**Guest blog by Prof Dejan Stojkovic, University of Buffalo**

Dear Lubos,

First, I would like to thank you very much for his kind invitation for a guest post. I am certainly honored by this gesture.

We recently published a paper titled “Radiation from a Collapsing Object is Manifestly Unitary” in PRL. The title was carefully chosen (note the absence of the term “black hole”) because of its potential implications for a very touchy issue of the information loss paradox. I will use this opportunity to explain our points of view.

The information loss paradox is one of the most persistent problems that we are currently facing in theoretical physics. For almost forty years, many possible solutions have been put forward, and practically whatever one could possibly say has already been said many times over. What we are missing are explicit calculations. However, it is difficult to start calculations if we do not agree what question to ask. From the standard field theory in curved space-time perspective, one has to solve a well-defined Cauchy problem. You start with some initial conditions on a complete space-like hypersurface, evolve the system in time using your dynamical equations, and you want to see if this evolution is unitary. Note that this formulation does not use the notion of an observer. You do not require that your hypersurfaces must be observable by a single observer. Like in cosmology, you start with a space-like hypersurface a fraction of a second after the Big Bang in order to avoid quantum gravity, and then you evolve it in time. You do not ask that a single observer must see the whole hypersurface (hat tip to Samir Mathur).

However, the complementarity picture advocated by Susskind puts an observer in the central position. You always ask what a certain observer would see. Since there is no non-trivial global slice in the black hole space-time which is simultaneously observed by both a static asymptotic observer and a freely falling one, then you introduce the principle of complementarity of the two pictures that they see. You assume that an asymptotic observer would see a normal unitary evolution (though to the best of my knowledge nobody yet solved the full time-dependent problem of the gravitational collapse and subsequent evaporation for an asymptotic observer), and then you ask what happens with information as seen by an infalling observer: did it get bleached at the horizon, doubled, transferred out by some non-local effects or something else. Though this picture sounds reasonable, it is much less defined in terms of the standard field theory in curved space-time. Perhaps, there could be many pictures in between these two extremes.

Let’s now explain what we calculated, without prejudice, and then discuss what implications for the information loss paradox are. For a while I was pondering on Don Page's result in Phys. Rev. Lett. 71, 1291 (1993), claiming that, if you divide a thermodynamic system into small subsystems, after some time there is very little or no information in small subsystems, and all the information is actually contained in subtle correlations between the subsystems. If you interpret subsystems as particles (with a few degrees of freedom), then it appears that particles do not carry information. All the information is encoded in subtle correlations between them. (Note: information within the system here is defined as the difference between the maximal possible and average entropy of the system. If this difference is zero, then system has no information in it.) Don Page showed that this is true in the absence of gravity, but I thought it might be true even in the presence of gravity. So we setup our calculations. We used the functional Schrödinger approach that we defined earlier with Lawrence Krauss and Tanmay Vachaspati.

Consider a thin shell of matter which collapses under its own gravity. We use Schwarzschild coordinates because we are interested in the point of view of an observer at infinity. The metric outside the shell can be written as \[

\eq{

{ds}^2 &= - \left(1- \frac{R_s}{r}\right) dt^2 + {\left(1- \frac{R_s}{r}\right)}^{-1} dr^2 + \\

&+ r^2 d{\Omega}^{2}

}

\] The interior of the shell is Minkowski\[

{ds}^2 = - dT^{2} + dr^2 + r^2 d{\Omega}^{2} .

\] We use the Gauss-Codazzi method (matching conditions at the shell) to find the relation between coordinates \( t \) and \( T \). An action of the massless scalar field propagating in the background of the collapsing shell can be written as\[

S = \int{ d^{4} x \sqrt{-g} \frac{1}{2} g^{\mu \nu } {\partial}_{\mu} \phi {\partial}_{\nu} \phi} .

\] We expand the field in modes\[

\phi = \sum_{\lambda} a_{\lambda} (t) f_{\lambda} (r) .

