Assume you have a metric in (+,-,-,-) signature, $$d s^2=e^{2 \Phi(r)} d t^2-\frac{d r^2}{1-\frac{b(r)}{r}}-r^2 d \Omega^2.$$ To embed it, we take $t=$Constant, $\theta=\pi/2$ slice, $$d s^2=-\frac{d r^2}{1-\frac{b(r)}{r}}-r^2 d \phi^2.$$ We need to embed it into Euclidean space $$ds^2=dz^2+dr^2+d\phi^2.$$ For the embedding surface, it becomes $$d s^2=\left[1+\left(\frac{d z}{d r}\right)^2\right] d r^2+r^2 d \phi^2$$ However, as we use the (+,-,-,-) convention, the metric on the embedding surface would have a different sign compared with our metric.
If I don't want to change the convention, how can I explain this -1 difference? It seems we need to define $r=i r_{Euclidean}$, which is a wired operation.
