What would happen if, beneath the event horizon, a photon was emitted outwards along the radius of the black hole? It's speed can't change to any observer in any reference frame, but it surely cannot escape the black hole either! Is there a flaw in my second assumption?
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Light cones within the event horizon are tilted such that they are directed towards the singularity at $r=0$. Any observer would see the photon moving away from them at $c$ locally, but it'd still be moving towards the singularity rather than away from it.
Here's an spacetime illustration which might help:
from http://www.damtp.cam.ac.uk/user/tong/gr.html, Figure 44. All geodesics are 'ingoing' within the horizon.
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