% Words:   1494
\chapter{The null geodesics in the Reissner-Nordstr\"om space-time}
\label{RNnull}
The equations governing the null geodesics are
\begin{equation}
    \left(\frac{{\mathrm d}r}{\mathrm{d}\tau}\right)^2 +
    L^2\frac{\varDelta}{r^4} = E^2 \,; \; \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{r68}
\end{equation}
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_*u^4 + 2Mu^3 - u^2 +
    \frac{1}{D^2} = {\mathrm f}(u) \quad \mbox{(say),}
    \label{r69}
\end{equation}
where $D(=L/E)$ is the impact parameter.


\section{The radial geodesics}
The equations governing the radial null-geodesics can be obtained
by setting \( L = 0 \) in equations (\ref{r68}). %thus,
%\begin{equation}%\label{r70}
%  \frac{\mbox dr}{\mbox d\tau} = \pm E \,, \qquad
%  \frac{\mbox dt}{\mbox d\tau} = \frac{r^2}\varDelta E \,, \qquad
%  \mbox{and} \qquad \frac{\mbox d\theta}{\mbox d\tau} =
%  \frac{\mbox d\varphi}{\mbox d\tau} = 0 \,.
%\end{equation}
Accordingly,
\begin{equation}\label{r71}
  \frac{\mbox dr}{\mbox dt} = \pm\frac\varDelta{r^2} \,.
\end{equation}
The solution of this equation is
\begin{equation}\label{r72}
  t = \pm r_* +\mbox{constant} \,,
\end{equation}
where
\begin{equation}\label{r42}
  r_* = \int\frac{r^2}\varDelta\,\mbox dr = r
  +\frac{r_+^2}{r_++r_-}\ln{\abs{r-r_+}}
  -\frac{r_-^2}{r_++r_-}\ln{\abs{r-r_-}}\,.
\end{equation}
Therefore, for an in-going null-ray, the coordinate time $t$
\emph{increases} from $-\infty$~to $+\infty$ as $r$ decreases from
$+\infty$ to $r_+$, \emph{decreases} from $+\infty$ to $-\infty$
as $r$ further decreases from $r_+$ to $r_-$, and \emph{increases}
again from $-\infty$ to a finite limit as $r$~decreases from $r_-$
to zero. All the while, the proper time decreases at a~constant
rate $E$ (cf. equation (\ref{s223null})). From a finite distance
\( r > r_+ \), a co-moving observer will arrive at the singularity
in a finite proper time.


\section{Black hole with $M^2 \geq Q_*^2$}
\label{rnnull1}
% jen jeden extrem protoze ten druhy je prostorupodobny
 The quartic equation (\ref{r69}), f$(u) = 0$,
always allows two real roots: one negative (which has no physical
significance) and one positive (which occurs for $r < r_-$). We
shall be concerned only with cases when the two remaining roots
are {\em either\/} both real (distinct or coincident) {\em or\/} a
complex-conjugate pair. Let the value of the impact parameter,
$D$, for which f$(u) = 0$ has a double root, be denoted by
$D_{\mathrm c}$. Then for all values of $D > D_{\mathrm c}$, we
shall have orbits of two kinds (as in the Schwarzschild
space-time): orbits of the {\em first kind\/} which lie entirely
outside the event horizon, coming from $+\infty$ and receding
again to $+\infty$ after a perihelion passage; and orbits of the
{\em second kind\/} which have two turning points, one outside the
event horizon and one inside the Cauchy horizon. For values of the
impact parameter $D < D_{\mathrm c}$ (when f$(u) = 0$ allows only
one real root, $> 1/r_-$, along the positive real $u$-axis) the
orbit coming from $+\infty$ will cross both horizons and have a
turning point for a~finite value of $r < r_-$. Thus, all these
orbits skirt the singularity at $r = 0$.


