The Senior Common Room and the Vanishing Black Hole

The Senior Common Room was unusually quiet.

Professor Quillibrace was reading the latest essay with a pencil in hand. Mr Blottisham had already reached the end and was looking dissatisfied.

Miss Elowen Stray was gazing into the middle distance in the way that usually meant she had noticed something nobody else had.

Mr Blottisham: I think we've made this unnecessarily complicated.

Professor Quillibrace: That is always an encouraging opening.

Mr Blottisham: The black hole evaporates. It loses energy. Its mass goes down. Its horizon gets smaller. Eventually the horizon disappears.

Professor Quillibrace: Yes.

Mr Blottisham: So what's the difficulty?

Professor Quillibrace: What happens next?

Mr Blottisham: Everything that was inside comes out.

Professor Quillibrace put down his pencil.

Professor Quillibrace: Why?

Mr Blottisham: Because there's no longer anything keeping it in.

Professor Quillibrace: What was keeping it in?

Mr Blottisham: The black hole.

Professor Quillibrace: That is not an answer.

Mr Blottisham: The gravity.

Professor Quillibrace: Also not quite an answer.

Mr Blottisham: The horizon, then.

Professor Quillibrace: Better. What is the horizon?

Mr Blottisham paused.

Mr Blottisham: The boundary of the black hole.

Professor Quillibrace: And what sort of boundary?

Mr Blottisham: The sort that stops things getting out.

Professor Quillibrace: You have just described a wall.

Mr Blottisham: I know what a horizon is.

Professor Quillibrace: I am beginning to wonder.

Miss Stray smiled.

Miss Stray: We seem to have arrived at the same problem again.

Mr Blottisham: Which problem?

Miss Stray: We keep turning a relationship into a thing.

Mr Blottisham: I don't see how that applies to evaporation.

Professor Quillibrace: It applies rather perfectly.

He picked up his pencil again.

Professor Quillibrace: We have inherited a very convenient picture. There is a black hole. It has an inside. Things fall inside. The horizon keeps them there. The black hole emits radiation. It gets smaller. Eventually the horizon disappears. Therefore the things inside are released.

He looked at Mr Blottisham.

Professor Quillibrace: It is almost a complete story.

Mr Blottisham: Yes.

Professor Quillibrace: And almost none of it should be taken literally.

There was a brief silence.

Miss Stray: Perhaps we should begin with the word itself.

Mr Blottisham: Evaporation?

Miss Stray: Yes.

Professor Quillibrace: An excellent place to start.

---

The puddle

Miss Stray: When water evaporates, we have a thing that remains recognisably the same kind of thing while losing some of its contents.

Mr Blottisham: Exactly.

Miss Stray: Molecules leave the puddle. The puddle gets smaller.

Professor Quillibrace: And eventually there is no puddle.

Miss Stray: So when we hear that a black hole evaporates, we naturally imagine the same sort of process.

Mr Blottisham: Which seems reasonable.

Professor Quillibrace: Up to a point.

Mr Blottisham: What point?

Professor Quillibrace: The black hole does lose mass.

Mr Blottisham: Good.

Professor Quillibrace: And for a simple non-rotating black hole, its characteristic horizon scale decreases as its mass decreases.

Mr Blottisham: Also good.

Professor Quillibrace: But that does not mean that there is a material black-hole substance which is gradually being removed from a container.

Mr Blottisham: I didn't say there was.

Professor Quillibrace: No. You merely keep talking as though there were.

Miss Stray laughed.

Mr Blottisham: All right. So what is actually changing?

Professor Quillibrace: The physical state of the system evolves. The spacetime geometry associated with that state evolves with it. And therefore the causal structure changes.

Mr Blottisham: That's an awfully elaborate way of saying the black hole gets smaller.

Professor Quillibrace: It is precisely the elaborate part that matters.

Miss Stray: Because "gets smaller" makes it sound as though there were a little black object whose boundary was contracting.

Professor Quillibrace: Whereas the horizon is not a material skin.

Mr Blottisham: Fine. The horizon isn't a skin.

Professor Quillibrace: Progress.

---

What is doing the trapping?

Mr Blottisham: But surely something still has to explain why things can't get out.

Professor Quillibrace: Certainly.

Mr Blottisham: So what is it?

Professor Quillibrace: The causal structure of spacetime.

Mr Blottisham: Which means?

Professor Quillibrace: Inside the event horizon, future-directed causal trajectories do not lead back out to distant infinity.

Mr Blottisham: So they're trapped.

Professor Quillibrace: In the relevant causal sense, yes.

Mr Blottisham: Then if the geometry changes enough, presumably they stop being trapped.

Professor Quillibrace: Now we have reached something interesting.

Miss Stray looked up.

Miss Stray: Because that sounds almost like what you were saying before.

