Lorentz surfaces, conformal metrics and eigenvalues

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<p>From what I understand of Lorentz surfaces (spacetimes of dimension 2), it seems that, according to <a href="http://rspa.royalsocietypublishing.org/content/401/1820/117"; rel="nofollow">Kulkarni's theorem</a>, two reasonable enough Lorentz surfaces (I am only interested in surfaces with topology $\Bbb R^2$) are conformally equivalent, that is, $g_1 = \Omega^2 g_2$. This includes Minkowski space, meaning that they must all be conformally flat.</p>

<p>To find the equivalent conformally flat metric, I assumed that since they are conformal, the metric's eigenvalues should be $-\Omega^2$ and $\Omega^2$. This would then mean that, given a real symmetric $2\times 2$ matrix with negative determinant, the eigenvalues should always be inverses of each other.</p>

<p>From some calculations, this seems not to be the case. Did I misunderstand Kulkani's theorem or is the method I tried incorrect for such a task?</p>
<font color="red"><small>This <a href="http://physics.stackexchange.com/questions/265713/lorentz-surfaces-conformal-metrics-and-eigenvalues">post</a>; imported from StackExchange <a href="http://physics.stackexchange.com">Physics</a> at 2017-02-28 18:21 (UTC), posted by SE-user <a href="http://physics.stackexchange.com/users/36941/slereah">Slereah</a></small></font>;
$\uparrow$ Which calculations? Which method you tried?
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