I Black hole and coordinate time from the perspective of Alice outside

[Bellaella]

TL;DR Summary

Bob falls into the black hole. What is the reality of Alice's world? It's not important to Alice what she sees. She can't see Bob fall through the horizon but did Bob actually fall through the horizon and reach the singularity in Alice's reality even though Alice can't observe it?

Hello everyone!:) This is my first post on here. I'm having trouble understanding what is real for Alice. I looked at different physics websites and there seem to be only a few questions addressing this some of which I don't understand. I thought maybe I'll get some clarity here if I ask my own question or my question just doesn't make sense.

Maybe this isn't a physics question but a philosophical one or one looking for a possible philosophical interpretation of my described event?

Alice is an external observer far way. Bob is falling into the black hole. Alice decided to use the Kruskal-Szekeres coordinate system to map the spacetime manifold. In that coordinate system the coordinate Kruskal time doesn't approach infinity for someone crossing the horizon and even after crossing it. So the Kruskal time is a finite value. Alice synchronizes her clock to the kruskal time which is also her proper time since she is far way.
Alice now takes a look at her clock. Then she looks at Bob falling into the black hole. She starts assigning some coordinate time to events she can see. She won't ever see him cross the EH because it'd take an infinite amount of time to observe it. BUT did that crossing happen in Alice's reality even tho she can't see it? Alice and Bob are in the same spacetime coordinates that describe that same manifold which is the same reality they both are part of? And looking at that global manifold we see it happens.

So is it true that there is some finite coordinate time value for Alice on her clock where Bob is in the black hole but that coordinate time could be any time since she can't ever see him cross and assign a time? But at the same time there has to be some unknown coordinate time because both are part of the same manifold that i.e. the same reality.

As far as my understanding is Alice herself can't assign a coordinate time to that event because in order to be able to do so she'd need to observe it and then look at her clock? It'd take an infinite amount of time for Alice to see it but it shouldn't take an infinite amount for Alice for it to actually happen but without Alice seeing it. But since the coordinates used have a finite Kruskal time there has to be a consensus that it happened for the outside observer as well in finite Kruskal time but the observer just doesn't know when. Alice synced her clock and it takes a finite kruskal time to cross so it'd be right for the outside observer to claim it happened at any of that finite time?

 

[Nugatory]

BUT did that crossing happen in Alice's reality even tho she can't see it?

There's no good answer until you have clearly defined what you mean by "Alice's reality".

We have a spacetime geometry. We can make statements about that geometry: Some of the points in that manifold are on Alice's worldline; some aren't. Some of them are included in the past or future lightcones of points on Alice's worldline; some aren't. What test do we use to determine which ones do or do not "happen in Alice's reality"?

 

[PeroK]

As far as my understanding is Alice herself can't assign a coordinate time to that event because in order to be able to do so she'd need to observe it and then look at her clock?

She could choose to use different coordinates. It could be Bob Eddington and Alice Finkelstein conducting the experiment!

 

[PeroK]

PS:

https://en.wikipedia.org/wiki/Eddington–Finkelstein_coordinates

 

[Dale]

in Alice's reality

What does “Alice’s reality” mean? What would make reality real if it belongs only to one person?

 

[Ibix]

Alice can never observe Bob crossing the event horizon. However, she can easily forecast Bob's horizon crossing - and there is most definitely an event on her worldline when she becomes too late to stop Bob crossing even in principle.

As others have commented, "reality" probably isn't a helpful concept here. The relevant concepts are causal future (events Alice could affect now), causal past (events Alice could have been affected by now), and whatever you call the region outside that. Bob's horizon crossing leaves Alice's causal future but never enters her causal past unless she follows him in. However, the event "Bob is now so close to the horizon that his rocket no longer has enough delta-v to stop him" does enter her causal past, as does "he's now so close the only way he's returning is as a smear on the back wall". So Alice can certainly set tight bounds on Bob's horizon crossing coordinates even if she can never see it.

 

[Bellaella]

There's no good answer until you have clearly defined what you mean by "Alice's reality".

Yeah good point! To be honest I found a post online that inspired me to ask this question because I didn't know exactly what to think of that myself.

Its difficult for me to define what I mean by "reality" because I thought of it more like a "thought experiment". I can't write a proper definition in mathematical terms but I can try to describe it.

