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## Main Question or Discussion Point

How to express electromagnetic field tensor in curvilinear coordinates, that is given a curvilinear coordinates [itex](t,\alpha,\beta,\gamma) [/itex]with metric tensor as follows:

[tex]

n_{\mu \nu }=

\left[ \begin{array}{cccc}h_0^2& 0 & 0 & 0 \\ 0 & -h_1^2 & 0 & 0 \\ 0 & 0 & -h_2^2 & 0 \\ 0 & 0 & 0 & -h_3^2 \end{array} \right]

[/tex]

How do we express electromagnetic field tensor [itex]F_{\mu \nu}[/itex] in terms of [itex]E_\alpha , E_\beta , E_\gamma , B_\alpha , B_\beta , B_\gamma[/itex]

I found in the internet that the [itex]F_{\mu \nu}[/itex] is given by:

[tex]

F_{\mu \nu }=

\left[ \begin{array}{cccc} 0 & -\frac{E_{\alpha}}{h_0 h_1} & -\frac{E_{\beta}}{h_0 h_2} & -\frac{E_{\gamma}}{h_0 h_3} \\ \frac{E_{\alpha}}{h_0 h_1} & 0 & \frac{B_{\gamma}}{h_1 h_2} &-\frac{B_{\beta}}{h_3 h_1} \\\frac{E_{\beta}}{h_0 h_2} & -\frac{B_{\gamma}}{h_1 h_2} & 0 & \frac{B_{\alpha}}{h_2 h_3} \\\frac{E_{\gamma}}{h_0 h_3} & \frac{B_{\beta}}{h_3 h_1} & -\frac{B_{\alpha}}{h_2 h_3}& 0 \end{array} \right]

[/tex]

Is it correct and how to derive it?

Thanks.

[tex]

n_{\mu \nu }=

\left[ \begin{array}{cccc}h_0^2& 0 & 0 & 0 \\ 0 & -h_1^2 & 0 & 0 \\ 0 & 0 & -h_2^2 & 0 \\ 0 & 0 & 0 & -h_3^2 \end{array} \right]

[/tex]

How do we express electromagnetic field tensor [itex]F_{\mu \nu}[/itex] in terms of [itex]E_\alpha , E_\beta , E_\gamma , B_\alpha , B_\beta , B_\gamma[/itex]

I found in the internet that the [itex]F_{\mu \nu}[/itex] is given by:

[tex]

F_{\mu \nu }=

\left[ \begin{array}{cccc} 0 & -\frac{E_{\alpha}}{h_0 h_1} & -\frac{E_{\beta}}{h_0 h_2} & -\frac{E_{\gamma}}{h_0 h_3} \\ \frac{E_{\alpha}}{h_0 h_1} & 0 & \frac{B_{\gamma}}{h_1 h_2} &-\frac{B_{\beta}}{h_3 h_1} \\\frac{E_{\beta}}{h_0 h_2} & -\frac{B_{\gamma}}{h_1 h_2} & 0 & \frac{B_{\alpha}}{h_2 h_3} \\\frac{E_{\gamma}}{h_0 h_3} & \frac{B_{\beta}}{h_3 h_1} & -\frac{B_{\alpha}}{h_2 h_3}& 0 \end{array} \right]

[/tex]

Is it correct and how to derive it?

Thanks.

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