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\begin{document}


\centerline{\large \sc On Useful Conformal Tranformations
In General Relativity}

\vskip 4mm

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Andre Gon\c{c}alves de Lima\ , 
%%%% Diego Ferreira Carneiro
%%%% Emanuel Ant\'onio Freiras,
% \centerline{
% Diego F. Carneiro\ , \quad 
% Emanuel A. Freiras\ , \quad  Bruno Gon\c{c}alves\ ,}
% \vskip 1mm
%
% \centerline{ 
% Andre G. de Lima\ ,  \ and \   Ilya L. Shapiro}
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\centerline{\bf 
D.F. Carneiro, \ 
E.A. Freiras,  \  B. Gon\c{c}alves,  \
A.G. de Lima, \ and \ I.L. Shapiro}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \centerline{
% D.F. Carneiro, \quad
% E.A. Freiras, \quad  B. Gon\c{c}alves, \quad
% A.G. de Lima, \quad I.L. Shapiro}
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\vskip 5mm

\centerline{{\small\sl
Departamento de F\'{\i}sica, ICE, 
Universidade Federal de Juiz de Fora}}
\vskip 1mm

\centerline{{\small\sl 
Juiz de Fora, CEP: 36036-330, MG,  Brazil}}

\vskip 5mm
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\begin{quotation}
\noindent
{\large\ Abstract.}
\quad
{Local conformal transformations are known as a useful 
tool in various applications of the gravitational theory, 
especially in cosmology. We describe some new aspects 
of these transformations, in particular using them for 
derivation of Einstein equations for the cosmological 
and Schwarzschild metrics. Furthermore, the conformal 
transformation is applied for the dimensional reduction 
of the Gauss-Bonnet topological invariant in $d=4$ to 
the spaces of lower dimensions.}  
\end{quotation}
\vskip 12mm

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\section{Introduction}

General Relativity (GR) is a successful  relativistic 
theory of gravitation. According to the existing data, 
its predictions fit nicely to the most of the tests and
the limits of its validity are likely to emerge only in 
the extreme situations when one has to account for, e.g. 
quantum effects \cite{LL-2,Weinberg72,MTW,wald}. The 
quantum theory 
of matter fields in curved space-time has achieved many 
solid results (see e.g. \cite{birdav,fulling,book}, and 
may be viewed as a source of important applications
in cosmology and black hole physics. Some of these 
achievements are related to the so-called conformal 
anomaly, which is nothing but the quantum violation of 
the local conformal symmetry which is imposed at the 
classical level. In order to illustrate this statement
it is sufficient to mention that this anomaly is 
related to such relevant phenomenon as the black hole 
evaporation (Hawking effect). The anomaly is also related 
to the derivation of the effective low-energy effective 
action of (super)string - the main candidate to be the 
unified theory of quantum gravity and other fundamental 
interactions. 

The importance of local conformal symmetry makes natural 
the idea to include it into the basic course of GR. 
However, the standard point of view is that such extension
of the program is difficult because of the shortage 
of time. In this article we shall suggest a new possibility
that including the conformal transformation into the 
program may be compatible with certain saving of time and 
efforts due to the relevant simplification in the derivation 
of Einstein equations for the metrics related to the most 
fundamental applications of GR such as cosmological and 
Schwarzschild solutions. Furthermore, we shall use the
conformal transformation to investigate a problem which 
goes slightly beyond the standard course of GR, that 
is establish the relations between the topological 
invariants in four and two dimensional spaces via the 
restricted dimensional reduction. It is supposed that this 
relation will be useful for those willing to learn more 
advanced aspects of the gravitational theory.

The paper is organized as follows. In section 2 we 
shall present a brief review of the main properties 
of the curvature tensor (see, e.g. 
\cite{petrov,Weinberg72,DNF} for the introduction). 
In particular, we shall 
prove a simple factorization theorem describing 
important properties of curvature for the metric
of the special factorized form. In section 3 the 
list of formulas concerning the local conformal 
transformation in the space of an arbitrary dimension 
is presented. For the sake of generality we shall 
include the transformations not only for the curvature
tensors but also for the quadratic invariants built 
from this tensor and its derivatives. In section 4
the derivation of the Einstein equations for the 
cosmological and spherically symmetric metrics is 
presented. Section 5 is devoted to establishing the 
relation between the topological invariants in 
dimensions 4, 2  and 3 via the factorization theorem 
from the section 2 and conformal transformations. 
Finally, in the last section we draw our conclusions.

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\section{Brief review of curvature tensor}

Let us start with the properties of the curvature tensor
which will be used below. A comprehensive introduction 
can be found in the monographs on GR and Differential 
Geometry \cite{petrov,Weinberg72,wald,MTW,DNF}.

The action of covariant derivative on the tensor 
$\,T^{\mu_1 \mu_2 ... \mu_k}\,_{\nu_1\nu_2 ... \nu_l}\,$
is defined by the relation 
\beq
\nabla_{\alpha} T^{\mu_1 \mu_2 ... }\,_{\nu_1\nu_2 ...} 
&=& \pa_{\alpha} 
T^{\mu_1 \mu_2 ... }\,_{\nu_1\nu_2 ... }
+ \Gamma^{\mu_1}_{\lambda \alpha}
T^{ \la\mu_2 ...}\,_{\nu_1\nu_2 ...}
+ \Gamma^{\mu_2}_{\lambda \alpha}
T^{\mu_1\la  ... \mu_k}\,_{\nu_1\nu_2 ... \nu_l}+...
\nonumber
\\
&-& \Gamma^{\lambda}_{\nu_1 \alpha} 
T^{\mu_1 \mu_2 ... \mu_k}\,_{\la\nu_2 ... \nu_l}
- \Gamma^{\lambda}_{\nu_2 \alpha} 
T^{\mu_1 \mu_2 ... \mu_k}\,_{\nu_1\la ... \nu_l}\,-\,...\,\,,
\label{man 11}
\eeq
providing tensor transformation rule under the general 
coordinate transformations.
The coefficients $\,\Gamma^{\lambda}_{\alpha \beta}\,$ are 
defined according to
\beq
\Gamma^{\lambda}_{\mu\nu} 
= \frac{1}{2}\,g^{\la\tau} \left(\pa_{\mu} g_{\tau\nu} 
+ \pa_{\nu}g_{\tau\mu} - \pa_{\tau} g_{\mu\nu}\right)\,,
\label{man 16}
\eeq
such that the covariant derivative $\,\na_\al\,$ satisfies 
the metricity condition $\,\na_\al g_{\mu\nu}=0\,$ 
and also is torsionless 
$\,\Gamma^{\lambda}_{\mu\nu}=\Gamma^{\lambda}_{\nu\mu}$.

Consider the commutator of the 
two covariant derivatives over the vector $\,T^\al$. 
By direct calculations we find
\beq
\left[\na_\mu\,,\,\na_\nu\right]\,T^\al 
\,=\, \na_\mu\na_\nu \,T^\al - \na_\nu\na_\mu \,T^\al
\,=\,-\,T^\la\cdot R^\al_{\, \cdot\,\la \,\nu\mu}\,,
\label{curve 7}
\eeq
where 
\beq
R^\al_{\, \cdot\,\la\nu\mu}\,=\, \pa_\nu \Ga^\al_{\la\mu} 
- \pa_\mu \Ga^\al_{\la\nu}
+\Ga^\tau_{\la \mu}\,\Ga^\al_{\tau \nu} 
-\Ga^\tau_{\la \nu}\,\Ga^\al_{\tau \mu} 
\label{curve 10}
\eeq
is called (Riemann) curvature tensor. 

