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If given $$g^{\mu\nu}=\pmatrix{\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{matrix}}$$

how does one find the covariant metric tensor, $$g_{\mu\nu}~?$$

Do I have to calculate it term by term using $$g^{\mu\nu}g_{\nu{i}}=\delta^{\mu}_{i}~?$$

Are there any other methods?

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    $\begingroup$ The downvotes seem harsh. It's a beginners question, but as beginner's qustion go it seems a reasonable one. It's easy to forget how confusing differential geometry was before we learned it. $\endgroup$ Commented Jun 10, 2015 at 14:40

2 Answers 2

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The contravariant metric tensor is the inverse metric tensor. If you have a general $g_{ab}$ you can find $g^{ab}$ by matrix inversion (which can usually be done in Mathematica or any other program of the kind).

In the special case of a diagonal metric tensor you can verify that $g^{ii} = 1/g_{ii}$.

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If it's diagonal, you can just focus on the diagonal elements, that is :

$g_{aa} g^{aa} = 1$

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