\] If we keep the only the dominant terms near the horizon, the action becomes\[

S = \int dt \left( - \frac{1}{2B} \frac{d{a}_{k}}{dt} A_{kk'}\frac{d{a}_{k'}}{dt} + \frac{1}{2} a_{k} B_{kk'}a_{k'} \right)

\] The factor \(B = 1 - R_s/R\), where \(R = R(t)\) is the radius of the shell, appears because of the matching conditions. Since matrices \(A\) and \(B\) are symmetric and real, the principal axis theorem guarantees that both can be diagonalized simultaneously with respective eigenvalues \(\alpha\) and \(\beta\). From the action one can find the Hamiltonian and then write the Schrödinger equation for the eigenmodes \(b\) (which are linear combinations of the original modes \(a\)) \[

\left[ - \frac{1}{2\alpha} \frac{{\partial}^2}{\partial b^2} + \frac{\alpha}{2} {\omega}^2 (\eta) b^2 \right] \psi(b, \eta) = i \frac{\partial \psi (b,\eta)}{\partial \eta}

\] This is an equation for a harmonic oscillator with time-dependent frequency. The time parameter is now \(\eta\) because we transferred the time dependence from the mass term to the frequency, but we can easily transform it back. The frequency of the modes is \({\omega}^2 =\left(\frac{\beta}{\alpha}\right) \frac{1}{B} \equiv \frac{{\omega_0}^2}{B} \), where \(\omega_0\) is the frequency of the mode at the time when it was created. Since the background spacetime is time-dependent this frequency will evolve in time. It is remarkable that we can solve the Schrödinger equation in question exactly to obtain the wave function \(\psi\) which now contains the complete information about all excitations in this space-time. The explicit form of the solution can be found either in the original paper with Lawrence and Tanmay or in our latest paper. We want to construct density matrix of the system so we need to expand the wavefunction in terms of a complete basis, say the simple harmonic oscillator \(\zeta_n (y)\): \[

\psi (b,t) = \sum_{n} c_n (t) \zeta_n (b)

\] Then the coefficients \(c_n (t)\) can be written as\[

c_n(t) = \int dy {\zeta_n}^{*}(b) \psi (b,t) .

\] which gives the probability of finding a particle in a particular state \(n\) as \({\mid c_n(t)\mid}^2\). The occupation number at eigenfrequency \(\bar{\omega}=\omega_0 e^{t/2 R_s} \) (which is the frequency of the mode at some finite Schwarzschid time \(t\)) is given by the expectation value\[

N(t, \bar{\omega}) = \sum_n n |c_n|^2 .

\] The process of the gravitational collapse takes infinite time for an outside observer, however, radiation is pretty close to Planckian \(N_{\rm Planck} (\bar{\omega}) =1/(e^{\bar{\omega}/T}-1)\) , when the collapsing shell approaches its own Schwarzschild radius at late times (see figure).

One could even perform the best fit through the late time curves and find the temperature which matches the Hawking temperature very well. It is very interesting that we obtained approximately thermal spectrum without tracing out any of the modes.

However, we are here interested in correlations between the emitted quanta, which is contained not in the diagonal spectrum, but actually in the total density matrix for the system. The density matrix is defined as \[

\hat{\rho} = \sum \left|\psi\right>\left<\psi\right| = \sum_{mn} c_{mn} \left|{\zeta}_{m}\right>\left<{\zeta}_{n}\right|.

\] where \(c_{mn} \equiv c_{m}c_{n}\). Original Hawking radiation density matrix, \(\rho_h\), contains only the diagonal elements \(c_{nn}\), while the cross-terms \(c_{mn}\) are absent. The off-diagonal terms represent interactions and correlations between the states. The rationale behind neglecting the cross-terms is that these correlations are usually higher order effects and will not affect the Hawking's result in the first order. However, the correlations may start off very small, but gradually grow as the process continues. It may happen at the end that these off-diagonal terms can modify the Hawking density matrix significantly enough to yield a pure sate. The time-dependent functional Schrödinger formalism is especially convenient to test this proposal since it gives us the time evolution of the system.

The main results are shown in the next two figures. First figure shows the magnitudes of the diagonal and off-diagonal terms.

We clearly see that the magnitudes of the modes start small, increase with time, reach their maximal value and then decrease. They must decrease to leave room for higher excitations terms, since the trace must remain unity. This implies that correlations among the created particles also increase with time. Since there are progressively more cross-terms than the diagonal terms, their cumulative contribution to the total density matrix simply cannot be neglected.

Information content in the system is usually given in terms of a trace of the squared density matrix. If the trace of the squared density matrix is one, then the state is pure, while the zero trace corresponds to a mixed state. In this figure we plot the traces of squares of two density matrices as functions of time for a fixed frequency. One is the Hawking radiation density matrix \(\hat{\rho}_{h}\) which contains only the diagonal terms \(c_{nn}\) and neglects correlations. The other one is the total density matrix \(\hat{\rho}\) which contains all the elements, including the off-diagonal correlations.

