Can You Always Reach an Object Falling into a Black Hole?
I know this question or similar ones have been asked before, but I want an answer based off of where my logic may be wrong, not a separate proof.
Premise:
For an outside observer, there is never a time $t_1$ for which there isn't a later time $t_2$ where light signals (travelling at speed $c$) confirm that the infalling object had not yet crossed the event horizon at $t_1$. → Therefore, from the outside frame, the infalling object never physically crosses the horizon.
At any given time $t_1$, the object has a measurable finite gravitational redshift $z$.
For any redshift $z$, there exists a velocity $v < c$ such that if you accelerate toward the infalling object at that speed, the redshift becomes a blueshift. → This is because blueshift approaches infinity as your velocity approaches the speed of light towards an emmiting source.
In general relativity, a blueshift indicates that the space between you and the emitter is closing.
Point 1 is true from every external frame outside the event horizon. → Therefore, the infalling object never disappears from view for any observer who remains outside, or from you at any point while approaching.
This means that the distance between you and the infalling object is closing, and you have an infinite amount of time to reach it (since it never crosses the horizon from your perspective, being outside the horizon).
Conclusion: Since you can always accelerate toward the object (at a speed less than $c$) from any moment after it began falling, and since its signals continue reaching you indefinitely, all external light signals — from the entire external universe — for all of time - will eventually reach the infalling observer before they cross the horizon. → Therefore, the infalling observer sees the entire future history of the universe unfold in front of them.
