The Senior Common Room at St Anselm's was unusually quiet.
Professor Quillibrace was standing at the fireplace, holding a sheet of paper at arm's length as though it had personally offended him.
Mr Blottisham was in his usual chair.
Miss Elowen Stray was by the window, reading the final page of the essay.
After a considerable silence, she looked up.
Miss Stray: Well.
Mr Blottisham: That's never a promising beginning.
Miss Stray: I think I've just read an essay in which a black hole has eaten our vocabulary.
Professor Quillibrace: An improvement, surely.
Mr Blottisham: On what?
Professor Quillibrace: Our vocabulary.
Blottisham looked at him.
Mr Blottisham: I was afraid you meant the black hole.
Professor Quillibrace: It is possible to mean both.
Stray smiled.
Miss Stray: The interesting thing is that the essay began with a problem that sounded almost embarrassingly straightforward.
Mr Blottisham: Matter falls into a black hole. The black hole radiates. The black hole disappears. Information is missing.
Professor Quillibrace: Precisely.
Mr Blottisham: And now we're being told that none of those sentences should be taken quite literally.
Professor Quillibrace: That is usually what happens when physicists become philosophers.
Miss Stray: Or philosophers become sufficiently suspicious of nouns.
Professor Quillibrace: An excellent habit.
Mr Blottisham: I don't know. At some point one has to be able to say what happened.
Professor Quillibrace: Yes.
Mr Blottisham: Good.
Professor Quillibrace: The difficulty is that we may have to learn what happened means before we can say what happened.
Blottisham sighed.
Mr Blottisham: There it is.
Miss Stray: What?
Mr Blottisham: The sentence that means we're going to be here all afternoon.
The black hole that isn't a box
Miss Stray: Let me try the simplest version.
At the beginning, we imagined the black hole as a container.
Something falls in.
It is stored inside.
The outside world cannot retrieve it.
Then the container evaporates.
Therefore something has disappeared.
Mr Blottisham: That sounds perfectly reasonable.
Professor Quillibrace: Which is precisely why it is dangerous.
Mr Blottisham: Why?
Professor Quillibrace: Because the picture contains an object called the black hole and another thing called the information, and the first is imagined as containing the second.
Mr Blottisham: But things do fall into black holes.
Professor Quillibrace: Certainly.
Mr Blottisham: And they don't come out.
Professor Quillibrace: Certainly.
Mr Blottisham: So what exactly is wrong with saying the information went inside?
Miss Stray: Perhaps nothing, provided we remember that inside is already doing conceptual work.
Blottisham looked from one to the other.
Mr Blottisham: You two have been conspiring.
Professor Quillibrace: No.
Miss Stray: Not consciously.
Professor Quillibrace: Which is generally the most effective form of conspiracy.
Mr Blottisham: So where is the information?
Quillibrace turned from the fireplace.
Professor Quillibrace: That may be the wrong question.
Mr Blottisham: Again?
Professor Quillibrace: Again.
Miss Stray: The deeper question is what happens to the distinctions and correlations represented by the initial quantum state.
Mr Blottisham: That sounds suspiciously like refusing to answer.
Professor Quillibrace: On the contrary. It is an attempt to answer a more precise question.
Mr Blottisham: I want the less precise one.
Professor Quillibrace: I had noticed.
What exactly is being preserved?
Stray placed the essay on the table.
Miss Stray: I think the most important sentence is the one about preservation of difference.
Mr Blottisham: Why?
Miss Stray: Because it strips away almost all the imagery.
Suppose two different initial quantum states collapse into black holes.
Call them A and B.
If the complete final state cannot distinguish A from B, then something has happened that is deeper than losing a message.
The distinction between two physically different possibilities has disappeared.
Professor Quillibrace: Precisely.
Miss Stray: So perhaps information is useful here because it gives us a mathematical vocabulary for the persistence of distinctions.
Mr Blottisham: You're making it sound terribly abstract.
Miss Stray: It is abstract.
Mr Blottisham: But the original problem isn't abstract. Something either preserves the information or it doesn't.
Professor Quillibrace: And that sentence is itself instructive.
Mr Blottisham: Why?
Professor Quillibrace: Because the information sounds like an object.
