
\documentstyle[sprocl,twoside]{article}

%%
%% A useful Journal macro
\def\Journal#1#2#3#4{{\em #1} {\bf #2}, #3 (#4)}

% Some useful journal names
\def\NCA{\em Nuovo Cimento}
\def\NPB{{\em Nucl. Phys.} B}
\def\PLB{{\em Phys. Lett.}  B}
\def\PRL{\em Phys. Rev. Lett.}
\def\PRD{{\em Phys. Rev.} D}
\def\ZPC{{\em Z. Phys.} C}

% Some other macros used in the sample text
\def\st{\scriptstyle}
\def\sst{\scriptscriptstyle}
\def\mco{\multicolumn}
\def\epp{\epsilon^{\prime}}
\def\vep{\varepsilon}
\def\ra{\rightarrow}
\def\ppg{\pi^+\pi^-\gamma}
\def\vp{{\bf p}}
\def\ko{K^0}
\def\kb{\bar{K^0}}
\def\al{\alpha}
\def\ab{\bar{\alpha}}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\CPbar{\hbox{{\rm CP}\hskip-1.80em{/}}}%temp replacemt due to no font

\def\citebk#1{\hspace{0.9mm}\raisebox{-1.85mm}[0mm][0mm]
  {\Large\cite{#1}}\hspace{-0.1mm}}

\def\citebkcap#1{\hspace{0.8mm}\raisebox{-1.5mm}[0mm][0mm]
  {\large\cite{#1}}\hspace{-0.2mm}}

\newcommand{\nit}{\noindent}
\newcommand{\nl}{\newline}
\newcommand{\np}{\newpage}
\newcommand{\dsp}{\displaystyle}
\newcommand{\vs}[1]{\vspace{#1 ex}}
\newcommand{\hs}[1]{\hspace{#1 em}}
\newcommand{\bfr}{\begin{flushright}}
\newcommand{\efr}{\end{flushright}}
\newcommand{\bc}{\begin{center}}
\newcommand{\ec}{\end{center}}
\newcommand{\ben}{\begin{enumerate}}
\newcommand{\een}{\end{enumerate}}
\newcommand{\gw}[1]{\bfr  GW #1 \efr}
\newcommand{\ea}{\end{array}}
\newcommand{\ct}{\cite}
\newcommand{\bit}{\bibitem}
\newcommand{\dd}[2]{\frac{\partial{#1}}{\partial{#2}}}
\newcommand{\ag}{\alpha}
\newcommand{\bg}{\beta}
\newcommand{\gam}{\gamma}
\newcommand{\del}{\delta}
\newcommand{\eps}{\epsilon}
\newcommand{\ve}{\varepsilon}
\newcommand{\zg}{\zeta}
\newcommand{\thg}{\theta}
\newcommand{\kg}{\kappa}
\newcommand{\lb}{\lambda}
\newcommand{\sg}{\sigma}
\newcommand{\rg}{\rho}
\newcommand{\fg}{\phi}
\newcommand{\vf}{\varphi}
\newcommand{\og}{\omega}
\newcommand{\Gam}{\Gamma}
\newcommand{\Del}{\Delta}
\newcommand{\Fg}{\Phi}
\newcommand{\Sg}{\Sigma}
\newcommand{\Og}{\Omega}
\newcommand{\Lb}{\Lambda}
\newcommand{\bxi}{\bar{\xi}}
\newcommand{\bfxi}{\mbox{{\boldmath $\xi$}}}
\newcommand{\tu}{\tilde{u}}
\newcommand{\tx}{\tilde{x}}
\newcommand{\tf}{\tilde{f}}
\newcommand{\tG}{\tilde{G}}
\newcommand{\tT}{\tilde{T}}
\newcommand{\tpr}{t^{\prime}}
\newcommand{\lh}{\left(}
\newcommand{\rh}{\right)}
\newcommand{\ld}{\left.}
\newcommand{\rd}{\right.}
\newcommand{\pl}{\partial}
\newcommand{\nb}{\nabla}
\newcommand{\tg}{\mbox{\,tg\,}}
\newcommand{\ctg}{\mbox{\,ctg\,}}
\newcommand{\lrder}{\stackrel{\leftrightarrow}{\nabla}}
\newcommand{\cL}{{\cal L}}
\newcommand{\bzg}{\overline{\zg}}
\newcommand{\of}{\overline{f}}
\newcommand{\slashed}{\hspace{-1.1ex}/}
\newcommand{\Slashed}{\hspace{-1.3ex}/\hspace{.2ex}}
\newcommand{\der}{\partial}
\newcommand{\Der}{D}
\newcommand{\sDer}{\Der\Slashed}
\newcommand{\sder}{\der\slashed}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\pagestyle{myheadings}  %%{Authors}{Title}
\markboth{\quad \small \em R. Kerner \hfill}{\hfill \small \em
Approximation methods \quad}

\thispagestyle{plain}
\setcounter{page}{676}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\sloppy

\title{APPROXIMATION  METHODS  IN  CLASSICAL  AND  QUANTUM  THEORIES}

 
\author{R. KERNER$^{*}$ \footnotetext{\,$^{*}$\,e-mail: rk@ccr.jussieu.fr}}

\address{Laboratoire de Physique Th\'eorique des Liquides, \\
Universit\'e Pierre et Marie Curie - CNRS URA 7600, \\
UPMC,  Tour 22, 4-\`eme \'etage, Boite 142, \\
4 Place Jussieu, 75005 Paris, France}


\maketitle\abstracts{ We show in this article how in certain cases Misha
Marinov's ideas and advices produced strong influence, which lasts even
after his passing away. In particular, his deep understanding of classical and
quantum description of particles and fields enabled us to find simple 
approximate solutions to a few interesting problems, e.g. the Cerenkov 
effect in a gravitational field and a new approach to the periodic
motion including the effects of spin in General Relativity.}

\vskip 0.4cm

\tableofcontents

\newpage
\section{Introduction}\label{introduction}
\subsection{Encounters with Misha Marinov}\label{subsec:Misha}

Although I had known some of Misha Marinov's work since long ago, especially
his celebrated paper with Felix Berezin\,\cite{BerMar77} and his original
explanation of path integral methods in Quantum
Theory,\cite{Marinov80} I met him personally for the first time
only in March 1993. It happened during the workshop {\it
``Mathematical Physics towards the 21-st Century''}  organized by R. Sen
at the Ben Gurion University of Beersheva. 

Before the end of the eighties it was quite difficult to travel to Russia
privately, and international meetings held there were much scarcer than
nowadays. And when came to visit Moscow in 1990 (and it was for me the first
time since 1958) on the occasion of the 18-th International Colloquium on Group
Theoretical Methods in Physics, Misha was already in Israel, where he could
finally go with his family after eight years of enduring hardships imposed on 
them by Soviet system just because they expressed their will to leave for
the only country in the world they could consider as their homeland.

Sharing Russian as mother tongue, we very quickly started to spend most
of the time discussing and making comments on everything, physics first,
of course, but also politics, history, Russian and Jewish literature and 
history. We listened to the radio news together, which I tried to understand 
with my quite rudimental knowledge of Hebrew, while Misha translated more 
difficult parts. I have identified the most frequently used words,
``piguah"  and ``bitakhon", and in his usual ironical style Misha told me
that these two words are fairly enough to understand the essence of the
present situation. 

Misha was only four years elder than myself, but his seniority was more than
what these few years would suggest. He had the privilege to learn physics
directly from Lev Landau and Evgeniy Lifshits,  and to start his career under
the guidance of Isaac Pomeranchuk. I was full of admiration for his deep and 
complete knowledge of theoretical physics; but what impressed me even more 
were his exceptional personal qualities, his integrity and firmness in opinions
which he was ready to defend and stand up for, combined with a rare tolerance
and openness towards the opinions of other people. He was always ready to
listen and to discuss, no matter how far the opinions of his opponent could 
be from his own ones.

Very soon have met again at one of the Group Theoretical Conferences organized 
by H.D.~Doebner in Goslar; then we have met in Poland, again in Germany,
later on  in Israel (where I visited him twice at the Technion, in 1996 and
in 1999). He  also visited me in Paris, as invited Professor to our
Relativity and Cosmology  Group (which I directed at that time) in
February 1998. At that time we have become friends, also with his
charming wife Lilia and his daughters Masha and Dina. Many memorable
days and evenings were spent together, often with other friends invited
to Marinov's home in Haifa, in particular Louis Michel and his wife
Th\'er\`ese, and Julius Wess. 

