\chapter{The null geodesics}\label{kerrnull}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/nulcirc1.eps}
         \caption{\color{blue}Critical null geodesic in ``spherical" coordinates with $r_s = 1.85$,
         $M = 1$ and $a = 0.8$.
         The orbit does not reach the singularity.}
         \label{nulcirc1}
\end{figure}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/nulcirc2.eps}
         \caption{The projections of the critical null geodesic with $r_s = 1.85$, $M = 1$ and
         $a = 0.8$ on $(r, \theta)$ and $\color{blue}(r, \varphi)$ respectively. The orbit does
         not reach the singularity. \textcolor{blue}{The first ``reversal" is actually where the orbits of the
         first and second kind coalesce. Other reversal are similar to those occurring for
         orbits in equatorial plane.}}
         \label{nulcirc2}
\end{figure}
In this section we follow \cite[section 63]{chandrasekhar} with  a few of our additions.

For the null geodesics $\delta_1 = 0$, and it is convenient to minimize the number of parameters by letting
\begin{equation}\label{k189}
  \xi = L/E \qquad \text{and} \qquad \eta = Q/E^2 \,,
\end{equation}
and writing R and $\Theta$ in place of R$/E^2$ and $\Theta/E^2$:
\bal
    \text R &= r^4 +\left(a^2-\xi^2-\eta\right)r^2 +2M\left[\eta+(\xi-a)^2\right]r -a^2\eta \label{k190}\\
    \intertext{and}
    \Theta &= \eta +a^2\cos^2\theta -\xi^2\cot^2\theta \label{k191}\,.
\eal

The equation governing the projection of the orbit in the $(r, \theta)$-plane is
\begin{equation}\label{k193}
  \int\limits_{r_i}^r\frac{\text dr}{\sqrt\text R} = \int\limits_{\theta_i}^\theta\frac{\text d\theta}{\sqrt\Theta} \,,
\end{equation}
where $r_i$ and $\theta_i$ are certain assigned initial values of $r$ and $\theta$. Also while the signs of $\sqrt\text R$ and $\sqrt\Theta$ can be chosen independently, they must be adhered to once the choice has been made.


\section{The $\theta$-motion}\label{nulteta}
In this section the $\theta$-motion, as specified by the integral over $\Theta^{-1/2}$ in equation \eqref{k193}, will be investigated. %\textcolor{blue}
{The condition on $\Theta$ being non negative restricts the constants $\xi$ and $\eta$ by the inequality
\begin{equation}\label{k194}
  \eta +\left(a-\xi\right) \geq 0 \,,
\end{equation}}
which follows from $\Theta$ written as follows:
\begin{equation}\label{k195}
  \Theta = \eta +\left(a-\xi\right)^2 -(a\sin\theta-\xi\csc\theta)^2 \,.
\end{equation}

 $\cos\theta = \mu$ replaces $\theta$ as the variable of integration. The integral
\begin{equation}\label{k196}
  I_\mu = \int\frac{\text d\mu}{\sqrt\Theta_\mu} \,,
\end{equation}
where
\begin{equation}\label{k197}
  \Theta_\mu = \eta -\left(\xi^2+\eta-a^2\right)\mu^2 -a^2\mu^4 \,
\end{equation}
is considered instead.
When $\eta > 0$, and for $\Theta \geq 0$, $\mu^2$ will lay between $\mu^2 = 0$ and a certain maximum ${\mu_\text{max}}^2:\ 0 \leq \mu^2 \leq {\mu_\text{max}}^2$.


\subsection{$\eta > 0$}
Explicit expression for the integral $I_\mu$ can be  written down as follows :
\begin{equation}\label{k199}
  \Theta_\mu = a^2 \left({\mu_-}^2+\mu^2\right) \left({\mu_+}^2+\mu^2\right) \qquad (0 \leq \mu^2 \leq {\mu_+}^2) \,,
\end{equation}
where
\bal
    {\mu_+}^2 &= \frac{\sqrt{(\xi^2+\eta-a^2)^2+4a^2\eta}-(\xi^2+\eta-a^2)}{2a^2} \label{k200+} \,,\\
    {\mu_-}^2 &= \frac{\sqrt{(\xi^2+\eta-a^2)^2+4a^2\eta}+(\xi^2+\eta-a^2)}{2a^2} \label{k200-} \,;
\eal
and
\begin{equation}\label{k201int}
  \int\limits_\mu^{\mu_+}\frac{\text d\mu}{\sqrt{\Theta_\mu}} = \frac{\F(\psi, k)}{a\sqrt{\mu_+^2+\mu_-^2}} \,,
\end{equation}
where
\begin{equation}\label{k201}
  k^2 = \frac{{\mu_+}^2}{{\mu_+}^2+{\mu_-}^2} \qquad \text{and} \qquad \cos\psi = \mu/\mu_+ \,.
\end{equation}


