## Friday, October 4, 2013

### Black Hole Information Paradox 3. Look for the information where you lost it

After I reviewed briefly the so-called black hole wars, and expressed my doubts about black hole complementarity, there are still many things to be said. However, I would like to skip over various solutions proposed in the last decades, and discuss the one that I consider most natural.

All the discussions taking place within the last year around black hole complementarity and firewall are concentrated near the event horizon. But why looking for the information at the event horizon, when it was supposed to be lost at the singularity?

Remember the old joke with the policeman helping a drunk man searching his lost keys under a streetlight, only to find later that the drunk man actually lost them in the park? When asked why did he search the keys under the streetlight, the drunk man replied that in the park was too dark. In science, this behavior is called the streetlight effect.

By analogy, the dark place is the singularity, because it is not well understood. The lightened place is the event horizon. This is Schwarzschild's equation describing the metric of the black hole:
$${d} s^2 = -(1-\frac{2m}{r}){d} t^2 +(1-\frac{2m}{r})^{-1}{d} r^2 + r^2{d}\sigma^2,$$
where ${d}\sigma^2 = {d}\theta^2 + \sin^2\theta {d} \phi^2$ is the metric of the unit sphere $S^2$, $m$ the mass of the body, and the units were chosen so that $c=1$ and $G=1$.

Schwarzschild's metric has two singularities, one at the event horizon, and the other one at the "center".

But in coordinates like those proposed by Eddington-Finkelstein, or by Kruskal-Szekeres, the metric becomes regular at the event horizon, showing that this singularity is due to the coordinates used by Schwarzschild. Fig. 1. represents the Penrose-Carter diagram of the Schwarzschild black hole. The yellow lines represent the event horizon, and we see that the metric is regular there.

 Figure 1. Penrose-Carter diagram of the Schwarzshild black hole.

While at the event horizon the darkness was dispersed by finding appropriate coordinates, it persisted at the central singularity, represented with red. This is a spacelike singularity, and it is not actually at the center of the black hole, but in the future. This kind of singularity could not be removed completely, because it was not due exclusively to the coordinates.

However, in my paper Schwarzschild Singularity is Semi-Regularizable, I showed that we can eliminate the part of the singularity due to coordinates, by the transformation $r = \tau^2$, $t = \xi\tau^4$. The Schwarzshild metric in the new coordinates becomes

$${d} s^2 = -\frac{4\tau^4}{2m-\tau^2}{d} \tau^2 + (2m-\tau^2)\tau^4(4\xi{d}\tau + \tau{d}\xi)^2 + \tau^4{d}\sigma^2.$$

The metric is still singular, because it is degenerate, but the coordinate singularity was removed. The metric extends analytically through the singularity $r=0$, and the Penrose-Carter diagram becomes as in Fig. 2.
 Figure 2. Penrose-Carter diagram of the extended Schwarzshild black hole.

In the new coordinates, the singularity behaves well. Although the metric is degenerate at the singularities, in arXiv:1105.0201 I showed that this kind of metric allows the construction of invariant geometric objects in a natural way. These objects can be used to write evolution equations beyond the singularity.

The Schwarzschild metric is eternal, but in the case relevant to the problem of information loss, the black hole is created and then evaporates. The analytic extension through the singularity presented earlier also works for this case, and the Penrose-Carter diagram is shown in Fig. 3.B.
 Figure 3. A. Penrose diagram for the evaporating black hole, standard scenario. B. Penrose diagram for the evaporating black hole, when the solution is analytically extended through the singularity (as in arXiv:1111.4837). In the new solution, the geometry can be described in term of finite quantities, without changing Einstein's equation. Fields can go through the singularity, beyond it.

Information is no longer blocked at the singularity. The physical fields can evolve beyond the singularity, carrying the information, which is therefore recovered if the black hole evaporates.

This is not a modification of General Relativity, it is just a change of variables. The proposed objects remain finite at singularity, and the standard equations can be rewritten in terms of these new, finite objects. These objects are natural, and don't require a modification of Einstein's General Relativity. The proposed fix is not made by changing the theory, but by changing our understanding of the mathematics expressing the theory.

One principal inspiration for me when finding this solution is the work of David Finkelstein, especially the brilliant solution to the problem of the apparent singularity on the event horizon. Imagine how happy I was when I received by email, at the end of December 2012, the following encouragements from him:

Dear Cristi Stoica,

I write concerning your paper "Schwarzschild Singularity is Semi-Regularizable" (arXiv 1111.4837v2).
I write first to thank you for the deep pleasure that this paper afforded me.
Your regularization of the central true singularity of the Schwarzschild metric is a remarkable and beautiful example of thinking outside the box. It is a natural, generally covariant, and deep result on a problem that has drawn wide attention, that of gravitational singularities. You found your solution easily once you conceived the idea, and yet it has been overlooked for these many decades by the truly great minds in the field.