We have discussed a lot, and I have learned a lot, but we had no time to 
develop 
close collaboration e.g. writing papers together. We just discussed his 
last papers 
with his Ph.D.\ student E.~Strahov, or my future projects, which ended up
as articles published with other collaborators. Misha was one of the last
students of Lev Landau, and my encounters with him did give me a
better insight in the methods of thinking of Landau's school, above all the
quest for simplicity and elegance. Several papers written since then are
an attempt to achieve these standards.   
\subsection{Simplicity as Art}\label{subsec:Simplicity}

Very often the most powerful and long lasting ideas in physics are also the 
simplest ones. being aware that in the majority of cases we have to content
ourselves with approximate solutions, the approximation methods become a
crucial issue in physical problems we are dealing with. 

 When I visited Misha in Israel last time (early 1999), he showed me 
a book by the Academician V.~Ginzburg with a beautiful personal dedication
(the Academician Ginzburg has visited Misha in Haifa a few months before
my short stay in the Technion). I have borrowed it and read almost all the
articles it contained. In particular, I have discovered how beautiful
was Ginzburg's explanation of the Cerenkov effect; then it gave me a few 
ideas, and with collaborators (one Russian, Sasha Balakin from Kazan,
another Portuguese, Jose Lemos from Lisbon) we wrote a paper on the 
possibility of a Cerenkov-type effect from the electromagnetic waves
interacting (very weakly, of course) via non-minimal terms of the type
$R_{ijkl} \, F^{ij}F^{kl}$ with gravitation.\cite{bkl} This paper has been 
published in ``Classical and Quantum Gravity" in 2000.
Another discussion with Misha helped me to better understand the
solitons  (see e.g. the paper\,\cite{DVGRK2000} written with D.V.~Gal'tsov
in 2000).
   
Finally, Ginzburg's technique of developing everything in a Fourier
series inspired our latest papers\cite{bhk,khc} on a new approach to 
the two-body
problem in General Relativity -- treating it as an infinite series
of approximations (geodesic deviations from a circular orbit) -- again,
these results, in collaboration with Jan-Willem van Holten from 
Amsterdam, Sasha Balakin and my Ph.D.\ student Roberto Colistete
appeared in ``Classical and Quantum Gravity.''

Another important influence can be traced back to Misha's visit to Paris 
as an Invited Professor in early 1998, and it also concerns an approximation 
technique. At that time Misha worked with his Ph.D.\  student Eugene Strahov 
on the adiabatic approximation for the description of spin in a variable 
magnetic field. The behavior of proton spin in varying magnetic field is 
an important problem which occurs in the analysis of nuclear magnetic 
resonance data. Its direct treatment turns out to be very difficult, but 
when the variations of the external field are not too rapid, the so-called
{\it adiabatic approximation} can be successfully used. The results of
this investigations have been published in \mbox{Ref.\,\protect\citebk{MarStra00}}.

The idea of separating slow and fast phenomena, or strong and weak ones,
is inherent in the Fourier expansion and in some sense was always present 
in the description of motions of planets in our Solar system. Indeed, the 
Ptolemaic system was already based on such a separation -- the first
approximation  given by the ``secular'' circular movement, then smaller
perturbations, called ``epicycles'', being added to it. Such separation often
becomes the only  reliable tool when one is dealing with non-linear
equations.  

Curiously enough, this approximation technique has not been much used in the
two-body problem in General Relativity. The most common approach was to start
with the approximation given by the Newtonian theory, and then to add up 
progressive corrections which take into account various relativistic effects.
A lot of important results have been obtained in, e.g.,
Refs.\,\protect\citebk{Damour} and\,\protect\citebk{Blanchet}.

But if we start from the geodesic equation in General Relativity, describing
the motion of a test particle with negligible mass, we can find an exact
solution in the case of special symmetry. Circular orbits with constant
four-velocity represent such exact solutions in the case of spherically
symmetric gravitaional fields, especially in Schwarzschild and 
Reissner-Nordstrom metrics. Small perturbations around this solution 
satisfy the well known {\it Jacobi equation}, or 
the {\it geodesic deviation
equation}. Higher-order generalizations of this equations are quite
straightforward, and have been known since a long time.

When applied to the particular case of circular orbits in spherically
symmetric gravitational field, these equations give rise to a series
of linear systems whose solutions are nothing else but harmonic oscillators
with frequencies given by the characteristic eigenvalues of the system.
Next approximations give further corrections in the form of ``epicycles'',
which are the consecutive terms in a Fourier series expansion of this
quasi-periodic motion.

These results have been obtained quite recently in collaboration with
A.Balakin, J.-W. van Holten and R. Colistete Jr.\,\cite{bhk,khc} Here I
shall expose the main lines of this work, stressing the similarity with
quasi-classical treatment of spin in the external field. This is my tribute
to the direct and indirect influence  Misha  Marinov's deep understanding
of Physics had on me during the last years of his life.

\section{Cherenkov radiation in gravitational background}\label{cerenkov}

The radiation emission stimulated by a particle moving with a 
supraluminal speed in a dielectric medium, first observed by Cherenkov and
Vavilov, and theoretically explained by Tamm and Frank, is one of the
cornerstones of classical electrodynamics (see, e.g.,
Refs.\,\protect\citebk{landau1}\,and\,\protect\citebk{ginzburg}).
This phenomenon, called the Cherenkov radiation, is currently used in measuring
devices of elementary particle physics, e.g. in high-energy collider detectors.

The Cherenkov radiation can be explained in simple terms in Minkowskian 
spacetime.
Although it can be
explained also in purely classical terms, the Quantum Oscillator approach
introduced by V.L.~Ginzburg is by far more elegant. It uses the fact that
the two interacting entities, the electromagnetic field and the charged
particle producing it, can be decomposed into an infinite sum of contributions
having the form of quantum oscillators. The resonance terms present on the
right-hand side of Maxwell equations (representing the source) give rise
to spontaneous emission of quanta, which are interpreted as Cherenkov's waves. 

Consider a charged
particle moving uniformly with velocity ${\bf V}$ in a static isotropic
dielectric medium with the refraction index $n$. This particle induces an
electromagnetic field which can be represented as a Fourier integral as
follows,
\begin{equation}
{\bf A} = {\displaystyle{\sum_{l}}} \, ( q_l {\bf A}_l + q^*_l {\bf A}_l^* )
{\rm \, \ \, with \;} l = 1,2,....\infty , {\rm \, \ \, and \; }
{\bf A}_l = \frac{c}{n} \, {\bf e}_l \, e^{(i {\bf k}_l \cdot {\bf r} )}\,.
\end{equation}
Then, in  Lorentz gauge $\partial_{\mu} A^{\mu} = 0$, and from 
\begin{equation}
\frac{n^2}{c^2} \, \frac{\partial^2  {\bf A}}{\partial t^2}\,-
\bigtriangleup {\bf A} = \frac{1}{c} \, {\bf j} \qquad {\rm and}
\qquad \frac{n^2}{c^2} \, \frac{\partial^2 \, {\varphi}}{\partial t^2}\,-
\bigtriangleup {\varphi} = \frac{1}{c} \, \rho \, ,
\end{equation}
we get, in terms of Fourier components,
\begin{equation}
\frac{d^2 q_l}{dt^2}\,+\,\omega^2q_l = \frac{1}{c}\int{\bf j} \cdot
{\bf A}^*_l  \, d V\,.
\end{equation}
For a point particle with electric charge $e$ moving with
constant velocity ${\bf V}$
\begin{equation}
{\bf j} = e {\bf V} \, \delta \, ( {\bf r} - {\bf V} t )\,,
\end{equation}
and we get the equation of an oscillator solicited by an external force:
\begin{equation}
\frac{d^2 q_l}{dt^2}\,+\,\omega^2  q_l = \frac{e}{n} \ ( {\bf e}_l \cdot
{\bf V} ) \ e^{- i\,({\bf k}_l \cdot {\bf V})\,t }\,,
\end{equation}
in which each component $l$ displays the frequency
$\Omega_l\!\!=\!  {\bf k}_l\!\cdot\!{\bf V}\! =\! k_l\, V \cos\theta$,
\pagebreak[3]where $\theta$ is the angle between the direction of the wave
three-vector ${\bf k}_l$ and the three-velocity of the particle ${\bf V}$. When
$\Omega_l =\omega$, a resonance occurs, making oscillator's amplitude grow 
linearly with $t$, which is interpreted as condition for radiation.
In order for this radiation to represent a real electromagnetic wave, the
relation between the frequency and the wave number should be $\omega = kc/n$.
Since one must always have $k\!>\!k\cos\theta$, one finds that this
condition is equivalent to
\begin{equation}
V>\frac{c}{n}\,,
\label{velocitycondition1}
\end{equation}
i.e. the velocity of the particle should be greater than the velocity 
of light in the medium $c/n$. It is also clear that the maximal angle of 
radiation, the Cherenkov angle, is given by the condition
\begin{equation}
\cos\theta_0=\frac{c}{nV}\,. 
\label{angle1}
\end{equation}
A possibility of the Cherenkov radiation emission by charged particles 
traveling
with a quasi-luminal speed in vacuum where the strong gravitational field
is present,  is suggested by a sentence of the book ``Classical Field Theory'' 
of Landau and Lifshitz,\cite{landau2} in the problem of paragraph 90,
where it is stated that the gravitational field plays a role of a medium
with electric  and magnetic permeabilities equal to $1/\sqrt{h}$, where
$h$ is the determinant  of the three-dimensional spatial induced metric.
Then one may ask the following  two questions: In which types of
gravitational fields can the Cherenkov  radiation be emitted? Is the
Cherenkov effect possible in vacuum hosting a gravitational field?

