Salam sejahtera.
\section{Menalar Turunan Lie dengan Analisis Tensor Biasa}
Andaikan $T := {T^{\mu_1\cdots\mu_m}}_{\nu_1\cdots\nu_n}e_{\mu_1}\otimes\cdots\otimes e_{\mu_m}\otimes e^{\nu_1}\otimes\cdots\otimes e^{\nu_n}$ adalah sebuah tensor yang bergantung pada $p$ buah koordinat umum, yaitu $x^1,\cdots,x^p \in \mathbb{R}$ sebagai $p$ buah parameter bagi manifold $M$ yang berdimensi $p$, serta $X := X^\lambda e_\lambda$ adalah sebuah vektor yang bergantung pada $p$ buah koordinat umum tersebut juga. Selanjutnya, akan dicari turunan Lie dari $T$ sepanjang $X$, yang didefinisikan sebagai
\[ L_XT := \left(\lim_{\epsilon\to 0}\frac{T_x(x + \epsilon X) - T}{\epsilon}\right)_{\partial_\rho X = 0} \]
untuk semua $\rho\in\{1,\cdots,p\}$, di mana $\partial_\rho := \partial/\partial x^\rho$.
Ternyata, pengolahan lebih lanjut dari persamaan terakhir, menghasilkan
\[ L_XT = \left(X^\lambda\partial_\lambda T\right)_{\partial_\rho X = 0}. \]
\[ L_XT = (X^\lambda(\partial_\lambda {T^{\mu_1\cdots\mu_m}}_{\nu_1\cdots\nu_n}e_{\mu_1}\otimes\cdots\otimes e_{\mu_m}\otimes e^{\nu_1}\otimes\cdots\otimes e^{\nu_n} \]
\[ + {T^{\mu_1\cdots\mu_m}}_{\nu_1\cdots\nu_n}\sum_{s=1}^m e_{\mu_1}\otimes\cdots\otimes e_{\mu_{s-1}}\otimes\partial_\lambda e_{\mu_s}\otimes e_{\mu_{s+1}}\otimes\cdots\otimes e_{\mu_m}\otimes e^{\nu_1}\otimes\cdots\otimes e^{\nu_n} \]
\[ + {T^{\mu_1\cdots\mu_m}}_{\nu_1\cdots\nu_n}e_{\mu_1}\otimes\cdots\otimes e_{\mu_m}\otimes\sum_{r=1}^n e^{\nu_1}\otimes\cdots\otimes e^{\nu_{r-1}}\otimes\partial_\lambda e^{\nu_r}\otimes e^{\nu_{r+1}}\otimes\cdots\otimes e^{\nu_n}))_{\partial_\rho X = 0}. \]
Dengan memanfaatkan identitas $\partial_\alpha e_\beta = {\Gamma^\gamma}_{\alpha\beta}e_\gamma$ dan $\partial_\alpha e^\beta = -{\Gamma^\beta}_{\alpha\gamma}e^\gamma$, diperoleh
\[ L_XT = (X^\lambda\partial_\lambda {T^{\mu_1\cdots\mu_m}}_{\nu_1\cdots\nu_n}e_{\mu_1}\otimes\cdots\otimes e_{\mu_m}\otimes e^{\nu_1}\otimes\cdots\otimes e^{\nu_n} \]
\[ + {T^{\mu_1\cdots\mu_m}}_{\nu_1\cdots\nu_n}X^\lambda\sum_{s=1}^m {\Gamma^\sigma}_{\lambda\mu_s} e_{\mu_1}\otimes\cdots\otimes e_{\mu_{s-1}}\otimes e_\sigma\otimes e_{\mu_{s+1}}\otimes\cdots\otimes e_{\mu_m}\otimes e^{\nu_1}\otimes\cdots\otimes e^{\nu_n} \]
\[ - {T^{\mu_1\cdots\mu_m}}_{\nu_1\cdots\nu_n}X^\lambda e_{\mu_1}\otimes\cdots\otimes e_{\mu_m}\otimes\sum_{r=1}^n {\Gamma^{\nu_r}}_{\lambda\tau} e^{\nu_1}\otimes\cdots\otimes e^{\nu_{r-1}}\otimes e^\tau\otimes e^{\nu_{r+1}}\otimes\cdots\otimes e^{\nu_n})_{\partial_\rho X = 0} \]
di mana $\Gamma$ adalah lambang Christoffel.
Dengan memanfaatkan identitas $\partial_\alpha X = D_\alpha X^\lambda e_\lambda$ di mana $D_\alpha X^\beta := \partial_\alpha X^\beta + X^\lambda{\Gamma^\beta}_{\alpha\lambda}$, maka diperoleh
\[ L_XT = X^\lambda\partial_\lambda {T^{\mu_1\cdots\mu_m}}_{\nu_1\cdots\nu_n}e_{\mu_1}\otimes\cdots\otimes e_{\mu_m}\otimes e^{\nu_1}\otimes\cdots\otimes e^{\nu_n} \]
\[ - {T^{\mu_1\cdots\mu_m}}_{\nu_1\cdots\nu_n}\sum_{s=1}^m \partial_{\mu_s}X^\sigma e_{\mu_1}\otimes\cdots\otimes e_{\mu_{s-1}}\otimes e_\sigma\otimes e_{\mu_{s+1}}\otimes\cdots\otimes e_{\mu_m}\otimes e^{\nu_1}\otimes\cdots\otimes e^{\nu_n} \]
\[ + {T^{\mu_1\cdots\mu_m}}_{\nu_1\cdots\nu_n}e_{\mu_1}\otimes\cdots\otimes e_{\mu_m}\otimes\sum_{r=1}^n \partial_\tau X^{\nu_r} e^{\nu_1}\otimes\cdots\otimes e^{\nu_{r-1}}\otimes e^\tau\otimes e^{\nu_{r+1}}\otimes\cdots\otimes e^{\nu_n}. \]
Dengan melakukan penukaran indeks boneka, akhirnya diperoleh bentuk eksplisit dari turunan Lie dari $T$ sepanjang $X$, yaitu
\[ L_XT = (X^\lambda\partial_\lambda {T^{\mu_1\cdots\mu_m}}_{\nu_1\cdots\nu_n} \]
\[ - \sum_{s=1}^m {T^{\mu_1\cdots\mu_{s-1}\sigma\mu_{s+1}\cdots\mu_m}}_{\nu_1\cdots\nu_n}\partial_\sigma X^{\mu_s} \]
\[ + \sum_{r=1}^n {T^{\mu_1\cdots\mu_m}}_{\nu_1\cdots\nu_{r-1}\tau\nu_{r+1}\cdots\nu_n}\partial_{\nu_r}X^\tau) \]
\[ e_{\mu_1}\otimes\cdots\otimes e_{\mu_m}\otimes e^{\nu_1}\otimes\cdots\otimes e^{\nu_n}. \]
Sanctus, Sanctus, Dominus Deus Sabaoth.
