Written by: Christoph Adami

Primary Source: Spherical Harmonics

For years now I have written about the quantum physics of black holes, and each and every time I have pushed a single idea: that if black holes behave as (almost perfect) black bodies, then they should be described by the same laws as black bodies are. And that means that besides the obvious processes of absorption and reflection, there is the quantum process of spontaneous emission (discovered to occur in black holes by Hawking), and this other process, called stimulated emission (neglected by Hawking, but discovered by Einstein). The latter solves the problem of what happens to information that falls into the black hole, because stimulated emission makes sure that a copy of that information is always available outside of the black hole horizon (the stories are a bit different for classical vs. quantum information. These stories are told in a series of posts on this blog:

Oh these rascally black holes (Part I)

Oh these rascally black holes (Part II)

Oh these rascally black holes (Part III)

Black holes and the fate of quantum information

The quantum cloning wars revisited

I barely ever thought about what happens to a black hole if nothing is falling in it. We all know (I mean, we have been told) that the black hole is evaporating. Slowly, but surely. Thermodynamic calculations can tell you how fast this evaporation process is: the rate of mass loss is inversely proportional to the square of the black hole mass.

But there is no calculation of the entropy (and hence the mass) of the black hole as a function of time!

Actually, I should not have said that. There are plenty of calculations of this sort. There is the CGHS Model, the JT Model, and several others. But these are models of quantum gravity in which the scalar field of standard curved space quantum field theory (CSQFT, the theory developed by Hawking and others to understand Hawking radiation) is coupled in one way or the other to another field (often the dilaton).You cannot calculate how the black hole loses its mass in standard CSQFT, because that theory is a free field theory! Those quantum fields interact with nothing!

The way you recover the Hawking effect in a free field theory is you consider not a mapping of the vacuum from time \(t=0\) to a finite time \(t\), you map from past infinity to future infinity. So time disappears in CSQFT! Wouldn’t it be nice if we had a theory that in some limit just becomes CSQFT, but allows us to explicitly couple the black hole degrees of freedom to the radiation degrees of freedom, so that we could do a time-dependent calculation of the S-matrix?

Well this post serves to announce that we may have found such a theory (“we” is my colleague Kamil Brádler and I). The link to the arXiv article will be below, but before you sneak a peek let me first put you in the right mind to appreciate what we have done.

In general, when you want to understand how a quantum state evolves forward in time, from time \(t_1\) to time \(t_2\), say, you write

\(|\Psi(t_2)\rangle=U(t_1,t_2)|\Psi(t_1)\rangle\\) (1)

where U is the unitary time evolution operator

\(U(t_2,t_1)=Te^{-i\int_{t_1}^{t_2}H(t’)dt’}\) (2)

The \(H\) is of course the interaction Hamiltonian, which describes the interaction between quantum fields. The \(T\) is Dyson’s time-ordering operator, and assures that products of operators always appear in the right temporal sequence. But the interaction Hamiltonian \(H\) does not exist in free-field CSQFT.

In my previous papers with Bradler and with Ver Steeg, I hinted at something, though. There we write this mapping from past to future infinity in terms of a Hamiltonian (oh, the wrath that this incurred from staunch relativists!) like so:

\(|\Psi_{\rm out}\rangle=e^{-iH}|\Psi_{\rm in}\rangle\\) (3)

where \(|\Psi_{\rm in}\rangle\) is the quantum state at past infinity, and \(|\Psi_{\rm out}\rangle\) is at future infinity. This mapping really connects creation and annihilation operators via a Bogoliubov transformation

\(A_k=e^{-iH}a_ke^{iH}\\) (4)

where the \(a_k\) are defined on the past null infinity time slice, and the \(A_k\) at future null infinity, but writing it as (3) makes it almost look as if \(H\) is a Hamiltonian, doesn’t it? Except there is no \(t\). The same \(H\) is in fact used in quantum optics a lot, and describes squeezing. I added to this a term that allows for scattering of radiation modes on the horizon in the 2014 article with Ver Steeg, and that can be seen as a beam spliter in quantum optics. But it is not an interaction operator between black holes and radiation.