This problem was taken up in Ref.\,\protect\citebk{gupta}, where the Cherenkov
radiation inside a material medium was considered. One may also
consider the Cherenkov  radiation in a gravitational wave background
with vacuum interacting with  curvature. More specifically, it was
possible to show that a covariant analysis  of this problem, based on exact
solutions for the coupled system of a charged particle and an
electromagnetic wave in a nonlinear gravitational wave  background,
makes it possible to answer these two questions.\cite{Balakin97,bkl}

One should distinguish between pure vacuum and vacuum interacting with
curvature.\cite{drummond} The latter behaves as a medium with
electric and magnetic properties, and can, therefore, be called a
quasi-medium.  The results of our investigation show that quasi-media
and true media can display the Cherenkov effect, while the pure vacuum
can not.

In order to adapt it to the curved space-time, the criterion for the Cherenkov 
radiation emitted by a charged particle must be reformulated in a
covariant  way. In a covariant formulation we cannot use the
three-velocity vector 
${\bf V}$, or the Cherenkov angle $\theta_0$, because they are not Lorentz 
invariant quantities. There are two covariant vectors in this problem, 
the time-like momentum four-vector of the charged particle $P^i$,
normalized according to $P_iP^i = m^2 c^2 > 0$, and the wave four-vector
$k_i$ characterizing the electromagnetic plane wave which can, in principle,
propagate inside the medium in a given space-time.  We say therefore that the
Cherenkov radiation can exist when the following equality is satisfied :
\begin{equation}
k_m P^m = 0\,,
\label{criterion}
\end{equation}
where Latin indices are spacetime indices running from 0 to 3, 
and where we use the metric with signature $(+\,-\,-\,-)\,$.

One can easily recover the relations 
(\ref{velocitycondition1}) and (\ref{angle1}).    
In a standard three-dimensional context in Minkowskian space-time the
relationship (\ref{criterion}) can be rewritten in the usual form 
\begin{equation}
k_m P^m = k_0 P^0 - {\bf k} \cdot {\bf P} =
\frac{P^0}{c}\,\left( \omega - {\bf k}\cdot {\bf V} \right) =
\frac{P^0 \omega}{c} \Big( 1- \frac{1}{c}\,{\bf n} \cdot {\bf V} \Big) = 0\,.
\label{3criterion}
\end{equation}
The classical definition of frequency is $\omega\!=\!c k^0$, the vector
refraction index is ${\bf n}\!=\!c {\bf k}/\omega$, and its square is
$n^2 \!=\!{\bf n}^2$. We can now reproduce the criterion to have Cherenkov's
radiation in flat space-time, i.e. we can reproduce
Eqs.\,(\ref{velocitycondition1})\,and\,(\ref{angle1}). For instance, using the 
definition of the angle $\theta$, i.e. 
$\cos \theta \!=\!  {\bf k} \cdot {\bf V}/ k V \!=\! {\bf n}\cdot {\bf V}/ nV$,
as well as the condition $\vert \cos \theta \vert \leq 1$, one can see from
(\ref{3criterion}) that $V  \geq c/ n$, i.e. particle's three-velocity
should be bigger than speed of light in the medium, recovering
(\ref{velocitycondition1}).
Equation (\ref{angle1}) also follows in a straightforward way.
Let us now proceed to find the necessary condition for the existence of
Cherenkov's radiation. It can be given in three equivalent ways, the first
involving the square $k_i k^i$ of the wave four-vector $k^i$, the second one
involving the square $n^2$ of the scalar refraction index, and the third
one involving the phase velocity of light in the medium, $v_{\rm ph}$. Note
that this necessary condition does not depend on particle's momentum $P^i$.

It is well known  that the frequency $\omega$ (considered as a scalar
quantity), and the four-vector $k_i^*$ (the spatial part of the wave
four-vector) may be defined by the following relations (see, e.g.,
Ref.\,\protect\citebk{syng}):
\begin{equation}
\frac{\omega}{c} \equiv k_i U^i \qquad {\rm and} \qquad
k^{*}_i \equiv k_l \Delta^l_i\ .
\label{3wavevector}
\end{equation}
Here $U^i$ is the four-velocity vector of the medium (or of the observer,
if we consider propagation in pure vacuum), and $\Delta^l_i$ is the projector
defined as
\begin{equation}
\Delta^i_j \equiv \delta^i_j - U^i U_j\,.
\label{projector}
\end{equation}
Following the standard definition (see p.~290 in 
Ref.\,\protect\citebkcap{landau1}), we
introduce now the four-vector index of refraction $n_i$
\begin{equation}
n_i = \frac{c}{\omega}\,k_i^*\,.
\label{vecopticalindex}
\end{equation}
The vector refraction index $n_i$ is in general a spacelike four-vector,
orthogonal to the four-velocity $U_i$, and only in an
isotropic medium it reduces to a scalar. Its absolute value is referred to as 
the scalar index of refraction, (or refraction index). Its 
square is given by  $n^2 \equiv -\,g^{ik} n_i n_k \,$.

We can now write the following useful identity satisfied by $k^i$, 
\begin{equation}
k_i = U_i\,(k_l U^l) + k^{*}_i\,.
\label{kdecomposition}
\end{equation}
Now, in order to use the criterion (\ref{criterion}) we note that $P^i$ and
$k^i$ must be orthogonal. This happens only when  $k_i$ is spacelike, since
the $P^i$ four-vector is timelike.
From the equation (\ref{kdecomposition}) we obtain the square of $K_i$, 
\begin{equation}
k^2 \equiv -\,g^{im}k_i k_m=-\,k_m k^m= -\,(k^m U_m)^2 - k^l k^s \Delta_{ls}\ .
\label{squareK}
\end{equation}
This equation can be written explicitly in two ways, 
\begin{equation}
k^2\!=\!{\left(\frac{\omega}{c}\right)}^2\!(n^2-1)
\qquad {\rm or} \qquad
k^2\!= {(k^*)}^2\!- \frac{\omega^2}{c^2} \ ,
\label{signksquare}
\end{equation}
where 
\begin{equation}
{(k^*)}^2 \equiv -\,k^l k^s \Delta_{ls} \ .
\label{ksquare}
\end{equation}
From the equation (\ref{ksquare}) we see that the sign of $k^2\!$ coincides
with that  of $(n^2\!-\!1)$. This means that  $k_i$ is spacelike for $n^2>1\,$.
From the equation (\ref{signksquare}) we also see that $k_i$ is spacelike
when $\omega^2/c^2<(k^*)^2$, i.e. when the phase velocity of light 
$v_{\rm ph}\equiv\omega/ k^* $ obeys $v_{\rm ph}<c\,$.

Three equivalent invariant forms of the necessary condition for Cherenkov's 
radiation can be used, namely,
\begin{equation}
k_m k^m < 0\,, \qquad {\rm or} \qquad n^2>1\,, \qquad {\rm or} \quad 
v_{\rm ph}<c\,.
\label{necessary}
\end{equation}  
All of them require the knowledge of the four-vector $k^i$, which is obtained
from the corresponding solution of Maxwell equations.

In order to use the criterion (\ref{criterion}) in a curved background we
have to resort now to a specific model. We have used as a background the 
pp-wave solution of Einstein equations in vacuum. First, the four-momentum $P^i$ of
a particle moving in this background is determined, then a specific solution
of Maxwell equations is found, in the same gravitational background. After
that, we can examine the criteria of existence of the Cherenkov radiation,
and establish its spatial properties in this background. In a curved
space-time different spatial directions are generally non-equivalent, which
implies that one should know the evolution of particle's four-momentum $P^i$
with arbitrary initial data in order to be able to use explicitly the
criterion (\ref{criterion}).

\section{Radiating charges in the gravitational wave }

\subsection{The gravitational wave background}

Let us consider the space-time described by the exact pp-wave solution of
Einstein's equations in vacuum.\cite{kramer} The metric describing a
gravitational wave propagating in the $x^1$ direction is supposed to take on
the following form:
\begin{equation}
ds^{2} =  2\,du\,dv -
L^{2} \left[e^{2\beta}{(dx^{2})}^{2} + e^{-2\beta}{(dx^{3})}^{2} \right],
\label{metric}
\end{equation}
where
\begin{equation}
u =  \frac{ct-x^{1}}{\sqrt{2}} \qquad {\rm and} \qquad
v = \frac{ct+x^{1}}{\sqrt{2}}
\label{times}
\end{equation}
are the retarded and the advanced times , respectively. The functions
$ L$ and $ \beta$ depend only on the variable $u$, i.e.\ $L=L(u)$ and
$\beta=\beta(u)$.

