% 1319 slov
\chapter{The geodesics in the equatorial plane}
\label{equator}
\sloppy
In the early
study of Kerr space-time equatorial geodesics were emphasized, because when compared with Schwarzschild geodesics the consequences of rotation stand out.

While geodesics in the equator are included in the broader class of
geodesics with Carter constant $Q = 0$, orbital
consequences are radically different for the equatorial and
non-equatorial cases.

Motion in the equatorial plane is described by Lagrangian
\begin{equation}\label{k62n}
  2{\cal L} = \left(1 -\frac{2M}r\right) \dot t^2 +
  \frac{4aM}r\dot t\dot\varphi -\frac{r^2}\varDelta\dot r^2 -
  \left[(r^2 +a^2) +\frac{2a^2M}r\right]\dot\varphi^2 \,;
\end{equation}
and we deduce from it that the generalized momenta are given by
\begin{equation} \label{k63n}
    p_t = \left(1-\frac{2M}r\right)\dot t +\frac{2aM}r\dot\varphi
    = E = \mbox{constant} \,,
\end{equation}
\begin{equation}\label{k64n}
  -p_\varphi = -\frac{2aM}r\dot t +\left(r^2 + a^2
  +\frac{2a^2M}r\right)\dot\varphi = L = \mbox{constant}
\end{equation}
and
\begin{equation}\label{k65n}
  -p_r = \frac{r^2}\varDelta\dot r \,.
\end{equation}
Stationary and axisymmetric character of the Kerr Geometry is evident from the fact that the Lagrangian does not depend on $t$ or $\varphi$. Therefore $p_t$ and $p_\varphi$ are constant.

The Hamiltonian is %given by
\begin{multline}\label{k66n}
  {\cal H} = p_t\dot t -p_\varphi\dot\varphi +p_r\dot r -{\cal L}
  = \\
  =\frac12\left(1 -\frac{2M}r\right)\dot t^2 +\frac{2aM}r\dot
  t\dot\varphi -\frac{r^2}{2\varDelta}\dot r^2 -\frac12\left(r^2
  +a^2 +\frac{2a^2M}r\right)\dot\varphi^2 \,; %\nonumber
\end{multline}
and from the independence of the Hamiltonian on $t$%, we deduce that
\begin{multline}\label{k67n}
  2{\cal H} = \\
  =\left[\left(1 -\frac{2M}r\right)\dot t
  +\frac{2aM}r\dot\varphi\right]\dot t -\left[\left(r^2 +a^2
  +\frac{2a^2M}r\right)\dot\varphi -\frac{2aM}r\dot
  t\right]\dot\varphi -\frac{r^2}\varDelta\dot r^2 \\
  = E\dot t
  -L\dot\varphi -\frac{r^2}{\varDelta}\dot r^2 =\delta_1 =
  \mbox{constant} \,.
\end{multline}

Solving equations (\ref{k63n}) and (\ref{k64n}) for $\dot\varphi$
and $\dot t$%, we obtain
\bal%egin{eqnarray}
    \dot\varphi &= \frac1\varDelta\left[\left(1
    -\frac{2M}r\right)L +\frac{2aM}rE\right] \,, \label{k69n}\\
    \dot t &=  \frac1\varDelta\left[\left(\dot r^2 + a^2
    +\frac{2a^2M}r\right)E -\frac{2aM}rL\right] \,; \label{k70n}
\eal%nd{eqnarray}
and inserting these solutions in the third line of equation
(\ref{k67n}), the radial equation can be deduced
\begin{equation}\label{k71n}
  r^2\dot r^2 = r^2E^2 +\frac{2M}r\left(aE -L\right)^2 +\left(a^2E^2
  -L^2\right)
  -\delta_1\varDelta \,.
\end{equation}
We write this as the following corollary
\begin{corollary}\label{4.14.1}
    For a geodesic $\gamma$ with $Q = 0$, we have \emph R$(r) =
    r\psi(r)$ where the function $\psi$ is given by
    \[ \psi(r) = E^2r^3 +\left(a^2E^2 -L^2\right)r +2M\left(L
    -aE\right)^2 -\delta_1r\varDelta \,.\]
\end{corollary}

When $Q = 0$, $r$-$L$ plots are determined solely by $E$, thus the $e$-$Q$ chart (figure \ref{4.8}) shows that time-like $r$-$L$ plots have either bay or continents configurations and that null and space-like $r$-$L$ plots have continents configurations (figures \ref{4.8null} and \ref{4.8space} respectively).

The previous results from section \ref{section 4.7} can be written down as follows.
\begin{corollary}\label{4.14.3}
    Let $\gamma$ be a $Q = 0$ geodesic, with $\psi$ as in Corollary \ref{4.14.1}.
\begin{enumerate}
  \item As a polynomial in $L$,
    \begin{equation}\label{4.14.3-1}
        \psi(r, L) = (2M -r)L^2 -4MaEL +\psi_0(r) \,,
    \end{equation}
    where $\psi_0(r) = \psi(r, L = 0) = E^2(r^3 +a^2r +2Ma^2) -\delta_1r\varDelta(r)$.
  \item If $r \neq 2M$, the discriminant of $\psi$ as a polynomial in $L$ is {\emph 4D}$(r) = r\varDelta(r)(E^2r -\delta_1r +2M\delta_1)$.
  \item The set $Z$ of zeros of $\psi(r, L)$ (or of \emph R$(r,L)$) is the union of the graphs of the two functions
    \begin{equation}
        L_\pm = \frac{2MaE \pm\sqrt{r\varDelta(E^2r -\delta_1r +2M\delta_1)}}{2M -r}
    \end{equation}
    defined on maximal $r$ intervals on which \emph D$(r) \geq 0$.
\end{enumerate}
\end{corollary}

Therefore, after the reduction of R to $\psi$, the discriminant polynomial $\phi$ of section \ref{section 4.7} is replaced  by the linear expression $(E^2 -\delta_1)r +2M\delta_1$. As in section \ref{section 4.7} the discriminant is essential in determining the configuration of the set $Z$, because (for $r \neq 2M$) the sign $(-,0,+)$ of D$(r)$ establishes whether there are zero, one, or two points, respectively, of R = 0 on the vertical line $L \rightarrow (r, L)$.

For the exceptional line $r = 2M$, the function $L \rightarrow \psi(2M, L)$ is linear with negative slope for $E > 0$. Therefore as $L$ increases, the line $r = 2M$ enters a forbidden region at $L_{2M} = \frac{\psi(r, 0)}{4MaE}$ and remains there. (When $E = 0$, $\psi$ is constant on $r= 2M$.)

\begin{proposition}
    In a $Q = 0$ $r$-$L$ plot with $E^2 -\delta_1 \neq 0$ and $E \neq 0$, the only vertical lines that contain a unique point $(r, L_r)$ of $Z$ are $r = 0$, $r_-$, $r_+$, $2M$, or $w = \frac{2M\delta_1}{\delta_1 -E^2}$. The corresponding angular momenta are $L_{2M}$ (as above) for $r = 2M$, and $L_r = \frac{2MaE}{2M-r}$ for the others. Explicitly,
    \begin{equation*}
        L_0 = aE \,, \qquad L_\pm = 2MaE/r_\mp \,, \qquad L_w = a\left(E^2 -\delta_1\right) \,.
    \end{equation*}
\end{proposition}
For proof see \cite[page 275]{onil}.


\section{The null geodesics}
For null geodesics $\delta_1 = 0$ and the radial equation
(\ref{k71n}) becomes
\begin{equation}\label{k72n}
  \dot r^2 = E^2 +\frac{2M}{r^3}\left(L -aE\right)^2 -\frac1{r^2}\left(L^2
  -a^2E^2\right)= \mbox f(r)
  \,.
\end{equation}

As in Chapters \ref{n} and \ref{RNnull}, it will be more convenient to
distinguish the geodesics by the impact parameter (\ref{ip})
rather than by $L$. So a single $r$-$D$ plot is sufficient to characterize the radial motion of all equatorial null geodesics.

