}[f_2'(\phi)-2]e^{2\phi -
f_2(\phi)} \equiv e^{-f_2(\phi)} \widetilde{A}_4 (\phi).
\label{12}
\end{eqnarray}
The structure of the counterterms shows that the gravitational
coupling is a really nice choice for parameter of the
perturbation theory.

By looking carefully at the relations (\ref{12}) we observe that
our theory is one-loop finite provided the following conditions
are fulfilled:
\begin{equation}
n=30, \ \ \ \ \Phi (\phi) = \sqrt{\frac{8}{n}} \, \phi, \ \ \ \
V(\phi) =-2 \phi, \ \ \ \ f_2(\phi) =2\phi.
\label{12a}
\end{equation}
(Notice, however, that perturbative finiteness is a more natural
property in 2-dimensional supergravity \cite{26}).

Renormalization is now performed in the standard way
 \begin{eqnarray}
S& =& S_{cl} + S_{count} =\int d^d x\,  \sqrt{-g}\,  \biggl[ {1
\over 16\pi G_0} R e^{-2\phi_0}
- {1 \over 2} g_0^{\mu\nu} \partial_\mu \chi_{0i} \partial_\nu
\chi_0^i e^{-2\Phi_0 (\phi_0)} \nonumber \\
&&+  \mu^\epsilon m_0^2 e^{-V_0(\phi_0 )}+ {1 \over 4} e^{-
f_{02}(\phi_0)} g_0^{\mu\alpha}g_0^{\nu\beta}F_{0\mu\nu}
F_{0\alpha\beta} \biggr]. \label{13}
\end{eqnarray}
The renormalization transformations can be defined as follows
\begin{eqnarray}
&& \phi_0 = \phi + f(\phi), \ \ \ g_{0\mu\nu} =g_{\mu\nu}e^{-
2\Lambda (\phi)}, \ \ \ \Phi_0 (\phi_0)= \Phi(\phi) +
F(\phi), \ \ \ V_0 (\phi_0)= V(\phi) + F_V(\phi), \nonumber \\
&&  f_{02} (\phi_0)= f_{2}(\phi) + F_f(\phi), \ \ \ \chi_{0i} =
Z_\chi^{1/2} \chi, \ \ \ A_{0\mu} = Z_A^{1/2} A_\mu, \ \ \ m_0^2
=Z_m m^2. \label{14}
\end{eqnarray}
The functions $\Lambda$, $f$, $F$,... are chosen so that
$\Lambda(0)$, $f(0)$, $F(0)$,... can be set equal to zero.
Substituting the renormalization transformations (\ref{14}) into
the renormalized action (\ref{13}) (in close analogy with ref.
\cite{6}), we obtain
 \begin{eqnarray}
S_0& =& \int d^d x\,  \sqrt{-g}\,  \biggl\{ {1 \over 16\pi G_0} R
e^{-2\phi-2f(\phi)-\epsilon \Lambda (\phi)} \nonumber \\ &&
+ {\epsilon +1 \over 16\pi G_0} [4\Lambda'(\phi) + \epsilon
\Lambda'(\phi)^2+
4f'(\phi)\Lambda'(\phi)]e^{-2\phi-2f(\phi)-
\epsilon \Lambda (\phi)} g^{\mu\nu} \partial_\mu \phi
\partial_\nu \phi \nonumber \\
&&- {1 \over 2}Z_\chi g^{\mu\nu} \partial_\mu \chi_{i}
\partial_\nu
\chi^i e^{-2\Phi(\phi)-2F(\phi)-\epsilon \Lambda (\phi)}+  \mu^\epsilon
m^2 Z_m e^{-V(\phi )-F_V(\phi)-(2+\epsilon) \Lambda (\phi)} \nonumber
\\ &&+ {Z_A \over 4} e^{-f_{2}(\phi)-F_f (\phi)+(2-\epsilon)
\Lambda (\phi) } F_{\mu\nu}^2 \biggr\}. \label{15}
\end{eqnarray}
 From here, one can easily obtain
\begin{eqnarray}
&& {1 \over 16\pi G_0}
e^{-2\phi-2f(\phi)-\epsilon \Lambda (\phi)} =   \mu^\epsilon
\left( {1 \over 16\pi G}  e^{-2\phi} -A_1 \right), \nonumber \\
&& {\epsilon +1 \over 16\pi G_0} [4\Lambda'(\phi) + \epsilon
\Lambda'(\phi)^2+ 4\epsilon f'(\phi)\Lambda'(\phi)]
e^{-2\phi-2f(\phi)-\epsilon \Lambda (\phi)} =  - \mu^\epsilon
A_2(\phi), \nonumber \\
&& Z_\chi e^{-2\Phi(\phi)-2F (\phi)-\epsilon
\Lambda (\phi) }= e^{-2\Phi(\phi)}, \nonumber \\
&& Z_A e^{-f_{2}(\phi)-F_f (\phi)+(2-\epsilon)
\Lambda (\phi) }= e^{-f_{2}(\phi)} [1- \widetilde{A}_4 (\phi)], \nonumber \\
&& Z_m e^{-V(\phi)-F_V (\phi)-(2+\epsilon)
\Lambda (\phi) }= e^{-V(\phi)} [1- \widetilde{A}_3 (\phi)]. \label{16}
\end{eqnarray}
Working to leading order in $G$ (notice that the functions $f$,
$\Lambda$, $F$,..., $F_f$ are of first order in $G$), we may
follow ref. \cite{6} and obtain the renormalized parameters as
\begin{eqnarray}
&& G_0^{-1}= \mu^\epsilon (G^{-1} -16 \pi A_1), \ \ \ Z_\chi =1,
\
\ \ \Lambda (\phi) = - \frac{4\pi G}{\epsilon +1} \int_0^\phi
d\phi' \, e^{2\phi'}A_2(\phi'), \ \ \ \epsilon \Lambda (\phi) =-
2F(\phi), \nonumber \\
&& f(\phi) = 8\pi A_1 G(e^{2\phi} -1)+\frac{2\pi\epsilon
G}{\epsilon +1} \int_0^\phi d\phi' \, e^{2\phi'}A_2(\phi'), \ \ \
Z_A =1-A_4 (0), \ \ \ Z_m =1-A_3 (0), \nonumber \\
&& F_f (\phi)= (2-\epsilon) \Lambda (\phi) + \widetilde{A}_4(\phi) -
\widetilde{A}_4(0), \ \ \ F_V (\phi)=-(2+\epsilon) \Lambda (\phi) +
\widetilde{A}_3(\phi) -\widetilde{A}_3(0). \label{17}
\end{eqnarray}
Hence, we see that the zero modes of the functions $f$, $f_2$ and
$V$ are indeed controlled by a constant renormalization of the
vector, the scalar and the mass.