The relevance of the Riemann tensor is due to its 
universality. One can easily show that the commutator 
of the two covariant derivatives acting on any tensor 
is a linear combination of the curvature tensors, e.g.
\beq
\left[\na_\mu\,,\,\na_\nu\right]\,W^\al_\be 
= R^\la_{\,\cdot\, \be\nu \mu}\,W^\al_\la
\,-\, R^\al_{\, \cdot\,\la\nu\mu}\,W^\la_\be \,\,.
\label{curve 12}
\eeq

The curvature tensor has following algebraic symmetries,
which can be better seen on the completely covariant 
version $\,R_{\al\be\mu\nu} 
\,=\, g_{\al\ga}\,R^\ga_{\,\cdot\,\be\mu\nu}\,$:
\beq
R_{\al\be\mu\nu}\,=\,-\,R_{\al\be\nu\mu}
\,=\,R_{\be\al\nu\mu}\,=\,R_{\mu\nu\al\be}\,,
\label{curve 18}
\eeq
where symmetry between the pairs of the indices 
supplements the antisymmetry in the indices of 
each pair, and
\beq
R_{\al\be\mu\nu}+R_{\al\nu\be\mu}+R_{\al\mu\nu\be}\,=\,0\,.
\label{curve 26}
\eeq

It proves useful to define the Ricci tensor
as the contraction of the Riemann curvature 
\beq
R_{\mu\al}\,=\,R^\be_{\,\cdot\,\al\be\mu}
\,=\,g^{\nu\be}\,R_{\nu\al\be\mu}\,.
\label{curve 29}
\eeq
The Ricci tensor is symmetric $\,R_{\mu\nu}\,=\,R_{\nu\mu}$.
Further contraction produces the scalar curvature
$\,R=R^\al_\al$\,.

The algebraic properties of the Riemann, Ricci tensors
and of the scalar curvature are 
the same in the Riemann or pseudo-Riemann spaces of any 
dimension. However, in the spaces of lower dimensions 
the curvature tensors are more restricted than in the 
general $D$-dimensional case. The extreme case is the 
$\,D=1\,$ space, 
where all curvatures vanish identically. 

In order to understand the possible role of $\,D\,$ better,
one has to calculate the number of independent components 
of the tensors $\,R_{\mu\nu\al\be}\,$ and $\,R_{\mu\nu}$. 
Due to the algebraic symmetries (\ref{curve 18}), the 
curvature tensor with 3 equal indices vanish. Therefore,
we have only three possible situations: 

{\it 1)}.\ 
Two couples of equal indices, that is the construction like
$\,R_{0101}$. The combinatorial calculus tells us that the 
number of distinct combinations of this type in a 
$D$-dimensional space is $\,n_1=C_D^2=\frac{D(D-1)}{2}$.

{\it 2)}.\ 
One couple of equal indices plus two distinct indices, 
that is the construction like $\,R_{0102}$. The 
number of distinct combinations here is 
$\,n_2=D\,C_{D-1}^2=\frac{D(D-1)(D-2)}{2}\,$.
%%but of course this possibility exists only for $D\geq 3$.

{\it 3)}.\ 
Four distinct indices, that is the construction like 
$\,R_{0123}$. In this case we have 
$$
n_3=\frac{2D(D-1)(D-2)(D-3)}{4!}\,.
$$ 
%% for $D\geq 4$..

Summing up the three expressions, we obtain the number
of independent components of the Riemann tensor in 
$\,D$-dimensional space.
\beq
N_D\,=\,n_1+n_2+n_3\,=\,\frac{D^2(D^2-1)}{12}\,.
\label{number 1}
\eeq

Now we are in a position to consider the particular cases 
of lower dimensions. Let us start from the $\,D=2\,$ case,
where only the option {\it 1)} takes place.
$\,N_2=1\,$ and we see that the number of independent 
components of the Riemann tensor equals to the one of the 
scalar curvature. Since both of them are linear combinations 
of the partial derivatives of the metric, they must be 
proportional. Taking the algebraic symmetries 
into account, we obtain 
$$
R_{\mu\nu\al\be}\,=\,t\,R\,\left(g_{\mu\al}\,g_{\nu\be}
- g_{\mu\be}\,g_{\nu\al}\right)\,,
$$
where $\,t\,$ is an unknown coefficient. Making a
contraction, we arrive at 
\beq
R_{\mu\al}\,=\,t\,R\,g_{\nu\be}\qquad
\mbox{and}\qquad R\,=\,2\,t\,R\,.
\label{number 2}
\eeq
Hence $t=1/2$ and we obtain well-known relations
\beq
R_{\mu\nu\al\be}\,=\,\frac12\,
R\,\left(g_{\mu\al}\,g_{\nu\be}-g_{\mu\be}\,g_{\nu\al}\right)
\,,\qquad 
R_{\mu\al}\,=\,\frac12\,R\,g_{\nu\be}\,.
\label{number 3}
\eeq

For $\,D=3\,$ space we have, according to (\ref{number 1}),
six independent components of both Riemann and Ricci tensors. 
The Riemann tensor is a linear combination of the  
scalar curvature and the Ricci tensor $\,R_{\mu\nu}$.
Taking into account the algebraic symmetries and 
making contractions we obtain the relation which always 
holds for $D=3$ but not necessary for $D\geq 3$
\beq
R_{\mu\nu\al\be}
\,=\,\left(R_{\mu\al}\,g_{\nu\be}-R_{\mu\be}\,g_{\nu\al}
+R_{\nu\be}g_{\mu\al}-R_{\nu\al}g_{\mu\be}\right)
-\frac12R\left(g_{\mu\al}g_{\nu\be}
-g_{\mu\be}g_{\nu\al}\right).
\label{number 5}
\eeq

Obviously, the relation (\ref{number 5}) means that there
is some part of the Riemann tensor which automatically 
vanish in $\,D=3\,$ but may be non-zero for $D>3$.
The corresponding object is called Weyl tensor and is
defined as follows
\beq
C_{\mu\nu\al\be}\,=\,R_{\mu\nu\al\be}
\,-\,\frac{1}{D-2}\,
\left(R_{\mu\al}\,g_{\nu\be}-R_{\mu\be}\,g_{\nu\al}
+R_{\nu\be}\,g_{\mu\al}-R_{\nu\al}\,g_{\mu\be}\right)
\nonumber
\\
\,+\,\frac{1}{(D-1)(D-2)}\,R\,\left(g_{\mu\al}\,g_{\nu\be}
-g_{\mu\al}\,g_{\nu\be}\right)\,.
\label{number 6}
\eeq
The Weyl tensor has, by construction,  the same algebraic
symmetries as the Riemann tensor and also two additional 
properties 
\beq
g^{\mu\al}\,C_{\mu\nu\al\be}(D)\,=\,0\,,\qquad
C_{\mu\nu\al\be}(D=3)\,\equiv\,0\,.
\label{Weyl}
\eeq
Another important feature of the Weyl tensor will be discussed 
in the next section.

The last sort of curvature which we have to 
define is called Einstein tensor 
\beq
G_{\mu\nu}\,=\,R_{\mu\nu}
\,-\,\frac{1}{2}\,R\,g_{\mu\nu}\,.
\label{number 7}
\eeq

The Einstein tensor is constructed, exactly as Riemann, 
Ricci, Weyl tensors and scalar curvature, from the metric
and its derivatives. All these tensors are homogeneous 
functions of the second order in the derivatives.
The importance of the Einstein tensor is due to the fact 
that the Einstein equations, which define the gravitational 
fields in General Relativity, have the form 
\beq
G_{\mu\nu}\,=\,8\pi G\,T_{\mu\nu}\,,
\label{number 8}
\eeq
where $\,G\,$ is the Newton constant and $\,T_{\mu\nu}\,$
is a source term which is called Energy-Momentum Tensor
(or stress tensor). In the next sections we shall develop 
the efficient method to calculate the Einstein tensor 
and, if necessary, also Riemann, Ricci and Weyl tensors,
for the particular metrics of special physical interest. 

The metric which provides $\,R_{\mu\nu\al\be}=0\,$
are necessary flat. However, there are situations 
when $\,R_{\mu\nu\al\be}\neq 0$, but some of the reduced
versions of curvature vanish. One can distinguish 
Ricci flat or Weyl flat metrics, which are not flat 
in the proper sense but provide the vanishing Ricci 
or Weyl tensors, correspondingly.
As we already know, any $\,D=3\,$ space is Weyl flat,
for example. 

