% Words:   1205
\chapter{The time-like geodesics in the Reissner-Nordstr\"om
space-time}\label{rt}
The equations governing the time-like
geodesics are
\begin{equation}
    \left(\frac{{\mathrm d}r}{{\mathrm d}\tau}\right)^2 +
    \frac\varDelta{r^2}\left(1+\frac{L^2}{r^2}\right) = E^2 \,; \quad
    \frac{{\mathrm d}t}{{\mathrm d}\tau} = E\frac{r^2}\varDelta \,; \quad
    \mbox{and} \quad \frac{{\mathrm d}\varphi}{{\mathrm d}\tau} =
    \frac{L}{r^2} \,.
    \label{r65}
\end{equation}

As before, we can interpret
\begin{equation}\label{RNtlpotent}\color{blue}
    {\cal V} = \frac\varDelta{r^2}\left(1+\frac{L^2}{r^2}\right)
    \,,
\end{equation}
as `potential energy'. (Cf. equations (\ref{s113time}),
(\ref{s113}) and (\ref{RNslpotent}).)

By considering $r$ as a~function of $\varphi$ and replacing $r$ by
$u = 1/r$ as the independent variable, we obtain the equation
(cf.~equation (\ref{s94}))
\begin{equation}
    \left(\frac{{\mathrm d}u}{{\mathrm d}\varphi}\right)^2 =
    -Q^2_*u^4+2Mu^3-u^2\left(1+\frac{Q^2_*}{L^2}\right) +
    \frac{2M}{L^2}u-\frac{1-E^2}{L^2} = {\mathrm f}(u) \quad
    \mbox{(say)}.
    \label{r67}
\end{equation}


