Schwarzchild spacetime singularity

[alle.fabbri]

Hi all!! I'm studying black holes and there's a point that I cannot understand. The book I'm reading is Modeling black hole evaporation, by Fabbri and Navarro Salas. The path is the following.
After introducing the Schwarzchild metric
ds^2 = \left(1 - \frac{2M}{r} \right) \ dt^2 - \left(1 - \frac{2M}{r} \right)^{-1} \ dr^2 - r^2 d \Omega^2
they get the radial null geodesic equation
dt^2 = \frac{dr^2}{\left(1 - \frac{2M}{r} \right)^2 }
that once solved gives, implicitly, the law of motion for a ray of light radially falling
t = r - 2M \ln \frac{|r-2M|}{2M}
Inspired by this one we can introduce the ingoing Eddington-Finkelstein coordinate by means of
v = t + r - 2M \ln \frac{|r-2M|}{2M}
and then switch to another system of coordinate, in order to remove the singularity in r=2M which is not physical. And here problems begin. One can make two choices for the coordinate system:
- the set (v,r,\Omega) for which the metric becomes
ds_r^2 = - \left(1 - \frac{2M}{r} \right) \ dv^2 - 2 dr dv - r^2 d \Omega^2
or
- the set (v,t,\Omega) for which
ds_t^2 = - \left(1 - \frac{2M}{r} \right) \left( dv^2 - 2 dt dv \right) - r^2 d \Omega^2
This is a straightforward calculation, so no probs. Then they say

" It is clear that only in the first case we can analytically continue the metric to all possible values of the radial coordinate r>0. In the second case we still have a singularity at r=2M. The coordinates (v,r,\Omega) are called the ingoing (or advanced) coordinates and because of the cross term drdv the metric is not singular at r=2M."


And I really don't understand the meaning of this. Any insight?

[haushofer]

Well, the essential problem I think is that at r=2M the determinant of the metric is 0 in the second case (just write down the matrix representing the metric). That's a problem, because you would like the vector spaces and their duals to be isomorphic. In the first case you don't have this problem, and you can do an analytic extension beyond this sphere r=2M. Soe everything is well-defined here.

[alle.fabbri]

So the point is that when the metric has zero determinant is not invertible so we can't use it to define covariant vectors (crudely speaking: you can't lower tensor indices?), i.e. the elements of the dual space?