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

XXyen

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 [tex]2M\ln{\left|\frac{r-2M}{2M}\right|}[/tex] part of [tex]r^*=r+2M\ln{\left|\frac{r-2M}{2M}\right|}[/tex]?

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