幾何学量の変分

作用から運動方程式を得るときや，摂動を考えるときなど，関数の変分を必要 とする．ここで一般相対論においてよく必要となる幾何学量について，１次変 分の表式をまとめておく．

$$\delta g^{\mu\nu} = - g^{\mu\alpha} g^{\nu\beta} \delta g_{\alpha\beta}$$ (B.2.113)

$$\delta \sqrt{-g} = \frac12 \sqrt{-g} g^{\mu\nu} \delta g_{\mu\nu}$$ (B.2.114)

$$\delta{\mit\Gamma}^{\mu}_{\nu\lambda} = \frac12 g^{\mu\alpha} \left( \delta g_{\alpha\nu;\lambda} + \delta g_{\alpha\lambda;\nu} - \delta g_{\nu\lambda;\alpha} \right)$$ (B.2.115)

$$\delta {R^\mu}_{\nu\alpha\beta} = \delta {\mit\Gamma}^\mu_{\nu\beta;\alpha} - \delta {\mit\Gamma}^\mu_{\nu\alpha;\beta}$$

$$= \frac12 g^{\mu\lambda} \left( \delta g_{\lambda\nu;\beta\alpha} -... ...+ \delta g_{\nu\alpha;\lambda\beta} - \delta g_{\nu\beta;\lambda\alpha} \right)$$ (B.2.116)

$$\delta R_{\mu\nu} = \delta {\mit\Gamma}^\lambda_{\mu\nu;\lambda} - \delta {\mit\Gamma}^\lambda_{\mu\lambda;\nu}$$

$$= \frac12 \left( {\delta g_{\lambda\mu;\nu}}^\lambda + {\delta g_{\... ... \delta g_{\lambda\rho;\mu\nu} - {\delta g_{\nu\mu;\lambda}}^{;\lambda} \right)$$ (B.2.117)

$$\delta R = {\delta g_{\mu\nu}}^{;\mu\nu} - g^{\mu\nu} \delta {g_{\mu\nu;\lambda}}^{;\lambda} - R^{\mu\nu} \delta g_{\mu\nu}$$ (B.2.118)