{\color{blue}\subsection{The orbits of the first kind}\label{rn1}
We now consider the case when all the roots of the biquadratic
equation f$(u) = 0$ are real and distinct. We rewrite equation
(\ref{r69}) in the form
\begin{equation}\label{r1kindn}
    \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{picture}). The solution is
\begin{multline}
    \varphi = \int\limits_0^y\frac{{\mathrm d}u}{\sqrt{{\mathrm f}(u)}} =
    \int\limits_{u_1}^y\frac{{\mathrm d}u}{\sqrt{{\mathrm f}(u)}} -
    \int\limits_{u_1}^0\frac{{\mathrm d}u}{\sqrt{{\mathrm f}(u)}} =
    \frac{g}{Q_*}{\mathrm F}(\psi, k) +
    \int\limits_0^{u_1}\frac{{\mathrm d}(u)}{\sqrt{{\mathrm f}(u)}}=\\
    =
    \frac{g}{Q_*}[{\mathrm F}(\psi, k) + {\mathrm K}(k)] \,,
    \label{252.00}
\end{multline}
where
\bal \label{252.00refA}
    g &= \frac2{\sqrt{(u_4-u_2)(u_3-u_1)}} \,,\\
    \psi &= \arcsin\sqrt{\frac{(u_4-u_2)(y-u_1)}{(u_2-u_1)(u_4-y)}}
    \,,\\
    k^2 &= \frac{(u_4-u_3)(u_2-u_1)}{(u_4-u_2)(u_3-u_1)} \,.
    \label{252.00refO}
\eal}
\begin{figure}
    \begin{picture}(360,142)
        \put(124,62){\line(1,0){155}}
        \put(150,24){\line(0,1){110}}
        \put(129,127){f($u$)}
        \put(116,0){\qbezier(9,52)(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(150,62){\line(1,0){23}}
        \put(208,62){\line(1,0){50}}
    \end{picture}
    \caption{The case when f$(u) = 0$ has all real roots.}
    \label{picture}
\end{figure}


\subsubsection{Geodesics with critical impact
parameter}\label{rncirc}
\begin{figure}
    \begin{picture}(360,142)
        \put(124,51){\line(1,0){155}}
        \put(150,24){\line(0,1){110}}
        \put(129,127){f($u$)}
        \put(116,0){\qbezier(9,40)(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(285,49){$u$}
        \put(191,42){$u_\mathrm{c}$}
        \put(193,50){\circle*{4}}
      %  \put(207,43){$u_3$}
        \put(129,42){$u_1$}
       \put(265,42){$u_4$}
        \put(121,0){$0 < y \leq u_\mathrm{c}$}
        \put(218,0){$u_3 < y \leq u_\mathrm{c}$}
        \linethickness{2pt}
        \put(150,51){\line(1,0){42}}
        \put(194,51){\line(1,0){67}}
    \end{picture}
    \caption{The case when f$(u) = 0$ has a double root.}
    \label{picture2}
\end{figure}
The value of the impact parameter for which f\((u) = 0 \) allows a
double root is determined by the equations
\begin{equation}\label{r73}
  \mbox f(u) = \frac1{D^2} -u^2\left(Q_*^2u^2 -2Mu +1\right) = 0
\end{equation}
and
\begin{equation}\label{r74}
  \mbox f'(u) = -2u\left(1-3Mu+2Q_*^2u^2\right) = 0 \,.
\end{equation}
Besides \(u=0\), equation (\ref{r74}) allows the roots
\begin{equation}\label{r75}
  u =
  \frac{3M}{4Q_*^2}\left(1\pm\sqrt{1-\frac{8Q_*^2}{9M^2}}\right)
  \,.
\end{equation}
At the larger of these two roots, f\((u)\) has a maximum. The
double root must occur at the minimum of f\((u)\) where
\begin{equation}\label{r76}
  u = \frac{3M}{4Q_*^2}\left(1-\sqrt{1-\frac{8Q_*^2}{9M^2}}\right) =
  u_\mathrm{c} \qquad \mbox{(say)}.
\end{equation}
The corresponding value of $r$ is
\begin{equation}\label{r77}
  r_\mathrm{c} = \frac32M\left(1 +\sqrt{1
  -\frac{8Q_*^2}{9M^2}}\right) \,.
\end{equation}
At this radius $r_\mathrm{c}$ the geodesic equation allows an
unstable circular orbit.