We clearly see that \({\rm Tr}(\hat{\rho}_h^2)\) goes to zero as time progresses which means that the system is going from a pure state to a maximally mixed thermal state. This would imply that information is lost in the process of radiation. However, if the plot the total \({\rm Tr}(\hat{\rho}^2)\) we see that it always remains unity, which means that the state always remain pure during the evolution and information does not get lost. This clearly tells us that correlations between the excited modes are very important, and if one takes them into account the information in the system remains intact.

For unitarity to be manifest we have to see the total density matrix, i.e. all the created modes and correlations between them. If some of the modes are lost (say into the singularity), then the incomplete density matrix may not look like that of the pure state. So what are implications of our results to the information loss paradox? We have to compare our results with the standard textbook Hawking result.

In his original calculations, Hawking used the Bogoliubov transformation between the initial (Minkowski) vacuum and final (Schwarzschild) vacuum at the end of the collapse. The vacuum mismatch gives the thermal spectrum of particles. In this picture, there is a negative energy flux toward the center of a black hole and positive energy flux toward infinity, and thus a black hole loses its mass. Note that the existence of the horizon is necessary for this, since it is not possible to have a macroscopic negative energy flux without the horizon (the fact that the time-like Killing vector becomes space-like within the horizon is responsible for this). Since an outside observer, which is the most relevant observer for the question of the information loss, never sees the formation of the horizon, it is not clear how this picture works for him. In fact, since the horizon is necessary for the Hawking radiation, and horizon is never formed for an outside observer, it is not clear if he will ever see Hawking radiation.

This is where the advantage of the time-dependent functional Schrödinger formalism becomes obvious. In this picture, it is the time-dependent metric during the collapse which produces particles. No horizon is needed for this. As the collapsing shell approaches its own Schwarzschild radius, this radiation becomes more and more Planckian.

We now emphasize that the Planckian spectrum of produced particles is not equivalent to a thermal spectrum. For a strictly thermal spectrum there should be no correlations between the produced particles. The corresponding density matrix should only have non-zero entries on its diagonal. In contrast, if subtle correlations exist, then the particle distribution might be Planckian, but the density matrix will have non-diagonal entries.

As mentioned, in his original calculations, Hawking used the Bogoliubov transformation between the initial (Minkowski) vacuum and final (Schwarzschild) vacuum at the end of the collapse in \( t \rightarrow \infty \) limit. He traced out the ingoing modes since they end up in the singularity and are inaccessible to an outside observer. As a consequence he obtained a thermal density matrix that leads to the information loss paradox. There are pretty convincing arguments by Samir Mathur in Class. Quant. Grav. 26, 224001 (2009) that corrections can’t purify this density matrix.

Note however that a static outside observer will never witness formation of the horizon since the collapsing object has only finite mass. He will observe the collapsing object slowly getting converted into Hawking-like radiation before horizon is formed. For him, no horizon nor singularity ever forms. That is why it was so important to solve the time-dependent problem rather than a problem in the \( t \rightarrow \infty \) limit. As a consequence, in the context of our calculations, if we treat our problem as a Cauchy problem, we DO NOT have to trace out the modes inside the collapsing shell. The total wavefunction, which we found as a solution to our time-dependent problem, contains all the excitations in this space-time and describes the modes inside and outside the shell. But both the modes outside and inside the shell are never lost for a static outside observer. Thus, in the foliation that an outside observer is using, we showed that the time-dependent evolution is unitary. If we extend the collapse all the way to \( t \rightarrow \infty \), the horizon will be formed for an outside observer, he will lose the modes inside the shell, and we would have to trace over them. In that limit the Hawking result would be recovered. But this situation never happens since the collapsing object has only finite mass. Thus, Hawking correctly solved the problem he was set to do, and his result is very robust. However, the question is whether this was the right problem to solve.

The next question is what happens in the foliation where the singularity forms, e.g. for an infalling observer. This is what we are currently working on, though the calculations are more involving. However, the crucial question here is whether the real singularity forms or not. Singularity at the center is a classical result. Most likely, it can be cured by quantization, just like we cured the hydrogen atom of the classical \(1/r\) singularity of the electrostatic potential. We analyzed that question in Phys. Rev. D89 (2014) 4, 044003, and the answer seems to be positive. We managed to solve the non-local equation governing the last stages of the gravitational collapse in the context of quantum mechanics and found that the wave function of the collapsing object is non-singular at the center (just like the wavefunction of the ground state of the hydrogen atom is non-singular at \(r=0\)). Of course, in the absence of the full quantum gravity, we can’t be sure about this result, but it is very suggestive. If there is no singularity at the center, and instead of it we find only a region of very strong but finite gravitational fields, then the black hole horizon cannot be a global event horizon. It could trap light for some finite time, but as the black hole losses its mass to evaporation, the trapped light will be eventually released out. Then the situation will be no different than that seen by a static outside observer - no modes are lost forever, and we can measure the whole density matrix. If the singularity indeed forms, then we need some more elaborate mechanisms. Perhaps, something like a non-local transfer of information from inside the horizon to outside. (Anecdotally, in the above mentioned paper we find that the last stages of the gravitational collapse are governed by a highly non-local equation, but I am not sure if this non-locality is of the right sort to transfer information.) However, singularity is just a signal that we have extrapolated our theory beyond its region of validity. If the singularity is really there, it represents the breakdown of the whole physics at that point, so I am not sure why people are so upset if unitarity breaks down too.