Mr Blottisham: You keep saying that.
Professor Quillibrace: Because you keep saying it.
Miss Stray: Perhaps that's the real achievement of the series.
We've been using ordinary grammar to describe something that may not have the structure ordinary grammar suggests.
Mr Blottisham: Grammar?
Professor Quillibrace: Yes.
Mr Blottisham: We're discussing black holes.
Professor Quillibrace: Exactly.
The grammar of information
Miss Stray: Think about the verbs.
Information is in the black hole.
Information comes out.
Information is lost.
Information is stored.
Information is transferred.
Every one of those expressions gives information the grammatical behaviour of a physical object.
Mr Blottisham: And perhaps it is useful to do so.
Miss Stray: Absolutely.
Professor Quillibrace: That qualification is important.
Miss Stray: The metaphors aren't bad because they're metaphors.
They're useful because they're metaphors.
The trouble begins when we forget the "as if".
Mr Blottisham: So the container metaphor isn't false?
Professor Quillibrace: It is inadequate if interpreted literally.
Mr Blottisham: Which is a philosopher's way of saying false.
Professor Quillibrace: No. It is a physicist's way of saying don't put that sentence into the ontology.
Miss Stray: The metaphor may organise a useful calculation without telling us what the world fundamentally consists of.
Mr Blottisham: That's rather like the membrane paradigm.
Professor Quillibrace: Indeed.
Mr Blottisham: The horizon behaves as if it were a membrane.
Professor Quillibrace: For certain purposes.
Mr Blottisham: But it isn't a membrane.
Professor Quillibrace: Correct.
Mr Blottisham: Yet the metaphor works.
Professor Quillibrace: Very well.
Mr Blottisham: Then perhaps the problem isn't metaphor.
Miss Stray: It's unexamined metaphor.
Quillibrace nodded.
Professor Quillibrace: Now we are getting somewhere.
The horizon that isn't there
Stray glanced towards the window.
Miss Stray: And the horizon is the perfect example.
We naturally imagine a line.
A surface.
A boundary.
Something between inside and outside.
But the event horizon isn't a material surface.
Mr Blottisham: We know that.
Professor Quillibrace: Do we?
Mr Blottisham: Yes.
Professor Quillibrace: Then why do you keep asking where the information is behind it?
Blottisham paused.
Mr Blottisham: Fair point.
Miss Stray: The horizon is better understood as a boundary in causal possibility.
It distinguishes events from which signals can reach a distant observer from events from which they cannot.
Mr Blottisham: So it's a boundary of communication.
Professor Quillibrace: More precisely, a boundary in the causal structure that determines possible communication.
Mr Blottisham: You couldn't just say yes.
Professor Quillibrace: I could.
Miss Stray: But then he'd be disappointed.
Mr Blottisham: I am already disappointed.
What the observer can know
Miss Stray: There is something rather beautiful here.
The horizon doesn't merely prevent us from seeing something.
It constrains what physical relationships can exist between events.
Mr Blottisham: That's a strong claim.
Professor Quillibrace: It is also the important one.
Miss Stray: If a signal cannot get from A to B, then no information can be transmitted from A to B by that signal.
So causal structure constrains information.
Mr Blottisham: Which means that accessibility is not merely epistemic.
Professor Quillibrace: Exactly.
Mr Blottisham: If I can't see something because I've misplaced my glasses, that's contingent.
Miss Stray: But if an event lies beyond an event horizon relative to you, no technological improvement will allow its signal to reach you.
Mr Blottisham: So spacetime itself determines the limits of information accessibility.
Professor Quillibrace: And that is why the information paradox is not simply an information problem.
Miss Stray: It is a problem about information, causality and geometry simultaneously.
Blottisham leaned back.
Mr Blottisham: I admit that's rather more interesting than a missing document.
The Page curve enters the room
Mr Blottisham: But surely this is where the Page curve becomes important.
Professor Quillibrace: At last.
Mr Blottisham: I knew I'd get there eventually.
Miss Stray: The Page curve gives us something remarkable.
It translates the vague idea of "information preservation" into a prediction about the changing entanglement structure of the radiation.
Mr Blottisham: Early radiation becomes increasingly entangled with the remaining black hole.
