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エネルギー・運動量テンソルの形（３）

<<   作成日時 ： 2012/06/18 00:01   >>

 計量を含むテンソル性の確認なので、計量の２階偏微分も求める必要があるので、それを検討します。 まず、 $f_{1}\equiv \frac{\partial^{2}x^{\alpha } }{\partial {x}'^{\lambda }\partial {x}'^{\mu }}\frac{\partial x^{\beta }}{\partial {x}'^{\nu }}\: g_{\alpha \beta }\left ( x \right )$ $f_{2}\equiv \frac{\partial x^{\alpha }}{\partial {x}'^{\mu }}\frac{\partial^{2}x^{\beta } }{\partial {x}'^{\lambda }\partial {x}'^{\nu }}\: g_{\alpha \beta }\left ( x \right )$ $f_{3}\equiv \frac{\partial x^{\alpha }}{\partial {x}'^{\mu }}\frac{\partial x^{\beta }}{\partial {x}'^{\nu }}\frac{\partial x^{\tau }}{\partial {x}'^{\lambda }}\frac{\partial g_{\alpha \beta }\left ( x \right )}{\partial x^{\tau }}$ とすると、前記事から $\frac{\partial {g}'_{\mu \nu }\left ( {x}' \right )}{\partial {x}'^{\lambda }}= f_{1}+ f_{2}+ f_{3}$ となります。 なので、 $\frac{\partial^{2} {g}'_{\mu \nu }\left ( {x}' \right )}{\partial {x}'^{\rho }\partial {x}'^{\lambda }}= \frac{\partial f_{1} }{\partial{x}'^{\rho }}+\frac{\partial f_{2} }{\partial{x}'^{\rho }}+\frac{\partial f_{3} }{\partial{x}'^{\rho }}$ です。よって右辺の項毎に計算すると、 $\left ( \frac{\partial f_{1} }{\partial{x}'^{\rho }} \right ) _{P}= \left ( \frac{\partial^{3}x^{\alpha } }{\partial {x}'^{\rho }\partial {x}'^{\lambda}\partial {x}'^{\mu }} \right )_{P}\left ( \frac{\partial x^{\beta} }{\partial {x}'^{\nu }} \right )_{P}\left ( g_{\alpha \beta }\left ( x \right ) \right )_{P}$ $+\left ( \frac{\partial^{2} x^{\alpha }}{\partial {x}'^{\lambda }\partial {x}'^{\mu }} \right )_{P}\left ( \frac{\partial^{2} x^{\beta }}{\partial {x}'^{\rho }\partial {x}'^{\nu }} \right )_{P}\left ( g_{\alpha \beta }\left ( x \right ) \right )_{P} + \left ( \frac{\partial^{2} x^{\alpha }}{\partial {x}'^{\rho }\partial {x}'^{\mu }} \right )_{P}\left ( \frac{\partial x^{\beta} }{\partial {x}'^{\nu }} \right )_{P}\left ( \frac{\partial g_{\alpha \beta }\left ( x \right )}{\partial {x}'^{\rho }} \right )_{P}$ $= a_{\rho \lambda \mu }^{\alpha }\delta _{\nu }^{\beta }\eta _{\alpha \beta }= \eta _{\alpha \nu }a_{\rho \lambda \mu }^{\alpha }$ $\left ( \frac{\partial f_{2} }{\partial{x}'^{\rho }} \right ) _{P}= \left ( \frac{\partial^{2} x^{\alpha }}{\partial {x}'^{\rho }\partial {x}'^{\mu }} \right )_{P}\left ( \frac{\partial^{2} x^{\beta }}{\partial {x}'^{\lambda }\partial {x}'^{\nu }} \right )_{P}\left ( g_{\alpha \beta }\left ( x \right ) \right )_{P}$ $+\left ( \frac{\partial x^{\alpha }}{\partial {x}'^{\mu }} \right )_{P}\left ( \frac{\partial^{3} x^{\beta }}{\partial {x}'^{\rho }\partial {x}'^{\lambda }\partial {x}'^{\nu }} \right )_{P}\left ( g_{\alpha \beta }\left ( x \right ) \right )_{P}+ \left ( \frac{\partial x^{\alpha } }{\partial {x}'^{\mu }} \right )_{P}\left ( \frac{\partial^{2} x^{\beta }}{\partial {x}'^{\lambda }\partial {x}'^{\nu }} \right )_{P}\left ( \frac{\partial g_{\alpha \beta }\left ( x \right )}{\partial {x}'^{\rho }} \right )_{P}$ $= \delta _{\mu }^{\alpha }a_{\rho \lambda \nu }^{\beta }\eta _{\alpha \beta }=\eta _{\mu \beta } a_{\rho \lambda \nu }^{\beta }$ $\left ( \frac{\partial f_{3} }{\partial{x}'^{\rho }} \right ) _{P}=\left ( \frac{\partial^{2}x^{\alpha } }{\partial {x}'^{\rho }\partial {x}'^{\mu }} \right )_{P}\left ( \frac{\partial x^{\beta }}{\partial {x}'^{\nu }}\right )_{P}\left ( \frac{\partial x^{\tau }}{\partial {x}'^{\lambda }}\right )_{P}\left ( \frac{\partial g_{\alpha \beta }\left ( x \right )}{\partial x^{\tau }}\right )_{P}$ $+\left ( \frac{\partial x^{\alpha } }{\partial {x}'^{\mu }} \right )_{P}\left ( \frac{\partial^{2} x^{\beta }}{\partial {x}'^{\rho }\partial {x}'^{\nu }}\right )_{P}\left ( \frac{\partial x^{\tau }}{\partial {x}'^{\lambda }}\right )_{P}\left ( \frac{\partial g_{\alpha \beta }\left ( x \right )}{\partial x^{\tau }}\right )_{P}$ $+\left ( \frac{\partial x^{\alpha } }{\partial {x}'^{\mu }} \right )_{P}\left ( \frac{\partial^{2} x^{\beta }}{\partial {x}'^{\nu }}\right )_{P}\left ( \frac{\partial^{2} x^{\tau }}{\partial {x}'^{\rho }\partial {x}'^{\lambda }}\right )_{P}\left ( \frac{\partial g_{\alpha \beta }\left ( x \right )}{\partial x^{\tau }}\right )_{P}$ $+\left ( \frac{\partial x^{\alpha } }{\partial {x}'^{\mu }} \right )_{P}\left ( \frac{\partial^{2} x^{\beta }}{\partial {x}'^{\nu }}\right )_{P}\left ( \frac{\partial x^{\tau }}{\partial {x}'^{\lambda }}\right )_{P}\left ( \frac{\partial x^{\sigma }}{\partial {x}'^{\rho }} \frac{\partial^{2} g_{\alpha \beta }\left ( x \right )}{\partial x^{\sigma }\partial x^{\tau }}\right )_{P}$ $=\delta _{\mu }^{\alpha }\delta _{\nu }^{\beta }\delta _{\lambda }^{\tau }\delta _{\rho }^{\sigma }\left ( \frac{\partial^{2} g_{\alpha \beta }\left ( x \right )}{\partial x^{\sigma }\partial x^{\tau }}\right )_{P}=\left ( \frac{\partial^{2} g_{\mu \nu }\left ( x \right )}{\partial x^{\rho }\partial x^{\lambda }}\right )_{P}$ なので、 $\left ( \frac{\partial^{2} {g}'_{\mu \nu }\left ( {x}' \right )}{\partial {x}'^{\rho }\partial {x}'^{\lambda }} \right )_{P}=\left ( \frac{\partial^{2} g_{\mu \nu }\left ( x \right )}{\partial x^{\rho }\partial x^{\lambda }}\right )_{P}+\eta _{\alpha \nu }a_{\rho \lambda \mu }^{\alpha }+\eta _{\mu \beta } a_{\rho \lambda \nu }^{\beta }$ となり、(　)P を省略して簡略化すると、 ${g}'_{\mu \nu ,\rho \lambda }=g_{\mu \nu ,\rho \lambda }+\eta _{\alpha \nu }a_{\rho \lambda \mu }^{\alpha }+\eta _{\mu \beta } a_{\rho \lambda \nu }^{\beta }$ ということになりました。

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