For the longest time, I didn’t know how to make time evolution possible for black holes, because I did not know how to write the interaction. Then I became aware of a paper by Paul Alsing from the Air Force Research Laboratory, who had read my paper on the classical capacity of the quantum black hole channel, repeated all of my calculations (!), and realized that there exists, in quantum optics, an extension to the Hamiltonian that explicitly quantizes the black hole modes! (Paul’s background is quantum optics, so he was perfectly positioned to realize this.)

Because you see, the CSQFT that everybody is using since Hawking is really a semi-classical approximation to quantum gravity, where the black hole “field” is static. It is not quantized, and it does not change. It is a background field. That’s why the black hole mass and entropy change cannot be calculated. There is no back-reaction from the Hawking radiation (or the stimulated radiation for that matter), on the black hole. In the parlance of quantum optics, this approximation is called the “undepletable pump” scenario. What pump, you ask?

In quantum optics, “pumps” are used to create excited states of atoms. You can’t have lasers, for example, without a pump that creates and re-creates the inversion necessary for lasing. The squeezing operation that I talked about above is, in quantum optics, performed via parametric downconversion, where a nonlinear crystal is used to *split photons into pairs *like so:

Fig. 1: Spontaneous downconversion of a pump beam into a “signal” and an “idler” beam. Source: Wikimedia |

Splitting photons? How is that possible? Well it is possible because of stimulated emission! Basically, you are seeing the quantum copy machine at work here, and this quantum copy machine is “as good as it gets” (not perfect, in other words, because you remember of course that perfect quantum copying is impossible). So now you see why there is such a tantalizing equivalence between black holes and quantum optics: the mathematics describing spontaneous downconversion and black hole physics is the same: eqs (3) and (4).

But these equations do not quantize the pump, it is “undepleted” and remains so. This means that in this description, the pump beam is maintained at the same intensity. But quantum opticians have learned how to quantize the pump mode as well! This is done using the so-called “tri-linear Hamiltonian”: it has quantum fields not just for the signal and idler modes (think of these as the radiation behind and in front of the horizon), but for the pump mode as well. Basically, you start out with the pump in a mode with lots of photons in, and as they get down-converted the pump slowly depletes, until nothing is left. This will be the model of black hole evaporation, and this is precisely the approach that Alsing took, in a paper that appeared in the journal “Cassical and Quantum Gravity” last year.

“So Alsing solved it all”, you are thinking, “why then this blog post?”

Not so fast. Alsing brought us on the right track, to be sure, but his calculation of the quantum black hole entropy as a function of time displayed some weird features. The entropy appeared to oscillate rather than slowy decrease. What was going on here?

For you to appreciate what comes now, I need to write down the trilinear Hamiltonian:

\(H_{\rm tri}=r(ab^\dagger c^\dagger-a^\dagger bc)\) (5)

Here, the modes \(b\) and \(c\) are associated with radiation degrees in front of and behind the horizon, whereas \(a\) is the annihilation operator for black hole modes (the “pump” modes). Here’s a pic so that you can keep track of these.

Fig. 2: Black hole and radiation modes b and c. |

In the semi-classical approximation, the a modes are replaced with their background-field expectation value, which morphs \(H_tri\) into \(H\) in eqs. (3) and (4), so that’s wonderful: the trilinear Hamiltonian turns into the Hermitian operator implementing Hawking’s Bogoliubov transformation in the semi-classical limit.

But how you do you use \(H_tri\) to calculate the S-matrix I wrote down long ago, at the very beginning of this blog post? One thing you could do is to simply say,

\(U_{\rm tri}=e^{iH_{\rm tri}t}\) ,

and then the role of time is akin to a linearly increasing coefficient \(r\) in eq. (5). That’s essentially what Alsing did (and Nation and Blencowe before him, see also Paul Nation’s blog post about it) but that, it turns out, is only a rough approximation of the true dynamics, and does not give you the correct result, as we will see.

Suppose you calculate \(|\Psi_{\rm out}\rangle=e^{-iH_{\rm tri}t}|\Psi_{\rm in}\rangle\), and using the density matrix \(\rho_{\rm out}=|\Psi_{\rm out}\rangle \langle \Psi_{\rm out}|\) you calculate the von Neumann entropy of the black hole modes as

\(S_{\rm bh}=-{\rm Tr} \rho_{\rm out}\log \rho_{\rm out}\) (6)

Note that this entropy is exactly equal to the entropy of the radiation modes b together with c, as the initial black hole is in a pure state with zero entropy.