I found this picture online. If you look at it you see spacetime described by kruskal coordinates. There's a blue horizontal line and an orange horizontal line. Both lines have an intersection with the green line marked by a dot in the corresponding color. Here we see that the blue dot is at around kruskal time 3 and the orange dot at around kruskal time 1. The green line is an external observer. It can be Alice but we said she's very far away which isn't properly represented here. The red line is Bob.

We see Bob merges with the singularity at around kruskal time 1 and I can draw a horizontal line in this global coordinate system which I assume is the global reality I'm talking about. That line shows Alice existing at a kruskal time 3 that is after the time according to that global manifold where Bob ended up in the singularity. It looks like Alice exists but Bob doesn't anymore at this coordinate time which Alice cannot assign or be aware of because she has no way of knowing it but the universe appears to know it as evident on here.

Isn't this what we see here? Can Alice therefore mention any coordinate time and there is a chance that it is true but she won't ever know if its true because the event is covered by an event horizon?

 

[Bellaella]

She could choose to use different coordinates. It could be Bob Eddington and Alice Finkelstein conducting the experiment!

Haha right. The thing is I was looking most of the time on the Kruskal-Szekeres coordinate system and that's the one I'm most familiar with at the moment. I want to keep that one for now to avoid confusion. After all I read it shouldn't matter what coordinate system I choose. I might choose a different one next time!

 

[Bellaella]

What does “Alice’s reality” mean? What would make reality real if it belongs only to one person?

I tried to describe my understanding of "reality" above. To quickly sum it up its the thing I can draw on my sheet of paper. The whole universe is on there and the manifold described by the coordinates I chose which includes the interior of the BH and the exterior. The fact that Bob and Alice can be represented on it. Bob might not be in Alice's light cone but he is still in the whole manifold where Alice as well. They're spacelike seperated but somehow there's 1 global kruskal time describing everything which is inside and outside the EH.

 

[Bellaella]

and whatever you call the region outside that.

Yes exactly. I'm talking about the region outside that as well. That'd mean they're spacelike separated but nevertheless they're still part of the same spacetime manifold. That region outside that is also the reality I'm referring to. Even though they're space like separated they're part of the same universe.

Bob's horizon crossing leaves Alice's causal future but never enters her causal past

What does this mean? What is the interpretation of this? What can Alice say about Bob now? I think that's what I'm asking about. As Bob crosses the horizon and Alice continues on her worldline outside the horizon the kruskal time keeps increasing. The whole universe which is everything in the diagram goes forward in time as this kruskal time increases and according to the coordinates both Alice and Bob increase in their kruskal time and their temporal evolution or however you want to call it keeps going on and they both keep following their worldines. Doesn't this mean there has to be a kruskal coordinate time for Alice that Alice is not aware of at which Bob is below the event horizon? Although it would be impossible to know for her. At some point "the universe" due to temporal evolution is going to "place" them such as Bob is annihilated and Alice is outside somewhere. The universe has to do it because there's just no other way around. Alice and Bob have to be somehow placed in this manifold. Haha okay does this make sense

This probably describes the best what I mean by "reality".

 

[Ibix]

Haha okay does this make sense

Yes. I don't think it's consistent with the physics, though.

Once Bob's horizon crossing is outside Alice's future lightcone she can choose to say it has already happened or it has not happened. There's no physical consequence to the claim, and it's just a matter of picking a different coordinate system - simply picking a different time to call the zero of the Kruskal-Szekeres coordinates is enough.

In principle, Alice cannot be completely certain that Bob entered the horizon until the horizon crossing event enters her past light cone - which it never does if she remains outside the hole. In practice, though, she can know that he does not have the capability to return from some point above the horizon, and that event does enter her past lightcone.

What all this means is that trying to divide the world into "will happen" and "has happened" is naive. There's a chunk of spacetime that doesn't fit into that classification - or, at least, where you have a free choice about how to classify it. This is already an issue in flat spacetime - black holes just magnify the problems.

 

[Dale]

I tried to describe my understanding of "reality" above. To quickly sum it up its the thing I can draw on my sheet of paper. The whole universe is on there and the manifold described by the coordinates I chose which includes the interior of the BH and the exterior. The fact that Bob and Alice can be represented on it. Bob might not be in Alice's light cone but he is still in the whole manifold where Alice as well. They're spacelike seperated but somehow there's 1 global kruskal time describing everything which is inside and outside the EH.

