%Riemann tensor ̑Ώ̐ %
%

# Riemann tensor ̎wW̑Ώ̐

%

### @

%LaTeX2.09
%Kiyoshi Shiraishi 1999
\documentstyle[12pt,graphics]{jarticle}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\nn}{\nonumber \\}

\newcommand{\vnabla}{{\bf \nabla}}
\newcommand{\vsigma}{\vec{\sigma}}
%\newcommand{\vsigma}{{\bf \sigma}}
\newcommand{\vA}{{\bf A}} %vector potential
\newcommand{\vB}{{\bf B}} %
\newcommand{\vD}{{\bf D}}
\newcommand{\vE}{{\bf E}}
\newcommand{\vF}{{\bf F}}
\newcommand{\vg}{{\bf g}}
\newcommand{\vH}{{\bf H}}
\newcommand{\vI}{{\bf I}}
\newcommand{\vi}{{\bf i}}
\newcommand{\vJ}{{\bf J}}
\newcommand{\vj}{{\bf j}}
\newcommand{\vk}{{\bf k}}
\newcommand{\vM}{{\bf M}}
\newcommand{\vn}{{\bf n}}
\newcommand{\vP}{{\bf P}}
\newcommand{\vp}{{\bf p}}
\newcommand{\vR}{{\bf R}}
\newcommand{\vr}{{\bf r}}
\newcommand{\vS}{{\bf S}}
\newcommand{\vs}{{\bf s}}
\newcommand{\vv}{{\bf v}}
\newcommand{\vx}{{\bf x}}
\newcommand{\vy}{{\bf y}}
\newcommand{\vzero}{{\bf 0}}
\newcommand{\tr}{{\rm tr}}
\newcommand{\Tr}{{\rm Tr}}
\newcommand{\bx}{{x\!\!\!\mbox{-~}}}

%%%%%%%%%%%%%%%
%\hfill {ver. 1.0}
%%%%%%%%%%%%%%%
\title{Riemann tensor ̑Ώ̐}
\author{
@}
\date{1999N0707}
\begin{document}
\maketitle
\begin{abstract}

\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Riemann tensor ̑Ώ̐}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Riemann tensor:
\be
R^{\mu}{}_{\sigma\beta\alpha}=
\partial_{\beta}\Gamma^{\mu}_{\sigma\alpha}-
\partial_{\alpha}\Gamma^{\mu}_{\sigma\beta}+
\Gamma^{\mu}_{\rho\beta}\Gamma^{\rho}_{\sigma\alpha}-
\Gamma^{\mu}_{\rho\alpha}\Gamma^{\rho}_{\sigma\beta}
\ee

\be
\Gamma^{\sigma}_{\mu\nu}=
\frac{1}{2}g^{\rho\sigma}\left(
\partial_{\mu}g_{\nu\rho}+
\partial_{\nu}g_{\mu\rho}-
\partial_{\rho}g_{\nu\mu}\right)
\ee

\bigskip

RWniǏnjł́C$\Gamma^{\sigma}_{\mu\nu}$
SĂ̐[ɂȂB

\be
\nabla_{\sigma}g_{\mu\nu}\equiv 0
\ee

\be
\Gamma^{\sigma}_{\mu\nu}=0\Leftrightarrow
\partial_{\sigma}g_{\mu\nu}=0
\ee

˂ɎRWnǏIɂƂ邱ƂłB

܂CǏI
\be
g_{\mu\nu}=\eta_{\mu\nu}
\ee

Ƃ邱ƂłB

\bigskip

̍Wn Riemann tensor iwẂj
\bea
R_{\lambda\sigma\beta\alpha}&=&
g_{\lambda\mu}R^{\mu}_{\sigma\beta\alpha} \nn
&=&
g_{\lambda\mu}\partial_{\beta}\Gamma^{\mu}_{\sigma\alpha}-
g_{\lambda\mu}\partial_{\alpha}\Gamma^{\mu}_{\sigma\beta} \nn
&=&
\frac{1}{2}\partial_{\beta}
\left(\partial_{\sigma}g_{\alpha\lambda}+
\partial_{\alpha}g_{\sigma\lambda}-
\partial_{\lambda}g_{\alpha\sigma}
\right)-
\frac{1}{2}\partial_{\alpha}
\left(\partial_{\sigma}g_{\beta\lambda}+
\partial_{\beta}g_{\sigma\lambda}-
\partial_{\lambda}g_{\beta\sigma}
\right)  \nn
&=&
-\frac{1}{2}\left(
\partial_{\lambda}\partial_{\beta}g_{\alpha\sigma}-
\partial_{\lambda}\partial_{\alpha}g_{\beta\sigma}+
\partial_{\sigma}\partial_{\alpha}g_{\lambda\beta}-
\partial_{\sigma}\partial_{\beta}g_{\lambda\alpha}
\right)
\eea

