% Words:   462
\part{Geodesics in the Reissner-Nordstr\"om solution}\label{r-n}
\chapter{Introduction to geodesics in the Reissner-Nordstr\"om space-time}
In this part of the thesis we will give solutions for geodesic
orbits in the Reissner-Nordstr\"om space-time (black hole
characterized by mass $M$ and charge $Q_*$.) We have extended the
discussion in \cite[chapter 5]{chandrasekhar}, which is concerned
only with circular time-like and null orbits, to general orbits
(with solutions in terms of the elliptic integrals) , and we have
also included space-like geodesics and black holes with \(M \leq
|Q_*|\). Since these kinds of black holes (with charge larger than
mass) are considered non-physical, interpretation of their orbits
are unclear. %We chose to extend the orbits from infinity to
%infinity rather than terminate them at a certain distance from the
%black hole.
Some orbits were completed manually (cyan lines), because, in some
regions, the program they were generated with (Maple) failed to
work adequately. In \cite{chandrasekhar} charged particle are
mentioned and their lagrangian given. We extend this discussion by
finding explicit solutions of their orbits.

We start with the introduction of horizons. Let
\begin{equation}\label{rn38}
  r_+ = M +\sqrt{M^2-Q_*^2} \qquad \mbox{and} \qquad
  r_- = M -\sqrt{M^2-Q_*^2} \,,
\end{equation}
be the roots of \( \varDelta = 0 \), where
\begin{equation}\label{r37}
  \varDelta = r^2 -2Mr +Q_*^2
\end{equation}
(cf. equation (\ref{r62})). These roots will be real and distinct
if \( M^2
> Q_*^2 \).

Lets assume that this inequality is satisfied. Since
\begin{equation}\label{rn39}
  \begin{array}{lcl}
    & \varDelta > 0 & \mbox{for} \qquad r > r_+ \qquad \mbox{and} \qquad
    0 < r < r_- \\
    \mbox{and} & \varDelta < 0 & \mbox{for} \qquad r_- < r < r_+
    \,,
  \end{array}
\end{equation}
it is clear that we must distinguish three regions:
\begin{equation}\label{rn40}
  \mbox{A:} \quad 0 < r < r_- \,; \qquad \mbox{B:} \quad r_- < r <
  r_+ \,; \qquad \mbox{and} \qquad \mbox{C:} \quad r > r_+ \,.
\end{equation}
While the surface \( r = r_+ \) is an event horizon in the same
sense that \( r = 2M \) is an event horizon in the Schwarzschild
space-time, the surface \( r = r_- \) is called a~{\em `Cauchy'
horizon\/} and it is a `horizon' in a different sense, see
\cite{chandrasekhar}. At the instant of crossing this horizon, the
person will witness, in a flash, the entire infinite history of
the external space-time, in  infinite blueshift. Once under this
horizon, the person's future is no longer determined by his or her
past.

For \( |Q_*| > M \), there is no horizon. The singularity at \( r =
0 \) is naked. Because of the cosmic censorship hypothesis this
case could not occur in gravitational collapse (\cite{townsend}).
According to \cite[footnote 9]{liang} this does not exclude the
possibility of such singularities being created primordially. In
\cite[chapter 31]{mtw} it is suggested that for the case of a large
ratio of charge to mass, the region near $r = 0$ is unphysical.

 Since the Reissner--Nordstr\"om metric
differs from the Schwarzschild metric only in the definition of
the `horizon function' $\varDelta$, it follows that the basic
equations governing geodesic motion are, with the redefinition of
$\varDelta$, the same as those considered in part~\ref{sch}.

Another difference between the equations governing geodesics in
Schwarzschild space-time and Reissner---Nordstr\"om space-time is
(as we shall see later) that f$(u)$ will be now a biquadratic
instead of a cubic. That makes an essential difference only for
the orbits which cross the horizon at \( r = r_+ \), i.e., for the
orbits, which in Schwarzschild geometry are terminated at
the singularity at \( r = 0 \).