I would like to end with a provocative analogy that my friend (a fellow scientist who would prefer not to mention his name) is using in this context. He says that this question is similar to the question of whether the afterlife exists. Horizon represents the moment of death of a person. An outside observer never sees what happens to a person beyond that point. But a person, as an infalling observer crosses the horizon and experiences either the end (singularity) or something more (or less) exotic. However, information about what he experiences seems to be lost to the outside world. Debating whether you can get information out or not is useless. Different religions have different answers.

I would end here. Thanks again for your kind invitation.

Best wishes,

Dejan Stojkovic

## snail feedback (57) :

Dear Lubos,

I think all of your observations are correct. If we extend our calculations all the way to t -> infinity, then the horizon will be formed, you have to trace over the interior modes and Hawking's result is back. But the point is that you can't do that. Collapse last only a finite amount of time in Schwarzschild coordinates if the mass of the collapsing object is finite. If you look at the Hawking original paper in 1974, he solved the problem of radiation from a collapsing object in t -> infinity, so you might get an impression that Hawking radiation starts only after the horizon is formed (and therefore the inner modes are lost into the singularity). But that is not what happens. The radiation starts as soon as the collapse starts, but it becomes thermal only when the objects is close to its own Schwarzschild radius. So the collapsing object loses its mass all the time during the collapse. If this mass is finite, collapse can't last infinite amount of time. But the horizon and singularity are formed only at infinity. Therefore an outside observer never needs to trace over the interior modes. They are always available to him, metric is Schwarzschild outside the shell and Minkowski inside the shell. No singularity means no need to trace over the inner modes. Note that I am not saying that an outside observer is seeing inside the black hole. Outside observer never deals with the black hole for the reasons I explained here.

So I think that in this foliation, evolution is always unitary.

Note however that this is not enough for the full solution of the information loss paradox. You have to ask what happens in the foliation which sees the horizon and singularity. We did not calculate this and are currently working on this. If the singularity indeed forms in the realistic gravitational collapse, the infalling observer would see it. Then the causal space-time diagram is exactly as you draw it, and then I expect information loss. Then you need some more elaborate solutions along the lines hat you mentioned. However, as I mentioned, if the singularity exists, the whole physics breaks down there, so why not unitarity? If the singularity does not exists (which I am almost sure about), then the global diagram is not the one that you drew, and again, one does not have to trace over the inner modes.

About why I am thinking that Hawking jumps right to t -> infinity, that is what he did explicitly. Moreover, he had no other choice when dealing with Bogoliubov transformations. You have to compare two static asymptotically flat vacua (Minkowski at the beginning of the collapse and Schwarzschild at the end of the collapse) because particles with definite energy are well defined only in that case. In time dependent background particles are not well defined, that is why we refer to them as modes (diagonal and off-diagonal) rather than particles. This is why I say that our method has some advantages. However, in the limit of t -> infinity, they match.

Please let me know if I am making sense.

Dejan

Dear Dejan, thanks for your extensive reply. Maybe one issue I can't quite comprehend it is that I don't understand what you mean by the "Schwarzschild coordinate" for the general collapsing and then evaporating black hole.

Then you say that there is no singularity etc. I don't understand these claims. It's interesting that you didn't get an approval from Sabine because this sounds almost exactly like the weird stuff she was writing with Smolin before.

Quantum gravity modifies physics of the singularity when the curvature becomes Planckian. But the claim that "the singularity is there anyway in GR" really means that the appearance of the region where the curvature is Planckian is inevitable. And before it's Planckian, the GR approximation is basically fine.

So if you take my diagram and erase a thin horizontal vicinity of the singularity, then you get the diagram that should be true according to GR in regimes where it should be trustworthy. And then, nearly the singularity, the conditions are already extreme, but there is no way to connect this (nearly) singular region to any other place of the diagram because any other place of the diagram is smooth and you can't connect a smooth and (nearly) singular regions. So there's no causal way for the information to get from the vicinity of the singularity outside.

