## Sunday, January 11, 2009 ... //

### Reincarnation of an infalling observer

In this essay, I would like to talk about physics and perceptions inside black holes.

The picture of reincarnation above was sketched by Prof Krishna :-). Note that the death and the birth are two faces of the same object, namely the infinity, through which they are connected.

While our treatment will try to be more serious than what you have seen so far, similar spiritual considerations will actually be unexpectedly important, especially when we get closer to the singularity. Why? Well, we should start with the question:

Why is physics of the black hole interior subjective?

The adjective "subjective" is probably not the most rational term we can use in this context. But my point is that from the viewpoint of billions of people who live outside a black hole, all the events inside the black hole may be believed not to exist at all. How is it possible?

If you look into one of the articles about similar issues, for example
Why can anything ever fall into a black hole?
you will find the following picture:

It is the so-called Penrose (causal) diagram of the spacetime that includes an electrically neutral black hole, the so-called Schwarzschild black hole. The absolute distances on the diagram can't be taken seriously but the "angles" can.

Repeating some features of the diagram

The "mostly vertical" directions are time-like while the "mostly horizontal" directions are space-like.

Besides the two (or one plus one) dimensions - two coordinates that you could call "r" and "t" (although the exact reparametrization is not canonically determined) - that you see on the picture, each point describes a two-dimensional sphere of some radius (that doesn't necessarily agree with any particular "r" coordinate that you chose on the two-dimensional diagram). Light rays propagate along diagonal lines on the diagram, under the 45° angle. On the other hand, massive objects are not allowed to surpass the speed of light which is why they move along "mostly vertical" world lines only.

If you believe me that the picture above can be proven to describe a black hole that is created out of a collapsing star in the past (the bottom of the diagram) and that will evaporate after a very long time in the future (the top of the diagram), you will surely get eager to derive the consequences of the picture.

The main lesson is that if your world line ever appears in the red triangle, the "black hole interior", you will never be able to return to the "infinite future" in the Northeast corner of the diagram. You're doomed, much like the point on the collapsing stellar surface, represented by the yellow line. Get ready for a singularity massage.

On the other hand, if you want to save your life for much longer a time, you have to turn your jets on and accelerate away from the black hole. Your world line will then resemble the orange line, the points where the internal two-sphere has the area corresponding to the "Schwarzschild radius".

Although your possible adventures in this spacetime can differ by many quantitative features, the orange line and the yellow line are the only two qualitatively different life journeys of massive observers. You are either swallowed by the black hole, like the yellow stellar surface, and then you inevitably hit the wiggly "singularity" at the left side of the top (future) of the picture, or you save your life and your future will approach the Northeastern "infinite future" where events resemble a normal spacetime without a black hole.

The infalling observers can't speak

It's important to realize that all the people inside the black hole (red triangle) will be unable to communicate their experience, measurements, and perceptions to the civilizations that live outside the black hole and that will survive the black hole. If the hapless astronauts wanted to send signals, these signals would have to propagate along "mostly horizontal" (i.e. space-like) world lines. But that's forbidden in general relativity. This restriction generalizes the usual restriction from special relativity that things can't move faster than light.

It is generally expected that normal observers who fall into the black hole - the infalling observers - will not experience any special things until the moment when they approach the singularity at the top. Once they're close to the singularity, the gravitational field (and especially tidal forces) becomes very strong and it eventually kills them.

But when the infalling observer just crosses the event horizon - the diagonal boundary of the red triangle - he shouldn't feel anything special. Why? Well, because the precise location of the event horizon shouldn't even be known at the moment. Note that if you want to determine where the event horizon is located, you must draw the whole spacetime, and sketch a 45° diagonal line back to the past, starting from the corner that will ultimately become the right vertex of the red triangle.

But the people who just crossed the horizon can't know the full future - and they can't know the location of the right vertex of the triangle which is necessary to reconstruct the locus of the event horizon. They can't even know whether or not someone will prevent the birth of a new black hole. So they can't know that the place that their spaceship visited a millisecond ago would be identified as the black hole horizon by the future historians. They can't know it, not even in principle. That's why they can't see any unusual phenomena there: otherwise they would know what happened which would be equivalent to seeing into the future.