\bigskip

̕\ňȉ̑Ώ͖̐炩B

\be
R_{\alpha\beta\gamma\delta}=R_{\gamma\delta\alpha\beta}
\label{eq:sym1}
\ee

\be
R_{\alpha\beta\gamma\delta}=-R_{\beta\alpha\gamma\delta}=
-R_{\alpha\beta\delta\gamma}
\label{eq:sym2}
\ee

\be
R_{\alpha\beta\gamma\delta}+R_{\alpha\delta\beta\gamma}+
R_{\alpha\gamma\delta\beta}=0
\label{eq:sym3}
\ee

\bigskip

$R_{\alpha\beta\gamma\delta}$̓e\Ȃ̂ŁC
̑Ώ͈̐ʂ̍WnɂĐ藧B

\begin{quote}
\noindent
\underline{exercise}

P
\be
\left[\nabla_{\lambda},
\left[\nabla_{\mu},\nabla_{\nu}\right]\right]\phi+
\left[\nabla_{\nu},
\left[\nabla_{\lambda},\nabla_{\mu}\right]\right]\phi+
\left[\nabla_{\mu},
\left[\nabla_{\nu},\nabla_{\lambda}\right]\right]\phi=0
\ee

C(\ref{eq:sym3}) 𓱂B
\end{quote}

\begin{quote}
\noindent
\underline{exercise}

_$X^{\mu}_0$ŋǏnƂB
̂Ƃ̋ߖT$X^{\mu}_0+\xi^{\mu}$ł̌vʂ
\be
g_{\mu\nu}(X^{\mu}_0+\xi^{\mu})=\eta_{\mu\nu}-\frac{1}{3}
R_{\mu\lambda\nu\sigma}(X^{\mu}_0)\xi^{\lambda}\xi^{\sigma}+O(\xi^3)
\ee

ƂȂBm߂B
\end{quote}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Bianchi identity}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Bianchi identity:
\be
\nabla_{\lambda}R_{\alpha\beta\mu\nu}+
\nabla_{\nu}R_{\alpha\beta\lambda\mu}+
\nabla_{\mu}R_{\alpha\beta\nu\lambda}=0
\label{eq:Bianchi}
\ee

̍PCRWnɂċ̓IɎƂłB

\begin{quote}
\noindent
\underline{exercise}

P
\be
\left[\nabla_{\lambda},
\left[\nabla_{\mu},\nabla_{\nu}\right]\right]A^{\alpha}+
\left[\nabla_{\nu},
\left[\nabla_{\lambda},\nabla_{\mu}\right]\right]A^{\alpha}+
\left[\nabla_{\mu},
\left[\nabla_{\nu},\nabla_{\lambda}\right]\right]A^{\alpha}=0
\ee

(\ref{eq:sym3})C(\ref{eq:Bianchi}) 𓱂B
\end{quote}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{divergence-free tensor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Bianchi identity ɂāC$g^{\alpha\mu}$ďkƂ
\be
\nabla_{\lambda}R_{\beta\nu}-\nabla_{\nu}R_{\beta\lambda}+
\nabla_{\mu}R^{\mu}{}_{\beta\nu\lambda}=0
\ee

\be
R_{\beta\delta}\equiv g^{\alpha\gamma}R_{\alpha\beta\gamma\delta}
\ee

Ricci tensorB

\bigskip

ɁC$g^{\beta\nu}$ďkƂ
\be
\nabla_{\lambda}R-\nabla_{\nu}R^{\nu}{}_{\lambda}-
\nabla_{\mu}R^{\mu}{}_{\lambda}=0
\ee

Ȃ킿
\be
\nabla_{\lambda}R-2\nabla_{\mu}R^{\mu}{}_{\lambda}=0
\ee

\be
R\equiv g^{\mu\nu}R_{\mu\nu}
\ee

̓XJ[ȗB

\bigskip

\be
\nabla_{\mu}\left(R^{\mu}{}_{\lambda}-
\frac{1}{2}\delta^{\mu}_{\lambda}R\right)=0
\ee

