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How Tim Maudlin "solved" the information loss puzzle

I believe that I have encountered the name "Tim Maudlin" of a self-described "philosopher" before 2011 but Fall 2011 was the first season when I was first fully exposed to his staggering arrogance combined with his utter stupidity. As discussed in Tom Banks and anti-quantum zealots, Maudlin was the most combative troll in the comment section of a guest blog about the foundations of quantum mechanics written by my former PhD adviser on Sean Carroll's blog.

Maudlin's name has appeared in the following years several times. But I honestly don't remember anything special about this particular "Gentleman's" opinions about quantum mechanics. He is just another anti-quantum zealot who accepts classical physics as a fact and says all the wrong things that "therefore the world is surely nonlocal" and the stuff that the anti-quantum zealots share. Mr Maudlin, don't you think that if it were enough to be a worthless peabrain like you that only understands the rough basics of classical physics to solve all problems in modern physics, the physicists would have already noticed?

Well, his "answers" to all questions in quantum mechanics based on the dictum that only classical physics is allowed wasn't enough for him. He decided to address a famous puzzle in contemporary decades, the black hole information paradox, too. The result was the fresh paper (Information) Paradox Lost whose content is equivalent to the following sentences:

The final slice after a black hole evaporated isn't a Cauchy surface – because some timelike trajectories don't quite get there (they end in the singularity). That is why this late surface shouldn't be expected to hold the whole information about the spacetime. Some information got clearly lost in the singularity. My solution is so straightforward that I refuse to call this trivial thing a "paradox" and all people working on complementarity, ER=EPR etc. have been idiots.
Maudlin is a stuttering moron so he needs 25 pages of rubbish to convey this point. The pages are full of trivial introductions to some aspects of the black hole geometry, repetitions, and variations of the basic claims that theoretical physicist are idiots.

So has Maudlin given us the right answer to the questions about the information loss so that we may stop thinking about it? Well, he hasn't. His answer is simple but a slight problem with it is that for some two decades, we have known for certain that it is wrong. The information is not lost.

OK, let's discuss the question in some more detail. First of all, the information loss issues are often called a "paradox" because a "paradox" is a situation in which we may use our axioms to derive two conflicting answers to a question. In the mid 1970s and some following, the information in black holes led to a "paradox" because quantum mechanics apparently implied (and it really does!) that the evolution has to be unitary and the information had to be preserved. General relativity seemed to imply (and we know it's wrong today) that the information cannot get from the black hole interior, and is therefore lost once the black hole shrinks to zero size.

These two conflicting answers represent a contradiction and that's why this situation is known as the "black hole information paradox".

In Nature, there are ultimately no paradoxes – some of the conflicting answers are simply incorrect. A problem is that Tim Maudlin has picked the wrong one.

The Penrose diagram of the evaporating Schwarzschild black hole looks like this:

A star collapses gravitationally. You may see where its surface (yellow world line) is located at different times. Every point in the diagram represents a two-sphere which is possible due to the rotational symmetry. The area of the stellar surface is shrinking and at some moment, it unavoidably shrinks to zero (the proper area of the "hidden two-sphere" is zero for all points on the vertical left boundary of the diagram as well as the horizontal line with the teeth). That moment is depicted as the violet teeth, the singularity.

Some world lines, like the yellow one for the surface of the star, end in this singularity, so it seems that the information carried by the atoms on the stellar surface is lost there once these atoms are destroyed by the diverging curvature in this doomed and "no longer usable" portion of the spacetime. It turns out that the singularity is spacelike (horizontal) so the moments or events when different objects in the star experience the infinite curvature for the first time are spacelike-separated from each other in this case. That's related to the statement that the "time and space are interchanged" in the interior of this black hole.

The green area, the black hole exterior, looks rather regular, and contains trajectories of observers who have escaped the black hole and who weren't fatally affected by it. But some observers, those in the purple triangle – the black hole interior – don't even have a chance to escape the black hole anymore and return to the green area because they would need to move faster than light and it's forbidden by relativity.

Above the "singularity" teeth in the diagram, the black hole has already evaporated and on the late horizontal slices (cut a small triangle by a horizontal cut at the top of the diagram) which looks just like a slice in the empty Minkowski spacetime again, there should be all the information that the spacetime can carry.

