% Words:   2332
\chapter{The time-like geodesics in the Schwarzschild space-time}
\label{schtl}
Remember that freely falling particles with positive rest-mass describe time-like geodesics. This chapter is taken from \cite[Chapter 3]{chandrasekhar} and shortened.

For time-like geodesics $2{\cal L}$ has the value $1$%. Now equation %(\ref{s89}) becomes
%\begin{equation}\label{s89time}
%    \frac{E^2}{1-2M/r}-\frac{\dot{r}^2}{1-2M/r}-\frac{L^2}{r^2} =
%    2{\cal L} = 1\,.
%\end{equation}
%This can be rewritten in the form
, and we can write
\begin{equation}
    \left(\frac{\mbox{d}r}{\mbox{d}\tau}\right)^2 +
    \left(1-\frac{2M}{r}\right) \left(1+\frac{L^2}{r^2}\right) = E^2\,.
    \label{s90time}
\end{equation}
%By considering $r$ as a function of $\varphi$, we obtain the equation
%(compare with equation (\ref{s92}))
%\begin{equation}
%    \left(\frac{\mbox{d}r}{\mbox{d}\varphi}\right)^2 =
%    \left(E^2-1\right)\frac{r^4}{L^2}+\frac{2Mr^3}{L^2}-r^2+2Mr\,.
%    \label{s92time}
%\end{equation}
As in chapters~\ref{n} and \ref{s} we obtain the basic equation of
the problem (cf. equations (\ref{s214null}) and (\ref{s94}))
\begin{equation} \label{s94time}
      \left(\frac{\mbox{d}u}{\mbox{d}\varphi}\right)^2 =
      2Mu^3-u^2+\frac{2M}{L^2}u-\frac{1-E^2}{L^2} \,.
\end{equation}

\section{The radial geodesics}
%Equations governing geodesics  are
%\begin{eqnarray}
%    \left(\frac{\mbox{d}r}{\mbox{d}\tau}\right)^2 &=& \frac{2M}r
%    -\left(1-E^2\right)\, \\
%    \frac{\mbox{d}t}{\mbox{d}\tau} &=& \frac{E}{1-2M/r}\,.
%\end{eqnarray}
Again we shall consider trajectories of particles with $L = 0$ which start from rest
at some finite distance $r_\mathrm i$ and drop towards the center. %The
%starting distance
%The starting distance \(r = r_\mathrm i\) (where \(\dot{r} = 0\)) is %related
%to
%the constant $E$ by
%\begin{equation}
%    r_\mathrm i = \frac{2M}{1-E^2} \,.
%\end{equation}

This time the change of variables will be
\begin{equation}
    r = \frac{M}{1-E^2}(1+\cos\eta) = r_\mathrm i\cos^2\frac\eta2 \,.
\end{equation}
%Clearly when $\eta = 0$ then
%$r = r_\mathrm i$;
%and the value of $\eta$ when $r$ crosses the horizon at $r = 2M$ is
%\begin{equation}
%    \eta_\mathrm{H} = 2\,\mbox{arcsin}E
%    \label{s99time}
%\end{equation}
%and
%\begin{equation}\label{s99time'}
%  \eta = \pi
%\end{equation}
%when $r = 0$ (at singularity).

