content_image/358740.jpg content_image/358738.jpg content_image/358739.jpg content_image/358737.jpg content_image/358735.jpg content_image/358736.jpg content_image/358734.jpg
is finite, then r$_{H}$ is a simple root. Since g$_{00}$ is differentiable on the interval [0 , ∞ ) we can expand it in a Taylor series and we obtain 

Hence, 

and in order to show that r$_{H}$ is a simple zero we need to prove that g ′ $_{00}$( r$_{H}$ ) ̸ = 0. Taking into account the fact that 

<img src='content_image/358740.jpg'>

<img src='content_image/358738.jpg'>

<img src='content_image/358739.jpg'>

where a prime denotes differentiation with respect to the horizon radius and comparing the above equation with the Eq. (5), we can see that g ′ $_{00}$( r$_{H}$ ) vanishes if and only if r$_{H}$ = r$_{0}$ . This implies that M = M$_{0}$ which is at variance with the initial assumption. As a result we can conclude that g ′ $_{00}$( r$_{H}$ ) ̸ = 0 for M > M$_{0}$ and r$_{H}$ ̸ = r$_{0}$ . 

## III. A NEW TRANSFORMATION 

We show that the singularities of (2) can be removed by a suitable coordinate transfor- mation as in the case of the Reissner-Nordstr¨ om solution. In order to do that we shall follow [15]. Like in the Kruskal approach [14] we introduce coordinates u ( t, r ) and v ( t, r ) such that the original metric goes over to 

<img src='content_image/358737.jpg'>

with the requirement that f 2 ̸ = 0. This will happen if u and v satisfy the non homogeneous system of first order nonlinear partial differential equations 

<img src='content_image/358735.jpg'>

<img src='content_image/358736.jpg'>

<img src='content_image/358734.jpg'>

The next step is to find a suitable transformation of the variable r such that the above system becomes a homogeneous system of PDEs. If we multiply (11) by g 2 00 and we introduce a new 