[...]

With good wishes for your future explorations,
David Finkelstein

## Thursday, October 3, 2013

### Black Hole Information Paradox 2. Stretched Complementarity

In this post, which continues Black Hole Paradox 1. Susskind vs. Hawking, I will explain my reasons for not accepting the black hole complementarity principle (BHC). I will argue that, in the process of inventing this principle, Susskind, Thorlacius and Uglum (STU) found an important result, but ignored it.

In the book L. Susskind, J. Lindesay, An introduction to black holes, information and the string theory revolution, World Scientific, 2005, is explained that three fundamental principles make the foundation of BHC:
1.  The principle of information conservation
2.  The equivalence principle
3.  The quantum xerox principle

The principle of information conservation was in fact what it had to be proven for the case of black hole information. To allow conservation, STU assumed that an external observer will see that  information in Hawking evaporation. They assumed implicitly that information remains outside the event horizon, at least for an external observer. So, they replaced implicitly 1. with

1'. The principle of information conservation by avoiding falling in the black hole.

The equivalence principle is the fundamental principle in General Relativity. It states that inertia and gravity are two faces of the same coin. Accelerated motion behaves as gravity, and gravity is due to the fact that spacetime is curved, so that reference frames cannot be without acceleration.

The quantum xerox principle is in fact the no-cloning theorem, which states that unknown quantum states cannot be copied. If we would be able to copy an unknown quantum state, then linearity of Quantum Mechanics would be violated. It is amazing that this simple but profound result was discovered only 30 years ago, given that the proof is so simple. There is a funny story about this. Asher Peres was anonymous reviewer for Foundations of Physics, and refereed a paper in which superluminal communication was predicted in Quantum Mechanics. He explained in the report that the result must be wrong, and even the author is aware. However, realizing that this mistaken result would stimulate the research, and a more important result would follow from this, he recommended publication. His intuition was right.
Assume Alice dives into the black hole. For an external observer Bob, she never reaches the event horizon. This is how the things look, according to General Relativity, from Bob's point of view. In Bob's coordinates, Alice never reaches the horizon, because GR predicts it gets closer slower and slower, like in a Zeno paradox. But in Alice's reference frame, she crosses the horizon in a finite time. This apparent contradiction is due to the different coordinates used by Alice and Bob. Bob's coordinates are singular at the horizon. So he is wrong, Alice crosses the horizon in finite time, but because he is accelerating continuously to avoid falling in the black hole, there is a redshift of the light coming from Alice, so that in Bob's frame, her time stops.

Moreover, because Bob sees the event horizon as being hot, he would see Alice being vaporized. This would be OK from Bob's point of view, because other wise he would experience violation of the no-cloning theorem. But this also takes place in Bob's coordinate system, which is singular at the event horizon. So, he should again be wrong. However, let's go with STU and assume that Bob is right.

But the equivalence principle implies that Alice would not experience something special when she would cross the horizon. So, in fact, the information describing her would cross the event horizon.

This amounts to an apparent contradiction between what Bob sees, and what Alice experiences. On the other hand, STU want that the information describing Alice remains outside the horizon. This can't be done, unless the information is cloned, one copy going with Alice, and the other remaining available to Bob in the Hawking radiation.

For me, this is a proof that 1' is wrong.  Admitting that 1' is true, we have to choose between no-cloning and the equivalence principle. Everybody agreed that we should not contradict these two principles. This means that the hypothesis that information survives by remaining outside the horizon, was wrong. Please note that this doesn't mean that 1 is wrong, only that 1' is wrong. 1 and 1' are not equivalent, although 1' implies 1. In other words, information may be preserved, but not as STU wanted.

I think this is a great result, found by STU, but they decided to ignore it. They didn't stop here. They didn't want to give up 1', because they believed that the only way to save information is this. In other words, he believed that 1 is equivalent to 1'. Because this led them to contradiction, they decided to accept 1' together with the contradiction. The way was to admit cloning of the information so that it is shared by Alice and Bob, but to claim in the same time that this is not violation of 2.

STU saw that there is a contradiction between Alice and Bob, so they decided to apply the solution from the Sufi joke with Mulla Nasrudin, and agree with both of them. But, unlike the dervish, they did not go beyond dualism, and proposed instead the black hole complementarity. Essentially, it said that, even though Alice has a copy, and Bob has a copy, this doesn't contradict the no-cloning theorem, because Alice can't see Bob's copy, and vice-versa.