Mr. Blottisham: Exactly.

Professor Quillibrace: Your intuition is pointing in a useful direction. But we have to formulate it carefully.

Mr Blottisham: I thought you were going to say I was wrong.

Professor Quillibrace: I generally prefer to say that you are approximately right in a dangerously misleading vocabulary.

Mr Blottisham: Thank you.

Professor Quillibrace: You're welcome.

---

The photon

Miss Stray: What happens to a photon that has fallen in?

Mr Blottisham: It can't get out.

Professor Quillibrace: For a sufficiently stationary black hole, once it is genuinely inside the event horizon, yes.

Miss Stray: And if the black hole is evaporating?

Professor Quillibrace paused.

Professor Quillibrace: Then we have to be more careful.

Mr Blottisham: Why?

Professor Quillibrace: Because an event horizon is not defined merely by asking whether light can escape at this particular moment.

Miss Stray: It is defined globally.

Professor Quillibrace: Precisely.

Mr Blottisham: Meaning?

Professor Quillibrace: Roughly, the event horizon separates events whose future can reach distant infinity from events whose future cannot.

Mr Blottisham: So it depends on the future.

Professor Quillibrace: Yes.

Mr Blottisham: That's rather inconvenient.

Professor Quillibrace: General relativity has never been particularly concerned with our convenience.

Miss Stray smiled.

Miss Stray: So if the complete future geometry says that the photon eventually reaches infinity, then strictly speaking it was not in the event-horizon region after all?

Professor Quillibrace: That is the subtle point.

Mr Blottisham: But how could the photon know that?

Professor Quillibrace looked at him.

Professor Quillibrace: It doesn't.

Mr Blottisham: Then why are we talking about the future?

Professor Quillibrace: Because the event horizon is a feature of the entire spacetime, not a local object announcing its presence to the photon.

Miss Stray: So the photon simply follows its worldline.

Professor Quillibrace: Exactly.

Mr Blottisham: Through a changing geometry.

Professor Quillibrace: Yes.

Mr Blottisham: Which means it isn't sitting there waiting for the horizon to move out of the way.

Professor Quillibrace: Now you are beginning to see the problem.

---

The door that isn't there

Mr Blottisham: But surely if the horizon eventually disappears, something changes.

Professor Quillibrace: Certainly.

Mr Blottisham: So why can't we say that the door opens?

Professor Quillibrace: Because there was no door.

Mr Blottisham: It's a metaphor.

Professor Quillibrace: A very persistent one.

Miss Stray: And perhaps an especially dangerous one here.

Mr Blottisham: Why?

Miss Stray: Because it makes us imagine that the objects inside remain in the same region while the boundary retreats around them.

Professor Quillibrace: Precisely.

Mr Blottisham: But isn't that roughly what happens?

Professor Quillibrace: No. The geometry itself is changing.

Mr Blottisham: So there isn't a fixed interior with a shrinking boundary?

Professor Quillibrace: Not in the simple sense your picture assumes.

Miss Stray: The distinction is important. We are not watching a room get smaller until its walls disappear.

Professor Quillibrace: Nor are we watching a room become larger until its occupants discover that the door is open.

Mr Blottisham: Then what are we watching?

Professor Quillibrace: A dynamical spacetime.

---

The room that isn't a room

Miss Stray looked thoughtful.

Miss Stray: Perhaps this is the same mistake we made with information.

Mr Blottisham: How?

Miss Stray: We asked where the information was.

Professor Quillibrace: And discovered that "where" may not be the right question.

Miss Stray: Now we ask where the objects inside the black hole go when the black hole disappears.

Professor Quillibrace: And again, perhaps the question contains too much of the answer.

Mr Blottisham: You mean "inside" is doing the same kind of work?

Professor Quillibrace: Exactly.

Mr Blottisham: But things really do fall into black holes.

Professor Quillibrace: Certainly.

Mr Blottisham: So surely they really are inside.

Professor Quillibrace: In the semiclassical spacetime description, yes.

Mr Blottisham: You keep saying "in the description".

Professor Quillibrace: Because we are discussing a problem at the boundary between general relativity and quantum theory. It would be unwise to assume that every feature of the semiclassical description survives unchanged into the ultimate quantum-gravitational account.

Miss Stray: So "inside" may be real without being fundamental.

Professor Quillibrace: That is an excellent way to put it.

---

But what actually happens?

Mr Blottisham: I still want to know what happens to the things that fell in.

Professor Quillibrace: So do I.

Mr Blottisham: Good.

Professor Quillibrace: Unfortunately, that is where our confidence ends.

Mr Blottisham: Surely physics knows what happens when the black hole evaporates.

Professor Quillibrace: Physics knows a great deal about the early and intermediate stages. It does not possess a universally accepted complete description of the final quantum-gravitational stage.