You got it, didn't you? ;-)

So whatever complete description of the black hole interior a quantum theory of gravity gives you, it should agree with the expectation of classical general relativity, explained above, that macroscopic infalling observers shouldn't detect any special phenomena right after the moment when they cross the horizon.

However, as the diagram shows, by crossing the event horizon, they are doomed in the sense that the observers in the large Universe outside the black hole will never learn about the infalling observers' feelings after they crossed the horizon. The rules of relativity make such a communication impossible.

The observers outside the black hole would only see the part of the infalling observers' life before they crossed the event horizon - the moments that are extremely close to the judgment day would be slowed down (and red-shifted) by an extremely high factor. But they won't learn what happened to the poor guys afterwords. On the other hand, the poor guys don't even know that something bad has just happened to them. They will only learn about their inevitable sad fate a few moments later, when they approach the singularity.

A few words from the external viewpoint

The exact, well-understood physics of quantum gravity and string theory almost always adopts the perspective of the external observers who avoid the black hole. They are going to live for a very long time - in an asymptotically flat space, they could in principle live forever. So they can make very accurate measurements, too.

The interior of the black hole doesn't directly influence the measurements by the external observers. But all objects and events outside the horizon - even those that are very close to the horizon - can influence and do influence the measurements performed by the external observers.

What do the external observers think about the black hole interior? For some of them, the black hole is black magic. For another group, the black hole is just a politically incorrect black ho that should become a white ho in order to sc*rew the whites.

For more rational observers outside the black hole, the black hole is just another black box. It behaves according to some rules. They don't have to look directly into its interior, especially because it seems forbidden by the classical rules of causality. Instead, they can imagine that the information from all the points of the event horizon at one moment is being scrambled and evolved into the information at the event horizon at a future moment.

These rules of evolution are arguably very complicated but if you're only interested in the macroscopic character of such evolution, semiclassical general relativity is enough. You will find out that the black hole emits structureless thermal radiation at a temperature you can calculate (to be proportional to the gravitational acceleration at the horizon).

If you want to know more accurate details than this semiclassical description, you need to create your own Hilbert space of the black hole microstates and study its evolution. But at this moment, it is probably not too useful to imagine that the microstates are associated with particular fields or properties located at well-defined points in space. The external observer may imagine that he deals with an abstract Hilbert space of microstates evolving according to an abstract Hamiltonian whose matrix elements may be in principle known. However, the states in the Hilbert space don't have to be given any explicit geometric interpretation - even though the fuzzball paradigm is trying to realize nothing less than this difficult task.

Let me summarize: the external observer may know general relativity and speculate about the fate of the infalling observer. But the external observer doesn't really have to care. And he may decide that the fate of the infalling observer is not science because it can't be measured (by those who survive). The rules of quantum gravity, as viewed from the external viewpoint, are qualitatively identical to the rules of normal particle physics. Black holes are just some new excited, very long-lived resonances, after all. That's why the string-theoretical (i.e. essentially particle-physics) description of the black holes has been so immensely successful and accurate.

But the infalling observer still exists, doesn't he? He will have some feelings until he is killed by the singularity (or by his crewmate), won't he? The positivist external observers may declare the infalling observers "legally dead" already at the moment when they crossed the event horizon - because their continuing life will never influence anyone in the "healthy" civilization outside the black hole (or demonstrably violate the treaties with others).

Can you resist to ask what the infalling observers actually feel and whether they can study physics and accurately compare their measurements and a theory? I can't.

What the infalling observers cannot do

You should realize that the infalling observers' ability to perform accurate measurements is limited because they are going to be killed after some time. For example, if they measure the frequency of a wave or a photon etc., the frequency can't be measured arbitrarily accurately.

Because the remaining lifetime before the hapless astronaut is killed is comparable to the black hole radius, "R/c" (where "c" is the speed of light), the inaccuracy of the measured frequency is inevitably at least "c/R" - also comparable to the typical frequency of the Hawking radiation (at least for highly non-extremal black holes). You can only measure frequencies exactly if you measure them for an infinitely long time. The unlucky astronaut can't do so.

There are other, related limitations of the infalling astronaut who tries to become an experimental physicist in the final moments or weeks of his life, depending on the size of the black hole. From an operational viewpoint, physics doesn't have to try to predict quantities more accurately than how they can be measured. Because the frequencies can't be measured with an error smaller than "c/R" or so, physics doesn't have to predict them more accurately.