In many problems of GR we do not need an arbitrary 
metric but instead just a particular form with a smaller 
number of independent components than the general one.
Sometimes it is possible to reduce the derivation 
of the curvature tensors for these particular 
metrics to the one in a lower dimensional case.
Let us now formulate the theorem about the factorized 
metrics which will prove very useful in what follows.
\vskip 1mm

{\bf Theorem.}\quad
Consider the $\,D\,$-dimensional Riemann or pseudo-Riemann
manifold with the coordinates 
$\,x^\mu=\left(x^a,\,x^i\right)$, where 
$\,a,b,c,...=1,2,...,n\,$ and $\,i,j,k,...=n+1,n+2,...,D$.
Let us assume that the metric is factorized 
\beq
g_{\mu\nu}\,=\,\left\|
\begin{array}{cc}
g_{ab}(x^a)  &  0           \\ 
0            &  g_{ij}(x^i) \end{array}
\right\|\,.
\label{number 9}
\eeq
For this metric the elements with the mixed
indices $\,g_{ai}\,$ are zero, and only the blocks with 
$\,g_{ab}\,$ and $\,g_{ij}\,$ do not vanish. 
Moreover, the metric elements in each block 
depend only on their ``own'' coordinates 
$\,g_{ab}=g_{ab}(x^a)\,$ and $\,g_{ij}=g_{ij}(x^i)$.
The statement is that, for the metric (\ref{number 9}),
the Riemann tensor is also factorized. This means that 
all its components with the mixed indices do vanish. 
\vskip 2mm

{\bf Proof.}\quad
First of all, it clear that the inverse metric will 
also have the factorized block structure
\beq
g^{\mu\nu}\,=\,\left\|
\begin{array}{cc}
g^{ab}(x^a)  &  0           \\ 
0            &  g^{ij}(x^i) \end{array}
\right\|\,.
\label{number 10}
\eeq
Consider the Christoffel symbol with the mixed 
indices, e.g. 
\beq
\Ga^a_{ib}\,=\,g^{ac}\left( \pa_i g_{bc}
+\pa_b g_{ic}-\pa_c g_{ib} \right)\,.
\label{number 11}
\eeq
Taking into account the relations $\,g_{ic}=0\,$ and
$\,\pa_i g_{bc}=0$, we obtain $\,\Ga^a_{ib}=0$. In the
same way one can prove that other mixed components 
vanish
\beq
\Ga^i_{jb}\,=\,\Ga^i_{ab}\,=\,\Ga^a_{ij}\,=\,0\,.
\label{number 12}
\eeq
Thus, only the ``pure'' components of the Christoffel 
symbol  $\,\Ga^a_{bc}(x^a)\,$ and $\,\Ga^k_{ij}(x^i)\,$ 
can be non-zero. 

Let us now consider the Riemann tensor with the 
mixed components. According to the formula (\ref{curve 10})
it is zero because {\it i)} the ``mixed'' derivatives
(e.g. $\,\pa_i\Ga^a_{bc}$) equal zero; \ 
{\it ii)} the $\,\Ga \Ga$-type terms with the mixed 
indices can not be constructed from the $\,\Ga$-symbols
with the ``pure'' indices.

The last observation is that one can easily extend the 
factorization theorem for the case of Ricci tensor. The 
scalar curvature is given by the sum 
$\,\,R(g_{\mu\nu}) = R(g_{ab}) + R(g_{ij})\,$ and the 
form of factorization of the Weyl tensor can be easily 
obtained from its definition (\ref{number 6}).


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\section{Local conformal transformations}

Many properties of the physically interesting metrics
become more apparent and easier to prove if we use the 
so called local conformal transformation
\beq
g_{\mu\nu}(x)\,=\,{\bar g}_{\mu\nu}(x)\,e^{2\si(x)}\,.
\label{c 1}
\eeq
(\ref{c 1}) is not the coordinate transformation, it 
can be seen as an alternative form of parametrizing the 
metric via the new variables $\,{\bar g}_{\mu\nu}(x)\,$ 
and $\,\si(x)$. Our purpose in this section will be
to derive the relevant quantities like curvatures in 
this new parametrization.
Since some of the formulas below are bulky, we shall 
use the condensed notations 
$\,(\na \si)^2=g^{\mu\nu}\na_\mu \si \na_\nu \si\,$
and $(\na^2 \si)=g^{\mu\nu}\na_\mu \na_\nu \si$. 
 
The transformation laws for the inverse metric and 
for the metric determinant have the form 
\beq 
g^{\mu\nu}={\bar g}^{\mu\nu}\,e^{-2\sigma}\,,
\qquad g={\bar g}\,e^{2D\sigma} \,. 
\label{c 2}
\eeq

For the Christoffel symbols we have 
\beq
{\Ga}^\la_{\al\be} = {\bar \Ga}^\la_{\al\be} 
+ \de{\Ga}^\la_{\al\be}
\,,\qquad
\de {\Ga}^\la_{\al\be}
\,=\,\de^\la_\al \,{\bar \na}_\be\si
\,+\, \de^\la_\be \,{\bar \na}_\al\si 
\,-\, {\bar g}_{\al\be} \,{\bar \na}^\la\si\,,
\label{c 3}
\eeq
where $({\bar \na}_\al\si)=\pa_\al\si$. Here and below
all quantities with bars (e.g. ${\bar \na}_\al$
or ${\bar R}_{\al\be}$ correspond to the metric 
$\,\,{\bar g}_{\mu\nu}$.
Let us remark that the quantity $\de{\Ga}^\la_{\al\be}$
is a tensor, while ${\Ga}^\la_{\al\be}$ and 
${\bar \Ga}^\la_{\al\be}$ are not. 

In order to derive the transformation for the Riemann
tensor one can start from the simple relation
\beq 
R^\al_{\,\cdot\,\be\mu\nu}
&=& {\bar R}^\al_{\,\cdot\,\be\mu\nu}
\,+\,{\bar \na}_\mu \de\Ga^\al_{\be\nu} 
- {\bar \na}_\nu \de\Ga^\al_{\be\mu}
+ \de\Ga^\tau_{\be \nu}\,\de\Ga^\al_{\tau \mu}
- \de\Ga^\tau_{\be \mu}\,\de\Ga^\al_{\tau \nu}\,.
\label{bunch 1}
\eeq 
Starting from (\ref{c 1}), after some algebra we obtain
\beq 
R^\al_{\,\cdot\,\be\mu\nu}
&=& {\bar R}^\al_{\,\cdot\,\be\mu\nu}
+\de^\al_\nu\,({\bar \na}_\mu{\bar \na}_\be\si)
-\de^\al_\mu({\bar \na}_\nu{\bar \na}_\be\si) 
+{\bar g}_{\mu\be}({\bar \na}_\nu{\bar \na}^\al\si)
\nonumber
\\
&-& {\bar g}_{\nu\be}({\bar \na}_\mu {\bar \na}^\al\si) 
+\de^\al_\nu {\bar g}_{\mu\be}({\bar \na} \si)^2 
-\de^\al_\mu {\bar g}_{\nu\be}({\bar \na} \si)^2 
+ \de^\al_\mu({\bar \na}_\nu\si)({\bar \na}_\be\si) 
\nonumber
\\
&-& \de^\al_\nu({\bar \na}_\mu\si)({\bar \na}_\be\si)  
+ {\bar g}_{\nu\be}({\bar \na}_\mu\si)({\bar \na}^\al\si) 
- {\bar g}_{\mu\be}({\bar \na}_\nu\si)({\bar \na}^\al\si)
\label{c Riemann}
\eeq 
and furthermore
\beq
R_{\mu\nu}\,=\,{\bar R}_{\mu\nu}
- {\bar g}_{\mu\nu}({\bar \na}^2 \si)
+ (D-2)\left[({\bar \na}_\mu\si)({\bar \na}_\nu\si) 
- ({\bar \na}_\mu {\bar \na}_\nu\si) 
-  {\bar g}_{\mu\nu}({\bar \na}\si)^2\right]
\label{c Ricci}
\eeq
and
\beq 
R\,=\,e^{-2\sigma}\,\Big[\,{\bar R}
- 2(D-1)({\bar \na}^2 \sigma) 
- (D-1)(D-2)({\bar \na} \sigma)^2\,\Big]\,. 
\label{c scalar}
\eeq

It is easy to see, using (\ref{c Riemann}), 
(\ref{c Ricci}) and (\ref{c scalar}), 
that the Weyl tensor (\ref{number 6}) transforms in a most 
simple possible way 
\beq
C^{\al}_{\,\cdot\,\be\mu\nu}
\,=\,{\bar C}^\al_{\,\cdot\,\be\mu\nu}\,, 
\qquad {\rm i.e.}\qquad 
C_{\al\be\mu\nu}\,=\,e^{2\si}\,{\bar C}_{\al\be\mu\nu}. 
\label{c 7}
\eeq

