Discriminant (Metric)

Given a Metric $g_{\alpha\beta}$ , the discriminant is defined by

$\begin{displaymath} g\equiv {\rm det}(g_{\alpha\beta}) = \left\vert\matrix{g_{11... ...}\cr g_{21} & g_{22}\cr}\right\vert = g_{11}g_{22}-(g_{12})^2. \end{displaymath}$

(1)

Let

be the discriminant and $\bar g$ the transformed discriminant, then

$\begin{displaymath} \bar g=D^2 g \end{displaymath}$

(2)

$\begin{displaymath} g={\bar D}^2\bar g, \end{displaymath}$

(3)

where

$\displaystyle D$	$\textstyle \equiv$	$\displaystyle {\partial (u^1,u^2)\over \partial ({\bar u}^1,{\bar u}^2)} =\left... ...ial {\bar u}^1} & {\partial u^2\over \partial {\bar u}^2}\end{array}\right\vert$	(4)
$\displaystyle \bar D$	$\textstyle \equiv$	$\displaystyle {\partial ({\bar u}^1,{\bar u}^2)\over \partial (u^1,u^2)} = \lef... ... \partial u^1} & {\partial {\bar u}^2\over \partial u^2}\end{array}\right\vert.$	(5)