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%Two points to note:
%  * The -G0 parameter passed to dvips is used to get around a bug in GhostScript which converts the "f" character to a pound sign in the final PDF.
%  * The -dPDFsettings parameter for ps2pdf is used to prevent downsampling of EPS images when they are converted to PDF. Without this switch, EPS graphics in the final PDF look very fuzzy, especially when viewed with a projector.
%Miscellaneous Features
%  * Since Prosper includes the hyperref package by default, you can easily set links and targets within your presentation with the \hyperlink and \hypertarget commands to enable easy navigation.
%  * PowerPoint allows you to embed animations within a presentation. This is also possible when using Prosper, since it uses the hyperref package. To embed an MPEG movie, you can include the following code snippet:
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%    * The resultant movie plays in its own window; it is not possible to actually "embed" the movie in the presentation itself (at least under Linux).
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%When converting the .ps output of dvips into .pdf using ps2pdf (thus ghostscript), all included bitmap graphics looked ugly, full of artefacts.
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%Even the hint "-dPDFsettings=/prepress" did not help. Funny enough, using ghostscript 5.50 produced good results, but lacking the transition effects between slides.
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%After a long search, I found that the options
%-dAutoFilterColorImages=false -dColorImageFilter=/FlateEncode
%together produced the intended result of good-looking bitmaps (which had to be converted into eps using ImageMagick's convert before). The .pdf files don't get too big.
%
%\documentclass[slideColor,colorBG,pdf,azureru]{prosper}
\documentclass[slideColor,colorBG,dvips,wjrua]{prosper}%white good
% \documentclass[slideColor,colorBG,dvips,wjru]{prosper}%white good
% \documentclass[slideColor,colorBG,pdf,lignesbleuru]{prosper}
\usepackage{textcomp}
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\usepackage{epsfig}
\usepackage{wrapfig}
\usepackage{amsmath}
%\usepackage{graphicx}
%\usepackage{graphicx,epsfig}
%\usepackage{fancybox,amssymb}
\usepackage{fancybox}
%\usepackage{seminar}
%\usepackage{sem-dem}
%\input seminar.con

%\usepackage{pstricks}

\definecolor{labelcolor}{RGB}{100,0,0}


% Definition of new colors
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\newrgbcolor{LightBlue}{0.68 0.85 0.9}
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% Default color
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%        (\SeminarPaperHeight,\SeminarPaperWidth)}

%\usepackage{pstcol} % To use the standard `color' package with Seminar
%\usepackage{semcolor}
%\input{seminar.bug}
%\input{seminar.bg2}
%\autoslidemarginstrue

\let\jnlstyle=\it
\def\refjnl#1{{\jnlstyle#1}}
\def\jetp{\refjnl{Sov.Physics JETP}}
\def\zhetf{\refjnl{Zh.Expt.Teor.Fiz.}}
\def\jgr{\refjnl{J.~Geophys.~Res.}}
\def\jqsrt{\refjnl{J.~Quant.~Spec.~Radiat.~Transf.}}
\def\memsai{\refjnl{Mem.~Soc.~Astron.~Italiana}}
\def\nphysa{\refjnl{Nucl.~Phys.~A}}
\def\nphysb{\refjnl{Nucl.~Phys.~B}}
\def\yaf{\refjnl{Sov.J.Nucl.Phys.}}

% Definition of new colors
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\newrgbcolor{LightBlue}{0.68 0.85 0.9}
\newrgbcolor{seclr}{0.7 0.0 1.}
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\newrgbcolor{lblue}{0.6 0.9 0.95}
\newrgbcolor{dblue}{0 0 0.6}
\newrgbcolor{darkgreen}{0.3 0.4 .2} %85 107 47
\newrgbcolor{darkred}{0.6 0.1 0.2}
\newrgbcolor{cream}{1 0.99 0.85}
\newrgbcolor{pink}{1. 0.6 0.65}
\newrgbcolor{purple}{0.67 0.15 0.95}
%
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\newrgbcolor{indigo}{.416 .372  .925}
\newrgbcolor{violet}{1 .63 1}
\newrgbcolor{dviolet}{.58 0 .827}



