読者です 読者をやめる 読者になる 読者になる

オタクof数理の共同ブログ

京大情報学科数理工学コースの学生4人による共同ブログです

クリストッフェル記号の変換則パート2

よねすけ 数学

こんにちは、よねすけです。
今日は2次試験二日目ですね。受験生の方は今日くらいゆっくり休んで下さい(受験生このブログ読んでなさそう)。

以前クリストッフェル記号の変換則について書きました。
otaku-of-suri.hatenablog.com

その時はすっきりと変換則を求める方法を書いたんですけど、そのあとややこしい方法でやってみたら出来たんで一応ここに記すことにします。
前回と同じ問題設定を考えます。
\begin{equation}
\displaystyle\Gamma^{\alpha}_{\beta\gamma}=\frac{1}{2}g^{\alpha\mu}\left(\frac{\partial g_{\mu\beta}}{\partial x^{\gamma}}+\frac{\partial g_{\mu\gamma}}{\partial x^{\beta}}-\frac{\partial g_{\beta\gamma}}{\partial x^{\mu}}\right)
\end{equation}
これがクリストッフェル記号で、これを\{x^{\mu}\}\to\{x^{\mu'}\}の変換に従って
\begin{equation}
\displaystyle\Gamma^{\alpha'}_{\beta'\gamma'}=\frac{1}{2}g^{\alpha'\mu'}\left(\frac{\partial g_{\mu'\beta'}}{\partial x^{\gamma'}}+\frac{\partial g_{\mu'\gamma'}}{\partial x^{\beta'}}-\frac{\partial g_{\beta'\gamma'}}{\partial x^{\mu'}}\right)
\end{equation}
と変換することを考えます。
まず計量テンソルの変換は
\displaystyle g_{\gamma'\mu'}=\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\frac{\partial x^{\mu}}{\partial x^{\mu'}}g_{\gamma\mu}
なのでこれの偏微分
\begin{eqnarray}
g_{\gamma'\mu',\beta'}&=&\frac{\partial}{\partial x^{\beta'}}\left(g_{\gamma'\mu'}\right)\\
&=&\frac{\partial}{\partial x^{\beta'}}\left(\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\frac{\partial x^{\mu}}{\partial x^{\mu'}}g_{\gamma\mu}\right)\\
&=&\frac{\partial^2 x^{\gamma}}{\partial x^{\beta'}\partial x^{\gamma'}}\frac{\partial x^{\mu}}{\partial x^{\mu'}}g_{\gamma\mu}+\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\frac{\partial^2 x^{\mu}}{\partial x^{\beta'}x^{\mu'}}g_{\gamma\mu}+\frac{\partial x^{\gamma}}{\partial x^{\mu'}}\frac{\partial x^{\mu}}{\partial x^{\mu'}}\underline{\frac{\partial g_{\gamma\mu}}{\partial x_{\beta'}}}\\
&=&\frac{\partial^2 x^{\gamma}}{\partial x^{\beta'}\partial x^{\gamma'}}\frac{\partial x^{\mu}}{\partial x^{\mu'}}g_{\gamma\mu}+\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\frac{\partial^2 x^{\mu}}{\partial x^{\beta'}x^{\mu'}}g_{\gamma\mu}+\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\frac{\partial x^{\mu}}{\partial x^{\mu'}}\underline{\frac{\partial x^{\beta}}{\partial x^{\beta'}}g_{\gamma\mu,\beta}}\\
\end{eqnarray}
になります。同様に考えると(添え字を入れ替えていく)、
\begin{eqnarray}
\displaystyle g_{\mu'\beta',\gamma'}&=&\frac{\partial^2 x^{\mu}}{\partial x^{\gamma'}\partial x^{\mu'}}\frac{\partial x^{\beta}}{\partial x^{\beta'}}g_{\mu\beta}+\frac{\partial x^{\mu}}{\partial x^{\mu'}}\frac{\partial^2 x^{\beta}}{\partial x^{\gamma'}x^{\beta'}}g_{\mu\beta}+\frac{\partial x^{\mu}}{\partial x^{\mu'}}\frac{\partial x^{\beta}}{\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}g_{\mu\beta,\gamma}\\
\displaystyle g_{\beta'\gamma',\mu'}&=&\frac{\partial^2 x^{\beta}}{\partial x^{\mu'}\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}g_{\beta\gamma}+\frac{\partial x^{\beta}}{\partial x^{\beta'}}\frac{\partial^2 x^{\gamma}}{\partial x^{\mu'}x^{\gamma'}}g_{\beta\gamma}+\frac{\partial x^{\beta}}{\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\frac{\partial x^{\mu}}{\partial x^{\mu'}}g_{\beta\gamma,\mu}\\
