So imagine a black hole the size of 100 km, and then there is a giant rod that is 50 km and pretends that the rod doesn't get broken due to tidal forces, and also assumes that the rod has a mini-robot on the other end of it. So once the robot end of the rod enters the event horizon, it can never escape. However, the robot can technically exert a force as pre-programmed, inward towards the singularity making the rest of the string or rod fall faster than normal. So if the rod is long enough, this extra acceleration can be interpreted as a message from the robot. And technically this doesn't violate the law that matter or energy can escape because the rod is still falling and in fact is falling faster now due to the added force. And then on top of that there could be something where sideways motion means something else, so some information about the black hole interior can be gotten until the robot and the rod completely get absorbed and broken apart. So how does this not violate the fact that information can not get out because the rate at which matter falls in changing can be seen as information leaving out?
2 Answers
This will not work. Any information that the robot tries to convey will travel up the rod at the speed of sound in the rod. This is far slower than the speed of light. So it will never escape the event horizon.
The inward force would propagate at the speed of longitudinal or compressive waves. The sideways force would propagate at the speed of transverse or shear waves. Longitudinal waves are usually faster than shear waves, but still far slower than light. It simply cannot catch up to the event horizon which moves outward at the speed of light in any local inertial frame.
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One thought experiment 'paradox' is to hit the end of very long rod with a very large hammer. If the far end moves at the same instant the near end was hit this would be an example of transmitting a signal faster than light, but in reality that does happen. The impulse signal of hitting one end of the rod has to propagate along the rod before the far end becomes 'aware' of the impulse. This impulse travels at the speed of sound of the material the rod is made of and this is slower than the speed of light, but for your thought experiment with the robot, if this impulse signal travelled at the speed of light, it would still not reach the event horizon. Even light can not travel 'upwards' inside a black hole.
Another thought experiment is to attach a rocket to one end of a very long 'rigid' rod. if the rocket accelerates very rapidly away from the far end, the rod will stretch while the far end remains stationary for a finite period, while the information that the rod has accelerated at the rocket end travels along the rod to the far end. For this reason relativity predicts no rod or material object can be perfectly rigid.
In your experiment, there is absolutely nothing the falling robot inside the black hole can do to affect the motion of the part of the rod still outside the event horizon.
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