Corollaries \ref{4.14.1} and \ref{4.14.3} lead to the following corollary.
\begin{corollary}
    For $Q = 0$ null geodesics with $E \neq 0$,
    \begin{equation}
        \psi = \text{\emph R}/r = r^3 +\left(a^2 -D^2\right)r +2M\left(D -a\right)^2 \,.
    \end{equation}
    The zero set $Z:\ \psi = 0$ is the union of the graphs of the two functions%\footnote{\cite[Corollary 4.14.5]{onil} restricts $\varDelta > %0$}
    \begin{equation}
        D_\pm(r) = \frac{2Ma \pm r\sqrt\varDelta}{2M -r} \, \qquad \text{where} \qquad \varDelta > 0.
    \end{equation}
\end{corollary}
Therefore there are two points of $Z$ on each vertical line in the strip $0 < r < r_-$ and in $r > 2M$, one on the lines $r = r_\pm$ and $r = 2M$, and none between the horizons. Like in the non-equatorial case (section \ref{continets}), the $r$-$D$ plot has two continental forbidden regions $N$ and $S$ in the exterior $r > r_+$, but in the strip $0 < r < r_-$ the forbidden region is a ``tongue" $W$, first observed by \cite{boyer-lindquist} (although their ``tongue" pointed ``southward" not ``northward"). $W$ starts from $r = 0$ at the $(0, a)$ and ends in $r = r_-$. The principal line $D = a$ (corresponding to $L = aE$) is tangent to $W$ from below (see figures \ref{4.26slow}, \ref{4.26extreme} and \ref{4.26fast} and cf. with \cite[figure 8]{boyer-lindquist}).
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/r-Leqslo.eps}
         \caption{The $r$-$D$ plot for equatorial null geodesics in the slow Kerr black hole. Forbidden regions are $N$, $S$ and $W$; their boundaries define the turning points for null rays. ($M = 1,\ a = 0.96$)}
         \label{4.26slow}
\end{figure}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/r-Leqext.eps}
         \caption{The $r$-$D$ plot for equatorial null geodesics in the extreme Kerr black hole. Forbidden regions are $N$, $S$ and $W$; their boundaries define the turning points for null rays. ($\color{blue}a = M = 1$)}
         \label{4.26extreme}
\end{figure}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/r-Leqfas.eps}
         \caption{The $r$-$D$ plot for equatorial null geodesics in the fast Kerr black hole. Forbidden regions are $N$, $S$ and $W$ (regions $W$ and $N$ merge); their boundaries define the turning points for null rays. ($M = 1,\ a = 1.04$)}
         \label{4.26fast}
\end{figure}

As figures \ref{4.26slow}, \ref{4.26extreme} and \ref{4.26fast} show, a null geodesics $\gamma$ coming in from $r = +\infty$ has flyby orbit if $\abs D$ is large ($\abs L \gg E$) and may hit the ring singularity if $\abs D$ is small. But because of the tongue, in the slow Kerr black hole, if $D$ is just larger than the principal level $D = a$, it meets the tongue in an $r$-turning point, and so the geodesic has a long flyby orbit (\cite{onil}); \textcolor{blue}{in the fast Kerr black hole the tongue meets the continent $N$, thus a null geodesics with $D > a$ does not reach the singularity}.



\subsection{The special case $D = a$}\label{keqnulrad}
Geodesics with impact parameter
\begin{equation}\label{k74n}
  D = a \,, \quad \mbox{when} \quad L = aE \,,
\end{equation}
have, in the rotating black hole, the same role as the radial
geodesics in the Schwarzschild and in the Reissner-Nordstr\"om
geometry. They meet the ring singularity and have crash-escape
orbit type. And so equations (\ref{k69n}),
(\ref{k70n}), and (\ref{k72n}) are reduced to
\begin{equation}\label{k75n}
  \dot r = \pm E \,, \qquad \dot t = \frac{(r^2 +a^2)E}\varDelta
  \,, \qquad \mbox{and} \qquad \dot\varphi = \frac{aE}\varDelta
  \,.
\end{equation}
The radial coordinate is described uniformly with respect to the
affine parameter, and the equations governing $t$ and $\varphi$
are
\begin{equation}\label{k76n}
  \frac{\mbox dt}{\mbox dr} = \pm\frac{r^2 +a^2}\varDelta \qquad
  \mbox{and} \qquad \frac{\mbox d\varphi}{\mbox dr} =
  \pm\frac a\varDelta \,.
\end{equation}


\subsubsection{Slow black hole}
The solutions of equations (\ref{k76n}) are
\bal%egin{eqnarray}
  \pm t &= r +\frac{r^2_+ +a^2}{r_+ -r_-}\ln\left(\frac r{r_+}
  -1\right) -\frac{r_-^2 +a^2}{r_+ -r_-}\ln\left(\frac r{r_-}
  -1\right) \,,\label{k77nst}\\
  \pm\varphi &= \frac a{r_+ -r_-}\ln\left(\frac r{r_+}
  -1\right) -\frac a{r_+ -r_-}\ln\left(\frac r{r_+}
  -1\right) \,.\label{k77nsfi}
\eal%nd{eqnarray}
They display the typical behaviors of $t$ and
$\varphi$ of approaching $\pm\infty$ near the horizons $r_+$ and
$r_-$.


\subsubsection{\color{blue}Extreme black hole}
\textcolor{blue}{The solutions of equations (\ref{k76n}) are
\bal\label{k77net}
    \pm t &= r -r_\text{in} +2M^2\left(\frac1{r_\text{in}-M}
    -\frac1{r-M}\right)
    +2M\ln\left|\frac{r-M}{r_\text{in}-M}\right|\,,\\
    \label{k77nefi}
    \pm \varphi &= M\left(\frac1{r_\text{in}-M}
    -\frac1{r-M}\right)
\eal with initial condition $r = r_\mathrm{in}$ for $t = \varphi =
0$.}


\subsubsection{Fast black hole}
\textcolor{blue}{The solutions of equations (\ref{k76n}) are
\begin{multline}\label{k77nft}
    \pm t
    = r -r_\mathrm{in}
    +M\ln\left|\frac\varDelta{\varDelta_\mathrm{in}}\right| +\\
    +\frac{2M}{\sqrt{a^2 -M^2}} \left(\arctan\frac{r -M}{\sqrt{a^2 -M^2}}
    -\arctan\frac{r_\mathrm{in} -M}{\sqrt{a^2 -M^2}}\right)
\end{multline}
and
\begin{equation}\label{k77nffi}
    \pm\varphi = a\left(\arctan\frac{r -M}{\sqrt{a^2 -M^2}}
    -\arctan\frac{r_\mathrm{in} -M}{\sqrt{a^2 -M^2}}\right)
\end{equation}
with initial condition $r = r_\mathrm{in}$ for $t = \varphi = 0$.}


\subsection{General orbits}\label{eqnulgen}
Unlike in the general $Q > 0$ case, it is now simple to find the horizontal tangents to $Z$, which represent the circular photons--and therefore to specify all null orbits types.

\begin{proposition}\label{4.14.6slow}
    In the slow Kerr equator there are, aside from restphotons, exactly three circular null orbits. For $E > 0$, these have radii and impact parameters $(r_\tau, D_\tau),\ (r_\rho, D_\rho),\ (r_\delta, D_\delta)$ with
\begin{gather*}
  0 < r_\tau < r_- < r_+ < r_\delta < 3M < r_\rho < 4M \,,\\ \intertext{and} D_\rho < -7a < 0 < a < D_\tau < 2a < D_\delta \,.
\end{gather*}
    The radii are the roots of the polynomial \emph p$(r) = r(r-3M)^2 -4Ma^2$, and the impact parameters are the roots of \emph q$(D) = (D +a)^3 -27M^2(D -a)$.
\end{proposition}
For proof see \cite[page 278]{onil}\footnote{with correction that p$(4M) = 4M^3 -4Ma^2$}.
\begin{proposition}\color{blue}\label{4.14.6extreme}
    In the extreme Kerr equator there are, aside from restphotons, exactly three circular null orbits. For $E > 0$, these have radii and impact parameters $(r_\tau, D_\tau),\ (r_\rho, D_\rho),\ (r_\delta, D_\delta)$ with
\begin{gather}\label{4.14.16r-extreme}
  0 < r_\tau = r_\pm = r_\delta < 3M < r_\rho = 4M \,,\\ \intertext{and} D_\rho = -7a < 0 < a < D_\tau = 2a = D_\delta \,.
\end{gather}
    The radii are the roots of the polynomial \emph p$(r) = r(r-3M)^2 -4Ma^2$, and the impact parameters are the roots of \emph q$(D) = (D +a)^3 -27M^2(D -a)$.
\end{proposition}

\begin{proof}
    The circular orbits occur at simultaneous solutions $(r, D)$ of the equations $\psi = 0,\ \frac{\partial\psi}{\partial r} = 0$. The second equations yields to
\begin{equation}\label{onil278-1}
  D^2 = 3r^2 +a^2 \,.
\end{equation}
Substituting $r^2 = \frac{D^2 -a^2}3$ into $\psi(r, D) = r^3 +(a^2 -D^2)R +2M(D -a)^2 = 0$ gives $r(a^2 -D^2) +3M(D -a)^2 = 0$. Therefore
\begin{equation}\label{onil278-2}
  r = \frac{3M(D -a)}{D +a} \,.
\end{equation}