Consider the transformation of the quadratic contractions 
of the curvature tensors
\beq
\sqrt{-g}\,R^{2}_{\mu\nu\al\be}
&=&\sqrt{-{\bar g}}
\,R^{\mu\nu\al\be}R_{\mu\nu\al\be}
=\sqrt{-{\bar g}}\,e^{(D-4)\sigma}
\,\Big\{\,{\bar R}^2_{\mu\nu\al\be}
\nonumber
\\
&+& 
2(D-2)\Big[\,2({\bar \na}_\mu{\bar \na}_\nu\si)^2
- 4({\bar \na}_\mu{\bar \na}_\nu\si)
({\bar \na}^\mu\si)({\bar \na}^\nu\si)
\nonumber
\\
&+& 
4({\bar \na}^2 \si)({\bar \na}\si)^2+(D-1)({\bar \na}\si)^4
\Big]
\nonumber
\\
&+& 8{\bar R}^{\mu\nu}
({\bar \na}_\mu\si{\bar \na}_\nu\si
-{\bar \na}_\mu{\bar \na}_\nu\si)
-4{\bar R}({\bar \na}\si)^2 
+4({\bar \na}^2 \si)^2\,\Big\} \,,
\label{111}
\eeq
where we multiplied the expression by $\sqrt{-g}$ for 
convenience.
\beq
\sqrt{-g}\,R^2_{\mu\nu}
&=& \sqrt{-{\bar g}}\,e^{(D-4)\sigma}
\,\Big\{\,{\bar R}^2_{\mu\nu}-2{\bar R}(\Box \si)
+(3D-4)({\bar \na}^2 \si)^2 
\nonumber
\\
&+& (D-2)\Big[\,2{\bar R}^{\mu\nu}
({\bar \na}_\mu\si{\bar \na}_\nu\si-{\bar \na}_\mu{\bar \na}_\nu\si)
-2{\bar R}({\bar \na}\si)^2
\nonumber
\\
&+& (D-1)(D-2)({\bar \na}\si)^4 
-2(D-2)({\bar \na}_\mu{\bar \na}_\nu\si)({\bar \na}^\mu\si)
({\bar \na}^\nu\si)
\nonumber
\\
&+& (D-2)({\bar \na}_\mu{\bar \na}_\nu\si)^2
+2(2D-3)({\bar \na^2} \si)({\bar \na}\si)^2\,\Big]\,\Big\}
\label{112}
\eeq
and
\beq
\sqrt{-g}\,R^2
=\sqrt{-g}\,e^{(D-4)\sigma}
\,\Big[\,{\bar R}-2(D-1)({\bar \na}\si)
-(D-1)(D-2)({\bar \na} \sigma)^2\Big]^2\,.
\label{113}
\eeq
For the square of the Weyl tensor the transformation is 
very simple
\beq
\sqrt{-g}\,C^{2}_{\mu\nu\al\be}
=\sqrt{-{\bar g}}\,e^{(D-4)\sigma}\,{\bar C}^2_{\mu\nu\al\be}\,.
\label{114}
\eeq

Another important combination of the quadratic in curvature 
invariants is
\beq
E = R_{\mu\nu\al\be}R^{\mu\nu\al\be} 
-4 \,R_{\al\be}R^{\al\be} + R^2\,.
\label{115}
\eeq
In $D=4$ it is the integrand of the Gauss-Bonnet topological 
term (or Euler characteristic, see, e.g. \cite{DNF})
\beq
\int d^4x\, \sqrt{-g}\,E\,.
\label{116}
\eeq
But, even for the $D\neq 4$ the expression 
does not contribute to the propagator of gravitons (traceless and 
completely transverse modes of the metric perturbations 
on the flat background). For this reason this term plays very 
special role in string theory \cite{GSW}. The conformal 
transformation of the Gauss-Bonnet integrand is
\beq
\sqrt{-g}E &=&\sqrt{-{\bar g}}e^{(D-4)\sigma}
\,\Big[\,{\bar E}
+8(D-3){\bar R}^{\mu\nu}({\bar \na}_\mu\si{\bar \na}_\nu\si
- {\bar \na}_\mu{\bar \na}_\nu\si)
\nonumber
\\
&-& 2(D-3)(D-4){\bar R}({\bar \na}\si)^2
+ 4(D-2)(D-3)^2({\bar \na}^2 \si)({\bar \na}\si)^2
\nonumber
\\
&-&  4(D-2)(D-3)({\bar \na}_\mu{\bar \na}_\nu\si)^2
+ 4(D-2)(D-3)({\bar \na}^2 \si)^2
\nonumber
\\
&+&  8(D-2)(D-3)({\bar \na}_\mu\si{\bar \na}_\nu\si)
({\bar \na}^\mu{\bar \na}^\nu\si)
-4(D-3)R({\bar \na}^2 \si)
\nonumber
\\
&+& (D-1)(D-2)(D-3)(D-4)({\bar \na}\si)^4
\,\Big] \,.
\label{117}
\eeq

The last remaining invariant is the surface term, which has the 
following transformation rule:
\beq
\sqrt{-g}\,({\na}^2 R)
&=&
\sqrt{-{\bar g}}\,e^{(D-4)\sigma}\,\Big[\,{\bar \na}^2 {\bar R}
-2(D-4)R({\bar \na}\si)^2 - 2R{\bar \na}^2 \si
\nonumber
\\
&-& 2(D-1)(D-6)({\bar \na}^\mu\si)({\bar \na}_\mu\Box\si)
+ (D-6)({\bar \na}^\mu\si)({\bar \na}_\mu {\bar R})
\nonumber
\\
&+& 2(D-1)(D-2)(D-4)({\bar \na}\si)^4
+2(D-1)(3D-10)(\Box \si)({\bar \na}\si)^2
\nonumber
\\
&-& 2(D-1)({\bar \na}^4 \si)
-(D-1)(D-2)(n-6)({\bar \na}^\mu\si){\bar \na}_\mu ({\bar \na}\si)^2 
\nonumber
\\
&+& 
4(D-1)(\Box \si)^2
-(D-1)(D-2)\Box ({\bar \na} \si)^2 
\,\Big] \,.
\label{118}
\eeq

In deriving the last expression we used the transformation
of the operator ${\Box}$ acting on scalars
$$
{\na}^2
=e^{-2\sigma}\,[\,{\bar \na}^2
+(D-2)({\bar \na}^\mu \si){\bar \na}_\mu\,]\,. 
$$

It is remarkable that in $D=4$ the following simple 
relation takes place
\beq
\sqrt{-g}\,\Big(E-\frac23\,{\Box} R\Big)\,=\,
\sqrt{-g}\,\Big({\bar E}-\frac23\,{\bar \na}^2 {\bar R}
+ 4{\bar \De_4}\si \Big)\,,
\label{119}
\eeq
where $\De_4$ is the fourth order Hermitian conformal invariant
operator acting on conformally invariant scalar field
\cite{pan,rei}
\beq
\De_4 = \na^4 + 2\,R^{\mu\nu}\na_\mu\na_\nu - \frac23\,R{\Box}
+ \frac13\,(\na^\mu R)\na_\mu\,.
\label{120}
\eeq

In section 4 we shall see that the formula (\ref{119})
is useful in understanding the relation between the 
topological terms in $D=4$ and $D=2$. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Practical use of conformal transformation}

Let us consider the practical application of the $D$-dimensional
conformal transformations derived in the previous section. We 
shall also use the factorization theorem of section 2. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Derivation of Einstein equations for the FRW metric}

Consider the derivation of the Einstein tensor for the 
Friedmann-Robertson-Walker (FRW) metric in terms of the 
conformal time $\,\eta$
\beq
ds^2 \,=\, g_{\mu\nu}\,dx^\mu \,dx^\nu
\,=\,a^2(\eta)\,\left(d\eta^2-dl^2\right)\,,
\label{FRW 1}
\eeq
where (see, e.g. \cite{Weinberg72})
\beq
dl^2\,=\,\frac{dr^2}{1-kr^2}+r^2d\th^2+r^2\sin^2 \th d\ph^2\,,
\label{FRW 2}
\eeq
where $\,k=0,\,\pm 1$.
In order to fit with our notations in section 3 we denote
\beq
a(\eta)=e^{\si(\eta)}\,,\qquad
g_{\mu\nu}\,=\,{\bar g}_{\mu\nu}\,e^{2\si}\,.
\label{FRW 3}
\eeq
It is easy to see that the new metric 
\beq
{\bar g}_{\mu\nu}\,=\,\mbox{diag}\,\left(1,\,
-\frac{1}{1-kr^2},\,-r^2,\,-r^2\sin^2 \th \right)
\label{FRW 4}
\eeq
is factorized and therefore satisfies the conditions
of the factorization theorem. One of the consequences 
is that the FRW metric is Weyl flat. This becomes 
clear if we remember the transformation rule for the
Weyl tensor (\ref{c 7}) and also that 
$\,C^\al\,_{\be\mu\nu}=0\,$ in $\,D=3\,$ dimensions
(see also \cite{tuk}).

