# T_NAKÄ[uO

## GlM[E^ʃe\̌iSj

<<   쐬 F 2012/06/20 00:01   >>

 OL܂łɋ߂֌WATʃ ̍Wϊ߂Ă܂B ܂OƂāA $\left ( T_{\mu \nu }\left ( x \right ) \right )_{P}= \left \{Ag_{\alpha \beta ,\mu \nu }+Bg_{\mu \nu,\alpha \beta } +C\left ( g_{\mu \alpha ,\nu \beta } +g_{\nu \alpha ,\mu \beta }\right ) \right \}\eta ^{\alpha \beta }$ $+\eta _{\mu \nu }\left ( Dg_{\alpha \beta ,\rho \sigma }+Eg_{\alpha \rho ,\beta \sigma } \right )\eta ^{\alpha \beta }\eta ^{\rho \sigma }+F\eta _{\mu \nu }$ łAWϊ邱Ƃl܂B܂A $\left ( {T}'_{\mu \nu }\left ( x \right ) \right )_{P}= \left \{A{g}'_{\alpha \beta ,\mu \nu }+B{g}'_{\mu \nu,\alpha \beta } +C\left ( {g}'_{\mu \alpha ,\nu \beta } +{g}'_{\nu \alpha ,\mu \beta }\right ) \right \}\eta ^{\alpha \beta }$ $+\eta _{\mu \nu }\left ( D{g}'_{\alpha \beta ,\rho \sigma }+E{g}'_{\alpha \rho ,\beta \sigma } \right )\eta ^{\alpha \beta }\eta ^{\rho \sigma }+F\eta _{\mu \nu }$ $= \big[A\left ( g_{\alpha \beta ,\mu \nu } +\eta _{\tau \beta }a_{\mu \nu \alpha }^{\tau }+\eta _{\tau \alpha }a_{\mu \nu \beta }^{\tau }\right )+B\left ( g_{\mu \nu,\alpha \beta} +\eta _{\tau \nu }a_{\alpha \beta \mu }^{\tau }+\eta _{\tau \mu }a_{\alpha \beta \nu }^{\tau }\right )$ $+C\left ( g_{\mu \alpha , \nu \beta } +\eta _{\tau \alpha }a_{ \nu \beta \mu }^{\tau }+\eta _{\tau \mu }a_{ \nu \beta \alpha }^{\tau }+g_{ \nu \alpha ,\mu \beta} +\eta _{\tau \alpha }a_{\mu \beta \nu }^{\tau }+\eta _{\tau\nu }a_{\mu \beta \alpha }^{\tau } \right )\big]\eta ^{\alpha \beta }$ $+\eta _{\mu \nu } \big[D\left ( g_{\alpha \beta ,\rho \sigma } +\eta _{\tau \beta }a_{\rho \sigma \alpha }^{\tau }+\eta _{\tau \alpha }a_{\rho \sigma \beta }^{\tau }\right )+E\left ( g_{\alpha \rho , \beta\sigma } +\eta _{\tau \rho }a_{\beta \sigma \alpha }^{\tau }+\eta _{\tau \alpha }a_{ \beta \sigma \rho }^{\tau }\right )\big]\eta ^{\alpha \beta }\eta ^{\rho \sigma }$ $+F\eta _{\mu \nu }$ $= \big[\left \{Ag_{\alpha \beta ,\mu \nu }+Bg_{\mu \nu,\alpha \beta } +C\left ( g_{\mu \alpha ,\nu \beta } +g_{\nu \alpha ,\mu \beta }\right ) \right \}\eta ^{\alpha \beta }$ $+\eta _{\mu \nu }\left ( Dg_{\alpha \beta ,\rho \sigma }+Eg_{\alpha \rho ,\beta \sigma } \right )\eta ^{\alpha \beta }\eta ^{\rho \sigma }+F\eta _{\mu \nu }\big]$ $+\big[A\left ( \eta _{\tau \beta }a_{\mu \nu \alpha }^{\tau }+\eta _{\tau \alpha }a_{\mu \nu \beta }^{\tau } \right )+B\left ( \eta _{\tau \nu }a_{ \alpha \beta\mu }^{\tau }+\eta _{\tau \mu }a_{\alpha \beta \nu }^{\tau } \right )$ $+C\left ( \eta _{\tau \alpha } a_{\nu \beta \mu }^{\tau }+\eta _{\tau \mu } a_{\nu \beta \alpha }^{\tau }+\eta _{\tau \alpha } a_{\mu \beta \nu }^{\tau }+\eta _{\tau \nu } a_{\mu \beta \alpha }^{\tau } \right )\big]\eta ^{\alpha \beta }$ $+\eta_{\mu \nu } \left \{ D\left ( \eta _{\tau \beta }a_{\rho \sigma \alpha }^{\tau } +\eta _{\tau \alpha }a_{\rho \sigma \beta }^{\tau }\right )+E\left ( \eta _{\tau \rho }a_{\beta \sigma \alpha }^{\tau } +\eta _{\tau \alpha }a_{ \beta \sigma\rho }^{\tau } \right ) \right \}\eta ^{\alpha \beta }\eta ^{\rho \sigma }$ $= \left ( T_{\mu \nu }\left ( x \right ) \right )_{P}$ $+A\left (\delta _{\tau }^{\alpha } a_{\mu \nu \alpha }^{\tau }+\delta _{\tau }^{\beta }a_{\mu \nu \beta }^{\tau } \right )+B\eta ^{\alpha \beta }\left ( \eta _{\tau \nu }a_{ \alpha \beta\mu }^{\tau }+\eta _{\tau \mu }a_{\alpha \beta \nu }^{\tau } \right )$ $+C\left ( \delta _{\tau }^{\beta } a_{\nu \beta \mu }^{\tau }+\delta _{\tau }^{\beta } a_{\mu \beta \nu }^{\tau } \right )+C\eta ^{\alpha \beta }\left ( \eta _{\tau \mu } a_{\nu \beta \alpha }^{\tau }+\eta _{\tau \nu } a_{\mu \beta \alpha }^{\tau } \right )$ $+ D\eta_{\mu \nu }\left ( \delta _{\tau }^{\alpha }a_{\rho \sigma \alpha }^{\tau } +\delta _{\tau }^{\beta }a_{\rho \sigma \beta }^{\tau }\right )\eta ^{\rho \sigma }+E\eta_{\mu \nu }\left ( \delta _{\tau }^{\sigma }\eta ^{\alpha \beta }a_{\beta \sigma \alpha }^{\tau } +\delta _{\tau }^{\beta }\eta ^{\rho \sigma }a_{ \beta \sigma\rho }^{\tau } \right )$ $= \left ( T_{\mu \nu }\left ( x \right ) \right )_{P}$ $+A\left ( a_{\mu \nu \alpha }^{\alpha }+a_{\mu \nu \beta }^{\beta } \right )+B\eta ^{\alpha \beta }\left ( \eta _{\tau \nu }a_{ \alpha \beta\mu }^{\tau }+\eta _{\tau \mu }a_{\alpha \beta \nu }^{\tau } \right )$ $+C\left ( a_{\nu \beta \mu }^{\beta }+ a_{\mu \beta \nu }^{\beta } \right )+C\eta ^{\alpha \beta }\left ( \eta _{\tau \mu } a_{\nu \beta \alpha }^{\tau }+\eta _{\tau \nu } a_{\mu \beta \alpha }^{\tau } \right )$ $+ D\eta_{\mu \nu }\left ( a_{\rho \sigma \alpha }^{\alpha } +a_{\rho \sigma \beta }^{\beta }\right )\eta ^{\rho \sigma }+E\eta_{\mu \nu }\left ( \eta ^{\alpha \beta }a_{\beta \sigma \alpha }^{\sigma } +\eta ^{\rho \sigma }a_{ \beta \sigma\rho }^{\beta } \right )$ $= \left ( T_{\mu \nu }\left ( x \right ) \right )_{P}$ $+A\left ( a_{\mu \nu \alpha }^{\alpha }+a_{\mu \nu \alpha }^{\alpha } \right )+B\eta ^{\alpha \beta }\left ( \eta _{\tau \nu }a_{ \alpha \beta\mu }^{\tau }+\eta _{\tau \mu }a_{\alpha \beta \nu }^{\tau } \right )$ $+C\left ( a_{\mu\nu \alpha }^{\alpha }+ a_{\mu \nu \alpha}^{\alpha } \right )+C\eta ^{\alpha \beta }\left (\eta _{\tau \nu } a_{\alpha \beta \mu}^{\tau } +\eta _{\tau \mu } a_{\alpha \beta \nu}^{\tau } \right )$ $+ D\eta_{\mu \nu }\left ( a_{\rho \sigma \alpha }^{\alpha } +a_{\rho \sigma \alpha }^{\alpha }\right )\eta ^{\rho \sigma }+E\eta_{\mu \nu }\left ( \eta ^{\rho \sigma }a_{\rho\alpha \sigma }^{\alpha } +\eta ^{\rho \sigma }a_{ \rho\alpha \sigma }^{\alpha } \right )$ $= \left ( T_{\mu \nu }\left ( x \right ) \right )_{P}$ $+2A a_{\mu \nu \alpha }^{\alpha } +B\eta ^{\alpha \beta }\left ( \eta _{\tau \nu }a_{ \alpha \beta\mu }^{\tau }+\eta _{\tau \mu }a_{\alpha \beta \nu }^{\tau } \right )+2C a_{\mu\nu \alpha }^{\alpha }+C\eta ^{\alpha \beta }\left (\eta _{\tau \nu } a_{\alpha \beta \mu}^{\tau } +\eta _{\tau \mu } a_{\alpha \beta \nu}^{\tau } \right )$ $+ 2D\eta_{\mu \nu } a_{\rho \sigma \alpha }^{\alpha }\eta ^{\rho \sigma }+2E\eta_{\mu \nu } a_{\rho\alpha \sigma }^{\alpha } \eta ^{\rho \sigma }$ $= \left ( T_{\mu \nu }\left ( x \right ) \right )_{P}$ $+2\left ( A+C \right ) a_{\mu \nu \alpha }^{\alpha } +\left ( B+C \right )\eta ^{\alpha \beta }\left ( \eta _{\tau \nu }a_{ \alpha \beta\mu }^{\tau }+\eta _{\tau \mu }a_{\alpha \beta \nu }^{\tau } \right )+ 2\left ( D+E \right )\eta_{\mu \nu } a_{\rho \sigma \alpha }^{\alpha }\eta ^{\rho \sigma }$ ƂȂ܂B ͂̕ӂŁBB

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