%% Created by Maple 6 (IBM INTEL NT)
%% Source Worksheet: RLEQSPAC.mws
%% Generated: Tue Jun 22 22:57:01 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{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}
Equatorial plane
\end{Heading 2}

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

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

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

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

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

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{popt := `portrait, noborder, color`:}{%
}
\end{mapleinput}

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

\end{maplegroup}
\begin{maplegroup}
\begin{Heading 4}
Slow Kerr solution
\end{Heading 4}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{a := .96:}{%
}
\end{mapleinput}

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

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{r_pm := 1.280000000, .7200000000;}{%
\[
\mathit{r\_pm} := 1.280000000, \,.7200000000
\]
%
}
\end{maplelatex}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{r_ := PLOT(CURVES([[r_pm[2],-16],[r_pm[2],8]],
[[r_pm[1],-16],[r_pm[1],8]]), LINESTYLE(3)):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{popis2 := textplot([[r_pm[2], 0, "r_-"], [r_pm[1], 0,
"r_+"]], 'align = BELOW'):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{popis3 := textplot([0, 2*M, "2M"], align = 'LEFT'):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{E1 := sqrt(r_pm[1]/r_pm[2]);}{%
}
\end{mapleinput}

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{E1 := 1.333333333;}{%
\[
\mathit{E1} := 1.333333333
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{E := 2: }{%
}
\vspace{-15pt}\hspace{18mm}\mapleinline{inert}{2d}{sqrt(r_plus/r_minus) < E;}{%
$\sqrt{\frac {\mathit{r\_plus}}{\mathit{r\_minus}}} < E$%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{minu := plot(L_minus, r = 0..5*M, color = blue, thickness
= 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus1 := plot(L_plus, r = 0..2*M, L = 0..8*M, color =
blue, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus2 := plot(L_plus, r = 2*M..5*M, L = -16..0, color =
blue, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{popis := textplot(\{[1.7, 7, N], [4, -16, S], [.6, 2.7,
W]\}, font = [TIMES, BOLDITALIC, 10]):}{%
}
\end{mapleinput}

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

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `rLeqsls1.eps`, plotoptions =
popt):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(plus1, plus2, minu, r_, popis, popis2, popis3,
title = "r-L plot in the slow Kerr solution: The
space-like geodesics\\n E > sqrt(r_+/r_-)", titlefont =
[TIMES, ROMAN, 15]);}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\emptyline
\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{E := E1:}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{minu := plot(L_minus, r = 0..5*M, color = blue, thickness
= 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus1 := plot(L_plus, r = 0..2*M, L = 0..8*M, color =
blue, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus2 := plot(L_plus, r = 2*M..5*M, L = -16..0, color =
blue, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{bod := PLOT(POINTS([r_pm[2], 2], SYMBOL(CIRCLE)), 
COLOR(RGB, 0, 0, 1)):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{dvaM := plot(2*M, r = 0..r_pm[2], linestyle = 2, color =
black):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `rLeqsls2.eps`, plotoptions =
popt):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(plus1, plus2, minu, bod, r_, dvaM, title = "r-L
plot in the slow Kerr solution: The space-like
geodesics\\n E = sqrt(r_+/r_-)", titlefont = [TIMES, ROMAN,
15], popis, popis2, popis3);}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\emptyline
\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{E2 := sqrt(r_pm[2]/r_pm[1]);}{%
}
\end{mapleinput}

