\documentclass[12pt]{uwthesis}
\usepackage{epsf, amsfonts, amsmath, moreverb, maple2e,
    color}
\usepackage[active]{srcltx}
\title{Geodesic Geometry of Black Holes}
\author{Gabriela Slez\'akov\'a}
\hyphenation{va-cu-um ne-ga-ti-ve effec-ti-ve para-me-ter}
\def\bal#1\eal{\begin{align}#1\end{align}}
\providecommand{\abs}[1]{\lvert#1\rvert}
\DeclareMathOperator{\F}{F} \DeclareMathOperator{\K}{K}
\DeclareMathOperator{\E}{E}
\DeclareMathOperator{\arctanh}{arctanh}
\DeclareMathOperator{\arcsinh}{arcsinh}
\DeclareMathOperator{\arcsech}{arcsech}
\DeclareMathOperator{\discrim}{discrim}
\DeclareMathOperator{\cn}{cn}
\DeclareMathOperator{\dn}{dn}
\DeclareMathOperator{\sn}{sn}
\DeclareMathOperator{\sd}{sd}
\DeclareMathOperator{\am}{am}
\allowdisplaybreaks[2]
\newcommand{\vek}[1]{\textbf{\textit{#1}}}
%\hfuzz1pc

\begin{document}
    \pagenumbering{roman}
    \maketitle
    \setcounter{page}{2}
    \begin{abstract}
    \sloppy
        The study of geodesics is of intrinsic significance in the study
        of the geometry of space-time. In this thesis null, space-like and
        time-like geodesics are studied in the case of the space-times of
        Schwarzschild, Reissner-Nordstr\"om and Kerr black holes. These
        space-times have been investigated with varying degrees of
        thoroughness in many articles and some books. However, there are
        some significant gaps in these treatments and the central aim of
        this thesis is to fill these gaps where necessary. Moreover, the
        following topics are covered for the first time.
\begin{enumerate}
  \item  In Chapter \ref{s} a thorough treatment of the space-like geodesics of the
  Schwarzschild solutions has been given. These geodesics are the
  trajectories of Tachyons (faster than light particles) and are treated
  in a~complete manner. This has been done by obtaining exact solutions
  and solving them numerically. \label{a1}
  \item In Part \ref{r-n} all solutions for geodesics for a Reissner-Nordstr\"om black hole
  have been given in complete detail, i.e. time-like, null and space-like
  geodesics and orbit of a charged particle. \label{a2}
  \item In Chapter \ref{equator} all solutions for geodesics in the equatorial plane of
  a~Kerr
  black hole have been given in complete detail, i.e. time-like, null and
  space-like geodesics.
    \label{a3}
    \item The study of special types of non-equatorial geodesics for a Kerr
    black hole have been given in complete detail, i.e. time-like (Chapter \ref{kerrtime}), null
    (Chapter \ref{kerrnull}) and
    space-like (Chapter \ref{kerrspace}). This has been done in order to distinguish the
    qualitatively
    different types of solutions.
    \label{a4}
\end{enumerate}
 Calculation of the explicit
formulas, which describe these geodesics, as well as numerically
computed diagrams representing the geodesics have been
incorporated in these studies. The following subjects have been
also treated:
\begin{enumerate}\addtocounter{enumi}{4}
    \item  Solutions for the geodesics in Reissner-Nordstr\"om black holes
    with $|Q_*| \geq M$, which are black holes with one ($|Q_*| = M$) or no horizon ($|Q_*| > M$)
    (Chapter \ref{RNnull}).
    \label{a5}
    \item Solutions of geodesics in extreme and fast Kerr black holes, i.e.
    black holes with $a = M$ (extreme) and $a > M$ (fast). As in the case of
    $|Q_*| > M$, fast black holes have naked singularities (Chapter \ref{equator}).
    \item Some general observations about orbit types of the Kerr black holes regarding
    relationships between parameters such as angular momentum, energy, Carter constant and mass
    and angular momentum of black holes (Chapter \ref{kerprefa}).
    \item Some corrections to errors found in the literature.
\end{enumerate}
While it has not been possible to cover all different cases which
occur for possible relations amongst the parameters specifying
a~general black hole, interesting geodesics have, however, been
studied and a more thorough  presentation of the properties of
geodesics has now been given.

\end{abstract}
    \begin{acknowledgements}
        I would like to present my deepest gratitude to:

\medskip
My supervisors from the University of Waikato, NZ - Prof. Ernest Kalnins, for his many
suggestions and constant support during my study in New Zealand, and Prof. Ian Craig, for his comments and suggestions.

Prof. Jan Horsk\'y and Dr. Zden\v ek Kopeck\'y, supervisors of my MSc. thesis, from the Masaryk University, CZ. What I learnt from them helped me in my study here.

And all others who taught me mathematics and physics.

All authors of the papers and books whose names are in the reference section. Each of them taught me something.

My husband for his support, help, comments and many hours spent fixing my computer.

My parents who supported me through my studies.

God who provided me with the resources that I use to understand a minute part of the complexities of His creation.

    \end{acknowledgements}
    \tableofcontents
    \listoffigures
    \chapter*{CD content}
        The content of the attached CD is listed in the {\tt read\_me.txt} file on the CD. This is the content of that file:
        \verbatiminput{read_me.txt}
     \newpage
    \pagenumbering{arabic}
    \include{preface2}
    \include{schprefa2}
    \include{schw_nul2}
    \include{schwarzs2}
    \include{schw_t-l2}
    \include{schsumma2}
    \include{r-nprefa2}
    \include{r-n_nul2}
    \include{r-n_spac2}
    \include{r-n_time2}
    \include{RNcharge2}
    \include{r-nsumma2}
    \include{kerprefa2}
    \include{equator2}
    \include{kerrnull2}
    \include{kerrspac}
    \include{kerrtime}
    \include{kersumma}
    \include{conclusi2}
    \include{appendix2}
    \include{introduc}
    \include{referenc3}
\end{document}