The  pp-wave metric (\ref{metric}) is invariant under the $G_{5}$ symmetry
group and admits the following set of five Killing vector fields $\xi_{(r)}$ 
(where the index $(r)$ takes on the values $(v)$, $(2)$, $(3)$, $(4)$,
$(5)$ and
characterizes each vector), 
$$ \xi^{i}_{(v)}= \delta ^{i}_{v} \,,  \qquad 
\xi^{i}_{(2)}= \delta ^{i}_{2} \,, \qquad 
\xi^{i}_{(3)}= \delta ^{i}_{3} \,, $$
\begin{equation}
\xi^{i}_{(4)}= x^{2}  \delta^{i}_{v}  -  \delta^{i}_{2} 
\!\int\!{g^{22}(u)\,du} \,, \qquad
\xi^{i}_{(5)} = x^{3}  \delta^{i}_{v}
-  \delta^{i}_{3}\!\int\!{g^{33}(u)\,du} \,. 
\label{killings}
\end{equation}
Here $g^{\alpha \beta}(u)$ ($\alpha, \beta = 2,3$) are the
contravariant components of the metric tensor. The vector
$\xi^{i}_{(v)}$ is isotropic, covariantly constant and orthogonal to
the other four ones, 
\begin{equation}
\nabla_{k} \, \xi^{i}_{(v)}\!=0\,, \qquad g_{ik}\,
\xi^{i}_{(v)} \, \xi^{k}_{(r)}\!=0\,.
\label{ortho}
\end{equation}
The three vectors
$\xi^{i}_{(v)}\,,\ \xi^{i}_{(2)}\,$, and $\,\xi^{i}_{(3)}$
%an alternative:
%$\xi^{i}_{(v),\,2,\,3}$
form the Abelian subgroup $G_{3}\,$.
The two functions  $L(u)$ and $\beta  (u)$  are coupled
by the Einstein equation
\begin{equation}
L^{''} + L \  (\beta^{'})^{2}= 0\,,
\label{einstein}
\end{equation}
which is unique in this case. The function $\beta (u)$ can be chosen at will,
and once given,  one can solve the equation (\ref{einstein}) for $L(u)$. The
curvature tensor has two non-vanishing components
\begin{equation}
-\,R^{2}_{\cdot u2u}\!= R^{3}_{\cdot u3u}\!=
L^{-2}\,{\left[L^{2} \,{\beta}^{\prime} \right]}^{\prime}. 
\label{riemann}
\end{equation}
Both the Ricci tensor $R_{ik}$ and the curvature scalar $R$ are equal to zero.

\subsection{Particle dynamics in the GW background}

The geodesic equation for a particle with mass $m$ in the GW 
field (\ref{metric})
reads
\begin{equation}
\frac{D P_i}{D\tau} = 0 \,, \qquad {\rm with} \quad
P^i = mc\,\frac{d x^i}{d\tau} \ .
\label{eqmotion}
\end{equation}
Using the well-known property of the Killing vectors (\ref{killings}),
we obtain the following expressions for the components of the momentum:
\begin{eqnarray}
P_v\!\equiv P_i \xi^i_{(v)}\!= {\rm const} \equiv  C_v\, ,\qquad
P_\alpha \equiv P_i\,\xi^i_{(\alpha)}\!= {\rm const} \equiv C_\alpha\,,
\label{momentum2}
\\[1.5mm]
P_u = \frac{1}{2 C_v}\left[ m^2 c^2 - g^{\alpha \beta} C_\alpha 
C_\beta \right], \hspace{2cm} \label{momentum3} 
\label{pu}
\end{eqnarray}
where the last equation for the  component $P_u$ of the momentum followed 
from the normalization condition.

Since $C_v$ and $C_\alpha$ are constants, they also represent the initial
values of the corresponding momentum components at the initial surface
defined by $u = 0$. These data determine the character of particle's motion.
For example, when $C_\alpha = 0\,$, i.e. when the particle moves initially
along the longitudinal direction (the direction of propagation of the GW
$x^1$), then it will always move along that direction without acceleration.
When $C_\alpha \neq 0\,$, the dynamical effects on the particle induced by the
GW appear in its longitudinal motion (see the expression (\ref{pu}) for
$P_u$)\,. Thus, in the field of GW, the criterion (\ref{criterion}) yields the
following equation:
\begin{equation}
K_mP^m=
K_u C_v + K_v\,\frac{1}{2 C_v}\left[ m^2 c^2 - g^{\alpha \beta}(u)\,
C_\alpha C_\beta \right]+ g^{\alpha \beta}(u)\,K_\alpha C_\beta = 0\,.
\label{gwcriterion}
\end{equation}
It remains to represent explicitly the components of the wave four-vector
$K_i$ and we also have to check the condition (\ref{necessary}), necessary
for the emission of the Cherenkov radiation.
The full solution of the combined field and particle motion equations has been
found in Ref.\,\protect\citebk{bkl} The main result is that although a plane 
gravitational wave modifies the dielectric
properties of pure vacuum, it does not modify its scalar refraction index,
and therefore there is no possibility of emission of the Cherenkov radiation
in this particular case.

On the other hand, vacuum interacting with curvature, considered as a sort of
quasi-medium, allows the possibility of the existence of the Cherenkov
radiation.  Cherenkov's radiation in vacuum interacting with curvature
can propagate along a non-circular cone, the spatial structure
of this cone being permanently modified with time. The Cherenkov
radiation emission exists alternatively for each polarization of the
electromagnetic wave.

Thus, we deal with a new phenomenon in Cherenkov's radiation, namely,
polarized radiation with oscillating polarization direction.  The Cherenkov
angle is predetermined by the value of the Riemann tensor.

We have performed an exact treatment in two particular cases, with two
privileged directions of particle motion, the longitudinal
and the transversal ones.  However, for arbitrary direction of particle
motion, the structure and the inclination of the cone axis with respect
to the GW front plane is given by more complicated modified expressions.

The dispersion phenomena for the Cherenkov radiation can not be ruled out,
as the effective refraction index induced by the ambient curvature is also
a function of the wavelength emitted. However, these effect seem to be
rather of the academic interest, as their order of magnitude is apparently
very small.

\section{Epicycles in General Relativity}\label{epicycles}

In General Relativity the problem of motion of planets, considered as test
particles moving along geodesic lines in the metric of Schwarzschild's
solution, has been solved in an approximate way by Einstein,\,\cite{AE}
who found that the perihelion advance during one revolution is given in
the near-Keplerian limit by the formula 
\begin{equation}
\Delta \phi =\frac{6\pi\,GM}{a(1-e^{2})}\ ,  \label{advance1}
\end{equation}
where $G$ is Newton's gravitational constant, $M$ the mass of the central
body, $a$ the greater half-axis of planet's orbit and $e$ its eccentricity. 
\newline
\indent
This formula is deduced from the exact solution of the General Relativistic
problem of motion of a test particle in the field of Schwarzschild metric,
which leads to the expression of the angular variable $\varphi $ as an
elliptic integral, which is then evaluated after expansion of the integrand
in terms of powers of the small quantity $GM\!/r\,$. \newline
\indent
The formula has been successfully confronted with observation, and represents 
one of the best confirmations of Einstein's theory of gravitation. 
In the case of small eccentricities the formula (\ref{advance1}) can be 
developed into a power series: 
\begin{equation}
\Delta \phi =\frac{6\pi GM}{a}\,(1+e^{2}+e^{4}+e^{6}+\dots\,)\ .
\label{advance2}
\end{equation}
One can note at this point that even for the case of Mercury ($e=0.251$), the
series truncated at the second term, i.e. taking into account only the
factor $(1+e^{2})$ will lead to the result that differs only by $0.18\%$
from the result predicted by relation (\ref{advance1}), which is below the
actual error bar.
\newline
\indent
This is why  it is useful to present an alternative way of treating
this problem, based on the use of geodesic deviation equations of
first and higher orders. Instead of developing the exact formulae of motion
in terms of powers of the parameter ${GM/r}$, we can start with
an exact solution of a particularly simple form (i.e. a circular orbit with
uniform angular velocity), and then generate  approximate solutions as
geodesics being close to this orbit. 
\newline
\indent
One of the advantages of this method is the fact that it amounts to treating
consecutively systems of linear equations with constant coefficients, all of
them being of harmonic oscillator type, eventually with an extra right-hand
side being a known periodic function of the proper time. The approximate
solution obtained in this manner has the form of a Fourier series and
represents the closed orbit as a superposition of epicycles with diminishing
amplitude as their circular frequencies grow as multiples of the basic one.
This approach is particularly well-suited for using numerical computations.