Equations \eqref{onil278-1} and \eqref{onil278-2} are the basic relations between the radius and the impact parameter of a~circular null geodesic. Substituting equation \eqref{onil278-2} into $\frac{\partial\psi}{\partial r} = 3r^2 +a^2 -D^2 = 0$ leads to $27M^2\frac{(D -a)^2}{(D +a)^2} +a^2 -D^2 = 0$. By omitting the principal null geodesics $D = a$, $D -a$ is cancelled from the preceding equation. That leaves
\begin{equation}\label{onil278-3}
  \text q(D) = \left(D +a\right)^3 -27M^2(D -a) = 0 \,.
\end{equation}
Solving equation \eqref{onil278-2} for $D = -a\frac{r +3M}{r -3M}$, and substituting this into equation \eqref{onil278-3} yields
\begin{equation}\label{onil278-4}
  \text p(r) = r\left(r -3M\right)^2 -4Ma^2 \,.
\end{equation}
This polynomial has no negative roots, and p$(0) < 0$, p$(3M) < 0$, \textcolor{blue}{p$(4M) = 0$}. The placing of $r_\pm$ derives from
\begin{equation}\label{onil278-5}\color{blue}
  \text p(r_\pm) = \text p(M) = 0 \,.
\end{equation}
It follows that p$(r)$ has three roots obeying equation \eqref{4.14.16r-extreme}. \textcolor{blue}{$D(r_\rho) = D(4M)$, which gives $D_\rho = -7a$.} The function $D(r) = a\frac{3M +r}{3M -r}$ is strictly increasing on $\{r < 3M\}$ and $\{r > 3M\}$, \textcolor{blue}{hence $D(0) < D(r_\tau) = D(M) = D(r_\delta)$, which gives $a < D_\tau = 2a = D_\delta$}.
\end{proof}

\begin{proposition}\color{blue}\label{4.14.6fast}
    In the fast Kerr equator there is, aside from restphotons, exactly one circular null orbit. For $E > 0$, this has radius and impact parameter $(r_\rho, D_\rho)$ with
\begin{gather}
   r_\rho > 4M \,,\\ \intertext{and} D_\rho > -7a \,.
\end{gather}
    The radius is the real root of the polynomial \emph p$(r) = r(r-3M)^2 -4Ma^2$, and the impact parameter is the real root of \emph q$(D) = (D +a)^3 -27M^2(D -a)$.
\end{proposition}

\begin{proof}\color{blue}
    Evidently the polynomial \eqref{onil278-4} is approaching $+\infty$ with $r \rightarrow +\infty$, and p$(4M) = 4M^3 - 4Ma^2 < 0$. The discriminant of p$(r)$ is $432M^2a^2(M^2 -a^2) < 0$, so there is only one real solution, and it obeys the stated inequality. With the same reasoning as in previous proof, $D(r_\rho) > D(4M)$ gives $D_\rho > -7a$.
\end{proof}

Once the circular photons are found in the null $r$-$D$ plot, the equatorial null orbit types can be described. Principal null geodesics hit the ring singularity (\cite[chapter 4]{onil}). There are no bound null orbits except for (unstable) circular orbits. For $E \neq 0$ the remaining flyby and crash orbits can be characterized in terms of impact parameter, as follows:
\begin{enumerate}
  \item Flyby orbits (for $r$ large) or crash orbits (for $r$ small) iff  the impact parameter $D = L/E$ is in one of the intervals: $(-\infty, D_\rho),\ (a, D_\tau),\ (D_\delta, \infty)$ \textcolor{blue}{(for the fast black holes that means $D > a$ or $D < D_\rho$)}.
  \item Crash-escape orbit iff $D$ lies in the interval $(D_\rho, a]$ or $(D_\tau, D_\delta)$ \textcolor{blue}{(for the fast black holes only the first interval applies)}.
  \item Circular orbit (or associated asymptotic orbit) iff $D = D_\rho,\ D_\tau$, or $D_\delta$.
\end{enumerate}

In the remaining case $E = 0$, $\gamma$ is a restphoton if $L = 0$, other than that it has crash-crash orbit in $0 < r \leq 2M$. When $L = 0$, all the first integrals of $\gamma$ are zero, which is possible only for restphotons (\cite[chaper 4]{onil}). If $L \neq 0$, then (from \ref{4.14.3-1}) $\psi(r, L) = (2M -r)L^2$, and so $\gamma$ is in the strip $0 < r \leq 2M$. The $r$-equation is (from \ref{k71n}) $r^3\dot r^2 = L^2(2M -r)$, therefore $\gamma$ hits $\varSigma$ at both ends, with a turn at $r = 2M$.

The null geodesics are \emph{central} in the sense that, for large $E$, both space-like and the time-like sets $Z$ resemble that of the null case (\cite[chapter 4]{onil}).% In fact, if $\psi^\pm$ denotes the function in \ref{4.14.1} when $\delta_1 = \pm1$, then for $E^2 \gg 1,\ \psi^\pm(r, L)/E^2$ is close to $\psi(r, D) = L/E$

%obrazky + dodelat poradne ruzne pripady
%In general it is clear that we must distinguish, as in
%Schwarzschild's geometry, orbits with impact parameters greater
%than or less than a certain critical value $D_{\mathrm c}$. For $D
%= D_\mathrm c$, the geodesics equations will allow an unstable
%circular orbit of radius $r_\mathrm c$ (say). For $D > D_\mathrm
%c$, we shall have orbits of two kinds: those of the \emph{first
%kind} which, arriving from infinity, have perihelion distances
%greater than $r_\mathrm c$; and those of the \emph{second kind}
%which, having aphelion distances less than $r_\mathrm c$ terminate
%at the singularity at $r = 0$ (and $\theta = \pi/2$). For $D =
%D_\mathrm c$, the orbits of the two kind coalesce: they both
%spiral, indefinitely, about the same unstable circular orbit at $r
%= r_\mathrm c$. For $D < D_\mathrm c$, there are only orbits of
%one kind: arriving from infinity, they terminate at the
%singularity. When $a^2 > M^2$, such distinction exists only for
%retrograde orbits: \emph{all direct orbits, arriving from infinity
%with $D > a$, recede again into infinity.}

\textcolor{blue}{Now we will find the general solution for all three types of orbits. From (\ref{k72n}) it is clear that turning points
occur at the zeros of the cube polynomial
\begin{equation}\label{bl5.1}
  f(u) = 2Mu^3\left(L -aE\right)^2 -u^2\left(L^2 -a^2E^2\right) +E^2 \,.
\end{equation}
Then for orbits of the first kind (flyby orbits)
\begin{multline}\label{234.17}
    \tau = \frac g{k^4 u_3^4\sqrt{2M}} \biggl\{u_2 \bigl[\F(\phi, k) +\K(k)\bigr]+ \biggr.\\
+2 \left(k^2 u_3^2 -u_2^2\right) \left[\Pi\left(\phi, \frac{k^2u_3^4}{u_2^2}, k\right) +\Pi\left(\frac{k^2u_3^4}{u_2^2}, k\right)\right]+\\
+\frac{u_2^2 -k^2u_3^2}{2(k^4u_3^2-u_2^2)} \biggl[k^2 u_3^2 \E(\phi, k) +\left(u_2^2 -k^2u_3^2\right) v+\biggr.\\
+\left(2 k^4 u_3^2 2 k^2 u_3^2 +\frac{k^6u_3^2}{u_2^2} -3 u_2^2\right) \Pi\left(\phi, \frac{k^2u_3^4}{u_2^2}, k\right) -\frac{k^6u_3^4\sn v\cn v\dn v}{u_2^2-k^4u_3^2\sn^2 v}+\\
+k^2 u_3^2 \E(k) +\left(u_2^2 -k^2u_3^2\right) \K(k)+\\ \left.\left.+\left(2 k^4 u_3^2 2 k^2 u_3^2 +\frac{k^6u_3^2}{u_2^2} -3 u_2^2\right) \Pi\left(\frac{k^2u_3^4}{u_2^2}, k\right)\right]\right\} \,,
\end{multline}
where
\bal%egin{eqnarray}
    g &= \frac2{\sqrt{u_3 -u_1}} \label{g}\,,\\
    \phi &= \am v = \arcsin\sqrt\frac{(u_3-u_1)(u_2-u)}{(u_2-u_1)(u_3-u)} \,,\label{fi}\\
    k^2 &= \frac{u_2 -u_1}{u_3 -u_1} \label{k}\,,
\eal%nd{eqnarray}
for orbits of the second kind (crash orbits)
\begin{multline}\label{238.17}
  \tau = \frac{\sqrt2}{\alpha^4\left(u_3-u_1\right)^{3/2}\sqrt M} \biggl\{\F(\phi, k) -2\Pi\left(\phi, \alpha^2, k^2\right)+\biggr.\\
+\frac1{2(\alpha^2-1)(k^2-\alpha^2)} \biggl[\alpha^2 \E(\phi, k) +\left(k^2 -\alpha^2\right)\F(\phi, k)+\biggr.\\
\left.\left.+\left(2 \alpha^2 k^2 +2 \alpha^2 -\alpha^4 -3 k^2\right) \Pi\left(\phi, \alpha^2, k^2\right) -\frac{\alpha^4\sn v\cn v\dn v}{1-\alpha^2\sn^2v}\right]\right\} \,,
\end{multline}
where $u_1$, $u_2$ and $u_3$ are the roots of f$(u) = 2M(u-u_1)(u-u_2)(u-u_3)$, $k$ is defined in \eqref{k}, $g$ in \eqref{g}, and
\bal
    \alpha^2 &= \frac{u_1}{u_1-u_3} \,,\label{237alfa}\\
    \phi &= \am v = \arcsin\sqrt\frac{u_3-u_1}{u-u_1} \label{237fi}\,.
\eal
The solution for the crash-escape orbits can be also expressed in terms of elliptic integrals, but we do not write it here because it is long and not essentially different from the previous two solutions.
%\begin{equation}\label{241.04}
%  \tau = \frac g{\alpha(A+u_1)\sqrt{2M}}\left[-v +\frac{\Pi\left(\phi, %\cfrac{\alpha^2}{\alpha^2-1}, k\right)}{1-\alpha} -\alpha f_1\right] \,,
%\end{equation}
%where
%\bal
%    \\
%    \alpha &= \frac{A-u_1}{A+u_1} \,,\\
%
%
%
%
%    f_1 &= \frac{\ln\cfrac{\sqrt{k^2 +k'^2\alpha^2}\dn u %+\sqrt{\alpha^2-1}\sn u}{\sqrt{k^2 +k'^2\alpha^2}\dn u %-\sqrt{\alpha^2-1}\sn u}} 2 \sqrt\frac{\alpha^2-1}{k^2 +k'^2\alpha^2} %\,,\\
%
%
%\eal
%
}