So I think that the solutions to the information paradox of the form "singularity is never formed and that's why..." are self-evidently incorrect. If your paper is one of those, then sorry but I think that then it should be discarded just like the Smolin-Hossenfelder paper here

http://motls.blogspot.com/2009/01/hippie-non-solutions-to-black-hole.html?m=1

among others.

Dear Dejan, Could you perhaps support what Hawking recently wrote about a different event horizon?

I like the 2014 firewall idea of S. Hawking about changing the “event horizon”- into an “apparent” horizon, where infalling matter is suspended and pushed back. See:

Information Preservation and Weather Forecasting for Black Holes" http://arxiv.org/pdf/1401.5761v1.pdf

Could you even imagine that a fermion push back could be originated if a black hole can split pushed back plasma into different electric charged layers ?

Dear Lubos,

You are right, singularity comment was kind off-topic. What you refer to as Schwarzschild coordinates is how we used them in a static black hole space-time. But Schwarzschild solution describes any spherically symmetric solution outside of the source, for example a star. Anyway, the context we are using this solution in is the space time of a collapsing shell. Metric outside of the shell is Schwarzschild and inside the shell is Minkowski (it has to be Minkwoski because of the Birkhoff theorem). So there is no singularity here.

Dejan

Dear Prof. Stojkovic,

I think you are correct. For the outside observer the singularity never forms within finite time. And after a finite time it will have evaporated again. So physics should stay nice.

But the shell does eventually collapse into the point with infinite density?

Dumb question, sorry. You say above that it loses the finite mass as it collapses.

Dear Giotis,

I did not go to Sabine's web page after I left my last response. Thank for repeating the question. I am not sure if I fully understand it, but I would agree with your statement:

"So basically this is what they are trying to do here i.e. to study the nature and strength of such corrections and how fast they can restore purity and that’s why they don’t feel the need to trace out anything; they assume that everything inside the BH will reach eventually the external observer."

except that I would replace the term BH with the "collapsing object" since an outside observer watching the collapse never sees a BH in finite time (though a BH would form in some other foliation). As I said, an outside observer describes the collapsing shell geometry as the Schhwarzschild metric outside and Minkowski inside. This is not my invention, see for example J. Ipser, P. Sikivie, Phys.Rev. D30 (1984) 712, eqs 3.1a and 3.1b. This is an exact solution of Einstein's equations for the collapsing shell configuration.

Best,

Dejan

Nice interesting article :-)

Are these considerations compatible with the holographic principle?

Let me rephrase the question: I am under the impression that something magical happens so that the rate of evaporation matches the rate of collapse in such a way that singularity is avoided. Where am I wrong with this mental image?

In its own frame the shell will collapse to form a singularity. However, in the frame of an outside observer who sees Hawking radiation it would disappear before horizon is formed. We calculate the evolution in Schwarzschild foliation. That is why I am saying that one would need to do the same calculations in a foliation where singularity and horizon are formed in finite time to complete the analysis.

They are certainly not against it. But I am not sure of any connection.

Thank you very much and best in your future research. I find the picture of particle production by time-varying gravitational field, even before the event horizon (or some other type of horizon) forms, more realistic.

Thank you very much for your kind words. I wish all the best to you too.

Dumb question but I have to ask.

Is it possible that gravity is not one of the four fundamental forces but the combined effect or result of the rotational/angular

interaction between electromagnetic, strong nuclear, and weak

nuclear forces?

Swing hard and make it hurt. I can take it.

I enjoyed reading your post. I very much agree with your approach.

It does seem contradictory to discuss evaporation from eternal black holes

and it is more physical to study gravitational collapse. Then the

reason for evaporation is simply due to the time-dependence

of the collapse. The finite rate of evaporation (as measured at infinity)

then implies that there won't be any singularity formation. Then the

evolution is unitary and the spacetime diagram is just like Minkowski

as you have in your earlier paper. I liked your computation of the off-diagonal

elements of the density matrix as it clarifies the whole picture.

Do you agree that the real problem with this picture is that the backreaction

of the quantum radiation on the collapse has been ignored? I think this is the

crucial problem that needs to be solved. Too bad that so few people are thinking

about it.

I also liked your analogy with the after-life question. We've all

asked similar questions and realized that they don't lead anywhere. It would help

for you and other experts to come up with some sharply defined questions

that can be tested by experiment. Even if such experiments are not

immediately realizable, they could help frame the discussion. Could gravitational waves emitted during binary black hole mergers be such an "experiment"?