With that clear explanation, then unambiguously yes Bob crossing the horizon is part of Alice’s reality. Just a part she doesn’t see.

 

[Nugatory]

They're spacelike seperated but somehow there's 1 global kruskal time describing everything which is inside and outside the EH.

Calling it Kruskal time is attaching too much weight to the concept. All time coordinates (and spatial coordinates as well, but our intuition handles them better) are best thought of as labels. Two spacelike-separated events may both be assigned the same time coordinate but that just means that we chose the labeling algorithm so that it came out that way.

The only “time” that is more than a label generated by a more or less arbitrarily chosen labeling algorithm (that is, choice of coordinate system) is proper time, the difference between two consecutive readings of the same clock. It is the elapsed time along the worldline the clock follows between the two measurement events.

We often choose the coordinate system in such a way that there is some worldline along which the labels do correspond to proper time; then we call that the worldline of the “observer”, in this case Alice. That can tempt us into thinking that the time coordinates of events off that worldline have some special significance to Alice. They don’t.

 

[Tomas Vencl]

I tried to describe my understanding of "reality" above. To quickly sum it up its the thing I can draw on my sheet of paper. The whole universe is on there and the manifold described by the coordinates I chose which includes the interior of the BH and the exterior. The fact that Bob and Alice can be represented on it. Bob might not be in Alice's light cone but he is still in the whole manifold where Alice as well. They're spacelike seperated but somehow there's 1 global kruskal time describing everything which is inside and outside the EH.

I’m not sure whether this definition of reality is physically useful. It is fulfilled by every event in a block universe, and I can’t imagine a physical process that wouldn’t qualify as reality in this sense.

 

[Dale]

I’m not sure whether this definition of reality is physically useful. It is fulfilled by every event in a block universe, and I can’t imagine a physical process that wouldn’t qualify as reality in this sense.

I think most definitions of reality are philosophical, so I doubt very many of them are physically useful. This one is the one the OP is using for the purposes of this question. We needn’t agree with it to use it for answering the question.

 

[Tomas Vencl]

I think most definitions of reality are philosophical, so I doubt very many of them are physically useful. This one is the one the OP is using for the purposes of this question. We needn’t agree with it to use it for answering the question.

Yes, agree.

 

[Bellaella]

In practice, though, she can know that he does not have the capability to return from some point above the horizon, and that event does enter her past lightcone.

Just to sum everything up and check if my understanding is right:

Are you saying this could be the case if Alice knew the specifications of Bobs spaceship (if he had one)? Otherwise I don't know how there could be a point still outside the EH that wouldn't allow Bob to avoid falling into the BH. I don't know what you mean by that.

Nevertheless now you basically said she can know he doesn't have the capability to return so that means that it "will happen" because he doesn't have the capability. Later you're saying assuming that is naive.

Once Bob's horizon crossing is outside Alice's future lightcone she can choose to say it has already happened or it has not happened.

This will be the case when Alice waits long enough and Bob is already far away enough so that it'd be impossible for her to save him because that's when he leaves her light cone?

There's a chunk of spacetime that doesn't fit into that classification

That's the spacelike separated "area"?

Two spacelike-separated events may both be assigned the same time coordinate but that just means that we chose the labeling algorithm so that it came out that way.

And those two spacelike events may be both assigned a different coordinate time? This possibility arises because of spacelike separation and wouldn't be the case for timelike separated events?

I'm asking because I though what you're saying only applies to reference frames not to a global coordinate system that's used to map the whole spacetime like Kruskal-Szekeres where reference frames would be placed in.

Now an important fact appears that I could draw the Kruskal-Szekeres coordinate system in such a way that in one Alice would be placed at the same kruskal time value as Bob when Bob hits the singularity and I could draw it differently such that Bob hits the singularity at some Kruskal time but Alice is at some greater Kruskal time then?

That can tempt us into thinking that the time coordinates of events off that worldline have some special significance to Alice. They don’t.

And because I could draw two different Kruskal-Szekeres coordinate systems like I described above the coordinate time has not much significance to Alice even though she synched her proper time to this coordinate time. Because I could just draw it a different way and Alice could assume that now at this coordinate time which is her proper time Bob met the singularity but she could also say he didn't. And neither statement would be valid nor wrong. She can say anything. And this arises due to spacelike separation caused by the event horizon.

