%% Created by Maple 6 (IBM INTEL NT)
%% Source Worksheet: orbits.mws
%% Generated: Wed May 26 02:54:33 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 space-time
\end{Heading 1}

\begin{Heading 2}
First integrals and orbits
\end{Heading 2}

\begin{Heading 3}
Continents configuration
\end{Heading 3}

\begin{Heading 4}
Time-like geodesics
\end{Heading 4}

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

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

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

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Dr := r*Delta*fir:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{fir := e*r^3 +2*M*delta1*r^2 -Q*r +2*M*Q:}{%
}
\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}{solve(e = E^2 -delta1, E);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{sqrt(e+delta1), -sqrt(e+delta1);}{%
\[\color{blue}
\sqrt{e + \delta 1}, \, - \sqrt{e + \delta 1}
\]
%
}
\end{maplelatex}

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

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{M := 1: a := sqrt(.84): e := .4: Q := 1: delta1 := 1:}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P1 := plot(L_plus, r = 0..2*M, L = -10..10, labelfont =
[TIMES, ITALIC, 16]):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P3 := plot(L_plus, r = 2*M..8*M):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P4 := plot(L_minus, r = 0..8*M):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{popis := textplot(\{[4, 8, N], [4, -9, S], [.2, 1, 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}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `t-orbits.eps`, plotoptions =
`portrait, noborder`):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(P1, P3, P4, popis, thickness = 2, title = "r-L 
plot: Continents configuration \\n The time-like
geodesics", titlefont = [TIMES, ROMAN, 15]);}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{Heading 4}
Null geodesics
\end{Heading 4}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{delta1 := 0:}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Q_plus := 27*M^2*e;}{%
}\smallskip
\mapleinline{inert}{2d}{Q < Q_plus;}{%
$Q < \mathit{Q\_plus}$%
}
\end{mapleinput}

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{Q_plus := 10.8;}{%
\[
\mathit{Q\_plus} := 10.8
\]
%
}
\end{maplelatex}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P1 := plot(L_plus, r = 0..2*M, L = -10..10, labelfont =
[TIMES, ITALIC, 16], color = blue):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P3 := plot(L_plus, r = 2*M..8*M, color = blue):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P4 := plot(L_minus, r = 0..8*M, color = blue):}{%
}
\end{mapleinput}

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

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(P1, P3, P4, popis, thickness = 2, title = "r-L 
plot: Continents configuration \\n The null geodesics",
titlefont = [TIMES, ROMAN, 15]);}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{Heading 4}
Space-like geodesics
\end{Heading 4}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{e := 1.04: delta1 := -1:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Q_plus := M^2*(27*e^2 -18*e -1 +(e -1)^(1/2)*(9*e 
-1)^(3/2))/2/e; Q_minus := M^2*(27*e^2 -18*e -1 -(e
-1)^(1/2)*(9*e -1)^(3/2))/2/e;}{%
}\smallskip
\mapleinline{inert}{2d}{Q < Q_minus;}{%
$Q < \mathit{Q\_minus}$%
}
\end{mapleinput}

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{Q_plus := 6.883444575;}{%
\[
\mathit{Q\_plus} := 6.883444575
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{Q_minus := 2.235016962;}{%
\[
\mathit{Q\_minus} := 2.235016962
\]
%
}
\end{maplelatex}

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

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P1 := plot(L_plus, r = 0..2*M, L = -10..10, labelfont =
[TIMES, ITALIC, 16], color = blue):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P3 := plot(L_plus, r = 2*M..8*M, color = blue):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P4 := plot(L_minus, r = 0..8*M, color = blue):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `s-orbits1.eps`, plotoptions =
`portrait, noborder, color`):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{popis1 := textplot([1.2, .5, U], font = [TIMES, 
BOLDITALIC, 10]):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(P1, P3, P4, popis, popis1, thickness = 2, title =
"r-L plot: Continents configuration \\n The space-like
geodesics", titlefont = [TIMES, ROMAN, 15]);}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Q := 5: }{%
}\smallskip
\mapleinline{inert}{2d}{Q_minus < Q < Q_plus;}{%
$\mathit{Q\_minus} < Q < \mathit{Q\_plus}$%
} %\TEXTsymbol{<} 
%\mapleinline{inert}{2d}{Q < Q_plus;}{%
%$Q < \mathit{Q\_plus}$%
%}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P1 := plot(L_plus, r = 0..2*M, L = -10..10, labelfont =
[TIMES, ITALIC, 16], color = blue):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P3 := plot(L_plus, r = 2*M..8*M, color = blue):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P4 := plot(L_minus, r = 0..8*M, color = blue):}{%
}
\end{mapleinput}