The relation between $R_{\mu\nu}$ and 
${\bar R}_{\mu\nu}$ is given by Eq. (\ref{c Ricci})
and the relation between $R$ and ${\bar R}$ by 
eq. (\ref{c scalar}). In both cases one has to set 
$D=4$. Consider, as an example, $R_{00}$. 
Using (\ref{c Ricci}) we arrive at
$$
R_{00}={\bar R}_{00}-3\si^{\prime\prime}\,,
$$ 
where the prime indicates the derivative with 
respect to $\eta$. Furthermore ${\bar R}_{00}=0$, 
because it is a Ricci tensor of a one-dimensional 
metric (remember factorization theorem). Therefore
\beq
R_{00}=-3\si^{\prime\prime}\,=\,-\,
\frac{3}{a^2}\,\Big(aa^{\prime\prime}-{a^{\prime}}^2\Big)\,.
\label{FRW 5}
\eeq

Further calculations involve ${\bar R}_{ij}$, where 
$i,j=1,2,3$. ${\bar R}_{i0}=0$ automatically.
So, in fact we reduced the $D=4$ calculations
to the simpler $D=3$ ones. The components of ${\bar R}_{ij}$
can be easily calculated directly (this is technically 
simpler) or through another 
conformal transformation in $D=3$
\beq
{\bar g}_{ij}\,=\,\mbox{diag}\,
\left(-\frac{1}{1-kr^2},\,-r^2,\,-r^2\sin^2 \th \right)
\,=\,\ga_{ij}\,e^{2\rho(r)}\,,\qquad \rho(r)=\ln r.
\label{FRW 6}
\eeq
It is easy to see that the new metric is again factorized 
and the calculation can be performed in a metric which is
a product of the one-dimensional metric $\ga_{11}\,$
and the two-dimensional metric 
$\,\diag \big(\ga_{22},\,\ga_{33}\big)$. Let us 
present only a final result ${\bar R}_{ij}=-2k\,{\bar g}_{ij}$. 
After performing contraction, we obtain ${\bar R}=-6k$. 
Now we are in a position to derive 
\beq
R_{i0}=0 \qquad  \mbox{and} \qquad
R_{ij}\,=\,-\,{\bar g}_{ij}\left(2k + \si^{\prime\prime}
+ 2{\si^{\prime}}^2\right)\,.
\label{FRW 7}
\eeq

Then, after
applying Eq. (\ref{c scalar}) with $D=4$ we arrive at 
\beq
R\,=\,-\,6\,e^{-2\si}\,\Big(k+\si^{\prime\prime}
+ {\si^{\prime}}^2\Big)
\,=\,-\,6\,\left(\frac{k}{a^2} 
+ \frac{a^{\prime\prime}}{a^3} \right)\,.
\label{FRW 8}
\eeq
Finally, the result for the Einstein tensor has the 
form
\beq
G_{\eta\eta} &=& R_{\eta\eta}-\frac12\,R\,g_{\eta\eta}
\,=\,3k + 3\left(\frac{a^\prime}{a}\right)^2\,,
\nonumber
\\
G_{\eta\,i}&=&0\,,
\nonumber
\\
G^i_j\,&=&\de_i^j\,\left( \frac{k}{a^2}
+\frac{ 2a^{\prime\prime} }{a^3}
-\frac{ {a^\prime}^2}{a^4}\right)\,.
\label{FRW 9}
\eeq
The transition to the physical time $t$ can be easily 
performed through the tensor transformation rule
and the standard relation between the two time units
$dt=a(\eta)d\eta$
\beq
G_t^t &=&G_{tt} 
= \left(\frac{d\eta}{dt}\right)^2\,G_{\eta\eta}
= \frac{3({\dot a}^2+k)}{a^2}\,.
\nonumber
\\
\nonumber
\\
G^i_j\,&=&\de_i^j\,\left( \frac{k}{a^2}
+\frac{ 2{\ddot a}}{a}
+ \frac{ {\dot a}^2}{a^2}\right)\,.
\label{FRW 10}
\eeq
Using the expressions (\ref{FRW 10}) one can 
easily write down the standard form of the 
Friedmann equations 
\beq
\frac{ {\dot a}^2 + k }{a^2}\,=\,\frac{8\pi G\,\rho}{3}\,,
\qquad\qquad
 \frac{2{\ddot a}}{a}
+ \frac{ {\dot a}^2}{a^2} + \frac{k}{a^2}\,=\,-8\,\pi G\,P\,,
\label{FRW 11}
\eeq
where $P$ and $\rho$ are pressure and energy density of
the ideal fluid filling the Universe. These two 
quantities are related
by the equation of state specific for the given type 
of fluid (matter, radiation, vacuum energy etc).
From our point of view, the method of deriving the 
Friedmann equations presented above is simpler than the
standard one. Of course, this is true only if one 
has calculated the conformal transformations for the 
Ricci tensor and scalar curvature first, but 
these calculations are not involved.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\subsection{Derivation for the Schwarzschild metric}

The Schwarzschild metric is another important particular 
example of the metrics used in GR. It is characterized by 
spherical symmetry and therefore applies to the description 
of the gravitational field in the exterior of numerous 
objects like starts at the different stages of their 
evolution. The extreme example is the point-like spherically 
symmetric mass distribution which corresponds to the black 
hole solution of GR. The formation and properties of the 
black holes is extremely interesting subject (see, e.g. 
the comprehensive monograph \cite{Frolov}). For this 
reason, and also due to its relative simplicity, the 
Schwarzschild solution is compulsory element of any course
of GR. The consideration of this solution always starts
with the derivation of Einstein equations for the corresponding 
metric
\beq
ds^2&=&g_{\mu\nu}dx^\mu dx^\nu
\,=\,e^{\nu(r,t)}{dt^2}-e^{\la(r,t)}{dr^2}-r^2d\Om\,,
\label{b h 1}
\\
\mbox{where}\qquad 
d\Om &=& \sin^2 \th d\ph^2 + d\th^2\,.
\nonumber
\eeq
The purpose of this section is the perform this calculation in 
a more economic way using the local conformal transformation
described in section 3. But, due to the relative simplicity 
of this way of calculations, we will be able to consider 
a little bit more complicated metric
\beq
ds^2 &=& g_{\mu\nu}dx^\mu dx^\nu
\,=\,e^{\nu(r,t)}{dt^2}-e^{\la(r,t)}{dr^2}-e^{2\Phi(r,t)}d\Om\,,
\label{b h 2}
\eeq
where $\,\Phi(r,t)\,$ is an arbitrary scalar function. Let us 
notice that the more general metric (\ref{b h 2}) is widely 
used in the black hole physics (see, e.g. \cite{Frolov}), in 
particular because it has some advantages in the theories
with vacuum quantum corrections to the Einstein GR
\cite{balsan}. The particular case (\ref{b h 1}) may be
always achieved by replacing $\,\,\Phi(r,t)=\ln r\,\,\,$ into 
the corresponding expressions. 