 

[PeroK]

Just to sum everything up and check if my understanding is right:

Let's start at the beginning, with good old projectile motion. We have a projectile that follows some parabolic path. That's the mathematical model. That's one part of physics: assume some initial conditions, apply the laws of physics and describe a trajectory mathematically. Now, of course, you want to test your theory. So, you do some experiments and, within reason, you can conclude that the projectile really did follow that trajectory. It's the same with the motion of the planets round the solar system.

In any case, we get used to the idea that one "observer" can experimentally confirm the entire mathematical model.

Now GR comes along and we find the concept of an event horizon. The first part of doing physics is the same: we develop the laws of physics and apply them to any given scenario. But, we find that a given observer cannot necessarily (directly and experimentally) confirm the entire trajectory of a particle. Initially we may find this unsettling and try to construct a philosophical aspect of the theory to make up for the loss of a global observational capability (if I can coin that phrase).

Things that cannot be directly observed can only be experimentally inferred from indirect observations. It's true that we have to rely on the theory to a greater extent and we have to be more wary that the theory might be wrong. For example, we cannot be certain of what happens beyond the EH of a black hole. As students of modern physics, we have to live with this. We have to live with the vagaries of Quantum Mechanics. And the modern mathematician has to live with Goedel's Incompleteness Theorem.

Alice cannot (even theoretically) directly observe every event along the worldline of a particle falling into a black hole. Okay. The physics goes on: unabashed and undeterred, as it were!

 

[Nugatory]

And those two spacelike events may be both assigned a different coordinate time? This possibility arises because of spacelike separation and wouldn't be the case for timelike separated events?

If they are timelike-separated they will always be assigned different coordinate times. It's not possible for them to be the same.

I'm asking because I though what you're saying only applies to reference frames not to a global coordinate system that's used to map the whole spacetime like Kruskal-Szekeres where reference frames would be placed in.

A reference frame is a rule for assigning cooordinates to events. Kruskal coordinates cover the entire Schwarzschild spacetime, Schwarzschild coordinates cover the spacetime outside the event horizon, Minkowski coordinates (which is what you mean when you say "reference frame" here) only cover a small regions of spacetime around a free-falling observer. All of these are conventions for assigning labels to events; if two events are assigned the same time coordinate it's because of the convention we chose to do the assigning.

Now an important fact appears that I could draw the Kruskal-Szekeres coordinate system in such a way that in one Alice would be placed at the same kruskal time value as Bob when Bob hits the singularity and I could draw it differently such that Bob hits the singularity at some Kruskal time but Alice is at some greater Kruskal time then?

There are mathematical difficulties here because "Bob hits the singularity" is not an event. However, "Bob crosses the event horizon" is an event and raises all the same issues, so let's consider that instead.
If Alice-one and Alice-two are using the same coordinate system to label events they will of course attach the same time and space coordinates to all events, including Bob crossing the horizon. However, the time coordinate assigned to the event "This point on Alice-one/two's worldline" may not be equal to the time displayed by their clock at that event.

And this arises due to spacelike separation caused by the event horizon.

The event horizon doesn't cause spacelike separation. Events on the same side of the horizon may be spacelike separated or not. The event horizon does mean that the future light cone of events at and inside the horizon will not include any region of spacetime outside the horizon.

 

[PeterDonis]

"Bob hits the singularity" is not an event.

Only because the singularity itself is not part of spacetime. But you can view it as a set of limit points, and there will be one unique such limit point on the locus $r = 0$ that Bob's worldline approaches. That point can be viewed as the event "Bob hits the singularity". In Kruskal coordinates this becomes even easier because the locus $r = 0$ is right there on the chart and you can pick out the particular point where Bob's worldline hits it.

I could draw the Kruskal-Szekeres coordinate system in such a way that in one Alice would be placed at the same kruskal time value as Bob when Bob hits the singularity and I could draw it differently such that Bob hits the singularity at some Kruskal time but Alice is at some greater Kruskal time then?

You're getting yourself confused here. If you're using Kruskal coordinates, Kruskal time defines "when" things happen according to those coordinates. Once you've picked out some event on Bob's worldline, whether it's the event where he crosses the horizon or the event where he hits the singularity (defined as a limit point per my statements above), there will be some event on Alice's worldline that has that same Kruskal coordinate time. Asking whether that event is "where Alice is when Bob experiences event X" is pointless; you've defined "where Alice is" by picking the coordinates.