But in fact, it could predict them more accurately. If people are unable to measure "something", it doesn't necessarily mean that the "something" doesn't objectively exist. In this particular case, the constraints preventing the people from doing exact physics seem to be "fundamental" properties of the Universe. But I can't offer you any proof that a more accurate prediction is impossible in theory. To decide this question, we would actually have to own the full theory appropriate for the infalling astronaut. And I don't claim to have one.

(This contrasts with the situation in ordinary quantum mechanics where I can show that our theory can't predict accurate values of "x" and "p" at the same time - because we actually know what quantum mechanics is and why it implies the uncertainty principle.)

From a pragmatic viewpoint, it is probably enough to imagine that the infalling observer spends the final moments of his life (inside the black hole) as a semiclassical perturbation on a well-defined black hole geometry calculated essentially according to the classical rules. It's clear that I don't quite know what I mean by the classical rules because classical fields can still be macroscopically influenced by seemingly tiny quantum phenomena (such as unexpected firings of a neural cell) but at some qualitative level, we could be satisfied with that. The doomed astronaut will think about different things than accuracy, after all.

Forget about the black hole microstates. The infalling observer clearly doesn't have the access to all the detailed degrees of freedom that could distinguish all the black hole microstates. He can't do enough measurements that would distinguish them and he probably can't access them even because of causal constraints. If the infalling observer uses some quantum microstates anyway, it is likely that each of them can be associated with some microstates outside the black hole.

But this "linking" is probably not quite a tensor product of two Hilbert spaces because the values of all the degrees of freedom inside the black hole are probably encoded, in a very complicated way, in some of the degrees of freedom naturally associated with the fields and objects outside the black hole. That's how I understand black hole complementarity. For example, multiply five electrons' x-coordinates outside the black hole that are the closest ones to the black hole and read the resulting product from the 385th decimal place. You will get the coordinate of a proton in the astronaut's eye. ;-)

I exaggerate a bit but you get the point. The degrees of freedom inside the black hole are not really new or commuting with or independent from the external degrees of freedom (complementarity!) but the way how they're encoded in the physics outside is so complicated that you may imagine that they're independent - and the semiclassical approximation actually makes this assumption.

Avoiding the doom at the singularity

As we're approaching the interior and ever deeper layers of it, we're getting more religious. The astronaut knows that the death is approaching. He may convert to Christianity and recall some of the moments in the childhood when he almost did it for the first time.

I suppose that this newly born Christian astronaut assumes that Jesus Christ or one of his servants whose name I forgot is collecting the souls from all places of the regular spacetime where people die as well as from the black hole singularities, in order to directly tunnel them into the Hell or the Heaven.

That's not a big deal for an omnipotent and omnipresent God. Even from the viewpoint of the limited infalling observer's physics, the singularity and/or the salvation brings a discontinuity to the physical treatment of the information. But the omniscient God sees no problem here, either, and some quantum xerox paradoxes are probably avoided because God can beat the rules of maths, too.

However, the astronaut may also turn his attention to Buddhism that believes in rebirth. I mentioned that the degrees of freedom of the astronaut are encoded in subtle numbers describing the objects outside the black hole. Because the rule was so complicated, the existence of the link wasn't that important. But it may become important when the astronaut approaches the singularity, at least for a qualitative picture what happens.

The astronaut is going to die but the details about his state are already encoded in some field-theoretical degrees of freedom outside the horizon: they are being imprinted into the Hawking radiation. The information is not lost. The astronaut and everything else that is destroyed by the singularity is reincarnated into a subtle pattern of the Hawking radiation.

This almost certainly happens - I just described it by words that some zealous atheists may find controversial. But this kind of reincarnation doesn't look terribly optimistic because we don't believe that patterns in the Hawking radiation can have a decent life and sophisticated consciousness, do we?

Saving the infalling observer

Can we save the infalling observer in a more satisfactory form? To do so, we would have to replace the vicinity of the singularity by a new piece of spacetime. If the rules of classical general relativity are obeyed, at least approximately, it seems impossible to avoid the singularity or at least a near-singularity. Whether you like it or not, the curvature around the astronaut will inevitably jump to levels that mean a certain death. Quantum corrections almost certainly can't be large enough to contribute in time and save his biological life. The singularities will develop just like the singularity theorems dictate.

