%% Created by Maple 6 (IBM INTEL NT)
%% Source Worksheet: null_2.mws
%% Generated: Tue Sep 20 13:10:08 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{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}
\mapleinline{inert}{2d}{Q_^2 <= M^2;}{%
$\mathit{Q\_}^{2}\leq M^{2}$%
}
\end{Heading 2}

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

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

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

\end{maplegroup}
\begin{maplegroup}
\begin{flushleft}
 
\hspace{18pt}\mapleinline{inert}{2d}{M;}{%
$M$%
} should be \TEXTsymbol{>} 
\mapleinline{inert}{2d}{.8 = Q_;}{%
$.8=\mathit{Q\_}$%
}.
\end{flushleft}

\begin{mapleinput}
\mapleinline{active}{1d}{evalf(12/14); Q_ := .8: is(12/14 > Q_);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{.8571428571;}{%
\[\color{blue}
.8571428571
\]
%
}
\end{maplelatex}

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

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{r_c := subs(M = 12/14, r_cr);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{r_c := 1.896501668;}{%
\[\color{blue}
\mathit{r\_c} := 1.896501668
\]
%
}
\end{maplelatex}

\end{maplegroup}
\vspace{-4mm}\begin{maplegroup}
\begin{flushleft}
\hspace{18pt}should be \TEXTsymbol{>} 
\mapleinline{inert}{2d}{r_c;}{%
$\mathit{r\_c}$%
}.
\end{flushleft}

\begin{mapleinput}
\mapleinline{active}{1d}{r2 := subs(M = 18/14, r_cr); is(r2 > r_c);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{r2 := 3.490425376;}{%
\[\color{blue}
\mathit{r2} := 3.490425376
\]
%
}
\end{maplelatex}

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

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{M := 12/14: subs(u = 1/r2, f);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{-.4607972106e-1+1/(De^2) = 0;}{%
\[\color{blue}
 - .04607972106 + {\displaystyle \frac {1}{\mathit{De}^{2}}} =0
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{flushleft}
\hspace{18pt}Impact parameter\end{flushleft} 
\mapleinline{active}{1d}{IP := \{solve(\%, De)\};}{%
}


\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{IP := \{-4.658489053, 4.658489053\};}{%
\[\color{blue}
\mathit{IP} := \{-4.658489053, \,4.658489053\}
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{subs(De = IP[2], f);}{%
}
\end{mapleinput}

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

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{-.64*u^4+12/7*u^3-u^2+.4607972106e-1 = 0;}{%
\[\color{blue}
 - .64\,u^{4} + {\displaystyle \frac {12}{7}} \,u^{3} - u^{2} + 
.04607972106=0
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{u := -.1854454261, .2864980317, .7358253611, 1.841693462;}{%
\[\color{blue}
u := -.1854454261, \,.2864980317, \,.7358253611, \,1.841693462
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{flushleft}
\hspace{18pt}Cauchy horizon
\end{flushleft}

\begin{mapleinput}
\mapleinline{active}{1d}{r_Ca := M- sqrt(M^2- Q_^2);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{r_Ca := .5494191538;}{%
\[\color{blue}
\mathit{r\_Ca} := .5494191538
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{flushleft}
\hspace{18pt}Is turning point of geodesics of second kind inside the Cauchy
horizon?
\end{flushleft}

\begin{mapleinput}
\mapleinline{active}{1d}{1/u[4]; is(1/u[4] < r_Ca);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{.5429785253;}{%
\[\color{blue}
.5429785253
\]
%
}
\end{maplelatex}

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

\end{maplegroup}
\begin{maplegroup}
\begin{flushleft}
\hspace{18pt}The turning point of the geodesic of the first kind
\end{flushleft}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{1/u[2];}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{3.490425376;}{%
\[\color{blue}
3.490425376
\]
%
}
\end{maplelatex}

\end{maplegroup}\vspace{5mm}
\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) +EllipticK(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{flushleft}
\hspace{18pt}\mapleinline{inert}{2d}{k;}{%
$k$%
} should be \TEXTsymbol{<} or = 1.
\end{flushleft}

\begin{mapleinput}
\mapleinline{active}{1d}{k := sqrt((u[4]-u[3])*(u[2]-u[1])/(u[4]-u[2])
/(u[3]-u[1])); is(k <= 1);}{%
}
\end{mapleinput}

\mapleresult\vspace{-8mm}
\begin{maplelatex}
\mapleinline{inert}{2d}{k := .6035461928;}{%
\[\color{blue}
k := .6035461928
\]
%
}
\end{maplelatex}

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

\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}{Fi1 := polarplot([r, ph1, y = 0..u[2]], -4..10, -10..10,
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}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{rP := polarplot(M+ sqrt(M^2- Q_^2), color = black):}{%
}
\end{mapleinput}

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

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

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `null2.eps`, plotoptions =
`color, portrait, noborder`):}{%
}
\end{mapleinput}

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

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

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