Has information been lost to a black hole?

[rjbeery]

[I originally posted this in Astronomy and Astrophysics but it didn't get much traction. Perhaps folks in this forum would have something to contribute...]

I'd like to understand if and when information is ever actually lost in a black hole; specifically, I'd like to analyze the statement:

Is there information, which existed in the past, that is theoretically unavailable to external observers today due to falling through the event horizon of a black hole?

I'd like to restrict this thread to GR and SR only (no QM).

I'm open to suggestions on how to analyze this question, but my thoughts are to follow. To the external observer an infalling object is never lost. He could continue to study such an object for all time, taking readings and measurements of the object, albeit from an asymptotically redshifted and time dilated view of it. The external observer could adjust his measurements to account for such redshifting and time dilation and record perfectly accurate data (with perfect instrumentation).

Does this answer the question? There does seem to be a view in the community that reality contains some sort of a physical disconnect between what the observer sees and what has "really happened". How can we probe this? One thought I had was to launch a mirror towards the event horizon, and let the external observer watch his own clock in that mirror. It seems to me that if there is a point of last communication (from the perspective of the mirror) then there would be a terminating time T after which the external observer could no longer see his own clock. It seems plausible, if not completely convincing, that the observer could proclaim information has been lost at time T.

From Reflections on Relativity:

Having discussed the prospects for hovering near a black hole, let's review the process by which an object may actually fall through an event horizon. If we program a space probe to fall freely until reaching some randomly selected point outside the horizon and then accelerate back out along a symmetrical outward path, there is no finite limit on how far into the future the probe might return. This sometimes strikes people as paradoxical, because it implies that the in-falling probe must, in some sense, pass through all of external time before crossing the horizon, and in fact it does, if by "time" we mean the extrapolated surfaces of simultaneity for an external observer. However, those surfaces are not well-behaved in the vicinity of a black hole. It's helpful to look at a drawing like this:

This illustrates schematically how the analytically continued surfaces of simultaneity for external observers are arranged outside the event horizon of a black hole, and how the in-falling object's worldline crosses (intersects with) every timeslice of the outside world prior to entering a region beyond the last outside timeslice.

If it's true that there exists no time T on an external clock which cannot be observed from that clock's location, after having been reflected on an infalling mirror, then I believe the answer to the question in this thread is no and information has never yet been lost to a theoretical event horizon. However it's supposedly a net redshifting of what the infalling mirror sees of the outside world when SR and GR effects are considered, and I don't believe that jibes with the Reflections on Relativity graph. The infalling worldline is finite and the book indicates that each point on that worldline has an extrapolated surface of simultaneity for all points on the external observer's infinite worldline. In order to fit an infinite number of points on to a finite length I don't see how a blueshifting is not required.

So does anyone have the answer? What I'd really like is the calculation for "when" an external observer could no longer see himself in the infalling mirror.

Thoughts?

 

[bcrowell]

By your definition, there will be a time T on the external observer's clock at which it will become impossible for him to continue observing his own reflection. This will happen because the redshift of the reflected light will become so great that the light becomes impossible to detect.

Note that your definition is observer-dependent. For example, if observer A jumps into a black hole, and observer B into a different black hole, then information about each is never going to be available to the other. This is an absolute thing (not a difficulty with excessive redshifts) and independent of the method of observation.

 

[PeterDonis]

To the external observer an infalling object is never lost. He could continue to study such an object for all time, taking readings and measurements of the object, albeit from an asymptotically redshifted and time dilated view of it.

The external observer can continue to study a finite portion of the history of the object in this way; but only a finite portion--the portion of its history that lies above the horizon. As the light signals from the object become more and more redshifted and time dilated, they also cover shorter and shorter segments of the object's history; and when you sum up the amount of the object's history contained in the theoretically infinite number of segments, they add up to only a finite amount of history (proper time) of the object.

Does this answer the question?

No, because the real question is whether the finite portion of the object's history that is accessible to the external observer is all that there is. The answer to that question is unequivocal: no, it isn't. There is a further portion of the object's history, containing what happens to it at and below the black hole's horizon, which is forever inaccessible to the external observer. So the correct answer to the question you pose is yes.

It seems to me that if there is a point of last communication (from the perspective of the mirror) then there would be a terminating time T after which the external observer could no longer see his own clock.

That's correct: the external observer can only see, by reflection, readings on his own clock up to a particular time. Readings after that time will never come back to him by reflection, because the mirror will have fallen below the horizon and the light reflected off it will never come back out.

