Farewell to Reality (40 page)

The positive energy particle created in the virtual pair may then escape, to all intents and purposes appearing as though it has been emitted by the black hole.

Bekenstein had been right all along. The entropy of a black hole is proportional to its surface area.
*
A black hole does have a temperature.
**
Black holes ain't so black, after all. They emit what has since become known as
Hawking radiation.
There is no spontaneous reduction in entropy; no spontaneous increase in the universe's information content.

But there
was
more trouble. Hawking showed that as negative energy (negative mass) particles spill through the event horizon, the black hole must lose mass overall and its surface area must therefore
decrease.
This apparent reduction in entropy is more than compensated for by the entropy of the emitted Hawking radiation. So, having demonstrated that the second law holds even when material is consumed by a black hole, there was no immediate threat of violating the law as a result of emitting Hawking radiation.

As the black hole emits radiation, its surface area decreases. Consequently, its temperature increases, as does the rate of emission. The black hole eventually ‘evaporates', disappearing altogether in an explosion.

It's important to hold on here to a simple fact. At no time in this evaporation process has anything come
out
of the black hole. Although its surface area has shrunk, and its temperature and the ‘glow' of Hawking radiation has increased, the whole process is driven by particles falling
into
the black hole.

And this is the problem. Think of everything that goes past the black hole's event horizon in terms of so many bits of information. What happens to all these bits when the black hole evaporates?

Hawking was unequivocal:

When a black hole evaporates, the trapped bits of information disappear from our universe. Information isn't scrambled. It is irreversibly, and eternally, obliterated.
¹⁰

This was
not
good. If information is indeed physical, then it should not be possible to destroy it in this way. But there was an even more immediate worry. In the absence of measurement, the physical state of a quantum object as it evolves in time is determined by the information carried in its wavefunction. It is a key postulate of quantum theory that this kind of information connects the future with the past and so must be conserved.

If, as Hawking was now arguing, black holes can destroy such information, the entire basis of quantum theory is threatened.

The black hole war

It was called the black hole ‘information paradox'.

Theorists Gerard 't Hooft and Leonard Susskind heard about Hawking's challenge directly from Hawking himself at a small private scientific conference in San Francisco in 1981. It was tantamount to a declaration of war.

Hawking is, arguably, a relativist. Both 't Hooft and Susskind are elementary particle theorists, for whom quantum theory — and the conservation of information it demands — is sacrosanct. Hawking just
had
to be wrong. But neither could provide an instant refutation.

Over the next twelve years, there were sporadic skirmishes, but battle was properly joined in 1993 at another conference organized at the Institute for Theoretical Physics
*
at the University of California at Santa Barbara. Susskind led the charge. At the start of his lecture, he announced: ‘I don't care if you agree with what I say. I only want you to remember that I said it.'
¹¹

What Susskind had to say seemed vaguely mad. There were not all that many options, and like Sherlock Holmes, Susskind figured that
when he had eliminated all the impossible options, what remained, however improbable, must be the truth.

If the scrambled bits of information were not to be lost for ever inside the evaporating black hole, then either they were somehow preserved on its surface, to be eventually emitted in the form of Hawking radiation, or they were preserved in some kind of remnant left behind after the black hole had evaporated completely. The latter seemed unlikely, so Susskind pitched for the former.

He argued that the processes involved in transferring information to a black hole must be subject to a curious kind of complementarity.

To an observer watching from a safe distance, high-entropy material (for some reason in these scenarios this is nearly always an unfortunate astronaut) approaches the event horizon. The astronaut encounters what Susskind called the ‘stretched horizon', a hot, Planck-length-thick layer surrounding the event horizon from which the Hawking radiation escapes, much as the very top of the earth's atmosphere evaporates into space. Here, he meets his inevitable fate. He is reduced to scrambled bits of information.

But, Susskind argued, the bits of information formerly known as the astronaut stay trapped on the surface of the black hole, each occupying a ‘cell' with an area equal to four times the Planck area. The bits are eventually emitted as Hawking radiation, which is what the distant observer sees. Although reconstituting the astronaut would be an extremely difficult (though not completely impossible) task, no bits are lost.

But this doesn't seem to square with what we think we know about black holes. Susskind explained that there is another, complementary, perspective. The astronaut himself observes something quite different. From his perspective, he passes through the stretched horizon and the event horizon without noticing anything particularly unusual. He passes the point of no return, possibly without even realizing it. He is eventually torn apart by gravitational tidal forces and destroyed by the singularity. The bits of information formerly known as the astronaut are irretrievably lost.

How can this make any sense? Susskind argued, much as Bohr had done in the 1920s, that despite appearances, these two very different perspectives are not actually contradictory. They are complementary. With the help of Canadian theorist Don Page, he was able to show that
the two perspectives are mutually exclusive, like the wave and particle perspectives of conventional quantum theory. We can observe what happens from a safe distance or we can join the astronaut on his journey through the event horizon. But we can't do both.

Page and Susskind were able to prove that it is not possible to recover information from the emitted Hawking radiation and then plunge with this into the black hole in search of the same information that the astronaut has carried into the interior. This turns out to be broadly analogous to showing that an electron cannot have both wave and particle properties simultaneously. By the time the information has been recovered from the Hawking radiation and transported into the interior of the black hole, the same information carried by the astronaut has already been destroyed by the singularity. The bits can't coexist.

A straw poll of the theorists gathered in Santa Barbara suggested that Susskind had won this round. More than half of those present agreed that information is not lost inside a black hole, but is recovered in the Hawking radiation that it emits.