Professor Quillibrace: And if evaporation is unitary, after the Page time the pattern must reverse.
Miss Stray: The radiation entropy eventually falls.
Mr Blottisham: Meaning the later radiation must contain correlations with the earlier radiation.
Professor Quillibrace: More precisely, the complete radiation state must eventually become pure if the initial state was pure and the total evolution is unitary.
Mr Blottisham: Which means the information doesn't emerge as a little message.
Miss Stray: Exactly.
Mr Blottisham: It emerges as correlation.
Professor Quillibrace: That is the crucial point.
Mr Blottisham: So when we say that the information comes out, we're already misleading ourselves.
Miss Stray: We are using the language of transport for a change in relational structure.
Quillibrace smiled faintly.
Professor Quillibrace: I believe that is the first sensible sentence you have spoken this afternoon.
Mr Blottisham: I shall have it engraved.
The parts don't know
Miss Stray: Consider the strange thing about quantum entanglement.
The individual components can look random.
The whole state can nevertheless contain enormous structure.
Mr Blottisham: So the information isn't necessarily in the parts.
Professor Quillibrace: Correct.
Mr Blottisham: It's in the relations.
Miss Stray: Yes.
Mr Blottisham: Which means that if I inspect each Hawking quantum separately, I may see nothing informative.
Professor Quillibrace: Yet the complete radiation state may still encode the initial distinctions.
Mr Blottisham: So the whole can contain something that none of the parts contains individually.
Professor Quillibrace: That is one way of putting it.
Miss Stray: Though I would be slightly careful with "contain".
Quillibrace raised an eyebrow.
Mr Blottisham: Of course.
Miss Stray: The word has become contaminated.
Professor Quillibrace: Quite.
Mr Blottisham: You two have ruined the entire English language.
Miss Stray: Only the nouns.
Professor Quillibrace: The verbs are still under investigation.
The holographic temptation
Blottisham picked up the essay.
Mr Blottisham: And then we arrive at the holographic principle.
Professor Quillibrace: Ah.
Mr Blottisham: You sound worried.
Professor Quillibrace: I am always worried when physicists borrow a word from a machine that produces pictures.
Miss Stray: Yet "holographic" is a wonderfully suggestive metaphor.
Mr Blottisham: Because a three-dimensional image can be encoded on a two-dimensional surface.
Professor Quillibrace: Yes.
Mr Blottisham: And black-hole entropy scales with area rather than volume.
Professor Quillibrace: Yes.
Mr Blottisham: So the boundary appears to carry enough information to describe the interior.
Professor Quillibrace: That is where I would ask you to stop.
Mr Blottisham: Why?
Professor Quillibrace: Because "carry enough information" sounds like a cosmic filing cabinet.
Miss Stray: The deeper idea is not that there is literally a little two-dimensional computer on the horizon.
Mr Blottisham: I wasn't imagining a computer.
Professor Quillibrace: You were about to.
Mr Blottisham: I was not.
Miss Stray: You were.
Mr Blottisham: Fine.
A pause.
Mr Blottisham: Slightly.
Professor Quillibrace: The important point is about equivalence of descriptions.
Miss Stray: A gravitational theory in a higher-dimensional bulk can, in certain settings, be equivalent to a nongravitational theory on a lower-dimensional boundary.
Mr Blottisham: So the boundary doesn't necessarily contain the bulk.
Professor Quillibrace: Correct.
Miss Stray: The boundary description may encode the same physical content in a different set of degrees of freedom.
Mr Blottisham: Which means that "inside" and "boundary" may not be two independently existing things.
Professor Quillibrace: Now you're learning.
The disappearing interior
Miss Stray: This may be the most radical possibility in the whole essay.
Perhaps the interior itself is emergent.
Mr Blottisham: You mean the inside isn't really there?
Professor Quillibrace: That formulation is precisely the sort of thing we should avoid.
Mr Blottisham: Why?
Professor Quillibrace: Because "really there" has become meaningless unless you specify the level of description.
Miss Stray: The interior could be perfectly real as an emergent spacetime description while not being fundamental in the underlying theory.
Mr Blottisham: Like temperature?
Professor Quillibrace: An excellent example.