We can now turn to the evaluation of the beta functions. We will
list below only the ones corresponding to the dilatonic couplings
and gravitational constant (for the purely dilatonic case they
were given in ref. \cite{6})
\begin{eqnarray}
\beta_G &=& \mu {\partial G \over \partial \mu} = \epsilon G - 16
\pi \epsilon A_1 G^2, \nonumber \\
&& \mu {\partial \Lambda (\phi_0) \over \partial \mu} =  -
\frac{4\pi
\epsilon G}{\epsilon +1} \int_0^\phi d\phi' \,
e^{2\phi'}A_2(\phi'), \nonumber \\
\beta_\Phi (\phi_0) &=& \mu {\partial \Phi (\phi_0)  \over
\partial \mu} = 8\pi \epsilon A_1 G (e^{2\phi_0}-1)\Phi'
(\phi_0) \nonumber \\
&&+ \frac{2\pi \epsilon^2 G}{\epsilon +1} [\Phi' (\phi_0) -1]
\int_0^{\phi_0} d\phi' \, e^{2\phi'}A_2(\phi'), \nonumber \\
\beta_{f_2} (\phi_0) &=& \mu {\partial f_2 (\phi_0)  \over
\partial \mu} = 8\pi \epsilon A_1 G (e^{2\phi_0}-1) f_2' (\phi_0)
\nonumber \\
&&+ \frac{2\pi \epsilon G}{\epsilon +1} [\epsilon f_2' (\phi_0)
+2(2-\epsilon)] \int_0^{\phi_0} d\phi' \, e^{2\phi'}A_2(\phi')-
\epsilon [\widetilde{A}_4(\phi_0) -\widetilde{A}_4(0)] , \nonumber \\
\beta_V (\phi_0) &=& \mu {\partial V (\phi_0)  \over \partial
\mu}
= 8\pi \epsilon A_1 G (e^{2\phi_0}-1) V' (\phi_0) \nonumber \\
&&+ \frac{2\pi \epsilon G}{\epsilon +1} [\epsilon V' (\phi_0)
-2(2+\epsilon)] \int_0^{\phi_0} d\phi' \, e^{2\phi'}A_2(\phi')-
\epsilon [\widetilde{A}_3(\phi_0) -\widetilde{A}_3(0)]. \label{18}
\end{eqnarray}
Similarly, one can obtain the $\gamma$-functions for the fields
and mass parameter, which are however of less importance to us,
due to the fact that they correspond to non-essential couplings.
Notice also that  the only conformal mode of the gravitational
field is renormalized in $2+\epsilon$ dimensions, what is quite
well known \cite{5,6} (for a study of conformal factor dynamics in four
dimensions, see \cite{29}). \bigskip

\section{Fixed points}

The $\beta$-function for the gravitational constant has the form
(\ref{18})
\begin{equation}
\beta_G = \epsilon G - \frac{2(30-n)}{3} G^2. \label{19}
\end{equation}
Hence, for $n<30$ we obtain the infrared stable fixed point
$G=0$. There is also an ultraviolet stable fixed point as in \cite{5}
\begin{equation}
G^* = \frac{3\epsilon}{2(30-n)}. \label{20}
\end{equation}
The theory is asymptotically free in the ultraviolet limit. The
inclusion of vector fields has increased the matter central
charge of our universe, when compared with the cases of pure
dilatonic gravity \cite{6}, Einstein gravity \cite{5}, or
Einstein gravity with a conformal scalar \cite{13}. Our theory
admits more matter than any of these previous models.

We start the search for fixed-point solutions corresponding to
the dilatonic couplings. To this end we choose the following {\it
Ansatz}:
\begin{equation}
\Phi (\phi) = \lambda \phi, \ \ \ \
f_2 (\phi) = \lambda_f \phi, \ \ \ \
V (\phi) = \lambda_V \phi.
\label{21}
\end{equation}
In that case the $\beta$-functions are (we study the functional
dependence of $\beta$-functions of the type (\ref{21}))
\begin{eqnarray}
\beta_\Phi &=& G(e^{2\phi} -1) \left[ \frac{30-n}{3} \lambda +
\frac{\epsilon}{4(1+\epsilon)} (\lambda -1) (n\lambda^2-8)
\right], \nonumber \\
\beta_{f_2} &=& G(e^{2\phi} -1) \left[ \frac{30-n}{3} \lambda_f +
\frac{n\lambda^2-8}{4(1+\epsilon)} (\epsilon\lambda_f+4 -
2\epsilon) -4\lambda_f +8 \right], \nonumber \\
\beta_V &=& G(e^{2\phi} -1) \left[ \frac{30-n}{3} \lambda_V +
\frac{n\lambda^2-8}{4(1+\epsilon)} (\epsilon\lambda_V-4 -
2\epsilon) -4\lambda_V -8 \right].
\label{22}
\end{eqnarray}

Equating these $\beta$-functions to zero we easily get the fixed-point
solutions.
For the dilatonic coupling, we find
\begin{equation}
\lambda^* = -\frac{6\epsilon}{30-n} + {\cal O} (\epsilon^2).
\label{23}
\end{equation}
There are also imaginary oscillating solutions for $\lambda \sim
\epsilon^{-1/2}$, which have been mentioned in ref. \cite{6} and
which are not physical solutions.

Using (\ref{23}), for the Maxwell-dilatonic coupling and dilatonic
potential, we obtain
\begin{equation}
\lambda_f^* = -\frac{36\epsilon}{18-n} + {\cal O} (\epsilon^2), \
\
\ \ \ \ \lambda_V^* = \frac{12\epsilon}{18-n} + {\cal
O} (\epsilon^2) \label{24}
\end{equation}
The fixed-point solutions (\ref{24}) are of the same nature as
solution (\ref{23}). Notice, however, that the denominator in
(\ref{24}) is different from the denominator for the case of a
purely dilatonic sector (\ref{23}). For the existence of the solutions
(\ref{24}) a new limitation appears: $n\neq 18$. Under a discontinuous
transition through the point $n=18$, the sign of the fixed points in
(\ref{24}) changes.

