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\documentclass{article}
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\usepackage{fullpage}
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\usepackage{amsmath}
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\usepackage[utf8x]{inputenc}
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\usepackage{amssymb}
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\usepackage{graphicx}
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\usepackage{etoolbox}
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\usepackage{url}
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\usepackage{hyperref}
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\usepackage[T1]{fontenc}
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\makeatletter
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\preto{\@verbatim}{\topsep=0pt \partopsep=0pt }
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\makeatother
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\usepackage{listings}
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\lstdefinelanguage{Sage}[]{Python}
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{morekeywords={True,False,sage,singular},
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sensitive=true}
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\lstset{
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  showtabs=False,
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  commentstyle={\ttfamily\color{dbrowncolor}},
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  keywordstyle={\ttfamily\color{dbluecolor}\bfseries},
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  stringstyle ={\ttfamily\color{dgraycolor}\bfseries},
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  backgroundcolor=\color{lightyellow},
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  language = Sage,
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  basicstyle={\ttfamily},
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  breaklines=true,
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  prebreak = \raisebox{0ex}[0ex][0ex]{\ensuremath{\backslash}},
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}
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\usepackage{color}
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\definecolor{lightyellow}{rgb}{1,1,.92}
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\definecolor{dblackcolor}{rgb}{0.0,0.0,0.0}
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\definecolor{dgraycolor}{rgb}{0.30,0.3,0.30}
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\definecolor{graycolor}{rgb}{0.35,0.35,0.35}
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\title{SM\_de\_Sitter.sagews}
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\author{Andy Huchala}
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\begin{document}
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\maketitle
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{
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\section{de Sitter spacetime}
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{This worksheet demonstrates a few capabilities of \url{SageManifolds} (version 0.8) in computations regarding de Sitter spacetime.}\\\\
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{It is released under the GNU General Public License version 3.}\\\\
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{(c) Eric Gourgoulhon, Michal Bejger (2015)}\\\\
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{The corresponding worksheet file can be downloaded from }{\url{here}}
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\subsection{Spacetime manifold}
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{We declare the de Sitter spacetime as a 4-dimensional differentiable manifold:}
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\begin{lstlisting}
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M = Manifold(4, 'M', r'\mathcal{M}')
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print M ; M
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\end{lstlisting}\begin{verbatim}4-dimensional manifold 'M'\end{verbatim}
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{$\mathcal{M}$}\\\\
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\iffalse
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\begin{lstlisting}
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%html
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<p>We consider hyperspherical coordinates $(\tau,\chi,\theta,\phi)$ on $\mathcal{M}$. Allowing for the standard coordinate singularities at $\chi=0$, $\chi=\pi$, $\theta=0$ or $\theta=\pi$, these coordinates cover the entire spacetime manifold (which is topologically $\mathbb{R}\times\mathbb{S}^3$). If we restrict ourselves to <em>regular</em> coordinates (i.e. to consider only mathematically well defined charts), the hyperspherical coordinates cover only an open part of $\mathcal{M}$, which we call $\mathcal{M}_0$, on which $\chi$ spans the open interval $(0,\pi)$, $\theta$ the open interval $(0,\pi)$ and $\phi$ the open interval $(0,2\pi)$. Therefore, we declare:</p>
