Om Swastyastu.
\section{Bentuk Kovarian dari Sistem Persamaan Maxwell}
Sistem persamaan Maxwell yang paling umum adalah
\[ \nabla\cdot\vec{D} = \rho, ~~~~~ \nabla\cdot\vec{B} = 0, ~~~~~ \nabla\times\vec{E} = -\frac{\partial\vec{B}}{\partial t}, ~~~~~ \nabla\times\vec{H} = \vec{J} + \frac{\partial\vec{D}}{\partial t} \]
di mana
\[ \vec{D} = \epsilon_0\vec{E} + \vec{P} ~~~~~ \text{dan} ~~~~~ \vec{B} = \mu_0\vec{H} + \vec{M}. \]
Dari persamaan $\nabla\cdot\vec{B} = 0$, diperoleh $\vec{B} = \nabla\times\vec{A}$, sehingga
\[ \nabla\times\left(\vec{E} + \frac{\partial\vec{A}}{\partial t}\right) = \vec{0} \]
yang dengan menggunakan teori tera $\nabla\times\nabla\varphi = \vec{0}$, persamaan terakhir identik dengan
\[ \vec{E} = -\nabla\varphi - \frac{\partial\vec{A}}{\partial t}. \]
Dari persamaan $\nabla\cdot\vec{D} = \rho$, diperoleh
\[ \epsilon_0\nabla\cdot\vec{E} + \nabla\cdot\vec{P} = \rho \]
alias
\[ -\epsilon_0\left(\nabla^2\varphi + \frac{\partial}{\partial t}\nabla\cdot\vec{A}\right) + \nabla\cdot\vec{P} = \rho \]
alias
\[ -\epsilon_0c(\nabla^2A^0 + \partial_0\nabla\cdot\vec{A}) + \nabla\cdot\vec{P} = \rho \]
alias
\[ (-1/\mu_0)(\nabla^2A^0 + \partial_0\nabla\cdot\vec{A}) + c\nabla\cdot\vec{P} = J^0 \]
alias
\[ (-1/\mu_0)(\partial_j\partial^jA^0 + \partial_0\partial_jA^j) + c\partial_jP^j = J^0 \]
di mana $A^0 := \varphi/c$, $A^1 := A_x$, $A^2 := A_y$, $A^3 := A_z$, $J^0 := \rho c$, $J^1 := J_x$, $J^2 := J_y$, $J^3 := J_z$, $\partial_0 := (1/c)\partial/\partial t$, $\partial_1 := \partial/\partial x$, $\partial_2 := \partial/\partial y$, $\partial_3 := \partial/\partial z$, $A_\mu := g_{\mu\nu}A^\nu$, $J_\mu := g_{\mu\nu}J^\nu$, $\partial^\mu := g^{\mu\nu}\partial_\nu$, $g_{00} = -1$, $g_{11} = g_{22} = g_{33} = 1$, dan $(g_{\mu\nu})_{\mu\neq\nu} = 0$.
Dari persamaan $\nabla\times\vec{H} = \vec{J} + \partial\vec{D}/\partial t$ dan $\vec{H} = (\vec{B} - \vec{M})/\mu_0$, diperoleh
\[ (1/\mu_0)(\nabla\times\vec{B} - \nabla\times\vec{M}) = \vec{J} + \epsilon_0\frac{\partial\vec{E}}{\partial t} + \frac{\partial\vec{P}}{\partial t} \]
alias
\[ (1/\mu_0)(\nabla(\nabla\cdot\vec{A}) - \nabla^2\vec{A} - \nabla\times\vec{M}) = \vec{J} - \epsilon_0\left(\nabla\frac{\partial\varphi}{\partial t} + \frac{\partial^2\vec{A}}{\partial t^2}\right) + \frac{\partial\vec{P}}{\partial t} \]
alias (dengan menerapkan identitas $\epsilon_0\mu_0c^2 = 1$)
\[ (1/\mu_0)(\partial^j\partial_kA^k - \partial_k\partial^kA^j - \epsilon^{jkl}\partial_kM_l) \]
\[ = J^j - \epsilon_0(c\partial^j\partial_0\varphi + c^2\partial_0\partial_0A^j) + c\partial_0P^j \]
\[ = J^j - \epsilon_0c^2(\partial^j\partial_0A^0 + \partial_0\partial_0A^j) + c\partial_0P^j \]
\[ = J^j - (1/\mu_0)(\partial^j\partial_0A^0 + \partial_0\partial_0A^j) + c\partial_0P^j \]
alias
\[ \partial^j\partial_kA^k - \partial_k\partial^kA^j - \epsilon^{jkl}\partial_kM_l = \mu_0J^j - \partial^j\partial_0A^0 + \partial_0\partial_0A^j + \mu_0c\partial_0P^j \]
alias
\[ \partial^j\partial_\mu A^\mu - \partial_\mu\partial^\mu A^j = \mu_0J^j + \epsilon^{jkl}\partial_kM_l + \mu_0c\partial_0P^j. \]
Karena dari persamaan terdahulu telah diperoleh
\[ \partial_j\partial^jA^0 - \partial^0\partial_jA^j - \mu_0c\partial_jP^j = -\mu_0J^0 \]
alias
\[ \partial^0\partial_jA^j - \partial_j\partial^jA^0 = \mu_0J^0 - \mu_0c\partial_jP^j \]
alias
\[ \partial^0\partial_jA^j + \partial^0\partial_0A^0 - \partial_j\partial^jA^0 - \partial_0\partial^0A^0 = \mu_0J^0 - \mu_0c\partial_jP^j \]
alias
\[ \partial^0\partial_\mu A^\mu - \partial_\mu\partial^\mu A^0 = \mu_0J^0 - \mu_0c\partial_jP^j, \]
maka diperoleh
\[ \partial^\nu\partial_\mu A^\mu - \partial_\mu\partial^\mu A^\nu = \mu_0J^\nu + C^\nu \]
alias
\[ \partial_\mu(\partial^\nu A^\mu - \partial^\mu A^\nu) = \mu_0J^\nu + C^\nu \]
alias
\[ \partial_\mu F^{\nu\mu} = \mu_0J^\nu + C^\nu \]
yang merupakan bentuk kovarian dari sistem persamaan Maxwell, di mana $C^j := \epsilon^{jkl}\partial_kM_l + \mu_0c\partial_0P^j$ dan $C^0 := -\mu_0c\partial_jP^j$ serta $F^{\nu\mu} := \partial^\nu A^\mu - \partial^\mu A^\nu$.
Arigatou ikimono gakari.