It is interesting to see in which way the model (\ref{5}) can be
rewritten at the fixed point. In particular, we will perform a
Weyl
rescaling (it is non-singular) in the manner suggested in ref.
\cite{6}
\begin{equation}
g_{\mu\nu} \longrightarrow g_{\mu\nu}  \, \exp \left( \frac{4
\lambda^*}{\epsilon} \phi \right). \label{25}
\end{equation}
Then, the classical action becomes
 \begin{eqnarray}
S& =& \int d^d x\,  \sqrt{-g}\,  \biggl\{ {\mu^\epsilon \over
16\pi G^*}
e^{-2(1-\lambda^*)\phi} \left[ R - \frac{4(1+\epsilon)}{\epsilon}
\lambda^* (2- \lambda^*) \right] g^{\mu\nu} \partial_\mu \phi
\partial_\nu \phi  \nonumber \\
&&- {1 \over 2} g^{\mu\nu} \partial_\mu \chi_{i} \partial_\nu
\chi^i +  \mu^\epsilon
m^2  e^{(2\lambda^* +4\lambda^*/\epsilon- \lambda^*_V)\phi}
+\frac{1}{4} F_{\mu\nu}^2 e^{(2\lambda^* -4\lambda^*/\epsilon-
\lambda^*_f)\phi} \biggr\}. \label{26}
\end{eqnarray}
As one can see from expression (\ref{26}), without the
electromagnetic sector and setting  $m^2=0$, it describes the
CGHS action \cite{14} (at the limit $\epsilon \rightarrow 0$ is finite
and scalars become non-interacting with the dilaton). As it stands,
action (\ref{26}) is of a  similar form as dilaton-Maxwell
gravity \cite{15,2} with the Liouville potential. Notice that
such  kind of dilaton-Maxwell gravity (which can also be
considered as a charged string-inspired model \cite{15}) admits charged
black hole solutions with multiple horizons, being in
this sense analogous to four or higher-dimensional
Einstein-Maxwell theories \cite{16}. Different forms of dilatonic
gravity can be easily obtained too, by transforming the metric and the
dilaton (see the appendix). For example, by transforming $g_{\mu\nu}
\rightarrow \exp
(\frac{2\epsilon}{2- \epsilon} \lambda_f^* \phi)  \, g_{\mu\nu}$, we can
present the theory at the critical point as having a free (i.e.,
non-interacting with the dilaton) Maxwell sector.

Notice also that, in order to fix the scale of the metric ---what
is certainly necessary for discussing the renormalization of the
gravitational constants \cite{5}--- one can use the same reference
operator as in ref. \cite{6}, e.g. a combination of the trace of
the metric and of the dilaton. As has been already explained in
detail in \cite{6}, this choice is enough to fix the scale of
$g_{\mu\nu}$ and the origin of $\phi$.

One can now study the
stability of the fixed points (\ref{20}), (\ref{23}), (\ref{24}),
along the same lines as in ref. \cite{6}. Now we can perform
variations along four different trajectories, but concerning the
existence of fixed points the situation is pretty similar to that
in ref. \cite{6}. In fact, a careful analysis of the the
beta-functions (\ref{18}) for the linear Ansatz (\ref{21}) shows
that the two last equations (i.e. those for $f_2$ and $V$) do not
produce a new multiplicity of solutions. In other words, for each
value of $G^*$ and $\lambda^*$ we just have one single value of
$\lambda_f^*$ and one of $\lambda^*_V$, that complete the four
coordinates of the fixed point. For $\lambda^*$ we obtain three
distinct solutions: the real one (\ref{23}) and two purely
imaginary ones, of order $\epsilon^{-1/2}$, namely
\begin{equation}
\lambda^*_\pm = \pm 2i\sqrt{\frac{30-n}{3n\epsilon}} + {\cal O}
(\epsilon^0), \label{261}
\end{equation}
that correspond to highly oscillating dilaton couplings and seem
not to have any sensible meaning.