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\end{lstlisting}\fi{We consider hyperspherical coordinates $(\tau,\chi,\theta,\phi)$ on $\mathcal{M}$. Allowing for the standard coordinate singularities at $\chi=0$, $\chi=\pi$, $\theta=0$ or $\theta=\pi$, these coordinates cover the entire spacetime manifold (which is topologically $\mathbb{R}\times\mathbb{S}^3$). If we restrict ourselves to {regular} coordinates (i.e. to consider only mathematically well defined charts), the hyperspherical coordinates cover only an open part of $\mathcal{M}$, which we call $\mathcal{M}_0$, on which $\chi$ spans the open interval $(0,\pi)$, $\theta$ the open interval $(0,\pi)$ and $\phi$ the open interval $(0,2\pi)$. Therefore, we declare:}
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\begin{lstlisting}
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M0 = M.open_subset('M_0', r'\mathcal{M}_0')
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X_hyp.<ta,ch,th,ph> = M0.chart(r'ta:\tau ch:(0,pi):\chi th:(0,pi):\theta ph:(0,2*pi):\phi')
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print X_hyp ; X_hyp
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\end{lstlisting}\begin{verbatim}chart (M_0, (ta, ch, th, ph))\end{verbatim}
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{$\left(\mathcal{M}_0,({\tau}, {\chi}, {\theta}, {\phi})\right)$}
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\begin{lstlisting}
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\end{lstlisting}\subsection{$\mathbb{R}^5$ as an ambient space}
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{The de Sitter metric can be defined as that induced by the embedding of $\mathcal{M}$ into a 5-dimensional Minkowski space, i.e. $\mathbb{R}^5$ equipped with a flat Lorentzian metric. We therefore introduce $\mathbb{R}^5$ as a 5-dimensional manifold covered by canonical coordinates:}
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\begin{lstlisting}
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R5 = Manifold(5, 'R5', r'\mathbb{R}^5')
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X5.<T,W,X,Y,Z> = R5.chart()
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print X5 ; X5
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\end{lstlisting}\begin{verbatim}chart (R5, (T, W, X, Y, Z))\end{verbatim}
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{$\left(\mathbb{R}^5,(T, W, X, Y, Z)\right)$}
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\begin{lstlisting}
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\end{lstlisting}{The embedding of $\mathcal{M}$ into $\mathbb{R}^5$ is defined as a differential mapping $\Phi$ from $\mathcal{M}$ to $\mathbb{R}^5$, by providing its expression in terms of $\mathcal{M}$'s default chart (which is X\_hyp = $(\mathcal{M}_0,(\tau,\chi,\theta,\phi))$ ) and $\mathbb{R}^5$'s default chart (which is X5 = $(\mathbb{R}^5,(T,W,X,Y,Z))$ ):}
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\begin{lstlisting}
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var('b')
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Phi = M.diff_mapping(R5, [sinh(b*ta)/b,
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                          cosh(b*ta)/b * cos(ch),
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                          cosh(b*ta)/b * sin(ch)*sin(th)*cos(ph),
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                          cosh(b*ta)/b * sin(ch)*sin(th)*sin(ph),
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                          cosh(b*ta)/b * sin(ch)*cos(th)],
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                  name='Phi', latex_name=r'\Phi')
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print Phi ; Phi.display()
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\end{lstlisting}
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{$b$}\begin{verbatim}differentiable mapping 'Phi' from the 4-dimensional manifold 'M' to the 5-dimensional
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manifold 'R5'\end{verbatim}
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{$\begin{array}{llcl} \Phi:& \mathcal{M} & \longrightarrow & \mathbb{R}^5 \\ \mbox{on}\ \mathcal{M}_0 : & \left({\tau}, {\chi}, {\theta}, {\phi}\right) & \longmapsto & \left(T, W, X, Y, Z\right) = \left(\frac{\sinh\left(b {\tau}\right)}{b}, \frac{\cos\left({\chi}\right) \cosh\left(b {\tau}\right)}{b}, \frac{\cos\left({\phi}\right) \cosh\left(b {\tau}\right) \sin\left({\chi}\right) \sin\left({\theta}\right)}{b}, \frac{\cosh\left(b {\tau}\right) \sin\left({\chi}\right) \sin\left({\phi}\right) \sin\left({\theta}\right)}{b}, \frac{\cos\left({\theta}\right) \cosh\left(b {\tau}\right) \sin\left({\chi}\right)}{b}\right) \end{array}$}
