Teorema tentang Tensor Elektromagnetik

[cotrans]

Hosana in excelcis.

\section{Teorema tentang Tensor Elektromagnetik}

Komponen kontravarian tensor elektromagnetik $F^{\mu\nu} := \partial^\mu A^\nu - \partial^\nu A^\mu$ memiliki padanan kovariannya, yaitu $F_{\mu\nu} := \partial_\mu A_\nu - \partial_\nu A_\mu$ untuk setiap $\mu, \nu \in \{ 0, 1, 2, 3 \}$, di mana $\partial_0 := (1/c)\partial/\partial t$, $\partial_i := \partial/\partial x^i$ untuk setiap $i \in \{ 1, 2, 3 \}$, di mana $\vec{r} := x^i\hat{x}_i$ adalah vektor posisi, $\hat{x}_1 := (1, 0, 0)$, $\hat{x}_2 := (0, 1, 0)$, $\hat{x}_3 := (0, 0, 1)$, $\partial^\mu := \eta^{\mu\nu}\partial_\nu$, di mana $\eta^{00} := -1$, $\eta^{11} = \eta^{22} = \eta^{33} = 1$, $(\eta^{\mu\nu})_{\nu \neq \mu} = 0$, $\eta^{\mu\nu}\eta_{\nu\rho} = {\delta^\mu}_\rho$, $A^0 := \varphi/c$, $A_\mu = \eta_{\mu\nu}A^\nu$, $c$ adalah kelajuan cahaya dalam ruang hampa, dan $t$ adalah waktu.
\[ F^{00} = \partial^0A^0 - \partial^0A^0 = 0. \]
\[ F^{0i} = \partial^0A^i - \partial^iA^0 = -\frac{1}{c}\frac{\partial A^j}{\partial x^i} - \frac{\partial}{\partial x^i}\frac{\varphi}{c} = \frac{1}{c}E^i = -F^{i0} \]
untuk setiap $i \in \{ 1, 2, 3 \}$.
\[ F^{ij} = \partial^iA^j - \partial^jA^i = \frac{\partial A^j}{\partial x^i} - \frac{\partial A^i}{\partial x^j} = \epsilon_{ijk}B^k = -F^{ji} \]
untuk setiap $i, j \in \{ 1, 2, 3 \}$.
\[ F_{00} = \partial_0A_0 - \partial_0A_0 = 0. \]
\[ F_{0i} = \partial_0A_i - \partial_iA_0 = \frac{1}{c}\frac{\partial A^i}{\partial t} + \frac{\partial}{\partial x^i}\frac{\varphi}{c} = -\frac{1}{c}E^i = -F_{i0} \]
untuk setiap $i \in \{ 1, 2, 3 \}$.
\[ F_{ij} = \partial_iA_j - \partial_jA_i = \frac{\partial A^j}{\partial x^i} - \frac{\partial A^i}{\partial x^j} = \epsilon_{ijk}B^k = -F_{ji} \]
untuk setiap $i, j \in \{ 1, 2, 3 \}$.
\[ F^{\mu\nu}F_{\mu\nu} = F^{00}F_{00} + F^{0i}F_{0i} + F^{i0}F_{i0} + F^{ij}F_{ij} \]
\[ = 0 + (E^i/c)(-E^i/c) + (-E^i/c)(E^i/c) + \epsilon_{ijk}B^k\epsilon_{ijl}B^l \]
\[ = 2(|\vec{B}|^2 - |\vec{E}|^2/c^2). \]

Selanjutnya, andaikan ada komponen tensor
\[ T^{\mu\nu} = F^{\mu\sigma}{F^\nu}_\sigma - \frac{1}{4}\eta^{\mu\nu}F^{\sigma\tau}F_{\sigma\tau} \]
untuk setiap $\mu, \nu \in \{ 0, 1, 2, 3 \}$.
\[ T^{0n} = F^{0\sigma}{F^n}_\sigma = \frac{1}{4}\eta^{0n}F^{\sigma\tau}F_{\sigma\tau} \]
untuk setiap $n \in \{ 1, 2, 3 \}$.  Karena $\eta^{0n} = 0$, maka
\[ T^{0n} = F^{0\sigma}\eta_{\sigma\nu}F^{n\nu} = F^{00}\eta_{0\nu}F^{n\nu} + F^{0i}\eta_{i\nu}F^{n\nu} \]
untuk setiap $n \in \{ 1, 2, 3 \}$.  Karena $F^{00} = 0$, maka
\[ T^{0n} = 0 + F^{0i}\eta_{i0}F^{n0} + F^{0i}\eta_{ij}F^{nj}. \]
Karena $\eta_{i0} = 0$, maka
\[ T^{0n} = \frac{1}{c}E^i\eta_{ij}\epsilon_{njl}B^l = \frac{1}{c}E^i\epsilon_{nil}B^l = \frac{1}{c}(\vec{E}\times\vec{B})\cdot\hat{x}_n. \]

Nderek langkung.