Expanding the beta functions near the (only real) fixed point, in
the way
\begin{equation}
G=G^*+ \delta G, \ \ \ \ \Phi=\lambda^*\phi+ \delta \Phi, \ \ \ \
f_2=\lambda_f^* \phi+ \delta f_2, \ \ \ \ V=\lambda^*_V \phi +
\delta V, \label{262}
\end{equation}
and assuming all the fluctuations to be small, we obtain
\begin{eqnarray}
\delta \beta_G &=&- \epsilon \delta G, \nonumber \\
\delta \beta_\Phi &=&\frac{ \epsilon}{2} \left(e^{2\phi} -
1\right) \frac{d}{d\phi} \delta \Phi + {\cal O} (\epsilon^2), \nonumber
\\
\delta \beta_{f_2}&=&\frac{ \epsilon}{2} \left( \frac{18-n}{30-n}
e^{2\phi} -1\right) \frac{d}{d\phi} \delta f_2 + {\cal O}
(\epsilon^2), \nonumber \\
\delta \beta_V&=&\frac{ \epsilon}{2} \left( \frac{18-n}{30-n}
e^{2\phi} -1\right) \frac{d}{d\phi} \delta V + {\cal O}
(\epsilon^2). \label{263}
\end{eqnarray}
As observed in ref. \cite{6}, we may take $e^\phi$ to play the
role of loop expansion parameter, and restrict ourselves to the
region $e^{2\phi} \leq 1$, that is $-\infty < \phi \leq 0$. The
change of variables
\begin{equation}
\eta_1 =\ln (e^{-2\phi} -1), \ \ \ \ \ \
\eta_2 =\ln \left[ \frac{30-n}{12} \, \left( e^{-2\phi} -
\frac{18-n}{30-n}\right) \right], \label{264}
\end{equation}
transform this region into the following ones
\begin{eqnarray}
\phi =0, & \eta_1 \rightarrow -\infty,  & \eta_2 =0,  \nonumber \\
\phi \rightarrow -\infty, & \eta_1 \rightarrow +\infty,  & \eta_2
\rightarrow +\infty,  \label{265}
\end{eqnarray}
respectively. They simplify expressions (\ref{263}), which now
read:
\begin{eqnarray}
\delta \beta_G &=&- \epsilon \delta G, \nonumber \\
\delta \beta_\Phi &=& \epsilon  \frac{d}{d\eta_1} \delta \Phi +
{\cal O} (\epsilon^2), \nonumber
\\
\delta \beta_{f_2}&=& \epsilon  \frac{d}{d\eta_2} \delta f_2 +
{\cal O} (\epsilon^2), \nonumber
\\
\delta \beta_V &=& \epsilon  \frac{d}{d\eta_2} \delta V + {\cal
O} (\epsilon^2).
 \label{266}
\end{eqnarray}
The eigenfunctions corresponding to the new differential
operators on the right hand side are, respectively,
\begin{eqnarray}
\delta \Phi &\sim& e^{\alpha_1 \eta_1} = (e^{-2\phi} -
1)^{\alpha_1}, \nonumber \\
\delta f_2 &\sim & e^{\alpha_2 \eta_2} = \left[ \frac{30-n}{12}
\, \left( e^{-2\phi} -\frac{18-n}{30-n}\right) \right]^{\alpha_2},
\nonumber \\
\delta V &\sim & e^{\alpha_3 \eta_2} = \left[ \frac{30-n}{12}
\, \left( e^{-2\phi} -\frac{18-n}{30-n}\right) \right]^{\alpha_3},
\label{267}
\end{eqnarray}
and the corresponding eigenvalues are
\begin{equation}
\delta \beta_\Phi =\epsilon \alpha_1 \delta \Phi, \ \ \ \
\delta \beta_{f_2}=\epsilon \alpha_2 \delta f_2, \ \ \ \
\delta \beta_V=\epsilon \alpha_3 \delta V.
 \label{268}
\end{equation}
By imposing the initial condition $\delta \Phi (\phi =0) =0$ we
see that $\alpha_1 >0$.
Hence, we observe that the fixed point for $G$ is not ultraviolet stable
in the direction $\delta \Phi$. At the same time, within such a picture
it is clear
that we can always choose $\alpha_2$ and $\alpha_3$ to be negative
---since, for the other two directions, $f_2$ and
$V$, there is no way to make $\delta f_2 =0$ nor $\delta V =0$ at
$\phi =0$. (By means of an adequate choice
of arbitrary constant, we can choose $\delta f_2 (\phi =0) $ and
$\delta V (\phi =0) $ as small as we like, but never zero ---with
the choice above we have set these two values equal to 1).
Therefore, the fixed points for $f_2$ and $V$ are ultraviolet stable in
the direction $\delta \Phi$ but are always infrared unstable in this
direction (since they are never zero at $\phi =0$). In this sense the
theory (four RG functions) possesses a saddle fixed point.
 \bigskip

\section{Gauge dependence and fixed points}

It is of interest to discuss in some detail how the results of
the previous sections would change if we made a different choice
of the gauge condition. This question is not trivial at all, as
was already mentioned in \cite{8,9}, where it was shown that at
some gauges dilatonic gravity could be rendered one-loop finite
(except for the conformal anomaly term). However, explicit
calculation of the divergences for dilatonic gravity with matter
in a parameter-dependent gauge are extremely cumbersome. Hence,
we shall here consider the simplified model of ref. \cite{8}
\begin{equation}
S= \int d^d x\,  \sqrt{-g}\,  \left\{ {1 \over
2}  g^{\mu\nu} \partial_\mu \varphi
\partial_\nu \varphi +C_1 \varphi R+ f_3(\varphi) g^{\mu\nu}
\partial_\mu \chi_{i} \partial_\nu \chi^i\right\},
\label{27}
\end{equation}
where $f_3(\varphi) $ is some dilatonic coupling and $C_1$ some
constant. The one-loop divergences of the theory (\ref{27}) can
be studied in the one-parameter dependent gauge
\begin{eqnarray}
S_{gf} &=& - \frac{C_1}{2\alpha} \int d^d x\,  \sqrt{-g}\,
\chi_\mu \varphi \chi^\mu, \nonumber \\
\chi_\mu &=& \nabla_\mu h^\nu_{\ \mu} - \frac{1}{2} \nabla_\mu h
- \frac{\alpha}{\varphi} \partial_\mu \widetilde{\varphi} + X(\varphi)
\widetilde{\varphi}  \partial_\mu \varphi \nonumber \\
&&+ h_{\rho\sigma} \left[ Y_1 (\varphi) \left( \delta^\rho_\mu
\nabla^\sigma +\delta^\sigma_\mu \nabla^\rho \right) \varphi +
Y_2 (\varphi)  g^{\rho\sigma} \partial_\mu \varphi \right],
\label{28}
\end{eqnarray}
where $\varphi$  is a background field ($\varphi \rightarrow
\varphi +\widetilde{\varphi}$, $g_{\mu\nu}  \rightarrow g_{\mu\nu}
+h_{\mu\nu}$), $\widetilde{\varphi}$ and $h_{\mu\nu}$ are quantum
fields, $\alpha$ a gauge parameter, and $X,Y_1,Y_2$ are arbitrary
dilatonic functions.

The calculation of the one-loop effective action for the theory
(\ref{27}) in the gauge (\ref{28}) (let aside from the
$f_3(\varphi)$-dependence) has been performed in ref. \cite{8}.
The result is now
\begin{equation}
\Gamma_{div} =  \frac{1}{2\pi\epsilon} \int d^d x\,  \sqrt{-g}\,
\left\{ \frac{24-n}{12}R + \left[ - \frac{\alpha}{\varphi^2} +
nV_1 (\alpha, f_3(\varphi)) \right] g^{\mu\nu} \partial_\mu \varphi
\partial_\nu \varphi \right\}.
\label{29}
\end{equation}
Notice that in ref. \cite{8} only the dilatonic contribution ---
the (-$\alpha/\varphi^2$)-term--- had been found. The explicit
calculation of the function $V_1$, which depends on $f_3$ and
$\alpha$, requires a huge algebra (see \cite{8}). Fortunately,
knowledge of the precise form of $V_1$ is not necessary for our
considerations here, as we will see below.

