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ビアンキ恒等式を、ひたすら計算で示す。

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この記事について

この記事は、人に見せるというよりも自分の計算を整理するために書いたようなものなので、人に読んでもらうということを全く考えていません。本を読んでてビアンキ恒等式が証明なしに出てきて、それを自分で何とか示したので、記録として残しておきたくなったのです。メモ程度のものだと思ってください。それでも読みたいという方がいらっしゃったら、ぜひ読んであげてください。

前提

数式にはアインシュタインの規約を用いる.
 $\delta$:クロネッカーのデルタ, $g$:計量, $\Gamma$:接続, $R$:曲率, $\nabla$:共変微分
$$ \Gamma^\mu_{\nu\lambda}=\Gamma^\mu_{\lambda\nu}\\ \nabla_\lambda g_{\mu\nu}=0\\ \Gamma^\mu_{\nu\lambda}=\frac{1}{2}g^{\mu\kappa}(\partial_\lambda g_{\kappa\nu}+\partial_\nu g_{\kappa\lambda}-\partial_\kappa g_{\lambda\nu})\\ R_{\mu\nu\lambda\kappa}=\frac{1}{2}(\partial_\nu\partial_\lambda g_{\mu\kappa}+\partial_\mu\partial_\kappa g_{\nu\lambda}-\partial_\mu\partial_\lambda g_{\nu\kappa}-\partial_\nu\partial_\kappa g_{\mu\lambda})+g_{\eta\tau}(\Gamma^\eta_{\mu\kappa}\Gamma^\tau_{\nu\lambda}-\Gamma^\eta_{\mu\lambda}\Gamma^\tau_{\nu\kappa})\\ R_{\mu\nu\lambda\kappa}=R_{\lambda\kappa\mu\nu}\\ R_{\mu\nu\lambda\kappa}=-R_{\nu\mu\lambda\kappa}\\ R_{\mu\nu\lambda\kappa}=-R_{\mu\nu\kappa\lambda} $$

目標

$$ \nabla_\lambda R_{\mu\nu\kappa\eta}+\nabla_\eta R_{\mu\nu\lambda\kappa}+\nabla_\kappa R_{\mu\nu\eta\lambda}=0 $$
を示します!!!!!

ひたすら計算して証明

$$ \nabla_\lambda R_{\mu\nu\kappa\eta}+\nabla_\eta R_{\mu\nu\lambda\kappa}+\nabla_\kappa R_{\mu\nu\eta\lambda}=B\\ $$
とおく. $B=0$を示す.

$$ B=\nabla_\lambda R_{\mu\nu\kappa\eta}+\nabla_\eta R_{\mu\nu\lambda\kappa}+\nabla_\kappa R_{\mu\nu\eta\lambda}\\ $$
$$ =\partial_\lambda R_{\mu\nu\kappa\eta}-\Gamma^\tau_{\mu\lambda}R_{\tau\nu\kappa\eta}-\Gamma^\tau_{\nu\lambda}R_{\mu\tau\kappa\eta}-\Gamma^\tau_{\kappa\lambda}R_{\mu\nu\tau\eta}-\Gamma^\tau_{\eta\lambda}R_{\mu\nu\kappa\tau}\\ +\partial_\eta R_{\mu\nu\lambda\kappa}-\Gamma^\tau_{\mu\eta}R_{\tau\nu\lambda\kappa}-\Gamma^\tau_{\nu\eta}R_{\mu\tau\lambda\kappa}-\Gamma^\tau_{\lambda\eta}R_{\mu\nu\tau\kappa}-\Gamma^\tau_{\kappa\eta}R_{\mu\nu\lambda\tau}\\ +\partial_\kappa R_{\mu\nu\eta\lambda}-\Gamma^\tau_{\mu\kappa}R_{\tau\nu\eta\lambda}-\Gamma^\tau_{\nu\kappa}R_{\mu\tau\eta\lambda}-\Gamma^\tau_{\eta\kappa}R_{\mu\nu\tau\lambda}-\Gamma^\tau_{\lambda\kappa}R_{\mu\nu\eta\tau} $$

