Thursday, August 24, 2017

How to create objects inside black holes by outside operators

An anonymous reader has encouraged me to write this short review that I wanted to write for a few days.

Schrödinger's cat is jumping to a black hole. Will it be reincarnated as Heisenberg's German shepherd?

Recall that Kyriakos Papadodimas of CERN – whom I remember as a very bright Harvard grad student – is one of the folks who have, in my humble opinion, understood the lessons from the last two decades of the research of string theory and quantum gravity and their implications for the black hole interior and its (incomplete) independence from the exterior world. In particular, he and Suvrat Raju have written papers analyzing how the local field operators seem to be approximately embedded into Hilbert spaces including finitely many black hole microstates. One of the surprising lessons was that the local operators, especially those inside the black holes, must be embedded in a space-dependent way.

Days ago, Kyriakos submitted his new, 56-page-long paper:
A class of non-equilibrium states and the black hole interior.
He is proposing a simple Ansatz for states of a black hole that don't quite look like a black hole with the empty interior. Instead, some objects are created inside the black hole. And he can create these objects just with the help of some operators that are naturally outside the black hole – specifically, using operators that have a simple enough form in the CFT description of the AdS/CFT correspondence.

The CFT side is often said to only see the black hole exterior – the interior of the black hole living in the AdS bulk seems inaccessible to the CFT degrees of freedom. Well, using his Ansatz, Kyriakos basically tears down this wall (or firewall) at the horizon and makes the objects inside accessible.

The key formula, equation (1.1), says\[

\ket\Psi = e^{-\beta H / 2} U({\mathcal O}) e^{+\beta H / 2} \ket{\Psi_0}.

\] The construction of \(\ket\Psi\) starts with a typical pure state \(\ket{\Psi_0}\) whose energy is \(E_0\). And he transforms it with a conjugated unitary operator. Here, the operator \(U({\mathcal O})\) is a unitary operator constructed out of some local fields at a slice \(t=t_0\) – some simple enough polynomials or near-polynomials that could also be simple enough in terms of the local fields in the boundary CFT, if I understand it well.

Kyriakos wants \(\ket{\Psi_0}\) to have a sharply peaked energy. I believe that he may only want some state in a microcanonical ensemble, not a true energy \(H\) eigenstate because unstable black holes' microstates shouldn't be precise energy eigenstates – am I right, Kyriakos? (Clarification: Kyriakos is telling me he's talking about the big, i.e. stable, AdS black holes, so they could be eigenstates.)

But this operator \(U\) is conjugated by \(\exp(\beta H / 2)\) where \(\beta\) is the inverse temperature corresponding to the black hole solution at that energy. This operator is the operator of "waiting for the imaginary time". If you think about the periodic imaginary time in the Euclideanized spacetime, it's the operator moving you on the opposite side of the event horizon, by the angle \(\pi\). That's why you're basically creating objects on the other side of the horizon.

Alternatively, I would say that you may think of the naturalness of the conjugation as follows. The density matrix \(\ket{\Psi_0}\bra{\Psi_0}\) is a "term" in the thermal density matrix \(\exp(-\beta H)\) that would describe a mixed ensemble for the black hole. You may interpret \(\exp(-\beta H)\) as the product of two copies of \(\exp(-\beta H/2)\), one for the ket, one for the bra. So the first \(\exp(+\beta H/2)\) is just cancelling the \(\exp(-\beta H/2)\) that is morally included in the ket vector, bringing you as close to the identity operators as you can get. That's where you may act with the regular unitary operators, which is what he does, but then you must return the state back to the states comparable to the original \(\ket{\Psi_0}\), which is why you multiply it by a \(\exp(-\beta H/2)\) from the left again.

The unitary group is just conjugated a little bit. Without the conjugation, you would be discussing the transformations of black hole microstates by some operations that look natural and local outside the black hole. With the conjugation, you're getting to the interior or the "evil twin" degrees of freedom of the black hole. And that's what physicists like Kyriakos find so irresistible.

OK, so his paper follows a typical strategy of the geniuses. First, in equation (1.1), guess the correct answer for the most important formula that settles everything. And in the rest of the paper, simply justify that it's really ingenious even to the people who are incapable of guessing the formula right away. ;-) Needless to say, some of his ingenious attitude is due to some heavy experience.

So how does the paper continue?

He simply studies how the expectation values of various operators – and their products – evolve in time. In the original state \(\ket{\Psi_0}\), we have\[

\frac{d}{dt} \langle A(t) \rangle = 0 , \quad
\frac{d}{dt} \langle A(t) H \rangle = 0 .

\] We divided the operators inside the expectation value into two groups: \(A(t)\) isn't allowed to include any factor \(H\), while the operators with the factor \(H\) are treated separately. But let's modify the original near-energy-eigenstate \(\ket{\Psi_0}\) by the unitary transformations.

If we consider the pure state \(U({\mathcal O}) \ket{\Psi_0}\), which I previously interpreted as objects added outside the black hole, the expectation values evolve in time as follows:\[

\frac{d}{dt} \langle A(t) \rangle \neq 0 , \quad
\frac{d}{dt} \langle A(t) H \rangle \neq 0 .

\] I copied and pasted the previous equation and changed \(=\) to \(\neq\). It was simple. When you add some objects outside the black hole, they violate the stationary character of the configuration. Things evolve in time, the black hole is trying to suck the objects in, and so on.

But most interestingly, consider the Papadodimas states \(\ket\Psi\) in which the unitary operator is conjugated by that \(\exp(\beta H/2)\) as we have previously explained. The expectation values will evolve as:\[

\frac{d}{dt} \langle A(t) \rangle = 0 , \quad
\frac{d}{dt} \langle A(t) H \rangle \neq 0 .

\] These two equations are a hybrid of the previous two. So the evolution of the operators themselves will be zero, due to the KMS condition on the underlying equilibrium state. There are cancellations between many microstates of different energies exactly because the exponential \(\exp(\beta H / 2)\) brings their contribution to the same order, regardless of the energy – my heuristic efforts of "trying to get as close to the identity operator" is basically the reason for this KMS cancellation. (Yes, it's the same KMS discussed in the papers by Dr Brothers Bogdanov LOL.) So the black hole with the "objects added inside" by this hypothetical procedure still looks stationary or in equilibrium for some purposes. However, when the operator in the expectation value includes \(H\) as a factor, the equilibrium is spoiled and you get nonzero. In other words, there is a way to detect some evolution inside if you add the Hamiltonian factor.

Some of the cancellations above result from "random phases" of the terms, some of them from special properties of the magnitude of the terms, and the random-phases cancellations may be "abolished" by introducing a correlation between the phases included inside \(U\) and inside the time evolution. It makes sense but you will need to read the full paper if this TRF sketch isn't enough. This sketch is designed to be short and therefore easy to read; but bringing you the right questions that may help you to study the paper and look for answers to questions you might find important.

He clarifies that this construction circumvents the need for state-dependent operators because he's basically constructing just the states, not operators. I think that the very fact that he uses a state like \(\ket{\Psi_0}\) at the beginning of the construction makes the need for state dependence of the operators manifest. Papadodimas also discusses the relationship between his construction and the traversable wormholes of Gao, Jafferis, and Wall, among other things.

Update: Well, the construction that starts with \(\ket{\Psi_0}\) seems basically independent of \(\ket{\Psi_0}\). So I think that the state dependence has been morphed to some big sensitivity of the operators (field operators located in the black hole interior) on some phases that are implicitly hiding in the microstates.

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