\chapter{The space-like geodesics}\label{kerrspace}
This chapter studies space-like geodesics in Kerr spacetime and it is our original work. As in Chapter \ref{kerrnull} we shall write $R$ and $\Theta$ instead 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 +\frac{r^2\varDelta}{E^2}\label{k190space}\\
    \intertext{and}
    \Theta &= \eta +a^2\left(1+\frac1{E^2}\right)\cos^2\theta -\xi^2\cot^2\theta \label{k191space}\,,
\eal
where $\xi$ and $\eta$ are defined in \eqref{k189}.


\section{The $\theta$-motion}\label{spaceteta}
In this section the $\theta$-motion specified by the integral over $\Theta^{-1/2}$ will be considered, and as in section \ref{nulteta}, the variable of integration shall be $\mu = \cos\theta$. The integral \eqref{k196} is considered with $\Theta_\mu$ in the form
\begin{equation}\label{k271space}
  \Theta = \eta -\left[\xi^2+\eta-a^2\left(1+\frac1{E^2}\right)\right]\mu^2 -a^2\left(1+\frac1{E^2}\right)\mu^4 \,.
\end{equation}

It is convenient to use substitution
\begin{equation}\label{k272space}
  \alpha^2 = a^2 \left(1+\frac1{E^2}\right) \,.
\end{equation}
and write $\Theta_\mu$ as
\begin{equation}\label{k273s[ace}
  \Theta_\mu = \eta -\left(\xi^2+\eta-\alpha^2\right)\mu^2 -\alpha^2\mu^4 \,.
\end{equation}
Now the discussion in section \ref{nulteta} applies with the replacement of $a^2$ by $\alpha^2$. The space-like orbits are very much like null geodesics.


\section{The $r$-motion}
In this section the radial motion specified by the integral over $\text R^{-1/2}$ is investigated.


\subsection{The unstable orbits of constant radius}
\label{kspacecirc}
These special orbits provide a basis for a classification of the orbits, and also illustrate the nature of the orbits. The conditions for the occurrence of orbits with constant radius are
\begin{equation}\label{k281space}
  \mbox R = 0 \qquad \mbox{and} \qquad \frac{\partial\text R}{\partial r} = 0\,.
\end{equation}
After following the same procedure as in section \ref{knullcirc}, only with the additional term, $+r^2\varDelta/E^2$, in the definition of R, we can replace the equations \eqref{k216} and \eqref{k217} by
\begin{equation}\label{k282space}
  3r^4 +r^2a^2 -\eta\left(r^2-a^2\right) +\frac{r^2}{E^2}\left(3r^2-4Mr+a^2\right) = r^2\xi^2
\end{equation}
and
\begin{equation}\label{k283space}
  r^4 -a^2Mr -\eta(a^2-Mr) +\frac{r^3}{E^2}(r-M) = rM\left(\xi^2-2a\xi\right)\,;
\end{equation}
and, as in section \ref{knullcirc}, these equations can be solved for $\xi$ and $\eta$.

Eliminating $\eta$ from the above equations yields (in place of equation \eqref{k218})
\begin{multline}\label{k284space}
  a^2(r-M)\xi^2 +2aM\left(r^2-a^2\right)\xi +\left(r^2+a^2\right)\left[r\left(r^2+a^2\right)-M\left(3r^2-a^2\right)\right] +\frac{r\varDelta}{E^2} \\
= 0 \,;
\end{multline}
and the solution of this equation is given by (cf. equation \eqref{k220})
\begin{equation}\label{k285space}
  \xi = \frac{M(r^2-a^2) \pm r\varDelta\sqrt{1+\dfrac1{E^2}\left(1-\dfrac Mr\right)}} {a(r-M)} \,;
\end{equation}
and the corresponding solution for $\eta$ is (cf. equation \eqref{k225})(see Fig.\ref{spacpara})
\begin{multline}\label{k286space}
  \eta a^2(r-M) = \\
  \frac{2r^3M\varDelta}{r-M} \left[1\mp\sqrt{1+\frac1{E^2}\left(1-\frac Mr\right)}\right] -r^4(r-M) - \frac{r^2}{E^2} \left[r\left(r-2M\right)^2-a^2M\right] \,.
\end{multline}
\begin{figure}
         \epsfxsize=13cm
         \epsfbox{kerr/spacpara.eps}
         \caption{\color{blue}The critical locus $(\xi_s, \eta_s)$ for space-like orbits with $E^2 = 1$. The chosen parameters for the  black hole are $M= 1$ and $a = 0.96$.}
         \label{spacpara}
\end{figure}

