- #1
ngkamsengpeter
- 195
- 0
Given a metric tensor gmn, how to calculate the inverse of it, gmn. 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
[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