\chapter{Conclusion}
There is a close relationship between Killing tensors and separable systems for the geodesic Hamilton-Jacobi equation  (conservation lemma). The existence of a separable system in Petrov type D space-times is a consequence of the algebraic properties of the Weyl tensor \cite{woodhouse}. \cite{carter} demonstrated the separability of the Hamilton-Jacobi equation (which led to the discovery of Carter constant). Then the basic equations of motions can be deduced, and their analytical solutions found.

Geodesics are of three types:
\begin{itemize}
  \item Time-like
  \item Null
  \item Space-like
\end{itemize}

\emph{Time-like} and \emph{null} geodesics in Schwarzschild
space-time have been studied before,
\cite{chandrasekhar, darwin,
mtw}. In this thesis space-like geodesics are studied and
solutions are provided.

In this thesis geodesics of Reissner-Nordstr\"om space-time are
studied and solutions for these
two types are provided:
\begin{itemize}
  \item $M^2 \geq Q_*^2$ (general solutions in terms of elliptic integrals)
  \item $ M^2 < Q_*^2$ (both general and circular orbits)
\end{itemize}

In this thesis equatorial geodesics of Kerr space-time are studied
and general solutions are provided.

Special cases of non-equatorial geodesics were also given. Doing
this we have corrected errors in various places (such as
\cite{chandrasekhar, onil}, see footnotes in the thesis).

\cite{hlr} claim there exist repulsive space-like orbits in the
Schwarzschild space-time and unstable, space-like, circular
    orbits in the Schwarzschild field for all radii in the range
    $0 < r < 3M$. By
investigating restrictions that should be applied to parameters
$L, E$, and $M$, in this thesis it is  concluded that there can be neither
repulsive orbits nor circular orbits with
 $r \leq
2M$.

\cite{chandrasekhar} gives two solutions for circular null
geodesics in the equator of the slow Kerr black hole. In this
thesis three solutions are presented that are in agreement with
\cite{onil} who claims there are three null circular orbits.

Let us recall that in classical mechanics flyby and bound orbits
are hyperbolas and ellipses respectively. While these orbits are
analogues of the orbits of the first kind, the associated orbits
of the second kind have no Newtonian analogue. Other differences
include the precession of bound orbits and dragging of the
inertial frame in the case of a rotating black hole.

It is clear that there are similarities between geodesics in
Schwarzschild black hole, Reissner-Nordstr\"om black hole and
equator of the Kerr black hole, because of spherical symmetry of
the first two and special choice of $\theta$ in the third case. To
find general solutions we had to find zeros of a cubic or quartic
polynomial and integrals of its square root, which is possible and
leads to elliptic integrals. The integration methods used were
standard and analytical solution were obtained sometimes using
Maple program for symbolic mathematics.

This thesis studies all geodesics regardless of their physical
significance. However, the results of this thesis could be used
for example to test whether the properties of orbits of a black
hole candidate have the characteristics of black hole orbits in
general relativity.

%In order to improve the thesis the following changes to the previous version were carried out
%\begin{itemize}
%  \item Chapter 3 (The null geodesics in the Schwarzschild space-time) and
% Chapter 5 (The time-like geodesics in the Schwarzschild space-time) were rewritten and shortened, and the derivation of the metrics has been shortened and put in Appendix \ref{metriky} (Derivation of the metrics).
%    \item Nonequatorial geodesics have been studied in greater detail. That is
%\begin{itemize}
%  \item in section 13.5 (First-integrals and orbits) $r$-$L$ plots for fast black hole were added;
%  \item in Chapter 15 (The null geodesics) solutions for $\theta$, $r$ and $\varphi$ coordinates for orbits with negative Carter constant (sections 15.1.3, 15.2.3 and 15.3.3 respectively) with figures 15.4 (($r, \theta$) plane), 15.7 (($r,\varphi$) plane) and 15.8 (3D space) of this orbits;
%  \item Chapter 16 (The space-like geodesics) which contains solutions for unstable circular orbits ($r, \varphi, \theta$ coordinates) and for geodesics with zero Carter constant ($r, \vartheta$ coordinates);
%  \item Chapter 17 (The time-like geodesics) which contains solutions for bound orbits ($\theta$ coordinate), marginally bound orbits ($\theta$ and $\varphi$ coordinates), unbound orbits ($\vartheta$ coordinate), circular orbits ($r$ coordinate) and orbits with negative Carter constant ($r$ coordinate).
%\end{itemize}
%\end{itemize}
%
%
Some cases that did not fit within the scope and time frame of
this PhD program, e.g. a complete analysis of non-equatorial
geodesics in rotating black holes, such as qualitative analysis of
geodesics with $Q < 0$ ($r$-$L$ plots), solutions for general
orbits and some special cases like axial and polar orbits and
geodesics in horizons.
 Further work could also include an analysis of geodesics in the charged black holes
 (including a magnetic monopole) and a~cosmological constant (\cite{stuchlik} discus equatorial
 photons in a Kerr-Newman black hole with non-zero cosmological constant).  Schwarzschild
 coordinates ($t, r, \theta, \varphi$), which were used
in this thesis, give pathological description at the horizons, so
that e.g. Kruskal-Szekeres coordinates could be used instead.

%within the scope and time frame of this PhD program