If it's true that there exists no time T on an external clock which cannot be observed from that clock's location

You are using vague ordinary language terminology which is particularly ill adapted for this scenario. Let me rephrase what the diagram you refer to is actually telling you (and more important, what it is not telling you), in language which will make it clearer what is going on:

Every event on the external clock's worldline has a time T associated with it. For every such event, we can draw a spacelike curve that passes through that event and also intersects the worldline of the infalling mirror, at some event above the hole's horizon. We can label that curve with the time T of the event where it intersects the external clock's worldline, and thereby label the corresponding event on the infalling mirror's worldline with the same time T.

However, now suppose that the external clock emits an ingoing light signal towards the mirror at some time T. This light signal does not travel on a spacelike curve; it travels on a null curve. And we cannot deduce, from the fact that there is an event on the infalling mirror's worldline labeled with the same time T that is above the horizon, that the light signal emitted from the external clock at time T will reach the infalling mirror while the mirror is still above the horizon. So we cannot deduce that every time T on the external clock's worldline will be visible to an observer at the external clock by reflection in the mirror. So the diagram you refer to does not show that no information is lost.

In order to fit an infinite number of points on to a finite length I don't see how a blueshifting is not required.

The labeling of events on the infalling mirror's worldline using spacelike curves, as described above, has nothing to do with blueshifting or redshifting of light signals. Those travel on null curves, not spacelike ones, as described above. And only a finite "length" of null curves from the external clock will intersect the infalling mirror's worldline while it is above the horizon, as above. So no fitting of an infinite length into a finite length is required.

(In fact, the actual math shows that, to an observer falling in with the mirror, the incoming light signals from the external clock are redshifted, not blueshifted. The blueshift due to the decrease in altitude is more than compensated for by the redshift due to the mirror's inward velocity.)

 

[rjbeery]

By your definition, there will be a time T on the external observer's clock at which it will become impossible for him to continue observing his own reflection. This will happen because the redshift of the reflected light will become so great that the light becomes impossible to detect.

Note that your definition is observer-dependent. For example, if observer A jumps into a black hole, and observer B into a different black hole, then information about each is never going to be available to the other. This is an absolute thing (not a difficulty with excessive redshifts) and independent of the method of observation.

Hi bcrowell, is the time T a definite time dependent on relative starting distances from the event horizon (and if so, how might I calculate that?) or are you saying that T is loosely bound by practical considerations of infinite redshifting? Please note that I mentioned "perfect instrumentation".

No, because the real question is whether the finite portion of the object's history that is accessible to the external observer is all that there is. The answer to that question is unequivocal: no, it isn't.

Hi PeterDonis, I think you're perhaps jumping ahead. I worded the question in this manner for a reason:

Is there information, which existed in the past, that is theoretically unavailable to external observers today due to falling through the event horizon of a black hole?

I'm not paying attention to infalling perspectives here because I restrict this to external observers (today). Your response was very well written and I completely appreciate what you're saying in regards to the difference between spacelike and null curves. This was my intuition because I was aware that the infaller sees a net redshifting of the outside world as he approaches the event horizon. I do have the formula for that redshifting, but I would like the formula for calculating the "termination point" after which an external observer could no longer see his own clock; is there any chance that you are familiar with this?

 

[PeterDonis]

I'm not paying attention to infalling perspectives here because I restrict this to external observers (today).

I wasn't talking about an "infalling perspective". The fact that only a finite portion of the infalling mirror's history will ever be visible to the external observer, and that there is a further portion of the infalling mirror's history that happens at and below the horizon and is therefore forever inaccessible to the external observer, is an invariant fact, independent of any "perspective". In other words, the answer to your question is "yes, there is information that is unavailable to external observers because it falls through the horizon of a black hole", and that answer is independent of any "perspective".

I would like the formula for calculating the "termination point" after which an external observer could no longer see his own clock; is there any chance that you are familiar with this?

I haven't seen this calculated explicitly in the literature. The calculation of how much proper time elapses by the infaller's clock before he reaches the horizon is a well known standard calculation that is often set as a homework problem (it's in MTW as an exercise, IIRC); but the calculation of how much time elapses by the external observer's clock before he can no longer send a light signal that will reach the infaller before he crosses the horizon is, as far as I can tell, not. I'll take a shot at working it out myself, but it might take some time.

 

[PeterDonis]

I'll take a shot at working it out myself

Here is a quick and dirty way of doing it. The answer comes out somewhat messy; I don't know if there is any cleaner way to express it.