In terms of $\eta$, the equations to be integrated are (compare with equations (\ref{s100}) and (\ref{s101}))
\begin{equation}
    \left(\frac{{\mathrm d}r}{\mathrm{d}\tau}\right)^2 = (1 -
    E^2)\tan^2\frac{\eta}{2} \quad \mbox{and} \quad \frac{{\mathrm
    d}t}{\mathrm{d}\tau} =
    \frac{E\cos^2\frac{\eta}{2}}{\cos^2\frac{\eta}{2} -
    \cos^2\frac{\eta_\mathrm{H}}{2}}\,,
    \label{s100time}
\end{equation}
together with
\begin{equation}
    \frac{{\mathrm d}r}{\mathrm{d}\eta} =
    -r_\mathrm i\sin\frac{\eta}{2}\cos\frac{\eta}{2}\,.
    \label{s101time}
\end{equation}
Because we are considering infalling particles,
\begin{equation}
    \frac{{\mathrm d}r}{\mathrm{d}\tau} = -\left(1 -
    E^2\right)^{1/2}\tan\frac{\eta}{2} =
    -\left(\frac{2M}{r_\mathrm i}\right)^{1/2}\tan\frac{\eta}{2}\,.
    \label{s102time}
\end{equation}
From equations (\ref{s101time}) and (\ref{s102time}) follows (compare with equation (\ref{s103}))
\begin{equation}
    \frac{\mathrm{d}\tau}{\mathrm{d}\eta} =
    \left(\frac{r_\mathrm i^3}{2M}\right)^{1/2}\cos^2\frac{\eta}{2} =
    (1 + \cos\eta)\sqrt{\frac{r_\mathrm i^3}{8M}}\,.
    \label{s103time}
\end{equation}
With the assumption that \(\tau = 0 \) at
\(\eta = 0\), we obtain
\begin{equation}\label{s104time}
    \tau = \left(\frac{r_\mathrm i^3}{8M}\right)^{1/2}(\eta+\sin\eta) \,.
\end{equation}
%From
%equation (\ref{s104time}) it follows that the particle crosses the horizon and arrives at the singularity at
%finite proper times,
%\begin{equation}
%    \tau_\mathrm{H} = (\sinh\eta_\mathrm{H} +
%    \eta_\mathrm{H})\sqrt{\frac{r_\mathrm i^3}{8M}}
%    \qquad \mbox{and} \qquad
%    \tau_0 = \pi\sqrt{\frac{r_\mathrm i^3}{8M}}\,.
%    \label{s105time}
%\end{equation}
Similarly to section~\ref{sr} the situation differs when we
consider the equation of the trajectory in coordinate time, $t$.
To obtain $t$, the equation (cf. equations
(\ref{s100time}) and (\ref{s103time}))
\begin{equation}
    \frac{{\mathrm d}t}{\mathrm{d}\eta} =
    E\frac{\cos^4\frac{\eta}{2}}{\cos^2\frac{\eta}{2} -
    \cos^2\frac{\eta_\mathrm{H}}{2}}\sqrt{\frac{r_\mathrm i^3}{2M}}\,
    \label{s106time}
\end{equation}
must be integrated.
After  integration
\begin{equation}\label{s107time}
    t = E\left(\frac{r_\mathrm i^3}{2M}\right)^{1/2}\left[\frac1
    2(\eta +\sin\eta)+\left(1-E^2\right)\eta\right]
    %\nonumber\\
    +2M\ln\left(\frac{\tan\frac12\eta_\mathrm{H}
    +\tan\frac1
    2\eta}{\tan\frac1 2\eta_\mathrm{H}-\tan\frac1
    2\eta}\right) \,.
\end{equation}
Solutions (\ref{s104time}) and (\ref{s107time}) are illustrated in
fig.~\ref{tm radia}.
Coordinate time for time-like geodesics displays the same behavior as for space-like geodesics as can be seen in figure~\ref{tm radia}.
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{schwarzs/tm_radia.eps}
         \caption{Proper time ($\tau$) and coordinate time ($t$)
         along a time-like radial geodesic with initial condition
         $r_\mathrm i=6M$.}
         \label{tm radia}
\end{figure}

\section{The bound orbits ($E^2 < 1$)}
\begin{figure} \label{fig2}
    \begin{picture}(398,122)
        \multiput(12,71)(73,0){5}{\line(1,0){69}}
        \put(2,16){\line(0,1){110}}
        \put(-10,135){f($u$)}
        \put(-1,71){\line(1,0){6}}
        \qbezier(11,50)(11,90)(26,90)
        \qbezier(26,90)(36,90)(43,70)
        \qbezier(43,70)(51,52)(61,52)
        \qbezier(61,52)(79,52)(81,100)
     %-------------------------------------
        \qbezier(82,50)(82,71)(102,71)
        \qbezier(102,71)(112,71)(119,62)
        \qbezier(119,62)(140,33)(152,100)
     %-------------------------------------
        \qbezier(160,50)(160,90)(180,90)
        \qbezier(180,90)(190,90)(197,80)
        \qbezier(197,80)(205,71)(215,71)
        \qbezier(215,71)(233,71)(235,100)
    %--------------------------------------
        \qbezier(230,50)(240,70)(270,71)
        \qbezier(270,71)(300,72)(300,100)
    %--------------------------------------
        \qbezier(300,50)(330,60)(370,100)
        \put(388,67){$u$}
        \multiput(15,60)(150,0){2}{$u_1$}
        \put(83,77){$u_1=u_2$}
        \put(250,60){$u_1=u_2$}
        \put(335,60){$u_1$}
        \put(35,60){$u_2$}
        \multiput(61,60)(67,0){2}{$u_3$}
        \put(265,50){$=u_3$}
        \put(190,60){$u_2=u_3$}
        \multiput(102,71)(114,0){2}{\circle*{4}}
        \put(270,71){\circle*{4}}
        \put(24,20){($\alpha$)}
        \put(104,20){($\beta$)}
        \put(190,20){($\gamma$)}
        \put(261,20){($\delta$)}
        \put(336,20){($\varepsilon$)}
        \linethickness{2pt}
        \put(13,71){\line(1,0){30}}
        \put(78,71){\line(1,0){3}}
        \put(145,71){\line(1,0){9}}
        \put(162,71){\line(1,0){64}}
        \put(272,71){\line(1,0){29}}
       \put(337,71){\line(1,0){40}}
    \end{picture}
    \caption{The location of the roots of a cubic f$(u)=0$ for $E^2 < 1$ .}
\end{figure}
The consideration of equation (\ref{s94time}), which governs the
geometry of the paths described in the invariant plane, is divided
into two sections. These two categories of orbits are
characterized by $E^2 < 1$ and $E^2 \geq 1$. This difference will
 decide whether orbits are bound or unbound. In this
section, bound orbits ($E^2 < 1$) shall be considered. These
geodesics are governed by the equation (\ref{s109}), where
\begin{equation}\label{s110time}
  \mbox{f}(u) = 2Mu^3-u^2+\frac{2M}{L^2}u-\frac{1-E^2}{L^2}\,.
\end{equation}