Mr Blottisham: Temperature is real.
Professor Quillibrace: Certainly.
Mr Blottisham: But it isn't a microscopic object.
Miss Stray: Exactly.
Mr Blottisham: So the interior could be real without being fundamental.
Professor Quillibrace: That is a possibility.
Mr Blottisham: And then asking how information gets from the interior to the exterior would be like asking how temperature gets from one level of description to another.
Miss Stray: Perhaps.
Professor Quillibrace: Though we should not confuse an analogy with an established theory.
Mr Blottisham: You really do have to ruin everything.
Professor Quillibrace: It is my vocation.
Complementarity
Miss Stray: And then there is complementarity.
Mr Blottisham: The outside observer sees one story.
The infalling observer sees another.
Professor Quillibrace: Yes.
Mr Blottisham: And both can be valid.
Professor Quillibrace: Within their respective descriptions.
Mr Blottisham: Which sounds contradictory.
Miss Stray: Only if we assume there must be one classical picture containing both accounts simultaneously.
Mr Blottisham: Relativity already made that assumption difficult.
Professor Quillibrace: And quantum theory makes it still more difficult.
Miss Stray: Perhaps the mistake is thinking that perspective merely means "what somebody happens to know".
Mr Blottisham: Whereas here the observer's causal position is part of the physical situation.
Professor Quillibrace: Exactly.
Miss Stray: Access is relational.
Mr Blottisham: Reality isn't subjective.
Professor Quillibrace: No.
Mr Blottisham: But what can physically be related to what depends upon causal structure.
Professor Quillibrace: Yes.
Mr Blottisham: That's a very different thing.
Miss Stray: And perhaps a very important one.
The firewall
Blottisham frowned.
Mr Blottisham: I still don't like the firewall.
Professor Quillibrace: Nor does anyone who enjoys falling into black holes.
Miss Stray: The firewall is interesting because it exposes the conflict among assumptions.
Mr Blottisham: Unitarity.
Smooth horizon.
Local quantum field theory.
Semiclassical geometry.
Professor Quillibrace: Exactly.
Mr Blottisham: And the trouble is that they may not all fit together.
Miss Stray: Which means the question isn't really whether there is literally a wall of fire.
Professor Quillibrace: The name is metaphorical.
Mr Blottisham: Naturally.
Professor Quillibrace: The underlying issue is whether the entanglement structure required for information preservation is compatible with a smooth horizon.
Mr Blottisham: So even the most dramatic metaphor in the debate is really shorthand for a structural conflict.
Miss Stray: Yes.
Mr Blottisham: That seems to be happening rather often.
Professor Quillibrace: One might almost call it a pattern.
What has actually survived?
There was a longer silence.
Stray looked again at the final page.
Miss Stray: I think there's something deeper here than the information paradox itself.
Mr Blottisham: Go on.
Miss Stray: We've spent the entire series asking what happens to information.
But perhaps we've gradually discovered that the more fundamental question is:
What does it mean for a distinction to survive a transformation?
Professor Quillibrace: Yes.
Miss Stray: A physical state changes.
Its organisation changes.
Its correlations change.
Its accessible structure changes.
Its geometry may change.
Yet we ask whether something about the distinction between possible initial states remains invariant.
Mr Blottisham: So information is functioning as a way of talking about invariance.
Professor Quillibrace: Among other things.
Miss Stray: And that makes the problem much larger than black holes.
Mr Blottisham: Larger than black holes is generally a bad sign.
Professor Quillibrace: In this case, it may be the point.
A question about identity
Mr Blottisham: Let me try something.
Suppose state A becomes state A-prime.
And state B becomes state B-prime.
What matters is that A-prime and B-prime remain distinguishable.
Professor Quillibrace: Yes.
Mr Blottisham: Even if neither A-prime nor B-prime looks remotely like A or B.
Miss Stray: Exactly.
Mr Blottisham: Then preservation doesn't mean preservation of form.
Professor Quillibrace: Very good.
Mr Blottisham: It means preservation of distinction.
Miss Stray: Or, more carefully, preservation of the relevant distinctions in the complete physical description.
Mr Blottisham: You had to add that.
Professor Quillibrace: She did.
Mr Blottisham: But I see the point.
