An observer passing the event horizon of a black hole

[DennisN]

TL;DR Summary

Are my 3 posted scenarios regarding an observer passing a black hole event horizon reasonable or incorrect?

Hi all, I've just read this entire thread and watched the videos about black holes posted by @PeroK, which I liked very much (thanks @PeroK! ).

I am not particularly well aquainted with GR and my questions are concerning the often mentioned statement that an observer that passes the event horizon won't notice it.

But is this really correct?

I have recently thought about different scenarios and I have three particular ones below that I would like others to consider. First of all, let's assume the black hole is massive enough that significant tidal forces (spaghettification) will not be of concern before or at the event horizon, and furthermore that there is no firewall at the event horizon.

Scenario 1: An observer and his hand

Very simple: An observer is falling quite slowly towards the event horizon with his arm and hand stretched out before him towards the horizon. When his hand suddenly passes the event horizon, does it disappear completely from his view, since the light can't reach his eyes? And as his arm continues to pass the event horizon, does the arm in the same way gradually disappear from his view?

Scenario 2: An observer in a long container (it's an extension of scenario 1, see attached picture below).

An observer is sitting at the right end of a long container that is passing through the event horizon at the left. Along the length of the container are 10 candle lights at different positions. Does the observer see one light after another from 1 to 10 get gradually dimmer (and maybe briefly distorted?) and then completely disappear from his view?
(in the picture below candle light 1 and 2 has passed the event horizon and cannot thus be seen by the observer).

Scenario 3: Invisibility beyond the event horizon

Beyond the event horizon it is my understanding that all paths lead to the singularity, which is a point in time and not in space. Is it therefore correct to assume that an observer will become invisible to himself, i.e. he won't be able to see his own body and limbs, since there is no way for the light from these to reach his eyes?

Thanks for you consideration, and I look forward to hearing if my scenarios are reasonable or incorrect. Cheers!

 

[PeterDonis]

is this really correct?

As far as classical GR is concerned, yes. There are speculations that quantum effects might create locally observable effects at the horizon, but those are just speculations at this point.

When his hand suddenly passes the event horizon, it ought to disappear completely from his view, since the light can't reach his eyes.

Sure the light can reach his eyes. It just won't reach them until his eyes have fallen below the horizon. From a global standpoint, his hand, his eyes, and the light are all falling inward; the light just falls more slowly than the hand and eyes, so from the standpoint of the hand and eyes it seems to be moving "upward". But globally the light is falling inward too--its radial coordinate when the eyes see it will be smaller than its radial coordinate when the hand emits it.

The observer ought to see one light after another from 1 to 10 get gradually dimmer (and maybe briefly distorted?) and then completely disappear from his view

No; once again, since everything is falling inward, the observer simply won't see light emitted below the horizon until he himself is below the horizon. But he won't be able to tell from any of his local observations exactly when he crosses the horizon or which light signals he sees were emitted below the horizon.

The "light gets gradually dimmer and then fades altogether" picture is what an observer hovering at a fixed altitude outside the horizon will see from an infalling object that passes the horizon. But that observer is not free-falling inward; he is accelerating outward in order to maintain a fixed altitude.

Is it therefore correct to assume that an observer will become invisible to himself, i.e. he won't be able to see his own body and limbs, since there is no way for the light from these to reach his eyes?

No. You are incorrect that there is no way for the light to reach his eyes; see above.

Strictly speaking, once the observer is close enough to the singularity, it is possible that light emitted from one end of his body will hit the singularity before it reaches the other end of his body. But his body will be torn apart by tidal gravity and he will be killed before that happens, so he won't observe it.

 

[Dale]

Another way to think about it is to consider the infalling observer’s local inertial frame. In that frame the event horizon is moving at c. Light emitted by the hand after it passes the horizon cannot catch up to the horizon, but after the horizon passes the observer’s eyes the light from the hand will reach it as normal.

 

[DennisN]

Thanks a lot @PeterDonis ! I had to read your post a couple of times to understand it . I think I get it now, and I have a follow up question:

Sure the light can reach his eyes. It just won't reach them until his eyes have fallen below the horizon.

Does this mean that during the brief time between when the hand passes the horizon (t_h) and the eye passes the horizon (t_e), i.e. the time interval (t_e - t_h) the hand gets briefly invisible? Or am I thinking incorrectly again?

 

[Nugatory]

An observer is falling quite slowly...

By “falling quite slowly” do you mean free fall, or are you considering some force acting on the observer to slow their fall?

 

[DennisN]

By “falling quite slowly” do you mean free fall, or are you considering some force acting on the observer to slow their fall?

Hi Nugatory, I didn't consider any further specification. The reason I said "quite slowly" was to allow some time for the observer to be able to notice any possible change happening with his hand or candle light as it passes the event horizon.