As in sections~\ref{nu} and \ref{su}, the geometry of the
orbits will be decided by the location of the roots of
the equation f$(u) = 0$. %Letting $u_1, u_2$, and $u_3$ denote the
%roots of the cubic equation f$(u)=0$, we have
%\begin{equation}
 %   u_1u_2u_3 = \frac{1 -E^2}{2ML^2} \,.
   % \label{s111time}
%\end{equation}
%and equation (\ref{s112}). Since we have assumed that $1 -E^2 >
%0$, the equation f$(u)=0$ must always allow a positive real root.
All the allowed possibilities, \cite[Chapter 3, \S 19(b)]{chandrasekhar}, are
distinguished in fig.~\ref{fig2}.

%\begin{itemize}
%    \item Case ($\alpha$): For every pair of values $E$ and $L$, which
%    allows for $u$ three real roots $0 < u_1<u_2<u_3$, there exists two
%    distinct orbits confined, respectively, to the intervals
%    $u_1 \leq u\leq u_2$ and $u\geq u_3$, i.e. orbits of the first
%    kind and
%    orbits of the second kind.
%   \item Case ($\beta$): The orbit of the first kind is a stable
%    circular orbit while the orbit of the second kind plunges into
%    singularity.
%    \item Case ($\gamma$): The orbit of the first kind starts at
%    certain aphelion distance, $u_1^{-1}$, and approaches the
%    circle of radius $u_3^{-1}$, asymptotically, by spiraling
%    around it an infinite number of times. The orbit of the second
%    kind spirals away from the same circle (towards the centre) to
%    plunge eventually into the singularity.
%    \item Case ($\delta$): In this case all three roots coincide.
%    We have only one class of orbits: they
%    all plunge into the singularity after starting from certain
%   finite aphelion distances. The equation also allow an unstable
%   circular orbit of radius $u_1^{-1} ( = u_2^{-1} = u_3^{-1}).$
%    \item Case ($\varepsilon$): Equation f$(u) = 0$ allows a pair of
%    complex-conjugate roots. Principal difference with the case
%    ($\delta$) is that they do not allow any circular orbit.
%\end{itemize}

The different cases can also be deduced by
interpreting\footnote{There is ${\cal V}^2$ in \cite[page 102]{chandrasekhar}.}
\begin{equation}
    {\cal V} = \left(1-\frac{2M}{r}\right)
    \left(1 +\frac{L^2}{r^2}\right)
    \label{s113time}
\end{equation}
as `potential energy' (see equation (\ref{s90time})).
\begin{figure} \label{time pot}
    \epsfxsize=13cm
    \epsfbox{schwarzs/time_pot.eps}
    \caption{The effective potential for time-like
         trajectory (comp. equation (\ref{s113time})). $L/M$ is a)~4.33,
         b)~4,
         c)~3.75, d)~3.464. The last stable
         circular orbit occurs at the point of inflection--case d).}
\end{figure}
(In fig.~\ref{time pot} there are displayed some examples of
potential energy.) The unstable circular orbits correspond to maximum
potential while the stable circular orbits correspond to minimum potential.