\section{The radial geodesics}
Considering first the radial geodesics, we can obtain governing
equations by setting $L = 0$ in equations (\ref{r65}). % thus,
%\begin{equation}
%    \left(\frac{{\mathrm d}r}{{\mathrm d}\tau}\right)^2 = E^2
%    -\frac\varDelta{r^2} \,, \quad \frac{{\mathrm d}t}{{\mathrm d}\tau} =
%    E\frac{r^2}\varDelta \,, \quad \mbox{and} \quad \frac{{\mathrm
%    d}\theta}{{\mathrm d}\tau} = \frac{{\mathrm d}\varphi}{{\mathrm
%    d}\tau} = 0 \,.
  %  \label{r70t}
%\end{equation}
\textcolor{blue}{Accordingly,
\begin{equation}
    \frac{{\mathrm d}r}{{\mathrm d}t} =
    \pm\left(\frac\varDelta{Er^2}\right) \sqrt{E^2 -\frac\varDelta{r^2}} \,.
    \label{r71t}
\end{equation}
The solution for the proper time, $\tau$, is
\begin{multline}
    \tau =\frac{r_\mathrm{i}\sqrt{E^2-\frac{\varDelta_\mathrm{i}}
    {r_\mathrm{i}^2}}-r\sqrt{E^2
    -\frac{\varDelta}{r^2}}}{E^2-1} +\\
    +\frac{M}{\left(E^2+1\right)^{3/2}}%\cdot\\
    %&&\cdot
    \ln\left(\frac r{r_\mathrm{i}}\cdot\frac{E^2-1+\frac{M}{r}
    +\sqrt{E^2-\frac\varDelta{r^2}}\sqrt{E^2-1}}
    {E^2-1+\frac M{r_\mathrm{i}}
    +\sqrt{E^2-\frac{\varDelta_\mathrm{i}}{r_\mathrm{i}^2}}
    \sqrt{E^2-1}}
    \right) %\nonumber
    \label{r72at}
\end{multline}
and for coordinate t,
\begin{multline}
    t = \frac{ME(2E^2-3)}{\left(E^2-1\right)^{3/2}}
    \ln\frac{r(E^2-1) +M +r \sqrt{E^2-1} \sqrt{E^2-\frac\varDelta{r^2}}}
    {r_\mathrm{i}(E^2-1) +M
    +r_\mathrm{i} \sqrt{E^2-1}
    \sqrt{E^2-\frac{\varDelta_\mathrm{i}}{r_{\mathrm i}^2}}}
    +\\
    +\frac{4M^3-3Q_*^2M-(Q_*^2-4*M^2)\sqrt{M^2-Q_*^2}
    }{2Q_*^2\sqrt{M^2-Q_*^2}} \cdot\\*
    \cdot\sqrt{2M^2-Q_*^2+2M\sqrt{M^2-Q_*^2}} %\cdot\nonumber\\
    %&&\cdot
    \ln\left\{\frac{r_\mathrm{i}-M+\sqrt{M^2-Q_*^2}}
    {r-M+\sqrt{M^2-Q_*^2}}\right. \cdot\\*
    \cdot\Biggl[M^2-Q_*^2-\left(M+r(E^2-1)\right)\sqrt{M^2-Q_*^2}\Biggr.
    +\\
    +\left.rE\left(ME+
    \sqrt{2M^2-Q_*^2-2M\sqrt{M^2-Q_*^2}}
    \sqrt{E^2-\frac\varDelta{r^2}}\right)\right]/ \\*
    /\Biggl[M^2-Q_*^2-\left(M+r_\mathrm{i}(E^2-1)\right)\sqrt{M^2
    -Q_*^2}\biggr.
    +\\
    \left.+\left.r_\mathrm{i}E\left(ME+
    \sqrt{2M^2-Q_*^2-2M\sqrt{M^2-Q_*^2}}
    \sqrt{E^2-\frac{\varDelta_\mathrm{i}}
    {r_\mathrm{i}^2}}\right)\right]\right\}
    +
    \\
    +\frac{3Q_*^2M-4M^3-(Q_*^2-4M^2)\sqrt{M^2-Q_*^2}
    }{2Q_*^2\sqrt{M^2-Q_*^2}} \cdot\\*
    \cdot\sqrt{2M^2-Q_*^2-2M\sqrt{M^2-Q_*^2}}
    \ln\left\{\frac{r_\mathrm{i}-M-\sqrt{M^2-Q_*^2}}
    {r-M-\sqrt{M^2-Q_*^2}}\right. \cdot\\*
    \cdot\Biggl[M^2-Q_*^2+\left(M+r(E^2-1)\right)\sqrt{M^2-Q_*^2}\Biggr.
    +\\
    +\left.rE\left(ME+
    \sqrt{2M^2-Q_*^2+2M\sqrt{M^2-Q_*^2}}
    \sqrt{E^2-\frac\varDelta{r^2}}\right)\right]/ \\*
    /\Biggl[M^2-Q_*^2+\left(M+r_\mathrm{i}(E^2-1)\right)\sqrt{M^2
    -Q_*^2}\Biggr.
    +\\
    \left.+\left.r_\mathrm{i}E\left(ME+
    \sqrt{2M^2-Q_*^2+2M\sqrt{M^2-Q_*^2}}
    \sqrt{E^2-\frac{\varDelta_\mathrm{i}}
    {r_\mathrm{i}^2}}\right)\right]\right\}
    +
    \\
    +\frac E{E^2-1}
    \left(r_\mathrm{i}\sqrt{E^2-\frac{\varDelta_\mathrm{i}}{r_\mathrm{i}^2}}
    -r\sqrt{E^2-\frac{\varDelta}{r^2}}\right) \,,
    \label{r72bt}
\end{multline}
where the initial condition for $r$ is given by
\begin{equation}
    E^2 -1 = -\frac{Q_*^2 +2Mr_\mathrm{i}}{r_\mathrm{i}^2}
    \label{r72ct}
\end{equation}}
for $r(0) = r_\mathrm{i}$ and $\dot{r}(0) = 0$.


\section{Circular orbits}
\cite{chandrasekhar} gives the solution for these orbits in case
of \( M^2
> Q_*^2 \) and \cite{liang} discusses these orbits in general. We
combine these two together to get a full description and, whenever
necessary, complete it.