Now, performing the background transformation in (\ref{27}) (in
exactly two dimensions)
\begin{equation}
g_{\mu\nu} \longrightarrow \exp \left( - \frac{\varphi}{4C_1}
\right) \, g_{\mu\nu}, \ \ \ \ \ \ \varphi =e^{-2\phi},
\label{30}
\end{equation}
we can rewrite (\ref{27}) as follows
\begin{equation}
S= \int d^d x\,  \sqrt{-g}\,  \left\{ {1 \over
16\pi G} R e^{-2\phi}+ f_3(\phi) g^{\mu\nu} \partial_\mu \chi_{i}
\partial_\nu \chi^i \right\},
\label{31}
\end{equation}
where $C_1$ is to be identified with $(16\pi G)^{-1}$ and
$f_3(\phi)$ with $\exp (-2 \Phi (\phi)$.
After proper transformation of the gauge-fixing Lagrangian
(\ref{28}), the $\Gamma_{div}$ corresponding to the theory
(\ref{27}) ---which is a particular case of the model
(\ref{5})--- in two dimensions is
\begin{equation}
\Gamma_{div}= \frac{1}{2\pi \epsilon} \int d^d x\,  \sqrt{-g}\,
\left\{ {24-n \over
12} R + 4 \left[ - \alpha + n V_1 (\alpha, \phi ) \right] g^{\mu\nu}
\partial_\mu \phi \partial_\nu \phi \right\}.
\label{32}
\end{equation}
For $\alpha =1$ the dilatonic gravity contribution in (\ref{32})
coincides with the result given in \cite{10,9,7,8} or in Eq. (\ref{10}).

Repeating the considerations of sects. 3 and 4 with $\Gamma_{div}$
(\ref{32}), we see that $A_2(\phi)$ changes  and hence
$\beta_\Phi$ changes also accordingly (but not $\beta_G$). Adding to the
central charge in (\ref{32}) the contribution from the Maxwell sector,
and looking for the fixed-point solution  (\ref{20}) and (\ref{21}), we
find
\begin{equation}
G^* = \frac{3\epsilon}{2 (30-n)}, \ \ \ \ \ \ \lambda^* =-
\frac{6\epsilon \alpha}{30-n}.
\label{33}
\end{equation}
The explicit form of $V_1$ (which is a combination of $\Phi$ and its
derivatives) is not necessary to find this fixed point, as it was also
the case with eq. (\ref{23})). The value of the gravitational coupling
constant at the ultraviolet fixed-point is gauge independent, as it
should be.

We realize that for any non-zero value of the gauge parameter a fixed
point $\lambda^*$ exists. (The case $\alpha =0$, which corresponds to a
gauge of Landau type, is in some sense degenerate and, hence, all the
considerations in this case must be given independently, including the
calculation of the explicit form of the function $V_1$). The change of
gauge parameter will affect the position of the fixed point through the
slope of the function $\Phi (\phi)$. However, the more physical issue of
the stability (or instability) of this fixed point will not be affected.
\bigskip

\section{Conclusions}

In this paper we have studied dilatonic-Maxwell gravity with matter near
two dimensions. The nice properties of this theory are: (i) its
renormalizability in $2+\epsilon$ dimensions and the asymptotically free
behavior in the ultraviolet regime shown by the gravitational coupling
constant. (ii) The fact that there is a  non-trivial fixed-point
solution for the dilatonic couplings and that, at the fixed point, the
theory may be represented in the standard form of the string-inspired
models that have been discussed recently. (iii) The increase of the
upper limit for the matter central charge ---due to the contribution of
the vector field--- from 24 (pure dilatonic gravity) or 25 (Einstein
gravity) to 30. This gives the possibility to extend the matter content
of the theory.

The investigation of the theory (\ref{5}) shows the way of considering
even more realistic toy models in $2+\epsilon$ dimensions. Indeed,
instead of the Maxwell term in (\ref{5}) we can insert its
non-abelian, Yang-Mills generalization corresponding to a gauge
potential $A_\mu^a$, $a=1,2, \ldots, N$. For simplicity, let us consider
the gauge group to be simple and compact and the structure constants
antisymmetric. Let us also add to  action (\ref{5}) the kinetic term for
$m$ fermions with a dilatonic coupling constant similar to $\Phi
(\phi)$. Such theory represents the unification of dilatonic QG with
matter (scalars, spinors and vectors) in $2+\epsilon$ dimensions.

The beta-function for the gravitational coupling constant is found to be
\begin{equation}
\beta_G = \epsilon G - \frac{1}{3} (48+ 12N - m - 2n) G^2.
\end{equation}
 From this expression we see that the matter central charge, $C=n+m/2$
is limited as follows:
\begin{equation}
0<C<24+6N,
\end{equation}
and for such a range of the central charge there is an ultraviolet
stable fixed point for the gravitational coupling constant, namely
\begin{equation}
G^* = \frac{3\epsilon}{48+12N-m-2n}, \ \ \ \ \epsilon >0.
\end{equation}
Hence, we still preserve asymptotic freedom in the gravitational
coupling constant. Moreover, we have now much less rigid restrictions to
the central charge: by increasing the dimension $N$ of the gauge
group we may increase the number of scalars and spinors in the theory.
The calculation of the one-loop effective action for this theory can be
carried out following ref. \cite{1} (and the second ref. of \cite{10})
and, at least qualitatively, the conclusions
about the existence of  non-trivial fixed point solutions of the form
(\ref{21}) remain true. In particular, we obtain for the above model:
\begin{equation}
\lambda^* = -\frac{12\epsilon}{48+12N-m-2n}, \ \ \
\lambda^*_f = -\frac{72\epsilon}{24+12N-m-2n}, \ \ \
\lambda^*_V = \frac{24\epsilon}{24+12N-m-2n}.
\end{equation}
The stability of the fixed-point solutions (as well as for the
corresponding fermion-dilaton coupling) can be studied similarly as in
sect. 4.