Dear Dejan, right, as long as it is a star, it has no singularity, but it also has no event horizon and no information loss paradox, so there is nothing to solve. Do I miss something?

Tony IMHO both can be true: collapse into a point of infinite density and losing its finite mass, if mass caryng fermions lose their spin inside the nucleus;

https://www.flickr.com/photos/93308747@N05/?details=1

Do I understand it correctly that the conclusion is that there is no information loss paradox for an observer at spatial infinity, whereas for any other observer the issue remains an open one ?

Dear Dejan,

thank you very much for your explanation.

I would like to use this occasion to also support you against the really rude and hostile kind of comments in the blog called

Backreaction.

I and other collegues use to read science blogs also on

different topics than our fields and I find Sabine reply really bad, with even open accusations from other commenters of

you being a lier or, in the best case, an incompetent.

I would like you to know that me and many of my collegues are on your side and think that that kind of blogging is definitely helping no one apart from blogger´s ego.

Best regards,

Alessio

Yes but they say that for the external observer no horizon forms so they treat it basically as a star.

Of course the whole point of the hawking radiation is that due to the horizon we have particle production even in static gravitational backgrounds of a static black hole.

The background does not have to be dynamical as it is during the collapse for particle production.

Dear Prof. Stojkovic,

Thank you very much for your interesting post. Although I’m not at all an expert in this field, to me your calculations in the paper look very convincing. Both your post and your paper are very well written and I wouldn’t know what could be wrong with your conclusion “we showed by explicit calculations that radiation coming from a collapsing object is manifestly unitary” for a static outside observer. It will be very interesting to see what the experts have to say about your paper. I really hope all will work out for the best.

Thanks! I really appreciate it.

Thanks for the suppot! I really appreciate it.

Yes. This is the bottom line.

Thanks for your opinion that our calculations clarify the picture. I also agree that backreaction during the collapse is a very important issue, however these calculations are very hard. I think the community is aware of its importance, it is just too hard to calculate it. And I agree, in the absence of concrete calculations, it becomes a war of opinions, which is more or less useless.

I also agree that we should devise a sharp experiment even if it is far from today's technology. But I am not sure about gravity waves from mergers. They probe the neighborhood of the event horizon, not the horizon itself (because anything which is emitted exactly from a horizon will be infinitely redshifted and unobservable). Then it is very difficult to distinguish between various scenarios. But we should definitely keep asking these questions.

Gerry, It sounds implausible.

Hawking proposal that a global event horizon might not form at all, just some sort of an apparent horizon is conceptually possible, but again, there are no concrete calculations supporting it. Note that if the event horizon never forms in any space-time foliation, then our calculations close the case of the information loss paradox.

If you take a star, there is no information loss paradox. Star has a static gravitational field so there is no time-dependent particle production. It has no horizon, so there is no Hawking effect.

We consider a gravitational collapse which will yield a black hole in some foliation. But in Schwarzschild foliation motivated by an outside observer, evolution will be unitary. To complete the argument, one has to study foliation which reveals the horizon and singularity: Kruskal, Eddington-Finkelstein...

I meant of course a collapsing star. Hawking radiation you can have even for static black holes due to the event horizon, you don't need the time dependent gravitational field of collapsing matter. That's the novel thing with Hawking radiation.

Dear Dejan,

I guess I am missing something here because I don't see how the result of your calculation could have been any different... You are basically solving a wave equation on a globally hyperbolic background (the region outside the horizon): isn't unitarity automatic?

As I see it, in order for your result to be nontrivial (except for the technical feat of actually solving the equation in this time-dependent background) and relevant to the information puzzle, you should show that the "infalling" modes, i.e. those with zero Cauchy data on scri+, do not appear in the basis you are using, and I don't see why that should be...

But again, I am probably missing something

Kind regards,

Andrea

Connection:

AN Information -> Area -> R^2 -> M^2

Mass * Mass -> Correlations between the subsystems (Each one with all others.)

"... and all the information is actually contained in subtle correlations between the subsystems."

Dear Andrea,

Thanks for the question. We start with some initial data on a complete Cauchy surface in Schwarzschild foliation, evolve it forward in time with the Schrodinger equation and stop at some later time. If we extend the process all the way to infinity, the horizon will be formed and I have to trace over the interior modes. But I gave the reason why I do not have to extend it to infinity - in this foliation the collapsing object disappears before the horizon is formed. I agree that unitarity is what we expect in this case. But we explicitly showed where information is hidden. Note that we got approximately thermal spectrum even without tracing out the interior modes, so one might think that the evolution is not unitary even at finite time, before he solved the full problem.