The value of impact parameter, $D_\mathrm{c}$, associated with the
double root $r_\mathrm{c}$ follows from equation (\ref{r73}). We
find
\begin{equation}\label{r78}
  D_\mathrm{c} = r_\mathrm{c}^2/\sqrt{\varDelta_\mathrm{c}} \,,
  \qquad \varDelta_\mathrm{c} = r_\mathrm{c}^2 -2Mr_\mathrm{c}
  +Q_*^2 \,.
\end{equation}

When \(D=D_\mathrm{c}\),
\begin{equation}\label{r79}
  \mbox f(u) = \left(u -u_\mathrm{c}\right)^2\left[-Q_*^2u^2
  +2\left(M-Q_*^2u_\mathrm{c}\right)u +u_\mathrm{c}\left(M
  -Q_*^2u_\mathrm{c}\right)\right]
  \,,
\end{equation}
and the solution for $\varphi$ is given by
\begin{equation}\label{r80}
  \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})}}\,.
\end{equation}
The substitution,
\begin{equation}\label{r81}
  \xi = \frac1{u-u_\mathrm{c}} \,,
\end{equation}
reduces $\varphi$ to the elementary integral,
\begin{equation}\label{r82}
  \varphi = \mp \int\frac{\mathrm d\xi}{\sqrt{-Q_*^2+b\xi+c\xi^2}}
  \,,
\end{equation}
where
\begin{equation}\label{r83}
  b = 2\left(M -2Q_*^2u_\mathrm{c}\right) \qquad \mbox{and} \qquad c =
  u_\mathrm{c}\left(3M -4Q_*^2u_\mathrm{c}\right) \,.
\end{equation}
We thus obtain the solution
\begin{equation}\label{r84}
  \varphi = \mp%\left\{
  %\begin{array}{cl}
    \displaystyle \frac{\ln\left[2c\xi+b+2c\sqrt{c(-Q_*^2+b\xi+c\xi^2)}\right]}{\sqrt c}% & (c > 0)\\
   % \displaystyle -\frac{\arcsin\frac{2c\xi+b}{\sqrt{4Q_*^2c+b^2}}}{\sqrt{-c}} & (c < 0) \
  %\end{array}
  %\right.
  \,,
\end{equation}
for values of the argument in the following ranges:
\begin{equation}\label{r85}
  -\infty < \xi < -\frac1{u_\mathrm c} \,, \qquad 0 \leq u \leq
  u_{\mathrm c} \,, \qquad \mbox{and} \qquad r_{\mathrm c} < r <
  \infty \,.
\end{equation}



{\color{blue}\subsection{The geodesics of the second kind}\label{rn2}
To obtain the null
geodesics, whose ranges are $u_3 < u \leq u_4$, we rewrite
equation (\ref{r69}) in the form
\begin{equation}\label{rn2rov}
    \left(\frac{{\mathrm d}u}{{\mathrm d}\varphi}\right)^2 = Q_*^2(u -
    u_1)(u - u_2)(u - u_3)(u_4 - u)
\end{equation}
(see fig.~\ref{picture}). The solution is
\begin{equation}
    \varphi = \int\limits_{u_3}^y\frac{{\mathrm d}u}{\sqrt{{\mathrm
    f}(u)}} = \frac{g}{Q_*}{\mathrm F}(\psi, k) \,,
    \label{257.00}
\end{equation}
where this time, unlike in the case of geodesics of the first
kind,
\begin{equation}
    \psi = \arcsin\sqrt{\frac{(u_4- u_2)(y- u_3)}{(u_4- u_3)(y- u_2)}}
    \,.
\end{equation}
\begin{figure}
    \epsfxsize=13cm
    \epsfbox{Reis-Nor/null2.eps}
    \caption{\textcolor{blue}{General null geodesics of the first and the
    second
    kind} with $Q_* = 0.8,
    {\color{blue}\ M = 6/7}$ {\color{blue}and $D =
    4.658489054$.} The inner
    and outer horizons are shown by the thin circles; the dotted circle
    represents the unstable circular-orbit. The orbit of the first kind starts from infinity,
    orbit of the second kind starts above the vent horizon and ends below the Cauchy horizon.}
    \label{RNNull}
\end{figure}

An example of orbits of the first and second kind can be
seen in picture~\ref{RNNull}.}