I use Painleve coordinates because they cover the region at and inside the horizon with no issues. In these coordinates, an object that falls in from rest at infinity has coordinate velocity $dr / dt = \sqrt{2M / r}$ (I'm using units in which $G = c = 1$ throughout). (Note that this is not exactly the desired initial condition; this object will have a nonzero inward velocity at the radius $R$ where the external observer is. But if $R >> 2M$, this inward velocity will be small and won't much affect the answer, and the math is easier the way we're doing it here.) So the Painleve coordinate time it will take for the object (the mirror in this case) to fall from radius $R$ to the horizon at $r = 2M$ will be

$$
T_\text{o} = \int_{2M}^R \sqrt{\frac{r}{2M}} dr = \frac{2}{3} \left[ \sqrt{\frac{R^3}{2M}} - 2M \right]
$$

(Note that Painleve coordinate time is the same as proper time for the infalling object, so the above is also the time that will elapse on the mirror's clock.)

The coordinate velocity of a radially ingoing light ray in Painleve coordinates is $dr / dt = 1 + \sqrt{2M / r}$, so the Painleve coordinate time it will take for an ingoing light ray to go from radius $R$ to the horizon will be

$$
T_\text{l} = \int_{2M}^R \frac{\sqrt{r}}{\sqrt{r} + \sqrt{2M}} dr = R + 2M - 2 \sqrt{2MR} + 4M \ln \left[ \frac{1}{2} \left( \sqrt{\frac{R}{2M}} + 1 \right) \right]
$$

If we suppose that the ingoing light ray and the infalling object meet at the horizon, then the Painleve coordinate time between the mirror falling inward from radius $R$ and the light ray being emitted from radius $R$ will be the difference of the two times above, or

$$
T_\text{o} - T_\text{l} = R \left( \frac{2}{3} \sqrt{\frac{R}{2M}} - 1 \right) + 2 \sqrt{2MR} - M \left[ \frac{10}{3} - 4 \ln \frac{1}{2} \left( \sqrt{\frac{R}{2M}} + 1 \right) \right]
$$

The actual proper time elapsed on the external observer's clock will be the above times the time dilation factor $\sqrt{1 - 2M / R}$ (which will be almost equal to 1 if $R >> 2M$, so it will make very little difference). The external observer must emit a light signal before that much time has elapsed on his clock after the mirror falls inward from his radius. if he wants to see that signal reflected in the mirror. After that, any light signal he emits will only reach the mirror after it has reached or fallen below the horizon, so he will never see it reflected back.

 

[rjbeery]

This is brilliant, PeterDonis, thanks for taking the time. Let me study it...

 

[bcrowell]

Please note that I mentioned "perfect instrumentation".

You need to distinguish between perfect and impossible.

 

[rjbeery]

You need to distinguish between perfect and impossible.

Yep, that was the intent of my clumsily worded follow-up question. It sounds like, and apparently looks like (from PeterDonis' analysis), T is definitely finite from a given R.

 

[rjbeery]

So I've continued thinking about this. It's problematic to claim that that an object doesn't exist just because its information is "lost" to an observer, and an obvious example is the universe outside of our Hubble volume; Observer A and Observer C may reside outside of the other's respective Hubble volume but there could be a third-party B who can "see" both of them. I need to make it clear that I'm not exploring existence in any absolute sense in this thread, only existence from the perspective of all outside observers.

Analogously, I think it isn't informative to conclude anything from the perspective of a single observer when we're trying to decide if information has "ever" been lost to a black hole. All outside perspectives need to be included before we can make such a declaration. I'm not yet sure the best way to do this but I suspect but one possibility is to analyze what an observer hovering just above the event horizon might claim about the infalling mirror.

 

[PeterDonis]

I need to make it clear that I'm not exploring existence in any absolute sense in this thread, only existence from the perspective of all outside observers.

That was certainly clear to me from your OP, and my responses have been made with that in mind.

All outside perspectives need to be included before we can make such a declaration.

The responses I have given are valid for all outside observers. For example, the equations I gave in post #6 are valid for any $R > 2M$, i.e., for any observer outside the horizon. The numerical values for the times are different, of course, but nothing I said depends on the times having any particular numerical value; all I claimed was that the times are finite for any finite $R$, and the equations clearly show that they are.

one possibility is to analyze what an observer hovering just above the event horizon might claim about the infalling mirror.

Plug a value of $R$ that is only a little greater than $2M$ into the equations I gave in post #6.