So this technical exercise was indeed necessary, if nothing else then to setup calculations for the foliation where singularity and horizon are revealed (say Kruskal).

Best wishes,

Dejan

"Prominent Hawaii high school teacher Walter Wagner has sued the LHC and proved that the risk of Earth's destruction was 50% because there were only two possibilities – destroyed or not destroyed."

The apotheosis of Bayesian subjective probability calculation?

2 outcomes, hard to assign priors, hence it *must* be 50-50... :-D

Andrea - You are correct. The authors of the PRL have done a nice calculation but have misunderstood what the issue is. This has been explained by Sabine Hossenfelder at Backreaction a week ago.

Dear Ramanujan,

I wrote an answer to Andrea hours ago but it got stuck somewhere in cyberspace. We understand the standard procedure. You trace out the modes that end up behind he horizon and into the singularity because that part of space-time is causally disconnected from the future infinity. That is how you end up with the thermal spectrum and information loss.

In our paper we first shed the light on the question where is information hidden. As Don Page showed in the absence of gravity, it is not hidden in particles, but in subtle correlations between them. We showed this is true even in the presence of gravity.

Note that we get the thermal spectrum without tracing out any of the modes. So one would naively think we have information loss even at finite time. We clarify that this is not the case.

Second we diverge from the standard procedure and do not trace out the inner modes. Why? Because in this foliation horizon is not formed in finite time, while collapsing object can radiate only for finite time. Thus, inner modes never get behind the horizon and there is no need to trace them out. This does not mean there is no problem in other foliations which reveal the horizon in finite time, like Kruskal, but we do not answer that question.

This Penrose diagram might help. An outside observer A will register a flux of quantum radiation even during collapse, and will be able to account for the entire energy of the shell by the time he gets to the line EF. From this point on, A will conclude that there is no energy left in the region of the collapsing shell. No horizon is encountered and no need to trace out any modes.

Dear Dejan,

You are claiming that black holes cannot be formed to an outside observer? Or you are claiming that black holes cannot be formed anyway? If this last question has a affirmative answering, then you "solve" the problem by saying that there is no problem.

This is a very strong claim. I'm not aware of your paper's calculations, but how we come so far with eternal black holes without note that? That quantum fields in curved space protecting the space-time from black holes (at least, for outside observer, that you show now).

Then you suggest that apply black hole's physics should be now only an approximation for large masses (large time-life), when the shell persist at large times near the Schwarzschild radius? And the information problem are indeed solved by the (pseudo-) Hawking's Radiation in some complementary picture?

If I understand right, your attitude here is that considering the process of a collapsing spherical body, we may note that the black hole is only a limit made by suitable relations of mass and time-scale under study. Like we do with hydrogen atom under energy/time scale.(?)

Kind regards,

Lucas

Kin

Dear Dejan,

what is the meaning of "If we keep the only the dominant terms near the horizon, the action becomes..." above?

I'm a bit confused since I am trying to reconcile this with the fact that external observer never sees any horizon formed in finite time.

Oddly enough, I happened to attend a federal advisory committee meeting at Livermore at which some local activists claimed more or less the same thing about the (then) proposed National Ignition Facility. In short, it was really a fiendish weapons project. Apparently, by pointing all of its 192 laser beams at a strategically located orbital mirror, one could cause fusion ignition of the water in a lake in the middle of Prague. The alpha particles released in the process would deposit their energy in a spherical "burn wave," consuming all the lake's water, resulting in a massive explosion that would annihilate all life up to a point just short of the Slovak border.

Lucky guy! With such odds he should be a multimillionaire by now. But not everything is in money so, in his place, I would fly to Hollywood and hit on all the most attractive female stars.

The problem isn't that the calculation or probability interpretation is subjective. The problem is that it's wrong - because it ignores all the evidence (that actually breaks the symmetry).

If you've never shown the TV piece where Wagner also makes this 50-50 claim, among many much more funny things, watch

http://motls.blogspot.com/2009/05/jon-stewart-john-ellis-lhc.html

Hi again.

Thanks for your reply, but I still don't get your point.

What does it mean that the horizon does not form? You are on a (patched) schw. background: the horizon is there already, and a complete set of modes, whatever your foliation might be, has to span the ingoing modes as well.

You seem to imply that the horizon will not form because of some energy conservation argument!

I had to read the 0609024v3 to get a better idea and see R(t) that exponentially approaches R(schwarzschild). So the expansion is in vicinity of Rs.