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\begin{lstlisting}
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\end{lstlisting}{The constant $b$ is a scale parameter. Considering de Sitter metric as a solution of vacuum Einstein equation with positive cosmological constant $\Lambda$, one has $b = \sqrt{\Lambda/3}$.}
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{Let us evaluate the image of a point via the mapping $\Phi$:}
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\begin{lstlisting}
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p = M.point((ta, ch, th, ph), name='p') ; print p
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\end{lstlisting}\begin{verbatim}point 'p' on 4-dimensional manifold 'M'\end{verbatim}
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\begin{lstlisting}
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p.coord()
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\end{lstlisting}
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{(${\tau}$, ${\chi}$, ${\theta}$, ${\phi}$)}
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\begin{lstlisting}
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q = Phi(p) ; print q
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\end{lstlisting}\begin{verbatim}point 'Phi(p)' on 5-dimensional manifold 'R5'\end{verbatim}\begin{verbatim}\end{verbatim}
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\begin{lstlisting}
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q.coord()
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\end{lstlisting}
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{($\frac{\sinh\left(b {\tau}\right)}{b}$, $\frac{\cos\left({\chi}\right) \cosh\left(b {\tau}\right)}{b}$, $\frac{\cos\left({\phi}\right) \cosh\left(b {\tau}\right) \sin\left({\chi}\right) \sin\left({\theta}\right)}{b}$, $\frac{\cosh\left(b {\tau}\right) \sin\left({\chi}\right) \sin\left({\phi}\right) \sin\left({\theta}\right)}{b}$, $\frac{\cos\left({\theta}\right) \cosh\left(b {\tau}\right) \sin\left({\chi}\right)}{b}$)}
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\begin{lstlisting}
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\end{lstlisting}{The image of $\mathcal{M}$ by $\Phi$ is a hyperboloid of one sheet, of equation $-T^2+W^2+X^2+Y^2+Z^2=b^{-2}$. Indeed:}
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\begin{lstlisting}
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(Tq,Wq,Xq,Yq,Zq) = q.coord()
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s = -Tq^2 + Wq^2 + Xq^2 + Yq^2 + Zq^2
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s.simplify_full()
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\end{lstlisting}
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{$\frac{1}{b^{2}}$}
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\begin{lstlisting}
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\end{lstlisting}{We may use the embedding $\Phi$ to draw the coordinate grid $(\tau,\chi)$ in terms of the coordinates $(W,X,T)$ for $\theta=\pi/2$ and $\phi=0$ (red) and $\theta=\pi/2$ and $\phi=\pi$ (green) (the brown lines are the lines $\tau={\rm const}$):}
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\begin{lstlisting}
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graph1 = X_hyp.plot(X5, mapping=Phi, ambient_coords=(W,X,T), fixed_coords={th:pi/2, ph:0}, nb_values=9, color={ta:'red', ch:'brown'}, thickness=2, max_value=2, parameters={b:1}, label_axes=False)
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graph2 = X_hyp.plot(X5, mapping=Phi, ambient_coords=(W,X,T), fixed_coords={th:pi/2, ph:pi}, nb_values=9, color={ta:'green', ch:'brown'}, thickness=2, max_value=2, parameters={b:1}, label_axes=False)
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show(set_axes_labels(graph1+graph2,'W','X','T'), aspect_ratio=1)
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\end{lstlisting}\url{https://cloud.sagemath.com/blobs/f97f4212-d46d-4ad8-b8f7-4979d61774e6.sage3d?uuid=f97f4212-d46d-4ad8-b8f7-4979d61774e6}
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{}
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\begin{lstlisting}
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\end{lstlisting}\subsection{Spacetime metric}
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{First, we introduce on $\mathbb{R}^5$ the Minkowski metric $h$:}
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\begin{lstlisting}
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h = R5.lorentz_metric('h')
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h[0,0], h[1,1], h[2,2], h[3,3], h[4,4] = -1, 1, 1, 1, 1
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h.display()
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\end{lstlisting}