\subsubsection{Geodesics with critical impact parameter}
The orbits of the second kind are described by the same solution
(\ref{r84}) for values of the argument in the ranges
\begin{equation}\label{r86}
    \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}
where $r_\mathrm{min}$ is the positive root of the equation
(cf.~equation(\ref{r79})),
\begin{equation}\label{r87}
  u_\mathrm{c}\left(M -Q_*^2u_\mathrm{c}\right)r^2 +2\left(M
  -Q_*^2u_\mathrm{c}\right)r
  -Q_*^2 = 0 \,.
\end{equation}
Both orbits of the first and second kind approach the unstable
circular orbit at \(r=r_\mathrm{c}\), asymptotically from opposite
sides, by spiralling around it an infinite number of times.




{\color{blue}\subsection{The orbits with imaginary eccentricities}\label{rni}
Finally, we consider the orbits when the equation f$(u) = 0$ has a
pair of complex-conjugate roots, besides two real roots. That is
true when the impact parameter, $D$, has values less than critical,
$D_\mathrm{c}$, with
\begin{equation}
    D_\mathrm{c} = \frac{r_\mathrm{c}^2}{\sqrt{\Delta_\mathrm{c}}}\,,
    \label{r78a}
\end{equation}
where the critical radius has value (\ref{r77})
%\begin{equation}
%    r_\mathrm{c} = \frac{3}{2}M\left(1+\sqrt{1-\frac{8Q_*^2}{9M^2}}\right)
%    \label{r77}
%\end{equation}
and
\begin{equation}
    \Delta_\mathrm{c} = r_\mathrm{c}^2-2Mr_\mathrm{c}+Q_*^2 = Mr_\mathrm{c}-Q_*^2\,.
    \label{r78b}
\end{equation}

We rewrite equation (\ref{r69}) in the form
\begin{equation}\label{rnirov}
    \left(\frac{{\mathrm d}u}{{\mathrm
    d}\varphi}\right)^2 = Q_*^2(u - u_1)(u - u_2)(u - \bar{u}_2)(u_4 -
    u)
\end{equation}
(see fig.~\ref{picture3}). The solution is
\begin{multline}
    \varphi = \int\limits_0^y\frac{{\mathrm d}u}{\sqrt{{\mathrm
    f}(u)}} = \int\limits_{u_1}^y\frac{{\mathrm d}u}{\sqrt{{\mathrm
    f}(u)}} - \int\limits_{u_1}^0\frac{{\mathrm d}u}{\sqrt{{\mathrm
    f}(u)}} = \frac{g}{Q_*}{\mathrm F}(\psi, k) +
    \int\limits_0^{u_1}\frac{{\mathrm d}u}{\sqrt{{\mathrm f}(u)}} =
    \\ = \frac{g}{Q_*}[{\mathrm F}(\psi, k) + {\mathrm
    K}(k)] \,,
    \label{259.00}
\end{multline}
where
\bal\label{rniconstA}
    g &=
    \frac{1}{\sqrt{AB}}\,,\\ \psi &= \arccos\frac{(u_4 - y)B - (y -
    u_1)A}{(u_4 - y)B + (y - u_1)A}\,,\\ A^2 &= \left(u_4 - b_1\right)^2 +
    a_1^2\
    ,\\ B^2 &= \left(u_1 - b_1\right)^2 + a_1^2\,,\\ b_1 &= \frac{(u_2 +
    \bar{u}_2)}{2}\,,\\ a_1^2 &= -\frac{\left(u_2 -
    \bar{u}_2\right)^2}{4}\,,\\
    k^2 &= \frac{\left(u_4 - u_1\right)^2 - \left(A- B\right)^2}{4AB}\,.
    \label{rniconstO}
\eal
\begin{figure}
    \begin{picture}(360,142)
        \put(124,52){\line(1,0){155}}
        \put(150,24){\line(0,1){110}}
        \put(129,127){f($u$)}
        \put(116,10){\qbezier(9,52)(15,90)(35,90)}
        \put(116,10){\qbezier(35,90)(45,90)(54,68)}
        \put(116,10){\qbezier(54,68)(60,52)(78,52)}
        \put(116,10){\qbezier(78,52)(88,52)(109,87)}
        \put(116,10){\qbezier(109,87)(130,120)(150,30)}
        \put(290,50){$u$}
        \put(125,62){\line(-1,-6){4}}
        \put(126,44){$u_1$}
        \put(252,44){$u_4$}
        \put(171,0){$0 < y < u_4$}
        \linethickness{2pt}
        \put(150,52){\line(1,0){113}}
    \end{picture}
    \caption{The case when f$(u) = 0$ has two complex-conjugate roots.}
    \label{picture3}
\end{figure}
\begin{figure}
    \epsfxsize=13cm
    \epsfbox{Reis-Nor/nuimagi1.eps}
    \caption{Null geodesic with imaginary eccentricity with
    $\textcolor{blue}{M = 6/7},\ Q_* = 0.8$ and
    $\color{blue}D = 3$ starts in infinity and ends below the Cauchy horizon. The inner
    and outer horizons are shown by the thin circles.}
    \label{RNnuima1}
\end{figure}
An example of an orbit with imaginary eccentricity is in
figure~\ref{RNnuima1}.}