Now, let the constants of the motion, $\xi$ and $\eta$, have the values $\xi_s$ and $\eta_s$ appropriate to a particle on a orbit with constant radius $r_s$. That means that in equations \eqref{k285space} and \eqref{k286space} $r$ is replaced by $r_s$. Then $r_s$ is a double root of R$\,= 0$, and R can be reduced to
\begin{equation}\label{k287space}
  \text R = \left(r-r_s\right)^2 \left\{r^2\left(1+\frac1{E^2}\right) +2rr_s\left[1+\frac1{E^2}\left(1-\frac M{r_s}\right)\right] -\frac{a^2\eta_s}{r_s^2}\right\} \,.
\end{equation}
After substitution \eqref{k233} the integral is
\begin{multline}\label{k290space}
  \int\frac{\text dr}{\sqrt\text R} = - \int\text dx \left\{\left(1+\frac1{E^2}\right) +\left[4r_s\left(1+\frac1{E^2}\right) -\frac{2M}{E^2}\right]x\: + \right.\\
\left. +\left[3r_s^2\left(1+\frac1{E^2}\right) -\frac{a^2\eta_s}{r_s^2} -\frac{2Mr_s}{E^2}\right]x^2\right\}^{-\frac12} \,.
\end{multline}
With the definition
\begin{equation}\label{k291space}
  \text X(x) = Ax^2 +Bx +C \,,
\end{equation}
where $A,B$ and $C$ are the coefficients of the quadratic in $x$ that appear in the integral \eqref{k290space}. Therefore
\bal
    \int\frac{\text dr}{\sqrt\text R} &= \frac{\ln\abs{2\sqrt{A \text X} +2Ax +B}} {\sqrt A} &\mbox{for}\ A > 0 \,, \label{k292space1}\\
%&= \frac{\arcsinh\dfrac{2Ax+B}{\sqrt D}}{\sqrt A} &\mbox{for}\ A > 0, D > 0 \,, \label{k292space2}\\
&=-\frac{\arcsin\dfrac{2Ax+B}{\sqrt{-D}}}{\sqrt{-A}} &\mbox{for}\ A < 0%, D < 0
\,, \label{k292space3}
\eal
where
\begin{equation}\label{k293space}
  D = -4\left\{\left[r_s\left(1+\frac1{E^2}\right) -\frac M{E^2}\right]^2 +\frac{\alpha^2\eta_s}{r_s^2}\right\} %\,.
\end{equation}
is negative.


\subsection{The case $\eta = 0$}
The $r$-motion is determined by (cf. equation
\eqref{k190space})
\begin{equation}\label{k236space}
  \int\frac{\text dr}{\sqrt\text R} = \int\frac{\text dr}{\sqrt{r\left\{\left(1+\frac1{E^2}\right)r^3 +\left[a^2\left(1+\frac1{E^2}\right) -\xi^2 -\frac{2Ma}{E^2}\right]r +2M\left(a-\xi\right)^2\right\}}} \,,
\end{equation}
while the $\theta$-motion is determined by the equation \eqref{k203}. By substituting $r$ with $u = 1/r$ as the independent variable, the above integral becomes
\begin{equation}\label{k236space_u}
  -\int\frac{\text du}{\sqrt{\left(1+\frac1{E^2}\right)
+\left[a^2\left(1+\frac1{E^2}\right) -\xi^2
-\frac{2Ma}{E^2}\right]u^2 +2M\left(\xi-a\right)^2u^3}} \,
\end{equation}
and can be reduced to a Jacobian elliptic integral.

When $a^2(1+1/E^2) > (L^2+2Ma)/E^2$, there is only one real root of the cubic polynomial, $u_1$, and the integration gives
\begin{equation}\label{k237space1}
  -g[\F(\varphi, k) +\K(k)] \,,
\end{equation}
where
\begin{equation}\label{k237spacefi}
  \varphi = \arccos\frac{A+u_1-u}{A-u_1+u}
\end{equation}
and $g$ is defined by equations \eqref{241g} \eqref{241A},
\eqref{241a1} and \eqref{241b1}, and $k$ is as in equation \eqref{241k}.

When $a^2(1+1/E^2) < (L^2+2Ma)/E^2$, there are three real roots of the cubic polynomial, $u_1 < 0 < u_2 < u_3$, and the above solution holds for $u \leq u_2$, with $\varphi$ as it is in equation \eqref{234fi}, $k$ in \eqref{k}, and $g$ in
\eqref{g}. For $ u_3 \leq u < \infty$, the solution becomes
\begin{equation}\label{k237space2}
  -g\F(\varphi, k) \,,
\end{equation}
where
\begin{equation}\label{238.00fi}
  \varphi = \arcsin\sqrt{\frac{u_3-u_1}{u-u_1}} \,,
\end{equation}
and $k$ and $g$ are defined by \eqref{k} and \eqref{g} respectively.

\section{The $\varphi$-motion}
The integral $I_\mu$ is solved in section \ref{nullphi}, with the replacement of $a^2$ by $\alpha^2$ in \eqref{213}.


\subsection{The unstable orbits of constant radius}
Substituting \eqref{k287space} for R yields to the integral over $r$ in \eqref{kch181} having form
\begin{equation}\label{kspaceIR}
  I_{\text R} = a \int^r\frac{(r^2+a^2)E -aL}{\varDelta(r-r_s)\sqrt{{r^2\left(1+\dfrac1{E^2}\right) +2rr_s\left[1+\dfrac1{E^2}\left(1-\dfrac M{r_s}\right)\right] -\dfrac{a^2\eta_s}{r_s^2}}}} \text dr  \,.
\end{equation}
Substitution \eqref{k233} and the partial fractions method transform this integral to three integrals
\begin{multline}\label{kspaceIRfin}
  I_x = \frac{a[aL-E(a^2+r_s^2)]}{\varDelta_s}\int\frac{\text dx}{\sqrt{\text X(x)}} \\ -\frac{a(aL-2MEr_s^2)}{\varDelta_s}\int\frac{\text dx}{[\varDelta_sx^2+2(r_s-M)x+1]\sqrt{\text X(x)}} \\ -\frac{2a[ME(a^2-r_s^2)+aL(r_s-M)]}{\varDelta_s}\int\frac{x\,\text dx}{[\varDelta_sx^2+2(r_s-M)x+1]\sqrt{\text X(x)}} \,,
\end{multline}
where \eqref{k291space} is the definition of X$(x)$ and $\varDelta_s = r_s^2 -2Mr_s +a^2$. 




