Black Hole horizons and the Equivalence Principle?

Question

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

Answer 1

An object inside a black hole horizon cannot send signals outside of the horizon, but something falling into the black hole can fall through the signal.

Take the example of the camera attached to the astronaut's foot. When the astronaut is halfway through the horizon, the camera is about a meter inside the horizon and the transmitter is about a meter outside the horizon. You say this would allow information to be sent out of the horizon, but that assumes instantaneous signal transmission from camera to antenna.

In actuality, the signal moves at the same speed as light - more generally, the speed of information, field disturbance propagation, massless particles, etc. - which, one meter inside the horizon, is still toward the center even when directed outward. So the camera sends a burst of data through a wire up the astronaut's leg, but the data is slowly falling down despite being sent "up" at the speed of light.

However, the astronaut is falling quickly toward the center of the black hole, so from her perspective the signal is still moving up the wire relative to her, and because the speed of electromagnetic perturbations (like light and changes in electric current) is locally invariant, she will measure it moving at the same 3x10^6 m/s as always. She can't tell she's already fallen through the horizon because there's nothing amiss about her measurements locally. By the time the signal from the camera reaches the transmitter, the transmitter is already inside the horizon.

Since all the other forces at play also move at the same speed, for example the electromagnetic forces holding her feet to her ankles and such, the same principle applies. The astronaut won't sense that her feet have passed the horizon, because in order for her brain to register any difference, the signal would have to travel from her ankles to her head faster than light.

Answer 2

The equivalence principle only holds for extremely small regions of spacetime. Which means they hold for short time intervals as well as small spatial regions.

Consider an event near the event horizon. If the event is outside the horizon there might be a frame moving away from it that cover a very small region that is also completely outside the horizon and it could all fall down, but with an upwards velocity so that for the short time interval it exists the whole thing stays outside the horizon.

If the event is inside the horizon then there is a small region and a small time interval where the whole frame starts out inside the horizon and all stays inside the horizon.

If the event is right on the horizon, then no matter what timelike tangent you select it is heading in. So you can pick a short time interval and a small region of space where the whole frame falls within the horizon, just like the worldline tangent to that event does. You could also do this for the example with the event outside if it is close enough and heading in enough.

Everything has to be small since general relativity doesn't necessarily have global frames (that is what the word general indicates).

So the only part that seems relevant are the frames that cross the horizon. But each of those frames is inertial and nothing interesting is happening. In those frames the horizon is like an expanding surface of light that rushes past them at the speed of light. It just is not an issue. The frame moves in a locally timelike manner and the horizon is a locally lightlike surface so it just rushes past.

It's like how if you lie down with your feet facing your favorite sports team how your feet might get the information about their victory before the rest of you. It's not going to look or feel weird. Imagine that by the time that information gets to your friend you can't make predictions about who's going to win.

An event horizon is a global feature it is a last call, last time to send a letter home and have it get there, last chance to order before they stop taking orders, last chance to change your mind. Its the "are you sure" button except its actually all invisible and it is relative to someone far far away.

You don't feel that the last mailman heading to a particular destination has passed you. And that is 100% all that happens in classical physics when there is an event horizon. You can imagine that someone else is already sending a signal out the signal that never get out. You can send something before it and it will get out, or you can send something with it and also fail.

It literally does not feel like something. And you head and your foot can be in different frames. That always happens in general relativity, frames are local. That just means it's like your body communicates by letter you send something and then wait to hear back.