It is well know that it takes pi*m time to go from rest at the surface of a Schwarzschild black to the singularity. How much time does an external observer at infinity experience then in that time? How do I show it?
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Hm, well in the first place you cannot "start from rest" from the Schwarzschild surface, since it is a null surface. Only an outward going null geodesic can be at rest there, and any timelike curve will cross it with a nonzero inward velocity.
The external time has already gone to plus infinity when you reach the Schwarschild surface. The inside is a different coordinate patch, in which the radial coordinate r has become the timelike coordinate. So whatever "time" it takes to fall from the horizon to the singularity, there is no way to correspond it to external time.