Does an infalling observer ever cross the black hole event horizon in finite proper time?

Question

I am interested in how an infalling observer perceives the plunge into a black hole. Let me assume that actually we have three spaceships: C being at a constant distance to the event horizon, and A and B in a free fall into the back hole, A being a short distance closer to it and B following behind. Let me also assume that the black hole is supermassive, so that A and B will have enough proper time left to do some analysis before being ripped apart by tidal forces (Even if sadly we will never be able to read their paper).

It is well known that from the point of view of C, A and B will never cross the event horizon as time dilation approaches infinity. But how do things look for A and B?

If A and B are close to each other, then the dilation of A's proper time, measured by B, will be less than what is measured by C. Seen from B, A will also never cross the event horizon, but get closer to it similar to what C sees. For an infalling observer like A, the event horizon seems to recede, so both will approach the singularity with increasing acceleration (and tidal forces will draw them apart), but nor will B see A cross, and neither will A experience crossing it.

So my assumptions are:

• an infalling observer B who is following another infalling observer A will see that their distance becomes larger, and A will approach the event horizon but will never cross it.
• an infalling observer will (according to his own perception) never cross the event horizon either (of course, he will eventually be ripped apart by the increasing tidal forces).

My question is whether these assumptions are correct?

Answer 1

Forget observer C, you seem to understand their scenario and situation.

For two observers falling (freely) into a black hole (A first, B following), one after the other, then indeed B can never receive light from A that was emitted from inside the event horizon, until B has also crossed the event horizon.

As regards your assumptions - you cannot "see a distance". I don't know what that means. B will witness A cross the horizon, but only when they cross the horizon too. An infalling observer arrives at $r=0$ in a finite time on their own clock (or at least their components do...).

Such scenarios are best visualised in Gullstrand-Painleve coordinates, where the time axis is that measured by freely falling observers (from infinity, i.e. with $E/mc^2 = 1$). Below is a bit of a wobbly diagram I've adapted from "Exploring Black Holes" by Taylor, Wheeler & Bertschinger (highly recommended).

The solid infalling world line would be A and the red line I've added would be B. The diagram shows inwardly and outwardly directed light pulses emitted by A. You will observe that "outgoing" pulses actually travel towards a smaller $r$ coordinates once A is inside the event horizon. (The event horizon is at a value of 2 on the x-axis of this plot).

B will receive light from A that is emitted from the points/events that are labelled A, B, C, D, E on the diagram. To B, this light will appear to come from the "forward" direction. Note that event E is actually inside the event horizon. Also note that B cannot receive light pulses emitted from inside the event horizon until they too are inside the event horizon.

The light pulse emitted at event F will never reach observer B.

The time on the y-axis is the time that would be measured on the falling observer's clocks.