\end{eqnarray}
です。ただし、偏微分の順序交換と計量テンソル対称性から3つめの式は
\displaystyle g_{\beta'\gamma',\mu'}=2\frac{\partial^2 x^{\beta}}{\partial x^{\mu'}\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}g_{\beta\gamma}+\frac{\partial x^{\beta}}{\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\frac{\partial x^{\mu}}{\partial x^{\mu'}}g_{\beta\gamma,\mu}
と書けます。
また
\displaystyle g^{\alpha'\mu'}=\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\mu'}}{\partial x^{\lambda}}g^{\alpha\lambda}
なので、これをはじめの式に代入していきましょう。
\begin{equation}
\displaystyle\Gamma^{\alpha'}_{\beta'\gamma'}=\frac{1}{2}g^{\alpha'\mu'}\left(\frac{\partial g_{\mu'\beta'}}{\partial x^{\gamma'}}+\frac{\partial g_{\mu'\gamma'}}{\partial x^{\beta'}}-\frac{\partial g_{\beta'\gamma'}}{\partial x^{\mu'}}\right)
\end{equation}
について、第1項は
\begin{eqnarray}
\displaystyle g^{\alpha'\mu'}g_{\gamma'\mu',\beta'}&=&\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\mu'}}{\partial x^{\lambda}}g^{\alpha\lambda}\Bigg{(}\frac{\partial^2 x^{\gamma}}{\partial x^{\beta'}\partial x^{\gamma'}}\frac{\partial x^{\mu}}{\partial x^{\mu'}}g_{\gamma\mu}\\
&\ &+\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\frac{\partial^2 x^{\mu}}{\partial x^{\beta'}x^{\mu'}}g_{\gamma\mu}+\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\frac{\partial x^{\mu}}{\partial x^{\mu'}}\frac{\partial x^{\beta}}{\partial x^{\beta'}}g_{\gamma\mu,\beta}\Bigg{)}\\
\end{eqnarray}
になります。この中の第1項は
\begin{eqnarray}
\displaystyle&\ &\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\underline{\frac{\partial x^{\mu'}}{\partial x^{\lambda}}g^{\alpha\lambda}}\frac{\partial^2 x^{\gamma}}{\partial x^{\beta'}\partial x^{\gamma'}}\underline{\frac{\partial x^{\mu}}{\partial x^{\mu'}}}g_{\gamma\mu}\\
&=&\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial^2 x^{\gamma}}{\partial x^{\beta'}\partial x^{\gamma'}}\underline{\frac{\partial x^{\mu}}{\partial x^{\lambda}}g^{\alpha\lambda}}g_{\gamma\mu}\\
&=&\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial^2 x^{\gamma}}{\partial x^{\beta'}\partial x^{\gamma'}}\underline{\delta^{\mu}_{\lambda}g^{\alpha\lambda}}g_{\gamma\mu}\\
&=&\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial^2 x^{\gamma}}{\partial x^{\beta'}\partial x^{\gamma'}}\underline{\underline{g^{\alpha\mu}}g_{\gamma\mu}}\\
&=&\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial^2 x^{\gamma}}{\partial x^{\beta'}\partial x^{\gamma'}}\underline{g^{\alpha\mu}g_{\mu\gamma}}\\
&=&\underline{\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}}\frac{\partial^2 x^{\gamma}}{\partial x^{\beta'}\partial x^{\gamma'}}\underline{\underline{\delta^{\alpha}_{\gamma}}}\\
&=&\underline{\frac{\partial x^{\alpha'}}{\partial x^{\gamma}}}\frac{\partial^2 x^{\gamma}}{\partial x^{\beta'}\partial x^{\gamma'}}\\
\end{eqnarray}
になります。第2項は
\begin{eqnarray}
\displaystyle&\ &\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\underline{\frac{\partial x^{\mu'}}{\partial x^{\lambda}}}g^{\alpha\lambda}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\underline{\frac{\partial^2 x^{\mu}}{\partial x^{\beta'}\partial x^{\mu'}}}g_{\gamma\mu}\\