How can a black hole that starts with zero entropy lose entropy, you ask?

That’s a good question. We begin at \(t=0\) with a black hole in a defined state of n modes (the state \(|\Psi_{\rm in}\rangle=|n\rangle\)) for convenience of calculation. We could instead start in a mixed state, but the results would not be qualitatively different after the black hole has evolved for some time, yet the calculation would be much harder. Indeed, after interacting with the radiation the black hole modes become mixed anyway, and so you should expect the entropy to start rising from zero quickly at first, and only after it approached its maximum value would it decay. That is a behavior that black hole folks are fairly used to, as a calculation performed by Don Page in 1993 shows essentially (but not exactly) this behavior.

Page constructed an entirely abstract quantum information-theoretic scenario: suppose you have a pure bi-partite state (like we start out with here, where the black hole is one half of the bi-partitite state and the radiation field bc is the other), and let the two systems interact via random unitaries. Basically he asked: “What is the entropy of a subsystem if the joint system was in a random state?” The answer, as a function of the (log of the) size of the dimension of the radiation subsystem is shown here:

Fig. 3: Page curve (from [1]) showing first the increase in entanglement entropy of the black hole, and then a decrease back to zero. |

People usually assume that the dimension of the radation subsystem (dubbed by Page the “thermodynamic entropy” (as opposed to the entanglement entropy) is just a proxy for time, so that what you see in this “Page curve” is how at first the entropy of the black hole increases with time, then turns around at the “Page time”, until it vanishes.

This calculation (which has zero black hole physics in it) turned out to be extremely useful, as it showed that the amount of information from the black hole (defined as the maximum entropy minus the entanglement entropy) may take a long time to come out (namely at least the Page time), and it would be essentially impossible to determine from the radiation field that the joint field is actually in a pure state. But as I said, there is no black hole physics in it, as the random unitaries used in that calculation were, well, random.

Say you use the \(U_tri\) instead for the interaction? This is essentially the calculation that Alsing did, and it turns out to be fairly laborious, because as opposed to the bi-linear Hamiltonian that can be solved analyically, you can’t do that with \(U_tri\). Instead, you have to either expand \(H_tri\) in \(rt\) (that really only works for very short times) or use other methods. Alsing used an approximate partial differential equation approach for the quantum amplitude \(c_n(t)=\langle n|e^{-iH_{\rm tri}t}|\Psi_{\rm in}\rangle\). The result shows the increase of the black hole entropy with time as expected, and then indeed a decrease:

Fig. 4: Black hole entropy using (6) for n=16 as a function of t |

Actually, the figure above is not from Alsing (but very similar to his), but rather is one that Kamil Brádler made, but using a very different method. Brádler figured out a method to calculate the action of \(U_tri\) on a vacuum state using a sophisticated combinatorial approach involving something called a “Dyck path“. You can find this work here. It reproduces the short-time result above, but allows him to go much further out in time, as shown here:

Fig. 5: Black hole entropy as in Fig. 4, at longer times. |

The calculations shown here are fairly intensive numerical affairs, as in order to get converging results, up to 500 terms in the Taylor expansion have to be summed. This result suggests that the black hole entropy is not monotonically decreasing, but rather is oscillating, as if the black hole was absorbing modes from the surrounding radiation, then losing them again. However, this is extremely unlikely physically, as the above calculation is performed in the limit of perfectly reflecting black holes. But as we will see shortly, this calculation does not capture the correct physics to begin with.

What is wrong with this calculation? Let us go back to the beginning of this post, the time evolution of the quantum state in eqs. (1,2). The evolution operator \(U(t_2,t_1)=Te^{-i\int_{t_1}^{t_2}H(t’)dt’}\) is applied to the initial state gives rise to an integral over the state space: a path integral. How did that get replaced by just \(e^{-iHt}\)?