$$ =\partial_\lambda R_{\mu\nu\kappa\eta}-\Gamma^\tau_{\mu\lambda}R_{\tau\nu\kappa\eta}-\Gamma^\tau_{\nu\lambda}R_{\mu\tau\kappa\eta}\\ +\partial_\eta R_{\mu\nu\lambda\kappa}-\Gamma^\tau_{\mu\eta}R_{\tau\nu\lambda\kappa}-\Gamma^\tau_{\nu\eta}R_{\mu\tau\lambda\kappa}\\ +\partial_\kappa R_{\mu\nu\eta\lambda}-\Gamma^\tau_{\mu\kappa}R_{\tau\nu\eta\lambda}-\Gamma^\tau_{\nu\kappa}R_{\mu\tau\eta\lambda}\\ -\Gamma^\tau_{\kappa\lambda}(R_{\mu\nu\tau\eta}+R_{\mu\nu\eta\tau})-\Gamma^\tau_{\lambda\eta}(R_{\mu\nu\tau\kappa}+R_{\mu\nu\kappa\tau})-\Gamma^\tau_{\eta\kappa}(R_{\mu\nu\tau\lambda}+R_{\mu\nu\lambda\tau}) $$

$$ =\partial_\lambda R_{\mu\nu\kappa\eta}-\Gamma^\tau_{\mu\lambda}R_{\tau\nu\kappa\eta}-\Gamma^\tau_{\nu\lambda}R_{\mu\tau\kappa\eta}\\ +\partial_\eta R_{\mu\nu\lambda\kappa}-\Gamma^\tau_{\mu\eta}R_{\tau\nu\lambda\kappa}-\Gamma^\tau_{\nu\eta}R_{\mu\tau\lambda\kappa}\\ +\partial_\kappa R_{\mu\nu\eta\lambda}-\Gamma^\tau_{\mu\kappa}R_{\tau\nu\eta\lambda}-\Gamma^\tau_{\nu\kappa}R_{\mu\tau\eta\lambda} $$

次に,
$$ \partial_\lambda R_{\mu\nu\kappa\eta} $$

$$ =\partial_\lambda \left(\frac{1}{2}(\partial_\nu\partial_\kappa g_{\mu\eta}+\partial_\mu\partial_\eta g_{\nu\kappa}-\partial_\mu\partial_\kappa g_{\nu\eta}-\partial_\nu\partial_\eta g_{\mu\kappa})+g_{\tau\theta}(\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa}-\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\eta})\right) $$

$$ =\frac{1}{2}(\partial_\lambda\partial_\nu\partial_\kappa g_{\mu\eta}+\partial_\lambda\partial_\mu\partial_\eta g_{\nu\kappa}-\partial_\lambda\partial_\mu\partial_\kappa g_{\nu\eta}-\partial_\lambda\partial_\nu\partial_\eta g_{\mu\kappa})\\ +(\partial_\lambda g_{\tau\theta})(\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa}-\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\eta})+g_{\tau\theta}\partial_\lambda(\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa}-\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\eta}) $$
を用いると,
$$ \partial_\lambda R_{\mu\nu\kappa\eta}+\partial_\eta R_{\mu\nu\lambda\kappa}+\partial_\kappa R_{\mu\nu\eta\lambda} $$

$$ =\frac{1}{2}(\partial_\lambda\partial_\nu\partial_\kappa g_{\mu\eta}+\partial_\lambda\partial_\mu\partial_\eta g_{\nu\kappa}-\partial_\lambda\partial_\mu\partial_\kappa g_{\nu\eta}-\partial_\lambda\partial_\nu\partial_\eta g_{\mu\kappa})\\ +\frac{1}{2}(\partial_\eta\partial_\nu\partial_\lambda g_{\mu\kappa}+\partial_\eta\partial_\mu\partial_\kappa g_{\nu\lambda}-\partial_\eta\partial_\mu\partial_\lambda g_{\nu\kappa}-\partial_\eta\partial_\nu\partial_\kappa g_{\mu\lambda})\\ +\frac{1}{2}(\partial_\kappa\partial_\nu\partial_\eta g_{\mu\lambda}+\partial_\kappa\partial_\mu\partial_\lambda g_{\nu\eta}-\partial_\kappa\partial_\mu\partial_\eta g_{\nu\lambda}-\partial_\kappa\partial_\nu\partial_\lambda g_{\mu\eta})\\ +(\partial_\lambda g_{\tau\theta})(\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa}-\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\eta})+g_{\tau\theta}\partial_\lambda(\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa}-\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\eta})\\ +(\partial_\eta g_{\tau\theta})(\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\lambda}-\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\kappa})+g_{\tau\theta}\partial_\eta(\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\lambda}-\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\kappa})\\ +(\partial_\kappa g_{\tau\theta})(\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\eta}-\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\lambda})+g_{\tau\theta}\partial_\kappa(\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\eta}-\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\lambda}) $$