Miss Stray: So the disappearance of the horizon is not something we can simply extrapolate all the way down to zero size with confidence.

Professor Quillibrace: Correct.

Mr Blottisham: Because the semiclassical approximation eventually breaks down.

Professor Quillibrace: Yes.

Mr Blottisham: That's annoying.

Professor Quillibrace: It is also scientifically respectable.

Mr Blottisham: You academics do enjoy saying "we don't know".

Professor Quillibrace: Only when we don't know.

---

Evaporation is not emptying

Miss Stray: There is another thing bothering me.

Professor Quillibrace: What is it?

Miss Stray: If Hawking radiation isn't simply matter leaking out from inside, then what exactly are we imagining when we say the black hole emits radiation?

Professor Quillibrace: A quantum-field-theoretic process associated with quantum fields in curved spacetime.

Mr Blottisham: That sounds less picturesque.

Professor Quillibrace: Physics is not obliged to be picturesque.

Miss Stray: So the radiation should not be thought of as the black hole's contents escaping?

Professor Quillibrace: Not in that simple sense.

Mr Blottisham: Then "evaporation" really is a rather misleading word.

Professor Quillibrace: It is useful, provided we remember that it is a metaphor.

Mr Blottisham: Which we have just spent an entire series learning not to forget.

Professor Quillibrace: Exactly.

Miss Stray: The black hole loses mass, but it doesn't necessarily lose mass in the manner of a puddle losing molecules.

Professor Quillibrace: Correct.

Mr Blottisham: And it doesn't become empty.

Professor Quillibrace: Not in the ordinary sense.

Mr Blottisham: It changes.

Professor Quillibrace: Now you have it.

---

The disappearing horizon

Miss Stray: There is something rather beautiful about that.

Mr Blottisham: What?

Miss Stray: At first the horizon seems to be the thing that keeps everything in.

Professor Quillibrace: And then we discover that the horizon isn't a thing.

Miss Stray: It is a causal boundary.

Professor Quillibrace: Yes.

Miss Stray: And when the black hole evaporates completely, that causal boundary disappears.

Mr Blottisham: So what disappears isn't a wall.

Professor Quillibrace: Precisely.

Miss Stray: A regime of causal possibility disappears.

Professor Quillibrace smiled.

Professor Quillibrace: That is very well put.

Mr Blottisham: But then what happens to the information that was inaccessible because of that regime?

Professor Quillibrace: Ah.

Mr Blottisham: I recognise that "ah".

Professor Quillibrace: It is the sound one makes when a geometrical question becomes a quantum-gravitational one.

---

The information comes back

Mr Blottisham: Suppose we say that the information eventually becomes accessible again.

Professor Quillibrace: That is one possibility.

Mr Blottisham: Then haven't we solved the problem?

Professor Quillibrace: No.

Mr Blottisham: Why not?

Professor Quillibrace: Because accessibility is not the same as preservation.

Miss Stray: We have been here before.

Professor Quillibrace: Indeed.

Mr Blottisham: Explain.

Professor Quillibrace: Suppose two different quantum states, A and B, collapse into black holes.

Mr Blottisham: All right.

Professor Quillibrace: Suppose the black holes eventually evaporate.

Mr Blottisham: Fine.

Professor Quillibrace: If the final radiation is exactly the same quantum state in both cases, then A and B have become physically indistinguishable.

Mr Blottisham: So the distinction has been destroyed.

Professor Quillibrace: Exactly.

Miss Stray: It isn't enough for something to come out.

Professor Quillibrace: Correct.

Miss Stray: What matters is whether the final state still carries the distinctions that made the initial states different.

Professor Quillibrace: Yes.

Mr Blottisham: So even if everything "gets out", we could still have information loss.

Professor Quillibrace: Now you are using the word "gets" rather more cautiously.

---

The black hole does not let anything out

Mr Blottisham: Let me try again.

If the black hole evaporates, it doesn't necessarily let anything out.

Professor Quillibrace: Good.

Mr Blottisham: The universe simply evolves.

Professor Quillibrace: Yes.

Mr Blottisham: The geometry changes.

Professor Quillibrace: Yes.

Mr Blottisham: The causal structure changes.

Professor Quillibrace: Yes.

Mr Blottisham: The correlations change.

Professor Quillibrace: Yes.

Mr Blottisham: And eventually what we called the black hole no longer exists as that particular causal structure.

Professor Quillibrace: Very good.

Mr Blottisham looked pleased.

Mr Blottisham: I believe I have finally understood it.

Professor Quillibrace: You have understood the sentence.

Mr Blottisham: Isn't that the same thing?

Professor Quillibrace: No.

Miss Stray laughed.

---

The strange fate of the inside

Miss Stray: There is still something I find difficult.

Professor Quillibrace: What?