&=&\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\underline{\frac{\partial x^{\mu'}}{\partial x^{\lambda}}\frac{\partial}{\partial x^{\mu'}}\left(\frac{\partial x^{\mu}}{\partial x^{\beta'}}\right)}g^{\alpha\lambda}g_{\gamma\mu}\\
&=&\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\underline{\frac{\partial}{\partial x^{\lambda}}\left(\frac{\partial x^{\mu}}{\partial x^{\beta'}}\right)}g^{\alpha\lambda}g_{\gamma\mu}\\
&=&\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\underline{\frac{\partial}{\partial x^{\beta'}}\left(\frac{\partial x^{\mu}}{\partial x^{\lambda}}\right)}g^{\alpha\lambda}g_{\gamma\mu}\\
&=&\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\underline{\frac{\partial}{\partial x^{\beta'}}\left(\delta^{\mu}_{\lambda}\right)}g^{\alpha\lambda}g_{\gamma\mu}\\
&=&0\\
\end{eqnarray}
になります。第3項は
\begin{eqnarray}
\displaystyle&\ &\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\underline{\frac{\partial x^{\mu'}}{\partial x^{\lambda}}g^{\alpha\lambda}}\frac{\partial x^{\beta}}{\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\underline{\frac{\partial x^{\mu}}{\partial x^{\mu'}}}g_{\gamma\mu,\beta}\\
&=&\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\beta}}{\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\underline{\delta^{\mu}_{\lambda}g^{\alpha\lambda}}g_{\gamma\mu,\beta}\\
&=&\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\beta}}{\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\underline{g^{\alpha\mu}}g_{\gamma\mu,\beta}\\
\end{eqnarray}
これでクリストッフェル記号の第1項の計算が終わりました。クリストッフェル記号の第2項は同様の計算でできます(添え字を変える)。
\displaystyle g^{\alpha'\mu'}g_{\mu'\beta',\gamma'}=\frac{\partial x^{\alpha'}}{\partial x^{\beta}}\frac{\partial^2 x^{\beta}}{\partial x^{\beta'}\partial x^{\gamma'}}+\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\beta}}{\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}g^{\alpha\mu}g_{\mu\beta,\gamma}
最後にクリストッフェル記号の第3項は
\displaystyle g^{\alpha'\mu'}g_{\beta'\gamma',\mu'}=\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\mu'}}{\partial x^{\lambda}}g^{\alpha\lambda}\left(2\frac{\partial^2 x^{\beta}}{\partial x^{\mu'}\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}+\frac{\partial x^{\beta}}{\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\frac{\partial x^{\mu}}{\partial x^{\mu'}}g_{\beta\gamma,\mu}\right)
ですが、この中の第1項は
\begin{eqnarray}
\displaystyle &\ &\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\underline{\frac{\partial x^{\mu'}}{\partial x^{\lambda}}}g^{\alpha\lambda}\underline{\frac{\partial^2 x^{\beta}}{\partial x^{\mu'}\partial x^{\beta'}}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}g_{\beta\gamma}\\
&=&\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\underline{\frac{\partial x^{\mu'}}{\partial x^{\lambda}}\frac{\partial}{\partial x^{\mu'}}\left(\frac{\partial x^{\beta}}{\partial x^{\beta'}}\right)}g^{\alpha\lambda}g_{\beta\gamma}\\
&=&\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\underline{\frac{\partial}{\partial x^{\lambda}}\left(\frac{\partial x^{\beta}}{\partial x^{\beta'}}\right)}g^{\alpha\lambda}g_{\beta\gamma}\\