\subsection{$\eta = 0$}
Integral $I_\mu$ can be written down explicitly:
\bal
  \Theta_\mu &= \left(a^2-\xi^2\right)\mu^2 -a^2\mu^4 \qquad &\left(0 \leq \mu^2 \leq {\mu_\text{max}}^2 \right) \,,\label{k202Teta}\\
    \intertext{where}
    {\mu_\text{max}}^2 &= 1-\xi^2/a^2 &\left(\abs{\xi}^2 \leq \abs{a}^2\right)\,,\label{k202max}
\eal
and
\begin{equation}\label{k203}
  \int\limits_\mu^{\mu_\text{max}}\frac{\text d\mu}{\sqrt\Theta} = -\frac{\arcsech\frac\mu{\mu_\text{max}}}{\sqrt{a^2-\xi^2}} \,.
\end{equation}

\subsection{$\eta < 0$}
Integral $I_\mu$ written explicitly is, in this case
\bal
    \Theta_\mu &= a^2\left(\mu^2 -\mu_-^2\right)\left(\mu_+^2 -\mu^2\right) \qquad \left(\mu_-^2 \leq \mu^2 \leq \mu_+^2\right) \,,\label{k204}\\
    \intertext{where}
    \mu_\pm^2 &= \frac{\abs{\eta}+a^2-\xi^2 \pm\sqrt{\left(\abs{\eta}+a^2-\xi^2\right)^2-4a^2\abs{\eta}}} {2a^2}\,,\label{k205}
\eal
and
\bal
  \int\limits_{\mu_-}^\mu \frac{\text d\mu}{\sqrt{\Theta_\mu}} &= \frac{\text F(\psi, k)}{a\mu_+}\,,\label{k206int}\\
    \intertext{where}  k^2 &= \frac{\mu_+^2 - \mu_-^2}{\mu_+^2}\,, \label{k206k}\\
    \abs\eta &\leq a^2\,,\label{k206eta}\\
    0 \leq \abs\xi &\leq \abs a -\sqrt{\abs\eta}\,,\label{k206range}\\
    \intertext{and}
    \sin\psi &= \sqrt\frac{\mu_+^2(\mu^2-\mu_-^2)}{\mu^2(\mu_+^2-\mu_-^2)}\,.\label{k206psi}
\eal



\subsection{The principal null-congruences}
From equations \eqref{k194} and \eqref{k195} it can be seen that the case
\begin{equation}\label{k207}
  \eta +\left(a-\xi\right)^2 = 0
\end{equation}
is distinguished because it is compatible with the condition $\Theta_\mu \geq 0$ only if
\bal
  \theta &= \theta_0 = \text{a constant}\,,\label{k208theta}\\
  \intertext{and}
  \xi &= a\sin^2\theta_0 \label{k208xi}\,,
\eal
when $\Theta_\mu = 0$ for $\theta = \theta_0$ (see equation \eqref{kch161} for equation for $\dot\theta$). Now the condition for equation \eqref{k207} is that
\begin{equation}\label{k209}
  \eta = -a^2\cos^4\theta_0 \,.
\end{equation}
Inserting the above values of $\xi$ and $\theta$ in equation \eqref{k190}, it yields
\begin{equation}\label{k210}
  \text R = \left(r^2+a^2\cos^2\theta_0\right)^2E^2 \,,
\end{equation}
where the previously suppressed factor $E^2$ was restored. Now  from equation \eqref{kch160} it follows that
\begin{equation}\label{k211}
  \dot r =\pm E \,.
\end{equation}
Similarly, from equation\eqref{kch163} it can be concluded that
\begin{equation}\label{k212fi}
  \dot\varphi = aE/\varDelta
\end{equation}
and from equation \eqref{kch162}
\begin{equation}\label{k212t}
  \dot t = \left(r^2+a^2\right)E/\varDelta \,.
\end{equation}
These special null geodesics are described by
\begin{equation}\label{k213n}
  \frac{\text dt}{\text d\tau} = E\frac{r^2+a^2}\varDelta \,, \quad \frac{\text dr}{\text d\tau} = \pm E \,, \quad \frac{\text d\theta}{\text d\tau} = 0 \,, \quad \text{and} \quad \frac{\text d\varphi}{\text d\tau} = E\frac a\varDelta \,.
\end{equation}
Solutions for $t(r)$ and $\varphi(r)$ can be found in section \ref{keqnulrad}.