Miss Stray: If we say that something fell inside, and later the black hole is gone, it is difficult not to imagine that the thing must have either remained there or escaped.

Professor Quillibrace: That is because ordinary language gives us two choices.

Miss Stray: Remain or escape.

Professor Quillibrace: Yes.

Miss Stray: But perhaps the deeper description has neither.

Professor Quillibrace: Precisely.

Mr. Blottisham: What would it have instead?

Professor Quillibrace: Transformation.

Miss Stray: The quantum state remains a quantum state, while the geometrical and causal description changes.

Professor Quillibrace: Exactly.

Miss Stray: So what we call "the interior" might not survive as an interior.

Professor Quillibrace: Nor need the information survive in the same form.

Mr. Blottisham: But it could survive as correlations.

Professor Quillibrace: If evolution is unitary, that is the essential possibility we are trying to understand.

---

A final return to the puddle

Mr Blottisham picked up the essay.

Mr. Blottisham: I think I see what bothered me about the puddle analogy.

Professor Quillibrace: Do tell.

Mr. Blottisham: With the puddle, the object remains the same sort of object while becoming smaller.

Professor Quillibrace: Yes.

Mr. Blottisham: With the black hole, the thing that is changing is partly the very structure that made it a black hole.

Professor Quillibrace: Precisely.

Mr. Blottisham: So saying "the black hole gets smaller" is useful, but it doesn't tell us what is fundamentally happening.

Professor Quillibrace: Correct.

Miss Stray: It tells us that the mass and characteristic horizon scale decrease.

Professor Quillibrace: But the deeper story concerns the evolving geometry and causal structure.

Mr. Blottisham: And eventually the causal structure no longer contains the black-hole region.

Professor Quillibrace: Yes.

Mr. Blottisham: Which means that asking what happens to what was "inside" is really asking how the earlier description relates to the later one.

Professor Quillibrace looked genuinely pleased.

Professor Quillibrace: I think you have finally caught up with the essay.

Mr. Blottisham: I thought the essay was supposed to catch up with me.

Miss Stray: Perhaps neither of you is where you started.

---

The last question

The room fell quiet for a moment.

Miss Stray looked towards the window.

Miss Stray: There is something rather strange about where this leaves us.

Professor Quillibrace: Go on.

Miss Stray: We began by asking what happens when things fall into a black hole.

Professor Quillibrace: Yes.

Miss Stray: Then we asked what happens to the information.

Professor Quillibrace: Yes.

Miss Stray: And now we are asking what happens when the very causal structure that made the question meaningful changes.

Professor Quillibrace: Precisely.

Mr. Blottisham: Which means we have spent an entire discussion trying to find out where things go, only to discover that "where" may itself be changing.

Professor Quillibrace: That is the difficulty.

Miss Stray: And perhaps the clue.

Mr. Blottisham: To what?

Miss Stray: That the black hole does not merely disappear.

She paused.

Miss Stray: The geometry by which we recognised it as a black hole disappears.

Professor Quillibrace nodded.

Professor Quillibrace: And that may be the most useful way to think about evaporation.

Mr. Blottisham: So nothing necessarily escapes.

Professor Quillibrace: Not necessarily.

Mr. Blottisham: Nothing necessarily remains inside.

Professor Quillibrace: Correct.

Mr. Blottisham: The universe simply changes.

Professor Quillibrace: The universe is always changing.

Miss Stray: But in this case, it changes the very structure of what counts as accessible, trapped, inside and outside.

Professor Quillibrace closed the essay.

Professor Quillibrace: And that is why "evaporation" is such a dangerous word.

Mr. Blottisham: Because it makes us think a puddle has disappeared.

Professor Quillibrace: Yes.

Miss Stray: When perhaps what has disappeared is not a thing at all.

She looked towards the empty fireplace.

Miss Stray: Perhaps what has disappeared is a way of organising the possible relations among events.

There was a silence.

Then Mr Blottisham reached for the wine.

Mr. Blottisham: I think I preferred the puddle.

Professor Quillibrace: Naturally.

Miss Stray: It was easier to understand.

Professor Quillibrace: That is why metaphors are so useful.

Mr. Blottisham: And so dangerous.

Professor Quillibrace raised his glass.

Professor Quillibrace: Precisely.

And for once, nobody had anything more to add.

The Senior Common Room: After the Metaphors

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.

The Senior Common Room: The Boundary That Isn’t There

The Senior Common Room at St Anselm’s was unusually quiet.

Professor Quillibrace was standing at the window, looking out across the quadrangle.

Mr Blottisham was reading the newspaper with the expression of a man who had discovered that the world was once again failing to behave sensibly.

Miss Elowen Stray was turning the pages of the latest essay.

She looked up.

Miss Stray: There is something rather odd about this business with the event horizon.

Mr Blottisham: Only something?