Of course coordinates have no physical meaning, and there's no actual observable that tells you "where Alice is" when something particular happens to Bob at a spacelike separated event. But that doesn't mean there is some other "true" answer to "where Alice is" that's different from the answer Kruskal coordinates (or whatever coordinates you've chosen) give you. It means there is no "true" answer at all; asking "where Alice is" when something particular happens to Bob is a meaningless question. Only questions involving invariants--things that are independent of the coordinates you choose--are meaningful questions.

For example, you could pick some event on Alice's worldline and ask, can Alice send a light signal to Bob from that event, that will reach Bob before he hits the singularity (or crosses the horizon)? That question has an invariant answer (and the Kruskal diagram makes it easier to find it because light signals travel on 45 degree lines, so it's easy to see where the light cone boundaries are).

 

[Nugatory]

Only because the singularity itself is not part of spacetime. But you can view it as a set of limit points, and there will be one unique such limit point on the locus r=0 that Bob's worldline approaches. That point can be viewed as the event "Bob hits the singularity". In Kruskal coordinates this becomes even easier because the locus r=0 is right there on the chart and you can pick out the particular point where Bob's worldline hits it.

Yes. I didn't want to go there (instead alluding to vague "mathematical difficulties") because the horizon crossing event works as well for pedagogical purposes and because no opportunity to disabuse people of the idea that the singularity is a point in space should ever be passed up.

 

[Bellaella]

Thank you everyone for your input. I understand everything you guys said except one single sentence is still confusing me which is the following:

However, the time coordinate assigned to the event "This point on Alice-one/two's worldline" may not be equal to the time displayed by their clock at that event.

Why may not the time coordinate assigned to an event be equal to the time displayed by their clock at that event? After all Alice has synchronized her clock with coordinate time.

I thought about it yesterday and what I suspect is that my mistake lies in the falsely assumption that what applies to the Schwarzschild coordinates also applies to the Kruskal coordinates. I read that at a "far away distance" so where r is infinity Alice can sync her clock to the time coordinate and her proper time will be equal the coordinate time on Alice's whole worldine in Schwarzschild coordinates. I assumed this is also valid for the Kruskal coordinates "just because" it's valid for the Schwarzschild ones which is wrong I think. So I thought after Alice synchronizing her clock with the Kruskal time the coordinate time will always be the proper time just as with Schwarzschild. It appears as if some shady stuff happens to the coordinates after a coordinate transformation from Schwarzschild to Kruskal so that the proper time can be the coordinate time at the moment she synchronizes it but after some time passes for some reason it (definitely?) won't because that's how the Kruskal coordinates work.
Is this reasoning right and is this the reason you wrote that Alices proper time may not be equal to kruskal time?

 

[Dale]

Given a timelike worldline and a coordinate time, it is always possible to transform the coordinate time to match the proper time along the worldline. This keeps the meaning of simultaneity from the original coordinate system, but adapts it to the chosen worldline.

I believe that @Nugatory is just saying that without such a transformation there is no guarantee that a particular coordinate time is adapted to a specific worldline.

 

[Bellaella]

@PeterDonis @Nugatory @PeroK @Ibix @Dale Thank you so much for bearing with me and answering my questions even though sometimes they weren't entirely physical

I liked all your posts because each one helped me understand the problem better. I really appreciate it!

You can close the thread now!

 

[PeterDonis]

Alice has synchronized her clock with coordinate time.

Not if you're using Kruskal coordinates. Kruskal coordinate time doesn't match either Alice's or Bob's clock time.

 

[PeterDonis]

what applies to the Schwarzschild coordinates also applies to the Kruskal coordinates.

You're correct that this assumption is false.

 

[PAllen]

Given a timelike worldline and a coordinate time, it is always possible to transform the coordinate time to match the proper time along the worldline. This keeps the meaning of simultaneity from the original coordinate system, but adapts it to the chosen worldline.

I believe that @Nugatory is just saying that without such a transformation there is no guarantee that a particular coordinate time is adapted to a specific worldline.

Right, but of course doing so may lead to less spacetime coverage than the original coordinate system. For example, in Kruskal spacetime, all radial free world lines are either future incomplete, past incomplete, or both. So, adapting to any of these will 'lose' part of the coverage of Kruskal coordinates.