\be
\nabla_{\mu}\left(R^{\mu\nu}-
\frac{1}{2}g^{\mu\nu}R\right)=0
\ee

\bigskip

Einstein tensor $G_{\mu\nu}$ ̂悤ɒB
\be
G_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R
\ee

̂Ƃ
\be
\nabla_{\mu}G^{\mu\nu}=0
\ee

𖞂Ƃ킩B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Riemann tensor ̓ƗȐ}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Riemann tensor ̓ƗȐ̐Fl̏ꍇ}

l̏ꍇCnɐĂ݂B

\bigskip

$R_{\alpha\beta\gamma\delta}$ɂāC
$\alpha$$\beta͔Ώ̂Ȃ̂ŁC \alpha$$\beta$̑ĝƂ肤ꍇ̐́C
\be
\frac{4\times 3}{2}=6
\ee

$\gamma$$\deltȃgɂĂlŁC6ʂB \bigskip R_{\alpha\beta\gamma\delta}$$R_{(\alpha\beta)(\gamma\delta)}$
Ƃ݂ƁC$(\alpha\beta)$$(\gamma\delta)$ɂđΏ̂Ȃ̂ŁC
\ȏꍇ̐
\be
\frac{6\times 7}{2}=21
\ee

\bigskip

ł肩H܂܂B

\bigskip

܂łŉ\ȎwW
\bea
(\alpha\beta\gamma\delta)&=&
(0101),(0102),(0103),(0112),(0113),(0123),\nn
& &(0202),(0203),(0212),(0213),(0223),
(0303),\nn
& &(0312),(0313),(0323),
(1212),(1213),(1223),\nn
& &(1313),(1323),(2323)
\eea

$21$ł邪Ĉ
\be
R_{0123}-R_{0213}+R_{0312}=0
\ee

Ȃ̂ŁC͓Ɨł͂Ȃ
i܂CȊOɂ͏]֌W͂ȂjB

\bigskip

āCl̏ꍇCRiemann tensor ̓ƗȐ̐$20$łB

\subsection{Riemann tensor ̓ƗȐ̐F$D$̏ꍇ}

\bigskip

(\ref{eq:sym2})́C\Ȑ
\be
\frac{D(D-1)}{2}\times\frac{D(D-1)}{2}=\frac{D^2(D-1)^2}{4}
\ee

B

(\ref{eq:sym3})̐́C
\be
D\times\frac{D(D-1)(D-2)}{3!}=\frac{D^2(D-1)(D-2)}{6}
\ee

BāCƗȐ̌
\be
\frac{D^2(D-1)^2}{4}-\frac{D^2(D-1)(D-2)}{6}=\frac{D^2(D^2-1)}{12}
\ee

łB

\bigskip

́C(\ref{eq:sym1})(\ref{eq:sym2})(\ref{eq:sym3})
ƂłB

\begin{quote}
\noindent
\underline{exercise}

̂ƂB
\end{quote}

ǁC$D$ɂẮCRiemann tensor ̓Ɨ̌
\be
\frac{D^2(D^2-1)}{12}
\ee

łB

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{2ɂRiemann tensor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$\mu, \nu=1, 2$ƂB

\bigskip

QɂēƗ Riemann tensor ̐
\be
R_{1212}
\ee

łB

\bigskip

Ricci tensor ̐
\bea
R_{11}&=&g^{22}R_{2121}=g^{22}R_{1212} \nn
R_{22}&=&g^{11}R_{1212} \nn
R_{12}&=&R_{21} \nn
&=&g^{21}R_{2112}=-g^{12}R_{1212}
\eea

XJ[ȗ
\bea
R&=&g^{11}R_{11}+g^{22}R_{22}+2g^{12}R_{12} \nn
&=&2g^{11}g^{22}R_{1212}-2g^{12}g^{12}R_{1212} \nn
&=&2\left(g^{11}g^{22}-g^{12}g^{12}\right)R_{1212}
\eea

Ƃ
\bea
g^{11}&=&\frac{1}{g}g_{22} \nn
g^{22}&=&\frac{1}{g}g_{11} \nn
g^{12}&=&-\frac{1}{g}g_{12}
\eea

\be
g=g_{11}g_{22}-g_{12}g_{12}
\ee

Ȃ̂
\bea
R_{11}&=&\frac{1}{g}g_{11}R_{1212} \nn
R_{22}&=&\frac{1}{g}g_{22}R_{1212} \nn
R_{12}&=&\frac{1}{g}g_{12}R_{1212}
\eea

