Given a metric tensor g(adsbygoogle = window.adsbygoogle || []).push({}); _{mn}, how to calculate the inverse of it, g^{mn}. For example, the metric

[tex]

g_{\mu \nu }=

\left[ \begin{array}{cccc} f & 0 & 0 & -w \\ 0 & -e^m & 0 &0 \\0 & 0 & -e^m &0\\0 & 0 & 0 & -l \end{array} \right]

[/tex]

From basic understanding, I would think of divided it, that is

[tex]

g^{\mu \nu }=

\left[ \begin{array}{cccc} 1/f & 0 & 0 & -1/w \\ 0 & -e^{-m} & 0 &0 \\0 & 0 & -e^{-m} &0\\0 & 0 & 0 & -1/l \end{array} \right]

[/tex]

But the author gave some different answer, that is

[tex]

g^{\mu \nu }=

\left[ \begin{array}{cccc} \frac{l}{fl+w^2} & 0 & 0 & -\frac{w}{fl+w^2} \\ 0 & -e^{-m} & 0 &0 \\0 & 0 & -e^{-m} &0\\0 & 0 & 0 & -\frac{f}{fl+w^2}\end{array} \right]

[/tex]

So how should I calculate the inverse metric tensor as shown by the author?

Thanks

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# How to caculate the inverse metric tensor

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