The conditions for the occurrence of circular orbits are f$(u) =
0$ and
\begin{equation}\label{rn90}
  \mbox{f}'(u) = -4Q_*^2u^3 +6Mu^2
  -2u\left(1+Q_*^2/L^2\right)+2M/L^2 = 0 \,.
\end{equation}
From these equations, it follows that the energy $E$ and the
angular momentum $L$ of a circular orbit of radius \(r_\mathrm{c}
= 1/u_\mathrm{c}\) is given by
\bal\label{rn91}
  E^2 &= \frac{\left(1-2Mu_\mathrm{c}+Q_*^2u_\mathrm{c}^2\right)^2}
  {1-3Mu_\mathrm{c}+2Q_*^2u_\mathrm{c}^2}
  %\end{equation}
  \intertext{and}
  \label{rn92}
  L^2 &= \frac{M-Q_*^2u_\mathrm{c}}
  {u_\mathrm{c}(1-3Mu_\mathrm{c}+2Q_*^2u_\mathrm{c}^2)} \,.
\eal
These equations require (\ref{rn93l}) and
\begin{equation}\label{rn93}
  1 -3Mu_\mathrm{c} +2Q_*^2u_\mathrm{c}^2 > 0
\end{equation}
(cf. inequality (\ref{r93})). This inequality shows that the
minimum radius for a~time-like circular orbit, in case \( M^2 \geq
Q_*^2 \), is the radius of the unstable circular photon-orbit.
Since for \( M^2 < Q_*^2 < 9M^2/8 \) we have two distinct circular
photon-orbits (one stable and one unstable), particle circular
orbits exist in the regions \( Q_*^2/M < r < r_\mathrm{cs} \) (all
stable) and \(r > r_\mathrm{cu} \) (some unstable). When \(Q_*^2 =
9M^2/8 \), particle orbits exist for all \(r \geq Q_*^2/M \)
except \( r = r_\mathrm{c} = 3M/2 \), which is an unstable
photon-orbit. When \(Q_*^2 > 9M^2/8 \), particle orbits are
allowed everywhere in the region \( r \geq Q_*^2/M \).


When $E$ and $L$ have the values (\ref{rn91}) and (\ref{rn92}),
appropriate for a circular orbit of radius \(r_\mathrm{c} =
1/u_\mathrm{c}\), equation (\ref{r67}) becomes (cf. equation
(\ref{r79}))
\begin{equation}\label{r94}
  \left(\frac{\text du}{\text d\varphi}\right)^2 = \left(u-u_\text
  c\right)^2\left[-Q_*^2u^2 +2\left(M-Q_*^2u_\text c\right)u
  +\left(M-Q_*u_\text c-M/L^2u_\text c^2\right)u_\text c\right] \,.
\end{equation}
Therefore, besides the circular orbit of radius $r_\text c =
1/u_\text c$, equation \eqref{r94} provides an orbit of the
second kind determined by (cf. equation \eqref{r80})
\begin{equation}\label{rn95}
  \varphi = \pm \int\frac{\mathrm du}{(u-u_\mathrm{c})
  \sqrt{-Q_*^2u^2
  +2(M-Q_*^2u_\mathrm{c})u
  +u_\mathrm{c}(M-Q_*^2u_\mathrm{c}-M/L^2u_\mathrm{c}^2)}}\,.
\end{equation}
With the substitution (\ref{r81}) we obtain the same solution
(\ref{r84}) with, however,
\begin{equation}\label{rn97}
    b = 2\left(M -2Q_*^2u_\mathrm{c}\right) \qquad \mbox{and} \qquad
    c = u_\mathrm{c}\left(3M
    -4Q_*^2u_\mathrm{c}-M^2/L^2u_\mathrm{c}^2\right) \,.
\end{equation}

When the circular orbit is unstable, the solution has the form
(\ref{r84-2}) for values of the argument in the following ranges:
\begin{equation}\label{rn86}
    \xi_\mathrm{min} \left(= \frac1{u_\mathrm{min}-u_\mathrm{c}}\right) < \xi < +\infty \,,
    \ u_\mathrm{c} < u \leq u_\mathrm{max}
    \left(=\frac1{r_\mathrm{min}}\right) \, \ \mbox{and} \
    r_\mathrm{min} < r < r_\mathrm{c} \,.
\end{equation}

