%% Created by Maple 6 (IBM INTEL NT)
%% Source Worksheet: nuimagin.mws
%% Generated: Tue Feb 17 13:20:26 2004
%\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}
\DefineParaStyle{Warning}
\DefineCharStyle{2D Comment}
\DefineCharStyle{2D Math}
\DefineCharStyle{2D Output}
%\begin{document}
%\pagestyle{empty}
\begin{maplegroup}
\begin{Heading 1}
Reissner-Nordstrom solution
\end{Heading 1}

\begin{Heading 2}
Null geodesics
\end{Heading 2}\nopagebreak[3]

\begin{Heading 3}
Complex roots
\end{Heading 3}

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

\end{maplegroup}\vspace{4mm}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{ph := g *(EllipticF(((u_4-y)*B-(y-u_1)*A)
/((u_4-y)*B+(y-u_1)*A), k) +EllipticK(k)) /Q_:}{%
}
\end{mapleinput}

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

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{k := sqrt(((u_4-u_1)^2-(A-B)^2)/4/A/B):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{A := sqrt((u_4-b_1)^2+a_1^2):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{B := sqrt((u_1-b_1)^2+a_1^2):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{b_1 := (u_2-_u_2)^2/2:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{a_1 := sqrt(-(u_2-_u_2)^2/4):}{%
}
\end{mapleinput}

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

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{f := -Q_^2*u^4+2*M*u^3-u^2+1/De^2 = 0:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{D_c := r_cr^2/sqrt(r_cr^2-2*M*r_cr+Q_^2):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Q_ := 0.8:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{with(plots):}{%
}
\end{mapleinput}
{\color{blue}
\mapleresult\vspace{-8mm}
\begin{maplettyout}
Warning, the name changecoords has been redefined
\end{maplettyout}}
\smallskip
\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{PO := `color, portrait, noborder`:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\emptyline
\end{maplegroup}
\begin{Heading 4}
\hspace{18pt}\mapleinline{inert}{2d}{Q_^2 <= M^2;}{%
$\mathit{Q\_}^{2}\leq M^{2}$%
}
\end{Heading 4}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{M := 12/14: r_cr := 1.5*M*(1+sqrt(1-8*Q_^2/9/M^2)):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{D_c;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{3.622947910;}{%
\[\color{blue}
3.622947910
\]
%
}
\end{maplelatex}

\end{maplegroup}\vspace{-4mm}
\begin{maplegroup}
\begin{flushleft}
\hspace{18pt}\mapleinline{inert}{2d}{De < D_c;}{%
$\mathit{De} < \mathit{D\_c}$%
}
\end{flushleft}

\begin{mapleinput}
\mapleinline{active}{1d}{De := 3:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{res := solve(f, u);}{%
}
\end{mapleinput}

\mapleresult\vspace{-4mm}
\begin{maplelatex}
\mapleinline{inert}{2d}{res := -.2711026131, .5402070456-.2253128891*I,
.5402070456+.2253128891*I, 1.869259951;}{%
\maplemultiline{
\textcolor{blue}{\mathit{res} := -.2711026131, \,.5402070456 - .2253128891\,I, \,
.5402070456 + .2253128891\,I,}  \\
\textcolor{blue}{1.869259951} }
%
}
\end{maplelatex}

\end{maplegroup}\vspace{4mm}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{u_1 := res[1]: _u_2 := res[2]: u_2 := res[3]: u_4 :=
res[4]:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{k;}{%
}
\end{mapleinput}

\begin{flushleft}
\vspace{-5mm}\hspace{18pt}should be \TEXTsymbol{<} or = 1.
\end{flushleft}

\mapleresult\vspace{-5mm}
\begin{maplelatex}
\mapleinline{inert}{2d}{.8589066470+0.*I;}{%
\[\color{blue}
.8589066470 + 0.\,I
\]
%
}
\end{maplelatex}

\end{maplegroup}\vspace{4mm}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{rI := polarplot([r, ph, y = 0..u_4], -10..2, -2..2,
thickness = 3, colour = blue):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{rP := polarplot(M+ sqrt(M^2- Q_^2), colour = black):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{rN := polarplot(M- sqrt(M^2- Q_^2), colour = black):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `nuimagi1.eps`, plotoptions =
PO):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(\{rI, rP, rN\}, axes = frame, scaling =
constrained);}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\emptyline
\end{maplegroup}
\begin{Heading 4}
\hspace{18pt}\mapleinline{inert}{2d}{M^2 < Q_^2;}{%
$M^{2} < \mathit{Q\_}^{2}$%
} \TEXTsymbol{<}
\mapleinline{inert}{2d}{9*M^2/8;}{%
$\frac {9\,M^{2}}{8}$%
}
\end{Heading 4}\nopagebreak[1]

