%% Created by Maple 6 (IBM INTEL NT)
%% Source Worksheet: timeplot.mws
%% Generated: Tue Feb 24 14:07:05 2004
%\documentclass{article}
%\usepackage{maple2e}
 \def\emptyline{\vspace{12pt}}
\DefineParaStyle{Heading 1}
\DefineParaStyle{Heading 3}
\DefineParaStyle{Heading 4}
\DefineParaStyle{Maple Output}
\DefineParaStyle{Warning}
\DefineCharStyle{2D Math}
\DefineCharStyle{2D Output}
%\begin{document}
%\pagestyle{empty}
\begin{maplegroup}
\begin{Heading 1}
Reissner-Nordstrom solution
\end{Heading 1}

\begin{Heading 3}
Time-like geodesics
\end{Heading 3}

\begin{Heading 4}
All real roots
\end{Heading 4}

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

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{ph1 := g *EllipticF(arcsin(sqrt((u_4-u_2) *(y-u_1)
/(u_2-u_1) /(u_4-y))), k)/Q_:}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{ph2 := g *EllipticF(arcsin(sqrt((u_4-u_2) *(y-u_3)
/(u_4-u_3) /(y-u_2))), k)/Q_:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{g := 2/sqrt((u_4-u_2)*(u_3-u_1)):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{k := sqrt((u_4-u_3)*(u_2-u_1)/(u_4-u_2)/(u_3-u_1)):}{%
}
\end{mapleinput}

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

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

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{u_c := 0.2225: Q_ := .8: M := 1.2: r_c := 1/u_c;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{r_c := 4.494382022;}{%
\[\color{blue}
\mathit{r\_c} := 4.494382022
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{r_r := 9*M^2/8; Q_^2; Q_^2/M;}{%
}
\end{mapleinput}

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{r_r := 1.620000000;}{%
\[
\mathit{r\_r} := 1.620000000
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{.64;}{%
\[
.64
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{.5333333333;}{%
\[
.5333333333
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{flushleft}
\hspace{18pt}Following must hold:
\end{flushleft}

\vspace{-8mm}\begin{mapleinput}
\mapleinline{active}{1d}{1 -3*M*u_c +2*Q_^2*u_c^2 > 0;}{%
}
\end{mapleinput}

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{0 < .2623680000;}{%
\[
0 < .2623680000
\]
%
}
\end{maplelatex}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{E_c := (1-2*M*u_c+Q_^2*u_c^2)/ sqrt(1-3*M*u_c
+2*Q_^2*u_c^2);}{%
}
\end{mapleinput}
\medskip
\begin{mapleinput}
\mapleinline{active}{1d}{L_c := sqrt(M-Q_^2*u_c)/ sqrt(u_c*(1-3*M*u_c
+2*Q_^2*u_c^2));}{%
}
\end{mapleinput}

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{E_c := .9716240286;}{%
\[
\mathit{E\_c} := .9716240286
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{L_c := 4.256378896;}{%
\[
\mathit{L\_c} := 4.256378896
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{L := 6: E := 0.98:}{%
}
\end{mapleinput}

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

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{u := .2671189456e-1, .4910213582e-1, .4002564198, 3.273929550;}{%
\[
u := .02671189456, \,.04910213582, \,.4002564198, \,3.273929550
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{u_1 := u[1]: u_3 := u[3]: u_4 := u[4]: u_2 := u[2]:}{%
}
\end{mapleinput}

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

\begin{flushleft}
\vspace{-8mm}\hspace{18pt}should be \TEXTsymbol{<} or = 1.
\end{flushleft}

\mapleresult\vspace{-5mm}\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{.2311126109;}{%
\[
.2311126109
\]
%
}
\end{maplelatex}

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

\mapleresult\vspace{-8mm}\color{blue}
\begin{maplettyout}
Warning, the name changecoords has been redefined
\end{maplettyout}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Fi1 := polarplot([r, ph1, y = u_1..u_2], -24..38, -30..30,
thickness = 3, color = blue):}{%
}
\end{mapleinput}

\end{maplegroup}\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Fi2 := polarplot([r, -ph1, y = u_1..u_2], thickness = 3,
color = blue):}{%
}
\end{mapleinput}

\end{maplegroup}\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Se := polarplot([r, ph2, y = u_3..u_4], thickness = 3,
color = blue):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{rC := polarplot(r_c, linestyle = 2, color = black):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{rR := polarplot(r_r, color = black):}{%
}
\end{mapleinput}

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

\end{maplegroup}\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(\{Fi1, Fi2, Se, rC, rR\}, axes = frame, scaling =
constrained);}{%
}
\end{mapleinput}

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

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