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{$h = -\mathrm{d} T\otimes \mathrm{d} T+\mathrm{d} W\otimes \mathrm{d} W+\mathrm{d} X\otimes \mathrm{d} X+\mathrm{d} Y\otimes \mathrm{d} Y+\mathrm{d} Z\otimes \mathrm{d} Z$}
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\begin{lstlisting}
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\end{lstlisting}{As mentionned above, the de Sitter metric $g$ on $\mathcal{M}$ is that induced by $h$, i.e.$g$ is the pullback of $h$ by the mapping $\Phi$:}
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\begin{lstlisting}
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g = M.metric('g')
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g.set( Phi.pullback(h) )
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\end{lstlisting}
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\begin{lstlisting}
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\end{lstlisting}{The expression of $g$ in terms of $\mathcal{M}$'s default frame is found to be}
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\begin{lstlisting}
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g.display()
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\end{lstlisting}
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{$g = -\mathrm{d} {\tau}\otimes \mathrm{d} {\tau} + \frac{\cosh\left(b {\tau}\right)^{2}}{b^{2}} \mathrm{d} {\chi}\otimes \mathrm{d} {\chi} + \frac{\cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2}}{b^{2}} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \frac{\cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \sin\left({\theta}\right)^{2}}{b^{2}} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$}
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\begin{lstlisting}
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g[:]
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\end{lstlisting}
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{$\left(\begin{array}{rrrr}
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-1 & 0 & 0 & 0 \\
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0 & \frac{\cosh\left(b {\tau}\right)^{2}}{b^{2}} & 0 & 0 \\
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0 & 0 & \frac{\cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2}}{b^{2}} & 0 \\
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0 & 0 & 0 & \frac{\cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \sin\left({\theta}\right)^{2}}{b^{2}}
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\end{array}\right)$}
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\begin{lstlisting}
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\end{lstlisting}\subsection{Curvature}
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{The Riemann tensor of $g$ is}
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\begin{lstlisting}
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Riem = g.riemann()
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print Riem
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Riem.display()
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\end{lstlisting}\begin{verbatim}tensor field 'Riem(g)' of type (1,3) on the 4-dimensional manifold 'M'\end{verbatim}
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{$\mathrm{Riem}\left(g\right) = \cosh\left(b {\tau}\right)^{2} \frac{\partial}{\partial {\tau} }\otimes \mathrm{d} {\chi}\otimes \mathrm{d} {\tau}\otimes \mathrm{d} {\chi} -\cosh\left(b {\tau}\right)^{2} \frac{\partial}{\partial {\tau} }\otimes \mathrm{d} {\chi}\otimes \mathrm{d} {\chi}\otimes \mathrm{d} {\tau} + \cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \frac{\partial}{\partial {\tau} }\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\tau}\otimes \mathrm{d} {\theta} -\cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \frac{\partial}{\partial {\tau} }\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\tau} + \cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \sin\left({\theta}\right)^{2} \frac{\partial}{\partial {\tau} }\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\tau}\otimes \mathrm{d} {\phi} -\cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \sin\left({\theta}\right)^{2} \frac{\partial}{\partial {\tau} }\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\tau} + b^{2} \frac{\partial}{\partial {\chi} }\otimes \mathrm{d} {\tau}\otimes \mathrm{d} {\tau}\otimes \mathrm{d} {\chi} -b^{2} \frac{\partial}{\partial {\chi} }\otimes \mathrm{d} {\tau}\otimes \mathrm{d} {\chi}\otimes \mathrm{d} {\tau} + \cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \frac{\partial}{\partial {\chi} }\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\chi}\otimes \mathrm{d} {\theta} -\cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \frac{\partial}{\partial {\chi} }\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\chi} + \cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \sin\left({\theta}\right)^{2} \frac{\partial}{\partial {\chi} }\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\chi}\otimes \mathrm{d} {\phi} -\cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \sin\left({\theta}\right)^{2} \frac{\partial}{\partial {\chi} }\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\chi} + b^{2} \frac{\partial}{\partial {\theta} }\otimes \mathrm{d} {\tau}\otimes \mathrm{d} {\tau}\otimes \mathrm{d} {\theta} -b^{2} \frac{\partial}{\partial {\theta} }\otimes \mathrm{d} {\tau}\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\tau} + \left( -\frac{\sin\left({\chi}\right)^{2} \sinh\left(b {\tau}\right)^{2} - \cos\left({\chi}\right)^{2} + 1}{\sin\left({\chi}\right)^{2}} \right) \frac{\partial}{\partial {\theta} }\otimes \mathrm{d} {\chi}\otimes \mathrm{d} {\chi}\otimes \mathrm{d} {\theta} + \cosh\left(b {\tau}\right)^{2} \frac{\partial}{\partial {\theta} }\otimes \mathrm{d} {\chi}\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\chi} + \cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \sin\left({\theta}\right)^{2} \frac{\partial}{\partial {\theta} }\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\phi} -\cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \sin\left({\theta}\right)^{2} \frac{\partial}{\partial {\theta} }\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\theta} + b^{2} \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} {\tau}\otimes \mathrm{d} {\tau}\otimes \mathrm{d} {\phi} -b^{2} \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} {\tau}\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\tau} + \left( -\frac{\sin\left({\chi}\right)^{2} \sinh\left(b {\tau}\right)^{2} - \cos\left({\chi}\right)^{2} + 1}{\sin\left({\chi}\right)^{2}} \right) \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} {\chi}\otimes \mathrm{d} {\chi}\otimes \mathrm{d} {\phi} + \cosh\left(b {\tau}\right)^{2} \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} {\chi}\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\chi} -\cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\phi} + \cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\theta}$}
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\begin{lstlisting}
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Riem.display_comp(only_nonredundant=True)
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\end{lstlisting}
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{$\begin{array}{lcl} \mathrm{Riem}\left(g\right)_{ \phantom{\, {\tau} } \, {\chi} \, {\tau} \, {\chi} }^{ \, {\tau} \phantom{\, {\chi} } \phantom{\, {\tau} } \phantom{\, {\chi} } } & = & \cosh\left(b {\tau}\right)^{2} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\tau} } \, {\theta} \, {\tau} \, {\theta} }^{ \, {\tau} \phantom{\, {\theta} } \phantom{\, {\tau} } \phantom{\, {\theta} } } & = & \cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\tau} } \, {\phi} \, {\tau} \, {\phi} }^{ \, {\tau} \phantom{\, {\phi} } \phantom{\, {\tau} } \phantom{\, {\phi} } } & = & \cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \sin\left({\theta}\right)^{2} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\chi} } \, {\tau} \, {\tau} \, {\chi} }^{ \, {\chi} \phantom{\, {\tau} } \phantom{\, {\tau} } \phantom{\, {\chi} } } & = & b^{2} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\chi} } \, {\theta} \, {\chi} \, {\theta} }^{ \, {\chi} \phantom{\, {\theta} } \phantom{\, {\chi} } \phantom{\, {\theta} } } & = & \cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\chi} } \, {\phi} \, {\chi} \, {\phi} }^{ \, {\chi} \phantom{\, {\phi} } \phantom{\, {\chi} } \phantom{\, {\phi} } } & = & \cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \sin\left({\theta}\right)^{2} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\theta} } \, {\tau} \, {\tau} \, {\theta} }^{ \, {\theta} \phantom{\, {\tau} } \phantom{\, {\tau} } \phantom{\, {\theta} } } & = & b^{2} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\theta} } \, {\chi} \, {\chi} \, {\theta} }^{ \, {\theta} \phantom{\, {\chi} } \phantom{\, {\chi} } \phantom{\, {\theta} } } & = & -\frac{\sin\left({\chi}\right)^{2} \sinh\left(b {\tau}\right)^{2} - \cos\left({\chi}\right)^{2} + 1}{\sin\left({\chi}\right)^{2}} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\theta} } \, {\phi} \, {\theta} \, {\phi} }^{ \, {\theta} \phantom{\, {\phi} } \phantom{\, {\theta} } \phantom{\, {\phi} } } & = & \cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \sin\left({\theta}\right)^{2} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\phi} } \, {\tau} \, {\tau} \, {\phi} }^{ \, {\phi} \phantom{\, {\tau} } \phantom{\, {\tau} } \phantom{\, {\phi} } } & = & b^{2} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\phi} } \, {\chi} \, {\chi} \, {\phi} }^{ \, {\phi} \phantom{\, {\chi} } \phantom{\, {\chi} } \phantom{\, {\phi} } } & = & -\frac{\sin\left({\chi}\right)^{2} \sinh\left(b {\tau}\right)^{2} - \cos\left({\chi}\right)^{2} + 1}{\sin\left({\chi}\right)^{2}} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\phi} } \, {\theta} \, {\theta} \, {\phi} }^{ \, {\phi} \phantom{\, {\theta} } \phantom{\, {\theta} } \phantom{\, {\phi} } } & = & -\cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \end{array}$}