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{E2 := .7500000000;}{%
\[
\mathit{E2} := .7500000000
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{E := 1: }{%
}
\vspace{-8.5pt}\hspace{18mm}\mapleinline{inert}{2d}{E2 < E;}{%
$\mathit{E2} < E$%
} \TEXTsymbol{<} 
\mapleinline{inert}{2d}{E1;}{%
$\mathit{E1}$%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{minu := plot(L_minus, r = 0..5*M, color = black, thickness
= 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus1 := plot(L_plus, r = 0..2*M, L = 0..8*M, color =
black, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus2 := plot(L_plus, r = 2*M..5*M, L = -16..0, color =
black, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{popis := textplot(\{[2.5, 6, N], [3.5, -13, S], [.85, 2,
W]\}, font = [TIMES, BOLDITALIC, 10]):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `rLeqsls3.eps`, plotoptions =
popt):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(plus1, plus2, minu, r_, title = "r-L plot in the
slow Kerr solution: The space-like geodesics\\n
sqrt(r_-/r_+) < E < sqrt(r_+/r_-)", titlefont = [TIMES,
ROMAN, 15], popis, popis2, popis3);}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\emptyline
\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{E := E2:}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{minu := plot(L_minus, r = 0..5*M, color = black, thickness
= 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus1 := plot(L_plus, r = 0..2*M, L = 0..8*M, color =
black, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus2 := plot(L_plus, r = 2*M..5*M, L = -16..0, color =
black, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{dvaM := plot(2*M, r = 0..r_pm[1], linestyle = 2, color =
black):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{popis := textplot(\{[2.4, 6, N], [3.5, -12, S], [1, 1.5,
W]\}, font = [TIMES, BOLDITALIC, 10]):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `rLeqsls4.eps`, plotoptions =
popt):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(plus1, plus2, minu, r_, dvaM, title = "r-L plot in
the slow Kerr solution: The space-like geodesics\\n E =
sqrt(r_-/r_+)", titlefont = [TIMES, ROMAN, 15], popis,
popis2, popis3);}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\emptyline
\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{E := .6: }{%
}
\vspace{-8pt}\hspace{21mm}\mapleinline{inert}{2d}{0;}{%
$0$%
} \TEXTsymbol{<} 
\mapleinline{inert}{2d}{E < E2;}{%
$E < \mathit{E2}$%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{minu := plot(L_minus, r = 0..5*M, color = black, thickness
= 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus1 := plot(L_plus, r = 0..2*M, L = 0..8*M, color =
black, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus2 := plot(L_plus, r = 2*M..5*M, L = -16..0, color =
black, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `rLeqsls5.eps`, plotoptions =
popt):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(plus1, plus2, minu, r_, title = "r-L plot in the
slow Kerr solution: The space-like geodesics\\n 0 < E <
sqrt(r_-/r_+)", titlefont = [TIMES, ROMAN, 15], popis,
popis2, popis3);}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\emptyline
\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{E := 0:}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{minu := plot(L_minus, r = 0..5*M, color = black, thickness
= 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus1 := plot(L_plus, r = 0..2*M, L = 0..8*M, color =
black, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus2 := plot(L_plus, r = 2*M..5*M, L = -16..0, color =
black, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{popis := textplot(\{[3, 6, N], [3.4, -12, S], [1, 0, W]\},
font = [TIMES, BOLDITALIC, 10]):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `rLeqsls6.eps`, plotoptions =
popt):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(plus1, plus2, minu, r_, title = "r-L plot in the
slow Kerr solution: The space-like geodesics\\n E = 0",
titlefont = [TIMES, ROMAN, 15], popis, popis2, popis3);}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{Heading 4}
Extreme Kerr solution
\end{Heading 4}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{a := M:}{%
}
\end{mapleinput}

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

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{r_pm := 1, 1;}{%
\[
\mathit{r\_pm} := 1, \,1
\]
%
}
\end{maplelatex}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{popis2 := textplot([[r_pm[2], 0, "r_+/-"]], 'align =
BELOW'):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{minu := plot(L_minus, r = 0..5*M, color = blue, thickness
= 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus1 := plot(L_plus, r = 0..2*M, L = 0..8*M, color =
blue, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus2 := plot(L_plus, r = 2*M..5*M, L = -16..0, color =
blue, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{popis := textplot(\{[3, 6, N], [3.4, -12, S], [.9, 0.5,
W]\}, font = [TIMES, BOLDITALIC, 10]):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{bod := PLOT(POINTS([r_pm[2], 0], SYMBOL(CIRCLE)), 
COLOR(RGB, 0, 0, 1)):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{r_ := PLOT(CURVES([[r_pm[2],-16],[r_pm[2],8]],
[[r_pm[1],-16],[r_pm[1],8]]), LINESTYLE(3)):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `rLeqexs1.eps`, plotoptions =
popt):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(plus1, plus2, minu, r_, bod, title = "r-L plot in
the extreme Kerr solution\\n The space-like geodesics with
E = 0", titlefont = [TIMES, ROMAN, 15], popis, popis2,
popis3);}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\emptyline
\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{E := .6: }{%
}
\vspace{-8pt}\hspace{21mm}\mapleinline{inert}{2d}{0;}{%
$0$%
} \TEXTsymbol{<} 
\mapleinline{inert}{2d}{E < 1;}{%
$E < 1$%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{minu := plot(L_minus, r = 0..5*M, color = blue, thickness
= 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus1 := plot(L_plus, r = 0..2*M, L = 0..8*M, color =
blue, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus2 := plot(L_plus, r = 2*M..5*M, L = -16..0, color =
blue, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{bod := PLOT(POINTS([r_pm[2], 2*M*E], SYMBOL(CIRCLE)),
COLOR(RGB, 0, 0, 1)):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{dvaME := plot(2*M*E, r = 0..r_pm[2], linestyle = 2, color
= black):}{%
}
\end{mapleinput}