\section{Black hole with $M^2 < Q_*^2$}
Once we allow \(Q_* > M \), $r_\pm$ disappear. We must now
distinguish more cases than in section~\ref{rnnull1} based on the
relations between the coefficients $M$ and $Q_*$.


\subsection{Case $M^2 < Q_*^2 < \frac98M^2$}
For \( M^2 < Q_*^2 < \frac98M^2 \) there are two circular photon
orbits, one stable and one unstable, with impact parameter
$D_\mathrm{cs}$ and $D_\mathrm{cu}$ respectively. For \( D <
D_\mathrm{cs}\) discussion is formally as in section
\ref{rnnull1}.%, except that critical radius is now

Case \( D
> D_\mathrm{cs} \) is pictured in
{fig.~\ref{picture4}}.
\begin{figure}
    \begin{picture}(360,142)
        \put(124,52){\line(1,0){155}}
        \put(150,24){\line(0,1){110}}
        \put(129,127){f($u$)}
        \put(116,10){\qbezier(9,-18)(15,50)(35,50)}
        \put(116,10){\qbezier(35,50)(45,50)(54,28)}
        \put(116,10){\qbezier(54,28)(60,12)(78,12)}
        \put(116,10){\qbezier(78,12)(88,12)(94,18)}
        \put(116,10){\qbezier(94,18)(130,60)(150,-10)}
        \put(290,50){$u$}
        \put(121,44){$u_1$}
        \put(172,44){$u_2$}
        \put(179,0){$0 < y < u_2$}
        \linethickness{2pt}
        \put(150,52){\line(1,0){13}}
    \end{picture}
    \caption{The case when f$(u) = 0$ has two complex-conjugate roots.}
    \label{picture4}
\end{figure}
\begin{figure}
    \epsfxsize=13cm
    \epsfbox{Reis-Nor/nuimagi2_2.eps}
    \caption{Null geodesic with imaginary eccentricity with
    $\textcolor{blue}{M = 11/14},\ Q_* = 0.8$ and $\color{blue}
    D = 5$.}
    \label{RNnuima2}
\end{figure}
We can see in \textcolor{blue}{figure~\ref{RNnuima2}}
that orbits
are coming from
infinity %to infinity %terminate at distance $1/u_2$ from singularity. %zeptat se, ale asi jdou z nekonecna do nekonecna, viz extended manifold
and have a turning point at $1/u_2$. The solution is formally the
same as in
section~\ref{rni}. %formally as in imaginary case