The limiting case in fig.~\ref{picture5t} (where the second
picture illustrates the last unstable circular orbit) happens for
(\ref{rn90})
%\begin{equation}\label{r90t}
%  {\mathrm f}'(u) = -4Q_*^2u^3 +6Mu^2 -2u\left(1
%  +\frac{Q_*^2}{L^2}\right)+\frac{2M}{L^2} = 0
%\end{equation}
and
\begin{equation}\label{rn90t'}
  {\mathrm f}''(u) = -6Q_*^2u^2 +6Mu - 1 -\frac{Q_*^2}{L^2} = 0
  \,.
\end{equation}
\textcolor{blue}{Since we require $M$ to be positive, the
condition for parameters $M$, $L$ and $Q_*$ is
\begin{equation}\label{rntcondM}
  M = \frac{\sqrt{6\left[12Q_*^2L^2 +\left(L^2 +Q_*^2\right)^2\right]
  \pm2\sqrt{(Q_*^2+3L^2)\left(L^2 -5Q_*^2\right)^3}}}
  {12L}
\end{equation}
or
\begin{multline}\label{rncondLt}
  L =
  \mbox{sqrt}\left\{3M^{2/3}\left[2Q_*^4\left(8Q_*^2-9M^2\right)
  \sqrt{(Q_*^2-M^2)\left(4Q_*^2-5M^2\right)^3}+ \right.\right.
  \\
  +8154Q_*^4M^6 -5177Q_*^6M^4 +1472Q_*^8M^2 -128Q_*^{10}+
  \\
  \biggl.+1728M^{10} -6040Q_*^2M^8
  \biggr]^{1/3} \\
  -3M^{4/3}\left(64Q_*^6 -257Q_*^4M^2 -144M^6 +336Q_*^2M^4\right)\\
  /\Bigl[2Q_*^4\left(8Q_*^2-9M^2\right)
  \sqrt{(Q_*^2-M^2)\left(4Q_*^2-5M^2\right)^3}\Bigr.\\
  +8154Q_*^4M^6 -5177Q_*^6M^4 +1472Q_*^8M^2 -128Q_*^{10} +1728M^{10}
  \\
  \biggl.\Bigl. -6040Q_*^2M^8
  \Bigr]^{1/3} +36M^4 +8Q_*^4 -42M^2Q_*^2\biggr\}
  /\sqrt{9M^2 -8Q_*^2} \,.
\end{multline}}
\begin{figure}
    \begin{picture}(360,284)
        \multiput(0,52)(129,0){3}{\line(1,0){109}}
        \multiput(-10,162)(129,0){3}{\line(1,0){99}}
        \multiput(14,24)(129,0){3}{\line(0,1){90}}
        \multiput(14,134)(129,0){3}{\line(0,1){90}}
        \multiput(7,237)(129,0){3}{f($u$)}
        \multiput(32,22)(129,20){3}{\qbezier(-10,-15)(-8,10)(12,10)}
        \multiput(0,132)(129,20){3}{\qbezier(-10,-15)(-8,10)(12,10)}
        \multiput(32,22)(129,20){3}{\qbezier(12,10)(22,10)(31,28)}
        \multiput(0,132)(129,20){3}{\qbezier(12,10)(22,10)(31,28)}
        %\multiput(32,20)(129,20){3}{\qbezier(31,28)(37,12)(55,12)}
        \multiput(32,22)(129,20){3}{\qbezier(31,28)(45,45)(68,0)}
        \multiput(0,132)(129,20){3}{\qbezier(31,28)(45,45)(68,0)}
        %\put(275,60){\line(-1,-6){5}}
        %\put(280,52){\line(1,0){50}}
        \multiput(358,62.5)(-32,110){2}{\line(1,-2){10}}
        \multiput(378,50)(0,110){2}{$u$}
        \multiput(45,44)(-30,110){2}{$u_3$}
        \multiput(285,44)(-30,110){2}{$u_1$}
        \multiput(92,44)(257,0){2}{$u_4$}
        \multiput(62,154)(257,0){2}{$u_4$}
        \multiput(175,44)(-30,110){2}{$u_\mathrm{c}$}
        \multiput(170,34)(-23,110){2}{$= u_1$}
        \multiput(212,44)(-30,110){2}{$u_4$}
        \multiput(35,0)(-20,120){2}{$u_3 \leq y \leq u_2$}
        \put(165,0){$u_\mathrm{c} < y \leq u_4$}
        \multiput(145,120)(130,0){2}{$0 < y \leq u_4$}
        \put(294,0){$u_1 \leq y \leq u_4$}
        \multiput(173,52)(-33,110){2}{\circle*{4}}
        \linethickness{2pt}
        \multiput(65,52)(-32,110){2}{\line(1,0){14}}
        \put(173,52){\line(1,0){51}}
        \put(143,162){\line(1,0){49}}
        \put(281,52){\line(1,0){82}}
        \put(272,162){\line(1,0){59}}
        \put(50,257){($\alpha$)}
        \put(166,257){($\beta$)}
        \put(282,257){($\gamma$)}
    \end{picture}
    \caption{Limiting case for ($\alpha$) $E <
    E_\mathrm{c}$, ($\beta$) $E = E_\mathrm{c}$ and ($\gamma$)
    $E > E_\mathrm{c}$. (See equation (\ref{rn91}).) The top row are orbits
    with $E^2 > 1$ and
    the
    bottom row are orbits with $E^2 < 1$.
    }
    \label{picture5t}
\end{figure}