Susskind wasn't satisfied, however. He realized he needed a firmer mathematical basis for his notion of black hole complementarity. This was provided by 't Hooft a year later, and championed by Susskind through the use of a startling visual metaphor.

The holographic principle

Here's a clue. If I want to work out how much information I can pack into the British Library in London, I would probably start by working out how many shelves I can get into the
volume
of space that the building contains. So how come all this talk about information and black holes has all been about the black hole's surface
area?

I guess the simple answer is that we really have no clue about what goes on inside a black hole's event horizon, and so we can say nothing really meaningful about its volume. In other words, volume is a measure that is by definition
interior.
To resolve the black hole information paradox, we need to work with the only measure that is still accessible to us, the measure that defines the black hole but remains firmly
exterior —
its area.

The really rather intriguing thing about the next step, however, is that it offered a generalization that takes us a long, long way from black
hole physics. Area, it turns out, is fundamentally connected with information in a way that has nothing to do with black holes.

In 1994, Susskind visited 't Hooft at the University of Utrecht in the Netherlands, 't Hooft told him of a paper he had written some months before. As he explained his most recent work, Susskind realized what was really going on. On his way back to California, he began working on what was to become known as the
holographic principle.

Simply put, this principle says that the information content of a bounded volume of space — for example, a black hole bounded by its event horizon — is equivalent to the information content held on the boundary. More generally, the information contained in an
n
-dimensional space is equivalent
*
to the information on its (
n
-1)-dimensional boundary surface. As 't Hooft had done in his paper, Susskind now compared this to the way a hologram works:

On the second floor of the Stanford [University] physics department, there used to be a display of a hologram. Light reflecting off a two-dimensional film with a random pattern of tiny dark and light spots would focus in space and form a floating three-dimensional image of a very sexy young woman who would wink at you as you walked past.
¹²

According to the maths, this is a general principle, not something that is specific to black holes. Susskind went on to speculate that the information content of the entire universe — in other words, everything in the universe, including me, you and Max Tegmark — is actually a low-energy projection of the information ‘encoded' on the universe's cosmic horizon.

In this interpretation of the holographic principle, our three-dimensional world is an illusion. It is really a hologram, like the three-dimensional image of the sexy young woman who winks at you as you walk by. Reality is actually information stored on the boundary of the universe.

This is Plato's allegory of the cave in reverse. In that story, the prisoner perceived reality as a two-dimensional shadow projection of a three-dimensional ‘real' world. The holographic principle says that our perceived three-dimensional reality is actually a projection from a two-dimensional hologram ‘painted' on the boundary of the universe.

This is where we find the source code of the cosmos.

Holography and superstring theory

The holographic principle might well have remained an interesting curiosity in information theory and a fascinating slice of metaphysics. But in 1998, Argentinian theorist Juan Maldacena announced a powerful new result. He showed that the physics described by a Type IIB superstring theory within an
n
-dimensional bulk spacetime is entirely equivalent to the physics described by low-energy quantum field theory applied to its (
n
-l)-dimensional boundary. This was a whole new superstring duality.

What makes this duality extraordinary is that low-energy quantum field theory does not include gravity. Yet its dual superstring theory does.

In his paper, Maldacena did not make explicit the connection between his result and the holographic principle. But just a few months later Witten posted a paper elaborating Maldacena's result. On seeing Witten's paper, Susskind understood that the black hole war had finally been won:

Quantum field theory is a special case of quantum mechanics, and information in quantum mechanics can never be destroyed. Whatever else Maldacena and Witten had done, they had proved beyond any shadow of a doubt that information would never be lost behind a black hole horizon. The string theorists would understand this immediately; the relativists would take longer. But the war was over.
¹³

Indeed, Witten used this new duality to show that a black hole in the bulk spacetime is equivalent to a relatively mundane hot ‘soup' of elementary particles, such as gluons, on the boundary surface.

There are caveats. Maldacena had considered a model consisting of a stack of D-branes sandwiched together. D-branes are places where
the ends of open strings ‘stick'. The open strings can wander all over the surface of a D-brane, but they can't escape it. Closed strings, on the other hand, are free to wander through the bulk.

The stack of D-branes form a slab. Open strings can now wander over the different layers of the slab, and their ends may be located on a single layer or on different layers. When only low-energy configurations (low-mass strings) are considered, closed strings (and hence gravity) are eliminated and the model can be represented by a quantum field theory such as QCD, in which the open strings are gluons.

Maldacena then changed perspective. We can consider the D-branes as surfaces on which open strings move about, but we can also consider them as physical entities in their own right, with their own energy and mass. Stack enough D-branes together and spacetime starts to warp, just as it does in the presence of any mass. Add more D-branes and we cross a threshold. We form a
black brane.
The curved spacetime created by the black brane has some singular properties. It is a so-called anti-de Sitter space (or AdS).

In 1917, Dutch physicist Willem de Sitter solved Einstein's gravitational field equations for a model universe empty of matter in which spacetime expands exponentially. We can think of such a universe as consisting only of dark energy, producing a positive cosmological constant. It therefore has positive curvature. As on the surface of a sphere, the angles of a triangle drawn in de Sitter space will add up to more than 180°.

In an anti-de Sitter space, the cosmological constant is negative, the spacetime curvature is negative and the angles of a triangle add up to less than 180°.
*
This is a hyperbolic universe shaped more like a saddle. Inject matter into an anti-de Sitter universe and the curvature of spacetime causes it to be pushed away from the boundary and drawn towards the centre, even in the absence of a conventional gravitational field.