\def\bes{\begin{slide}}
\def\ens{\end{slide}}


\def\etal{et al. }
\def\nifsx{${}^{56}$Ni}
\def\msun{M_\odot}

\def\Msun{$\msun$}
\def\rsun{R_\odot}
\def\Rsun{$R_\odot$}
\def\elem#1#2{{}^{#1}{\rm{#2}}}
\def\elema#1#2{{}^{#1}\!{\rm{#2}}}
\def\fefsx{$^{56}$Fe}
\def\cofsx{$^{56}$Co}
\def\ltsima{$\; \buildrel < \over \sim \;$}
\def\ltsim{\lower.5ex\hbox{\ltsima}}
\def\gtsima{$\; \buildrel > \over \sim \;$}
\def\gtsim{\lower.5ex\hbox{\gtsima}}
%\def\lesssim{\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}}
%\def\gtrsim{\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}}}
\let\la=\ltsim
\let\ga=\gtsim
\def\p{\mathrm p}
\def\n{\mathrm n}


%\def\bes{\ } %{\begin{slide}}
%\def\ens{\ } %{\end{slide}}
%\def\bes #1 {\begin{slide}{#1}}
\def\bes{\begin{slide}}
\def\ens{\end{slide}}


\def\FIG #1 #2 [#3] #4\par{%
  \begin{figure}[!h] \begin{center}%
    \includegraphics*[#3]{#2}%
    \\ \parbox{10cm}{%
    \caption{\label{#1}#4}}%
  \end{center}\end{figure}%
}
\def\rfig#1{Fig.\ref{#1}}


\renewcommand{\t}{\mathrm}
\newcommand{\PB}[2]{\parbox[b]{#1}{#2}}
\newcommand{\PT}[2]{\parbox[t]{#1}{#2}}
%\newcommand{\PB}[2]{\parbox[b]{#1}{\phantom{#2}}}
\newcommand{\IG}[2]{\includegraphics*[#1]{#2}}
\newcommand{\CIG}[2]{\begin{center}\doublebox{\includegraphics*[#1]{#2}}\end{center}}

\newcommand{\pv}[1]{\par\vspace*{\stretch{#1}}\null}
\newcommand{\isn}[2]{\mbox{$^{#2}${#1}}}

\def\G#1 #2 #3 {\Gamma^#1_{#2#3}}
\def\R#1 #2 #3 #4 {R^#1_{#2#3#4}}


%differentials
\renewcommand{\d}[1]{\;\textsf{d}#1}
\newcommand{\pd}[1]{\partial #1}
\newcommand{\D}{\;\textsf{D}}

%dot and double dot for D_t en D_t^2
\newcommand{\dt}[1]{\dot{#1}} %dot notation for d/dt
\newcommand{\dtt}[1]{\ddot{#1}} % double dot notation for d^2 / dt^2
\def\ddt{\frac{\pd}{\pd t}}

%accent (acute) for D_s and D_s^2
\newcommand{\ds}[1]{#1 \acute{}\,} % accent notation for d/ds
\newcommand{\dss}[1]{#1 \acute{}\, \acute{}\,} % double accent notation for d^2 / ds^2

%accent notation for arbitrrary derivative of order 1 or 2
%\newcommand{\dx}[1]{#1 \grave{}\,} % accent notation for arbitrary d / dx
\newcommand{\dx}[1]{ \grave{#1}\,} % accent notation for arbitrary d / dx
\newcommand{\dxx}[1]{ #1 \hspace*{-0.6em} \raisebox{.8ex}{$ \grave{} \; \grave{}\,$}} % double accent
\newcommand{\dxxa}[1]{ #1  \hspace*{-0.3em} \raisebox{.4ex}{$ \grave{} \; \grave{}\,$}}% double accent for one symbol
%\newcommand{\dxx}[1]{ \raisebox{-.8ex}{$#1$} \hspace*{-0.6em} \grave{} \; \grave{}\,} % double accent notation for d^2 / dx^2



\title{Геодезические, кривизна и вывод метрики Шварцшильда}
% \subtitle{в нашей Вселенной}
\author{С.И.Блинников}
\email{sergei.blinnikov@itep.ru,  sblinnikov@gmail.com}
\institution{ИТЭФ, ГАИШ, НГУ}
\slideCaption{Novosibirsk - \texttt{Prosper}}

