Black Hole Time Dilation: Light-ray Interpretation

[Alexrey]

I know that a black hole creates infinite curvature in spacetime and hence infinite time dilation. I was wondering though, if I could think of this stopping of time due to the fact that a light ray moving radially towards the centre of a black hole would have to travel infinitely far along the coordinate system's proper radial axis to reach the black hole's centre, as the infinite curvature would produce a proper radial distance tending to infinity as the light ray gets closer to the singularity?

 

[Simon Bridge]

Probably best for you to look at:
http://www.jimhaldenwang.com/black_hole.htm
... introduction to space-time geometry inside and close to a black hole.

Don't be too intimidated by the math - scroll till it starts talking about the Schwarzschild metric.