A paradox in an accelerated flexible object and the horizon?

Suppose that we accelerate a charge for some time at a constant acceleration. Part of its infinite electric field falls behind the apparent horizon of the accelerated frame. How does the field behind the horizon "tell" to the accelerating charge that it should send some momentum to keep the field accelerating?

In the Minkowski coordinates, the far behind parts cannot send a light signal to the accelerating charge as long as the charge keeps accelerating. Does this pose a problem?

https://physics.stackexchange.com/questions/440877/why-do-accelerating-bodies-have-an-event-horizon

The apparent horizon is at a distance c^2/a from the accelerating particle, as seen by a static Minkowski observer. For an acceleration a = 10 m/s^2 that is one light year.

If we have any large mechanical, interacting system, then parts of the system may fall behind the apparent horizon of an accelerating part of the system. Thus, this is quite a general question of special relativity. Is that a problem that some far parts of the system cannot send signals to the accelerating part?

If the accelerating object is pulling behind it an inelastic rope, an observer hanging on the rope close to the apparent horizon will feel an enormous acceleration. The rope will break before it reaches the horizon. But an electric field cannot break? What happens to it?

Let us look at the problem in the static Minkowski frame. Let us start accelerating the charge at 10 m/s^2. A year after that, the field one light year away will learn that the charge has started moving. It at first moves at a moderate pace. The electric field of the charge will start deforming in a smooth manner. There are no singularities or infinities at the horizon, from the point of view of the Minkowski observer.

If there were another observer hanging from an inelastic rope which is attached to the charge, that observer would have an infinite acceleration at the horizon, and he would observe something singular or infinite at the horizon. This is like the black hole horizon: it looks like a singularity for a static observer, while for a freely falling observer there is no drama. The natural way to look at the field is the static observer, not the infinitely accelerating one.

If we can find a solution of Maxwell's equations in the Minkowski space, then an apparent horizon poses no problem. The accelerating charge does not need to "know" what happens behind the horizon. It is the global solution of the field equations which has all the necessary information.

The apparent horizon may hide lots of things from the accelerating observer. Earth might fall behind the horizon. There is no problem or contradiction in this if we can find a global solution which respects what is in the incoming light cone of a spacetime point.

In summary, there is no paradox.

A static charge and an accelerating observer in a Minkowski space

We may imagine that the charge has painted the electric lines of force to the entire Minkowski space. The accelerating observer can see the painted lines. What does he see?

Nothing special. His view is like an inertial observer who moves at the same speed as the accelerating observer. If his speed is close to light, he will see the painted space contracted in the direction of his velocity. The field in most cases appears to be almost normal to his velocity vector.

Suppose that our observer is pulling behind him an inelastic rope. Suppose that we have another observer hanging on the rope, close to the apparent horizon of the first observer. This second observer has an enormous acceleration relative to a static Minkowski observer. The second observer moves almost at the speed of light. For him, the field will look almost "singular", but that is just because his own motion is so extreme. The field looks very smooth for a static observer.

We see that the "singular" conditions which an accelerating observer close to the horizon encounters, are just an artifact of the "singular" movement of the observer.

An accelerating observer sees light coming from close to his apparent horizon as enormously red-shifted. He cannot see past the horizon. His inability to see is not because there is something special at the horizon but because of his movement. If there is another observer next to him who moves at the same speed, but does not accelerate, that other observer does not see any apparent horizon. He sees a huge redshift in light coming from the horizon, but he can see also behind the apparent horizon of the accelerating observer.