%% Created by Maple 6 (IBM INTEL NT)
%% Source Worksheet: null-Cart.mws
%% Generated: Sat Dec 03 18:39:11 2005
%\documentclass{article}
%\usepackage{maple2e}
 \def\emptyline{\vspace{12pt}}
\DefineParaStyle{Heading 1}
\DefineParaStyle{Heading 2}
\DefineParaStyle{Heading 3}
\DefineParaStyle{Heading 4}
\DefineParaStyle{Maple Output}
\DefineParaStyle{Maple Output}
\DefineCharStyle{2D Comment}
\DefineCharStyle{2D Math}
\DefineCharStyle{2D Output}
%\begin{document}
%\pagestyle{empty}
\begin{maplegroup}
\begin{Heading 1}
Kerr space-time
\end{Heading 1}

\begin{Heading 2}
Non-equatorial geodesics
\end{Heading 2}

\begin{Heading 3}
Null geodesics
\end{Heading 3}

\begin{Heading 4}
\mapleinline{inert}{2d}{eta < 0;}{%
$\eta  < 0$%
}
\end{Heading 4}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{restart:}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{res := 1/a/mu_plus *EllipticF(sqrt(mu_plus^2
*(mu^2-mu_minus^2)/mu^2/(mu_plus^2-mu_minus^2)), k_theta)
= g*(EllipticF(sin(ph), k_r) +EllipticK(k_r)):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{k_theta := sqrt(mu_plus^2 - mu_minus^2)/mu_plus:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{mu := cos(teta):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{mu_plus := sqrt((abs(eta)+a^2-xi^2
+sqrt((abs(eta)+a^2-xi^2)^2-4*a^2*abs(eta))) /a^2/2):}{%
}
\end{mapleinput}
\medskip
\begin{mapleinput}
\mapleinline{active}{1d}{mu_minus := sqrt((abs(eta)+a^2-xi^2
-sqrt((abs(eta)+a^2-xi^2)^2-4*a^2*abs(eta))) /a^2/2):}{%
}
\end{mapleinput}

\end{maplegroup}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{g:= 2/(A+B):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{ph := arctan((r-b1+a1*g1)/(a1+b1*g1-g1*r)):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{g1 := sqrt((4*a1^2-(A-B)^2)/((A+B)^2-4*a1^2)):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{k_r := sqrt(4*A*B/(A+B)^2):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{A := sqrt((b1-b2)^2+(a1+a2)^2):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{B := sqrt((b1-b2)^2+(a1-a2)^2):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{a1 := sqrt(-(u1-u1bar)^2/4): b1 := (u1+u1bar)/2:}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{b2 := (u3+u3bar)/2: a2 := sqrt(-(u3-u3bar)^2/4):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{R := r^4 +(a^2-xi^2-eta)*r^2 +2*M*(eta+(xi-a)^2)*r
-a^2*eta:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{M:=1: a:= .8: xi:= -.05: eta:= -.3:}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{is(abs(eta) < a^2); is(abs(xi) < abs(a)-sqrt(abs(eta)));
is(a^2 > xi^2); is(sqrt(abs(eta)) < abs(a-xi));}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{true;}{%
\[\color{blue}
\mathit{true}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{true;}{%
\[\color{blue}
\mathit{true}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{true;}{%
\[\color{blue}
\mathit{true}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{true;}{%
\[\color{blue}
\mathit{true}
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{res_r := solve(R);}{%
}
\end{mapleinput}
\vspace{-15pt}
\mapleresult\textcolor{blue}{
\begin{maplelatex}
\mapleinline{inert}{2d}{res_r := -.3616054637-.1237265849*I, -.3616054637+.1237265849*I,
.3616054637-1.087983803*I, .3616054637+1.087983803*I;}{%
\maplemultiline{
\mathit{res\_r} := -.3616054637 - .1237265849\,I, \\
    -.3616054637 + .1237265849\,I,\, 
.3616054637 - 1.087983803\,I,\\
    .3616054637 + 1.087983803\,I }
%
}
\end{maplelatex}}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{u1 := res_r[1]: u1bar := res_r[2]: u3 := res_r[3]:
u3bar := res_r[4]:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{evalf(k_theta); k_r;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{.7240394697;}{%
\[\color{blue}
.7240394697
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{.9969020174+0.*I;}{%
\[\color{blue}
.9969020174 + 0.\,I
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{r := 0.825: res_mu := solve(res, mu);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{res_mu := -.7924506676, .7924506676;}{%
\[\color{blue}
\mathit{res\_mu} := -.7924506676, \,.7924506676
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{evalf(arccos(mu_plus)*360/2/Pi);  # minimum angle}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{4.935416764;}{%
\[\color{blue}
4.935416764
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{evalf(arccos(mu_minus)*360/2/Pi); # maximum angle}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{46.59103978;}{%
\[\color{blue}
46.59103978
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{evalf(mu_plus); evalf(mu_minus);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{.9962923063;}{%
\[\color{blue}
.9962923063
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{.6872011280;}{%
\[\color{blue}
.6872011280
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\end{mapleinput}

\end{maplegroup}
%\end{document}
%% End of Maple 6 Output
