Every Time You Accelerate, You Create An Event Horizon
To begin with, that just slows down their approach speed. Then that constant acceleration brings them to a halt, and further, they start moving back in the opposite direction. They then accelerate back up to close to the speed of light and keep accelerating.
If I fire a proton at the point of closest approach, say to send a message, that proton can never catch up to the other person as long as they stay on that hyperbolic trajectory.
The proton will always be inching closer, but it will never overtake.
In reality, eternal constant acceleration would take infinite energy. So, after draining all of the energy in the Universe, they’d finally have to stop accelerating, and the sent message would overtake the person.
They also stay ahead of any photon emitted from any point on the left of the diagonal line of speed of light.
This means that the events happening to the left of that diagonal will never affect the observer, which sounds pretty horizon like.
In fact, the active acceleration does create a type of an Event Horizon called Rindler Horizon. This Rindler Horizon flows at a fixed distance behind a constantly accelerating observer.
The distance of a Rindler Horizon is inversely proportional to acceleration.
All parts of the Universe beyond that horizon are out of casual connection with the Rindler observer.
And just as with Hawking Radiation, that Horizon cuts off your access to certain fundamental frequency modes of the Quantum Vaccum.
This Horizon also leads to a mixing of positive and negative frequency modes in the accelerating frame which in turn leads to the creation of real particles known as the “Unruh Particles”.
These particles have the same type of spectrum as the Hawking Radiation. The vaccum itself appears warm with a temperature proportional to the acceleration.
This is the Unruh Effect.