In the reference frame of a freefalling observer, does crossing the event horizon not cause a contradiction between two classical principles that supposedly apply at the event horizon: the equivalence principle and the principle that no signal can cross from inside the horizon to outside?
Consider the following:
Let us first suppose that the equivalence principle holds. In this case, consider an astronaut falling into a very large black hole such that the tidal forces at the horizon are negligible. The equivalence principle states that the astronaut will not even be aware that she is crossing the horizon (the spacetime there is smooth and ‘uninteresting’ from her perspective, and she will feel the same as she would if she were floating in empty space). If this is true, then she could have a camera on her foot connected to a transmitter on her head, and as she crosses the horizon she will be able to transmit the images captured by her camera outside the black hole via the transmitter on her head. If she detects nothing physically different as she crosses the horizon, then when her feet are inside the horizon while her head is outside the horizon, there should be nothing preventing her from transmitting the pictures out to the Unvierse from her transmitter. But this violates the principle that no signals can leave a black hole. Now, if we rather take as true the principle that no signals can pass from inside to outside the horizon, then as her feet cross the horizon, the forces holding her feet to her ankles will be changed because while her ankles can send force signals to her feet, her feet cannot send force signals back to her ankles, and she will therefore ‘feel’ herself cross the horizon, but this in tern violates the equivalence principle.
How are these contradictions resolved in the current understanding of black hole physics