\begin{document}

\maketitle

\begin{slide}{ }
% \begin{center}

%
    \textit{S.I.Blinnikov${}^{1,2}$}
\pv{1}
    \begin{minipage}[c]{6cm}
{${}^1$\it ИТЭФ = ITEP = Institute for Theoretical and  Experimental Physics,
Moscow}%\pv{1}
    \end{minipage}
    \begin{minipage}[c]{4cm}
 \IG{width=.6\textwidth}{itepm.ps}%\pv{1}
    \end{minipage} \hfill    %\pv{1}
\vskip 0.3cm
    \begin{minipage}[c]{6cm}
{${}^2$\it  ГАИШ = SAI = Sternberg Astronomical \\Institute, Moscow}\pv{1}
    \end{minipage}
    \begin{minipage}[c]{4cm}
   \IG{width=.6\textwidth}{sai-main.ps}\pv{1}
    \end{minipage}\hfill %\pv{1}
%     \begin{minipage}[c]{6cm}
% {${}^3$\it MPA, Garching}\pv{1}
%     \end{minipage}
%     \begin{minipage}[c]{4cm}
%    \IG{width=.6\textwidth}{MPA.ps}\pv{1}
%     \end{minipage}\hfill \pv{1}

%\end{center}

  \centerline{НГУ 06 и 10 октября 2012}

\end{slide}


\bes{Символы Кристоффеля}

\begin{center}

Если обозначить
$$
\Gamma_{mik} \equiv {1\over 2} \left(
  \partial_k g_{im} + \partial_i g_{km}-\partial_m g_{{ik}}
  \right)
$$
и
\begin{equation}
  \Gamma^n_{ik} \equiv g^{mn} \Gamma_{mik} \equiv {1\over 2} g^{mn}
  (\partial_k g_{im} + \partial_i g_{km}-\partial_m g_{{ik}}) \ ,
  \label{Crst}  %\label{3.21}
\end{equation}
-- коэффициенты связности (символы Кристоффеля 1-го и 2-го рода), то окончательно получим
\begin{equation}
  \dxx{x^n} + \Gamma^n_{ik} \grave{x^i} \grave{x^k}=0\ .
\label{geod}%  \label{3.47}
\end{equation}
Это и есть {\bf уравнение геодезической}.

\end{center}
\ens

\bes{К теоремам Грина и Стокса}

\CIG{width=.7\textwidth}{GreenTh.eps}

\ens


\bes{Тензоры Римана и Риччи}
Riemann tensor:
$$
  R^i_{klm} = \frac{\partial \Gamma^i_{km}}{\partial x^l}-
          \frac{\partial \Gamma^i_{kl}}{\partial x^m}
          +\Gamma^i_{nl}\Gamma^n_{km} - \Gamma^i_{nm}\Gamma^n_{kl} \; ,
$$
and Ricci tensor:
$$
 R_{ik}=R^m_{imk} \ .
$$
Т.е. тензор Риччи есть
$$
 R_{ik}=  \frac{\partial \Gamma^m_{ik}}{\partial x^m}-
          \frac{\partial \Gamma^m_{im}}{\partial x^k}
          +\Gamma^m_{nm}\Gamma^n_{ik} - \Gamma^m_{nk}\Gamma^n_{im} \; ,
$$

Скаляр Риччи
$$
 R=g^{ik}R_{ik} \ .
$$
\ens

\bes{Ansatz метрики Шварцшильда}
$$
  ds^2 = \mathrm{e}^{\nu(t,r)}dt^2 - \mathrm{e}^{\lambda(t,r)}dr^2 - r^2 d\theta^2 - r^2 \sin^2\theta d\varphi^2\ ,
$$

Все ненулевые кристоффели:
$$
\G 0 0 0 =  \frac{1}{2} \dot\nu , \quad
\G 0 0 1 = \frac{1}{2} \nu'  =\Gamma^0_{10} , \quad
\G 0 1 1 = \frac{1}{2}\mathrm{e}^{\lambda-\nu} \dot\lambda \ ,
$$
$$
\G 1 0 0 = \frac{1}{2}\mathrm{e}^{\nu-\lambda} \nu' ,   \quad
\G 1 0 1 =  \frac{1}{2} \dot\lambda = \G 1 1 0 ,  \quad
\G 1 1 1 =  \frac{1}{2} \lambda' ,  \quad
$$
$$
\G 1 2 2 =  -r \mathrm{e}^{-\lambda} ,  \quad
\G 1 3 3 =  -r\sin^2\theta \mathrm{e}^{-\lambda} \ ,
\G 2 1 2 = \frac{1}{r} = \G 2 2 1 ,
$$
$$
\G 2 3 3 =  -\sin\theta\cos\theta, \quad
\G 3 1 3 = \frac{1}{r} =\G 3 3 1  ,  \quad \G 3 2 3 = \ctg\theta = \G 3 3 2.
$$