$$ =(\partial_\lambda g_{\tau\theta})(\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa}-\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\eta})+g_{\tau\theta}\partial_\lambda(\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa}-\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\eta})\\ +(\partial_\eta g_{\tau\theta})(\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\lambda}-\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\kappa})+g_{\tau\theta}\partial_\eta(\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\lambda}-\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\kappa})\\ +(\partial_\kappa g_{\tau\theta})(\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\eta}-\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\lambda})+g_{\tau\theta}\partial_\kappa(\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\eta}-\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\lambda}) $$
よって,
$$ B\\ =(\partial_\lambda g_{\tau\theta})(\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa}-\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\eta})+g_{\tau\theta}\partial_\lambda(\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa}-\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\eta})\\ +(\partial_\eta g_{\tau\theta})(\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\lambda}-\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\kappa})+g_{\tau\theta}\partial_\eta(\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\lambda}-\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\kappa})\\ +(\partial_\kappa g_{\tau\theta})(\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\eta}-\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\lambda})+g_{\tau\theta}\partial_\kappa(\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\eta}-\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\lambda})\\ -(\Gamma^\tau_{\mu\lambda}R_{\tau\nu\kappa\eta}+\Gamma^\tau_{\nu\lambda}R_{\mu\tau\kappa\eta}+\Gamma^\tau_{\mu\eta}R_{\tau\nu\lambda\kappa}+\Gamma^\tau_{\nu\eta}R_{\mu\tau\lambda\kappa}+\Gamma^\tau_{\mu\kappa}R_{\tau\nu\eta\lambda}+\Gamma^\tau_{\nu\kappa}R_{\mu\tau\eta\lambda}) $$
次に,
$$ g_{\tau\theta}\partial_\lambda(\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa}-\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\eta})+g_{\tau\theta}\partial_\eta(\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\lambda}-\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\kappa})+g_{\tau\theta}\partial_\kappa(\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\eta}-\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\lambda}) $$
$$ =g_{\tau\theta}(\partial_\lambda\Gamma^\tau_{\mu\eta})\Gamma^\theta_{\nu\kappa}+g_{\tau\theta}\Gamma^\tau_{\mu\eta}(\partial_\lambda\Gamma^\theta_{\nu\kappa})-g_{\tau\theta}(\partial_\lambda\Gamma^\tau_{\mu\kappa})\Gamma^\theta_{\nu\eta}-g_{\tau\theta}\Gamma^\tau_{\mu\kappa}(\partial_\lambda\Gamma^\theta_{\nu\eta})\\ +g_{\tau\theta}(\partial_\eta\Gamma^\tau_{\mu\kappa})\Gamma^\theta_{\nu\lambda}+g_{\tau\theta}\Gamma^\tau_{\mu\kappa}(\partial_\eta\Gamma^\theta_{\nu\lambda})-g_{\tau\theta}(\partial_\eta\Gamma^\tau_{\mu\lambda})\Gamma^\theta_{\nu\kappa}-g_{\tau\theta}\Gamma^\tau_{\mu\lambda}(\partial_\eta\Gamma^\theta_{\nu\kappa})\\ +g_{\tau\theta}(\partial_\kappa\Gamma^\tau_{\mu\lambda})\Gamma^\theta_{\nu\eta}+g_{\tau\theta}\Gamma^\tau_{\mu\lambda}(\partial_\kappa\Gamma^\theta_{\nu\eta})-g_{\tau\theta}(\partial_\kappa\Gamma^\tau_{\mu\eta})\Gamma^\theta_{\nu\lambda}-g_{\tau\theta}\Gamma^\tau_{\mu\eta}(\partial_\kappa\Gamma^\theta_{\nu\lambda}) $$
