Is it possible to transmit information from within a black hole via the momentum of a rocket activated after passing the event horizon?

Question

Imagine a simple, non-rotating black hole and a massive rocket that is free-falling past the event horizon of the black hole, linearly towards the singularity. The rocket is massive enough that it carries an appreciable momentum, and the linear velocity of the black hole will be measurably changed by addition of the rocket's momentum to that of the black hole.

An observer outside of the black hole is carefully, continuously monitoring the position and velocity of the black hole event horizon, and is able to detect the change in its motion dynamics caused by the free-falling rocket combining with the black hole.

However the rocket also has very powerful engines, and the captain of the rocket is able to make a decision after passing the event horizon as to whether to engage these engines to accelerate (briefly) at full power towards the singularity. The captain is instructed to only engage the engines in the case that some specific condition inside the event horizon is observed, and the external observer is also aware of this plan.

So my question is, will the dynamics of two cases differ in any way, from the perspective of the external observer? In the first case (no engine), there is a single collision between the rocket and the singularity at a predictable point in time. In the second case the initial collision will take place at a different (earlier) time from both the rocket and the external observer's perspectives, and with higher initial momentum - noting this is a more complex dynamic scenario as the rocket exhaust gases will also interact with the singularity via both gravitational attraction and eventually their own secondary collisions.

If the position of the black hole event horizon in the two scenarios varies at all for any instance, then the observer can accurately detect the signal that the captain of the ship issued and information can be transmitted out of the black hole via momentum signals. Even if the final momentum of the system may be the same in both scenarios, the intermediate dynamics may vary and this can be detectable.

The scenario could be simplified from a rocket to two masses held together with an explosive charge between them, without loss of generality. In this case the decision is whether the explosive charge is detonated or not.

It seems to me that the main transfer of momentum from the falling object to the black hole must happen when it collides with the singularity. So if the timing of the singularity collision(s) can be changed, then the timing of the momentum change(s) of the black hole will also vary and this should be visible to an external observer.

What do I miss in this scenario, that prevents information being transmitted externally?

Answer 1

The point that you are missing is that the geometry of spacetime behind the event horizon is such that the light cones of any observer that cross it are "tilted" in such a way that there are no causal worldlines originating at the observer that point away from the singularity, cf. this question. Any detectable signal (In this case, gravitational radiation) must travel along a causal worldline and can therefore never cross the horizon from the inside. That being said, if the rocket is massive enough to impart measurable momentum on the black hole, this momentum transfer must happen when the rocket crosses the event horizon. This causes the horizon to slightly distort, and as it returns to a spherical shape, the black hole "rings", emitting gravitational radiation. Furthermore, conservation of momentum holds within black holes as well. If the rocket were to fire its thrusters after crossing the horizon, any momentum gained by the rocket is compensated by gas being ejected into the opposite direction, so that the total energy-momentum density inside the horizon stays constant (This is obvious from the fact that nothing can escape), thus yielding no measurable change in the shape or momentum of the black hole after the rocket has crossed the horizon.

Answer 2

Event Horizon is a global notion. Technically speaking, a BH is a region from which no causal signal can travel out into the spacetime , to the infinite future. EH which is the boundary of this region is therefore defined in a way taking into account all events which are going to happen in the future. This includes the choice of captain to accelerate and his intention to communicate the observer outside. In other words, if captain can communicate to the observer far away, then he is not inside an BH by definition.

Further, physically an isolated rocket can only accelerate in one direction if it ejects some mass in the opposite direction. However, at the very instant the velocity of center of mass doesn't change following conservation of momentum. Similar argument for angular momentum holds true. The observer outside can only measure the total momentum, angular momentum of the BH + rocket system, both of which will remain conserved in this process.

However, there is still a way to infer that some event has indeed taken place by exploiting how quantum fields can evolve under the effect of curvature or relativistic motion leading to a change in entropy. For eg you can devise some strategy along the lines discussed in these articles here and here. The observer and captain can share some entangled state and any attempt by the captain to accelerate will result in interference and particle creations thereby changing the entanglement entropy. However, this doesn't imply that you are sending new information out to the observer. The entropy is just a measure of correlation : you simply know since the start of the experiment how and when a state is going to evolve if there is such an acceleration.