%\vspace{-8mm}
\textcolor{blue}{We can write the solution of
\begin{equation}
    \varphi = \int\frac{D -2Mu(a -D)}{\varDelta(u)\sqrt{2M(a-D)^2u^3 +(a^2-D^2)u^2+1}}\text du \,,
\end{equation}
in the form
\begin{equation}\label{233.02}
  \varphi = \frac{gp}{(M^2-a^2)\sqrt{8M}} \left[\frac{\Pi\left(\phi, \alpha_+^2, k\right) +\Pi\left(\alpha_+^2, k\right)}{u_+ -u_1} -\frac{\Pi\left(\phi, \alpha_-^2, k\right) +\Pi\left(\alpha_-^2, k\right)}{u_--u_1}\right]
\end{equation}
for the orbits of the first kind (flyby orbits), where
\bal
    p &= 2M(a -D)\left(M^2 -a^2\right) +\left[a^2D -2M^2(a-D)\right]\sqrt{M^2 -a^2} \,,\\
    g  &= \frac2{u_3-u_1} \,,\label{234g}\\
    \phi &= \arcsin\sqrt\frac{u-u_1}{u_2-u_1} \label{234fi}\,,\\
    \alpha_+^2 &= \frac{u_2-u_1}{u_+-u_1} \,,\\
    \alpha_-^2 &= \frac{u_2-u_1}{u_--u_1} \,,
\eal
$k^2$ is defined in \eqref{k}, $u_+$ and $u_-$ are the roots of
\begin{equation}\label{Delta_u}
  \varDelta(u) = a^2u^2 -2M +1 \,;
\end{equation}
and
\begin{equation}\label{238.06}
  \varphi = \frac{gp}{(M^2-a^2)\sqrt{8M}} \left[\frac{\Pi\left(\phi, \alpha_-^2, k\right) -\F(\phi, k)}{u_- -u_1} -\frac{\Pi\left(\phi, \alpha_+^2, k\right) -\F(\phi, k)}{u_+-u_1}\right]
\end{equation}
for the orbits of the second kind (crash orbits), where $g$, $p$ and $k^2$ are same as before and
\bal
    \alpha_+^2 &= \frac{u_+-u_1}{u_3-u_1}\label{alfa+} \,,\\
    \alpha_-^2 &= \frac{u_--u_1}{u_3-u_1}\label{alfa-} \,,\\
    \phi &= \arcsin\sqrt\frac{u_3-u_1}{u-u_1}\label{238fi} \,,
\eal
and
\begin{equation}\label{keqnullim}
  \varphi = \frac{gp}{(M^2-a^2)\sqrt{8M}} \left[\frac{-v +\frac{\Pi\left(\phi, \frac{\alpha_-^2}{\alpha_-^2-1}, k\right) -\alpha_-f_-}{1+\alpha_-}} {\alpha_-(A+a-u_-)} -\frac{-v +\frac{\Pi\left(\phi, \frac{\alpha_+^2}{\alpha_+^2-1}, k\right) -\alpha_+f_+}{1+\alpha_+}} {\alpha_+(A+a-u_+)}\right]
\end{equation}
for crash-escape orbits, where $p$ is the same as before, and
\bal
    g &= 1/\sqrt A \,, \label{241g} \\
    \cn v &= \cos\phi \,,\label{241v}\\
    \phi & = \arccos\frac{u-u_1-A}{u-u_1+A} \,,\label{241phi}\\
    k^2 &= \frac{A +b_1-u_1}{2A} \,,\label{241k}\\
    A^2 &= (b_1-u_1)^2 +a_1^2 \,,\label{241A}\\
    b_1 &= \frac{u_2+\bar u_2}2 \label{241b1}\,,\\
    a_1^2 &= -\frac{\left(u_2 -\bar u_2\right)^2}4\label{241a1} \,,\\
    \alpha_- &= \frac{A-u_1+u_-}{A+u_1-u_-}\label{241alfa-} \,,\\
    \alpha_+ &= \frac{A-u_1+u_+}{A+u_1-u_+}\label{241alfa+} \,,\\
    f_\pm &=
    \begin{cases}
        \arctan\left(\sqrt{\cfrac{k^2 +k'^2\alpha_\pm^2}{1-\alpha_\pm^2}}\sn u\right)\sqrt{\cfrac{1-\alpha_\pm^2}{k^2+k'^2\alpha_\pm^2}} \,,& \text{if}\ \cfrac{\alpha_\pm^2}{\alpha_\pm^2-1} < k^2 \,;\\
        \sd u \,,& \text{if}\ \cfrac{\alpha_\pm^2}{\alpha_\pm^2-1} = k^2 \,;\\
        \cfrac{\sqrt{\cfrac{\alpha_\pm^2-1}{k^2+k'^2\alpha_\pm^2}} \ln\cfrac{\sqrt{k^2+k'^2\alpha_\pm^2}\dn u +\sqrt{\alpha_\pm^2-1}\sn u}{\sqrt{k^2+k'^2\alpha_\pm^2}\dn u -\sqrt{\alpha_\pm^2-1}\sn u}}2 \,,& \text{if}\ \cfrac{\alpha_\pm^2}{\alpha_\pm^2-1} > k^2 \,,
    \end{cases}\label{241f}
\eal
where $k'$ is
\begin{equation}
  k' = \sqrt{1-k^2} \label{241k'}
\end{equation}
and $u_1$, $u_2$, $\bar u_2$ are the roots of f$(u) = 2M(u-u_1)(u-u_2)(u-\bar u_2)$.
\begin{figure}
         %\epsfxsize=13cm
         \epsfbox{kerr/eqnuplot.eps}
         \caption{\color{blue}The flyby and the associated crash null orbits in the equatorial plane.
         Numerical values of points '+' are listed in section \ref{eqnuplot.m}. The flyby orbit is
         coming from infinity and after circling around singularity several times it goes off to
         infinity again. Crash orbit starts  from a certain distance from singularity and after twice
         reversing direction of its rotation it plunges into singularity.}
         \label{keqnull}
\end{figure}
An example of orbits of the first and second kind is in figure \ref{keqnull}.}

\subsubsection{Unstable circular orbits}%viz Boyer 1967 opravit podle nej nebo %onil??
The radius $r_\mathrm c$ of the unstable
circular orbit is determined by the equations (cf.~equation (\ref{k72n}))
\begin{equation}\label{k78n}
  E^2 +\frac{2M}{r_\mathrm c^3}\left(L -aE\right)^2
  -\frac1{r_\mathrm c^2}\left(L^2
  -a^2E^2\right) = 0
\end{equation}
and
\begin{equation}\label{k79n}
  -\frac{6M}{r_\mathrm c^4}\left(L -aE\right)^2
  +\frac2{r_\mathrm c^3}\left(L^2
  -a^2E^2\right) = 0 \,.
\end{equation}
From equation (\ref{k79n})%, we conclude that
\begin{equation}\label{k80n}
  r_\mathrm c = 3M\frac{L -aE}{L +aE} = 3M\frac{D_\mathrm c -a}{D_\mathrm c
  +a}\,.
\end{equation}
Inserting this relation in equation (\ref{k78n}), we get
\begin{equation}\label{k81n}
  E^2 = \frac1{27M^2}\frac{\left(L +aE\right)^3}{L -aE} \,
\end{equation}
or
\begin{equation}\label{k82n}
  \left(D_\mathrm c +a\right)^3 = 27M^2(D_\mathrm c -a) \,.
\end{equation}
Substitution
\begin{equation}\label{k83n}
  y = D_\mathrm c +a \,,
\end{equation}
leads to the cubic equation
\begin{equation}\label{k84n}
  y^3 -27M^2y +54aM^2 = 0\,.
\end{equation}


\begin{itemize}
  \item Slow and \textcolor{blue}{extreme} black hole: Cases with $a
  >
  0$ and $ a < 0$ corresponding to the direct and retrograde
  orbits are distinguished. For $a > 0$,
  \begin{equation}\label{k85ns}
    %\left.
    \begin{aligned}%{c}
        \color{blue}y &\color{blue}= 6M\cos\left(\frac\pi 3 \pm\frac\vartheta
        3\right)
        \qquad
        &\color{blue}\mbox{where} \qquad
        \cos\vartheta = a/M \,,\\
        D_\mathrm c &= y -a &\mbox{and} \qquad r_\mathrm c =
        3M(1 -2a/y) \,;
    \end{aligned}
    %\right\}
  \end{equation}
  and for $-a = \abs{a} > 0$,
  \begin{equation}\label{k86ns}
    %\left.
    \begin{aligned}%{c}
        \color{blue}y = 6M\cos\frac\vartheta 3 \qquad \mbox{where} \qquad
        \cos\vartheta = |a|/M \,\\
        D_\mathrm c = y +|a| \qquad \mbox{and} \qquad r_\mathrm c
        = 3M(1 +2|a|/y) \,.
    \end{aligned}
    %\right\}
  \end{equation}