&=&\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\underline{\frac{\partial}{\partial x^{\beta'}}\left(\frac{\partial x^{\beta}}{\partial x^{\lambda}}\right)}g^{\alpha\lambda}g_{\beta\gamma}\\
&=&\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\underline{\frac{\partial}{\partial x^{\beta'}}\left(\delta^{\beta}_{\lambda}\right)}g^{\alpha\lambda}g_{\beta\gamma}\\
&=&0\\
\end{eqnarray}
第2項は
\begin{eqnarray}
\displaystyle&\ &\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\underline{\frac{\partial x^{\mu'}}{\partial x^{\lambda}}g^{\alpha\lambda}}\frac{\partial x^{\beta}}{\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\underline{\frac{\partial x^{\mu}}{\partial x^{\mu'}}}g_{\beta\gamma,\mu}\\
&=&\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\beta}}{\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\underline{\delta^{\mu}_{\lambda}g^{\alpha\lambda}}g_{\beta\gamma,\mu}\\
&=&\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\beta}}{\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\underline{g^{\alpha\mu}}g_{\beta\gamma,\mu}\\
\end{eqnarray}
すべて計算ができました!!あとは足し合わせるだけです。
\begin{eqnarray}
\Gamma^{\alpha'}_{\beta'\gamma'}&=&\frac{1}{2}\left(g^{\alpha'\mu'}g_{\gamma'\mu',\beta'}+g^{\alpha'\mu'}g_{\mu'\beta',\gamma'}-g^{\alpha'\mu'}g_{\beta'\gamma',\mu'}\right)\\
&=&\frac{1}{2}\Bigg{(}\frac{\partial x^{\alpha'}}{\partial x^{\gamma}}\frac{\partial^2 x^{\gamma}}{\partial x^{\beta'}\partial x^{\gamma'}}+\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\beta}}{\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}g^{\alpha\mu}g_{\gamma\mu,\beta}+\frac{\partial x^{\alpha'}}{\partial x^{\beta}}\frac{\partial^2 x^{\beta}}{\partial x^{\beta'}\partial x^{\gamma'}}\\
&\ &+\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\beta}}{\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}g^{\alpha\mu}g_{\mu\beta,\gamma}-\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\beta}}{\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}g^{\alpha\mu}g_{\beta\gamma,\mu}\Bigg{)}\\
&=&\frac{\partial x^{\alpha'}}{\partial x^{\gamma}}\frac{\partial^2 x^{\gamma}}{\partial x^{\beta'}\partial x^{\gamma'}}+\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\beta}}{\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\frac{1}{2}g^{\alpha\mu}\left(g_{\gamma\mu,\beta}+g_{\mu\beta,\gamma}-g_{\beta\gamma,\mu}\right)\\
&=&\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\beta}}{\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\Gamma^{\alpha}_{\beta\gamma}+\frac{\partial x^{\alpha'}}{\partial x^{\gamma}}\frac{\partial^2 x^{\gamma}}{\partial x^{\beta'}\partial x^{\gamma'}}\\
\end{eqnarray}
よってクリストッフェル記号の変換則
\displaystyle\Gamma^{\alpha'}_{\beta'\gamma'}=\frac{\partial x^{\alpha'}}{\partial x^{\alpha}}\frac{\partial x^{\beta}}{\partial x^{\beta'}}\frac{\partial x^{\gamma}}{\partial x^{\gamma'}}\Gamma^{\alpha}_{\beta\gamma}+\frac{\partial x^{\alpha'}}{\partial x^{\gamma}}\frac{\partial^2 x^{\gamma}}{\partial x^{\beta'}\partial x^{\gamma'}}

以上です。どちゃくそ計算がめんどくさいですね。