% 780 slov
\chapter{Introduction}
\input{collapse2}
\section{Outline of thesis}
\sloppy
It is known (e.g. \cite{townsend} or \cite{mtw}) that gravitational collapse in a general relativity context ultimately leads to the formation of a black hole where the external solution is characterized
by the generalized Kerr-Newman four parameter
family of vacuum solutions. The parameters that specify this family are identified with the \emph{mass},
the \emph{angular momentum}, the \emph{electric charge,} and the
\emph{magnetic charge}. This thesis considers
only the cases of zero magnetic charge.  The
derivation of these metrics is given in Appendix \ref{metriky} and in
greater detail in \cite{chandrasekhar}.

One of the best ways to grasp the essence of the geometry of
a~space-time is to study geodesics. They are the analogues of
straight lines in curved space geometries, and they are second in
importance only to curvature. A~curve $\lambda$ on $M$ is a
geodesic if its tangent vector, after parallel propagation,
remains a~multiple of itself. There are two types of geodesics
described by freely falling particles, \emph{time-like} and
\emph{null}. If they have non-zero positive rest-mass they are time-like and
if they are massless they are called null. \emph{Space-like}
geodesics can be perceived as trajectories of particles faster
than light (tachyons) \cite{fayngold}. Geodesic motion is interesting in an astrophysical context as many astrophysical models assume that in-falling
matter moves along time-like geodesics, and because radiation
propagates to a distant observer along null geodesics (as long as
the wavelength is small compared to the radius of the black hole)
\cite{connors1}, \cite{connors2}.


Geodesics are examined in many papers (for a list see
e.g.~\cite{sharp}), but many solutions are omitted, usually
because they lack clear physical significance (e.g.~space-like
geodesics or naked singularities). The intention of this thesis is
to ``fill these gaps", and as a~result, give a comprehensive study of the
geodesics of black holes.


In the first part of the thesis solutions are
given for geodesic orbits in
a black hole characterized only by mass (Schwarzschild). Radial orbits (null, time-like and space-like) are depicted
(schematically only) in \cite[Chapter 31]{mtw}. General space-like
orbits are discussed and  numerically obtained trajectories shown in \cite{hlr}. In this thesis analytical
solutions of general space-like orbits are  presented (refer to Abstract--section \ref{a1}).

%black holes soubor praci
The chapters dealing with time-like and null geodesics are based
on exhaustive discussion in \cite[chapter~3]{chandrasekhar}. The
complete analytical solution for $\tau$ and $t$ coordinates of
general orbits are added to that. These were generated by Maple
software and then manually simplified. Time-like and null orbits
are also discussed in \cite{hagihara} and \cite{darwin}, who
restricts the analysis to the time-like circular geodesics.
\cite[Chapter 25]{mtw} describe qualitative features of orbits of
particles with zero or finite mass.

The second part of the thesis examines geodesics of a charged black
hole (Reissner-Nordstr\"{o}m--refer to Abstract--section \ref{a2}). Time-like circular orbits for a naked
singularity ($\abs{Q_*} > M$, where $Q_*$ is charge of the black
holes) are discussed by \cite{liang}. \cite[chapter
5]{chandrasekhar} examines null and time-like radial and circular
orbits of black holes with $\abs{Q_*} < M$. In this thesis
\cite[chapter 5]{chandrasekhar} is extended by solving equations
of motion for general time-like and null orbits. The space-like
orbits and black holes with $\abs{Q_*} \geq M$ are included (refer to Abstract--section \ref{a5}).
The solution for $t = t(\tau)$ in Schwarzschild space-time for general
geodesics did not bring any new features compared to solutions for $t = t(\tau)$ for the space-like radial
geodesics, therefore, we do not present explicit formulas in this case (that is why there
are not large formulas in chapters after Schwarzschild
space-time). Motion of charged particles is mentioned briefly.

In the third part geodesics of rotating black holes are
discussed. General features are thoroughly examined in
\cite[chapter 4]{onil}. Discussion of general features of the time-like and the null geodesics
can be found in \cite[chapter 7]{chandrasekhar} and
\cite{steward}. There is a small discussion of general null
geodesics by \cite{bardeen}, but his emphasis is on equatorial
geodesics. Equatorial geodesics of a black hole with $1.088M > a >
M$ are investigated by \cite{instabil}. Null geodesics in the
equatorial plane are used in \cite{gravmiro} to study the black
hole as a gravitational mirror. Null geodesics of naked
singularity are covered by \cite{calvani}. An account of null
geodesics for the range $0 \leq a^2 \leq M^2$ is given in \cite{43}.
General time-like geodesics appear in \cite{39} with emphasis on
the bound geodesics; time-like geodesic of naked singularity
appear in \cite{64}; \cite{cfms} are interested in geodesics in
the naked singularity which (do not) violate causality;
\cite{fyzika27} uses numerical methods to obtain criteria for the
existence of bound particle orbits. The space-like orbits in the
equatorial plane  are added, and time-like and null geodesics
where analytical solutions are not given are completed in the form of elliptic integrals (refer to Abstract--section \ref{a3}). We do not restrict ourselves with the condition $a^2 \leq M^2$. Finally we
give several examples of orbits out of the equatorial plane (refer to Abstract--section \ref{a4}).

Throughout the thesis $c = G = 1$, where $c$ is velocity of light
in vacuum and $G$ the gravitational constant. Metrics with the
signature $(+, -, -, -)$ are used. Unless stated otherwise, we
assume that $E$ and $M$ are positive.

\textcolor{blue}{For easy orientation the original work of this
thesis is printed blue.}