Miss Stray: We keep calling it a boundary.

Mr Blottisham: It is a boundary.

Miss Stray: Yes. But what is it a boundary of?

Mr Blottisham lowered the newspaper.

Mr Blottisham: Of the black hole.

Professor Quillibrace: That answer merely moves the question one noun to the right.

Mr Blottisham: I was afraid you were going to say that.

Professor Quillibrace turned from the window.

Professor Quillibrace: We have a remarkably persistent habit of assuming that every boundary must have something on either side of it which the boundary separates.

Miss Stray: Like a wall.

Mr Blottisham: Or a fence.

Professor Quillibrace: Or a border.

Miss Stray: But the event horizon isn't a wall.

Mr Blottisham: No. It is a one-way door.

Professor Quillibrace: Which is considerably better.

Mr Blottisham: Thank you.

Professor Quillibrace: Unfortunately, it is still wrong.

Mr Blottisham sighed.

Mr Blottisham: I knew there was a catch.

The boundary that isn't a thing

Miss Stray: The essay says that the event horizon is a boundary in causal structure rather than a material surface.

Mr Blottisham: Yes.

Miss Stray: I think that's the part I find most interesting.

Mr Blottisham: Why?

Miss Stray: Because it means the boundary isn't defined by what is there.

Professor Quillibrace: Excellent.

Miss Stray: It is defined by what can happen.

There was a brief silence.

Mr Blottisham: That's an unnecessarily philosophical way of saying that things can't get out.

Professor Quillibrace: On the contrary. It is a physically precise way of saying something that our ordinary language encourages us to misunderstand.

He picked up a pencil.

Professor Quillibrace: A wall is a thing which imposes a constraint. The event horizon is a feature of the structure that determines which future-directed causal paths are available.

Mr Blottisham: So the horizon doesn't stop you leaving.

Professor Quillibrace: Correct.

Mr Blottisham: It merely means that leaving is not among the possibilities.

Professor Quillibrace: Precisely.

Miss Stray: Which makes the phrase "trapped inside" slightly misleading.

Professor Quillibrace: Very.

Mr Blottisham: But surely if you're inside a black hole and can't get out, you're trapped.

Professor Quillibrace: In ordinary conversation, yes.

Miss Stray: But physically?

Professor Quillibrace: Physically, the more interesting statement is that the causal structure does not permit an outward-directed future trajectory that reaches the exterior.

Mr Blottisham stared at him.

Mr Blottisham: That sounds remarkably like being trapped.

Professor Quillibrace: It is.

Mr Blottisham: Thank you.

Professor Quillibrace: But without the unnecessary image of a cosmic jailer.

The cosmic jailer

Miss Stray: I hadn't thought of it that way.

Mr Blottisham: I hadn't either. But now that you've mentioned it, I rather like the cosmic jailer.

Professor Quillibrace: Naturally.

Mr Blottisham: Perhaps the black hole simply refuses parole.

Miss Stray: On what grounds?

Mr Blottisham: Irreversible causal misconduct.

Professor Quillibrace: I see that the morning is going to be difficult.

Miss Stray smiled.

Miss Stray: But there is something important about the metaphor, isn't there?

Professor Quillibrace: Yes. The jailer metaphor makes the horizon sound like an agent which prevents things from escaping.

Miss Stray: Whereas the geometry itself determines what is possible.

Professor Quillibrace: Exactly.

Mr Blottisham: So there isn't anything doing the trapping.

Professor Quillibrace: Not in the ordinary sense.

Miss Stray: The structure of spacetime is enough.

Professor Quillibrace: And that is one of the genuinely remarkable conceptual achievements of general relativity.

He put down the pencil.

Professor Quillibrace: Geometry does not merely describe where things are. It participates in determining what can happen.

Inside and outside

Miss Stray returned to the essay.

Miss Stray: There's another thing I noticed.

Mr Blottisham: The fact that you noticed it suggests that it will soon become my problem.

Miss Stray: We keep saying "inside" and "outside."

Professor Quillibrace: Yes.

Miss Stray: But those words sound much more straightforward than they really are.

Mr Blottisham: Inside means inside.

Professor Quillibrace: Does it?

Mr Blottisham: Surely.

Professor Quillibrace: Inside a box?

Mr Blottisham: Yes.

Professor Quillibrace: Inside a country?

Mr Blottisham: More or less.

Professor Quillibrace: Inside a black hole?

Mr Blottisham paused.

Mr Blottisham: Ah.

Miss Stray: That's the problem.

Professor Quillibrace: The word "inside" encourages us to imagine spatial enclosure. But the decisive distinction at an event horizon is causal.

Miss Stray: So "inside" really means something like "in a region from which signals cannot reach the relevant exterior."

Professor Quillibrace: Better.

