Hi all!! I'm studying black holes and there's a point that I cannot understand. The book I'm reading is(adsbygoogle = window.adsbygoogle || []).push({}); Modeling black hole evaporation, by Fabbri and Navarro Salas. The path is the following.

After introducing the Schwarzchild metric

[tex] ds^2 = \left(1 - \frac{2M}{r} \right) \ dt^2 - \left(1 - \frac{2M}{r} \right)^{-1} \ dr^2 - r^2 d \Omega^2 [/tex]

they get the radial null geodesic equation

[tex] dt^2 = \frac{dr^2}{\left(1 - \frac{2M}{r} \right)^2 } [/tex]

that once solved gives, implicitly, the law of motion for a ray of light radially falling

[tex]t = r - 2M \ln \frac{|r-2M|}{2M}[/tex]

Inspired by this one we can introduce theingoingEddington-Finkelstein coordinate by means of

[tex]v = t + r - 2M \ln \frac{|r-2M|}{2M}[/tex]

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 [tex](v,r,\Omega)[/tex] for which the metric becomes

[tex]ds_r^2 = - \left(1 - \frac{2M}{r} \right) \ dv^2 - 2 dr dv - r^2 d \Omega^2 [/tex]

or

- the set [tex](v,t,\Omega)[/tex] for which

[tex]ds_t^2 = - \left(1 - \frac{2M}{r} \right) \left( dv^2 - 2 dt dv \right) - r^2 d \Omega^2 [/tex]

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 [tex](v,r,\Omega)[/tex] are called the ingoing (or advanced) coordinates and because of the cross term [tex]drdv[/tex] the metric is not singular at r=2M."

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

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# Schwarzchild spacetime singularity

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