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On page 8 of the paper "Soft Hair on Black Holes" the authors say:

Let $|M\rangle$ denote the incoming quantum state of a black hole defined on $\mathcal{H}$. We take it to be formed with neutral matter so that $j =0$ on $\mathcal{H}$. Then $$Q_\varepsilon^+|M\rangle=\left(\dfrac{1}{e^2}\int_{\mathcal{H}}d\varepsilon\wedge \star F\right)|M\rangle\neq0,$$

is $|M\rangle$ with an additional soft photon of polarization $d\varepsilon$.

Now the authors are clearly talking about "quantum states of a black hole". Furthermore, they say that the state "is defined on the horizon".

Classically a black hole is a region of spacetime. How does it make sense to talk about "quantum states of a region of spacetime"? Or are they actually taking about the quantum state of the matter that formed the black hole by collapse?

I'm confused with that. What are these quantum states of a black hole? Why they are defined on the horizon? What is the corresponding Hilbert space and how does that get constructed?

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I'm not very well versed on quantum gravity, but I think the following is basically the right idea in case nobody more knowledgeable can provide an answer.

In classical GR, we have a set of theorems that are in perfect analogy with the laws of thermodynamics. The counterpart of the second law is a statement that the total area of black holes always increases.

Because entropy as understood in ordinary thermodynamics has a statistical interpretation as the log of the number of accessible states, this makes it natural to hypothesize that the information about the quantum state of a black hole can somehow be treated as living on the event horizon.

Classically a black hole is a region of spacetime. How does it make sense to talk about "quantum states of a region of spacetime"?

In a theory of quantum gravity, we would expect a vacuum spacetime to have just as much structure as the QFT vacuum, which is a complicated object. In semiclassical terms, we expect there to be things like virtual graviton pairs and such.

Or are they actually taking about the quantum state of the matter that formed the black hole by collapse?

The question of what happens to the information content of the infalling matter is the black hole information paradox, which is an open question and above my pay grade.

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