## Eddington-Finkelstein, Regge-Wheeler, Kruskal-Szerekes coordinates

how do Eddington-Finkelstein, Regge-Wheeler and Kruskal-Szerekes coordinates work? i mean, what is their run, for example, what is the idea of the $$2M\ln{\left|\frac{r-2M}{2M}\right|}$$ part of $$r^*=r+2M\ln{\left|\frac{r-2M}{2M}\right|}$$?

an issue is also the correct form of the Kruskal-Szekeres coordinates. in wikipedia, there are oppositional definitions of $$v$$ and $$u$$: KS(en) KS(de).
for $$r\leq2M$$ the definition for $$v$$ is either $$T = \left(\frac{r}{2GM} - 1\right)^{1/2}e^{r/4GM}\sinh\left(\frac{t}{4GM}\right)$$ or $$T = \left(\frac{r}{2GM} - 1\right)^{1/2}e^{r/4GM}\cosh\left(\frac{t}{4GM}\right)$$.

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