\section{The $r$-motion}
In this section we turn to a consideration of the radial motion as specified by the integral over $\text R^{-1/2}$.

In the non-planar case there must be a distinction of whether the constants of the motion, $\xi$ and $\eta$, are on one side or the other of a critical locus, $(\xi_s, \eta_s)$, in the $(\xi, \eta)$-plane, with the orbits, with the constants of the motion on the locus itself, representing unstable orbits with $r =$ constant.


\subsection{The unstable orbits of constant radius}\label{knullcirc}
The equations which determine the unstable orbits of constant radius are
\bal
  \text R &= r^4 +\left(a^2-\xi^2-\eta\right)r^2 +2M\left[\eta+(\xi-a)^2\right]r -a^2\eta = 0 \,,\label{k214}\\
\intertext{and}
\label{k215}
  \frac{\partial\text R}{\partial r} &= 4r^3 +2\left(a^2-\xi^2-\eta\right)r +2M\left[\eta+(\xi-a)^2\right] = 0\,.
\eal
They can be combined to give
\begin{equation}\label{k216}
  3r^4 +r^2a^2 -\eta\left(r^2-a^2\right) = r^2\xi^2
\end{equation}
and
\begin{equation}\label{k217}
  r^4 -a^2Mr +\eta\left(a^2-Mr\right) = Mr\left(\xi^2-2a\xi\right) \,.
\end{equation}
The above equations can be solved for $\xi$ and $\eta$. Therefore, eliminating $\eta$ between them yields
\begin{equation}\label{k218}
  a^2(r-M)\xi^2 -2aM\left(r^2-a^2\right)\xi -r\left(r^2+a^2\right)\left[r\left(r^2+a^2\right) -M\left(3r^2-a^2\right)\right] = 0 \,.
\end{equation}
We find the discriminant of this equation to be
\begin{equation}\label{k219}
  a^2r^2\varDelta^2
\end{equation}
and the solution of equation \eqref{k218} is% given by
\begin{equation}\label{k220}
  \xi = \frac{M\left(r^2-a^2\right)\pm r\varDelta}{a(r-M)} \,.
\end{equation}


To be consistent with our present requirements, we must choose the lower sign in this equation (\cite[section 63] {chandrasekhar}). Then
\bal\label{k224}
  \xi &= \frac{M\left(r^2-a^2\right)- r\varDelta}{a(r-M)}\\
    \intertext{and}
    \label{k225}
  \eta &= \frac{r^3\left[4a^2M-r(r-3M)^2\right]}{a^2(r-M)^2} = \frac{r^3\left[4M\varDelta -r(r-M)^2\right]}{a^2(r-M)^2} \,.
\eal
These equations parametrically determine the critical locus $(\xi_s, \eta_s)$.

\emph{There are no orbits of constant radius with $\eta < 0$} and when $\eta = 0$, the orbits lie entirely in the equatorial plane (\cite[section 63\S(c)]{chandrasekhar}).

For an orbit to have constant radius $r_s$, the constants of motions, $\xi$ and $\eta$, have the values $\xi_s$ and $\eta_s$. That means $\xi_s$ and $\eta_s$ have the values given by equations \eqref{k224} and \eqref{k225} with substitution $r = r_s$. Then $r_s$ is a double root of R $= 0$; and R is reduced to
\begin{equation}\label{k229}
  \text R = \left(r-r_s\right)^2\left(r^2+2rr_s-a^2\eta_s/{r_s}^2\right) \,.
\end{equation}
Then the integral to be considered is
\begin{equation}\label{k230}
  \int\frac{\text dr}{\sqrt\text R} = \int\frac{\text dr}{(r-r_s) \sqrt{r^2 +2rr_s -a^2\eta_s/{r_s}^2}} \,.
\end{equation}
From this equation, it can be seen that the trajectory consists of two parts: a~part exterior to $r = r_s$ and a part interior to $r = r_s$. Because $\eta_s$ is positive the orbit will terminate before $r = 0$ (\cite[section 63\S(c)]{chandrasekhar}.

