Manifest unitarity (simplified template)

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