In this respect, it is similar to the treatment\,\cite{MarStra00} of the 
spin precession in slowly varying magnetic field.

\subsection{World-line deviation equations}\label{deviations}  

According to the equivalence principle, structureless test bodies 
(sometimes referred to as point masses) in a gravitational field move 
on geodesics of space-time. Their world-line $x^{\mu}(\tau)$ is 
a solution of the geodesic equation 
\be 
\frac{D^2 x^{\,\mu}}{D\tau^2}\, =\, \frac{d^2 x^{\,\mu}}{d\tau^2}\, 
 + \Gam_{\lb\nu}^{\;\;\;\;\mu}\,\frac{dx^{\lb}}{d\tau}\,
 \frac{dx^{\nu}}{d\tau}\, =\, 0\,, 
\label{1}
\ee 
where the world-line parameter $\tau$ is the proper 
time. Introducing the four-velocity as the time-like tangent unit 
vector to the world-line: $u^{\,\mu} = dx^{\,\mu} / d\tau$, the equation 
can be written in geometrical language as 
\be 
u \cdot \nabla u = 0\,, \hs{2} u^2\!= -1\,. 
\label{2}
\ee 
with $\nabla$ the covariant derivative. It is easily observed 
from Eq.\,(\ref{1}), that the proper acceleration $a^{\,\mu} = d^2
x^{\,\mu} / d\tau^2$ is not a covariant object. In particular, its 
vanishing or non-vanishing has no observer-independent meaning. 

In contrast, the {\em relative} acceleration between world lines is
a covariant quantity, and its vanishing or non-vanishing does not 
depend on the frame of reference.\cite{syng}
Consider a one-parameter congruence of geodesics $x^{\,\mu}(\tau;\lb)$,
where $\lb$ labels the geodesics and $\tau$ is the proper-time 
parameter along the geodesic. We suppose the parametrization to 
be smooth, hence we can construct the tangent vector fields $u^{\,\mu} 
= \der x^{\,\mu}/\der \tau$, and $n^{\mu} = \der x^{\,\mu}/\der \lb$. 
It is straightforwardly established that 
\be 
{(u \cdot\nabla n)}^{\mu} = \frac{\der^2 x^{\,\mu}}{\der \tau\,\der \lb}\, 
 + \Gam_{\lb\nu}^{\;\;\;\;\mu}\,\dd{x^{\lb}}{\tau}\,\dd{x^{\nu}}{\lb} 
 = {(n \cdot\nabla u)}^{\mu}. 
\label{3}
\ee 
As a corollary, we obtain
\be 
u \cdot {\nabla(u \cdot \nabla n)}   =  n \cdot \nabla{(u \cdot \nabla u)} + 
 u^{\mu} n^{\nu}\,[\nabla_{\mu}, \nabla_{\nu}]\,u = u^{\mu} n^{\nu} 
 R_{\mu\nu}[u,\cdot\,]\ . 
\label{4} 
\ee
In component notation this reads
\be 
\frac{D^2n^{\mu}}{D\tau^2}\, =\, R_{\kg\lb\nu}^{\;\;\;\;\;\;\mu} 
 u^{\kg} u^{\nu} n^{\lb}\,.
\label{5}
\ee 
If $x^{\,\mu}_0(\tau) = x^{\,\mu}(\tau;\lb_0)$ is 
a solution of the geodesic equation (\ref{1}), then (to first order) 
$x_1^{\,\mu}\!= x_0^{\,\mu} + n^{\mu} \Del \lb$ is a solution as well:
\be 
x^{\,\mu}(\tau;\lb_1) = x^{\,\mu}(\tau;\lb_0) + \Del \lb\, 
 \dd{x^{\,\mu}}{\lb}\,(\tau,\lb_0) \approx x^{\,\mu}(\tau\,;\,\lb_0 
 + \Del \lb)\,. 
\label{6}
\ee 
It follows, that Eq.\,(\ref{5}) describes the covariant relative acceleration
between these world lines. Of course, $n^{\mu}$ is only a first approximation
to the neighboring geodesic at $\lb_1\!=\!\lb_0\!+\! \Del \lb\,$. To increase
the precision of the approximation, 
one has to compute higher-order derivatives w.r.t.\ $\lb$, by 
solving higher-order versions of Eq.\,(\ref{5}), involving not only 
the Riemann curvature tensor, but its derivatives as well. A 
systematic procedure of this type can be found in Ref.\,\protect\citebk{khc}\,.

\subsection{Application: 
the Coulomb-Reissner-Nordstrom field}\label{application1} 
 
World-line deviation equations can be used to compute the relative 
motion between particles in given background fields, or to obtain 
an approximation to solutions for orbits close to a known one. 
We illustrate the general results with an application to the study 
of the motion of charged particles in a central gravitational and 
electric Coulomb-Reissner-Nordstrom field.

The vector potential and electric field strength for the Coulomb
part of this solution of the Einstein-Maxwell equations are
given by the one- and two-forms
\be 
A = -\, \frac{Q}{4\pi r}\ dt\,, \hs{2} 
F = dA = \frac{Q}{4\pi r^2}\ dr\!\wedge dt\,,
\label{2.1}
\ee 
whilst the metric for the gravitational field can be taken as 
\be 
- d\tau^2 = -B(r)\,dt^2 + \frac{1}{B(r)}\, dr^2 + r^2\!\lh d\thg^2
 + \sin^2 \thg\, d\vf^2 \rh,
\label{2.2}
\ee 
where $B(r) = 1  - (2M/r) + (Q^2/r^2)\,$, $Q$ and $M$ are the charge and
the mass of the central body\pagebreak[3] which is the source of the field.

The orbits of particles with mass $m$ and charge $q$ in this 
background can be computed in closed form in terms of elliptic 
integrals.\cite{bhk} More precisely, the orbits are given by 
\be 
r(\vf) = \frac{r_0}{1 + e \cos y(\vf)}\;, 
\label{2.3}
\ee 
where $y(\vf)$ is the solution of the differential equation
\be 
\frac{dy}{d\vf}\, =\, \sqrt{A + B \cos y + C \cos^2 y}\;,
\label{2.4}
\ee 
with coefficients given by  
\be 
\begin{array}{ll} 
A =  \dsp{ 1 + \frac{Q^2}{\ell^2}\!\lh 1 - {\lh \frac{q}{4\pi m} 
 \rh}^2 \rh\!- \frac{6M}{r_0} + \frac{Q^2}{r_0^2}\,(6 + e^2)\,, }\\[6mm]
B =  \dsp{ -\,\frac{2e}{r_0} \lh M - \frac{2Q^2}{r_0} \rh, }\qquad
C =  \dsp{ \frac{e^2 Q^2}{r_0^2}\,. } 
\end{array} 
\label{2.5}
\ee 
Here $\ell$ is the constant angular momentum per unit of mass. As the 
periastra of the orbit are at the points $y(\vf) = 2\pi n$, one can now 
compute the angular distance $\Del \vf$ between successive periastra. 
From $\Del \vf = 2\pi + \del \vf$, it follows that the periastron 
shift per orbit is 
\be 
\del\,\vf = 2\pi \lh \frac{3M}{r_0} - \frac{Q^2}{2Mr_0} \rh +\,\ldots\ ,
\label{2.6}
\ee 
the dots denoting terms of higher order in $e$, $M/r_0$ or $Q/r_0$. 

Eqs.\,(\ref{2.3}) and (\ref{2.4}) describe a general orbit in the 
exterior region of the central body. However, they do not provide all 
information about the orbit. In particular, as the time coordinate 
has been eliminated from these equations, the solution does not 
tell us where in its orbit the test particle is at any moment. 
Such information can be relevant for some important applications, 
e.g.\ to compute estimates of the amount of electro-magnetic and 
gravitational radiation emitted by the system. The method of 
world-line deviations is useful to obtain parametrized expressions
of orbits $(r(t), \vf(t))$. 

