%% Created by Maple 6 (IBM INTEL NT)
%% Source Worksheet: tim_Cart.mws
%% Generated: Thu Mar 30 23:26:49 2006
%\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}
Timelike 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}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{res := 1/alpha/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)+alpha^2-xi^2
+sqrt((abs(eta)+alpha^2-xi^2)^2-4*alpha^2
*abs(eta))) /alpha^2/2):}{%
}
\end{mapleinput}
\medskip
\begin{mapleinput}
\mapleinline{active}{1d}{mu_minus := sqrt((abs(eta)+alpha^2-xi^2
-sqrt((abs(eta)+alpha^2-xi^2)^2-4*alpha^2
*abs(eta))) /alpha^2/2):}{%
}
\end{mapleinput}

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

\begin{mapleinput}
\mapleinline{active}{1d}{ph := arccos(((A-B)*r+u2*B-u1*A)/((A+B)*r-u2*B-u1*A)):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{k_r := sqrt(((A+B)^2-(u2-u1)^2)/A/B)/2:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{A := sqrt((u2-b1)^2+a1^2):}{%
}
\end{mapleinput}

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

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{a1 := sqrt(-(u3-u3bar)^2)/2: b1 := (u3+u3bar)/2:}{%
}
\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 -r^2*Delta/E^2:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{alpha := a*sqrt(1-E^(-2)): Delta := r^2-2*M*r+a^2:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{M:=1: a:= sqrt(.84): xi:= -.5: eta:= -1/9: E:=3:}{%
}
\end{mapleinput}

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

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{res_r := -1.551956533, -.2471812107e-1, .6633373271-1.515620811*I,
.6633373271+1.515620811*I;}{%
\maplemultiline{\color{blue}
\mathit{res\_r} := -1.551956533, \,-.02471812107, \,.6633373271
 - 1.515620811\,I,  \\\color{blue}
.6633373271 + 1.515620811\,I }
%
}
\end{maplelatex}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{u1 := res_r[1]: u2 := 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}{.6953171656;}{%
\[\color{blue}
.6953171656
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{.9631517120+0.*I;}{%
\[\color{blue}
.9631517120 + 0.\,I
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{r_p := M+sqrt(M^2-a^2); r_m := M-sqrt(M^2-a^2);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{r_p := 1.400000000;}{%
\[\color{blue}
\mathit{r\_p} := 1.400000000
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{r_m := .6000000000;}{%
\[\color{blue}
\mathit{r\_m} := .6000000000
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{r := 1.8: res_mu := solve(res, mu);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{res_mu := -.7304418066, .7304418066;}{%
\[\color{blue}
\mathit{res\_mu} := -.7304418066, \,.7304418066
\]
%
}
\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}{.7326270425;}{%
\[\color{blue}
.7326270425
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{.5265412721;}{%
\[\color{blue}
.5265412721
\]
%
}
\end{maplelatex}

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

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