Weyl

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\title{Weyl変換，Weyl tensor}
\author{
　}
\date{1999年07月07日改訂}
\begin{document}
\maketitle
\begin{abstract}

\end{abstract}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{曲率の定義}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

ここで
\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

Ricci tensor:
\be
R_{\sigma\alpha}\equiv R^{\mu}{}_{\sigma\mu\alpha}
\ee

scalar curvature:
\be
R\equiv g^{\sigma\alpha}R_{\sigma\alpha}
\ee

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Weyl変換}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Weyl変換:
\be
g_{\mu\nu}\rightarrow\bar{g}_{\mu\nu}=e^{2\omega}g_{\mu\nu}
\ee

\bigskip

このとき

\bea
\bar{R}^{\mu}{}_{\sigma\beta\alpha}&=&R^{\mu}{}_{\sigma\beta\alpha}+
\delta^{\mu}_{\alpha}\nabla_{\beta}\nabla_{\sigma}\omega-
\delta^{\mu}_{\beta}\nabla_{\alpha}\nabla_{\sigma}\omega \nn
&-&g_{\sigma\alpha}\nabla_{\beta}\nabla^{\mu}\omega+
g_{\sigma\beta}\nabla_{\alpha}\nabla^{\mu}\omega-
\delta^{\mu}_{\alpha}\nabla_{\beta}\omega\nabla_{\sigma}\omega \nn
&+&\delta^{\mu}_{\beta}\nabla_{\alpha}\omega\nabla_{\sigma}\omega+
g_{\sigma\alpha}\nabla_{\beta}\omega\nabla^{\mu}\omega-
g_{\sigma\beta}\nabla_{\alpha}\omega\nabla^{\mu}\omega \nn
&-&\delta^{\mu}_{\beta}g_{\sigma\alpha}\nabla_{\lambda}\omega\nabla^{\lambda}\omega+
\delta^{\mu}_{\alpha}g_{\sigma\beta}\nabla_{\lambda}\omega\nabla^{\lambda}\omega
\eea

\bea
\bar{R}_{\mu\nu}&=&R_{\mu\nu}-
(D-2)\nabla_{\mu}\nabla_{\nu}\omega-
g_{\mu\nu}\nabla^{\lambda}\nabla_{\lambda}\omega \nn
&+&(D-2)\nabla_{\mu}\omega\nabla_{\nu}\omega-
(D-2)g_{\mu\nu}\nabla_{\lambda}\omega\nabla^{\lambda}\omega
\eea

\be
\bar{R}=e^{-2\omega}\left[R-
2(D-1)\nabla^{\lambda}\nabla_{\lambda}\omega-
(D-1)(D-2)\nabla_{\lambda}\omega\nabla^{\lambda}\omega\right]
\ee

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

scalar field $\phi$がWeyl変換の下で
\be
\phi\rightarrow\bar{\phi}=e^{-\frac{D-2}{2}\omega}\phi
\ee
と変換するものとする。

このとき
\be
-\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}
\phi\right)+\frac{1}{4}\frac{D-2}{D-1}R\phi=0
\ee

は，Weyl「不変」な方程式であることを示せ。
\end{quote}

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

action
\be
S=\int d^Dx \sqrt{-g}\left[-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi
\partial_{\nu}\phi-\frac{1}{8}\frac{D-2}{D-1}R\phi^2\right]
\ee

は，Weyl不変であることを示せ。

また，Weyl不変な$\phi$の自己相互作用項をつくれ。
\end{quote}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Weyl tensor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Weyl tensor:
\bea
C_{\mu\sigma\beta\alpha}&=&R_{\mu\sigma\beta\alpha} \nn
&+&\frac{1}{D-2}\left(g_{\mu\alpha}R_{\sigma\beta}+
g_{\sigma\beta}R_{\mu\alpha}-g_{\mu\beta}R_{\sigma\alpha}-
g_{\sigma\alpha}R_{\mu\beta}\right) \nn
&+&\frac{1}{(D-1)(D-2)}\left(g_{\mu\beta}g_{\sigma\alpha}-
g_{\mu\alpha}g_{\sigma\beta}\right) R
\eea

\bigskip

\be
\bar{C}_{\mu\sigma\beta\alpha}=e^{2\omega}C_{\mu\sigma\beta\alpha}
\ee

\bigskip

対称性:
\be
C_{\alpha\beta\gamma\delta}=C_{\gamma\delta\alpha\beta}
\label{eq:sym1}
\ee

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

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

\bigskip

\be
C^{\alpha}{}_{\beta\alpha\delta}\equiv 0
\label{eq:trfree}
\ee

\bigskip

Weyl tensorの独立な成分の個数は，
Riemann tensor の独立成分の個数から
(\ref{eq:trfree})の制限の個数をひいたものであるので，
\be
\frac{D^2(D^2-1)}{12}-\frac{D(D+1)}{2}=\frac{D(D+1)(D+2)(D-3)}{12}
\ee

である。（$D\ge 3$）

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

３次元では
\be
C_{\alpha\beta\gamma\delta}\equiv 0
\ee
となることを示せ。
\end{quote}


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%\begin{thebibliography}{99}

%\bibitem{Sasaki} 佐々木節　一般相対論　産業図書

%\end{thebibliography}

\end{document}
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