\subsection{Orbits of the first kind}
%Orbits of the first kind occur in cases ($\alpha$), ($\beta$),
%($\gamma$), and ($\delta$); and we shall write real roots of
%equation f$(u) = 0$ as
%\begin{equation}
%    u_{1} = \frac{1}{l}(1-e)\,, \ u_{2} = \frac{1}{l}(1+e)\,,
%    \ \mbox{and}\
%    u_{3} = \frac{1}{2M}-\frac{2}{l}\,,
%   % \label{s114time}
%\end{equation}
%where latus rectum, $l$, is some positive constant and the
%eccentricity $e$ is less than 1.

%To be in conformity with ordering, $u_{1} \leq u_{2} \leq u_{3}$,
%we must require inequality (\ref{s116}) to be true. Defining
%$\mu$ in equation (\ref{s117}) we have the same important
%inequality (\ref{s118}) as in section~\ref{su}.

The requirement, that
\begin{equation}
    \mbox{f}(u) = 2M\left(u-\frac{1-e}{l}\right)
    \left(u-\frac{1+e}{l}\right)
    \left(u-\frac{1}{2M}+\frac{2}{l}\right)
    \label{s119time}
\end{equation}
agrees with its definition (\ref{s110time}), gives equations
%\begin{equation}
 %   \frac{M}{L^2} =
 %   \frac{1}{l^2}\left[l -M\left(3+e^2\right)\right]\quad\mbox{and}\quad
 %   \frac{1-E^2}{L^2} =
 %   \frac{1}{l^3}(l-4M)\left(1-e^2\right)\,,
   % \label{s120time}
%\end{equation}
%or, in terms of $\mu$,
(cf. equations (\ref{s121})),
\begin{equation}
    \frac{1}{L^2} =
    \frac{1}{lM}\left[1 -\mu\left(3+e^2\right)\right]\quad\mbox{and}\quad
    \frac{1-E^2}{L^2}  =
    \frac{1}{l^2}(1-4\mu)\left(1-e^2\right)\,.
    \label{s121time}
\end{equation}
%It follows from these relations that
%\begin{equation}%\label{s122time}
%  \mu < \frac1{(3+e^2)} \qquad \mbox{and} \qquad \mu < \frac14
%\end{equation}
%These inequalities are guarantied by (\ref{s118}).

A useful alternative expression for $E^2$ is
\begin{equation}
    \frac{E^2}{L^2} = \frac{1}{lM}\left[\left(2\mu-1\right)^2
    -4\mu^2e^2\right]\,.
    \label{s123time}
\end{equation}

According to the substitution (\ref{s199})
\begin{equation} \label{s125time}
    %\left.
    \begin{aligned}%{lrcr}
        &  \mbox{at aphelion, where} & u = \frac{1-e}l \,, &\quad& \chi = \pi\ \\
        \mbox{and} &&&\\
        & \mbox{at perihelion, where} & u = \frac{1+e}l \,, && \chi
        = 0 \,.
    \end{aligned}
    %\right\}
\end{equation}
Further analysis is the same as in section~\ref{su1}; and the
solution for $\varphi$ can be expressed in terms of the elliptic
integral (\ref{s130}). With the restriction (\ref{s125time}) in
the range of $\chi$, we may now write the solution for $\varphi$
in the form (cf. equation (\ref{s202})).
\begin{equation}
    \varphi = \frac{2}{\sqrt{1-6\mu+2\mu
    e}}\ \F\left(\frac{\pi}{2}-\frac{\chi}{2},k\right)\,,
    \label{s202time}
\end{equation}
where $\varphi$ starts at aphelion passage
where $\chi = \pi$. %The perihelion passage occurs at $\chi = 0$
%where $\psi = \pi/2$.

%The solution can be now completed by direct integration of the
%equation (\ref{s95}) for $t$ and $\tau$.