Mr Blottisham: Which means that "outside" is not simply another place.

Professor Quillibrace: Precisely. It is a region from which those signals can still reach the exterior.

Miss Stray: So the boundary separates different possibilities rather than different substances.

Professor Quillibrace: Now you have it.

Mr Blottisham folded the newspaper.

Mr Blottisham: I dislike this.

Miss Stray: Why?

Mr Blottisham: Because I was perfectly happy with inside and outside.

Professor Quillibrace: Most metaphors are perfectly happy until you ask them to do physics.

A horizon made of future

Miss Stray: There's something even stranger.

Mr Blottisham: Of course there is.

Miss Stray: The event horizon is defined in relation to the future.

Professor Quillibrace: Yes.

Miss Stray: So we draw it as though it were a thing sitting there in space, but its definition depends upon what can ultimately happen.

Mr Blottisham: You mean the horizon somehow knows the future?

Professor Quillibrace: No.

Mr Blottisham: Good.

Professor Quillibrace: The mathematical definition depends upon the global causal structure of spacetime. That is not the same thing as an object possessing foreknowledge.

Mr Blottisham: I was about to object.

Miss Stray: But it does mean that the horizon isn't simply a local surface.

Professor Quillibrace: Correct.

Miss Stray: Which is extraordinary.

Professor Quillibrace: It is one of those features of relativity that become stranger the more precisely you state them.

Mr Blottisham: I preferred it when it was a dark sphere.

Professor Quillibrace: That was an image.

Mr Blottisham: It was a good image.

Professor Quillibrace: It was.

Mr Blottisham: Thank you.

Professor Quillibrace: And it was not the thing itself.

Mr Blottisham sighed.

The astronaut

Miss Stray: What about the astronaut crossing the horizon?

Mr Blottisham: Now there's a reassuring subject.

Miss Stray: From the astronaut's perspective, nothing locally dramatic has to happen at the horizon.

Professor Quillibrace: Correct.

Mr Blottisham: But from far away, the astronaut appears to freeze near it.

Professor Quillibrace: In an appropriate sense, yes.

Mr Blottisham: So which is actually happening?

Professor Quillibrace: Both descriptions are valid within their respective frames and coordinates.

Mr Blottisham: That sounds suspiciously diplomatic.

Professor Quillibrace: It is mathematical.

Miss Stray: Perhaps the point is that there isn't one ordinary object called "the horizon" whose behaviour everyone is watching.

Professor Quillibrace: Exactly.

Miss Stray: The horizon is a feature of the causal structure connecting events and observers.

Professor Quillibrace: Yes.

Mr Blottisham: So the observer matters.

Professor Quillibrace: The observer's position in the causal structure matters.

Miss Stray: But that doesn't make the horizon subjective.

Professor Quillibrace: An important distinction.

He returned to the window.

Professor Quillibrace: The horizon is not created by somebody looking at it. But what information is accessible to an observer depends upon the observer's causal relationship to events.

Miss Stray: So accessibility is relational without reality becoming subjective.

Professor Quillibrace: Precisely.

The horizon and information

Mr Blottisham: Which brings us back to information.

Professor Quillibrace: Inevitably.

Miss Stray: We said in the previous essay that information is relational.

Professor Quillibrace: Yes.

Miss Stray: And now the horizon turns out to be relational too.

Mr Blottisham: Are you suggesting that the horizon and information are secretly dating?

Professor Quillibrace: I am suggesting that they are both concepts whose significance lies partly in relationships rather than substances.

Mr Blottisham: Much less entertaining.

Miss Stray: But quite interesting.

Professor Quillibrace: Information concerns distinctions and correlations among physical states. The horizon constrains which causal relations can exist between events.

Miss Stray: So the horizon constrains the possible organisation of information.

Professor Quillibrace: Exactly.

Mr Blottisham frowned.

Mr Blottisham: So a causal boundary becomes an information boundary.

Professor Quillibrace: That is a very useful way of putting it.

Mr Blottisham: Without being an information wall.

Professor Quillibrace: Correct.

Mr Blottisham: This is becoming unnecessarily subtle.

Miss Stray: I think that's because the physics is subtle.

Mr Blottisham: I prefer theories with furniture.

The missing information

Miss Stray: But now I see why the disappearance of the horizon matters.

Professor Quillibrace: Go on.

Miss Stray: If information falls across the horizon, then from outside it becomes inaccessible. But if the black hole eventually evaporates, the horizon disappears.

Mr Blottisham: And there goes the cosmic filing cabinet.

Professor Quillibrace: There was never a cosmic filing cabinet.

Mr Blottisham: We have established that.

Miss Stray: But that's exactly the problem. If we imagined information as being stored inside, then evaporation leaves us asking where the stored information went.

Professor Quillibrace: And that question already assumes too much.