It is also of interest to consider supergravity models in $2+\epsilon$
dimensions. Some attempt in this direction has already been started in
ref. \cite{27}. Finally, an important issue is to study
quantum cosmology in frames of our ($2+\epsilon$)-dimensional model and,
in particular, to check carefully the claim of ref. \cite{13} that
RG considerations in $2+\epsilon$ dimensions may indeed help to solve
the spacetime singularity problem. We plan to return to some of these
questions in the near future.
 \vspace{5mm}

\noindent{\large \bf Acknowledgments}

We  would like to thank N. Sakai and Y. Tanii for very helpful
discussions.
This work has been supported by DGICYT (Spain), project Nos.
PB93-0035
and SAB93-0024, and by CIRIT (Generalitat de Catalunya).

\begin{appendix}

\renewcommand{\theequation}{{\mbox A}.\arabic{equation}}

\section{Appendix: Jackiw-Teitelboim model with matter in $2+\epsilon$
dimensions}

\setcounter{equation}{0}

One of the most popular models of 2D QG is the so-called
Jackiw-Teitelboim (JT) model \cite{18}
\begin{equation}
S_{cl}= \int d^d x\,  \sqrt{-g}\,  \left[ \mu^\epsilon e^{-2\phi} \left(
\frac{R}{16\pi G} + m^2 \right) \right].
\label{a1}
\end{equation}
The classical solution of this theory is
\begin{equation}
R= -16\pi G m^2
\label{a2}
\end{equation}
and shows the quite remarkable fact that there is only possibility for
existence of constant curvature geometries in the model. It also gives
the way to construct non-critical string theories without limitations in
the matter central charge of the theory \cite{18}.

By interaction of the theory (\ref{a1}) with scalars and vectors under
the form of a dilatonic coupling, as in (5), the property (\ref{a2}) is
lost. Dilatonic field equations become more complicated, involving also
contributions from the matter sector. By performing renormalization as
it was done before, we observe that the choice of $V(\phi)$ as in
(\ref{a1}) breakes down at the quantum level (25), since $\lambda_V^*
\neq 0$. Hence, the JT model with matter is not a non-trivial fixed
point of the RG in $2+\epsilon$ dimensions. Of course, starting from
action (5) and carring out a non-singular gauge transformation of the
metric as in (26) one can obtain in the action ---at the fixed point---
a term of the form (\ref{a1}). However, the price for this will be the
appearence of a dilatonic kinetic term, what destroys the crucial
property of the JT model (\ref{a2}) already in the sector of pure
dilatonic gravity.

\end{appendix}

\newpage


\section{Introduction}

 \noindent
 Les projections st\'er\'eographiques montrent que les galaxies
 ne sont pas distribu\'ees au hasard, mais qu'elles sont agglom\'er\'ees
 en groupes ou en amas
 de plusieurs centaines de galaxies s'\'etendant sur quelques
 dizaines de m\'egaparsecs.
 En 1953-1956, l'astronome G\'erard de Vaucouleurs a
 montr\'e que les galaxies proches (vitesse de r\'ecession
 plus petite que $3000 km.s^{-1}$) forment une structure plane connue
 maintenant comme le Super Amas Local. En coordonn\'ees galactiques le p\^ole
 de ce plan est $l=47 \deg$; $b=6 \deg$.
 En 1982 Zel'dovich, Einasto et Shandarin notent
 que les amas de galaxies lointains (vitesse sup\'erieure \`a 7000
$km.s^{-1}$),
 sont situ\'es pr\'ef\'erentiellement dans un plan. Ce r\'esultat est
confirm\'e
 par Shectman en 1985 puis par Tully (1986 et 1987) qui note que
 ce plan pr\'ef\'erentiel est voisin de celui du Super Amas Local.
 Au m\^eme moment, Bottinelli et al. (1986) signalent une structure
 aplatie visible sur une projection st\'er\'eographique de Flamsteed
(projection
 conservant les aires) r\'ealis\'ee avec des galaxies individuelles. Ils
donnent
 la position du p\^ole $l=57\deg$; $b=26\deg$, valeur r\'evis\'ee plus tard en
 $l=57\deg$; $b=22\deg$ (Paturel et al. 1988).

 \section{Recherche du plan privil\'egi\'e pour l'\'etude de la cin\'ematique}

 \noindent Dans l'objectif de se ramener \`a un plan dans lequel on
 puisse \'etudier plus facilement la cin\'ematique de l'Univers
 local, nous avons cherch\'e de mani\`ere syst\'ematique le plan
 sur lequel se situe le plus grand nombre de galaxies de l'Univers proche.
 Nous avons construit un \'echantillon de 24324 galaxies \`a partir de la
 base de donn\'ees extragalactiques des observatoires de Lyon-Meudon
 (LEDA), en extrayant pour chaque
 galaxie les param\`etres suivants: -- les coordonn\'ees galactiques
 $l$ et $b$ correspondant \`a la longitude et \`a
 la latitude respectivement -- la vitesse radiale de r\'ecession v
 qui sera utilis\'ee comme mesure de la distance avec la loi de Hubble.
 \begin{equation}
 d=v/H
 \end{equation}
 o\`u $d$ est la distance en m\'egaparsecs, $v$ la vitesse radiale
en $km.s^{-1}$ et $H$ la constante de Hubble
 en $km.s^{-1}.Mpc$. Dans tout ce travail la valeur de cette constante
 sera prise arbitrairement \'egale \`a 75 $km.s^{-1}.Mpc$ (c'est \`a
 dire une valeur moyenne entre les valeurs extr\^emes publi\'ees).

 \noindent Par ailleurs, afin que le r\'esultat ne d\'epende pas de
 l'\'echantillon
 utilis\'e, nous nous sommes assur\'es de la compl\'etude de l'\'echantillon,
 pour travailler avec toutes les galaxies
 de diam\`etre sup\'erieur \`a une certaine limite.
 Pour cela, nous avons construit la courbe

 \begin{equation}
logN_{t}=f(logD),
 \end{equation}

 o\`u $N_{t}$ est le nombre de galaxies de diam\`etre isophotal
 sup\'erieur \`a D.
 L'isophote adopt\'e pour la d\'efinition du diam\`etre est par convention
 de 25 magnitudes par seconde de degr\'e au carr\'e
 \footnote {L'isophote est tel que le flux par
  seconde de degr\'e au carr\'e correspondrait au flux d'une \'etoile
  de magnitude 25}
 et l'unit\'e de mesure
 est 0,1 minute de degr\'e.
 Pour les galaxies n'ayant pas de mesure de diam\`etre mais seulement une
 mesure de magnitude, nous avons d\'eduit le diam\`etre de la relation
       $logD = -0,2 m + 4$
 o\`u m est la magnitude apparente totale.