$$ =g_{\tau\theta}\Gamma^\theta_{\nu\kappa}(\partial_\lambda\Gamma^\tau_{\mu\eta}-\partial_\eta\Gamma^\tau_{\mu\lambda})+g_{\tau\theta}\Gamma^\theta_{\nu\lambda}(\partial_\eta\Gamma^\tau_{\mu\kappa}-\partial_\kappa\Gamma^\tau_{\mu\eta})+g_{\tau\theta}\Gamma^\theta_{\nu\eta}(\partial_\kappa\Gamma^\tau_{\mu\lambda}-\partial_\lambda\Gamma^\tau_{\mu\kappa})\\ +g_{\tau\theta}\Gamma^\tau_{\mu\eta}(\partial_\lambda\Gamma^\theta_{\nu\kappa}-\partial_\kappa\Gamma^\theta_{\nu\lambda})+g_{\tau\theta}\Gamma^\tau_{\mu\kappa}(\partial_\eta\Gamma^\theta_{\nu\lambda}-\partial_\lambda\Gamma^\theta_{\nu\eta})+g_{\tau\theta}\Gamma^\tau_{\mu\lambda}(\partial_\kappa\Gamma^\theta_{\nu\eta}-\partial_\eta\Gamma^\theta_{\nu\kappa}) $$
となる. さらに,
$$ g_{\tau\theta}\Gamma^\theta_{\nu\kappa}(\partial_\lambda\Gamma^\tau_{\mu\eta}-\partial_\eta\Gamma^\tau_{\mu\lambda}) $$
$$ =g_{\tau\theta}\Gamma^\theta_{\nu\kappa}(\partial_\lambda(g^{\tau\phi}g_{\rho\phi}\Gamma^\rho_{\mu\eta})-\partial_\eta(g^{\tau\phi}g_{\rho\phi}\Gamma^\rho_{\mu\lambda})) $$
$$ =g_{\tau\theta}\Gamma^\theta_{\nu\kappa}((\partial_\lambda g^{\tau\phi})g_{\rho\phi}\Gamma^\rho_{\mu\eta}+g^{\tau\phi}\partial_\lambda(g_{\rho\phi}\Gamma^\rho_{\mu\eta})-(\partial_\eta g^{\tau\phi})g_{\rho\phi}\Gamma^\rho_{\mu\lambda}-g^{\tau\phi}\partial_\eta(g_{\rho\phi}\Gamma^\rho_{\mu\lambda})) $$
$$ =g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\kappa}((\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\mu\eta}-(\partial_\eta g^{\tau\phi})\Gamma^\rho_{\mu\lambda})\\ +\Gamma^\theta_{\nu\kappa}\delta^\phi_\theta\left(\partial_\lambda\left(\frac{1}{2}(\partial_\eta g_{\phi\mu}+\partial_\mu g_{\phi\eta}-\partial_\phi g_{\eta\mu})\right)-\partial_\eta\left(\frac{1}{2}(\partial_\lambda g_{\phi\mu}+\partial_\mu g_{\phi\lambda}-\partial_\phi g_{\lambda\mu})\right)\right) $$
$$ =g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\kappa}((\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\mu\eta}-(\partial_\eta g^{\tau\phi})\Gamma^\rho_{\mu\lambda})\\ +\frac{1}{2}\Gamma^\theta_{\nu\kappa}(\partial_\lambda\partial_\mu g_{\theta\eta}+\partial_\eta\partial_\theta g_{\lambda\mu}-\partial_\lambda\partial_\theta g_{\eta\mu}-\partial_\eta\partial_\mu g_{\theta\lambda}) $$
$$ =g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\kappa}((\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\mu\eta}-(\partial_\eta g^{\tau\phi})\Gamma^\rho_{\mu\lambda})\\ +\Gamma^\theta_{\nu\kappa}(R_{\mu\theta\eta\lambda}-g_{\tau\rho}(\Gamma^\tau_{\theta\eta}\Gamma^\rho_{\mu\lambda}-\Gamma^\tau_{\mu\eta}\Gamma^\rho_{\theta\lambda})) $$
$$ =g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\kappa}((\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\mu\eta}-(\partial_\eta g^{\tau\phi})\Gamma^\rho_{\mu\lambda})\\ +\Gamma^\tau_{\nu\kappa}R_{\mu\tau\eta\lambda}-g_{\tau\rho}\Gamma^\theta_{\nu\kappa}(\Gamma^\tau_{\theta\eta}\Gamma^\rho_{\mu\lambda}-\Gamma^\tau_{\mu\eta}\Gamma^\rho_{\theta\lambda}) $$
以上より,
$$ B $$
$$ =(\partial_\lambda g_{\tau\theta})(\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa}-\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\eta})+(\partial_\eta g_{\tau\theta})(\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\lambda}-\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\kappa})+(\partial_\kappa g_{\tau\theta})(\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\eta}-\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\lambda})\\ +g_{\tau\theta}\Gamma^\theta_{\nu\kappa}(\partial_\lambda\Gamma^\tau_{\mu\eta}-\partial_\eta\Gamma^\tau_{\mu\lambda})+g_{\tau\theta}\Gamma^\theta_{\nu\lambda}(\partial_\eta\Gamma^\tau_{\mu\kappa}-\partial_\kappa\Gamma^\tau_{\mu\eta})+g_{\tau\theta}\Gamma^\theta_{\nu\eta}(\partial_\kappa\Gamma^\tau_{\mu\lambda}-\partial_\lambda\Gamma^\tau_{\mu\kappa})\\ +g_{\tau\theta}\Gamma^\tau_{\mu\eta}(\partial_\lambda\Gamma^\theta_{\nu\kappa}-\partial_\kappa\Gamma^\theta_{\nu\lambda})+g_{\tau\theta}\Gamma^\tau_{\mu\kappa}(\partial_\eta\Gamma^\theta_{\nu\lambda}-\partial_\lambda\Gamma^\theta_{\nu\eta})+g_{\tau\theta}\Gamma^\tau_{\mu\lambda}(\partial_\kappa\Gamma^\theta_{\nu\eta}-\partial_\eta\Gamma^\theta_{\nu\kappa})\\ -(\Gamma^\tau_{\mu\lambda}R_{\tau\nu\kappa\eta}+\Gamma^\tau_{\nu\lambda}R_{\mu\tau\kappa\eta}+\Gamma^\tau_{\mu\eta}R_{\tau\nu\lambda\kappa}+\Gamma^\tau_{\nu\eta}R_{\mu\tau\lambda\kappa}+\Gamma^\tau_{\mu\kappa}R_{\tau\nu\eta\lambda}+\Gamma^\tau_{\nu\kappa}R_{\mu\tau\eta\lambda}) $$
$$ =(\partial_\lambda g_{\tau\theta})(\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa}-\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\eta})+(\partial_\eta g_{\tau\theta})(\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\lambda}-\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\kappa})+(\partial_\kappa g_{\tau\theta})(\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\eta}-\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\lambda})\\ +g_{\tau\theta}\Gamma^\theta_{\nu\kappa}(\partial_\lambda\Gamma^\tau_{\mu\eta}-\partial_\eta\Gamma^\tau_{\mu\lambda})-\Gamma^\tau_{\nu\kappa}R_{\mu\tau\eta\lambda}\\ +g_{\tau\theta}\Gamma^\theta_{\nu\lambda}(\partial_\eta\Gamma^\tau_{\mu\kappa}-\partial_\kappa\Gamma^\tau_{\mu\eta})-\Gamma^\tau_{\nu\lambda}R_{\mu\tau\kappa\eta}\\ +g_{\tau\theta}\Gamma^\theta_{\nu\eta}(\partial_\kappa\Gamma^\tau_{\mu\lambda}-\partial_\lambda\Gamma^\tau_{\mu\kappa})-\Gamma^\tau_{\nu\eta}R_{\mu\tau\lambda\kappa}\\ +g_{\tau\theta}\Gamma^\tau_{\mu\eta}(\partial_\lambda\Gamma^\theta_{\nu\kappa}-\partial_\kappa\Gamma^\theta_{\nu\lambda})-\Gamma^\tau_{\mu\eta}R_{\tau\nu\lambda\kappa}\\ +g_{\tau\theta}\Gamma^\tau_{\mu\kappa}(\partial_\eta\Gamma^\theta_{\nu\lambda}-\partial_\lambda\Gamma^\theta_{\nu\eta})-\Gamma^\tau_{\mu\kappa}R_{\tau\nu\eta\lambda}\\ +g_{\tau\theta}\Gamma^\tau_{\mu\lambda}(\partial_\kappa\Gamma^\theta_{\nu\eta}-\partial_\eta\Gamma^\theta_{\nu\kappa})-\Gamma^\tau_{\mu\lambda}R_{\tau\nu\kappa\eta} $$
$$ =(\partial_\lambda g_{\tau\theta})(\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa}-\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\eta})+(\partial_\eta g_{\tau\theta})(\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\lambda}-\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\kappa})+(\partial_\kappa g_{\tau\theta})(\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\eta}-\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\lambda})\\ +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\kappa}((\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\mu\eta}-(\partial_\eta g^{\tau\phi})\Gamma^\rho_{\mu\lambda}) -g_{\tau\rho}\Gamma^\theta_{\nu\kappa}(\Gamma^\tau_{\theta\eta}\Gamma^\rho_{\mu\lambda}-\Gamma^\tau_{\mu\eta}\Gamma^\rho_{\theta\lambda})\\ +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\lambda}((\partial_\eta g^{\tau\phi})\Gamma^\rho_{\mu\kappa}-(\partial_\kappa g^{\tau\phi})\Gamma^\rho_{\mu\eta}) -g_{\tau\rho}\Gamma^\theta_{\nu\lambda}(\Gamma^\tau_{\theta\kappa}\Gamma^\rho_{\mu\eta}-\Gamma^\tau_{\mu\kappa}\Gamma^\rho_{\theta\eta})\\ +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\eta}((\partial_\kappa g^{\tau\phi})\Gamma^\rho_{\mu\lambda}-(\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\mu\kappa}) -g_{\tau\rho}\Gamma^\theta_{\nu\eta}(\Gamma^\tau_{\theta\lambda}\Gamma^\rho_{\mu\kappa}-\Gamma^\tau_{\mu\lambda}\Gamma^\rho_{\theta\kappa})\\ +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\mu\eta}((\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\nu\kappa}-(\partial_\kappa g^{\tau\phi})\Gamma^\rho_{\nu\lambda}) -g_{\tau\rho}\Gamma^\theta_{\mu\eta}(\Gamma^\tau_{\theta\kappa}\Gamma^\rho_{\nu\lambda}-\Gamma^\tau_{\nu\kappa}\Gamma^\rho_{\theta\lambda})\\ +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\mu\kappa}((\partial_\eta g^{\tau\phi})\Gamma^\rho_{\nu\lambda}-(\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\nu\eta}) -g_{\tau\rho}\Gamma^\theta_{\mu\kappa}(\Gamma^\tau_{\theta\lambda}\Gamma^\rho_{\nu\eta}-\Gamma^\tau_{\nu\lambda}\Gamma^\rho_{\theta\eta})\\ +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\mu\lambda}((\partial_\kappa g^{\tau\phi})\Gamma^\rho_{\nu\eta}-(\partial_\eta g^{\tau\phi})\Gamma^\rho_{\nu\kappa}) -g_{\tau\rho}\Gamma^\theta_{\mu\lambda}(\Gamma^\tau_{\theta\eta}\Gamma^\rho_{\nu\kappa}-\Gamma^\tau_{\nu\eta}\Gamma^\rho_{\theta\kappa})\\ $$
次に,
$$ g_{\tau\rho}\Gamma^\theta_{\nu\kappa}\Gamma^\tau_{\theta\eta}\Gamma^\rho_{\mu\lambda}+g_{\tau\rho}\Gamma^\theta_{\mu\lambda}\Gamma^\tau_{\theta\eta}\Gamma^\rho_{\nu\kappa} $$
$$ =\Gamma^\theta_{\nu\kappa}\Gamma^\rho_{\mu\lambda}(g_{\tau\rho}\Gamma^\tau_{\theta\eta}+g_{\tau\theta}\Gamma^\tau_{\rho\eta}) $$
$$ =\Gamma^\theta_{\nu\kappa}\Gamma^\rho_{\mu\lambda}(\partial_\eta g_{\theta\rho}-\nabla_\eta g_{\theta\rho}) $$
$$ =\Gamma^\theta_{\nu\kappa}\Gamma^\rho_{\mu\lambda}\partial_\eta g_{\theta\rho} $$
を用いると,
$$ -g_{\tau\rho}\Gamma^\theta_{\nu\kappa}(\Gamma^\tau_{\theta\eta}\Gamma^\rho_{\mu\lambda}-\Gamma^\tau_{\mu\eta}\Gamma^\rho_{\theta\lambda})-g_{\tau\rho}\Gamma^\theta_{\nu\lambda}(\Gamma^\tau_{\theta\kappa}\Gamma^\rho_{\mu\eta}-\Gamma^\tau_{\mu\kappa}\Gamma^\rho_{\theta\eta})-g_{\tau\rho}\Gamma^\theta_{\nu\eta}(\Gamma^\tau_{\theta\lambda}\Gamma^\rho_{\mu\kappa}-\Gamma^\tau_{\mu\lambda}\Gamma^\rho_{\theta\kappa})\\ -g_{\tau\rho}\Gamma^\theta_{\mu\eta}(\Gamma^\tau_{\theta\kappa}\Gamma^\rho_{\nu\lambda}-\Gamma^\tau_{\nu\kappa}\Gamma^\rho_{\theta\lambda})-g_{\tau\rho}\Gamma^\theta_{\mu\kappa}(\Gamma^\tau_{\theta\lambda}\Gamma^\rho_{\nu\eta}-\Gamma^\tau_{\nu\lambda}\Gamma^\rho_{\theta\eta})-g_{\tau\rho}\Gamma^\theta_{\mu\lambda}(\Gamma^\tau_{\theta\eta}\Gamma^\rho_{\nu\kappa}-\Gamma^\tau_{\nu\eta}\Gamma^\rho_{\theta\kappa}) $$
$$ =-(g_{\tau\rho}\Gamma^\theta_{\nu\kappa}\Gamma^\tau_{\theta\eta}\Gamma^\rho_{\mu\lambda}+g_{\tau\rho}\Gamma^\theta_{\mu\lambda}\Gamma^\tau_{\theta\eta}\Gamma^\rho_{\nu\kappa}) -(g_{\tau\rho}\Gamma^\theta_{\nu\lambda}\Gamma^\tau_{\theta\kappa}\Gamma^\rho_{\mu\eta}+g_{\tau\rho}\Gamma^\theta_{\mu\eta}\Gamma^\tau_{\theta\kappa}\Gamma^\rho_{\nu\lambda}) -(g_{\tau\rho}\Gamma^\theta_{\nu\eta}\Gamma^\tau_{\theta\lambda}\Gamma^\rho_{\mu\kappa}+g_{\tau\rho}\Gamma^\theta_{\mu\kappa}\Gamma^\tau_{\theta\lambda}\Gamma^\rho_{\nu\eta})\\ +(g_{\tau\rho}\Gamma^\theta_{\nu\kappa}\Gamma^\tau_{\mu\eta}\Gamma^\rho_{\theta\lambda}+g_{\tau\rho}\Gamma^\theta_{\mu\eta}\Gamma^\tau_{\nu\kappa}\Gamma^\rho_{\theta\lambda}) +(g_{\tau\rho}\Gamma^\theta_{\nu\lambda}\Gamma^\tau_{\mu\kappa}\Gamma^\rho_{\theta\eta}+g_{\tau\rho}\Gamma^\theta_{\mu\kappa}\Gamma^\tau_{\nu\lambda}\Gamma^\rho_{\theta\eta}) +(g_{\tau\rho}\Gamma^\theta_{\nu\eta}\Gamma^\tau_{\mu\lambda}\Gamma^\rho_{\theta\kappa}+g_{\tau\rho}\Gamma^\theta_{\mu\lambda}\Gamma^\tau_{\nu\eta}\Gamma^\rho_{\theta\kappa}) $$