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\begin{lstlisting}
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\end{lstlisting}{The Ricci tensor:}
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\begin{lstlisting}
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Ric = g.ricci()
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print Ric
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Ric.display()
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\end{lstlisting}\begin{verbatim}field of symmetric bilinear forms 'Ric(g)' on the 4-dimensional manifold 'M'\end{verbatim}
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{$\mathrm{Ric}\left(g\right) = -3 \, b^{2} \mathrm{d} {\tau}\otimes \mathrm{d} {\tau} + 3 \, \cosh\left(b {\tau}\right)^{2} \mathrm{d} {\chi}\otimes \mathrm{d} {\chi} + 3 \, \cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + 3 \, \cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$}
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\begin{lstlisting}
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Ric[:]
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\end{lstlisting}
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{$\left(\begin{array}{rrrr}
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-3 \, b^{2} & 0 & 0 & 0 \\
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0 & 3 \, \cosh\left(b {\tau}\right)^{2} & 0 & 0 \\
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0 & 0 & 3 \, \cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} & 0 \\
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0 & 0 & 0 & 3 \, \cosh\left(b {\tau}\right)^{2} \sin\left({\chi}\right)^{2} \sin\left({\theta}\right)^{2}
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\end{array}\right)$}
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\begin{lstlisting}
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\end{lstlisting}{The Ricci scalar:}
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\begin{lstlisting}
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R = g.ricci_scalar()
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print R
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R.display()
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\end{lstlisting}\begin{verbatim}scalar field 'r(g)' on the 4-dimensional manifold 'M'\end{verbatim}
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{$\begin{array}{llcl} \mathrm{r}\left(g\right):& \mathcal{M} & \longrightarrow & \mathbb{R} \\ \mbox{on}\ \mathcal{M}_0 : & \left({\tau}, {\chi}, {\theta}, {\phi}\right) & \longmapsto & 12 \, b^{2} \end{array}$}
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\begin{lstlisting}
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\end{lstlisting}{We recover the fact that de Sitter spacetime has a constant curvature. It is indeed a {maximally symmetric space}. In particular, the Riemann tensor is expressible as}
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{\[ R^i_{\ \, jlk} = \frac{R}{n(n-1)} \left( \delta^i_{\ \, k} g_{jl} - \delta^i_{\ \, l} g_{jk} \right) \]}
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{where $n$ is the dimension of $\mathcal{M}$: $n=4$ in the present case. Let us check this formula here, under the form $R^i_{\ \, jlk} = -\frac{R}{6} g_{j[k} \delta^i_{\ \, l]}$:}
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\begin{lstlisting}
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delta = M.tangent_identity_field()
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Riem == - (R/6)*(g*delta).antisymmetrize(2,3)  # 2,3 = last positions of the type-(1,3) tensor g*delta
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\end{lstlisting}
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{$\mathrm{True}$}
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\begin{lstlisting}
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\end{lstlisting}{We may also check that de Sitter metric is a solution of the vacuum {Einstein equation} with (positive) cosmological constant:}
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\begin{lstlisting}
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Lambda = 3*b^2
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Ric - 1/2*R*g + Lambda*g == 0
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\end{lstlisting}
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{$\mathrm{True}$}
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\end{document}
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%sagemathcloud={"latex_command":"pdflatex -synctex=1 -interact=nonstopmode 'fixed.tex'"}
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