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

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(P1, P3, P4, popis, thickness = 2, title = "r-L 
plot: Continents configuration \\n The space-like
geodesics", titlefont = [TIMES, ROMAN, 15]);}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{Heading 3}
Barrier configuration
\end{Heading 3}

\begin{Heading 4}
Time-like geodesics
\end{Heading 4}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{e := .4: Q := 28: delta1 := 1:}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P1 := plot(L_plus, r = 0..2*M, L = -10..10, labelfont =
[TIMES, ITALIC, 16]):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P3 := plot(L_plus, r = 2*M..10*M):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P4 := plot(L_minus, r = 0..10*M):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `t-orbits2.eps`, plotoptions =
`portrait, noborder`):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(P1, P3, P4, popis, thickness = 2, title = "r-L 
plot: Barrier configuration \\n The time-like geodesics",
titlefont = [TIMES, ROMAN, 15]);}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{Heading 4}
Null geodesics
\end{Heading 4}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{delta1 := 0:}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P1 := plot(L_plus, r = 0..2*M, L = -15..10, labelfont =
[TIMES, ITALIC, 16], color = blue):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P3 := plot(L_plus, r = 2*M..10*M, color = blue, numpoints
= 5000):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P4 := plot(L_minus, r = 0..10*M, color = blue, numpoints =
2000):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `n-orbits2.eps`, plotoptions =
`portrait, noborder, color`):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(P1, P3, P4, popis, thickness = 2, title = "r-L 
plot: Barrier configuration \\n The null geodesics",
titlefont = [TIMES, ROMAN, 15]);}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{Heading 4}
Space-like geodesics
\end{Heading 4}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{e := 1.04: delta1 := -1:}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P1 := plot(L_plus, r = 0..2*M, L = -10..10, labelfont =
[TIMES, ITALIC, 16], color = blue):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P3 := plot(L_plus, r = 2*M..8*M, color = blue):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P4 := plot(L_minus, r = 0..8*M, color = blue):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `s-orbits2.eps`, plotoptions =
`portrait, noborder, color`):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(P1, P3, P4, popis, thickness = 2, title = "r-L 
plot: Barrier configuration \\n The space-like geodesics",
titlefont = [TIMES, ROMAN, 15]);}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{Heading 3}
Bay configuration
\end{Heading 3}

\begin{Heading 4}
Time-like geodesics
\end{Heading 4}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{delta1 := 1: e := -.2: Q := 1:}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P1 := plot(L_plus, r = 0..2*M, L = -10..10, labelfont =
[TIMES, ITALIC, 16]):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P3 := plot(L_plus, r = 2*M..10*M):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P4 := plot(L_minus, r = 0..10*M):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{popis := textplot(\{[2, -2, "R > 0"], [4, -9, "R < 0"]\},
font = [TIMES, BOLDITALIC, 10]): }{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `t-orbits3.eps`, plotoptions =
`portrait, noborder`):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(P1, P3, P4, popis, thickness = 2, title = "r-L 
plot: Bay configuration \\n The time-like geodesics",
titlefont = [TIMES, ROMAN, 15]);}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{Heading 3}
Lake configuration
\end{Heading 3}

\begin{Heading 4}
Time-like geodesics
\end{Heading 4}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{e := -.08: Q := 14.2:}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P1 := plot(L_plus, r = 0..2*M, L = -10..10, labelfont =
[TIMES, ITALIC, 16]):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P3 := plot(L_plus, r = 2*M..20*M):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P4 := plot(L_minus, r = 0..20*M, numpoints = 2000):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{popis1 := textplot([12, .5, "lake"], font = [TIMES, BOLD,
10]):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `t-orbits4.eps`, plotoptions =
`portrait, noborder`):}{%
}
\end{mapleinput}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(P1, P3, P4, popis, popis1, thickness = 2, title =
"r-L plot: Lake configuration \\n The time-like
geodesics", titlefont = [TIMES, ROMAN, 15]);}{%
}
\end{mapleinput}

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

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