We can start by discretizing the integral into a sum, so that \(\int_0^t H(t’)dt’\approx\sum_{i=0}^NH(t_i)\Delta t\), where \(\Delta t\) is small, and \(N\Delta t=t\). And because that sum is in the exponent, \(U\) actually turns into a product:

\(U(0,t)\approx \Pi_{i=0}^N e^{-i\Delta t H(t_i)}\) (7)

Because of the discretization, each Hamiltonian \(H(t_i)\) acts on a different Hilbert space, and the ground state that \(U\) acts on now takes the form of a product state of time slices

\(||0\rangle_{bc}=|0\rangle_1|0\rangle_2\times\dots\times |0\rangle_N\)

And because of the time-ordering operator, we are sure that the different terms of \(U(0,t)\) are applied in the right temporal order. If all this seems strange and foreign to you, let me assure you that this is a completely standard approximation of the path integral in quantum many-body physics. In my days as a nuclear theorist, that was how we calculated expectation values in the shell model describing heavy nuclei. I even blogged about this approach (the Monte Carlo Path Integral approach) in the post about nifty paper that nobody is reading. (Incidentally, nobody is reading those posts either).

And now you can see why Alsing’s calculation (and Bradler’s initial recalculation of the same quantity with very different methods, confirming Alsing’s result) was wrong: it represents an approximation of (7) using a single time-slice only (\(N=1\)). This approximation has a name in quantum many-body physics, it is called the “Static Path Approximation” (SPA). The SPA can be accurate in some cases, but it is generally only expected to be good at small times. At larger times, it ignores the self-consistent temporal fluctuations that the full path integral describes.

So now you know what we did, of course: we calculated the path integral of the S-matrix of the black hole interacting with the radiation field using many many time slices. Kamil was able to do several thousand time slices, just to make sure that the integral converges. And the result looks very different from the SPA. Take a look at the figure below, where we calculated the black hole entropy as a function of the number of time slices (which is our discretized time)

This plot shows that the entropy quickly increase as the pure state decoheres, and then starts to drop because of evaporation. Obviously, if we would start with a mixed state rather than a pure state, the entropy would just drop. The rapid increase at early times is just a reflection of our short-cut to start with a pure state. It doesn’t look exactly like Page’s curves, but we cannot expect that as our x-axis is indeed time, while Page’s was thermodynamic entropy (which is expected to be linear in time). Note that Kamil repeated the calculation using an even smaller \(\Delta t=1/25\), and the results do not change.

I want to throw out some caution here. The tri-linear Hamiltonian is not derived from first principles (that is, from a quantum theory of gravity). It is a “guess” at what the interaction term between quantized black hole modes and radiation modes might look like. The guess is good enough that it reproduces standard CSQFT in the semi-classical limit, but it is still a guess. But it is also very satisfying that such a guess allows you to perfrom a straightforward calculation of black hole entropy as a function of time, showing that the entropy can actually get back out. One of the big paradoxes of black hole physics was always that as the black hole mass shrunk, all calculations implied that the entanglement entropy steadily increases and never turns over as in Page’s calculation. This was not a tenable situation for a number of physical reasons (and this is such a long post that I will spare you these). We have now provided a way in which this can happen.

So now you have seen with your own eyes what may happen to a black hole as it evaporates. The entropy can indeed decrease, and within a simple “extended Hawking theory”, all of it gets out. This entropy is not information mind you, as there is no information in a black hole unless you throw some in it (see my series “What is Information?” if this is cryptic for you). But Steve Giddings convinced me (on the beach at Vieques no less, see photo below) that solving the infomation paradox was not enough: you’ve got to solve the entropy paradox also.

A quantum gravity session at the beach in Vieques, Puerto Rico (January 2014). Steve Giddings is in sunglasses watching me explain stimulated emission in black holes. |

I should also note that there is a lesson in this calculation for the firewall folks (who were quite vocal at the Vieques meeting). Because the entanglement between the black hole and radiation involves three entities rather than two, monogamy of entanglement can never be violated, so this argument provides another (I have shown you two others in earlier posts) arguments against those silly firewalls.

The paper describing these results is on arXiv:

K. Brádler and C. Adami: One-shot decoupling and Page curves from a dynamical model for black hole evaporation

[1] Don Page. Average entropy of a subystem. Phys. Rev. Lett. 71 (1993) 1291.

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