$$ =-\Gamma^\theta_{\nu\kappa}\Gamma^\rho_{\mu\lambda}\partial_\eta g_{\theta\rho} -\Gamma^\theta_{\nu\lambda}\Gamma^\rho_{\mu\eta}\partial_\kappa g_{\theta\rho} -\Gamma^\theta_{\nu\eta}\Gamma^\rho_{\mu\kappa}\partial_\lambda g_{\theta\rho} +\Gamma^\theta_{\nu\kappa}\Gamma^\rho_{\mu\eta}\partial_\lambda g_{\theta\rho} +\Gamma^\theta_{\nu\lambda}\Gamma^\rho_{\mu\kappa}\partial_\eta g_{\theta\rho} +\Gamma^\theta_{\nu\eta}\Gamma^\rho_{\mu\lambda}\partial_\kappa g_{\theta\rho} $$
$$ =(\partial_\lambda g_{\tau\theta})(\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa}-\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\eta})+(\partial_\eta g_{\tau\theta})(\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\lambda}-\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\kappa})+(\partial_\kappa g_{\tau\theta})(\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\eta}-\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\lambda}) $$
したがって,
$$ B $$
$$ =2(\partial_\lambda g_{\tau\theta})(\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa}-\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\eta})+2(\partial_\eta g_{\tau\theta})(\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\lambda}-\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\kappa})+2(\partial_\kappa g_{\tau\theta})(\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\eta}-\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\lambda})\\ +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\kappa}((\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\mu\eta}-(\partial_\eta g^{\tau\phi})\Gamma^\rho_{\mu\lambda})\\ +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\lambda}((\partial_\eta g^{\tau\phi})\Gamma^\rho_{\mu\kappa}-(\partial_\kappa g^{\tau\phi})\Gamma^\rho_{\mu\eta})\\ +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\eta}((\partial_\kappa g^{\tau\phi})\Gamma^\rho_{\mu\lambda}-(\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\mu\kappa})\\ +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\mu\eta}((\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\nu\kappa}-(\partial_\kappa g^{\tau\phi})\Gamma^\rho_{\nu\lambda})\\ +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\mu\kappa}((\partial_\eta g^{\tau\phi})\Gamma^\rho_{\nu\lambda}-(\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\nu\eta})\\ +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\mu\lambda}((\partial_\kappa g^{\tau\phi})\Gamma^\rho_{\nu\eta}-(\partial_\eta g^{\tau\phi})\Gamma^\rho_{\nu\kappa})\\ $$
となる. 次に,
$$ g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\mu\eta}((\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\nu\kappa}-(\partial_\kappa g^{\tau\phi})\Gamma^\rho_{\nu\lambda})\\ +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\mu\kappa}((\partial_\eta g^{\tau\phi})\Gamma^\rho_{\nu\lambda}-(\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\nu\eta})\\ +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\mu\lambda}((\partial_\kappa g^{\tau\phi})\Gamma^\rho_{\nu\eta}-(\partial_\eta g^{\tau\phi})\Gamma^\rho_{\nu\kappa}) $$
$$ =g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\mu\eta}(\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\nu\kappa}-g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\mu\eta}(\partial_\kappa g^{\tau\phi})\Gamma^\rho_{\nu\lambda}\\ +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\mu\kappa}(\partial_\eta g^{\tau\phi})\Gamma^\rho_{\nu\lambda}-g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\mu\kappa}(\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\nu\eta}\\ +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\mu\lambda}(\partial_\kappa g^{\tau\phi})\Gamma^\rho_{\nu\eta}-g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\mu\lambda}(\partial_\eta g^{\tau\phi})\Gamma^\rho_{\nu\kappa} $$
