Summing the above equations we obtain Hence, once we solve for h the unknown function g can be determined from Since the variables y and r ∗ in (25) can be regarded as independent variables, we can set both sides in (25) equal to a separation constant 2 γ . The factor 2 has been introduced in order to simplify the left hand side of (25). Hence, the solutions read with γ = 0. However, this condition on γ is always satisfied as we shall see in the next section. Guided by the principle that we wish to derive the most simple expressions for u and v we can choose without loss of generality, c$_{3}$ = c$_{5}$ = 0. Taking into account the definition of the tortoise coordinate r ∗ we obtain, Let ( v$_{I}$, u$_{I}$ ) denote the specific coordinates ( v, u ) specialized for the region I characterized by r > r$_{+}$ . In this region F ( r ∗ ) > 0 and equation (23) requires that we choose h and g with a positive relative sign. This implies that the constants c$_{2}$ and c$_{4}$ have to be chosen with the same sign. The simplest choice is c$_{2}$ = γ/ 2 = c$_{4}$ . Thus, we have and In the next section we shall see that the requirement f 2 = 0 fixes the value of the separation constant γ . The constant c$_{1}$ can be fixed by requiring that the inverse transformation from the coordinates ( v$_{I}$, u$_{I}$, ϑ, ϕ ) to the coordinates ( t, r, ϑ, ϕ ) gives (2) again and we find that c$_{1}$ = γ $^{2}$. Thus, we conclude that f 2 = g$_{00}$ ( r ) γ 2 e − 2 $^{γr$_{∗}$}$. 8 f 2 = g$_{00}$ ( r ) c$_{1}$ e − 2 $^{γr$_{∗}$}$. v$_{I}$ ( t, r ∗ ) = e γr$_{∗}$ sinh ( γt ) , u$_{I}$ ( t, r ∗ ) = e γr$_{∗}$ cosh ( γt ) v ( t, r ∗ ) = e γr$_{∗}$ γ ( c$_{2}$ e γt − c$_{4}$ e − γt ) , u ( t, r ∗ ) = e γr$_{∗}$ γ ( c$_{2}$ e γt + c$_{4}$ e − γt ) . F ( r ∗ ) = c$_{1}$e 2 $^{γr$_{∗}$}$, h ( y ) = c$_{2}$ γ e γy + c$_{3}$, g ( z ) = c$_{4}$ γ e γz + c$_{5}$, 1. Regions I and III ( r > r $_{+}$) 2 d dy ( ln dh dy ) = d dr ∗ (ln F ) . (25) d dz ( ln dg dz ) = d dy ( ln dh dy ) . (26)