As the reference orbit, the zeroth order approximation to the real orbit,
we take a circular one with constant radial coordinate $R$. Constants 
of motion on all orbits are the angular momentum per unit of mass, 
$\ell = \og R^2$, with the angular velocity $\og = \dot{\vf}$, and 
the energy $\ve$ per unit of mass defined by 
\be
\frac{dt}{d\tau} = \frac{\ve - \,q\,Q/4\pi mR}{1 - 2M\!/R\, + Q^2\!/R^2}\;.
\label{2.7}
\ee 
Then, on circular orbits the constants $R$, $\ell$ and $\ve$ are related by  
\be
{\lh \ve -\, \frac{q\,Q}{4\pi mR} \rh}^2\!\! = \lh 1 - \frac{2M}{R} 
 + \frac{Q^2}{R^2} \rh \lh 1 + \frac{\ell^2}{R^2} \rh, \vspace{-2.5mm}
\ee
\begin{eqnarray}
%\begin{array}{l}
{\rm and} \qquad\qquad\quad
 {\left[ \frac{\ell^2}{R}\,- M\!\lh 1 + \frac{3\ell^2}{R^2} \rh
+\hs{-.1} \frac{Q^2}{R} \hs{-.1} 
 \lh 1 + \frac{2\ell^2}{R^2} \rh \right]}^2 \hspace{2.7cm}\\[2.5mm]\label{2.8}
={\lh \frac{q\,Q}{4\pi m} \rh}^{\!\!2} \!\!\lh 1 + \frac{\ell^2}{R^2}\rh\!
 \lh 1 - \frac{2M}{R} + \frac{Q^2}{R^2} \rh. \nonumber
\end{eqnarray} 
As all orbits are planar, we can always choose the orientation of
the coordinate system such that $\thg = \pi/2$ for the reference orbit. 
For orbits tilted  w.r.t.\ this one, we then find from Eq.\,(\ref{14}) that 
\be 
\ddot{n}^{\thg} + \og^2 n^{\thg} = 0\,,
\label{2.9}
\ee 
from which it follows, as the physics dictates, that the distance 
perpendicular to the plane of the reference orbit oscillates with the 
period of the circular orbit $T = 2\pi/\og = 2\pi R^2/\ell$. 

Considering orbits in the plane of the reference orbit, the world-line
deviation equations (\ref{14}) for the other components $n^i = (n^t, 
n^r, n^{\vf})$ become
\be 
\ddot{n}^i + \gam^i_{\;j} \dot{n}^j + m^i_{\;j} n^j = 0\,,
\label{2.10}
\ee 
where the coefficient matrices take the form 
\be 
\gam = \lh \begin{array}{ccc} 0 & \gam^t_{\;r} & 0 \\[0.6mm]
                    \gam^r_{\;t} & 0 & \gam^r_{\;\vf} \\[0.6mm]
                    0 & \gam^{\vf}_{\;r} & 0 \end{array} \rh, \hs{2} 
m = \lh \begin{array}{ccc} 0 & 0 & 0 \\[0.6mm]
                 0 & m^r_{\;r} & 0 \\[0.6mm]
                 0 & 0 & 0 \end{array} \rh. 
\label{2.11}
\ee 
This represents a system of coupled linear oscillators, which 
has solutions 
\be 
n^t(\tau) = n^t_0 \sin \og_1 \tau\,, \hs{2} 
n^r(\tau) = n^r_0 \cos \og_1 \tau\,, \hs{2} 
n^{\vf}(\tau) = n^{\vf}_0 \sin \og_1 \tau\,,
\label{2.12}
\ee 
where $\og_1$ is the solution of the characteristic equation for 
(\ref{2.10}). The detailed form of this equation, using explicit 
expressions for the elements of the matrices $\gam$ and $m$ 
were given in Ref.\,\protect\citebk{bhk}. The resulting expression for the 
characteristic frequency is 
\begin{equation} 
\omega_1 = \omega \Big( 1 - \frac{3M}{R} + \frac{Q^2}{2MR} + ...\; \Big)\,,
\label{2.13}
\end{equation}
where the dots represent terms of higher order in $M\!/R$, $Q/M$ or
$q/m$. We also observe, that the amplitudes $n_0^i$ are not
all independent: as $u^2\! = -1$ both on the original orbit and
on the displaced world-line, it follows that $n$ is space-like
and $u \cdot n = 0\,$. In the present case this general result 
translates to the constraint
\be 
\lh \ve - \frac{q\,Q}{4\pi mR} \rh\!n_0^t - \frac{q\,Q}{4\pi m\,\og_1 R^2}
\,u^t n_0^r - \ell\,n_0^{\vf} = 0\,.
\label{2.14}
\ee 
Although the components $n^{\mu}$ define the direction of the deviation,
they do not determine the actual distance between neighboring world 
lines; this is given by equation (\ref{12}) as $\Del x^{\,\mu} = n^{\mu} 
\Del \lb$. Therefore, for any particular orbit specified by the circular
reference orbit (zeroth order approximation) and a world-line deviation 
vector $n$ (first order approximation), we must determine in addition 
the scale factor $\Del \lb$ to be applied. This can be done as follows. 
Comparing the approximate solution (\ref{2.10}) with the exact solution 
(\ref{2.3}), we observe that 
\be 
r(\vf) = R + \Del r \approx R - e\,R \cos y(\vf)\,. 
\label{2.15}
\ee 
Hence at the periastron, one has 
\be 
\Del r = - \,\Del \lb\,n_0^r = -\,e\,R\,. 
\label{2.16}
\ee 
Thus the scale is set by the eccentricity of the orbit. 
\nit
Finally, we can determine the shift in angular coordinate between 
successive periastra, i.e.\ the advance of the periastron per 
orbital period. First observe, that the periastron occurs at the 
minima of $n^r(\tau)$, i.e.\ for $\tau_n = (2n + 1) \pi /\og_1$. 
Thus the amount of proper time elapsing between periastra is 
$\Del \tau = 2 \pi / \og_1$; the corresponding period of observer
time is 
\be 
T =\!\int_0^{2\pi /\og_1} \!\!\! d\tau\, \frac{dt}{d\tau} 
 = \!\int_0^{2\pi /\og_1}\!\!\! d\tau\, (u^t + \dot{n}^t\,\Del \lb)
 = \frac{2\pi}{\og_1}\, u^t.
\label{2.17}
\ee 
Here $u^t$ is the rate of change of $t$ per unit of proper time 
along the circular reference orbit. Next we observe, that at 
the proper times $\tau_n$ the angular coordinates at the 
reference orbit and the true orbit coincide: $n^{\vf}(\tau_n) = 0\,$.
Hence the change in angular coordinate $\vf$ between successive 
periastra is the same as the change of this coordinate along the
circular reference orbit after time $T$. This we can easily 
compute. Defining
\be 
\del \vf = \vf\,(t_0 + T) - \vf\,(t_0) - 2 \pi\,,
\label{2.18}
\ee 
and using  the expression $d\vf/dt = \dot{\vf}\,d\tau/dt = \og/u^t$
for the angular velocity, we find 
\be 
\del \vf = \frac{\og\,T}{u^t} - 2 \pi = 2 \pi \lh \frac{\og}{\og_1} 
 - 1 \rh \approx 2 \pi \lh \frac{3M}{R} - \frac{Q^2}{2M\!R} \rh. 
\label{2.19}
\ee 
This is in perfect agreement with the expression (\ref{2.6}) 
obtained from the analytical\pagebreak[3] form of the orbit. 

It is of interest to consider the geometrical interpretation of
the approximation scheme we have used in a little more detail. 
The zeroth order approximation to the orbit we have constructed
is a fully relativistic circular solution of the Einstein-Lorentz
equation in a Coulomb-Reissner-Nordstrom field, with period 
$T_0 = 2\pi /\og_0$. Included in this result is of course the 
simpler case of a circular geodesic in a Schwarzschild field. The 
first-order correction is a geodesic deviation which oscillates in 
all its components in the same plane with period $T_1 = 2 \pi/\og_1$.
Geometrically this represents another circular movement on the 
background of the zeroth-order solution, i.e.\ an {\em epicycle}, 
with period slightly different from the zeroth-order approximation. 
This has two immediate consequences: the orbit becomes 
eccentric, and the period between extrema of the orbit differs 
from the period of the average (zeroth order) circular motion.
This is in contrast with Newtonian gravity, where the periods 
are equal. Thus the extrema of the orbit (periastron and apastron) 
are shifted compared to the Newtonian approximation, by the amount 
predicted by the analytic description of the orbit. 

It can be easily shown,\cite{khc} that higher-order world-line 
deviations all satisfy linear harmonic-oscillator type 
equations. Thus, computing higher-order corrections to our
result amounts to the construction of higher-order epicycles. 
For the case of orbits in a central field, the method of 
world-line deviations then becomes a fully relativistic version 
of the Ptolemaean scheme,\cite{Ptolemaios} which differs genuinely from 
the standard post-Newtonian approximation scheme because it uses 
the eccentricity of the orbit and the quantities $M\!/R$, $Q/M$ as 
expansion parameters, rather than $v/c$. As such this scheme 
offers an alternative to post-Newtonian calculations of 
binary systems in a different physical regime, e.g.\ in the 
calculation of radiative effects. 