Eliminating \(L^2\) in equation (\ref{rn90t'}) with the aid of
equation (\ref{rn92}), we obtain the equation
\begin{equation}\label{r99}
  4Q_*^4u_\mathrm{c}^3 -9MQ_*^2u_\mathrm{c}^2 +6M^2u_\mathrm{c} -M
  = 0 \,,
\end{equation}

When all three equations f\((u) = 0\), (\ref{rn90}), and
(\ref{rn90t'}), are satisfied, equation (\ref{r67}) takes form
\begin{equation}\label{r101}
  \left(\frac{\mathrm du} {\mathrm d\varphi}\right)^2 = \left(u
  -u_\mathrm{c}\right)^3 (2M -3Q_*^2u_\mathrm{c} -Q_*^2u) \,;
\end{equation}
and the solution of this equation is (cf. equation
(\ref{s165time}))
\begin{equation}\label{r102}
  u = u_\mathrm{c} +\frac{2(M -2Q_*^2u_\mathrm{c})}{(M -2Q_*^2u_\mathrm{c})^2
  (\varphi -\varphi_0)^2 +Q_*^2} \,.
\end{equation}



\section{\color{blue}The bound orbits ($E^2 < 1$)}
The consideration of equation (\ref{r67}), governing the geometry
of the orbits described in the invariant plane, is separated to
two parts (similarly as it was in chapter~\ref{schtl}). These two
classes of orbits are characterized by $E^2 < 1$ and $E^2 \geq 1$.
This distinction is one which will determine whether orbits are
bound or unbound. In this section, we shall restrict ourselves to
bound orbits ($E^2 < 1$). The geometry of the geodesics will be
determined by the disposition of the roots of the equation f$(u) =
0$ (see figure~\ref{tlpicture}).






\subsection{\color{blue}Orbits of the first kind}\label{r1kindt}
We now consider the case when all the roots of the biquadratic
equation f$(u) = 0$ are real and distinct. Letting $u_1, u_2, u_3$
and $u_4$ denote the roots, we rewrite equation (\ref{r67}) in the
form \eqref{r1kindn}
%\begin{equation}\label{
%    \left(\frac{{\mathrm d}u}{{\mathrm d}\varphi}\right)^2 =
%    Q_*^2(u_1-u)(u-u_2)(u-u_3)(u-u_4) = {\mathrm f}(u)
%\end{equation}
(see fig.~\ref{tlpicture} ($\alpha$)). The solution is
%\begin{figure}
%    \begin{picture}(360,142)
%        \put(94,62){\line(1,0){185}}
%        \put(120,24){\line(0,1){110}}
%        \put(99,127){f($u$)}
%        \put(116,0){\qbezier(1,33)(15,90)(35,90)}
%        \put(116,0){\qbezier(35,90)(45,90)(54,68)}
%        \put(116,0){\qbezier(54,68)(60,52)(78,52)}
%        \put(116,0){\qbezier(78,52)(88,52)(109,87)}
%        \put(116,0){\qbezier(109,87)(130,120)(150,30)}
%        \put(290,60){$u$}
%        \put(165,54){$u_2$}
%        \put(207,54){$u_3$}
%        \put(129,54){$u_1$}
%        \put(262,54){$u_4$}
%        \put(121,0){$0 < y \leq u_2$}
%        \put(218,0){$u_3 < y \leq u_4$}
%        \linethickness{2pt}
%        \put(126,62){\line(1,0){47}}
%        \put(208,62){\line(1,0){50}}
%    \end{picture}
%    \caption{The case when f$(u) = 0$ has all real roots.}
%    \label{tpicture}
%\end{figure}
\begin{equation}\label{t252.00}\color{blue}
  \varphi =
  \int\limits_{u_1}^y\frac{{\mathrm d}u}{\sqrt{{\mathrm f}(u)}} =
  \frac g{Q_*}{\mathrm F}(\psi, k) \,,
\end{equation}
where $g$, $\psi$ and $k^2$ are defined in equations
(\ref{252.00refA}) -- (\ref{252.00refO}).
\begin{figure}
    \begin{picture}(360,242)
        \put(0,62){\line(1,0){10}}
        \put(85,172){\line(1,0){10}}
        \put(5,24){\line(0,1){90}}
        \put(90,134){\line(0,1){90}}
        \multiput(15,62)(140,0){3}{\line(1,0){100}}
        \multiput(100,172)(140,0){2}{\line(1,0){100}}
        \multiput(0,127)(85,110){2}{f($u$)}
        %\put(85,237){f($u$)}
        \multiput(16,0)(140,10){3}
        {\qbezier(9,32)(15,90)(25,90)}
        \multiput(16,0)(140,10){3}
        {\qbezier(25,90)(35,90)(44,68)}
        \multiput(16,0)(140,10){3}
        {\qbezier(44,68)(50,52)(58,52)}
        \multiput(16,0)(140,10){3}
        {\qbezier(58,52)(68,52)(79,87)}
        \multiput(16,0)(140,10){3}
        {\qbezier(79,87)(90,120)(95,30)}
        \multiput(150,152)(140,-30){2}
        {\qbezier(-30,0)(0,100)(50,0)}
        \put(400,60){$u$}
        \put(365,170){$u$}
        \put(50,54){$u_2$}
        \put(127,158){$u_1$}
        \put(180,158){$u_2$}
        \put(90,54){$u_3$}
        \multiput(30,54)(140,0){3}{$u_1$}
        \put(115,54){$u_4$}
        \multiput(235,54)(140,0){2}{$u_4$}
        \put(5,15){$u_1 < y \leq u_2$}
        \put(80,15){$u_3 < y \leq u_4$}
        \put(319,15){$u_1 < y \leq u_4$}
        \put(160,15){unstable circular orbit}
        \put(126,112){$u_1 < y \leq u_2$}
        \put(246,112){stable circular orbit}
        \put(60,-5){($\alpha$)}
        \put(200,-5){($\beta$)}
        \put(340,-5){($\gamma$)}
        \put(150,220){($\delta$)}
        \put(290,220){($\epsilon$)}
        \put(215,62){\circle*{4}}
        \put(295,172){\circle*{4}}
        \linethickness{2pt}
        \put(29,62){\line(1,0){33}}
        \put(85,62){\line(1,0){24}}
        \put(168,62){\line(1,0){82}}
        \put(306,62){\line(1,0){84}}
        \put(127,172){\line(1,0){63}}
    \end{picture}
    \caption{The case when f$(u)$ has ($\alpha$) four real roots, ($\beta$)
    double root, ($\gamma$) two complex conjugate roots, ($\delta$) two real
    roots, and ($\epsilon$) one real root.}
    \label{tlpicture}
\end{figure}