\end{maplegroup}
\vspace{-5pt}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{popis4 := textplot([0, 2*M*E, "2ME"], align = 'LEFT'):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `rLeqexs2.eps`, plotoptions =
popt):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(plus1, plus2, minu, r_, bod, dvaME, title = "r-L
plot in the extreme Kerr solution\\n The space-like
geodesics with 0 < E < 1", titlefont = [TIMES, ROMAN, 15],
popis, popis2, popis3, popis4}{%
}
%\mapleinline{active}{1d}{);}{%
%}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\emptyline
\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{E := 1:}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{minu := plot(L_minus, r = 0..5*M, color = blue, thickness
= 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus1 := plot(L_plus, r = 0..2*M, L = 0..8*M, color =
blue, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus2 := plot(L_plus, r = 2*M..5*M, L = -16..0, color =
blue, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{popis := textplot(\{[3, 6, N], [3.4, -12, S]\}, font =
[TIMES, BOLDITALIC, 10]):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{dvaM := plot(2*M, r = 0..r_pm[2], linestyle = 2, color =
black):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `rLeqexs3.eps`, plotoptions =
popt):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(plus1, plus2, minu, r_, dvaM, title = "r-L plot in
the extreme Kerr solution\\n The space-like geodesics with
E = 1", titlefont = [TIMES, ROMAN, 15], popis, popis2,
popis3);}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\emptyline
\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{E := 2: }{%
}
\vspace{-8pt}\hspace{21mm}\mapleinline{inert}{2d}{1 < E;}{%
$1 < E$%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{minu := plot(L_minus, r = 0..5*M, color = blue, thickness
= 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus1 := plot(L_plus, r = 0..2*M, L = 0..8*M, color =
blue, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus2 := plot(L_plus, r = 2*M..5*M, L = -16..0, color =
blue, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{dvaME := plot(2*M*E, r = 0..r_pm[2], linestyle = 2, color
= black):}{%
}
\end{mapleinput}

\end{maplegroup}
\vspace{-5pt}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{popis4 := textplot([0, 2*M*E, "2ME"], align = 'LEFT'):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{popis := textplot(\{[1.6, 7, N], [3.6, -16, S], [.6, 3,
W]\}, font = [TIMES, BOLDITALIC, 10]):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `rLeqexs4.eps`, plotoptions =
popt):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(plus1, plus2, minu, r_, dvaME, title = "r-L plot
in the extreme Kerr solution\\n The space-like geodesics
with E > 1", titlefont = [TIMES, ROMAN, 15], popis,
popis2, popis3, popis4);}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{Heading 4}
Fast Kerr solution
\end{Heading 4}

\end{maplegroup}
\vspace{-5pt}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{a := 1.04:}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{minu := plot(L_minus, r = 0..5*M, color = blue, thickness
= 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus1 := plot(L_plus, r = 0..2*M, L = 0..8*M, color =
blue, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus2 := plot(L_plus, r = 2*M..5*M, L = -16..0, color =
blue, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{popis := textplot(\{[1.6, 7, N], [4, -16, S]\}, font =
[TIMES, BOLDITALIC, 10]):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `rLeqfas1.eps`, plotoptions =
popt):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(plus1, plus2, minu, title = "r-L plot in the fast
Kerr solution\\n The space-like geodesics with E > 0",
titlefont = [TIMES, ROMAN, 15], popis, popis3);}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\emptyline
\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{E := 0:}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{minu := plot(L_minus, r = 0..5*M, color = blue, thickness
= 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus1 := plot(L_plus, r = 0..2*M, L = 0..8*M, color =
blue, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plus2 := plot(L_plus, r = 2*M..5*M, L = -16..0, color =
blue, thickness = 2):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{popis := textplot(\{[3, 5, N], [3, -10, S]\}, font = [TIMES,
BOLDITALIC, 10]):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `rLeqfas2.eps`, plotoptions =
popt):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(plus1, plus2, minu, title = "r-L plot in the fast
Kerr solution\\n The space-like geodesics with E = 0",
titlefont = [TIMES, ROMAN, 15], popis, popis3);}{%
}
\end{mapleinput}

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

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