\section{Quasiclassical spinning bodies}\label{spin}

The generalized geodesic deviation equations can be extended to the case
of the particles carrying an electric charge and/or quasi-classical spin.
In these cases particles do not move on geodesics, but on more general 
world lines.\cite{krip} For the case of charged particles in 
a combined electro-magnetic and gravitational field, the resulting 
world line deviation equation was derived in Ref.\,\protect\citebk{bhk}.
\newline
\indent
To this end, it is useful to consider the Lagrangian formulation of the
geodesic deviation equations.We first observe, that this equation is linear and
homogeneous in $n^{\,\mu}$. It is therefore  not very difficult to construct an
action from which it can be derived. The Lagrangian of interest reads 
\begin{equation} 
{\cal{L}} \, (n) = \frac{1}{2}\, g_{\mu\nu}\,\frac{Dn^{\mu}}{D\tau} 
\,\frac{Dn^{\nu}}{D\tau}\, + \frac{1}{2}\, R_{\mu\rho\nu\lambda}\,
 u^{\rho}\,u^{\lambda}\,n^{\mu}\,n^{\nu}\,. 
\label{7}
\end{equation} 
In this Lagrangian the metric, connection and curvature are
those on the given reference geodesic $x_0^{\mu}(\tau)$, with $u^{\mu}(\tau) 
= \dot{x}^{\,\mu}_0$ representing the four-velocity. These quantities will be 
treated as background variables. Only the $n^{\mu}(\tau)$ should be considered
as independent Lagrangian coordinates which are to be varied in the action. 
The action (\ref{7}) can be derived independently by starting 
from the geodesic Lagrangian
\begin{equation} 
{\cal{L}} \, (x) = \frac{1}{2}\, g_{\mu \nu} \, (x)\, \frac{dx^{\,\mu}}{d \tau}
\,\frac{d x^{\nu}}{d \tau}\;, 
\label{8}
\end{equation} 
and expanding $x^{\,\mu}(\tau)$ near the given background geodesic
solution in the form $x^{\,\mu} = x_0^{\,\mu} + n^{\mu} \Delta \lambda$. 
The term independent of $\Delta \lambda$ does not contain $n^{\mu}$, 
and contributes, after integration, only a constant term to the action. 
Next all terms linear in $\Delta \lambda$ drop out of the result because 
$x_0$ is a solution of the geodesic equation. Finally, the terms quadratic 
in $\Delta \lambda$ reproduce the expression (\ref{7}), up to a total
proper-time derivative and terms which vanish because of the geodesic 
equation for $x_0^{\,\mu}(\tau)$. Thus the Lagrangian (\ref{7}) 
represents the lowest-order non-trivial term in a systematic expansion 
of our action integral:
\begin{equation}
S\,[x] = m\!\int\!\!d\tau L(x_0)+m\,{(\Delta \lambda)}^2\!\!\int\!\!d\tau L(n) + 
 {\cal O}\left[{(\Delta \lambda)}^3\right].
\label{9}
\end{equation}  
The higher-order approximations can also be derived in this way. 
\newline
\indent 
This derivation of the deviation equations can be also applied to the
case of charged particles. One should start from the action 
\begin{equation} 
S_q[x] =\!\int\!\!d\tau\, \biggl[ \frac{m}{2}\, g_{\mu\nu}(x)\, 
 \dot{x}^{\,\mu} \dot{x}^{\nu} + q\, A_{\mu}(x)\, \dot{x}^{\,\mu} \biggr] ,
\label{10}
\end{equation} 
with the overdot being the usual short-hand for proper-time derivatives. 
The world-lines generated by this action are solutions of the 
Einstein-Lorentz equation 
\begin{equation} 
\frac{D^2 x^{\,\mu}}{D\tau^2}\, = \frac{q}{m}\; F^{\mu}_{\;\;\nu}\,
 \frac{dx^{\nu}}{d\tau}\;. 
\label{11}
\end{equation} 
Now given a solution $x_0^{\,\mu}(\tau)$ of this equation, and 
expanding the path in $S_q[x]$ as
\begin{equation} 
x^{\,\mu}(\tau) = x_0^{\,\mu}(\tau) + \Delta \lambda \, n^{\mu}(\tau)\,, 
\label{12}
\end{equation}
one can expand to second order in $\Delta \lambda$: 
%\marginpar{\small There is something wrong with two open round parentheses
%in this equation! (underlined)}
\begin{eqnarray}
S_q[x] \!\!\!& = & \!\!\! S_q[x_0] + \frac{{(\Delta \lambda)}^2}{2} 
 \int\!d\tau \Bigg[\,m 
%%%%%%%%%%%%%%%%%%
{\biggl(}
%%%%%%%%%%%%%%%%%%
g_{\mu\nu}\, \frac{Dn^{\mu}}{D\tau}\,
 \frac{Dn^{\nu}}{D\tau}\, + R_{\mu\rho\nu\sigma} u^{\rho} u^{\sigma} 
 n^{\mu} n^{\nu}{\biggr)} \nonumber \\
\!\!\!&+&\!\!\! \, q \biggl( 
F_{\mu\nu}\, n^{\mu}\, 
 \frac{Dn^{\nu}}{D\tau}\, + \nabla_{\mu} F_{\nu\rho}\, u^{\rho} 
 n^{\mu} n^{\nu} \biggr)\Bigg] \, + {\cal O}\left[{(\Delta \lambda)}^3\right].
\label{13}
\end{eqnarray}
To this order we then find that other (close to $x_0^{\,\mu}(\tau)$)
solutions of the world-line equation (\ref{11}) are given by 
(\ref{12}), with $n^{\,\mu}$ being the solution of the world line deviation 
equation\,\cite{bhk,kmmh}
\begin{equation} 
\frac{D^2 n^{\mu}}{D\tau^2} =\, R_{\rho\nu\sigma}^{\;\;\;\;\;\mu}
 u^{\rho} u^{\sigma} n^{\nu} + \frac{q}{m}\, F^{\mu}_{\;\;\nu}\, 
 \frac{Dn^{\nu}}{D\tau}\, +\, \frac{q}{m}\, \nabla_{\rho} 
 F^{\mu}_{\;\;\nu}\, u^{\nu} n^{\rho}\,. 
\label{14}
\end{equation} 
The alternative interpretation of $n^{\mu}$, as parametrizing the distance 
between two particles on neighboring world lines, holds in this case 
as well, provided the particles have the same charge-to-mass ratio $q/m$. 

The same result can be obtained (\ref{14}) from reduction of the geodesic 
equation and geodesic deviation equation in five-dimensional space-time, 
as particles with different $q/m$ ratio in four dimensions correspond to 
particles with different momenta in five-dimensional space-time.\cite{kmmh}

Similarly, pseudo-classical spinning particles can be described by 
the supersymmetric Lagrangian\,\cite{bcl,bdvh}
\begin{equation} 
{\cal{L}}_{spin}(x,\psi) = \frac{1}{2}\, g_{\mu\nu} \dot{x}^{\,\mu} 
 \dot{x}^{\nu} + \frac{i}{2}\, \psi_a \frac{D\psi^a}{D\tau}\,,
\label{15}
\end{equation} 
with $\psi^a$ an anti-commuting tangent-space vector\,\footnote{\,Note that
the transition between base-space and tangent-space vectors is made, 
as usual, by the vierbein $e_{\mu}^{\;\;a}$ and its inverse.} such 
that the pseudo-classical spin is described by $S^{ab}\!=\!-i\,\psi^a 
\psi^b$. The corresponding equations of motion for spinning particles 
can be written as 
\begin{eqnarray} 
&&\frac{D^2 x^{\,\mu}}{D\tau^2} = \frac{1}{2}\, S^{ab} 
 R^{\mu}_{\;\;\nu ab} u^{\nu}\,,\nonumber\\[1mm] 
&&\frac{DS^{ab}}{D\tau ~} = 0\,.
\label{15.1}
\end{eqnarray} 
Starting from a one-parameter congruence of solutions 
$(x^{\,\mu}(\tau;\lambda), \psi^a(\tau;\lambda))$ we define the deviation 
vectors 
\begin{equation} 
n^{\mu} = \frac{\partial x^{\,\mu}}{\partial \lambda}\,,\quad
\xi^a\!= \frac{D\psi^a}{D\lambda} 
 = \frac{\partial \psi^a}{\partial \lambda} -\,n^{\mu}\, 
\omega_{\mu\,\,b}^{\,\;a}\;\psi^b\,, 
\label{15.2}
\end{equation} 
where $\omega_{\mu\,\,b}^{\,\;a}$ is the spin connection. Then, the
covariant change in the spin-tensor is
\begin{equation} 
J^{ab} = \frac{DS^{ab}}{D\lambda} = -i [ \psi^a \xi^b + \xi^a \psi^b ] \, .
\label{15.2.1}
\end{equation}
\newpage
\noindent
These vectors satisfy the world line deviation equations
\begin{eqnarray}
\frac{D^2n^{\mu}}{D\tau^2}  \!\!\!\!&=&\!\!\!\!  
R_{\sigma\nu\rho}^{\quad\;\,\mu}\,u^{\sigma}u^{\rho}n^{\nu} +
\frac{1}{2}\,S^{ab}R^{\,\mu}_{\;\,\nu ab}\,\frac{Dn^{\nu}}{D\tau}
\nonumber\\[1mm]
\!\!\!\!&+&\!\!\!\! \frac{1}{2}\,\left[ S^{ab}\,\nabla_{\rho} 
 R^{\mu}_{\;\nu ab}\, u^{\nu} n^{\rho} + J^{ab} R^{\mu}_{\;\nu ab}\, 
 u^{\nu}\right] \, , \\[1mm]
 \frac{DJ^{ab}}{D\tau ~} \!\!\!\!&=&\!\!\!\!  {\left[ S, 
R_{\mu\nu} \right]}^{ab}  u^{\mu} n^{\nu}. \nonumber
\end{eqnarray} 
They define the stationary points of the quadratic deviation action 
\begin{eqnarray} 
{\cal{L}}_{spin}(n,\xi)\!\!\!\!&=&\!\!\!\! \frac{1}{2}\, g_{\mu\nu} 
\frac{Dn^{\mu}}{D\tau}\, \frac{Dn^{\nu}}{D\tau}\, + \frac{i}{2}\, 
\xi_a\,\frac{D\xi^a}{D\tau}   + \frac{1}{2}\,
R_{\mu\rho\nu\sigma} \, u^{\rho} u^{\sigma} n^{\mu} n^{\nu}
\label{19}  \\[3mm]
\!\!\!\!&-&\!\!\!\! \frac{i}{4}\, \psi^a  \psi^b\,\left[ R_{\mu\nu ab} \,
n^{\mu}\frac{Dn^{\nu}}{D\tau}\,+ \nabla_{\!\mu} R_{\nu \sigma ab}\, 
u^{\sigma} n^{\mu} n^{\nu}\right]  
 -\, i\,R_{\mu\nu ab}\, n^{\mu} u^{\nu} \xi^a \psi^b.\nonumber 
\end{eqnarray}
This set of coupled equations is of interest for the analysis of fine
gyroscopic effects, namely in the field of Kerr's metric.

