content_image/374946.jpg content_image/374947.jpg content_image/374944.jpg content_image/374943.jpg content_image/374942.jpg content_image/374941.jpg content_image/374939.jpg
Summing the above equations we obtain 

<img src='content_image/374946.jpg'>

Hence, once we solve for h the unknown function g can be determined from 

<img src='content_image/374947.jpg'>

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 

<img src='content_image/374944.jpg'>

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, 

## 1. Regions I and III ( r > r $_{+}$) 

<img src='content_image/374943.jpg'>

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 

<img src='content_image/374942.jpg'>

<img src='content_image/374941.jpg'>

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 

<img src='content_image/374939.jpg'>