Our strategy in calculating the Einstein equations for 
the metric (\ref{b h 2}) will be the following. The first 
step is the conformal transformation 
\beq
g_{\mu\nu}\,=\,{\bar g}_{\mu\nu}\,e^{2\Phi}\,.
\label{b h 3}
\eeq
The new metric $\,{\bar g}_{\mu\nu}\,$ is diagonal and 
factorized 
$$
{\bar g}_{ab}
\,=\,\diag \left({\bar g}_{00} ,\,{\bar g}_{11}\right)
\,,\qquad
{\bar g}_{ij}
\,=\,\diag \left({\bar g}_{22} ,\,{\bar g}_{33}\right)\,,
$$
where
\beq
{\bar g}_{00}=e^{A}\,,\quad
{\bar g}_{11}=-e^{B}\,,\quad
{\bar g}_{22}= -1 \,,\quad
{\bar g}_{22}= -\sin^2\th \,,
\label{b h 4}
\eeq
and 
\beq
A=A(r,t)=\nu(r,t)-2\Phi(r,t)
\,,\qquad 
B=B(r,t)=\la(r,t)-2\Phi(r,t)\,.
\label{b h 5}
\eeq
The relations between the two Ricci tensors and the two 
scalar curvatures are given by the particular form of 
(\ref{c Ricci})
\beq
R_{\mu\nu}&=&{\bar R}_{\mu\nu}
- {\bar g}_{\mu\nu}({\bar \na}^2 \si)
+ 2({\bar \na}_\mu\si)({\bar \na}_\nu\si) 
- 2({\bar \na}_\mu {\bar \na}_\nu\si) 
- 2{\bar g}_{\mu\nu}({\bar \na}\si)^2\,, 
\nonumber
\\ 
R&=&e^{-2\sigma}\,\Big[\,{\bar R}
- 6({\bar \na}^2 \sigma) 
- 6({\bar \na} \sigma)^2\,\Big]\,. 
\label{c scalar 4}
\eeq

According to the factorization theorem of section 2,
all components of the curvature tensor with the mixed 
indices vanish and we need to calculate only the 
curvature tensors and other quantities which emerge in 
(\ref{c scalar 4}) for the two-dimensional metrics 
in (\ref{b h 4}). For the curvatures we have
\beq
K_{abcd}=\frac12\,K\,
\left({\bar g}_{ac}\,{\bar g}_{bd}
-{\bar g}_{ad}\,{\bar g}_{bc}\right)\,,\qquad
a,b,... = t,r\,\,,
\label{b h 6}
\eeq
where $K_{abcd}$ is the Riemann tensor for the metric 
$\,{\bar g}_{ab}\,$ and 
\beq
k_{ijkl}=\frac12\,k\,
\left({\bar g}_{ik}\,{\bar g}_{jl}
-{\bar g}_{il}\,{\bar g}_{jk}\right)
\,,\qquad i,j,... = \th,\ph\,,
\label{b h 7}
\eeq
where $\,k\,$ is nothing but the scalar curvature of the
$D=2$ sphere $\,k=-2$, the negative sign appears due to 
the negative sign of the metric components in (\ref{b h 4}).

For the metric ${\bar g}_{ab}$ we obtain, by direct calculation
\beq
{\bar \Ga}^t_{tt}=\frac{{\dot A}}{2}
\,,\quad
{\bar \Ga}^r_{tt}=\frac{A^\prime}{2}\,e^{A-B}
\,,\quad
{\bar \Ga}^t_{rt}=\frac{A^\prime}{2}
\nonumber
\\
{\bar \Ga}^r_{tr}=\frac{{\dot B}}{2}
\,,\quad
{\bar \Ga}^t_{rr}=\frac{{\dot B}}{2}\,e^{B-A}
\,,\quad
{\bar \Ga}^r_{rr}=\frac{B^\prime}{2}\,,
\label{b h 8}
\eeq
where the dot stands for $d/dt$ and the prime for the $d/dr$.
After a small algebra we obtain 
\beq
K=\frac12\,e^{-A}\,\big({\dot A}{\dot B}-2{\ddot B} -{\dot B}^2\big)
+\frac12\,e^{-B}\,\big(2A^{\prime\prime}-A^\prime B^\prime
+ {A^\prime}^2\big)\,,
\label{b h 9}
\eeq
and check that 
$\, K_{tt}=\frac{K}{2}\,e^A \,\, \mbox{and} \,\,
K_{rr}=-\frac{K}{2}\,e^B
\,\, \mbox{according to (\ref{b h 6})}$.
Furthermore
\beq
{\bar \na}_t{\bar \na}_t\Phi &=& {\ddot \Phi} 
- \frac12\,{\dot A}{\dot \Phi}
- \frac12\,A^\prime \Phi^\prime\,e^{A-B}\,,
\nonumber
\\
{\bar \na}_r{\bar \na}_r\Phi &=& \Phi^{\prime\prime} 
- \frac12\,B^\prime \Phi^\prime
- \frac12\,{\dot B}{\dot \Phi}\,e^{B-A}\,,
\nonumber
\\
{\bar \na}_t{\bar \na}_r\Phi &=& {\dot \Phi}^\prime 
- \frac12\,A^\prime{\dot \Phi}
- \frac12\,{\dot B} \Phi^\prime\,.
\label{b h 10}
\eeq
and 
\beq
\left({\bar \na \Phi}\right)^2 
&=& e^{-A}\,{\dot \Phi}^2 \,-\, e^{-B}\,{\Phi^\prime}^2\,,
\nonumber
\\
{\bar \na}^2\Phi &=& e^{-A}\,\Big(
{\ddot \Phi}-\frac12\,{\dot A}{\dot \Phi} +
\frac12\,{\dot B}{\dot \Phi}\Big)
\,+\,
e^{-B}\,\Big(
\frac12\,B^\prime\Phi^\prime - \frac12\,A^\prime\Phi^\prime 
-\Phi^{\prime\prime}\Big)\,.
\label{b h 11}
\eeq
Now we are in a position to calculate $R$ and $R_{\mu\nu}$.
Using the second equation (\ref{c scalar 4}), relation 
$\,{\bar R}=K+2\,$ and formulas (\ref{b h 11}), we arrive at 
\beq
R&=&-2e^{-2\Phi}+
e^{-\nu}
\Big[(\dot{\nu}-\dot{\la})\Big(2\dot{\Phi}
+\frac{\dot{\la}}{2}\Big) 
-\ddot{\la}
-4\ddot{\Phi}-6\dot{\Phi}^2\Big]
\nonumber
\\
&+& e^{-\la}\Big[{\nu^{\prime \prime}}
+ \Big(2\Phi^\prime + \frac{\nu^{\prime}}{2}\Big)
(\nu^{\prime}-\la^{\prime})
+ 4\Phi^{\prime \prime}
+6\Phi^{\prime\,2}\Big]\,.
\label{b h 12}
\eeq
Similarly, using the first equation (\ref{c scalar 4})
we obtain the components of the Ricci tensor and finally 
of the Einstein tensor. For example,
\beq
R_{\th\th}\,=\,-2e^{-2\Phi}+
e^{-\nu}
\Big[ \dot{\Phi}
\Big(\frac{\dot{\nu}}{2}-\frac{\dot{\la}}{2} -2\dot{\Phi}\Big)
-\ddot{\Phi}\Big]
% \nonumber
% \\
+ e^{-\la}\Big[\Phi^{\prime \prime} +  \Phi^{\prime}
\Big(\frac{\nu^{\prime}}{2}-\frac{\la^{\prime}}{2}
+2\Phi^{\prime}\Big) \Big]
\label{b h 13}
\eeq
leads to 
\beq
G^\th_\th
&=& R^\th_\th-\frac12\,R\,\de^\th_\th 
\,=\, e^{-\nu}\,
\Big[\ddot{\Phi}+\ddot{\la}+\dot{\Phi}^2
+ \frac{\dot{\Phi}\,(\dot{\la}-\dot{\nu})}{2}
+ \frac{\dot{\la}\,\big(\dot{\la}-\dot{\nu}
+ 2\dot{\Phi}\big)}{4}\Big]
\nonumber
\\
&+& e^{-\la}\,\Big[
\frac{\Phi^{\prime}\,(\la^{\prime}-\nu^{\prime})}{2}
- \Phi^{\prime \prime} - {\Phi^{\prime}}^2
-\frac{\nu^{\prime \prime}}{2}
+\frac{\nu^{\prime}\,\big(\la^{\prime}-\nu^{\prime}\big)}{4}
\Big]\,.
\label{b h 14}
\eeq
In the special case $\,\Phi=\ln r\,$ (\ref{b h 14}) boils
down into
\beq
G^\th_\th
&=& e^{-\nu}\,
\Big(
  \frac{\dot{\la}^2}{4}
- \frac{\dot{\nu}\dot{\la}}{4}
+ \frac{\ddot{\la}}{2}\Big)
%%%%
+ e^{-\la}\,\Big(\frac{\nu^\prime\la^\prime}{4}
-\frac{\nu^{\prime\, 2}}{4}
-\frac{\nu^{\prime\prime}}{2}
+ \frac{\la^{\prime}-\nu^{\prime} }{r}\Big)\,,
\label{b h 15}
\eeq
in a perfect fit with the well-known result \cite{LL-2}.

