- Radiation from a collapsing object is manifestly unitary
Anshul Saini, Dejan Stojkovic
Phys.Rev.Lett. 114 (2015) 11, 111301
What do they do to arrive at this groundbreaking result that solves the black hole information loss problem in 4 PRL-approved pages? The authors calculate the particle production due to the time-dependent background of the gravitational field of a collapsing mass-shell. Using the mass-shell is a standard approximation. It is strictly speaking unnecessary, but it vastly simplifies the calculation and is often used. They use the functional Schrödinger formalism (see eg section II of this paper for a brief summary), which is somewhat unusual, but its use shouldn’t make a difference for the outcome. They find the time evolution of the particle production is unitary.
In the picture they use, they do not explicitly use Bogoliubov transformations, but I am sure one could reformulate their time-evolution in terms of the normally used Bogoliubov-coefficients, since both pictures have to be unitarily equivalent. There is an oddity in their calculation which is that in their field expansion they don’t seem to have anti-particles, or else I am misreading their notation, but this might not matter much as long as one keeps track of all branch cuts.
Due to the unusual picture that they use one unfortunately cannot directly compare their intermediate results with the standard calculation. In the most commonly used Schrödinger picture, the operators are time-independent. In the picture used in the paper, part of the time-dependence is pushed into the operators. Therefore I don’t know how to interpret these quantities, and in the paper there’s no explanation on what observables they might correspond to. I haven’t actually checked the steps of the calculation, but it all looks quite plausible as by method and functional dependence.
What’s new about this? Nothing really. The process of particle production in time-dependent background fields is unitary. The particles produced in the collapse process do form a pure state. They have to because it’s a Hamiltonian evolution. The reason for the black hole information loss is not that the particle production isn’t unitary – Bogoliubov transformations are by construction unitary – but that the outside observer in the end doesn’t get to see the full state. He only sees the part of the particles which manage to escape. The trouble is that these particles are entangled with the particles that are behind the horizon and eventually hit the singularity.It is this eventual destruction of half of the state at the singularity that ultimately leads to a loss of information. That’s why remnants or baby-universes in a sense solve the information loss problem simply by preventing the destruction at the singularity, since the singularity is assumed to not be there. For many people this is a somewhat unsatisfactory solution because the outside observer still doesn’t have access the information. However, since the whole state still exists in a remnant scenario the time evolution remains unitary and no inconsistency with quantum mechanics ever arises. The new paper is not a remnant scenario, I am telling you this to explain that what causes the non-unitarity is not the particle production itself, but that the produced particles are entangled across the horizon, and part of them later become inaccessible, thereby leaving the outside observer with a mixed state (read: “information loss”).
The authors in the paper never trace out the part behind the horizon, so it’s not surprising the get a pure state. They just haven’t done the whole calculation. They write (p. 3) “Original Hawking radiation density matrix contains only the diagonal elements while the cross-terms are absent.” The original matrix of the (full!) Hawking radiation contains off-diagonal terms, it’s a fully entangled state. It becomes a diagonal, mixed, matrix only after throwing out the particles behind the horizon. One cannot directly compare the both matrices though because in the paper they use a different basis than one normally does.
So, in summary, they redid a textbook calculation by a different method and claimed they got a different result. That should be a warning sign. This is a 30+ years old problem, thousands of papers have been written about it. What are the odds that all these calculations have simply been wrong? Another warning sign is that they never explain just why they manage to solve the problem. They try to explain that their calculation has something in common with other calculations (about entanglement in the outgoing radiation only) but I cannot see any connection, and they don’t explain it either.
The funny thing about the paper is that I think the calculation, to the extent that they do it, is actually correct. But then the authors omit the last step, which means they do not, as stated in the quote above, calculate what the asymptotic observer sees. The conclusion that this solves the black hole information problem is then a classical non-sequitur.