Substituting
\begin{equation}\label{k233}
  x = \frac 1{r-r_s} \,,
\end{equation}
and integrating \eqref{k230}, the equation of  the projection of the orbit on the $(r, \theta = \arccos\mu)$-plane is found. Thus,
\begin{equation}\label{k234}
  \int\frac{\text d\mu}{\sqrt{\Theta_\mu}} = \pm\frac{\ln\left[4r_s +2cx +2\sqrt{c(1+4r_sx+cx^2)}\right]}{\sqrt c} \,,
\end{equation}
where (\cite[page 353]{chandrasekhar})
\begin{equation}\label{k235}
  c = 3{r_s}^2 -a^2\eta_s/{r_s}^2 > 0 \,.
\end{equation}
The integral over $\mu$ on the left-hand\footnote{see \cite[page 353]{chandrasekhar}} side is given by equation \eqref{k201}.

In the first picture of the figure \ref{nulcirc2} there is an example of a trajectory derived from equation \eqref{k234}.


\subsection{The case $\eta = 0$}
All orbits of constant radius with $\eta = 0$ lie entirely in the equatorial plane. Also, all orbits in the equatorial plane have the Carter constant being zero. But \emph{not all orbits with $\eta = 0$ are in the equatorial plane}, because when $\eta = 0$, the $r$-motion is generally determined by (cf. equation \eqref{k214})
\begin{equation}\label{k236}
  \int\frac{\text dr}{\sqrt\text R} = \int\frac{\text dr}{\sqrt{r\left[r^3+(a^2-\xi^2)r+2M\left(a-\xi\right)^2\right]}} \,,
\end{equation}
and the $\theta$-motion is determined by equation \eqref{k203}. The above integral can be reduced to a Jacobian elliptic integral. Then the trajectory in the $(r, \theta)$-plane is determined by
\begin{equation}\label{k237}
  \frac{\arcsech\frac\mu{\mu_\text{max}}}{\sqrt{a^2-\xi^2}} = \pm\frac{\F(\psi, k)}{\sqrt{AB}} \,,
\end{equation}
where
\bal
    k^2 &= \frac{\left(A+B\right) -r_1^2}{4AB} \,,\label{k237k}\\
    \cos\psi &= \frac{(A-B)r+r_1A}{(A+B)r+r_1A} \,,\label{k237psi}\\
    A^2 &= r_1^2 +\left(a^2-\xi^2\right) \,,\label{k237A}\\
    B^2 &= 3r_1^2 +\left(a^2-\xi\right) \,,\label{k237B}\\
    r_1 &= -\sqrt{\frac43\left(a^2-\xi^2\right)}\sinh\vartheta \label{k237r1}\\
    \intertext{and}
    \sinh(3\vartheta) &= \sqrt[3]{27M^2\frac{a-\xi}{\left(a+\xi\right)^3}} \,.\label{k237teta}
\eal
These trajectories exist only for $\color{blue}\eta < a$. They terminate at the singularity (see figure \ref{nul1} where there is an example of a trajectory derived from equation \eqref{k237}).
\begin{figure}[h]
         \epsfxsize=13cm
         \epsfbox{kerr/null1.eps}
         \caption{An example of a null geodesic with $\eta = 0$ in the $(r, \theta)$-plane (projection on the plane). These are the only orbits, out of the equatorial plane, which reach the singularity. Parameters are $M = 1$, $E=1$, $a=0.8$ and  $L=0.4$.}
         \label{nul1}
\end{figure}

