%% Created by Maple 6 (IBM INTEL NT)
%% Source Worksheet: null2.mws
%% Generated: Fri Jul 02 04:36:32 2004
%\documentclass{article}
%\usepackage{maple2e}
 \def\emptyline{\vspace{12pt}}
\DefineParaStyle{Heading 1}
\DefineParaStyle{Heading 2}
\DefineParaStyle{Heading 3}
\DefineParaStyle{Maple Output}
\DefineParaStyle{Warning}
\DefineCharStyle{2D Comment}
\DefineCharStyle{2D Math}
\DefineCharStyle{2D Output}
%\begin{document}
%\pagestyle{empty}
\begin{maplegroup}
\begin{Heading 1}
Kerr black hole
\end{Heading 1}

\begin{Heading 2}
Null geodesics
\end{Heading 2}

\begin{Heading 3}
 
\vspace{-10mm}\hspace{-15pt}
\mapleinline{inert}{2d}{eta = 0;}{%
$\eta =0$%
}
\end{Heading 3}

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

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{fii := a*E*g*(EllipticF(sin(ph), k)+EllipticK(k))
+a*g*(B-A)/(r_p+r_m)*
(((a^2+r_p^2)*E-a*L)*(alfa_2*(EllipticF(sin(ph),
k)+EllipticK(k))
+(alfa_p-alfa_2)/(1-alfa_p)*(EllipticPi(sin(ph),
alfa_p^2/(alfa_p^2-1),
k)+EllipticCPi(alfa_p^2/(alfa_p^2-1),
k)-2*alfa_p*f_p))/(A*r1-(A+B)*r_p)
-((a^2+r_m^2)*E-a*L)*(alfa_2*(EllipticF(sin(ph),
k)+EllipticK(k))
+(alfa_m-alfa_2)/(1-alfa_m)*(EllipticPi(sin(ph),
alfa_m^2/(alfa_m^2-1),
k)+EllipticCPi(alfa_m^2/(alfa_m^2-1),
k)-2*alfa_m*f_m))/(A*r1-(A+B)*r_m))
+(L-a*E)/sqrt(a^2-xi^2)
*ln(abs((sqrt(1-xi^2/a^2)+sqrt(1-xi^2/a^2-mu^2))/mu))
+L/2/xi
*arcsin((1-mu-xi^2/a^2)/(mu-1)/sqrt(1-xi^2/a^2))
+L/2/xi *arcsin((mu+1-xi^2/a^2)/(mu+1)/sqrt(1-xi^2/a^2)):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{k_ := sqrt(1-k^2):}{%
}
\end{mapleinput}

\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}

\begin{mapleinput}
\mapleinline{active}{1d}{alfa_2 := (B+A)/(B-A):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{alfa_p := (A*r1-A*r_p-B*r_p)/(-A*r1+A*r_p-B*r_p):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{alfa_m := (A*r1-A*r_m-B*r_m)/(-A*r1+A*r_m-B*r_m):}{%
}
\end{mapleinput}

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

\begin{mapleinput}
\mapleinline{active}{1d}{ph := arccos(((A-B)*r-A*r1)/((A+B)*r-A*r1)):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{r1 := -sqrt(4*(a^2-xi^2)/3)*sinh(mateta/3):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{mateta := arcsinh(M*sqrt((a-xi)/(a+xi)^3)):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{psi := arccos((r*A-r*B+r_1*A)/(r*A+r*B+r_1*A)):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{k := 1/2*sqrt(-A*B*(-A^2-2*A*B-B^2+r_1^2))/A/B:}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{B := sqrt(3*r_1^2+a^2-xi^2):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{A := sqrt(r_1^2+a^2-xi^2):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{r_1 := -sqrt(4*(a^2-xi^2)/3)*sinh(teta):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{teta := 1/3*arcsinh(3*(M^2*(a-xi)/((a+xi)^3))^(1/3)):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{mu := cos(theta): mu_max := sqrt(1-xi^2/a^2): xi := L/E:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{E:=1: a:=.8: L:=.4:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{evalf(subs(M = 1, alfa_p^2/(alfa_p^2-1) -k^2));}{%
}
\end{mapleinput}

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{.418448815e-1;}{%
\[
.0418448815
\]
%
}
\end{maplelatex}

\end{maplegroup}
\vspace{-15pt}
\begin{maplegroup}
\begin{flushleft}
\hspace{17pt}because above is positive
\end{flushleft}

\begin{mapleinput}
\mapleinline{active}{1d}{f_p := sqrt((alfa_p^2-1)/(alfa_p^2*k_^2+k^2)) /2
*ln(abs((sqrt(k^2+k_^2*alfa_p^2)*JacobiDN(sin(ph),
k)+sqrt(alfa_p^2-1)*JacobiSN(sin(ph),
k))/(sqrt(k^2+k_^2*alfa_p^2)*JacobiDN(sin(ph),
k)-sqrt(alfa_p^2-1)*JacobiSN(sin(ph), k)))):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{evalf(subs(M = 1, alfa_m^2/(alfa_m^2-1) -k^2));}{%
}
\end{mapleinput}

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{.402442245e-1;}{%
\[
.0402442245
\]
%
}
\end{maplelatex}

\end{maplegroup}
\vspace{-15pt}
\begin{maplegroup}
\begin{flushleft}
\hspace{17pt}because above is positive
\end{flushleft}

\begin{mapleinput}
\mapleinline{active}{1d}{f_m := sqrt((alfa_m^2-1)/(alfa_m^2*k_^2+k^2)) /2
*ln(abs((sqrt(k^2+k_^2*alfa_m^2)*JacobiDN(sin(ph),
k)+sqrt(alfa_m^2-1)*JacobiSN(sin(ph),
k))/(sqrt(k^2+k_^2*alfa_m^2)*JacobiDN(sin(ph),
k)-sqrt(alfa_m^2-1)*JacobiSN(sin(ph), k)))):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{vysl := subs(M = 1, arcsech(mu/mu_max)/sqrt(a^2-xi^2) =
EllipticF(sin(psi), k)/sqrt(A*B)):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{with(plots):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{addcoords(rtheta,[e1,e2],[e1,e1*cos(e2)]):}{%
}
\end{mapleinput}

\mapleresult\color{blue}\vspace{-15pt}
\begin{maplettyout}
Warning, the name changecoords has been redefined
\end{maplettyout}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `null1.eps`, plotoptions =
`portrait, noborder`):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{obr := implicitplot(vysl, r = 0..3, theta=0..Pi/2,
numpoints=500, color = black, thickness = 3):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{changecoords(obr, rtheta);}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{r := 1.107:}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{theta_fin := solve(vysl, theta);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{evalf(subs(\{M = 1, theta = theta_fin\}, fii));}{%
}
\end{mapleinput}

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{theta_fin := 1.175652122;}{%
\[
\mathit{theta\_fin} := 1.175652122
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{-14.51054171;}{%
\[
-14.51054171
\]
%
}
\end{maplelatex}

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

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