Clearly, we still have the possibility to choose the functions h and g both with negative signs so that their relative sign is again positive. In this case we get a new set of coordinates representing a portion of the manifold (we call it region III ) which is isometric to region I . In particular, the coordinates which map such a region are Note that u$_{III}$ is always negative as it should be according to the present choice of the relative sign of the functions h and g whereas v$_{III}$ can assume both negative and positive values. Region II is obtained by deriving the corresponding transformation when the radial coordinate varies in the interval ( r − , r$_{+}$ ). In this case F ( r $^{∗}$) < 0 and the relative sign of the functions h and g must be negative. If we choose c$_{4}$ = − γ/ 2 = − c$_{2}$ we get Clearly, we are also free to make the opposite choice. This is equivalent to the transformation ( v, u ) −→ ( − v, − u ). Thus, region IV is isometric to region II and it is described by the coordinates In the next section we shall discuss that the overlapping conditions at r = r ± are satisfied. It is evident that the inverse transformations can only be given implicitly since from (13) we see that r ∗ ( r ) cannot be inverted in terms of elementary functions. However, from the following relations we see that in the ( v, u )-plane,the lines t =const are straight lines v/u =const whereas lines r =const are represented by the hyperbolae u 2 − v 2 =const. Let us first consider the singularity at r$_{+}$ . Lemma 1 ensures that r$_{+}$ is a simple zero of the metric coefficient g$_{00}$ . Thus, the following Taylor expansion holds in a neighborhood of 9 IV. DETERMINATION OF THE SEPARATION CONSTANT γ u 2 − v 2 = e 2 $^{γr$_{∗}$}$, 1 γ tanh − 1 ( v u ) = t, (27) v$_{IV}$ ( t, r ∗ ) = − e γr$_{∗}$ cosh ( γ t ) , u$_{IV}$ ( t, r ∗ ) = − e γr$_{∗}$ sinh ( γ t ) . v$_{II}$ ( t, r ∗ ) = e γr$_{∗}$ cosh ( γ t ) , u$_{II}$ ( t, r ∗ ) = e γr$_{∗}$ sinh ( γ t ) . v$_{III}$ ( t, r ∗ ) = − e γr$_{∗}$ sinh ( γt ) , u$_{III}$ ( t, r ∗ ) = − e γr$_{∗}$ cosh ( γt ) . 2. Regions II and IV ( r − < r < r $_{+}$)