\noindent Il est facile de montrer que pour un univers
homog\`ene le nombre de galaxies augmente avec la distance selon une loi
en puissance (Paturel et al., 1994).
La pente de la
 relation 2 est alors constante tant que l'\'echantillon est complet.
 Au-del\`a, la courbe s'infl\'echit brusquement et le point de d\'ecrochage
 donne la limite de compl\'etude.
 Le r\'esultat de ce test est que la compl\'etude est bonne jusqu'\`a
 $logD=1,2$. Finalement, notre \'echantillon complet, contenant toutes
 les galaxies dont la vitesse de r\'ecession est inf\'erieure \`a
 $15000 km.s^{-1}$ et telles que $logD>1,2$, se restreint \`a
 5683 galaxies.

 \noindent
 Pour un plan d\'efini par les coordonn\'ees de son p\^{o}le: $lp$ et $bp$
 (longitude et latitude galactique respectivement), nous avons compt\'e
 le nombre de galaxies se trouvant dans une zone de $\pm 15\deg$ de part
 et d'autre du plan.
 Nous avons fait une recherche syst\'ematique du meilleur plan en donnant des
 valeurs de 0 \`a $90\deg$ \`a $b$ et de 0 \`a $360\deg$ \`a $l$.\\

 \noindent Nous pouvons voir dans la Table ~\ref{Tpole} le r\'esultat
 du comptage de galaxies:
 $d$ est la distance limite du comptage (donn\'ee en m\'egaparsec),
 $Ngal$ est le nombre total de galaxies situ\'ees dans cette limite,
 $lp$ et $bp$ sont les coordonn\'ees galactiques du p\^{o}le d\'efinissant
 le plan regroupant le plus grand nombre de galaxies et
 $\% gal$ est le pourcentage de galaxies se trouvant \`a $\pm 15\deg$ du
 plan pr\'ec\'edent.
 \noindent Si les galaxies \'etaient r\'eparties uniform\'ement dans la
 sph\`ere de comptage de rayon $d$, en regardant dans une zone de $30\deg$,
 c'est \`a dire d'angle solide $\pi$ st\'eradians, on devrait y trouver
 25\% du nombre total de galaxies de cette sph\`ere (proportion des
 angles solides).
 Dans la Table~\ref{Tpole}, on voit que le nombre de galaxies trouv\'ees
 dans le plan est bien sup\'erieur (44\%), ce qui montre que la r\'epartition
 des galaxies dans l'Univers local est plut\^ot aplatie.

\noindent Une position du p\^{o}le \`a $lp=52\deg$ et $bp=16\deg$ semble
 particuli\`erement stable entre 80 Mpc et 200 Mpc. Ce p\^ole est assez proche
 de celui trouv\'e en 1986 et 1988. Ce p\^ole d\'efinira
 le plan hypergalactique.
 D'apr\`es la Table~\ref{Tpole}, on peut \'egalement donner une nouvelle
 mesure de la position du p\^ole du Super Amas Local.
 En 1976 de Vaucouleurs donnait
 $lp=47\deg$ et $bp=6\deg$ pour les galaxies jusqu'\`a environ
 3000 $km.s^{-1}$, ici nous trouvons $lp=46\deg$ et $bp=14\deg$,
 pour la m\^eme r\'egion.

 \section{Repr\'esentation de l'Univers local dans le plan hypergalactique}

 \noindent Par une transformation de coordonn\'ees, on peut repr\'esenter
 les galaxies de la zone $\pm 15\deg$ en projection
 sur le plan hypergalactique. La transformation est la suivante:
 \begin{equation}
        X= d.cos(hgb).cos(hgl)
 \end{equation}
 \begin{equation}
        Y= d.cos(hgb).sin(hgl)
 \end{equation}
 \begin{equation}
        Z= d.sin(hgb)
 \end{equation}
 Les coordonn\'ees hypergalactiques (hgl,hgb) sont calcul\'ees
 par un changement de
 rep\`ere \`a partir des coordonn\'ees galactiques du p\^ole et de l'origine
des
 longitudes.
 Cette origine des longitudes hypergalactiques est d\'efinie arbitrairement
 dans le plan hypergalactique aux coordonn\'ees:
 $l(origine)=l(pole)$ et $b(origine)=b(pole)-90\deg$.
 On peut voir cette repr\'esentation sur la figure Fig.~\ref{planhyper}.

 Cette repr\'esentation appelle plusieurs remarques:
\begin{itemize}
\item Des structures radiales tr\`es allong\'ees sont visibles, par exemple \`a
 la p\'eriph\'erie du graphique.
 Ces structures r\'esultent de l'incertitude sur la distance
 d\'eduite de la vitesse radiale. En effet la vitesse radiale
 mesur\'ee n'est pas seulement la vitesse cosmologique (fonction de la
 distance). Elle compte aussi la composante radiale de la dispersion
al\'eatoire
 des vitesses. Il appara\^{\i}t donc une dispersion sur les distances, alors
 que pratiquement aucune erreur n'existe en direction. Les amas de galaxies
 apparaissent donc \'etir\'es le long de la ligne de vis\'ee. La cons\'equence
 pratique est qu'il ne faudra pas accorder trop de r\'ealit\'e aux structures
 radiales qui peuvent provenir de cet artefact.

\item On voit \'egalement une zone (verticale sur le graphique) o\`u aucune
 galaxie n'appara\^{\i}t. Cette zone r\'esulte de l'absorption due \`a la
 poussi\`ere du disque de notre Galaxie qui cache les autres galaxies de
 l'Univers dans cette direction. Les relev\'es dans l'infrarouge devraient
 permettre de combler partiellement cette zone d'absence.

\item Le r\'esultat le plus int\'eressant visible sur la Fig.~\ref{planhyper}
 est la structure en forme d'anneau centr\'ee sur le Super Amas Local
(amas Virgo) et d'un rayon d'environ
 90 Mpc (pour $H=75 km.s^{-1}.Mpc^{-1}$).
On ne peut pas invoquer l'artefact vu pr\'ecedemment puisque
 cette structure n'est pas radiale.
 Une projection perpendiculaire au plan hypergalactique (Fig.~\ref{planperpan})
 permet de dire que cet anneau est la trace d'une
 coquille ellipso\"{\i}dale, sensiblement centr\'ee sur le Super amas local.
 La structure connue sous le nom de "grand mur" (Huchra et al., 1990)
 appara\^{\i}t \^{e}tre une portion de cette coquille.
\end{itemize}

 \section{Conclusion}
 \noindent Avec un \'echantillon complet en diam\`etre apparent, nous avons
 trouv\'e un plan privil\'egi\'e dans l'Univers local. Ce plan est
 identifi\'e au plan hypergalactique trouv\'e avec la base de donn\'ees
 de Lyon-Meudon  (Bottinelli et al. 1986; Paturel et al. 1988).
 Ce plan s'identifie \`a plus grande \'echelle avec le plan trouv\'e par
 Tully (1986).
 La repr\'esentation des galaxies dans ce plan fait
 appara\^{\i}tre la trace d'une coquille
 ellipso\"{\i}dale aplatie approximativement centr\'ee sur le Super Amas Local
 (c'est \`a dire approximativement sur le centre de l'amas Virgo).
 Ce r\'esultat confirme clairement l'existence de la tr\`es grande
 structure {\it hypergalactique} mentionn\'ee par Paturel et al. (1988).