I must admit it is tough to visualize and one is immediately compelled to wonder how does the solution look inside the shell, how does it evolve there and how does it connect to the one across the boundary.

No, per:

http://arxiv.org/pdf/gr-qc/0609024v3.pdf

he means:

"This solution implies that,

from the classical point of view, the asymptotic observer

never sees the formation of the horizon of the black hole,

since R(t) = RS only as t → ∞. This result is similar to

the well-known result (for example, see [5]) that it takes

an infinite time for objects to fall into a pre-existing black

hole as viewed by an asymptotic observer [6]. In our case

there is no pre-existing horizon, which is itself taking an

infinite amount of time to form during collapse"

As the authors show: if you do their calculation the Hawking way there is an issue, it you do the calculation the authors way there is no issue. That’s what I get out of it and there is your issue I would say.

I hope this is not the case, as the statement above is both (spectacularly) trivial and irrelevant...

Hey man, good point! I just PM-ed Megan Fox to tell her she can send a private jet to pick me up this evening...

Only 2 possible outcomes, right?

:D :D

Dear Dejan,

do you then think from the conclusions of your work, that is should in principle be possible to construct some kind of an S-matrix formalism to describe the transition from matter falling into a black hole to Hawking radiation, as I once asked here?

http://www.physicsoverflow.org/28179/formalism-describe-transition-falling-hawking-radiation?show=28179#q28179

Dear Dilaton,

Interesting question. Since an asymptotic observer never really probes what happens inside, if you treat the problem quantum mechanically and you care just about the in and out states far from the black hole, you should be able to construct an S-matrix which is completely unitary. The problem is that you would have to integrate over all classically possible geometries, including the ones which are singular. Then it is not clear that you would get a unitary result.

Best,

Dejan

Dear Tony,

It is a good question. We performed our calculations in the near horizon limit when interesting things start happening. Note that Hawking did his calculation at t= infinity limit, so horizon is formed already. We are near the horizon but still outside, that is the shell is near its own Schwarzschild radius, but still outside. Then the above sentence becomes justified.

Best,

Dejan

As explained above, as long as the shell is outside its own Schwarzschild radius, one can perform the above calculations clearly.

Dejan

Dear Lucas,

What you are describing is certainly a logical possibility, but there are no calculations supporting it. Black hole event horizon is a very non-local object. Its formation is the question of the final rather than initial conditions. Why? Because you have to wait infinite amount of time to make sure that light can't come out of it.

Thus, you have to know the whole future history of the system in order to know the location of the event horizon. So we as outside observers, looking at the central galactic black hole, having only finite time at our disposal, can never be sure if that this is really an event horizon. But from observations you can infer that there is so much mass there within certain radius that no known form of matter can counter gravitational collapse. So you infer that in its own frame, this thing must have formed a singularity.

Backreaction due to Hawking radiation is perhaps too weak to prevent this conclusion. But there is always a possibility that we are doing something wrong.Black holes might be macroscopic quantum objects, and classical calculations might fail even for large mass black holes. Just like superconductivity and superfluidity are macroscopic quantum phenomena, and classical Maxwell equation fail at the scales where we were convinced they must be true. But note that this only a logical possibility, I am not sure if there are any calculations supporting it.

Again, our claim is simpler that it looks. An outside observer A in the Penrose diagram above, watching the collapse labeled by I, will always see unitary evolution. The ingoing modes are there, but never cross the horizon because the object disappeared before horizon is formed. He does not need trace them out.

Does that mean that in the full global causal structure of the space time there is no paradox? No. You still have to deal with the horizon and singularity.

Best,

Dejan

Interesting controversy ! I do not know enough about the subject to make any scientific comment. But I would say that in general controversies are healthy for science. There could be sacrificial lambs in the process! But in the long run this is good for science. Let the best theory win! Because of the "40 years controversy" I would like to see comments of people who have actually published refereed papers on the information paradox. Is it possible to ask them to express their opinion?

Some people, who do not agree with the conclusions of the paper, blame PRL for quality going down. This is not fair. It just means that it went to a referee who thought this was a valuable contribution which deserved urgent publication.

The real problem I have with Bayesian probability calculation (although I was pretty much persuaded against it by the writing of Richard von Mises; gotta love the von Mises brothers) is precisely the possibility (to which I would assign high a priori probability, if I were inclined to do such things) that it will be (and in fact is being) used by some fool somewhere to confidently make bad decisions, not just on his own behalf but for me too (when the fool has political power).

Post a Comment