Miss Stray: Because information isn't a substance.

Professor Quillibrace: And the horizon isn't a container.

Mr Blottisham: So we have a non-substance in a non-container.

Professor Quillibrace: You are learning.

A boundary without a surface

Miss Stray looked thoughtful.

Miss Stray: Perhaps the most interesting phrase in the essay is "a boundary without a surface."

Mr Blottisham: It sounds like something the Dean would say about a committee.

Professor Quillibrace: An excellent example.

Miss Stray: A committee can have a boundary without there being a physical line around it.

Mr Blottisham: Although sometimes there is a physical door.

Professor Quillibrace: And sometimes the door is locked.

Miss Stray: But the point is that a boundary can be a difference in organisation.

Professor Quillibrace: Yes.

Miss Stray: It doesn't have to be a thing.

Professor Quillibrace: That is worth dwelling upon.

He took the pencil again.

Professor Quillibrace: We tend to think that reality consists of things, and that relations merely connect the things. But modern physics repeatedly complicates that picture.

Mr Blottisham: Objects first, relationships second?

Professor Quillibrace: That is the habitual picture.

Miss Stray: And perhaps sometimes the relationship is the more fundamental feature.

Professor Quillibrace: Perhaps.

The membrane

Mr Blottisham: But physicists themselves sometimes talk about the horizon as a membrane.

Miss Stray: The membrane paradigm.

Professor Quillibrace: Indeed.

Mr Blottisham: So surely we're allowed to think of it as a membrane.

Professor Quillibrace: We are allowed to use the model.

Mr Blottisham: Ah.

Professor Quillibrace: That does not mean we should mistake the model for ontology.

Miss Stray: The membrane is useful because it captures certain relationships.

Professor Quillibrace: Exactly.

Mr Blottisham: So when the physicist says "membrane", the physicist may mean "behaves mathematically in certain respects as though there were a membrane."

Professor Quillibrace: Yes.

Mr Blottisham: And when the popular article says "membrane", it means "there is a membrane."

Professor Quillibrace: Frequently.

Mr Blottisham: I see.

He looked pleased.

Mr Blottisham: So the problem isn't metaphor.

Professor Quillibrace: No.

Mr Blottisham: It's forgetting that it is metaphor.

Professor Quillibrace: Now you are learning very quickly.

The horizon acquires entropy

Miss Stray turned another page.

Miss Stray: Then there's the really strange part.

Mr Blottisham: The part involving entropy?

Miss Stray: Yes.

Professor Quillibrace: Black-hole thermodynamics.

Miss Stray: The entropy is proportional to the area of the horizon.

Mr Blottisham: Why not the volume?

Professor Quillibrace: Precisely.

Mr Blottisham: I knew you would say that.

Professor Quillibrace: Our ordinary intuition about storage is volumetric.

Miss Stray: A larger room contains more things.

Mr Blottisham: A larger hard drive stores more data.

Professor Quillibrace: Yet black-hole entropy scales with area.

Miss Stray: Which makes the boundary itself suddenly look informationally important.

Professor Quillibrace: Very much so.

Mr Blottisham: Perhaps the universe has discovered shelving.

Professor Quillibrace: Or perhaps we should resist that metaphor too.

Mr Blottisham: I was afraid of that.

Does the boundary contain the interior?

Miss Stray: The temptation is to say that the information is stored on the horizon.

Professor Quillibrace: Yes.

Miss Stray: But that sounds almost as though we've simply moved the storage metaphor from the inside to the surface.

Professor Quillibrace: Exactly.

Mr Blottisham: We have upgraded the filing cabinet to a filing membrane.

Professor Quillibrace: Which is not a conceptual improvement.

Miss Stray: The deeper question is why the number of possible states associated with the black hole scales with its boundary area.

Professor Quillibrace: Yes.

Miss Stray: Not "what is written on the surface?"

Professor Quillibrace: Precisely.

Mr Blottisham: So the area is a clue, not an answer.

Professor Quillibrace: An excellent formulation.

Holographic

Miss Stray: And this is where holography comes in.

Mr Blottisham: I knew we were heading there.

Professor Quillibrace: The holographic principle is one of those cases where the metaphor is extraordinarily suggestive and extraordinarily dangerous.

Miss Stray: Because a hologram is an image.

Professor Quillibrace: Yes.

Mr Blottisham: So saying the universe is holographic makes one imagine that reality is a projection.

Professor Quillibrace: Which is not what the slogan, by itself, establishes.

Miss Stray: The deeper idea is that a description formulated in terms of degrees of freedom associated with a boundary can encode the same physical content as a description involving a volume.

Professor Quillibrace: Very roughly, yes.

Mr Blottisham: So the boundary doesn't necessarily contain a little picture of what's inside.

Professor Quillibrace: Correct.