\subsubsection{The case $e = 0$} \label{sst}
In this scenario geodesic is a circle with the radius
\begin{equation}\label{s139time}
  r_\mathrm c = l \,. %\qquad \mbox{and} \qquad \mu = \frac M{r_\mathrm c}.
\end{equation}
%Equation (\ref{s121time}) now gives
%\begin{equation}%\label{s140time}
%  \frac 1{L^2} = \frac{1-3M/r_\mathrm c}{r_\mathrm cM}
%\end{equation}
%Rewriting this equation in the form
%\begin{equation}%\label{s141time}
%  r_\mathrm c^2 -\frac{L^2}Mr_\mathrm c +3L^2 = 0 \,,
%\end{equation}
It is evident from fig.~\ref{time pot} that an
orbit of zero eccentricity is compatible with one of the two roots
\begin{equation}\label{s142time}
  r_\mathrm c = \frac{L^2}{2M}\left(1 \pm \sqrt{1-\frac{12M^2}{L^2}}\right)
  \,.
\end{equation}
%and, further, that no circular orbit is possible for
%\begin{equation}%\label{s143time}
%  \frac LM < 2\sqrt{3} \,.
%\end{equation}
%For the minimum allowed value of $L/M$,
%\begin{equation}%\label{s144time}
%  r_\mathrm c = 6M \qquad \mbox{and} \qquad E^2 = \frac89 \qquad %\mbox{for}
%  \qquad \frac LM = 2\sqrt{3} \,.
%\end{equation}

%The larger of the two roots of equation (\ref{s142time}) locates
%the minimum of the potential-energy curve $\mathcal{V}(r)$, while
%the smaller root locates the maximum. Therefore, the circular
%orbit of the larger radius will be stable in contrast to the
%circular orbit of the smaller radius which will be unstable.
The
permitted ranges for the radii of the stable and unstable orbits are
\begin{equation}\label{s144'time}
  6M < r_\mathrm c(\mbox{stable}) < \infty \qquad \mbox{and} \qquad 3M
  \leq r_\mathrm c(\mbox{unstable}) \leq 6M \,.
\end{equation}


\subsubsection{The case $2\mu(3+e) = 1$}
The perihelion, $r_{\scriptstyle\mathrm p}$, and
the aphelion $r_{\scriptstyle\mathrm{ap}}$, distances are given
by (\ref{s146}) and
\begin{equation}\label{s146time}
  r_{\scriptstyle\mathrm{ap}} = 2M\frac{3+e}{1-e}\,.
\end{equation}
Perihelion distances of these geodesics are confined
to the range
\begin{equation}\label{s147time}
  4M \leq r_{\scriptstyle\mathrm p} < 6M \,.
\end{equation}
For these orbits (cf. equations \ref{s195})
\begin{equation}
    \frac{L^2}{M^2} = \frac{4\left(3+e\right)^2}{(3-e)(1+e)} \qquad \mbox{and}
    \qquad
    1-E^2 = \frac{1-e^2}{9-e^2}\,.
    \label{s148time}
\end{equation}
The discussion in section~\ref{ss1} applies in this case except
that $\varphi = 0$ when $\chi = \pi$. Equation (\ref{s150}) shows
that $\varphi \rightarrow \infty$ when $\chi \rightarrow 0$.


\subsection{Orbits of the second kind}
%Since \(u_1 +u_2 +u_3 = \frac 1{2M}\) and \(u_1 +u_2 > 0\), \(u_3
%< \frac1{2M}\); and therefore all these orbits start outside the
%horizon.

\begin{figure}
    \epsfxsize=13cm
    \epsfbox{schwarzs/timelike2.eps}
    \caption{Orbits of the first and the second kind. Orbit of the first time precesses around
    the black hole. Orbit of the second, starting
    from a certain
    distance from the black hole (aphelion), falls into the
    singularity. Eccentricity is
    $e = 1/2$ and latus rectum $l = 3$. (${\color{blue} M = 1/7}$ in the scale along the
    coordinate axes.)}
    \label{timelike}
\end{figure}
Except for different conditions for $e$ and $\mu$, discussion is
 the  same as in section~\ref{su2}. An example of bound
orbits is illustrated in fig.~\ref{timelike}.