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{evalf(11/14);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{.7857142857;}{%
\[\color{blue}
.7857142857
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{M := 11/14:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{is(9*M^2/8 > Q_^2);}{%
}
\end{mapleinput}

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

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{r_cr := 1.5*M*(1-sqrt(1-8*Q_^2/9/M^2)):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{D_c;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{4.414717576;}{%
\[\color{blue}
4.414717576
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{flushleft}
\vspace{-5mm}\hspace{18pt}\mapleinline{inert}{2d}{D_cs < De;}{%
$\mathit{D\_cs} < \mathit{De}$%
}
\end{flushleft}

\begin{mapleinput}
\mapleinline{active}{1d}{De := 5:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{res := solve(f, u);}{%
}
\end{mapleinput}

\mapleresult\vspace{-5mm}
\begin{maplelatex}
\mapleinline{inert}{2d}{res := -.1756926702, .2481796423, 1.191435085-.1177231404*I,
1.191435085+.1177231404*I;}{%
\maplemultiline{
\textcolor{blue}{\mathit{res} := -.1756926702, \,.2481796423, \,1.191435085 -
.1177231404\,I,}  \\
\textcolor{blue}{1.191435085 + .1177231404\,I} }
%
}
\end{maplelatex}

\end{maplegroup}\vspace{4mm}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{u_1 := res[1]: u_4 := res[2]: _u_2 := res[3]: u_2 :=
res[4]: }{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{k;}{%
}
\end{mapleinput}

\begin{flushleft}
\vspace{-5mm}\hspace{18pt}should be \TEXTsymbol{<} or = 1
\end{flushleft}

\mapleresult\vspace{-5mm}
\begin{maplelatex}
\mapleinline{inert}{2d}{.8589066470+0.*I;}{%
\[\color{blue}
.8589066470 + 0.\,I
\]
%
}
\end{maplelatex}

\end{maplegroup}\vspace{4mm}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `nuimagi2.eps`, plotoptions =
PO):}{%
}
\end{mapleinput}

\end{maplegroup}\vspace{2mm}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{polarplot(\{[r, ph, y = 0..u_4]\},
-20..6, -10..10, scaling = constrained, thickness = 3,
axes = frame, color = blue, numpoints = 100);}{%
}
\end{mapleinput}

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

\end{maplegroup}\vspace{-2mm}
\begin{Heading 4}
\bigskip\hspace{18pt}\mapleinline{inert}{2d}{Q_^2 = 9*M^2/8;}{%
$\mathit{Q\_}^{2}=\frac {9\,M^{2}}{8}$%
}
\end{Heading 4}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{M:= sqrt(8*Q_^2/9): r_cr :=
1.5*M*(1+sqrt(1-8*Q_^2/9/M^2)):}{%
}
\end{mapleinput}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{D_c;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{D_c := 2.771281294;}{%
\[\color{blue}
\mathit{D\_c} := 2.771281294
\]
%
}
\end{maplelatex}
\end{maplegroup}
\end{maplegroup}
\begin{maplegroup}
\begin{flushleft}
\vspace{-3mm}\hspace{18pt}\mapleinline{inert}{2d}{D_c < D;}{%
\boldmath$\mathit{D\_c} < \mathit{D}$%
}
\end{flushleft}

\end{maplegroup}
\begin{maplegroup}
\begin{flushleft}
\vspace{-2mm}\hspace{18pt}\mapleinline{inert}{2d}{D_c < De;}{%
$\mathit{D\_c} < \mathit{De}$%
}
\end{flushleft}

\begin{mapleinput}
\mapleinline{active}{1d}{De := 3:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{res := solve(f, u);}{%
}
\end{mapleinput}

\mapleresult\vspace{-3mm}
\begin{maplelatex}
\mapleinline{inert}{2d}{res := -.2754794469, .5566080169,
1.037947017-.2343222916*I,
1.037947017+.2343222916*I;}{%
\maplemultiline{\color{blue}
\mathit{res} := -.2754794469, \,.5566080169, \,1.037947017 -
.2343222916\,I,  \\ \color{blue}
1.037947017 + .2343222916\,I }
%
}
\end{maplelatex}
\bigskip
\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{u_1 := res[1]: u_2 := res[4]: _u_2 := res[3]: u_4 :=
res[2]:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{k;}{%
}
\end{mapleinput}

\begin{flushleft}
\vspace{-5mm}\hspace{18pt}should be \TEXTsymbol{<} or = 1.
\end{flushleft}
\vspace{-5mm}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{.7980466640+0.*I;}{%
\[\color{blue}
.7980466640 + 0.\,I
\]
%
}
\end{maplelatex}
\bigskip
\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `nuimagi3_2.eps`, plotoptions =
\quad PO):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{polarplot(\{[r, ph, y = 0..u_4]\},
\quad-10..10, -10..10, thickness = 3, axes = frame, scaling =
\quad constrained, color = blue);}{%
}
\end{mapleinput}

