Schwarzschild Black hole and an infalling object from the perspective of an outside observer

Question

Person A is falling into a black hole. Person B is an outside stationary observer. I know according to person A they fall into a black hole and die in the singularity. Easy.

Now let's talk about the perspective of person B.

Can an outside observer (person B) assign a moment in time on their wristwatch and say "in x minutes on my watch person A crossed the event horizon" or "in x minutes person A is going to be at the event horizon"? What would be the difference in "crossed" and "be at the horizon" be in terms of the proper time of person B if any? I know person B wont be able to assign a moment in time on their wirst watch that'd correspond to A dying in the singularity because it'd take an infinite amount of time on B's watch for A to meet the singularity because on B's watch the black hole singularity doesn't exist. It forms infinitely far in the future.

The question rephrased: Will person A cross or be at the event horizon in a finite amount of proper time on persons B wristwatch? Or would it take an infinite amount of time on B's watch for A to cross/ be at the event horizon? I don't care if B can see A doing it. It's not a visibility issue. I want to know if A crossing the event horizon is happening in a form of an event on B's watch. I'm aware that light gets redshifted and B won't ever see person A cross the horizon but is it going to happen on B's timeline/ can B assign a time on their watch when? As far as I'm aware they won't even see A at the horizon. The last "place" where B sees A is just immediately right above the horizon. They won't see A at the horizon anymore since light would take an infinite amount of time on B's watch to reach an outside observer (B) from the event horizon. But is B going to be able to say that for example at 2pm on their watch (even tho B can't "see" it!) person A crossed the horizon no matter if B sees A doing it or not? B knows they can't see it because of the redshift but can B say it "happened"?

Being able to "see" something happening is not the same as something actually happening? Just because I don't see something doesn't mean it isn't? "Being" and "seeing" shouldn't be equal? Or is it the same in relativity? So does not being able to see something happening equal to it not happening on my timeline?

When I look at the Penrose diagram I can see that the singularity from the point of view of an outside observer forms infinitely far in the future. A singularity doesn't exist for an outside observer? So on B's wristwatch it takes an infinite amount of time for A to reach the singularity. So B can never say A died. Cause as long as B's universe exists A never dies? Person A only dies in the infinite future (according to B's watch). Does the event horizon also take an infinite amount of time to form according to B's watch/ does it take an infinite amount of time on B's watch for A to reach the horizon? (It'll take an infinite amount of time to see it but how much time for it to happen as an event on B's watch if it can even happen?) If it takes an infinite time to form how is this evident on the Penrose diagram? (It is evident in the case of the singularity).

Is this a logical conclusion?: If the event horizon for B exists then B has to be able to assign a time on their watch when A is there/ crosses? If the event horizon for B forms infinitely far in the future meaning it doesn't exist for B then and only then B won't be able to assign a time/ event on their watch/timeline that'd correspond to A crossing it? So that'd mean that not only singularities don't exist for outside observers but also event horizons don't exist for outside observers. The event horizon only comes into being for an observer as they cross it?

Question doesn't seem to be a duplicate of: Can black holes form in a finite amount of time?

nor of

How can anything ever fall into a black hole as seen from an outside observer?

The "Can black holes form in a finite amount of time?" question has an accepted answer but some below are saying that the accepted answer is just false and "completely wrong".

Apparently the blue and red "time slices" showed in question 1 are wrong and "misinformation".

Same goes for the second question. Everyone seems to be saying different stuff. Even some answers with a couple of upvotes are questionable.

Anyways the point is there's no precise answer to this question. People are arguing, discrediting each other or just giving wrong answers/ contracting answers. So what's right?

Answer 1

The question is somewhat ill-posed due to relativity of simultaneity:

Can an outside observer (person B) assign a moment in time on their wristwatch and say "in x minutes on my watch person A crossed the event horizon"

An observer cannot uniquely assign time "on their wristwatch" to any distant event, because there is no absolute notion of simultaneity. Two events that occur at the same time in one coordinate systems, will occur at different times using a different coordinate system.

The only thing we can generally say about two events is whether they are spacelike-, timelike- or lightlike-separated, and what is the proper time/distance of a line connecting them. In the case described in the question, person A will cross the horizon at some event $P$. If A and B start from the same point at an event $P'$, then $P$ and $P'$ are timelike-separated and connected by a path of finite proper time. Future events along person B's worldline might be spacelike-separated from $P$, so you can construct a coordinate system in which they occur "at the same time".

PS: This animation demonstrates the scenario in question, using both Schwarzschild and Kruskal–Szekeres coordinates. While in Schwarzschild coordinates the infalling observer (blue) never reaches the horizon, in Kruskal coordinates (starting at 02:13 in the video) it does so after finite time. The simulation shows both observers emitting light signals at fixed intervals of their respective proper times, which can demonstrate how they "see" each other. Note that this is exactly the same in both coordinates systems, i.e. the physical effects experienced by the observers are the same, even though they can be described using different coordinate systems.

Answer 2

People are arguing, discrediting each other or just giving wrong answers/ contracting answers. So what's right?