\subsubsection{The case $e = 0$ and \(\mu \neq 1/6\)}
In this situation %equation (\ref{s156})
%integrates to give
%\begin{equation}\label{s161time}
%  \xi = (\varphi-\varphi_0)\sqrt{1-6\mu} \,,
%\end{equation}
%
the %corresponding
solution for $u$ is
\begin{equation}\label{s162time}
  u = \frac1l
  +\left(\frac1{2M}-\frac3l\right)
  \sec^2\left[\frac12 (\varphi-\varphi_0) \sqrt{1-6\mu}\right]\,,
\end{equation}
where $\varphi_0$ is a constant of integration.
%Starting at an aphelion distance $u_3^{-1}$, when \(\varphi =
%\varphi_0\), it arrives at the singularity at \(r = 0\), when
%\begin{equation}%\label{s163time}
%  \varphi - \varphi_0 = \frac\pi{\sqrt{1-6\mu}} \,,
%\end{equation}
%after circling one or more time depending on how close $\mu$ is to
%1/6.
See fig.~\ref{timelike b} for an example of this orbit and an
associated stable circular orbit.
\begin{figure}
    \epsfxsize=13cm
    \epsfbox{schwarzs/tm_bnd_b2.eps}
    \caption{A circular orbit $e = 0,\ l = 3/2$ and the associated orbit
    of the second
    kind. Orbit of the first kind is a stable circular orbit. Orbit of the second, starting from
    a certain
    distance from the black hole (aphelion), falls into the
    singularity. (${\color{blue}M = 1/7}$ in the scale along the
    coordinate axes.)}
    \label{timelike b}
\end{figure}


\subsubsection{The case $e = 0$ and $\mu = 1/6$}
In this case, all three roots coincide (case ($\delta$) of
fig~\ref{fig2}) and \(u_1 = u_2 = u_3 = \frac1{6M}\). As was
shown in section~\ref{sst}, a circular orbit of radius $6M$ is
permitted by the equations of motion as a special solution; it is the lower bound for the radii of the stable circular
orbit (see equations (\ref{s144'time}) and (\ref{s146time})). %The
%general solution can be obtained from the equation
%\begin{equation}%\label{s164time}
%  \left(\frac{\mbox{d}u}{\mbox{d}\varphi}\right)^2 =
%  2M\left(u-\frac1{6M}\right)^3 \,;
%\end{equation}
The general solution is
\begin{equation}\label{s165time}
  u = \frac1{6M} +\frac2{M\left(\varphi-\varphi_0\right)^2} \,.
\end{equation}
\begin{figure}
    \epsfxsize=13cm
    \epsfbox{schwarzs/tm_bnd_d2.eps}
    \caption{The last unstable circular orbit. The orbit of the second
    kind spirals out of the orbit of the first kind and falls into the
    singularity.($e = 0,
    \ \mu =
    1/6$).
    ($\color{blue} M = 1/7$ in the scale along the
    coordinate axes.)}
    \label{timelike d}
\end{figure}
For example of this orbit and associated unstable circular orbit
see fig.~\ref{timelike d}.


\subsubsection{The case $2\mu(3+e) = 1$}
Formally the discussion and solution in this case has the same form
as in section~\ref{ss2}. For this orbit, \(\varphi = 0\) when
\(\xi = \pi\) and \(r \rightarrow 0\); and \(\varphi \rightarrow
\infty\) when \(\xi \rightarrow 0\) and $r$ approaches the aphelion at
\(r = \frac l{1+e}\). The circle at \(\frac l{1+e}\) is the
perihelion for the orbits of the first kind, and it is the
aphelion for the orbits of the second kind. See
fig.~\ref{timelike g}.
\begin{figure}
    \epsfxsize=13cm
    \epsfbox{schwarzs/tm_bnd_g2.eps}
    \caption{An example of a trajectory in which the orbits of the first and
    the second
    kind coalesce ($e = 1/2$). Orbit of the second, starting from a certain
    distance from the black hole, falls into the
    singularity.
    ($M = 1/7$ in the scale along the
    coordinate axes.)}
    \label{timelike g}
\end{figure}


\subsection{The orbits with imaginary eccentricities} \label{ti}
In this case f$(u) = 0$ allows only
one real root and pair of complex-conjugate roots (see the case
($\varepsilon$) of fig.~\ref{fig2}). The
corresponding orbits start at some finite aphelion
distances, and fall into the singularity. As in
sections~\ref{ni} and \ref{si}, these orbits will be characterized
by imaginary eccentricities and roots of f$(u)$ = 0 will be
written as in equation (\ref{s171}). Again we can obtain the
equations which replace equations (\ref{s121time}) and
(\ref{s123time}) by writing $-e^2$ wherever $+e^2$ occurs
and conversely. Thus (cf. equation(\ref{s172}))
\begin{equation}
    %\left.
    \begin{aligned}%{l}
        &{\displaystyle\frac{1}{L^2} = \frac{1}{lM}\left[1 -\mu\left(3
        -e^2\right)\right]\,, \quad
        \frac{1-E^2}{L^2} = \frac{1}{l^2}(1-4\mu)\left(1+e^2\right)}\,,\\
        \mbox{and}\\
        &{\displaystyle\frac{E^2}{L^2} =
        \frac{1}{lM}}\left[\left(2\mu-1\right)^2 -4\mu^2e^2\right]\,.
    \end{aligned}
    %\right\}
    \label{s172time}
\end{equation}
Because bound orbits with \(1 -E^2 > 0\) are considered,  \(\mu <
1/4\); and this inequality guarantees
\begin{equation}\label{s173time}
  1 -3\mu +\mu e^2 > 0 \,.
\end{equation}
There is no upper limit for $e^2$.