\textcolor{blue}{Special case \( D = D_\mathrm{cs} \)} (see
picture~\ref{picture4c})
\textcolor{blue}{allows a stable circular photon orbit with radius %(\ref{r77}).
\begin{equation}\label{r77n}
    r_\mathrm{cs} =
    \frac{3}{2}M\left(1-\sqrt{1-\frac{8Q_*^2}{9M^2}}\right)\,.
\end{equation}}
For orbits of the first kind the discussion is formally as in
section~\ref{rncirc} with the solution in the form
\begin{equation}\label{r84-2}
  \varphi = \pm
   \frac{\arcsin\cfrac{2c\xi+b}{\sqrt{4Q_*^2c+b^2}}}{\sqrt{-c}}
  \,,
\end{equation}
for values of the argument in the ranges
\begin{equation}\label{r85-2}\color{blue}
  \frac1{u_2-u_\mathrm c} < \xi < -\frac1{u_\mathrm c} \,, \qquad 0 \leq u \leq
  u_2 \,, \qquad \mbox{and} \qquad r_2 < r <
  \infty \,.
\end{equation}
For \(b\) and \(c\) see equation (\ref{r83}).
\begin{figure}
    \epsfxsize=13cm
    \epsfbox{Reis-Nor/nullcst_2.eps}
    \caption{{\color{blue}Stable circular orbit with associated orbit of the
    first kind. Orbit of the first kind starts from infinity. $M = 11/14$} and $Q_* =
    0.8$.}
    \label{RNncst}
\end{figure}
\textcolor{blue}{An example of such orbits is in
figure~\ref{RNncst}.}

\begin{figure}
    \begin{picture}(360,142)
        \put(124,52){\line(1,0){155}}
        \put(150,24){\line(0,1){110}}
        \put(129,127){f($u$)}
        \put(116,18){\qbezier(9,-18)(15,50)(35,50)}
        \put(116,18){\qbezier(35,50)(45,50)(54,28)}
        \put(116,18){\qbezier(54,28)(60,12)(78,12)}
        \put(116,18){\qbezier(78,12)(88,12)(94,18)}
        \put(116,18){\qbezier(94,18)(130,60)(150,-10)}
        \put(290,50){$u$}
        \put(121,44){$u_1$}
        \put(173,44){$u_2$}
        \put(179,0){$0 < y < u_2$}
        \put(235,52){\circle*{4}}
        \put(250,44){$u_\mathrm{cs}$}
        \linethickness{2pt}
        \put(150,52){\line(1,0){17}}
    \end{picture}
    \caption{The case when f$(u) = 0$ has a double root.}
    \label{picture4c}
\end{figure}


\subsection{Case $Q_*^2 = \frac98M^2$} When \( Q_*^2 = \frac98M^2
\), f\( (u) = 0 \) allows only one real positive root. In case of
\( D = D_\mathrm{c} \) there is a (unstable) circular orbit at \(
r_\mathrm{c} = \frac32M \) and the solution is
\begin{equation}\label{r84-3}\color{blue}
  \varphi = \pm 2 \sqrt{b\xi-Q_*^2} /b \,,
\end{equation}
where \(b\) is defined in equation (\ref{r83}).