܂
\be
g^{11}g^{22}-g^{12}g^{12}=\frac{1}{g}
\ee

Ȃ̂
\be
R=\frac{2}{g}R_{1212}
\ee

\bigskip

ȏ̂Ƃ

\be
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=0
\ee

\be
R_{\mu\nu\rho\sigma}=\frac{1}{2}
R\left(g_{\mu\rho}g_{\nu\sigma}-
g_{\mu\sigma}g_{\nu\rho}\right)
\ee

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{3ɂRiemann tensor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$\mu, \nu=1, 2, 3$ƂB

\bigskip

RɂēƗ Riemann tensor ̐
\be
R_{1212},~~R_{1223},~~R_{1231},~~
R_{2323},~~R_{2331},~~R_{3131}
\ee

$6$łB

\bigskip

Ɨ Ricci tensor ̐
\bea
R_{11}&=&g^{22}R_{2121}+g^{33}R_{3131}+g^{23}R_{2131}+g^{32}R_{3121} \nn
&=&g^{22}R_{1212}+g^{33}R_{3131}-2g^{23}R_{1231} \nn
R_{22}&=&g^{11}R_{1212}+g^{33}R_{3232}+g^{31}R_{3212}+g^{13}R_{1232} \nn
&=&g^{11}R_{1212}+g^{33}R_{2323}-2g^{31}R_{1223} \nn
R_{33}&=&g^{11}R_{1313}+g^{22}R_{2323}+g^{12}R_{1323}+g^{21}R_{2313} \nn
&=&g^{11}R_{3131}+g^{22}R_{2323}-2g^{12}R_{2331} \nn
R_{12}&=&g^{21}R_{2112}+g^{23}R_{2132}+g^{31}R_{3112}+g^{33}R_{3132} \nn
&=&-g^{12}R_{1212}+g^{23}R_{1223}+g^{13}R_{1231}-g^{33}R_{2331} \nn
R_{23}&=&g^{12}R_{1223}+g^{11}R_{1213}+g^{32}R_{3223}+g^{31}R_{3213} \nn
&=&g^{12}R_{1223}-g^{11}R_{1231}-g^{23}R_{2323}+g^{13}R_{2331} \nn
R_{31}&=&g^{22}R_{2321}+g^{12}R_{1321}+g^{13}R_{1331}+g^{23}R_{2331} \nn
&=&-g^{22}R_{1223}+g^{12}R_{1231}-g^{13}R_{3131}+g^{23}R_{2331}
\eea

\bigskip

RWniǏnjpB
ɁC$g_{\mu\nu}(P)=\delta_{\mu\nu}$ƂB

\be
\left(\begin{array}{c}
R_{11} \\
R_{22} \\
R_{33} \\
R_{12} \\
R_{23} \\
R_{31}
\end{array}\right)
=
\left(\begin{array}{cccccc}
0 & 1 & 1 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & -1
\end{array}\right)
\left(\begin{array}{c}
R_{2323} \\
R_{3131} \\
R_{1212} \\
R_{2331} \\
R_{3112} \\
R_{1223}
\end{array}\right)
\ee

\be
\left(\begin{array}{c}
R_{2323} \\
R_{3131} \\
R_{1212} \\
R_{2331} \\
R_{3112} \\
R_{1223}
\end{array}\right)
=
\left(\begin{array}{cccccc}
-\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 \\
\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 \\
\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & -1
\end{array}\right)
\left(\begin{array}{c}
R_{11} \\
R_{22} \\
R_{33} \\
R_{12} \\
R_{23} \\
R_{31}
\end{array}\right)
\ee

\bigskip

̂ƂĈƂ킩邩H
\bea
R_{\alpha\beta\gamma\delta}&=&
R_{\alpha\gamma}g_{\beta\delta}+
R_{\beta\delta}g_{\alpha\gamma}-
R_{\alpha\delta}g_{\beta\gamma}-
R_{\beta\gamma}g_{\alpha\delta} \nn
&-&\frac{1}{2}R\left(
g_{\alpha\gamma}g_{\beta\delta}-
g_{\alpha\delta}g_{\beta\gamma}
\right)
\eea

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}

\bibitem{Sasaki}
Xؐ߁@ʑΘ_@YƐ}

\end{thebibliography}

\end{document}
`

߂