Speed of stuff falling into a black hole, beyond the event horizon

Question

We know the escape velocity from the event horizon (EH) $= c$. Anything that crosses the EH is destined to reach the singularity. Whether the escape velocity (beyond the EH) can be calculated, or not, can we say that something falling into a BH, would move faster than $c$ after crossing the event horizon? Or, does the acceleration ends at EH? Why? This is when something is free-falling into a BH, ignoring resistance. I am using the fact that the speed of (hypothetical) absolute free fall will be same as the escape velocity at any point.

Answer 1

You do not exceed or even reach the speed of light when crossing an event horizon. The easiest way to check this is to carry a flashlight with you. When you turn it on the photons emitted from the flashlight will still move away from you. This "flashlight" analogy can be carried out in GR by calculating the trajectory of a photon with the right initial position and velocity.

To reconcile your intuition it is easiest to think in terms of curved spacetime rather than forces. Once you are inside the event horizon the reason you cannot escape is that spacetime is severely warped and all trajectories with speeds less than $c$ move further into the black hole. See wiki for an illustration.

Answer 2

Escape velocity is a hypothetical cut off, where something with a greater velocity might escape (depending on its direction) but something with less will not.

So if you can't move at speed $c$ or above and someone says that $c$ (or above) is the escape velocity they are just saying you can't escape.

I am using the fact that speed of free fall will be same as escape velocity at any point.

This is not a true fact, don't use it. That would be like using 0=1 to deduce something. Don't do it. Your free fall speed depends on how fast you were going in the past. It could be lower than escape velocity (for example, if you are bound, like an elliptical orbit such as the earth going around the sun) it could exceed the escape velocity if you are unbound (for instance if you loaded up a rocket with lots of fuel and got up to a big speed by burning lots of fuel) or it could be equal (like above, but use less fuel).