\section*{References}
\addcontentsline{toc}{section}{\numberline{}References}

\begin{thebibliography}{99}

\bibitem{BerMar77}F.A.~Berezin and M.S.~Marinov, 
{\em Ann.\ Phys.}\ {\bf 104}, 386 (1977).

\bibitem{Marinov80} M.S.~Marinov, {\em Phys.\ Rept.} {\bf 60}, 1--57 (1980).

\bibitem{MarStra00} M.S.~Marinov and E.~Strahov, {\em Comptes
Rendus Acad.\ Sci.\ Paris}, Ser.\,IIb {\bf 326}, {245} (2000).

\bibitem{landau1} L.D.~Landau, E.M.~Lifchitz and L.P.~Pitaevskii,  
{\it Electrodynamics of Continuous Media} (Butterworth Heinemann, 
Oxford 1996).

\bibitem{ginzburg} V.L.~Ginzburg, {\it Physique Th\'eorique et
Astrophysique} (in French) (Mir, Moscow, 1978).

\bibitem{landau2} L.D.~Landau and  E.M.~Lifchitz,  {\it Classical Theory of
Fields} (Butterworth Heinemann, Oxford, 1996).

\bibitem{DVGRK2000} D.V.~Gal'tsov and R.~Kerner,
{\em Phys.\ Rev.\ Lett.}\ {\bf 84}, {5955--5958} (2000).

\bibitem{gupta}  A.~Gupta, S.~Mohanty and M.~K.~Samal,
{\em Class.\ Quant.\  Grav.}\ {\bf 16}, {291} (1999).

\bibitem{drummond} I.~T.~Drummond and  S.~J.~Hathrell, 
{\em Phys.\ Rev.}\ D {\bf 22}, {343} (1980). 

\bibitem{syng} J.~L.~Synge, {\it Relativity: the general theory}
(Amsterdam,  North Holland, 1966).

\bibitem{kramer} D.~Kramer, H.~Stephani, M.~McCallum, E.~Herlt and
E.~Schmutzer,  {\it Exact solutions of the Einstein field   equations}
(Cambridge University Press, Cambridge, 1980). 

%\bibitem{mtw} C.~W.~Misner, K.~S.~Thorne and J.~A.~Wheeler, {\it
%Gravitation}, (Freeman, San Francisco 1973).

%\bibitem{sbytov} Yu.G.~Sbytov, \Journal{JETP}{71}{2001}{1976}.

%\bibitem{calura} M.~Calura, E.~Montanari,  Class.\ Quant.\ Grav.\
%{\bf{16}}, 643 (1998).

\bibitem{Balakin97} A.B.~Balakin, {\em Class.\ Quantum
Grav.}\ {\bf 14}, {2881} (1997).

%\bibitem{maugin} G.~A.~Maugin, J. Math. Phys. {\bf{19}}, 1198 (1978).

\bibitem{bkl}
A.~B.~Balakin, R.~Kerner and J.~P.~Lemos,
%``Cherenkov radiation in a gravitational wave background,''
{\em Class.\ Quant.\ Grav.}\  {\bf 18}, 2217 (2001)
[gr-qc/0012101].
%%CITATION = GR-QC 0012101;%%
%A.~Balakin, R.~Kerner and J.P.S.~Lemos, \Journal{Class.
%Quant. Gravity}{..}{..}{2000}

\bibitem{khc} 
R.~Kerner, J.~W.~van Holten and R.~Colistete~Jr,
%``Relativistic Epicycles : another approach to geodesic deviations,''
{\em Class.\ Quant.\ Grav.}\  {\bf 18}, 4725 (2001)
[gr-qc/0102099].
%%CITATION = GR-QC 0102099;%%


%\bibitem{jwvh} J.W.~van Holten, Proc.~Sem.~Math.\ Structures in 
% Field Theories 1986-87, eds.\ E.A.\ de Kerf and H.G.J.\ Pijls 
% (CWI Syllabus Vol.\ 26), 109  

\bibitem{krip} I.B.~Kriphlovich, {\em Sov.\ Phys.\ JETP}, {\bf 69}, 217 
(1989).

\bibitem{bhk} A.\ Balakin, J.W.\ van Holten and R.\ Kerner, {\em Class.\
Quant.\ Grav.}\ {\bf 17}, 5009 (2000).

\bibitem{kmmh}
R.~Kerner, J.~Martin, S.~Mignemi and J.~W.~van Holten,
%``Geodesic deviation in Kaluza-Klein theories,''
{\em Phys.\ Rev.}\ D {\bf 63}, 027502 (2001)
[gr-qc/0010098].
%%CITATION = GR-QC 0010098;%%

\bibitem{bcl} A.~Barducci, R.~Casalbuoni and L.~Lusanna, 
{\em Nuovo Cim.}\ A {\bf 35},  377 (1976).

\bibitem{bdvh} L.~Brink, P.~Di Vecchia and P.~Howe, {\em Nucl.\ Phys.}\ B
{\bf 118}, 76 (1977).

\bibitem{AE}  A.~Einstein, {\em Ann.\ Physik}, {\bf 49},  769 (1916).
%Einstein A\ 1916 {\em Ann.\ Physik} {\bf 49} 769

\bibitem{Ptolemaios}  Ptolemaios ca.\ 145 AD {\em Almagest}; original 
title `$%
HM\alpha \theta \epsilon \mu \alpha \theta \iota \kappa \eta $ $\Sigma
\upsilon \nu \tau \alpha \xi \iota \varsigma\,$' [{\em The mathematical
syntax}\,].%

%\bibitem{Chandra} Chandrasekhar S\ 1983 {\em The Mathematical Theory of
% Black Holes} (New York: Oxford University Press)

%\bibitem{Weinberg}  Weinberg S\ 1972 {\em Gravitation and Cosmology} (New
% York: Academic Press)

%\bibitem{Kepler}  J.~Kepler, {\em Astronomia Nova} [{\em The New
%Astronomy}] (1609).
%Kepler J\ 1609 {\em Astronomia Nova} (The New Astronomy)


%\bibitem{Landau} L.~Landau and E.~Lifshitz,  {\em Lectures in
%Theoretical Physics, Vol. 2: Classical Theory of Fields} (Pergamon, 1969)

\bibitem{Damour} T.~Damour and N.~Deruelle, {\em  Ann.\ Inst.\
H.\ Poincar\'{e} (Phys.\ Th\'{e}or.)}\ {\bf 43}, 107 (1985); {\em ibid.}\ 
{\bf 44}, 263 (1986).

\bibitem{Blanchet} L.~Blanchet and G.~Sch\"{a}fer {\em Class.\
Quant. Grav.}\ {\bf 10}, 2699 (1993), and references therein.

\end{thebibliography}

\end{document}