%\begin{figure}
%    \begin{picture}(360,142)
%        \put(124,52){\line(1,0){155}}
%        \put(150,24){\line(0,1){110}}
%        \put(129,127){f($u$)}
%        \put(124,10){\qbezier(9,0)(15,58)(35,58)}
%        \put(124,10){\qbezier(35,58)(45,58)(54,36)}
%        \put(124,10){\qbezier(54,36)(60,20)(78,20)}
%        \put(124,10){\qbezier(78,20)(88,20)(94,26)}
%        \put(124,10){\qbezier(94,26)(130,68)(150,-2)}
%        \put(290,50){$u$}
       % \put(129,44){$u_1$}
       % \put(180,44){$u_2$}
       % \put(179,0){$0 < y < u_4$}
       % \linethickness{2pt}
       % \put(150,52){\line(1,0){121}}
%    \end{picture}
%    \caption{The case when f$(u) = 0$ has two complex-conjugate roots.}
%    \label{picture5}
%\end{figure}
\begin{figure}
    \begin{picture}(360,142)
        \multiput(0,52)(100,0){3}{\line(1,0){90}}
        \multiput(26,24)(110,0){3}{\line(0,1){90}}
        \multiput(19,127)(110,0){3}{f($u$)}
        \multiput(14,20)(110,20){3}{\qbezier(-15,0)(-8,50)(12,50)}
        \multiput(14,20)(110,20){3}{\qbezier(12,50)(22,50)(31,28)}
        \multiput(14,20)(110,20){3}{\qbezier(31,28)(37,12)(55,12)}
        \multiput(14,20)(110,20){3}{\qbezier(55,12)(65,12)(68,0)}
        \put(219,60){\line(-1,-6){5}}
        \put(280,52){\line(1,0){50}}
        \put(302,61){\line(1,-6){5}}
        \put(340,50){$u$}
        \multiput(12,44)(105,0){3}{$u_1$}
        \put(33,44){$u_2$}
        \put(173,44){$u_\mathrm{c}$}
        %\put(165,34){$= u_4$}
        \put(312,44){$u_4$}
        \put(19,0){$0 < y < u_2$}
        \put(129,0){$0 < y < u_\mathrm{c}$}
        \put(239,0){$0 < y < u_4$}
        \put(178,52){\circle*{4}}
        \linethickness{2pt}
        \put(26,52){\line(1,0){18}}
        \put(136,52){\line(1,0){43}}
        \put(246,52){\line(1,0){57}}
    \end{picture}
    \caption{The case when $Q_*^2 = \frac98M^2$ for $D >
    D_\mathrm{c}$, $D = D_\mathrm{c}$ and $D < D_\mathrm{c}$ respectively.}
    \label{picture5}
\end{figure}
\begin{figure}
    \epsfxsize=13cm
    \epsfbox{Reis-Nor/nuimagi3_2.eps}
    \caption{\textcolor{blue}{Null geodesic with $Q_*^2 = \frac98M^2$ and
    $D > D_{\mathrm c}$. It starts from infinity.} ($Q_* =
    0.8$ and $\color{blue}D = 3$.)}
    \label{RNnuima3}
\end{figure}
\begin{figure}
    %\epsfxsize=13cm
    \begin{center}
        \epsfbox{Reis-Nor/nullcst2.eps}
        \caption{\textcolor{blue}{Null geodesics with $Q_*^2 = \frac98M^2$
        and $D = D_{\mathrm c}$
        associated with unstable circular orbit. It starts from infinity.} ($Q_* =
        0.8$.)}
        \label{RNnucst2}
    \end{center}
\end{figure}
\begin{figure}
    %\epsfxsize=13cm
    \begin{center}
        \epsfbox{Reis-Nor/nuimagi4_2.eps}
        \caption{\textcolor{blue}{Null geodesic with $Q_*^2 = \frac98M^2$ and
        $D < D_{\mathrm c}$. It starts from infinity.} ($Q_* =
        0.8$ and $\color{blue}D = 2$.)}
        \label{RNnuimagi4}
    \end{center}
\end{figure}
You can see the different possible cases in fig.~\ref{picture5}.
Solutions for general cases are formally as in section~\ref{rni},
but with different ranges of arguments. \textcolor{blue}{For
examples see figures \ref{RNnuima3}, \ref{RNnucst2} and
\ref{RNnuimagi4}.}


\subsection{Case $Q_*^2 > \frac98M^2$}
In this case there is no circular orbit and f\((u)\) has only one
positive root (see fig.~\ref{picture6}).
\begin{figure}
    \begin{picture}(360,142)
        \put(105,52){\line(1,0){150}}
        \put(180,24){\line(0,1){90}}
        \put(176,127){f($u$)}
        \put(175,30){\qbezier(-30,0)(0,100)(50,0)}
        \put(270,50){$u$}
        \put(138,44){$u_1$}
        \put(220,44){$u_2$}
     %   \put(150,44){$u_2 = u_3$}
     %   \put(165,34){$= u_4$}
     %  \put(312,44){$u_4$}
        \put(160,0){$0 < y < u_2$}
        \linethickness{2pt}
        \put(180,52){\line(1,0){33}}
    \end{picture}
    \caption{The case when $Q_*^2 > \frac98M^2$.}
    \label{picture6}
\end{figure}
Again the solution is formally as in section~\ref{rni} with a~different
range of argument.
\begin{figure}
    %\epsfxsize=13cm
    \begin{center}
        \epsfbox{Reis-Nor/nuimagi5.eps}
        \caption{\textcolor{blue}{Null geodesic with $Q_*^2 > \frac98M^2$.
        It starts from infinity and after reaching perihelion goes to infinity again.}
        ($Q_* =
        0.8$ and $\color{blue}M = 0.7$.)}
        \label{RNnuimagi5}
    \end{center}
\end{figure}
\textcolor{blue}{For example of this orbit see
figure~\ref{RNnuimagi5}}.