Further discussion is the same as in section~\ref{si}, except now
the range of \(u\) is
\begin{equation}\label{s176time}
  \frac1{2M} -\frac2l \leq u < \infty \,,
\end{equation}
and so the corresponding range of \(\xi\) is
\begin{equation}\label{s177time}
  \xi_0 \leq \xi < \pi \,,
\end{equation}
where
%\begin{equation}%\label{s178time}
%  \tan\frac{\xi_0}2 = - \frac{6\mu-1}{2\mu e}\,,
%\end{equation}
%or, equivalently,
\begin{equation}\label{s179time}
  \sin\frac{\xi_0}2 = -\frac{6\mu-1}\triangle \qquad \mbox{and}
  \qquad \cos\frac{\xi_0}2 = \frac{2\mu e}\triangle \,,
\end{equation}
where \(\triangle\) is defined in equation~(\ref{s180}). From
equations (\ref{s179time}) and (\ref{s184}), it follows that
\begin{equation} \label{s185time}
  \begin{array}{crrlc}
    \sin^2\psi = 1 & \mbox{both} & \mbox{when} & \xi = \xi_0 & \mbox{(at
    aphelion)}\\
    &&\mbox{and} & \xi = \pi & \mbox{(at singularity)} \,.
  \end{array}
\end{equation}
And
\begin{equation}\label{s186time}
  \sin^2\psi = 0 \qquad \mbox{when} \qquad \xi = \arctan\frac{2\mu
  e}{6\mu-1}\,.
\end{equation}
%Therefore, \(\psi\) assumes the value zero within the range
%(\ref{s177time}) of \(\xi\). We conclude that
The range of
\(\psi\) associated with the range of \(\xi\) is
\begin{equation}\label{s187time}
  -\frac\pi2 \leq \psi \leq +\frac\pi2 \qquad (\xi_0 \leq \xi \leq
  \pi)\,.
\end{equation}
Accordingly, the solution for \(\varphi\) may be written as
(\ref{s209}), where we have assumed \(\varphi = 0\) at the
singularity where \(\xi = \pi\) and \(\psi = \pi/2\).  \(\varphi\) at aphelion, where \(\xi = \xi_0\) and \(\psi =
-\pi/2\), is
\begin{equation}\label{s189time}
  \varphi_{\scriptstyle\mathrm{ap}} = 2\frac{\K(k)}{\sqrt\triangle}\,.
\end{equation}
\begin{figure}
    \epsfxsize=13cm
    \epsfbox{schwarzs/tm_bnd_i2.eps}
    \caption{Orbit with $l = 1$ and with imaginary eccentricity $e =
        $ i  ($M = 1/7$) falls into the
    singularity.}
    \label{tm_bnd_i}
\end{figure}
Example of an orbit with imaginary eccentricity is in fig.~\ref{tm_bnd_i}.


\section{The unbound orbits ($E^2 > 1$)}
In this case the constant term in f\((u)\) is positive. Therefore, the
equation f\((u) = 0\) allows a negative root; and
the different cases to be distinguished are those shown in fig.~\ref{fig}.
They look essentially as null and space-like orbits.


\subsection{Orbits of the first and second kind}
Because all three roots are real, they shall be expressed the same way
as in section~\ref{su1}, only now \(e
\geq 1\). Again the inequality (\ref{s118}) continues to hold.
The relations (\ref{s121time}) also continue to hold, again with
the difference that \(e \geq 1\). Therefore
\begin{equation}
    \frac{1}{L^2} =
    \frac{1}{lM}\left[1 -\mu\left(3+e^2\right)\right]\quad\mbox{and}\quad
    \frac{E^2-1}{L^2}  =
    \frac{1}{l^2}(1-4\mu)\left(e^2-1\right)\,.
    \label{s192time}
\end{equation}
Since we assume that\(L^2 > 0\) and \(E^2 -1 \geq 0\), in
addition to (\ref{s193}) we have
%\begin{equation}%\label{s194time}
%  \mu \leq \frac14
%\end{equation}
the requirement, \(\mu \leq \frac14\), guaranteed by both
inequalities (\ref{s118}) and (\ref{s193}). (\ref{s193}) ensures
(\ref{s118}) for \(e < 3\) while (\ref{s118}) ensures(\ref{s193})
for \(e \geq 3\).