In a similar way, using (\ref{c scalar 4}) and
(\ref{b h 10}) we obtain
 \beq
G_t^r &=& g^{rr}G_{tr}\,=\,g^{rr}R_{tr}-\frac{1}{2}\de_t^r\,R
\,=\,g^{rr}R_{tr}
\nonumber
\\
 &=&  
2e^{-\la}\,\Big(\dot{\Phi}^{\prime}-\frac{\nu^{\prime}\dot{\Phi}}{2}
-\frac{\dot{\la}\Phi^{\prime}}{2}+\Phi^{\prime}\dot{\Phi}\Big)\,,
\label{b h 16}
\eeq
\beq
G_r^r &=& g^{rr}R_rr-\frac{1}{2}\de_r^r\,R
\,=\,-e^{-\la}R_r^r-\frac{1}{2}\,R
\nonumber
\\
 &=& e^{-2\ph}-e^{-\la}
\left(\nu^{\prime}\Phi^{\prime}+\Phi^{\prime\,2}\right)
\,+\,
e^{-\nu}\left(2\ddot{\Phi}+3\dot{\Phi}^2-\dot{\nu}\dot{\Phi}\right)
\,,
\label{b h 17}
\eeq
\beq
G_t^t &=& g^{tt}R_tt-\frac{1}{2}\,R
\nonumber
\\
 &=& e^{-2\Phi}+e^{-\nu}\,\Big(\dot{\la}\dot{\Phi}+\dot{\Phi}^2)
\,+\,e^{-\la}\,[-2\Phi^{\prime \prime}+\la^{\prime}\Phi^{\prime}
-3\Phi^{\prime\,2}\Big)\,,
\label{b h 18}
\eeq
In the special case $\,\Phi=\ln r\,$ these expressions become
the standard ones \cite{LL-2}
\beq
G_t^r &=& -e^{-\la}\dot{\la}\ph^{\prime}
\,=\,-\frac{\dot{\la}}{r}e^{-\la}\,,
\nonumber
\\
\nonumber
\\
G_r^r &=& -e^{-\la}\,\Big(\frac{\nu^{\prime}}{r}
+\frac{1}{r^2}\Big)+\frac{1}{r^2}\,,
\nonumber
\\
\nonumber
\\
G_t^t &=& e^{-\la}\Big(\frac{\la^{\prime}}{r}
-\frac{1}{r^2}\Big)+\frac{1}{r^2}\,.
\label{b h 19}
\eeq
Let us remark that if the derivation is performed 
directly for the usual metric (\ref{b h 1}), it is 
technically very simple and definitely less cumbersome 
than the standard one without conformal transformations. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Dimensional reduction of the topological 
invariant}

The purpose of this section is to illustrate the 
calculational power of the conformal transformation 
for the theories beyond the GR. Let us use this 
transformation for deriving the form of the $\,D=4\,$ 
Gauss-Bonnet 
topological invariant (\ref{116}) in the particular 
metric 
\beq
g_{\mu\nu} \,=\,\left\| \begin{array}{cc}
 g_{ab}(x^a) & 0 \\
0 & e^{2\Phi(x^a)}h_{ij}(x^i) \end{array}\right\|\,.
\label{g b 0}
\eeq
Here $g_{ab}(x^a)$ and $h_{ij}(x^i)$ are both 
two-dimensional metrics, but $h_{ij}(x^i)$
has constant scalar curvature $\,\,k\,\,$ while 
$g_{ab}(x^a)$ is arbitrary. $\Phi(x^a)$ is a scalar 
field in the two-dimensional space. The result of this 
reduced dimensional reduction is 
supposed to be the topological or surface term 
in the two-dimensional space with the metric $g_{ab}$,
$\,a,b=0,1$. Taking into account the simplicity of 
the conformal transformation for the combination 
(\ref{119}), we shall consider not the proper 
Gauss-Bonnet term, but instead its combination with 
the surface term 
\beq
S_{top}\,=\,\int d^4x\sqrt{-g}
\,\left(E - \frac23\,\na^2 R\right)\,.
\label{g b 1}
\eeq
After the conformal transformation 
$\,g_{\mu\nu}={\bar g}_{\mu\nu}\cdot\exp (2\Phi)$, 
according to (\ref{119}), we obtain
\beq
S_{top}\,=\,\int d^4x
\sqrt{-{\bar g}}\left({\bar E} 
- \frac23\,{\bar \na}^2 {\bar R} + 4{\bar \De}_4\Phi\right)\,.
\label{g b 2}
\eeq
Since the metric ${\bar g}_{\mu\nu}$ is factorized
\beq
{\bar g}_{\mu\nu} \,=\,\left\| \begin{array}{cc}
 g_{ab}(x^a) & 0 \\
       0 & h_{ij}(x^i) \end{array}\right\|\,,
\label{g b 3}
\eeq
we need to calculate the complicated curvature-squared 
expressions
only for the two-dimensional metrics $\,g_{ab}(x^a)\,$ and 
$\,h_{ij}$. The quantities based on the metric of the 
two-dimensional sphere $\,h_{ij}\,$ are very simple
due to Eq. (\ref{number 3}), and the whole problem can 
be resolved immediately. Let us denote the curvature tensor 
corresponding to the metric $\,g_{ab}\,$ by 
$\,{\cal K}_{abcd}\,$
and the curvature tensor corresponding to the metric 
$\,{\bar g}_{ab}\,$ by $\,{\bar {\cal K}}_{abcd}$. Indeed, 
as always in $\,D=2$, we have (\ref{number 3}), that is
\beq
{\bar {\cal K}}_{abcd} &=& \frac12\,{\bar {\cal K}}
\,\big( {\bar g}_{ac}{\bar g}_{bd}-
{\bar g}_{ad}{\bar g}_{bc}\big)\,,\qquad
{\bar {\cal K}}_{ab} = \frac12\,{\bar {\cal K}}\,{\bar g}_{ab}
\nonumber
\\
{\cal K}_{abcd} &=& \frac12\,{\cal K}
\,\big( g_{ac}g_{bd} - g_{ad}g_{bc}\big)\,,\qquad
{\cal K}_{ab} = \frac12\,{\cal K}\,g_{ab}\,.
\label{g b 4}
\eeq
Furthermore, in the $h_{ij}$ sector the Riemann and Ricci
tensors are 
$$
k_{ijkl} = \frac12\,k
\,\big( h_{ik}h_{jl} - h_{il}h_{jk}\big)\,,\qquad
k_{ij} = \frac12\,k\,h_{ij}\,.
$$
Let us calculate the elements of the expression 
(\ref{g b 2}). 
\beq
{\bar E}\,=\,{\bar {\cal K}}_{abcd}{\bar {\cal K}}^{abcd} 
- 4 {\bar {\cal K}}_{ab}{\bar {\cal K}}^{ab} 
+ k_{ijkl}k^{ijkl}-4k_{ij}k^{ij}
+ \left({\bar {\cal K}} + k\right)^2 \,=\,2k\bar{\cal K}
\label{g b 6}
\eeq
The covariant derivatives corresponding to the metrics 
$\,g_{ab}\,$ and $\,{\bar g}_{ab}\,$ will be denoted by 
$\,{\cal D}_a\,$ and $\,{\bar {\cal D}}_a\,$ correspondingly.
When acting on scalar (e.g. $\Psi$), the transformation 
rule for the d$^\prime$Alembert operator looks like
\beq
{\cal D}^2\Psi\,=\,e^{-2\Phi}{\bar {\cal D}}^2\Psi\,.
\label{g b 7}
\eeq

