Tortoise-like coordinate transform for interior metric

[FunkyDwarf]

Hello!

When using the Schwarzschild exterior metric in the klein-gordon equation one can perform the standard tortoise(E-F) coordinate transform to yield a wave equation which has a well defined potential that is independent of the energy term. My understanding is that the motivation for this coordinate transform was the removal of the coordinate singularity at the horizon of this metric. If one wanted to generate a wave equation from an INTERIOR metric, say the Schwarzschild interior, that also had a well defined potential independent of energy, on what premise would one start, given that if we want to consider a fluid sphere there is no singularity at the horizon?

Certainly i can take the wave equation and put it in a Schrödinger-like form but this yields an effective potential with the energy term coupled to coordinate terms which I must admit i don't know how to transform away without re-instating the first derivative (ie no longer in Schrödinger form).

Hope that makes sense!

 

[Bill_K]

Don't know if it's what you want, but one way of arriving at Eddington-Finkelstein coordinates is to find a retarded time that serves to eliminate the dr² term in the metric. This can be done in general. For

ds2 = A(r)² dt² - B(r)² dr² + ...

just let u = t ± integral(B/A dr)