Miss Stray: It may be that what we call the interior can be described through another set of degrees of freedom.

Professor Quillibrace: And that possibility is far more radical than simply putting information on a surface.

The trouble with location

There was a longer silence.

Miss Stray looked back at the first page.

Miss Stray: I think this changes the question we've been asking.

Mr Blottisham: Which question?

Miss Stray: "Where is the information?"

Professor Quillibrace: Yes.

Miss Stray: We have been assuming that information must have a location.

Mr Blottisham: Inside or outside.

Miss Stray: Exactly.

Professor Quillibrace: And if information is fundamentally relational, that assumption may already be too strong.

Miss Stray: A correlation isn't necessarily located in one of the systems it relates.

Professor Quillibrace: Correct.

Miss Stray: And a boundary isn't necessarily a thing located between two things.

Professor Quillibrace: Correct.

Miss Stray: So perhaps the question "Where is the information?" is itself built from the wrong ontology.

Mr Blottisham looked at Professor Quillibrace.

Mr Blottisham: You know, she has a point.

Professor Quillibrace: She does.

Mr Blottisham: That's becoming quite a habit.

What if inside and outside are emergent?

Miss Stray: Suppose, just for a moment, that spacetime itself isn't fundamental.

Mr Blottisham: Must we?

Miss Stray: Suppose the geometry emerges from deeper relationships.

Professor Quillibrace: A perfectly respectable possibility.

Miss Stray: Then perhaps the horizon emerges too.

Professor Quillibrace: Yes.

Miss Stray: And if the horizon emerges, then perhaps "inside" and "outside" emerge with it.

Mr Blottisham: Meaning that the distinction between them isn't fundamental either?

Professor Quillibrace: That would be one possible implication.

Mr Blottisham: Then what is fundamental?

Professor Quillibrace smiled.

Professor Quillibrace: That, Mr Blottisham, is generally what one discovers at the end of an essay rather than the beginning.

Mr Blottisham: Very inconvenient.

The shape of the problem

Miss Stray: It seems to me that the argument has shifted.

Professor Quillibrace: How so?

Miss Stray: We began by asking what happens to information at a black hole.

Mr Blottisham: Then we asked what "lost" means.

Miss Stray: Then we asked what the horizon is.

Professor Quillibrace: And?

Miss Stray: Now I'm wondering whether the real question is what we mean by where.

Professor Quillibrace looked pleased.

Professor Quillibrace: Yes.

Mr Blottisham: That sounds ominous.

Professor Quillibrace: It is promising.

Miss Stray: Because if information can be encoded in correlations, and if correlations are not necessarily localised in ordinary spatial objects, then asking where information is may be like asking where a relationship is.

Mr Blottisham: In the room?

Miss Stray: Between the people.

Mr Blottisham: That seems annoyingly sensible.

The Common Room gets quiet

For a while nobody spoke.

Outside, the gardener was pushing a wheelbarrow across the quadrangle.

Mr Blottisham watched him.

Mr Blottisham: I have just realised something.

Professor Quillibrace: This should be interesting.

Mr Blottisham: We started with a line.

Miss Stray: The horizon.

Mr Blottisham: Yes.

Miss Stray: And now the line has disappeared.

Professor Quillibrace: Not disappeared.

Mr Blottisham: Become less substantial.

Professor Quillibrace: Better.

Miss Stray: It was never a wall.

Mr Blottisham: It was a difference in possible futures.

Professor Quillibrace: Yes.

Mr Blottisham: And the more we investigate it, the more the boundary seems to belong to the relations rather than the things.

Professor Quillibrace nodded.

Professor Quillibrace: That is the interesting lesson.

Miss Stray: A boundary can be real without being a thing.

Professor Quillibrace: And information can be real without being a substance.

Mr Blottisham: Which means the two things we thought we were locating—

He paused.

Mr Blottisham: —may not be the sort of things that have locations.

Miss Stray smiled.

Miss Stray: Then perhaps the black hole isn't hiding information somewhere.

Professor Quillibrace: Perhaps.

Miss Stray: Perhaps it is changing the relationships in which information can be found.

Professor Quillibrace looked once more through the window.

Professor Quillibrace: And if that is so, then the horizon has done something rather remarkable.

Mr Blottisham: What?

Professor Quillibrace: It has turned a question about location into a question about structure.

Mr Blottisham considered this.

Mr Blottisham: I suppose that's progress.

Professor Quillibrace: It may be.

Mr Blottisham: Although I still think a filing cabinet would have been easier.

Miss Stray: Perhaps.

She closed the essay.

Miss Stray: But I don't think the universe is obliged to be easy to file.

There was a pause.

Professor Quillibrace: An unfortunate omission in the original design.

Mr Blottisham: I shall raise it with the appropriate committee.

And the Common Room returned, with some relief, to its tea.