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

\end{maplegroup}
\begin{maplegroup}
\begin{flushleft}
\vspace{-5mm}\hspace{18pt}\mapleinline{inert}{2d}{D < D_c;}{%
\boldmath$\mathit{D} < \mathit{D\_c}$%
}
\end{flushleft}

\end{maplegroup}
\begin{maplegroup}
\begin{flushleft}
\vspace{-3mm}\hspace{18pt}\mapleinline{inert}{2d}{D_c < De;}{%
$\mathit{D\_c} < \mathit{De}$%
}
\end{flushleft}
\begin{mapleinput}
\mapleinline{active}{1d}{De := 2:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{res := solve(f, u);}{%
}
\end{mapleinput}

\mapleresult\vspace{-3mm}
\begin{maplelatex}
\mapleinline{inert}{2d}{res := -.3860244110, .6880983387-.5165741872*I,
.6880983387+.5165741872*I, 1.366850338;}{%
\maplemultiline{
\color{blue}\mathit{res} := -.3860244110, \,.6880983387 - .5165741872\,I, \,
.6880983387 + .5165741872\,I,  \\ \color{blue}
1.366850338 }
%
}
\end{maplelatex}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{u_1 := res[1]: u_2 :=res[3]: _u_2 := res[2]: u_4 :=
res[4]:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{k;}{%
}
\end{mapleinput}

\begin{flushleft}
\vspace{-5mm}\hspace{18pt}should be \TEXTsymbol{<} or = 1.
\end{flushleft}

\mapleresult\vspace{-3mm}
\begin{maplelatex}
\mapleinline{inert}{2d}{.4912118011+0.*I;}{%
\[\color{blue}
.4912118011 + 0.\,I
\]
%
}
\end{maplelatex}
\bigskip
\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `nuimagi4_2.eps`, plotoptions =
PO):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{polarplot(\{[r, ph, y = 0..u_4]\},
-10..2, -30..30, thickness = 3, axes = frame, scaling =
constrained, color = blue);}{%
}
\end{mapleinput}

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

\end{maplegroup}

\begin{Heading 4}
\hspace{18pt}\mapleinline{inert}{2d}{9*M^2/8 < Q_^2;}{%
$\frac {9\,M^{2}}{8} < \mathit{Q\_}^{2}$%
}
\end{Heading 4}

\begin{maplegroup}
\begin{flushleft}
\hspace{18pt}\mapleinline{inert}{2d}{M;}{%
$M$%
} should be \TEXTsymbol{<}
\mapleinline{inert}{2d}{sqrt(8*Q_^2)/3;}{%
$\frac {\sqrt{8\,\mathit{Q\_}^{2}}}{3}$%
}.
\end{flushleft}

\begin{mapleinput}
\mapleinline{active}{1d}{evalf(sqrt(8*Q_^2)/3);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{.7542472333;}{%
\[\color{blue}
.7542472333
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{M := .7: De := 5:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{res := solve(f, u);}{%
}
\end{mapleinput}

\mapleresult\vspace{-3mm}
\begin{maplelatex}
\mapleinline{inert}{2d}{res := -.1775581062, .2386464340, 1.063205836-.5869991701*I,
1.063205836+.5869991701*I;}{%
\maplemultiline{
\color{blue}\mathit{res} := -.1775581062, \,.2386464340, \,1.063205836 -
.5869991701\,I,  \\
\color{blue}1.063205836 + .5869991701\,I }
%
}
\end{maplelatex}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{u_1 := res[1]: u_2 := res[4]: _u_2 := res[3]: u_4 :=
res[2]:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{k;}{%
}
\end{mapleinput}

\begin{flushleft}
\vspace{-5mm}\hspace{18pt}should be \TEXTsymbol{<} or = 1.
\end{flushleft}

\mapleresult\vspace{-3mm}
\begin{maplelatex}
\mapleinline{inert}{2d}{.1444101517+0.*I;}{%
\[\color{blue}
.1444101517 + 0.\,I
\]
%
}
\end{maplelatex}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `nuimagi5.eps`, plotoptions =
PO):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{polarplot(\{[r, ph, y = 0..u_4], [r, -ph, y = 0..u_4]\},
-80..10, -160..160, scaling = constrained, thickness = 3,
axes = frame, color = blue);}{%
}
\end{mapleinput}

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

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