\subsection{The case $\eta < 0$}
In this section the class of orbits with $\eta < 0$ are  considered. These orbits extend to regions of negative $r$. They have very strict conditions of occurrence: $\abs\eta < a^2$ and $0 < \abs\xi \leq \abs a -\sqrt{\abs\eta}$. \textcolor{blue}{The result of integration is
\bal
    \int\limits_0^r \frac{\text dr}{\sqrt R} &= g[\text F(\phi,k)+\text K(k)] \,, \label{k236neg}\\
    \intertext{where}
    g &= \frac2{A+B} \,, \label{k236negg}\\
    \phi &= \arctan\frac{r-b_1+a_1g_1}{a_1+b_1g_1-g_1r} \,, \label{k236negfi}\\
    g_1^2 &= \frac{4a_1^2-\left(A-B\right)^2} {\left(A+B\right)^2-4a_1^2} \,, \label{k236negg1}\\
    k^2 &= \frac{4AB}{\left(A+B\right)^2} \,, \label{k236negk}\\
    A^2 &= \left(b_1-b_2\right)^2+\left(a_1+a_2\right)^2 \,, \label{k236negA}\\
    B^2 &= \left(b_1-b_2\right)^2+\left(a_1-a_2\right)^2 \,, \label{k236negB}\\
    a_1^2 &= -\frac{\left(u_1-\bar{u_1}\right)^2}4 \,, \label{k236nega1}\\
    b_1 &= \frac{u_1+\bar{u_1}}2 \,, \label{236negb1}\\
    b_2 &= \frac{u_3+\bar{u_3}}2 \,, \label{k236negb2}\\
    a_2^2 &= -\frac{\left(u_3-\bar{u_3}\right)^2}4 \,, \label{k235nega2}
\eal
and $u_1,\ \bar{u_1},\ u_3$ and $\bar{u_3}$ are the roots of the polynomial R$ = r^4 +(a^2-\xi^2-\eta)r^2 +2M[\eta+(\xi-a)^2]r -a^2\eta$.} The $r$- and the $\theta$- motions are shown in Figure \ref{null_Cart}\footnote{Cf. \cite[Figure 37]{chandrasekhar}}.
\begin{figure}[b]
         \epsfxsize=13cm
         \epsfbox{kerr/null_Cart.eps}
         \caption{An example of a null geodesic with $\eta = -0.3$ in the $(r, \theta)$-plane (projection on the plane). The orbit is confined between two angles. Parameters are $M = 1$, $\xi = -0.05=1$ and $a=0.8$.}
         \label{null_Cart}
\end{figure}



\textcolor{blue}{\section{The $\varphi$-motion}\label{nullphi}
In the second integral in the equation \eqref{kch181} we replace $\theta$ by $\mu = \cos\theta$ as the variable of integration (as in the section \ref{nulteta} before) and consider, instead, the integral
\begin{equation}\label{knulImu}
  I_\mu = \int\frac{L+aE(\mu^2-1)}{(\mu^2-1)\sqrt{\Theta_\mu}} \text d\mu \,,
\end{equation}
where $\Theta_\mu$ is the same as in \eqref{k197}.}


\textcolor{blue}{\subsection{$\eta > 0$}
With $\Theta_\mu$ in the form \eqref{k199}, explicit expression for the integral \eqref{knulImu} can be written down:
\begin{equation}\label{213}
  \int\limits_\mu^{\mu_+} \frac{L+aE(\mu^2-1)}{(\mu^2-1)\sqrt{\Theta_\mu}} \text d\mu = g\left[E\F(\psi, k) -\frac L{a(1-\mu^2)}\Pi\left(\psi, \alpha^2, k\right)\right] \,,
\end{equation}
where
\begin{equation}
    g = 1/\sqrt{{\mu_+}^2+{\mu_-}^2} \,,
\end{equation}
$k$ and $\psi$ are the same as in \eqref{k201}, and $\mu_\pm$ are defined in equations \eqref{k200+}, \eqref{k200-}.}

