We are given $ds^2 =- \frac{du\,dv}{M - uv},$ where $$v=t+x\, \qquad u= t - x\qquad -\infty < t, x < \infty$$ and $M$ is a positive constant.

The Riemann curvature tensor is proportional to $\frac{1}{(M - uv)^3}$, which means that the physical singularity is actually a line $M = uv$. The problem here is that I don't how to construct the Kruskal diagram, because when I try to do it "by the book", e.g. introduce coordinates in the form $u' \propto \exp{u}$, likewise for $v$, it doesn't seem to produce a metric that is conformally Minkowski (the conformal factor isn't positive at all times, unless we constrain the coordinates). I know that there should be 4 separate regions, 2 that are asymptotically flat and 2 with a singularity. I need to find coordinates in which this will be manifested.


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