Every Time You Accelerate, You Create An Event Horizon
Extending light ray world lines backwards from our observer defines what we call “Past Light Cones”, the region of Spacetime that can have a casual influence on the observer.
That’s because photons fired from anywhere in the “Past Light Cone” can reach our observe at the current point.
As our observer moves forward in time, his ‘Past Light Cone’ should actually contain the entire Universe.
It means, if you wait long enough, photons from anywhere in the Universe can catch up to you.
But actually, there is a sub-light speed world-line that can out-space light or at least keep ahead of it. That’s a world-line of an observer undergoing constant acceleration.
Acceleration means ‘Change in Speed’. Hence, an accelerating world line is curved.
Imagine a person traveling towards you initially at close to the speed of light. For some reason, they change their mind, fire their transported and begin a constant acceleration in the opposite direction.
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.
That constant acceleration world line traces out a hyperbola, and it has a very interesting property.