\textcolor{blue}{\subsubsection{The unstable orbits of constant radius}
After substituting \eqref{k229} for R, the first integral in the equation \eqref{kch181} will have the form
\begin{equation}\label{knulIR}
  I_{\text R} = a \int^r\frac{(r^2+a^2)E -aL}{\varDelta(r-r_s)\sqrt{r^2-2rr_s-a^2\eta_s/{r_s}^2}} \text dr  \,.
\end{equation}
This integral can be reduced to three simple integrals by the method of partial fractions. After letting
\begin{equation}\label{knullsubs}
  y_\pm = r_\pm -r \qquad \text{and} \qquad y_s = r -r_s \,,
\end{equation}
we can write
\begin{multline}\label{knulres}
  \varphi = I_\mu + a\left[\frac{({r_+}^2+a^2)E-aL}{(r_+-r_-)(r_+-r_s)}f_+ -\frac{({r_-}^2+a^2)E-aL}{(r_+-r_-)(r_--r_s)}f_-+\right. \\ \left.+\frac{({r_s}^2+a^2)E-aL}{(r_+-r_s)(r_--r_s)}f_s\right] \,,
\end{multline}
where $I_\mu$ is given by \eqref{213} and
\bal
    f_\pm &=
    \begin{cases}
        -\cfrac1{\sqrt c}\ln\left|\cfrac{2\sqrt c}{y_\pm}\sqrt{{y_\pm}^2+2(r_s-r_\pm)y_\pm+c} +\cfrac{2c}{y_\pm} +2(r_s-r_\pm)\right| \ \text{for} \ c > 0 ,\\
        \cfrac1{\sqrt{-c}} \arcsin\cfrac{2 [(r_s-r_\pm)y_\pm+c]}{y_\pm\sqrt{{r_s}^2+\cfrac{a^2\eta}{{r_s}^2}}} \qquad \text{for }c < 0,\\
-\cfrac1{r_s-r_\pm}\sqrt{\cfrac{y_\pm+2(r_s-r_\pm)}{y_\pm}} \qquad \text{for }c = 0,
    \end{cases}\\
    f_s &= \frac{r_s}{\sqrt{r_s^4+a^2\eta}} \arccos\frac{\sqrt{r_s^4+a^2\eta}}{r_s\abs{y_s}} \,,
\eal
where
\begin{equation}\label{knulcirc_c}
  c = {r_\pm}^2 -\frac{a^2\eta}{{r_s}^2} -2r_\pm r_s \,.
\end{equation}}

\textcolor{blue}{An example of a trajectory derived from equation \eqref{knulres} is illustrated in the second picture of figure \ref{nulcirc2}. The trajectory derived from equations \eqref{k234} and \eqref{knulres} is illustrated in figure \ref{nulcirc1}.}


\textcolor{blue}{\subsection{$\eta = 0$}
With $\Theta_\mu$ in the form \eqref{k202Teta}, an explicit expression for the integral \eqref{knulImu} can be written down:
\begin{multline}\label{Imu}
  \int\limits_\mu^{\mu_\text{max}} \frac{L+aE(\mu^2-1)}{(\mu^2-1)\sqrt{\Theta_\mu}} \text d\mu = \frac{L-aE}{\sqrt{a^2-\xi^2}} \ln\left|\frac{\mu_\text{max}+\sqrt{{\mu_\text{max}-\mu^2}}}\mu\right|+ \\ +\frac L{2\xi}\left[\arcsin\frac{{\mu_\text{max}}^2-\mu}{\mu_\text{max}(\mu-1)} +\arcsin\frac{{\mu_\text{max}}^2+\mu}{\mu_\text{max}(\mu+1)}\right] \,,
\end{multline}
where $\mu_\text{max}$ is defined in \eqref{k202max}.}