  It can be directly verified that the solution for $r_\mathrm c$,
  expressed directly in terms of $\vartheta$, is given
  by
  \bal\color{blue}
    r_\mathrm c &\color{blue}= 4M\cos^2\frac{\pi \pm\vartheta}3
    \label{k87nsd} = 4M\cos^2\frac{\pi \pm\arccos\frac aM}3\\
    \intertext{for direct orbits, and}
    \label{k87nsr}
    r_\mathrm c &= 4M\cos^2\frac\vartheta 3 = 2M\left[1
    +\cos\left(\frac23\arccos\frac{\abs{a}}M\right)\right]
  \eal
  for retrograde orbits.

  \item Fast black hole:
  Circular orbits appear only for retrograde orbits. We have
  \begin{equation}
    %\left.
    \begin{aligned}%{c}
        \color{blue}y &\color{blue}= 6M\cosh(\vartheta/3) &\color{blue}\qquad
        \mbox{where}
        \qquad
        \cosh\vartheta = \abs{a}/M \,\\
        D_\mathrm c &= y +\abs{a} &\mbox{and} \qquad r_\mathrm c
        = 3M(1 +2\abs{a}/y) \,,
    \end{aligned}
    %\right\}
  \end{equation}
  \textcolor{blue}{with condition
  \begin{equation}\label{fccond}
    \abs{a}/M < \pi/2 \,.
  \end{equation}}


  \textcolor{blue}{Then
  \begin{equation}
    r_\mathrm c = 4M\cosh^2\frac\vartheta3 = 2M\left[1
    +\cosh\left(\frac23\mbox{arccosh}\frac{\abs{a}}M\right)\right]
    \,.
  \end{equation}}
\end{itemize}

When the impact
parameter has the critical value $D_\mathrm c$, and f$(r) = 0$
allows a double root, the governing equation (\ref{k72n}) can
be reduced to
\begin{equation}\label{k90n}
  \dot u^2 = ME^2\left(D_\mathrm c -a\right)^2u^4\left(u
  -u_\mathrm c\right)^2(2u+u_c)
  \,,
\end{equation}
where
\begin{equation}\label{k91n}
  u = \frac1r \qquad \mbox{and} \qquad u_\mathrm c =
  \frac1{r_\mathrm c} = \frac{D_\mathrm c +a}{3M(D_\mathrm c -a)}
  \,.
\end{equation}
Equation (\ref{k90n}) directly integrated gives
\begin{multline}\label{k92n}
  [E(D_\mathrm c -a)\sqrt M]\tau = \pm \int\frac{\mbox du}{u^2(u -u_\mathrm c)
  \sqrt{2u +u_\mathrm c}} = \\
  = \pm\frac1{u_\mathrm
  c^2}\left(\frac{\sqrt{2u +u_\mathrm c}}u + \frac1{\sqrt{3u_\mathrm c}}
  \ln\left|\frac{\sqrt{2u +u_\mathrm c} -\sqrt{3u_\mathrm c}}
  {\sqrt{2u +u_\mathrm c} +\sqrt{3u_\mathrm c}}\right|\right) \,.
\end{multline}
To show the orbit in the equatorial plane, we
combine it with equation
\begin{equation}\label{k93n}
  \dot\varphi = \frac{Eu^2}{3(a^2u^2 -2Mu +1)u_\mathrm
  c}[3D_\mathrm cu_\mathrm c -2(D_\mathrm c +a)u]
\end{equation}
(which follows directly from equation (\ref{k69n})). We obtain
\begin{equation}\label{k94n}
  \frac{\mbox du}{\mbox d\varphi} = \pm\frac{D_\mathrm c +a}{\sqrt
  M}(a^2u^2 -2Mu +1)\frac{(u - u_\mathrm c)\sqrt{2u +u_\mathrm
  c}}{3D_\mathrm cu_\mathrm c -2(D_\mathrm c +a)u} \,,
\end{equation}
or
\begin{equation}\label{k95n}
  \varphi = \pm\frac{\sqrt M}{a^2(D_\text{c}
  +a)}\int\frac{3D_\text{c}u_\text{c}-2(D_\text{c}+a)u}
  {(u-u_+)(u-u_-)(u-u_\text{c})\sqrt{2u+u_\text{c}}} \,.
\end{equation}
The explicit expression of this integral is
\begin{multline}\color{blue}
  \label{k95nres}
  \varphi = \pm\frac{\sqrt M}{a^2(D_\text c+a)}
  \left\{\frac{[3D_\text{c}u_\text{c}-2u_+(D_\text{c}+a)]
  \ln\left|\frac{\sqrt{2u+u_\text{c}}-\sqrt{2u_++u_\text{c}}}
  {\sqrt{2u+u_\text{c}}+\sqrt{2u_++u_\text{c}}}\right|}
  {(u_+-u_-)(u_+-u_\text c)\sqrt{2u_++u_\text c}}+\right.\\\color{blue}
  +\frac{[3D_\text{c}u_\text{c}-2u_-(D_\text{c}+a)]
  \ln\left|\frac{\sqrt{2u+u_\text{c}}-\sqrt{2u_-+u_\text{c}}}
  {\sqrt{2u+u_\text{c}}+\sqrt{2u_-+u_\text{c}}}\right|}
  {(u_+-u_-)(u_\text c-u_-)\sqrt{2u_-+u_\text c}}+\\
  \textcolor{blue}{\left.+\sqrt{\frac{u_\text
  c}3}\frac{(2a-D_\text c)\ln\left|\frac{\sqrt{2u+u_\text
  c}-\sqrt{3u_\text c}}{\sqrt{2u+u_\text
  c}+\sqrt{3u_\text c}}\right|}{(u_\text c-u_-)(u_+-u_\text c)}
  \right\}}\,.
\end{multline}


\section{Space-like geodesics}
For space-like geodesics $\delta_1 = -1$, therefore we have the radial equation
\begin{equation}\label{eqsprad}\color{blue}
  r^2\dot r^2 = r^2E^2 +\frac{2M}r\left(aE -L\right)^2 +\left(a^2E^2 -L^2\right) +\varDelta
\end{equation}