Using (\ref{g b 4}), (\ref{c scalar}) and (\ref{g b 7}) 
we obtain
\beq
\bar{\De}_4 \Phi &=& \left[{\bar {\cal D}}^4 
- \frac13\,{\bar {\cal K}}\,{\bar {\cal D}}^2
- \frac13\,\left( {\bar {\cal D}}^a {\bar {\cal K}}\right)
{\bar {\cal D}}_a\right]\, \Phi
\nonumber
\\
&=& e^{2\Phi}{\cal D}^2\left(e^{2\Phi}{\cal D}^2 \Phi\right)
- \frac13\,e^{2\Phi}\,\left({\cal K} + 2{\cal D}^2\Phi\right)
e^{2\Phi} \left({\cal D}^2\Phi\right)
\nonumber
\\
&-& \frac13\,e^{2\Phi}\,{\cal D}^a
\left[e^{2\Phi} \Big({\cal K} + 2{\cal D}^2\Phi\Big)\right]
\,{\cal D}_a \Phi
\label{g b 8}
\eeq
and 
\beq
{\bar \na}^2 {\bar R}\,=\,e^{2\Phi}\,{\cal D}^2
\left[ e^{2\Phi}\, \left({\cal K} 
+ 2{\cal D}^2\Phi\right)\right]\,.
\label{g b 9}
\eeq
Finally, for the modified topological term (\ref{g b 2}) 
we find
\beq
S_{top} &=& \int d^2x
\sqrt{-g(x^a)} \,\left\{\,2k\,{\cal K}
\,+\,
{\cal D}^2\left(\frac83\,e^{2\Phi}{\cal D}^2\Phi
- \frac23\,e^{2\Phi}{\cal K} + 4k\Phi\right)
\right.
\nonumber
\\
&-& \left.
\frac13\,{\cal D}_a \Big[e^{2\Phi} \left({\cal K} 
+ 2{\cal D}^2\Phi\right)\,\left({\cal D}_a \Phi\right)\Big]
\right\}\,.
\label{g b 10}
\eeq
The first term here is the Einstein-Hilbert action 
(with the coefficient $2k$) which is indeed topological 
term in $D=2$. Other terms are surface integrals which 
depend on the boundary conditions only. Therefore the 
expression (\ref{g b 10}) is a kind of a direct relation 
between the topological invariants in $D=4$ and $D=2$.

If we consider similar dimensional reduction to $D=3$
space, the topological invariant does not emerge, and 
we meet only surface integrals. In order to see
this one does not need to perform calculations. It is 
enough to look at (\ref{g b 2}) and remind that the 
potential source of the topological terms ${\bar E}$
vanish in $D=3$, because it coincides with the square
of the Weyl term $\,{\bar C}^2_{abcd}$. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}

We have pedagogically reviewed the local conformal 
transformations and demonstrated their utility in 
relatively easy derivation of various quantities 
related to the gravitational theory. In particular,
using these transformations and the factorization 
theorem of section 2 one can essentially reduce the
amount of work needed to obtain the Einstein 
equations for the FRW and Schwarzschild metrics. From 
our point of view, this way of calculation provides
certain advantages compared to the direct one, and
can be used in the basic courses of GR. 

Furthermore, we applied the same scheme of calculation
to the fourth order terms in the gravitational 
action. These terms emerge frequently as quantum 
corrections to GR, in particular in the frameworks 
of semiclassical approach to gravity and in string 
theory. The application of conformal 
transformations made the calculations fairly easy, while 
the direct way of deriving the same quantities would 
require an enormous amount of work. As a result  
of this calculus we have obtained a direct relation
between the topological invariants in $D=4$ and $D=2$ 
spaces. In $D=3$ case the same procedure reduces the
Gauss-Bonnet term to the integral of total derivative
and there is no relation to the corresponding 
topological (Chern-Simons) action.  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip 8mm

\noindent
{\large\bf Acknowledgements.} \
Authors are grateful to CNPq for reserach fellowship 
(I.L.Sh.) 
and also to CNPq (B.G.), FAPEMIG (D.F.) and UFJF (A.de L.)
for the scientific initiation schoolarships.

%% \newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}


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\bibitem{Weinberg72}
S. Weinberg, ``Gravitation and Cosmology'',
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\bibitem{MTW} C.W.Misner, K.S.Thorn and J.A.Wheeler,
``Gravitation'', Freeman, San Francisco, 1973.

\bibitem{wald} R.M. Wald, ``General Relativity'',
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\bibitem{birdav}
N.D. Birrell and P.C.W. Davies, ``Quantum fields in 
curved space'', Cambridge University Press, Cambridge, 1982. 

\bibitem{fulling} S.A. Fulling, 
``Aspects of Quantum Field Theory in Curved Space-Time''
Cambridge Univ. Press, Cambridge, 1989.

\bibitem{book}
I.L. Buchbinder, S.D. Odintsov and I.L. Shapiro,
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\bibitem{petrov}
A.Z. Petrov, ``Einstein Spaces'', Pergamon, Oxford, 1969.

\bibitem{DNF}
B.A. Dubrovin, A.T. Fomenko and S.P. Novikov,
``Modern Geometry-Methods and Applications'',
Springer-Verlag, 1984.

\bibitem{GSW} 
M.B. Green, J.H. Schwarz and E. Witten, 
``Superstring Theory'', 
Cambridge University Press, Cambridge, 1987.

\bibitem{pan} 
S. Paneitz, 
``A Quartic Conformally Covariant Differential
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MIT preprint, 1983 (unpublished).

\bibitem{rei}
R.J. Reigert, 
%% A non-local action for the trace anomaly.
{\it Phys. Lett.} {\bf 134B} 56 (1980).

\bibitem{tuk} 
A.P. Lightman, W. H. Press, R. H. Price and
S. A. Teukolsky, ``Problem Book in relativity and Gravitation'',
Princeton University Press, Princeton, 1975.

\bibitem{Frolov} 
V.P. Frolov and I.D. Novikov,
``Black Hole Physics - Basic Concepts and New Developments'',
Kluwer Academic Publishers, 1989.

\bibitem{balsan}
R. Balbinot, A. Fabbri and I.L. Shapiro,
% {\it Anomaly induced effective actions and Hawking radiation,} 
{\it Phys. Rev. Lett.} {\bf 83} (1999) 1494;
% {\it Vacuum polarization in Schwarzschild space-time
% by anomaly induced effective actions and Hawking radiation,}
{\it Nucl. Phys.} {\bf B559} (1999) 301.

\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


gr-qc0412113.extracted_bib
\begin{thebibliography}{99}


\bibitem{LL-2}
L.D. Landau and E.M. Lifshits, ``Field Theory'',
Elsevier, 1995.

\bibitem{Weinberg72}
S. Weinberg, ``Gravitation and Cosmology'',
Wiley, New York, 1972.

\bibitem{MTW} C.W.Misner, K.S.Thorn and J.A.Wheeler,
``Gravitation'', Freeman, San Francisco, 1973.

\bibitem{wald} R.M. Wald, ``General Relativity'',
University of Chicago Press, Chicago, 1984.

\bibitem{birdav}
N.D. Birrell and P.C.W. Davies, ``Quantum fields in 
curved space'', Cambridge University Press, Cambridge, 1982. 

\bibitem{fulling} S.A. Fulling, 
``Aspects of Quantum Field Theory in Curved Space-Time''
Cambridge Univ. Press, Cambridge, 1989.

\bibitem{book}
I.L. Buchbinder, S.D. Odintsov and I.L. Shapiro,
``Effective Action in Quantum Gravity'',
IOP Publishing, Bristol, 1992.

\bibitem{petrov}
A.Z. Petrov, ``Einstein Spaces'', Pergamon, Oxford, 1969.

\bibitem{DNF}
B.A. Dubrovin, A.T. Fomenko and S.P. Novikov,
``Modern Geometry-Methods and Applications'',
Springer-Verlag, 1984.

\bibitem{GSW} 
M.B. Green, J.H. Schwarz and E. Witten, 
``Superstring Theory'', 
Cambridge University Press, Cambridge, 1987.

\bibitem{pan} 
S. Paneitz, 
``A Quartic Conformally Covariant Differential
Operator for Arbitrary Pseudo-Riemannian Manifolds'',
MIT preprint, 1983 (unpublished).

\bibitem{rei}
R.J. Reigert, 
%% A non-local action for the trace anomaly.
{\it Phys. Lett.} {\bf 134B} 56 (1980).

\bibitem{tuk} 
A.P. Lightman, W. H. Press, R. H. Price and
S. A. Teukolsky, ``Problem Book in relativity and Gravitation'',
Princeton University Press, Princeton, 1975.

\bibitem{Frolov} 
V.P. Frolov and I.D. Novikov,
``Black Hole Physics - Basic Concepts and New Developments'',
Kluwer Academic Publishers, 1989.

\bibitem{balsan}
R. Balbinot, A. Fabbri and I.L. Shapiro,
% {\it Anomaly induced effective actions and Hawking radiation,} 
{\it Phys. Rev. Lett.} {\bf 83} (1999) 1494;
% {\it Vacuum polarization in Schwarzschild space-time
% by anomaly induced effective actions and Hawking radiation,}
{\it Nucl. Phys.} {\bf B559} (1999) 301.

\end{thebibliography}