Now, if you draw a horizontal (spacelike) slice \(\Sigma_1\) near the bottom of the diagram, it is a "Cauchy slice" (one that may be interpreted as "the whole spacetime at a single moment"), while a similar horizontal slice \(\Sigma_2\) near the top of the diagram is not a Cauchy slice. The reason is that a Cauchy surface is defined as
a surface that has exactly 1 point in the intersection with every timelike or null trajectory in the spacetime that cannot be extended.
The yellow world line of the stellar surface counts as one of the "timelike or null [causal] trajectories" which are "inextensible". It is inextensible because in the past, it reaches the end of the spacetime (which is an infinite proper time away, in this case), and in the future, it ends at the singularity so it cannot be extended, either.

Well, this yellow line is different from the "inextensible curves in the Minkowski spacetime". Those are infinite in both directions. In the black hole case, the yellow trajectory is inextensible because of the singularity. Such curves are sometimes called "incomplete inextensible curves" and the spacetime containing them – and/or the singularities which truncate them – are sometimes called "incomplete".

OK, so Maudlin's statement that the late surface isn't a Cauchy surface is technically true. But his "similar" statement sold as a "consequence", namely that it means that we shouldn't expect this late surface \(\Sigma_2\) after the black hole evaporated to carry the whole initial information, is false.


Just think about the situation with some common sense. In the past, in the very real Universe surrounding us, some tiny black holes have been randomly created by collisions of high-energy cosmic rays and they quickly already evaporated. Does it mean that the slices at the present should be considered incomplete – assumed not to know everything we should know? That would be pretty bad because the creation and evaporation of tiny black holes in the past is nearly unavoidable which means that it would be unavoidable that we can't describe any physics completely now.

But there's another reason to be convinced that the final surface must act as the complete storage of the spacetime's information. It's a reason that depends on quantum mechanics so anti-quantum zealot Mr Tim Maudlin has zero chance to understand it. But I am convinced that the average TRF reader is significantly smarter than Mr Tim Maudlin so let me mention it. Look at the diagram once again.

Imagine that there are the two horizontal slices \(\Sigma_1\) near the bottom and \(\Sigma_2\) at the top. There is some spacetime in between – with most of the black hole, the whole event horizon, the whole singularity etc. Do these aspects of the geometry exist? How sure can you be about the answer?

Well, according to quantum mechanics, things only exist to the extent to which they have been measured. If the observer makes no measurement in between the slices \(\Sigma_1\) and \(\Sigma_2\), then there exists no "completely sharp" answer to the question whether some places with a singularity etc. have existed at all.

Something like a black hole has "probably" existed because classical physics is a good approximation. But the whole process is analogous to the quantum tunneling. The "region inside the wall" should be compared to a "configuration that looks like the black hole spacetime". The Universe temporarily finds itself in this region or configuration but it returns back to a configuration that looks like a modest excitation of the empty spacetime.

It means that in quantum mechanics, it should always be possible to ignore the details of the intermediate state. In other words, in quantum mechanics, it's possible to compute the Feynman integral over all possible intermediate histories that may be inserted between the initial slice \(\Sigma_1\) and the final slice \(\Sigma_2\). Because both of these slices look like (and have all properties like) slices in the empty spacetime, we have exactly the same reasons as in the empty spacetime to expect that the evolution operator in between them, namely the S-matrix \(S\), should be unitary, i.e. \(SS^\dagger=1\).

And indeed, since the 1990s, we have known that this statement is correct. Matrix theory, AdS/CFT, but also some more general arguments independent of string theory imply that the information is preserved. Indeed, in Matrix theory or AdS/CFT, we don't even get any sharp answer to the question whether an intermediate state is a black hole or not. In the boundary CFT, for example, it becomes rather difficult to distinguish the histories with and without black holes.

Even though it may be demonstrated that the CFT description is equivalent to quantum gravity in the AdS space which does contain black holes, the CFT description is as unitary as any non-gravitational quantum field theory. So it's obvious that the evolution from the initial slice \(\Sigma_1\) which must be encoded by some initial data of the CFT to \(\Sigma_2\) which is encoded in some late CFT data has to be unitary. So regardless of the singularity-containing bulk interpretation of some intermediate moments, no information can be lost in between. The CFT is defined on a topologically trivial space (the AdS boundary) so there's no place where the information could be lost.

What Tim Maudlin has done is nothing else than to repeat one side of the paradox – the side that ends up with "and the information is therefore lost". Too bad, he completely misunderstands the other side which happens to be the side producing the right answer – the information is not lost – the opposite answer to his. He's a guy who only understands one-half of some basic material, the easier one-half, and sells his more-than-half-empty skull as a skull of a brilliant guy immersed among idiots because he's a wonderful dimwit.

He is a textbook representative of a pompous fool and they drive me up the wall for the same reasons why they drove Feynman up the wall.

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