\ens

\bes{$\boldsymbol{\Gamma^n_{}ik, \; R_{ik}} $ в Maxima}
\begin{center}
%   \fbox{\href{run:/home/seb/prg/gitWork/Lectures/GR/MyITEP/maxima/schwarzschild4.wxm}{Click}}
 \fbox{\href{http://dau.itep.ru/lections/schwarzschild4.wxm}{\dred wxmaxima -- click here for web}}
 \fbox{\href{run:schwarzschild4.wxm}{\dred wxmaxima -- for files saved  locally}}

\end{center}
\ens

\bes{Скаляр Риччи на {\tt Maple}}
{\tiny
% \begin{eqnarray*}
%\lefteqn{
$$
 R = \left[ - r^{2}\,\left({\frac {\partial^{2}}{\partial r
^{2}}}\,\nu (r)\right)
- \frac{r^{2}}{2}\,\left({\frac {\partial }{\partial r}}\,\nu (r)\right)^{2} \right.
+ \frac{r^{2}}{2}\,\left({\frac {\partial }{\partial r}}\,\nu(r)\right)\,
\left({\frac {\partial }{\partial r}}\,\lambda (r)\right)
$$
$$
 - 2 \left({\frac {\partial }{\partial r}}\,\nu (r)\right)r
+ \left. 2\left({\frac {\partial }{\partial r}}\,\lambda (r)\right)r
 + 2\mathrm{e}^{\lambda (r)} - 2 \right] \left/ {\vrule
height0.56em width0em depth0.56em} \right. \!  \! (r^{2}\,
\mathrm{e}^{\lambda (r)}) %\mbox{\hspace{272pt}}
$$
% \end{eqnarray*}
}

Если проделать вручную и
сравнить  с Ландау-Лифшицем или Вихлининым, то сразу видно,
что ответ Мапла ({\tt Maple}) правильный, только знак $R$ другой.

\ens

\bes{ wxMaxima $\boldsymbol R$ }

Для случая $\lambda(t,r)\ , \nu(t,r)$:

{\tiny

$${e^{-{\it \nu}-{\it \lambda}}\,\left(\left(e^{{\it \lambda}}\,
 \left({{d}\over{d\,t}}\,{\it \lambda}\right)\,\left({{d}\over{d\,t
 }}\,{\it \nu}\right)+2\,e^{{\it \nu}}\,\left({{d^2}\over{d\,r^2}}
 \,{\it \nu}\right)+e^{{\it \nu}}\,\left({{d}\over{d\,r}}\,
 {\it \nu}\right)^2-{{d}\over{d\,r}}\,{\it \lambda}\,e^{{\it \nu}}
 \,\left({{d}\over{d\,r}}\,{\it \nu}\right) \right. \right.	}
$$
$$
\left. \left.
-2\,e^{{\it \lambda}}\,
 \left({{d^2}\over{d\,t^2}}\,{\it \lambda}\right)-e^{{\it \lambda}}
 \,\left({{d}\over{d\,t}}\,{\it \lambda}\right)^2\right)\,r^2+\left(
 4\,e^{{\it \nu}}\,\left({{d}\over{d\,r}}\,{\it \nu}\right)-4\,
 \left({{d}\over{d\,r}}\,{\it \lambda}\right)\,e^{{\it \nu}}\right)
 \,r+\left(4-4\,e^{{\it \lambda}}\right)\,e^{{\it \nu}}\right)
$$
$$
 /{2\,r^2}
$$
}
\ens


\bes{Скаляр Риччи на {\tt Maxima}}

Для случая $\lambda(r)\ , \nu(r)$:


{\tiny

\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o23) }
% \frac{
{e}^{-\lambda}\,\left[ \left( 2\,\left( \frac{{d}^{2}}{d\,{r}^{2}}\,\nu\right) \right. \right.
\end{math}
\begin{math}\displaystyle
\left. \left. +{\left( \frac{d}{d\,r}\,\nu\right) }^{2}-\left( \frac{d}{d\,r}\,\lambda\right) \,\left( \frac{d}{d\,r}\,\nu\right) \right) \,{r}^{2}+\left( 4\,\left( \frac{d}{d\,r}\,\nu\right) -4\,\left( \frac{d}{d\,r}\,\lambda\right) \right)
\,r-4\,{e}^{\lambda}+4\right] /
{2\,{r}^{2}}
\end{math}
}

\vskip 1cm

Знаки в {\tt Maxima} совпадают с Ландау-Лифшицем.


\ens

\end{document}