A melody can be transformed beyond recognition and yet its structure can survive.
Miss Stray: Yes.
Professor Quillibrace: Provided we remember that the musical analogy is only an analogy.
Mr Blottisham: I refuse to answer.
Translation
Miss Stray: Which brings us to the final metaphor.
Mr Blottisham: Translation.
Professor Quillibrace: A surprisingly good one.
Mr Blottisham: I thought you'd object.
Professor Quillibrace: I object selectively.
Miss Stray: Translation is different from transport.
A translated sentence does not carry the original words into another language.
It reorganises the expression while attempting to preserve something structurally important.
Mr Blottisham: Meaning.
Professor Quillibrace: In the appropriate sense.
Miss Stray: And perhaps the analogy helps here.
The quantum state need not remain in the same form.
The geometry need not remain in the same form.
The causal organisation changes.
The correlations change.
But perhaps certain distinctions remain invariant across the transformation.
Mr Blottisham: So black-hole evaporation might be thought of as a translation between physical descriptions.
Professor Quillibrace: As a metaphor, yes.
Mr Blottisham: Quantum state into spacetime.
Spacetime into causal structure.
Interior into boundary.
Black hole into radiation.
Miss Stray: Local description into global relation.
Professor Quillibrace: That last one is particularly interesting.
The question underneath the question
Mr Blottisham: Then perhaps the original question really was badly phrased.
Professor Quillibrace: Which original question?
Mr Blottisham: "Where did the information go?"
Miss Stray: Yes.
Mr Blottisham: Because it assumes information is a thing that can go somewhere.
Professor Quillibrace: Precisely.
Mr Blottisham: The better question is something like:
"What remains invariant when the physical organisation changes?"
Miss Stray: Or:
"How are the distinctions encoded in one description related to the distinctions encoded in another?"
Professor Quillibrace: Or, if we wish to be even more severe:
"Can gravitational evolution erase distinctions that unitary quantum theory requires to remain encoded in the complete state?"
Blottisham considered this.
Mr Blottisham: That is much less catchy.
Professor Quillibrace: Most accurate questions are.
Miss Stray: But perhaps the catchiness of the original question was part of the problem.
The paradox survives
Mr Blottisham: There's one thing I want to insist upon.
We haven't actually solved the paradox.
Professor Quillibrace: Correct.
Mr Blottisham: We've just stopped saying silly things about it.
Miss Stray: That is not insignificant.
Mr Blottisham: No.
He looked down at the essay.
Mr Blottisham: But the physics still has a problem.
Professor Quillibrace: A profound one.
Mr Blottisham: We still don't have one universally accepted account that reconciles quantum theory, gravity, Hawking radiation, entanglement, causal structure and the apparent smoothness of the horizon.
Professor Quillibrace: Correct.
Mr Blottisham: So after all this, the black hole remains mysterious.
Miss Stray: Yes.
Mr Blottisham: Good.
Professor Quillibrace: Why "good"?
Mr Blottisham: Because otherwise we'd have spent all this time removing metaphors only to discover that someone had already solved it.
Quillibrace gave him a rare smile.
The metaphors were not the enemy
Miss Stray: I don't think the lesson is that we should stop using metaphors.
Professor Quillibrace: Quite the contrary.
Mr Blottisham: We couldn't have begun without them.
Miss Stray: The container gave us a way to think about what falls in.
The message gave information physical significance.
The storage metaphor connected information with entropy.
The wall made causal inaccessibility vivid.
The membrane translated aspects of horizon physics into familiar terms.
The hologram opened the possibility of boundary descriptions.
Professor Quillibrace: Each metaphor made something thinkable.
Miss Stray: And each eventually showed us something it could not explain.
Mr Blottisham: So the metaphor succeeds by eventually becoming inadequate.
Professor Quillibrace: Sometimes.
Mr Blottisham: That's a rather peculiar definition of success.
Professor Quillibrace: Not in science.
Miss Stray: A metaphor is often most valuable when it gets us far enough to discover its own limits.
Blottisham nodded slowly.
Mr Blottisham: So the danger isn't metaphor.
Professor Quillibrace: No.
Mr Blottisham: It's forgetting that the metaphor is there.