\begin{corollary}
    For space-like equatorial geodesics\footnote{cf. \cite[Corollary 4.14.12]{onil}},
\begin{equation}\label{onil285psi}
  \psi(r, L) = (E^2 +1) -2Mr^2 +\left[a^2\left(E^2 +1\right) -L^2\right]r +2M\left(L -aE\right)^2 \,,
\end{equation}
    and the set $Z:\ \psi = 0$ is the union of the graphs of
\begin{equation}\label{onil285L}
  L\pm (r) = \frac{2MaE \pm \sqrt{\text{\emph D}(r)}}{2M -r} \,,
\end{equation}
    defined on intervals where \emph D$(r) = r\varDelta(r)[(E^2 +1)r -2M]$ is non-negative.
\end{corollary}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/rLeqsls1.eps}
         \caption{Space-like $r$-$L$ plots for equatorial geodesics. As $E$ decrease from $+\infty$, the island $W$ shrinks. \textcolor{blue}{($M = 1$, $a = 0.96$ and $E = 2$.)}}
         \label{rLeqsls1}
\end{figure}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/rLeqsls2.eps}
         \caption{Space-like $r$-$L$ plots for equatorial geodesics. The island $W$ is reduced to a single point $(r_-, 2M)$. \textcolor{blue}{($M = 1$, $a = 0.96$ and $E = \sqrt\frac{r_+}{r_-}$.)}}
         \label{rLeqsls2}
\end{figure}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/rLeqsls3.eps}
         \caption{Space-like $r$-$L$ plots for equatorial geodesics. As $E$ decrease from $+\infty$, the island $W$ moves through region $B$. ($M = 1$, $a = 0.96$ and $E = 1$.)}
         \label{rLeqsls3}
\end{figure}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/rLeqsls4.eps}
         \caption{Space-like $r$-$L$ plots for equatorial geodesics. The island $W$ meets with the north continent $N$. ($M = 1$, $a = 0.96$ and $E = \sqrt\frac{r_-}{r_+}$.)}
         \label{rLeqsls4}
\end{figure}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/rLeqsls5.eps}
         \caption{Space-like $r$-$L$ plots for equatorial geodesics. The island $W$ approaches the $r$-axis. ($M = 1$, $a = 0.96$ and $E = 0.6$.)}
         \label{rLeqsls5}
\end{figure}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/rLeqsls6.eps}
         \caption{Space-like $r$-$L$ plots for equatorial geodesics. The island $W$ lays symmetrically around the $r$-axis. ($M = 1$, $a = 0.96$ and $E = 0$.)}
         \label{rLeqsls6}
\end{figure}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/rLeqexs1.eps}
         \caption{Space-like $r$-$L$ plots for equatorial geodesics. \textcolor{blue}{The island $W$ is reduced to a single point $(r_\pm, 0)$. ($M = a = 1$ and $E = 0$.)}}
         \label{rLeqexs1}
\end{figure}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/rLeqexs2.eps}
         \caption{Space-like $r$-$L$ plots for equatorial geodesics. \textcolor{blue}{The island $W$, reduced to a single point $(r_\pm, 2ME)$, is moving north from the $r$-axis. ($M = a = 1$ and $E = 0.6$.)}}
         \label{rLeqexs2}
\end{figure}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/rLeqexs3.eps}
         \caption{Space-like $r$-$L$ plots for equatorial geodesics. \textcolor{blue}{The point $W$ joins the north continent $N$. ($M = a = 1$ and $E = 1$.)}}
         \label{rLeqexs3}
\end{figure}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/rLeqexs4.eps}
         \caption{Space-like $r$-$L$ plots for equatorial geodesics. \textcolor{blue}{The island $W$ emerges from the north continent $N$ staying connected with $N$ in the point $(r_\pm, 2ME)$. $W$ grows with increasing $E$. ($M = a = 1$ and $E = 2$.)}}
         \label{rLeqexs4}
\end{figure}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/rLeqfas1.eps}
         \caption{Space-like $r$-$L$ plots for equatorial geodesics. \textcolor{blue}{The island $W$ and the north continent $N$ form a single continent. ($M = 1$, $a = 1.04$ and $E = 2$.)}}
         \label{rLeqfas1}
\end{figure}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/rLeqfas2.eps}
         \caption{Space-like $r$-$L$ plots for equatorial geodesics. \textcolor{blue}{With decreasing $E$ the continent $N$ shrinks. ($M = 1$, $a = 1.04$ and $E = 0$.)}}
         \label{rLeqfas2}
\end{figure}

Outside the region $B$ the $r$-$L$ plots usually look like the null case (figures \ref{4.26slow}, \ref{4.26extreme} and \ref{4.26fast}). But the vertical tangent $w(E) = \frac{2M}{E^2 +1}$ is positive, and its interaction with the horizon radii $r_\pm$ creates many different types of orbits.

When $E$ is large, the only important difference from the null case is that $W$ has moved from the $L$-axis at the short distance $w(E) > 0$. In the slow case it has become an island. In the fast case $W$ is connected to $N$ (see figure \ref{rLeqfas1}). But as $E$ decreases, this distance increases, while (in the slow and extreme cases) the right side of $W$ remains pinned at $(r_-, L_-)$ (see figures \ref{rLeqsls1} and \ref{rLeqexs4}). Thus the island diminishes, until it is reduced to the single point $(r_-, L_-)$ at $w = r_-$. This happens when $E = \sqrt{r_+/r_-}$, with $L_- = L_w = 2M$. Therefore, in the slow case, $(r_-, 2M)$ is an isolated maximum of $\psi(r, L)$. This point represents an unstable
circular orbit in the intersection of the equator and the horizon $H_-$ (see figure \ref{rLeqsls2}).

In the slow case, when $E$ decreases below $\sqrt{r_+/r_-}$, $w(E)$ increases past $r_-$ and the island reemerges, now running from $r = r_-$ to $r = w(E) > r_-$. Hence, unlike null and time-like geodesics, space-like ones can turn in region $B$.
As $w$ approaches $r_+$ the island expands toward $N$. When $E = \sqrt{r_-/r_+}$, $w$ reaches $r_+$, and the island meets $N$ at the critical point $(r_+, 2M)$ (see figure \ref{rLeqsls4}). This is the unstable circular orbit in $H_+$. In the extreme case there is only one unstable orbit in $H$ at the critical point $(r_\pm, 2ME$) (see figures \ref{rLeqexs1}, \ref{rLeqexs2} and \ref{rLeqexs3}).

Further decrease of $E$ moves $w$ past $r_+$ where it drives $N$ eastward (see figures \ref{rLeqsls5}, \ref{rLeqexs2} and \ref{rLeqfas2}). In the slow case the island is left behind, extending from $r_-$ to $r_+$. Thereafter, it moves south until at $E = 0$ a symmetry is achieved (see figures \ref{rLeqsls6} and \ref{rLeqexs1}).


\subsection{\color{blue}The special case, $L =aE$}
\textcolor{blue}{In this case equations \eqref{eqsprad} reduces to
\begin{equation}\label{k97s}
  r^2\dot r = (E^2+1)r^2 -2Mr +a^2 \,,
\end{equation}
while the equations for $\dot\varphi$ and $\dot t$ are the same as for the null geodesics (cf.~equations \eqref{k75n}).
Equation \eqref{k97s} on integration gives
\begin{multline}\label{eqspaEtau}
  \tau = \frac1{E^2+1} \left[\sqrt{(E^2+1)r^2-2Mr+a^2}+\right. \\*\left.+\frac M{\sqrt{E^2+1}} \arcsinh\frac{\left(E^2+1\right)r -M} {\sqrt{(E^2+1)a^2 +M^2}}\right] \,.
\end{multline}}

\vspace{-18pt}
\textcolor{blue}{Alternatively, we may combine equations governing $\dot r$ and $\dot\varphi$ in the manner
\begin{equation}\label{k101s}
  \frac{\text du}{\text d\varphi} = \frac{1-2Mu+a^2u^2}{aE} \sqrt{E^2+1-2Mu+a^2u^2} \,,
\end{equation}
and obtain the solution
\begin{multline}\label{k102s}
  \varphi = \frac{E\sqrt{M^2-a^2}}2 \left[\arctanh\frac{a^2(E^2+1) -(a^2u-M)\sqrt{M^2-a^2} -M^2} {a^2 E \sqrt{E^2+1-2Mu+a^2u^2}} \right.\\ \left.-\arctanh\frac{a^2(E^2+1) +(a^2u-M)\sqrt{M^2-a^2} -M^2} {a^2 E \sqrt{E^2+1-2Mu+a^2u^2}}\right] \,,
\end{multline}
where we assume $E$ to be positive. An example of an orbit derived with the aid of the equation \eqref{k102s} is illustrated in figure \ref{eqspacsp}.}
\begin{figure}
         %\epsfxsize=13cm
         \epsfbox{kerr/eqspacsp.eps}
         \caption{\textcolor{blue}{An example of a space-like geodesic with $L = aE$,
         described in the equatorial plane of a Kerr black hole with $M = 1$ and $a = 0.8$.
         For the orbit illustrated, $E = 2$. An equivalent of radial
         geodesic this orbit comes from infinity to singularity,
         reversing its direction on both horizons.}
         The two horizons are represented by dash-dotted
         circles.}
         \label{eqspacsp}
\end{figure}


\textcolor{blue}{\subsection{General orbits}
In this section we will find general solutions for all three types of orbits. From equation \eqref{eqsprad} it is clear that turning points occur at the zeros of the cubic polynomial
\begin{equation}\label{eqspacepoynomial}
  f(u) = 2Mu\left(aE-L\right)^2u^3 +\left(a^2e-L^2\right)u^2 -2Mu +e.
\end{equation}
With this polynomial instead of \eqref{bl5.1} the solutions for $\tau$ and $\varphi$ are formally the same as in section \ref{eqnulgen}.}