 

[DennisN]

@PeterDonis , regarding scenario 3 I understand your reply about light being able to reach the observer within the event horizon. Hehe, I actually knew this before I read the other thread, but one of the videos posted there made me forget it and think wrongly about the inside of the event horizon. It was a brief misunderstanding of mine.

 

[Nugatory]

Does this mean that during the brief time between when the hand passes the horizon (t_h) and the eye passes the horizon (t_e), i.e. the time interval (t_e - t_h) the hand gets briefly invisible?

No. The light that leaves the hand as the hand passes the horizon reaches the eye as the eye passes through the horizon.

 

[PeterDonis]

Does this mean that during the brief time between when the hand passes the horizon (th) and the eye passes the horizon (te), i.e. the time interval (te - th) the hand gets briefly invisible?

No. If the hand emits a continuous stream of light signals, the eye will receive a continuous stream of light signals.

Light signals emitted by the hand above the horizon will be received by the eye at a slightly larger radial coordinate than that at which they were emitted; but the difference will get smaller as the hand gets closer to the horizon.

Light signals emitted by the hand exactly at the horizon will be received by the eye exactly at the horizon--the light is radially outgoing light so it just stays at the horizon and "waits" there for the eye to catch up to it.

Light signals emitted by the hand below the horizon will be received by the eye at a slightly smaller radial coordinate than that at which they were emitted; the difference will get larger as the hand falls further below the horizon.

 

[Ibix]

Does this mean that during the brief time between when the hand passes the horizon (th) and the eye passes the event horizon (te), i.e. the time interval (te - th) the hand gets briefly invisible? Or am I thinking incorrectly again?

Right now, as you are reading this post, you can't see your hand now - you see it as it was a nanosecond or two ago.

In your local inertial frame as you fell into a black hole, the event horizon would be a null surface passing up your arm at the speed of light - the same speed as the light coming from your fingertip. So the light emitted from your hand before it crossed the horizon reaches your eyes before the horizon reaches your eyes; light emitted as it crosses the horizon reaches you as you cross the horizon; light emitted after it crosses the horizon reaches your eye after your eye crosses the horizon.

Note that this local inertial frame is an approximation - we ignore the effects of curvature in a small enough region. This means we've approximated away the possibility of you turning on your rocket pack and escaping the hole. But once your hand is in the hole it cannot escape - so the thrust necessary for you to escape must literally be strong enough to rip your hand off. In that case, you'd never see light from your fingertips at or after they crossed the horizon - but that is not paradoxical, merely painful.

Edit: got interrupted, and therefore beaten to it multiple times, I see.

 

[DrGreg]

An observer is falling quite slowly towards the event horizon...

Just to spell out explicitly what several people have already implied, but you might not have noticed, when you fall through the horizon, it passes you at the speed of light (no matter how hard you try to accelerate away from it).

 

[DennisN]

Hi all, and thanks for all your replies! And I just wanted to say that I've got it with regards to my three scenarios. But black holes can sure get weird when you think about them, and I'm pretty sure this is not the last time they mess with my brain.

 

[pervect]

Scenario 1: An observer and his hand

Very simple: An observer is falling quite slowly towards the event horizon with his arm and hand stretched out before him towards the horizon. When his hand suddenly passes the event horizon, does it disappear completely from his view, since the light can't reach his eyes? And as his arm continues to pass the event horizon, does the arm in the same way gradually disappear from his view?

It's impossible for the observer to fall slowly through the event horizon. A ood way of looking at it is that the observer is standing still in his own frame. The event horizon approaches the observer at the speed of light. Only something moving at the speed of light can stay at the event horizon, so we can say formally that the event horizon is lightlike. The relative velocity between an something lightlike, and "an observer", who must be timelike, is always the speed of light.

The argument can't be reversed, beause a lightlike object can't be "an observer" in the usual sense. So the only possible candidate for an observer is an infalling object, and such an observer will always see the event horizon pass him at the speed of light.

Scenario 2: An observer in a long container (it's an extension of scenario 1, see attached picture below).

An observer is sitting at the right end of a long container that is passing through the event horizon at the left. Along the length of the container are 10 candle lights at different positions. Does the observer see one light after another from 1 to 10 get gradually dimmer (and maybe briefly distorted?) and then completely disappear from his view?
(in the picture below candle light 1 and 2 has passed the event horizon and cannot thus be seen by the observer).

If the container is long enough, it may or may not be possible that there would be some graviational redshifting due to tidal forces. Though if the observer observes the specific moment when the front of the box crossed the event horizon, he will observe it when he raches the event horizon himself, So I don't think there would be any tidal force induced redshift in that particular case, but there ould be in general. It'd probably be better to do a detaile calculation.

Scenario 3: Invisibility beyond the event horizon

Beyond the event horizon it is my understanding that all paths lead to the singularity, which is a point in time and not in space. Is it therefore correct to assume that an observer will become invisible to himself, i.e. he won't be able to see his own body and limbs, since there is no way for the light from these to reach his eyes?

No.