\textcolor{blue}{The first integral in the equation \eqref{kch181} will have the form
\begin{equation}\label{260}
  I_{\text R} = a\int^r\frac{(r^2+a^2)E-aL}{\varDelta \sqrt{r\left[r^3+(a^2-\xi^2)r+2M\left(a-\xi\right)^2\right]}}\text dr \,.
\end{equation}
This integral can be reduced, with the help of the partial fractions method, to Jacobian elliptic integrals. We can write
\begin{multline}\label{261}
  \varphi = I_\mu +aEg[\F(\phi, k) +\K(k)]+\\ +\frac{ag(B-A)}{r_++r_-}\Biggl\{\left[\left(a^2 +{r_+}^2\right)E -aL\right]\cdot\biggr.\\
\cdot \frac{\alpha_2[\F(\phi, k) +K(k)] +\cfrac{\alpha_+-\alpha_2}{1-\alpha_+}\Biggl[\Pi\Biggl(\phi, \cfrac{{\alpha_+}^2}{{\alpha_+}^2-1}, k\Biggr) +\Pi\Biggl(\cfrac{{\alpha_+}^2}{{\alpha_+}^2-1}, k\Biggr) -2\alpha_+f_+\Biggr]}{Ar_1-r_+(A+B)} \\
-\left[\left(a^2 +{r_-}^2\right)E -aL\right]\cdot\\
\left.\cdot\frac{\alpha_2[\F(\phi, k) +K(k)] +\cfrac{\alpha_--\alpha_2}{1-\alpha_-}\Biggl[\Pi\Biggl(\phi, \cfrac{{\alpha_-}^2}{{\alpha_-}^2-1}, k\Biggr) +\Pi\Biggl(\cfrac{{\alpha_-}^2}{{\alpha_-}^2-1}, k\Biggr) -2\alpha_-f_-\Biggr]}{Ar_1-r_-(A+B)}\right\},
\end{multline}
where $k$ is defined in \eqref{k237k}, $A$ in \eqref{k237A}, B in \eqref{k237B}, $f_\pm$ in \eqref{241f}, $r_\pm$ are the roots of $\varDelta$, and
\bal
    g &= 1/\sqrt{AB} \,,\\
    \phi & = \arccos\frac{(A-B)r-Ar_1}{(A+B)r-Ar_1} \,\\
    r_1 &= -\sqrt{\frac43\left(a^2-\xi^2\right)}\sinh\frac\vartheta3 \,,\\
    \sinh\vartheta &= M\sqrt{\frac{a-\xi}{\left(a+\xi\right)^3}} \,,\\
    \alpha_\pm &= \frac{Ar_1-Ar_\pm-Br_\pm}{-Ar_1+Ar-\pm_Br_\pm} \,,\\
    \alpha_2 &= \frac{B+A}{B-A} \,.
\eal}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/null2.eps}
         \caption{An example of a null geodesic with $\eta = 0$ in the $\color{blue}(r, \varphi)$-plane. The orbit spirals to the singularity. Parameters are $M = 1$, $E=1$, $a=0.8$ and  $\color{blue}L=0.4$.}
         \label{nul2}
\end{figure}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/null3.eps}
         \caption{\color{blue}An example of a null geodesic in ``spherical" coordinates with
         $\eta = 0$. The orbit spirals to the singularity. Parameters are $M = 1$, $E=1$, $a=0.8$ and  $L=0.4$.}
         \label{nul3}
\end{figure}

\textcolor{blue}{An example of a trajectory derived from equation \eqref{261} is illustrated in the figure \ref{nul2}. The trajectory derived from equations \eqref{k237} and \eqref{261} is illustrated in figure \ref{nul3}.
}


{\color{blue}\subsection{$\eta < 0$}
The function $\Theta_\mu$ has form \eqref{k204}. Therefore
\begin{equation}\label{218.00}
    \int\limits_\mu^{\mu_+} \frac{L+aE(\mu^2-1)}{(\mu^2-1)\sqrt{\Theta_\mu}} \text d\mu = g\left[E\text F(\psi, k) +\frac L{a(\mu_+^2-1)}\Pi\left(\psi, \alpha^2, k\right)\right] \,,
\end{equation}
where
\bal
    g &= \frac1{\mu_+} \,,\\
    \psi &= \arcsin\sqrt\frac{\mu_+^2-\mu^2}{\mu_+^2-\mu_-^2} \,,\\
    k^2 &= \frac{\mu_+^2-\mu_-^2}{\mu_+^2}\\
    \intertext{and}
    \alpha^2 &= \frac{\mu_+^2-\mu_-^2}{\mu_+^2-1} \,.
\eal

The first integral in \eqref{kch181} can be written in the form
\begin{equation}\label{267.05}
  E \int\limits_0^r \frac{\text dr}{\sqrt R} +\int\limits_0^r \frac{(2EMr-aL)\text dr}{\varDelta\sqrt R} \,,
\end{equation}
where R is defined in \eqref{k190} and $\varDelta$ is the horizon function \eqref{k45}. %Therefore
%\begin{equation}\label{276.05}
%  \varphi = ag\left\{E[\text F(phi, k) +K(k)
%\end{equation}
The first integral in the expression above is \eqref{k236neg}. An example of an orbit with negative $\eta$ derived numerically is in figures \ref{nuetapr} and \ref{nuneta3D}.
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/nuetapr.eps}
         \caption{\color{blue}An example of a null geodesic with $\eta < 0$. Projection on the $r$-$\varphi$ plane. Parameters are $M = 1$, $E=1$, $a=0.8$ , $\eta = -0.3$ and  $\xi = 0.4$.}
         \label{nuetapr}
\end{figure}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/nuneta3D.eps}
         \caption{\color{blue}An example of a null geodesic with $\eta < 0$ in ``spherical"
         coordinates. Parameters are $M = 1$, $E=1$, $a=0.8$ , $\eta = -0.3$ and  $\xi = 0.4$.}
         \label{nuneta3D}
\end{figure}