\subsubsection{\color{blue}Unstable circular orbits}
We look for the values of $L$ and $E$ which a circular orbit at some assigned radius, $r = 1/u$, will have. When $L$ and $E$ have these values, the cubic polynomial \eqref{eqspacepoynomial} will have a double root. The conditions for a double root are
\begin{equation}\label{k105s}\color{blue}
  a^2u^2 -2Mu +1 +E^2 +2Mx^2u^3 -\left(x^2 +2aEx\right) = 0
\end{equation}
and
\begin{equation}\label{k106s}\color{blue}
  a^2u -M +3Mx^2u^2 -\left(x^2 +2aEx\right)u = 0 \,,
\end{equation}
where we have written
\begin{equation}\label{k107}
  x = L -aE \,.
\end{equation}
Equations \eqref{k105s} and \eqref{k106s} can be combined to give
\begin{equation}\label{k108s}\color{blue}
  E^2 = (Mu -1) +Mx^2u^3
\end{equation}
and
\begin{equation}\label{k109s}\color{blue}
  2axEu = x^2(3Mu -1)u +a^2u -M \,.
\end{equation}
By eliminating $E$ between these equations, we obtain the following quadratic equation for $x$:\textcolor{blue}{
\begin{multline}\label{k110s}
  x^4u^2\left[\left(3Mu -1\right)^2 -4a^2Mu^3\right] \\ +2x^2u\left[\left(3Mu -a\right)\left(a^2u-M\right) -2a^2u(Mu -1)\right] +\left(a^2u -M\right)^2 = 0 \,.
\end{multline}
The discriminant of this equation is
\begin{equation}\label{k111s}
  -4a^2M\varDelta^2(u)u \,,
\end{equation}}
where $\varDelta(u)$ is defined in \eqref{Delta_u}; and the solution of equation \eqref{k110s} takes a~simple form by writing
\begin{equation}\label{k112}
  \left(3Mu -1\right)^2 -4a^2Mu^3 = Q_+Q_- \,,
\end{equation}
where
\begin{equation}\label{k113}
  Q_\pm = 1 -3Mu \pm 2a\sqrt{Mu^3} \,.
\end{equation}
Thus, we find
\begin{equation}\label{k114s}\color{blue}
  x^2u^2 = -\frac{Q_\pm\varDelta(u)-Q_+Q_-}{Q_+Q_-} = -\frac{\varDelta(u)-Q_\mp}{Q_\mp} \,.
\end{equation}
But
\begin{equation}\label{k115}
  \varDelta(u) -Q_\mp = u\left(a\sqrt u \pm\sqrt M\right)^2 \,.
\end{equation}
Therefore the solution for $x$ takes the form
\begin{equation}\label{k116s}
  x = -\frac{a\sqrt u \pm\sqrt M}{\sqrt{uQ_\mp}} \,.
\end{equation}
Inserting the solution \eqref{k116s} for $x$ in equation \eqref{k108s}, we find
\begin{equation}\label{k117s}\color{blue}
  E = \sqrt\frac{4Mu-1-2M^2u^2+Mu^3a^3\pm a\sqrt{Mu^3}}{Q_\mp} \,;
\end{equation}
and the value of $L$ to be associated with this value of $E$ is
\begin{equation}\label{k118s}
  L = aE = \textcolor{blue}{\frac{\pm\sqrt M+a\sqrt u\left(\sqrt{4Mu-1-2M^2u^2+Mu^3a^3\pm a\sqrt{Mu^3}}-1\right)}{\sqrt{uQ_\mp}}} \,.
\end{equation}
$E$ and $L$ given by equations \eqref{k117s} and \eqref{k118s} are the energy and the angular momentum (per unit mass) of a~particle describing a circular orbit of reciprocal radius $u$.


\section{The time-like geodesics}
For time-like geodesics, equations \eqref{k69n} and \eqref{k70n}
for \(\dot\varphi\) and \(\dot t\) remain the same. On the other hand, equation
\eqref{k72n} is replaced by
\begin{equation}\label{k96t}
  r^2\dot r^2 = -\varDelta +r^2E^2 +\frac{2M}r\left(L -aE\right)^2
  -\left(L^2 -a^2E^2\right) \,,
\end{equation}
where \(E\) is now the energy per unit mass
of the particle.


\subsection{The special case, $L =aE$}
The time-like geodesics with \(L = aE\), are like null geodesics with
\(D= a\), in that that their behavior as they cross the
horizons is characteristic of the orbits in general.

For \(L = aE\), equation \eqref{k96t} reduces to
\begin{equation}\label{k97t}
  r^2\dot r^2 = (E^2 -1)r^2 +2Mr -a^2 \,,
\end{equation}
while the equations for \(\dot\varphi\) and \(\dot t\) remain the
same as for the null geodesics (cf.~equations \eqref{k75n}):
\begin{equation}\label{k98t}
  \dot\varphi = aE/\varDelta \qquad \text{and} \qquad \dot t =
  E\left(r^2 +a^2\right)/\varDelta \,.
\end{equation}

Equation \eqref{k97t} yields
\begin{equation}\label{k97taut}
  \tau = \pm\int\frac{r\,\text dr}{\sqrt{(E^2 -1)r^2 +2Mr
  -a^2}} \,.
\end{equation}
Since we are considering infalling particles, we take equation
\eqref{k97taut} with the negative sign. After integration we
get\footnote{Compare \cite[page 333, equation
(100)]{chandrasekhar} and \eqref{k100t}.}
\begin{align}
  \begin{split}\label{k99t}
    \tau =& \frac1{1 -E^2} \biggl\{\sqrt{(E^2 -1)r^2 +2Mr -a^2}\\
    &\left.-\frac M{\sqrt{E^2 -1}}\ln\left[\sqrt{(E^2 -1)r^2 +2Mr -a^2}
    +r\sqrt{E^2 -1} +\frac M{\sqrt{E^2 -1}}\right]\right\} \\
    &\qquad \text{(for \(E^2 > 1\))\,,} \raisetag{26pt}
  \end{split}\\\color{blue}
  \tau =& \color{blue}-\frac{Mr +a^2}{3M^2}\sqrt{2Mr -a^2} \qquad \text{(for
  \(E^2 = 1\))}\label{k99.5t}\\
  \intertext{and}
  \begin{split}\label{k100t}
    \tau =&\frac1{1 -E^2} \Biggl[\sqrt{(E^2 -1)r^2 +2Mr -a^2}+\\
    &\left.+\frac M{\sqrt{1 -E^2}}\arcsin\frac{M -(1 -E^2)r}{\sqrt{M^2
    -a^2(1 -E^2)}}\right] %\\
    \qquad\text{(for \(\textcolor{blue}{1
  -M^2/a^2 < E^2} < 1\)).}
  \end{split}
  \raisetag{72pt}
\end{align}

Alternatively, for a slow black hole, equations
governing \(\dot r\)~and \(\dot\varphi\) can be combined as follows
\begin{equation}\label{ks101t}
  \frac{\text du}{\text d\varphi} = \pm \frac aE(u
  -u_+)(u-u_-)\sqrt{(E^2-1) +2Mu -a^2u^2} \,,
\end{equation}
where \(u = 1/r\) and \( u_\pm = 1/r_\pm\). The
solution is
\begin{multline}\label{ks102t}
    \varphi = \frac1{a(u_+ -u_-)}\left\{\ln\left[2E\sqrt{E^2\xi_+^2
    +2(M -a^2u_+)\xi_+
    -a^2} +2E^2\xi_+ +\right.\right.\\
    +2\left(M -a^2u_+\right)\biggr] -\ln\left[2E\sqrt{E^2\xi^2_- +2(M
    -a^2u_-)\xi_- -a^2}\right. +2E\xi_- +\\
    +2M\left(M -a^2u_-\right)\biggr]\biggr\} \,,
\end{multline}
where
\begin{equation}\label{k103t}
  \xi_\pm = \left(u -u_\pm\right)^{-1} \,.
\end{equation}

\textcolor{blue}{For an extreme or fast black hole, we may combine equations governing $\dot r$ and $\dot\varphi$ in the manner
\begin{equation}\label{kf101t}
  \frac{\text du}{\text d\varphi} = \pm\frac{aE}{1-2Mu+a^2u^2} \sqrt{E^2-1+2Mu-a^2u^2} \,,
\end{equation}
and obtain the solution
\begin{multline}\label{kf102t}
  \varphi = \frac{-aE}{2u_2(a^2u_2^2+1+2Mu_2)\sqrt A}\left\{2Bu_2\left(a^2Mu_2^2 +2M^2u_1 -a^2 +2M^2u_2+ \right.\right.\\
\left.+a^2Mu_1u_2 -a^2u_1\right) +2\sqrt A\left(a^2u_2^2 +1+\right.\\ \left.+2Mu_2\right)\ln\left(u_1z^2-u_2\right) -u_2\sqrt A\left(2M +a^2u_2\right)\ln\left[z^4\left(1-2Mu_1+a^2u_1^2\right)+\right.\\ \left.\left.+2z^2\left(Mu_2-Mu_1-a^2u_1u_2+1\right)+a^2u_2^2+2Mu_2+1\right]\right\} \\ \text{(for $ E^2 > 1 -M^2/a^2$),}
\end{multline}
where $u_1$ and $u_2$ are the solutions of polynomial $a\bigl(\frac{E^2-1}{a^2} +\frac{2M}{a^2}u -u^2\bigr)$, and
\bal
    z^2 &= \frac{u_2-u}{u-u_1} \,,\\
    A &= \left(a^2 -M^2\right)\left(u_2 -u_1\right)^2 \,,\\
    B &= \arctan\frac{a^2u_1^2z^2+z^2-2Mu_1z^2-a^2u_1u_2+Mu_2-Mu_1+1}{\sqrt A} \,.
\eal}