$$ =g_{\tau\theta}g_{\rho\phi}\Gamma^\rho_{\nu\kappa}((\partial_\lambda g^{\tau\phi})\Gamma^\theta_{\mu\eta}-(\partial_\eta g^{\tau\phi})\Gamma^\theta_{\mu\lambda})\\ +g_{\tau\theta}g_{\rho\phi}\Gamma^\rho_{\nu\lambda}((\partial_\eta g^{\tau\phi})\Gamma^\theta_{\mu\kappa}-(\partial_\kappa g^{\tau\phi})\Gamma^\theta_{\mu\eta})\\ +g_{\tau\theta}g_{\rho\phi}\Gamma^\rho_{\nu\eta}((\partial_\kappa g^{\tau\phi})\Gamma^\theta_{\mu\lambda}-(\partial_\lambda g^{\tau\phi})\Gamma^\theta_{\mu\kappa}) $$
より,
$$ B $$
$$ =2(\partial_\lambda g_{\tau\theta})(\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa}-\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\eta}) +2(\partial_\eta g_{\tau\theta})(\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\lambda}-\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\kappa}) +2(\partial_\kappa g_{\tau\theta})(\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\eta}-\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\lambda})\\ +2g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\kappa}((\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\mu\eta}-(\partial_\eta g^{\tau\phi})\Gamma^\rho_{\mu\lambda})\\ +2g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\lambda}((\partial_\eta g^{\tau\phi})\Gamma^\rho_{\mu\kappa}-(\partial_\kappa g^{\tau\phi})\Gamma^\rho_{\mu\eta})\\ +2g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\eta}((\partial_\kappa g^{\tau\phi})\Gamma^\rho_{\mu\lambda}-(\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\mu\kappa})\\ $$
最後に,
$$ (\partial_\lambda g_{\tau\theta})\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa} +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\kappa}(\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\mu\eta} $$
$$ =\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa}(\partial_\lambda g_{\tau\theta}+g_{\rho\theta}g_{\tau\phi}(\partial_\lambda g^{\rho\phi})) $$
$$ =\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa}(g_{\rho\theta}g^{\rho\phi}(\partial_\lambda g_{\tau\phi})+g_{\rho\theta}g_{\tau\phi}(\partial_\lambda g^{\rho\phi})) $$
$$ =\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa}g_{\rho\theta}(\partial_\lambda \delta^\rho_\tau) $$
$$ =0 $$
が成り立つので,
$$ B $$
$$ =2((\partial_\lambda g_{\tau\theta})\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\kappa} +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\kappa}(\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\mu\eta})\\ +2((\partial_\eta g_{\tau\theta})\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\lambda} +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\lambda}(\partial_\eta g^{\tau\phi})\Gamma^\rho_{\mu\kappa})\\ +2((\partial_\kappa g_{\tau\theta})\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\eta} +g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\eta}(\partial_\kappa g^{\tau\phi})\Gamma^\rho_{\mu\lambda})\\ -2((\partial_\lambda g_{\tau\theta})\Gamma^\tau_{\mu\kappa}\Gamma^\theta_{\nu\eta}+g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\eta}(\partial_\lambda g^{\tau\phi})\Gamma^\rho_{\mu\kappa})\\ -2((\partial_\eta g_{\tau\theta})\Gamma^\tau_{\mu\lambda}\Gamma^\theta_{\nu\kappa}+g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\kappa}(\partial_\eta g^{\tau\phi})\Gamma^\rho_{\mu\lambda})\\ -2((\partial_\kappa g_{\tau\theta})\Gamma^\tau_{\mu\eta}\Gamma^\theta_{\nu\lambda}+g_{\tau\theta}g_{\rho\phi}\Gamma^\theta_{\nu\lambda}(\partial_\kappa g^{\tau\phi})\Gamma^\rho_{\mu\eta})\\ $$
$$ =0 $$

以上より, $B=0$が示された!やったあ!

終わりに

もしこの記事をここまで読んでくださった方がいらっしゃったら、本当にありがとうございました!こんな記事にもなってない記事投稿して申し訳ございませんでした(土下座)!!!

投稿日:20201124

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