Apart from the restriction on the range of \(e\) and \(\mu\), the
analysis in section~\ref{su1} is applicable as it stands.
\begin{figure}
    \epsfxsize=13cm
    \epsfbox{schwarzs/tm_ubn_a2.eps}
    \caption{Orbits of the first and
    the second
    kind.   Orbit of the first kind  arrives from infinity, and after reaching
    the perihelion,
    goes
    again to infinity. Orbit of the second, starting from a certain
    distance from the black hole (aphelion), falls into the
    singularity. Eccentricity is $e = 3/2$, and latus rectum 4.5
    ($M = 1/7$).}
    \label{tm ubn a}
\end{figure}
An example of unbound orbits is in fig~\ref{tm ubn a}.


\subsubsection{The case $2\mu(3+e) = 1$}
%The relations (\ref{s192time}) become (cf. equation
%(\ref{s148time}))
%\begin{equation}%\label{s195time}
%    \frac{L^2}{M^2} = \frac{4\left(3+e\right)^2}{(3-e)(1+e)} \qquad %\mbox{and}
%    \qquad
%    E^2 -1 = \frac{e^2-1}{9-e^2}\,.
%\end{equation}
For these special orbits, the allowed range of $e$ is
\begin{equation}\label{s196}
    1 \leq e < 3 \,;
\end{equation}
and the corresponding perihelion-distances must lie in the range
(cf. the range (\ref{s147time}) for the bound orbits and
(\ref{s197}) for space-like orbits)
\begin{equation}\label{s197time}
    3M < r_p \leq 4M \,.
\end{equation}

This case is described by the same equation (\ref{s150}) (except,
again, for the restriction in the range of $\chi$). As was stated
in the context of the null and space-like orbits, these unbound
time-like orbits approach the circle at $r_p$, asymptotically, by
spiralling around it an infinite number of times.

\begin{figure}
    %\epsfxsize=13cm
    \epsfbox{schwarzs/tm_ubn_b2.eps}
    \caption{An example of a trajectory in which the orbits of the first and
    the second
    kind coalesce ($e = 3/2$) for which $2\mu(3+e) = 1$
    ($M = 1/7$). Orbit of the first kind arrives from infinity
    and reaches perihelion by spinning infinitely around it.
    Orbit of the second, starting from a certain
    distance from the black hole (aphelion), falls into the
    singularity.}
    \label{tm ubn b}
\end{figure}

The discussion of the space-like orbits in section \ref{ss2}
applies to these unbound orbits without any change; only now  we are
concerned with values of \(e \geq 1\).


\subsection{The orbits with imaginary eccentricities}
%The bound orbits which fall into the singularity are best
%characterized by imaginary eccentricities (section~\ref{ti}). The
%unbound orbits, which similarly fall into the singularity, can
%likewise be characterized by imaginary eccentricities.

%The relations (\ref{s172time}) for unbound orbits are
%\begin{equation}
%    {\displaystyle\frac{1}{L^2} = \frac{1}{lM}\left[1 -\mu\left(3
%    -e^2\right)\right]
%    \qquad \mbox{and} \qquad
%    \frac{E^2-1}{L^2} = \frac{1}{l^2}(4\mu-1)\left(1+e^2\right)}\,.
%   % \label{s204time}
%\end{equation}
We must now require
\begin{equation}\label{s205time}
    \mu \geq \frac1 4
\end{equation}
and (\ref{s173time}). %For \(\frac{1}{3} \geq \mu \geq
%\frac{1}{4}\), (\ref{s173time}) imposes no restriction on $e^2$;
For \(\mu > \frac1 3\), it is necessary to impose the
restriction
\begin{equation}\label{s206time}
    e^2 > 3 - 1/\mu \qquad \left(\mu > \frac1 3\right) \,.
\end{equation}

Discussion of section~\ref{si} can be now applied without any
change. %(Note that for \(\mu > \frac14\), the right hand side of
%equation~(\ref{s210}) is less than 1; and that it is equal to 1
%for \(\mu = \frac14\), and \(\psi_\infty = -\pi/2\) as in the case
%for the bound orbits.)
