Staticboson

In most cases, this space-like separation disappears as you move the clock forward from either event. But in this case, OO can never get a signal from the crossing event - so that crossing can never be in OO's "causal past", and thus once the crossing becomes "space-like" separated, it stays that way for as long as OO remains outside of the black hole.

I hope this helps.

But then there will be a moment in time when a signal sent from the OO would reach the EHX just as he was crossing. Let's say that is 1pm on OO's clock. Up until OO's 1pm, everyone would agree that the crossing was in OO's future - it was invariant. After that, in depends. So the events of OO's 2pm and EHX crossing would appear simultaneous from some reference frames, the crossing happens earlier in other frames (including EHX's) and the 2pm happens first is others (including OO's).

So the EHX departs and heads in. Right away "simultaneous" becomes an issue as the distance between you and your clock rates diverge. For a while, the EH crossing will still be within the OO's "causal future" - the OO could send a message and the EHX would receive it before the crossing. In this sense, the crossing would still be an "invariant" part of the OO's future - meaning that no one could dispute it. It appears to be in OO's future no matter what reference frame you use.

The phrase that was used for this was "causal future" - meaning it happens in your future as evidenced by the fact that you could potentially "cause" it (or stop it from happening, or change the way it happens, or maybe just get a message to it before it does it). This ability to get a message to an event before it happens is called "causal future" and it is the litmus test for determining whether something is in you future no matter what reference frame is used.

In the case of an outside observer (OO) and an event horizon crosser (EHX), the situation is not quite as straight forward. If the OO and EHX start out together, then they can both agree at that moment that the crossing is in their future. In particular, the OO can tell the EHX whether he should really go through with this and the EHX will get that message immediately (thus before the crossing occurs).

If both events are instantly signaled at light speed to the other location and observers at both locations see their event happen before the other signal arrives, then the events are "space-like" separated. This means that which event happened first will depend on you reference frame. But, if both observers can agree based on the exchanged signals that the same event happened before the other, then the events are time-like separated.

Since the thread was closed before I had a chance to rephrase what was being said - I'll do it here.
But I am limited to 500 characters - so I may have to do it in installments ...

When comparing two instantaneous events, such as a touchdown in Boston and a lift off in Florida, there are three possibilities, 2 invariant and one that depends on your frame of reference: 1) The touchdown happened first, 2) the lift off happened first, or 3) the events were "space-like separated".