\textcolor{blue}{\subsection{General orbits}
As before in the case of the null and space-like geodesics, we will find solutions for all three types of orbits. We have the same polynomial \eqref{eqspacepoynomial} as in the case of the space-like geodesics with redefinition of the effective energy $e$. (In the space-like case $e = E^2 +1$, now $e = E^2 -1$.) Solutions for the unbound orbits ($e \leq 1$) look formally the  same as in section \ref{eqnulgen}, but in the case of $e < 1$ we have bound orbits. For them
\begin{multline}\label{234.17t}
  \tau = \frac{gk^4}{{u_2}^2\alpha^4\abs{L-aE}\sqrt{2M}}\biggl\{\F(\phi, k)+\biggr.\\
+\frac{2(\alpha^2-k^2)}{k^2}\Pi\left(\phi, \alpha^2, k\right) -\frac{\alpha^2-k^2}{2k^4(\alpha^2-1)}\biggl[\alpha^2\E(\phi, k) -\left(\alpha^2-k^2\right)\F(\phi, k)+\biggr.\\
\left.\left. +\left(2\alpha^2k^2 +2\alpha^2 -\alpha^4-3k^2\right)\Pi\left(\phi, \alpha^2, k\right) -\frac{\alpha^4\sn v\cn v\dn v}{1-\alpha^2\sn^2v}\right]\right\} \,,
\end{multline}
where $g$ is defined in \eqref{g}, $k$ in \eqref{k}, $\phi$ and $v$ in \eqref{fi}, $u_1 < u_2 < u_3$ are roots of $f(u)$, and
\begin{equation}\label{234.17alfa}
  \alpha^2 = \frac{u_3k^2}{u_2} \,;
\end{equation}
\begin{multline}\label{234.18}
  \varphi = \frac g{u_+-u_-} \left\{\frac{L-2Mu_-(L-aE)}{\alpha_-^2(u_--u_2)} \left[k^4\F(\phi,k)+\right.\right.\\ \left.+\left(\alpha_-^2-k^2\right)\Pi\left(\phi, \alpha_-^2, k\right)\right] -\frac{L-2Mu_+(L-aE)}{\alpha_+^2(u_+-u_2)} \left[k^4\F(\phi,k)+\right.\\ \biggl.\left. +\left(\alpha_+^2-k^2\right)\Pi\left(\phi, \alpha_+^2, k\right)\right]\biggr\} \,,
\end{multline}
where $u_1$, $u_2$, $u_3$, $\phi$, $v$ and $k$ are the same as for solution for $\tau$, $g$ is defined in \eqref{234g}, and
\begin{equation}\label{234.18alfa}
  {\alpha_\pm}^2 = \frac{k^2(u_\pm-u_3)}{u_\pm-u_2} \,,
\end{equation}
where $u_\pm$ are the solutions of $\varDelta(u) = 1 -2Mu +a^2u^2$.}

\textcolor{blue}{For the orbits of the second kind, which are associated with the bound orbits
\begin{multline}\label{238.17t}
  \tau = \frac g{\alpha^4(u_3-u_1)^2\abs{L-aE}\sqrt{2M}} \biggl\{\F(\phi, k) -2\Pi\left(\phi, \alpha^2, k\right)+\biggr.\\
 +\frac 1{2(\alpha^2-1)(k^2-\alpha^2)}\biggl[\alpha^2\E(\phi, k) +\left(k^2-\alpha^2\right)\F(\phi, k)+\biggr.\\
\left.\left. +\left(2\alpha^2k^2 +2\alpha^2 -\alpha^4-3k^2\right)\Pi\left(\phi, \alpha^2, k\right) -\frac{\alpha^4\sn v\cn v\dn v}{1-\alpha^2\sn^2v}\right]\right\} \,,
\end{multline}
where we have the same $u_1$, $u_2$, $u_3$, $k$ and $g$ as in the case of orbits of the first kind, $\alpha$ is defined in \eqref{237alfa}, and $\phi$ and $v$ in \eqref{237fi};
\begin{multline}\label{keqtime2fi}
  \varphi = \frac{g}{(u_+-u_-)(u_3-u_1)} \left\{\frac{L-2Mu_+(L-aE)}{{\alpha_+}^2} \left[\Pi\left(\phi, {\alpha_+}^2, k\right) -\F(\phi, k)\right]\right.\\ \left.-\frac{L-2Mu_-(L-aE)}{{\alpha_-}^2} \left[\Pi\left(\phi, {\alpha_-}^2, k\right) -\F(\phi, k)\right]\right\} \,,
\end{multline}
where $\phi$, $v$, $u_1$, $u_2$, $u_3$ are the same as in the  above solution for $\tau$, $g$ is defined in \eqref{g}, $\alpha_+$ is in \eqref{alfa+}, $\alpha_-$ in \eqref{alfa-}, $k$ in \eqref{241k}, and $u_\pm$ have the same meaning as in the case of orbits of the first kind.}

\textcolor{blue}{The last type of orbit is the crash-escape orbit. Solution for $\tau$ can be again written in terms of elliptic integrals, but we will not do it here, because it is long and does not bring any new features. Solution for $\varphi$ is
\begin{multline}\label{keqtime3fi}
  \varphi = \frac{g[L-2Mu_+(L-aE)]}{\alpha_+(u_+-u_-)(A+u_3-u_+)} \left\{\frac1{1-\alpha_+}\left[\Pi\left(\phi, \frac{{\alpha_+}^2}{{\alpha_+}^2-1}, k\right) -\alpha_+f_+\right]\right\}\\ -\frac{g[L-2Mu_+(L-aE)]}{\alpha_-(u_+-u_-)(A+u_3-u_-)} \left\{\frac1{1-\alpha_-}\left[\Pi\left(\phi, \frac{{\alpha_-}^2}{{\alpha_-}^2-1}, k\right) -\alpha_-f_-\right]\right\} \,,
\end{multline}
where $g$ is defined in \eqref{241g}, $k$ as above in \eqref{241k}, $\phi$ in \eqref{241phi}, $\alpha_+$ in \eqref{241alfa+}, $\alpha_-$ in \eqref{241alfa-}, and $A$ in \eqref{241A}.}


\subsubsection{The circular and associated orbit}
We will follow \cite[chapter 7]{chandrasekhar} to find conditions for $E$ and $L$ for circular orbits. With the reciprocal radius $u = 1/r$ as the independent variable, the radial equation \eqref{k96t} becomes
\begin{equation}\label{k104t}
  u^{-4}\dot u^2 = -\left(a^2u^2 -2Mu\right) +E^2 +2M\left(L -aE\right)^2u^3 -\left(L^2 -a^2E^2\right)u^2 \,.
\end{equation}

When $L$ and $E$ have the values which a circular orbit at some assigned radius, $r = 1/u$, will have, the cubic polynomial on the right-hand side of equation
 \eqref{k104t} will have a double root. The conditions for a~double root are
\begin{equation}\label{k105t}
    -\left(a^2u^2 -2Mu +1\right) +E^2 +2Mx^2u^3 -\left(x^2 +2aEx\right)u^2 = 0
\end{equation}
and
\begin{equation}\label{k106t}
  -\left(a^2u -M\right) +3Mx^2u^2 -\left(x^2 +2aEx\right)u = 0 \,,
\end{equation}
where $x$ is defined in \eqref{k107}. The equations above can be combined to give
\begin{equation}\label{k108t}
  E^2 = (1-Mu) +Mx^2u^3
\end{equation}
and
\begin{equation}\label{k109t}
  2axEu = x^3(3Mu -1)u -\left(a^2u -M\right) \,.
\end{equation}
Eliminating $E$ between these equations leads to the following quadratic equation for $x$:
\begin{multline}\label{k110t}
  x^4u^2\left[\left(3Mu-1\right)^2 -4a^2Mu^3\right]\\ -2x^2u\left[(3Mu-1)\left(a^2u-M\right) -2a^2u(Mu-1)\right] +\left(a^2u-M\right)^2 = 0 \,.
\end{multline}
Its discriminant is
\begin{equation}\label{k111t}
  4a^2M\varDelta^2(u)u \,,
\end{equation}
where $\varDelta(u)$ is defined in \eqref{Delta_u}; and by using \eqref{k112} and \eqref{k113}, the solution of equation \eqref{k110t} takes a simple form. Hence
\begin{equation}\label{k114t}
  x^2u^2 = \frac{Q_\pm\varDelta(u)-Q_+Q_-}{Q_+Q_-} = \frac{\varDelta(u)-Q_\mp}{Q_\mp} \,.
\end{equation}
With the use of \eqref{k115} the solution for $x$ takes a simple form \eqref{k116s}. Inserting it in equation \eqref{k108t}, we find
\begin{equation}\label{k117t}
  E = \frac{1-2Mu\mp a\sqrt{Mu^3}}{\sqrt{Q_\mp}} \,;
\end{equation}
and the value of $L$ associated with this value of $E$ is
\begin{equation}\label{k118t}
  L = aE +x = \mp\left(a^2u^2 +1 \pm2a\sqrt{Mu^3}\right)\sqrt\frac M{uQ_\mp} \,.
\end{equation}
$E$ and $L$ given by the above equations are the energy and the angular momentum (per unit mass) of a particle in a~circular orbit of reciprocal radius $u$.