Miss Stray: Exactly.
The black hole that swallowed categories
The fire had burned low.
Stray closed the essay.
Miss Stray: There's one line near the end I rather like.
Mr Blottisham: Which one?
Miss Stray: That the black hole has swallowed our ordinary categories.
Professor Quillibrace: A good line.
Mr Blottisham: Slightly dramatic.
Professor Quillibrace: Therefore suitable for publication.
Miss Stray: But I think it captures something.
At first we had:
objects,
locations,
containers,
inside,
outside,
messages,
movement,
loss.
Now we have:
states,
relations,
correlations,
entanglement,
causal structure,
duality,
emergence,
invariance.
Mr Blottisham: That's quite a change of vocabulary.
Professor Quillibrace: More importantly, a change of ontology.
Miss Stray: Or perhaps a change in what we take the ontology to require.
Professor Quillibrace: Quite.
Mr Blottisham: And the strange thing is that none of the new concepts is really a "thing".
Miss Stray: Exactly.
Mr Blottisham: Relations.
Structures.
Correlations.
Possible transformations.
Professor Quillibrace: Which brings us back to the beginning.
Mr Blottisham: Does it?
Professor Quillibrace: Yes.
Miss Stray: The information was never quite the thing we thought it was.
A boundary of thought
Stray stood and walked towards the fireplace.
Miss Stray: Perhaps there's a final symmetry.
The black hole has an event horizon.
A boundary beyond which a particular causal description cannot reach.
And perhaps our theories have something analogous.
A conceptual horizon.
A point beyond which the categories that worked perfectly well before no longer organise the problem adequately.
Mr Blottisham: So the information paradox is a boundary between conceptual regimes.
Professor Quillibrace: Perhaps.
Miss Stray: On one side:
things,
places,
containers,
messages,
movement.
On the other:
states,
relations,
entanglement,
duality,
emergence.
Mr Blottisham: And we're trying to describe the second using the vocabulary of the first.
Professor Quillibrace: Which may explain some of the confusion.
Mr Blottisham: Then perhaps physics doesn't need another metaphor.
Miss Stray: It probably does.
Mr Blottisham: I thought you said metaphors were the problem.
Professor Quillibrace: She said unexamined metaphors were the problem.
Miss Stray: And besides, we can't stop making metaphors.
Mr Blottisham: Why not?
Miss Stray: Because we have to think somehow.
The last question
For a while nobody spoke.
Then Blottisham looked at the empty fireplace.
Mr Blottisham: There's something I still can't get out of my head.
Professor Quillibrace: That is usually a dangerous sign.
Mr Blottisham: Suppose the universe changes the language in which its relationships are expressed.
Suppose the same physical content can appear as geometry in one description and quantum correlations in another.
Suppose an interior can emerge from relationships that, at the fundamental level, don't look like an interior at all.
Then what exactly does it mean to say that information has been preserved?
Quillibrace did not answer immediately.
Stray looked towards him.
Miss Stray: Perhaps it means that something remains distinguishable.
Mr Blottisham: Even though everything about its form has changed?
Miss Stray: Yes.
Professor Quillibrace: That would be one way of putting it.
Mr Blottisham: And what is that "something"?
Quillibrace looked at the dying fire.
Professor Quillibrace: Perhaps that is the question.
Miss Stray: Not a thing.
Mr Blottisham: A relation?
Professor Quillibrace: Perhaps.
Miss Stray: A pattern of possible distinctions.
Professor Quillibrace: Perhaps.
Mr Blottisham: And if we knew what that was, we'd know how information survives gravitational transformation.
Professor Quillibrace: We might.
There was another silence.
Then Blottisham stood.
Mr Blottisham: Well.
Miss Stray: That sounds ominous.
Mr Blottisham: I think I need a drink.
Professor Quillibrace: At last, a physical process we understand.
Miss Stray: And one with an event horizon?
Mr Blottisham: Several, if Professor Quillibrace continues talking.
Quillibrace picked up his glass.
Professor Quillibrace: To the persistence of difference.
Miss Stray: To new relations.
Mr Blottisham: To metaphors, provided nobody mistakes them for reality.
They raised their glasses.
All three: Chin-chin.