 Cette structure est int\'eressante  car c'est actuellement la
plus grande structure connue.
Son \'etude tant cin\'ematique que dynamique pose quelques probl\`emes.
Si la gravitation est responsable de la forme \`a sym\'etrie centrale,
on  comprend mal comment l'\'equilibre d'une telle structure pourrait
\^etre assur\'e.

\vspace{1.5cm}

\small
 REFERENCES BIBLIOGRAPHIQUES \\

   [1]     L. BOTTINELLI, P. FOUQUE, L. GOUGUENHEIM, G. PATUREL,
 {\it Structure \`a grande \'echelle visible avec la base de donn\'ees
 extragalactiques}, 1986, La dynamique des structures gravitationnelles.

   [2]    J.P. HUCHRA, J.P. HENRY, M. POSTMAN et M.J. GELLER,
 1990, The Astrophysical Journal, 365, 66

   [3]   G. PATUREL, L. BOTTINELLI, L. GOUGUENHEIM et P. FOUQUE,
 {\it New determination of the pole of a "hypergalactic" large-scale
 system}, Astronomy and Astrophysics, 1988, 189, 1

   [4]   G. PATUREL, H. DI NELLA, L. BOTTINELLI, P. FOUQUE et L. GOUGUENHEIM
 {\it Interpretation of a completeness curve}, Astronomy and Astrophysics,
 1994 (Sous-presse)

   [5]   S.A. SHECTMAN, 1985, The Astrophysical Journal Supplement Series,
 57, 77

   [6]   R.B. TULLY,
 {\it Alignment of clusters and galaxies on scales up to 0.1c},
  1986, The Astrophysical Journal, 303, 25

   [7]   R.B. TULLY,
 {\it More about clustering on a scale of 0.1c},
 1987, The Astrophysical Journal, 323, 1

   [8]   G. DE VAUCOULEURS, 1953, Astronomical Journal, 58, 30

   [9]   G. DE VAUCOULEURS, 1956, Vistas Astr., 2, 1584

   [10]   G. DE VAUCOULEURS, A. DE VAUCOULEURS, JR. H.G. CORWIN,
 {\it Second Reference Catalogue of Bright Galaxies}, 1976, University of Texas
 Press, Austin

   [11]   YA. B. ZEL'DOVICH, J. EINASTO et S.F. SHANDARIN, 1982, Nature, 300,
407

\vspace{1cm}

{\it
\hspace {8cm}  \noindent Observatoire de Lyon

\hspace {8cm}  \noindent 69561 Saint-Genis Laval CEDEX, France

\hspace {8cm} \noindent t\'el\'ephone: 78.86.83.83

\hspace {8cm} \noindent t\'el\'ecopie: 78.86.83.86

\hspace {8cm} \noindent e-mail: <EMAIL>
}

\newpage
 \begin{table}
 \caption{
Nombre de galaxies situ\'ees \`a une distance inf\'erieure \`a d et
pourcentage (\%gal) de galaxies localis\'ees
 \`a $\pm 15\deg$ du plan d\'efini par le p\^ole de coordonn\'ees
 $lp$ et $bp$.
Ce pourcentage est tr\`es sup\'erieur \`a ce que laisse attendre
une distribution homog\`ene des galaxies (voir le texte).
 -- {\it Number of galaxies located at a distance smaller than d and
 percentage (\%gal) of galaxies located
 at $\pm 15\deg$ of the plane defined by the pole of coordinates
$lp$ and $bp$. This percentage is much higher than the one expected
for an homogeneous distribution of galaxies (see text).}
}
\vspace{1cm}
\label{Tpole}
\begin{tabular}{rllllllllll}
\hline
 $d (Mpc)$   : &  20 &  40 &  60 &  80 & 100 & 120 & 140 & 160 & 180 & 200 \\
 $Ngal$        : & 1037& 2774& 3945& 4939& 5310& 5507& 5593& 5643& 5661& 5683\\
 $lp (\deg)$ : &  46 &  46 &  46 &  51 &  52 &  52 &  52 &  52 &  52 &  52 \\
 $bp (\deg)$ : &  14 &  14 &  17 &  16 &  16 &  16 &  16 &  16 &  16 &  16 \\
 $\% gal $     : &  55 &  46 &  46 &  45 &  45 &  44 &  44 &  44 &  44 &  44 \\
\hline
\end{tabular}
\end{table}

\newpage

 \begin{figure}
 \caption{
 Vue du plan hypergalactique. La trace de la coquille est visible.
 Sa position est approximativement centr\'ee sur l'amas Virgo.
 -- {\it Face-on view of the hypergalactic plane. The trace of
 the shell is visible. Its position is nearly centered on
 the Virgo cluster.}
 }
\epsfxsize=11cm
\epsfysize=8cm
\epsffile{fig1.ps}
 \label{planhyper}
 \end{figure}

 \begin{figure}
 \caption{
 Vue perpendiculaire au plan hypergalactique. Une trace similaire
 \`a celle que nous avions dans le plan hypergalactique est encore visible,
 ce qui fait penser que la structure \`a la forme d'une coquille
 ellipso\"{\i}dale.
 --{\it View perpendicular to the hypergalactic plane. A trace
similar to the one seen on the hypergalactic plane is still visible.
This suggests that the structure has the shape of an ellipsoidal shell.}
}
\epsfxsize=11cm
\epsfysize=8cm
\epsffile{fig2.ps}
 \label{planperpan}
 \end{figure}

 \end{document}