\subsection{\color{blue}Orbits of the second kind} To obtain the
time-like geodesics, whose ranges are $u_3 < u \leq u_4$, we
rewrite equation (\ref{r67}) in the form (\ref{rn2rov}). Solution
is then formally as in section~\ref{rn2}. \textcolor{blue}{An
example of the orbit of the first kind and associated orbit of
the second kind is in figure~\ref{RNtimepl}.}
\begin{figure}
    \epsfxsize=13cm
    \epsfbox{Reis-Nor/timeplot.eps}
    \caption{\textcolor{blue}{Time-like geodesics of the first and the second
    kind} with $Q_* =
    0.8,\ \textcolor{blue}{M = 1.2, \ L = 6}$ \textcolor{blue}{and $E =
    0.98$.}  The
    thin blue circle
    represents the circular-orbit $r_{\mathrm c}$ ($u_{\mathrm c} =
    1/r_{\mathrm c} =
    0.2225$); the thin black circle
    represents
    radius below which no circular orbits are allowed. Orbit of the first kind precesses around
    the black hole. Orbit of the second kind starts above the event horizon and ends below the
    Cauchy horizon.}
    \label{RNtimepl}
\end{figure}





\subsection{\color{blue}The orbits with imaginary
eccentricities}\label{rimagit}
Finally, we consider the orbits
when the equation f$(u) = 0$ has a pair of complex-conjugate
roots, besides two real roots. We rewrite equation (\ref{r67}) in
the form (\ref{rnirov}). Unlike in section~\ref{rni} this time
\textcolor{blue}{solution is (\ref{t252.00}) where constants $g$,
$\psi$, $A$, $B$, $b_1$, $a_1$ and $k$ are defined by
(\ref{rniconstA}) -- (\ref{rniconstO})}. The case ($\delta$) in
figure~\ref{tlpicture} has the same formal solution.


\section{\color{blue}The unbound orbits ($E^2 > 1$)}
When \(E^2 > 1\), the constant term in f\((u)\) is positive. The
equation f\((u) = 0\) must, therefore, allow a negative root. The
cases we must distinguish look essentially as space-like and null
orbits. \textcolor{blue}{Analytical solution are formally the
same as in Chapter~\ref{RNnull}.}