But let's forget about biology. We want to preserve at least something. At the singularity, the very time seems to terminate (which makes the singularity analogous to the Big Crunch). Is there any sense in which the time should continue?

This is a very subtle quesion. We may merge the singularity with the Big Bang - the initial singularity - of another Universe. But do we have the right to say that these two Universes should actually be connected through the black hole? It seems clear that the astronauts can't survive to testify that they came from the black hole of an older Universe.

Gluing two Universes at a singular point seems very problematic. In fact, I have doubts about the well-definedness of a similar "continuity of time" in the context of the vacuum bubbles in the eternal inflation, too. It makes sense to ask how quickly our Universe decays but it may be meaningless to imagine that the time in a newer Universe "continues" with the same time as we knew it in the paternal world. I have some quasi-technical arguments - based on the Lorentz invariance and the different definitions of space in various dual (T-dual etc.) descriptions of the same physics - that indicate that you shouldn't imagine that it's the "same time" in both Universes.

In fact, you can ask: why do you still think that as a conscious being, you're the same subject as you were in 2007? It is surely not due to the sodium atoms' in your brain being the same: they are being exchanged rapidly - and moreover, all sodium atoms in the world are strictly identical, so you couldn't use them to distinguish different people, anyway. Instead, it is due to some patterns in your brain that change slowly. If all the information changes too quickly in an object, the subject doesn't remember things and can't have the same perception of its identity as it had in the past. A degree of constancy is surely essential for consciousness to operate in the usual way. And I claim that some - although more modest - degree of constancy of physics is needed even for a continuous and continuing time coordinate to make sense. If you can't find any degrees of freedom that are approximately and nontrivially continuous as a function of your time at a critical moment, you shouldn't be talking about a continuing time.

On the other hand, we also know some examples of physical phenomena that indicate that "full physics", if defined properly, could be completely smooth.

For example, when a conifold - a singular deformation e.g. of a Calabi-Yau manifold - truly degenerates, there are spheres that shrink to zero. The Riemann tensor blows up near the singular point of the conifold geometry. But that's not a problem. There are new light degrees of freedom - carried by branes wrapped around the shrunk spheres - and when you add these degrees of freedom, you will discover that the transition of the shape moduli through the seemingly "singular" point of the moduli space is actually smooth. The mirror description (as in mirror symmetry) looks smooth, after all.

Can we identify a kind of mirror symmetry that makes the transition through the singularity smooth but forces us to redefine the "spatial" degrees of freedom, much like when we change the topology at the conifold point? The answer is not obvious, at least not to me.

Reincarnation through mirror symmetry

If something like that is possible, it might make sense to talk about physics that doesn't terminate at the singularity but it is connected to the time in a new Universe. However, the link between these two Universes wouldn't be completely geometrical in character. In some sense, you would need to perform a kind of T-duality, or mirror symmetry, to switch to your new description.

Imagine that the convergence towards the singularity is analogous to a circular direction in spacetime whose radius shrinks to zero. As you know, there exists a new, T-dual description in which the size of the circle is infinite. So if the astronaut wants to save his soul in some unusual but decent way, he should adopt Buddhism and switch his degrees of freedom to the T-dual, or mirror-symmetric, description of physics. "He" will suddenly find himself in a very large, and possibly smooth, space.

And we should all wish him good luck.

The main reason why I believe that nothing like this can actually happen is that the entropy of the black hole is finite and it shouldn't allow one to switch into any new description that can in principle support an infinite-dimensional Hilbert space, as you would expect in a space with a non-compact dimension. So I propose to treat all the speculations above as somewhat sophisticated jokes and accept the hypothesis that the astronaut is going to die, whether he or we like it or not.

The physics of the external observers seems pretty well-defined and sharp while the physics of the infalling observer - and even the very questions that should be allowed to be asked - seem to be shrouded in chronic mystery. It would be great to uncover more of this mystery. But we should always realize that some of the questions could be asking for too much and there might actually exist no answers to some specific questions, not even in principle, because of the inherently inaccurate nature of the finite-time physics of a doomed astronaut.

And that's the memo.