Now, call this however you want, but to me, it's a contradiction. Susskind even claimed that in fact this is just Bohr's complementarity, applied to this new case. It is true that Bohr stretched his idea of complementarity, until he saw it everywhere, and others stretched it more. But there is no connection between Bohr's and STU's complementarity. In Bohr's complementarity, there is no contradiction. Sometimes light behaves like waves, sometimes like point particles, but this is not a contradiction. If in a particular experiment, light behaves like waves for Alice, it does the same for Bob.

STU said that Alice and Bob can never meet, to compare their notes, hence there will be no proof that the no-cloning was violated. In other words, Nature can break her own laws whenever she wants, if we can't catch her in the act.

But, a question was raised, what if Bob dives into the black hole, following Alice, to compare their observations? Susskind found relatively quickly an answer to this: before Bob meeting Alice, they will be destroyed by the singularity. Indeed, calculations for Schwarzschild black holes show that Susskind is right about this.

But what if the black hole has the tiniest electric charge or rotation? In this case, the singularity is not spacelike, as in the Schwarzshild black hole. The singularity is timelike, and Alice and Bob can, in principle, avoid for indefinite time to reach it. So, there is plenty of time to meet and compare their notes. For some reason, this situation is never mentioned, only the Schwarzschild black hole case, for which there is an answer.

There is another reason why I disagree with BHC: it violates the equivalence principle. I explained this already in 2011. Ironically, although BHC was invented to allow 2 coexist with 1' and 3, it actually contradicts 2. Here is why. According to the equivalence principle, an experiment involving gravity should give the same result as an experiment in which we replace gravity with acceleration. Consider for example that Alice is moving inertially (free-fall motion), and Bob's frame is accelerated. This can happen at the black hole, when Bob sees Alice crossing the event horizon, while his accelerated motion helps him avoid falling. But it can happen somewhere far from any black holes. In this case, due to his acceleration, Bob will see something similar to the event horizon - the Rindler horizon. If he will see Alice crossing the Rindler horizon, he will see her evaporating. This is the equivalent of what happens in the case of a black hole, according to the equivalence principle. There is one big difference from the case when Alice falls in a Schwarzschild black hole: if Bob goes after her, he will find her alive and in good health. He will realize that she was not destroyed when she crosses the Rindler horizon. So, the equivalence principle tells us that even though Bob sees Alice being destroyed near the event horizon, he is again wrong, as it was in the case of the Rindler horizon. Hence, we have to choose between BHC and the equivalence principle.

Last year (in 2012), Almheiri, Marolf, Polchinski, and Sully (AMPS) wrote the paper Black Holes: Complementarity or Firewalls?, in which they show, by a different argument, that BHC doesn't solve the problem. They propose instead that Alice is actually destroyed at the horizon, by a firewall (formerly considered by Susskind, who called it "brick wall"). The price paid is that this sacrifices the equivalence principle.

So, if AMPS are right, and the solution is to admit the firewall, then why should we keep BHC? It is sometimes answered that BHC is still needed, to explain why Bob sees Alice never crossing the horizon, while she actually crosses her, in a finite proper time. But, as I explained, this is just an effect of GR, due to the fact that Bob's coordinates are singular at the horizon.

All the discussions taking place within the last year around black hole complementarity and firewall are concentrated near the event horizon. Information is supposed to be destroyed by the singularity, but it is hoped that, somehow, the event horizon plays the major role in recovering it. Black hole complementarity is based on the idea that Nature makes a backup of the information on the stretched horizon. The firewall proposal suggests that the event horizon is a shield that burns whatever may fall in the black hole, in order to make the information immortal.

To me, these are a Deus ex machina kind of explanations; it appears as if the supporters of these ideas see a purpose in the universe, and that purpose is eternal life for the information falling into the black hole, at any costs. It looks like God found a problem after he patched together General Relativity and Quantum Mechanics, and decided to fix it somehow. For instance, if God was a programmer, he would make a backup of the information on the horizon, to fix the memory leak caused by the singularities. Or, if God was a plumber, he would connect a pipe at the event horizon, to deviate information and prevent it leaking through the singularity. Fixing a bug, or a leak, would reveal intention in creating  the universe, a watchmaker who made an imperfect work and then repaired it using an improvisation.

Most part of this post I explained why I don't buy BHC. I also said that, during the process, STU found that 1' contradict 2 and 3, and that I consider this the correct result, and the attempt to remove the contradiction by embracing and giving it a name, did not actually remove it. So, my main point was to explain that in fact to save the lost information, copying it at the horizon is not the solution. I also don't think it is a solution to break the principle of equivalence, by building a firewall in a place where GR and QFT work well. In fact, as I will explain in a future post, I think that all this endeavor was misguided: why search the lost information in another place than that where was lost? Giving up this assumption will reveal that there is no contradiction between 1, 2, and 3 on the event horizon, without having to invoke mystical principles like no contradiction is a contradiction, until it is an observed contradiction. We will see this in the next post, named Look for the information where you lost it.