That is unavoidable on the internet. You need to evaluate the credibility of information you get from any source. The accepted answer of the first question is good, IMO. But you need to form your own opinion. You will never get consensus online.

Can an outside observer (person B) assign a moment in time on their wristwatch and say "in x minutes on my watch person A crossed the event horizon" or "in x minutes person A is going to be at the event horizon"? What would be the difference in "crossed" and "be at the horizon" be in terms of the proper time of person B if any?

No. Wristwatches, or clocks in general, measure proper time. Proper time is only defined at the location of the clock itself.

However, what B can do is that B can choose a coordinate time that matches their proper time on their worldline. There is a lot of freedom in choosing a coordinate time. Essentially, B has the freedom to select a synchronization convention. Any foliation of spacelike hypersurfaces can be used. The Schwarzschild coordinates are the most commonly used foliation, but they are not mandatory and any other foliation of spacelike hypersurfaces may be used.

At the moment that A crosses the horizon according to A, you can draw a lightlike line backwards in time from A to B. Let $t_0$ be the proper time on B's watch at that event. If B sends a flash of light at $t_0$ that light would reach A just as A crosses the horizon. For any proper time $t>t_0$ there exists some valid choice of coordinate time for B such that A crosses the horizon at $t$.

So, for any $t>t_0$ there is some valid choice of simultaneity where B can say "A is crossing the horizon right now" according to B's coordinate time. But there is also a valid choice of simultaneity where B can say “A is still outside the horizon right now” at the same $t$.

Being able to "see" something happening is not the same as something actually happening? Just because I don't see something doesn't mean it isn't? "Being" and "seeing" shouldn't be equal? Or is it the same in relativity? So does not being able to see something happening equal to it not happening on my timeline?

Being and seeing are not the same. This is an unfortunate confusion that is due to the usual teaching approach relying heavily on "thought experiments". Students often get this mistaken impression. It is not the case.

If the event horizon for B exists then B has to be able to assign a time on their watch when A is there/ crosses?

Again, proper time is only defined at the location of the clock itself. So there is no proper time on B's watch when A crosses the horizon simply because A is not at the location of B's watch.

The event of A crossing the horizon is not something measurable by B's proper time, but only by B's coordinate time, which is quite arbitrary as described above.

Answer 3

Your question inspired me to ask a similar one on another Physics forum!! I can only reiterate what the previous answers already stated but maybe you'll understand it if I word it differently and add some examples?

For the sake of simplicity I'm assigning some pronouns here so I can refer to A as "her" and B as "him" in my text.

Can an outside observer (person B) assign a moment in time on their wristwatch and say "in x minutes on my watch person A crossed the event horizon" or "in x minutes person A is going to be at the event horizon"?

If you ask it this way the quick answer is: No.

Nevertheless I see what you're trying to ask so the answer might be actually a Yes if asked differently. The issue here is that you're using the wrong terminology to describe what you mean.

The "type of time" that is on B's wristwatch is called proper time. B's proper time means the time on B's clock in his own pocket. B can't assign proper time to anything else than himself. But what kind of time could B assign to A? He could assign some coordinate time to A! The coordinate time value is going to depend on the coordinates B is going to choose. By definition B can't assign proper time to A. If you had replaced proper time and coordinate time in the relevant passages of your question and rephrased it a little you would have gotten a valid-sounding one.

B is going to use the aforementioned Kruskal-Szekeres coordinates which also cover the place of the manifold beyond the event horizon that is not covered by the Schwarzschild coordinates. After drawing that B could draw A's worldline and his own worldline. Let's say that B has drawn a map now.

B knows they can't see it because of the redshift but can B say it "happened"?

B looks at his map and sees events on A's worldine that correspond to crossing the event horizon. A's worldline continues inside the black hole and reaches the singularity in some finite coordinate time. Let's say on this map A meets the singularity at coordinate time 1. From there we draw a horizontal line and see where B is at that coordinate time. B also sees events on his worldline that "happen" at greater values than 1 of coordinate time. For example B can tell they're still hovering next to the black hole at coordinate time 5. Does this mean that A already died in the singularity and B is still alive and hovering next to the black hole at that coordinate time? Well yes but it's only because that's the way B has drawn the map. B clearly drew those worldlines in his favorite coordinate system in a specific way where B can claim this. Does this have some physical meaning? No it doesn't. Coordinate time has no real physical meaning like proper time does. This coordinate time doesn't have any special significance to B. He could've chosen some different coordinates to map spacetime with where those events could happen at a different coordinate times and the answers would be therefore different. There's no coordinate system you could choose that is more right then the other one. There's no "real" or right answer to this question nor is there a wrong answer. By drawing those horizontal lines you're connecting two space-like separated events. That's why some here called it "meaningless" because clearly it's difficult to assign some specific meaning to it in the framework of the theory of relativity. A "meaningful" question in this context would be asking about something invariant that isn't dependent on the coordinate system you choose. That would be for example a time-like separated event because if those events were time-like separated those would be always assigned a different coordinate time and it wouldn't matter what kind of coordinates you choose.