by Jim Baggott
What does this mean? Suppose we agree a simple code to communicate a particular state of affairs, such as ‘my plane took off on time’. We code this event as ‘1’. The alternative, ‘my plane did not take off on time’, is coded as ‘0’. Knowing airline punctuality as you do, you expect to receive the message ‘0’ with 85 per cent probability. If you do indeed receive the message ‘0’, then Shannon’s formula implies that its information content is very low.* But if you receive the message ‘1’, then the information content is high. In this context, information is a measure of the ‘unexpectedness’ of the message, or the extent to which you’re surprised by it.
So, the entropy of Macbeth’s soliloquy is low,** and its information content is high (the particular sequence of words is of low probability and therefore unexpected or ‘surprising’, as well as very moving). Heat the soliloquy into a steam of bits. The entropy increases and the information content is, if not lost, then hidden very, very deeply.
In 1961, this kind of logic led IBM physicist Rolf Landauer to declare that ‘information is physical’. He was particularly interested in the processing of information in a computer. He concluded that when information is erased during a computation, it is actually dumped into the environment surrounding the processor, adding to the entropy. This increase in entropy results in an increase in temperature: the environment surrounding the processor heats up. Anyone who has ever run a complex computation on their laptop computer will have noticed how, after a short while, the computer starts to get uncomfortably hot.
Landauer’s famous statement requires some careful interpretation, but it’s enough for now to note the direct connection between the processing of information and physical quantities such as entropy and temperature. It seems that ‘information’ is not an abstract concept invented by the human mind. It is a real, physical thing with real, physical consequences.
Now we’re coming to it. The second law of thermodynamics insists that in a spontaneous change, entropy will always increase or information will always be diluted or degraded and so ‘hidden’. So what then happens when we throw stuff — a volume of steam, for example — into a black hole? By definition, when the material crosses the black hole’s event horizon, there’s no coming back. If we dispose of a lot of high-entropy material in a black hole, this seems to imply that the net entropy of the universe has somehow reduced. Material with high entropy has disappeared from the universe. The change is spontaneous, so this appears to contradict the second law, which says that entropy can never decrease.
And if the entropy of the universe has reduced, this implies that its information content has somehow increased.
It is estimated that the total information in the universe is of the order of 10120 bits.8 If we regard information to be an elementary, physical constituent of the universe, then this implies that, like energy, it can be neither created nor destroyed. How, then, can throwing high-entropy material into a black hole increase the information content of the universe?
Black holes and the second law
Gravity might be the weakest of nature’s forces, but it is ultimately irresistible. Gravity binds together and compresses clouds of gas drifting in space. Compressing clouds of gas with sufficient mass sparks fusion reactions at their cores, and stars are born. The pressure of radiation released by the fusion reactions holds back further compression, and the star enters a period of relative stability.
However, as the fuel is expended, the force of gravity grips tighter. For any mass greater than about 1.4 M (1.4 solar masses),* the force of gravity is ultimately irresistible. It crushes the body of matter into the obscurity of a black hole, a name coined by John Wheeler.
For a time it was thought that black holes would indeed invalidate the second law. The only way to preserve the law would be to ensure that the entropy of the material that was consumed by a black hole was somehow transferred to the black hole itself.
But does it make any sense to think of a black hole as something that has entropy?
In the late 1960s, young Cambridge University physicist Stephen Hawking produced a series of papers on black hole physics in collaboration with mathematician Roger Penrose, then at Birkbeck College in London. General relativity, they claimed, predicted that at the heart of a black hole there beats a singularity, a region of infinite density and spacetime curvature where the laws of physics break down. Of course, what goes on in the region of a singularity is completely hidden from observation by the black hole’s event horizon, a fact that Penrose elevated to the status of a principle, which he called the cosmic censorship hypothesis.
Working with Canadian Werner Israel, Australian Brandon Carter and British physicist David Robinson, Hawking demonstrated that, in terms of the mathematics needed to describe them, holes are surprisingly simple. Their properties and behaviour depend only on their mass, angular momentum and electric charge, a conjecture called the ‘no hair’ theorem. In this context, ‘hair’ means all other kinds of information apart from mass, angular momentum and electric charge. Beyond these basic properties, a black hole is featureless — it has no hair. All other kinds of information are judged to be lost behind the black hole’s event horizon.
In a moment of inspiration one night in November 1970, Hawking realized that the properties of the event horizon meant that this could never shrink — the surface area of a black hole (meaning the area bounded by the event horizon) could in principle never decrease. If a black hole consumes an amount of material, then its surface area increases by a proportional amount.
We do a mental double-take. Isn’t there one other well-known physical property that in a spontaneous change can never decrease? So, could there be a connection between the surface area of a black hole and its entropy?
In his 1972 Princeton PhD thesis, Israeli theorist Jacob Bekenstein (another of Wheeler’s students) claimed precisely this. He identified the surface area of a black hole with its entropy. So, when a black hole consumes some high-entropy material, its surface area increases (as Hawking had observed), and this indicates that its entropy increases too, in accordance with the second law.
He was shouted down from all sides.
Hawking was irritated. While this appeared to offer a neat solution, it dragged along with it a number of implications which he felt Bekenstein hadn’t properly addressed. For one thing, a body with entropy also has to have a temperature. And a body with a temperature has to emit radiation. This made no sense at all. How could a black hole, with properties and behaviour determined only by its mass, angular momentum and electric charge, possess a temperature and emit radiation?
A black hole is supposed to be ‘black’. Right?
Hawking radiation
A few years later, Hawking set out to refute Bekenstein’s hypothesis. Lacking a fully fledged quantum theory of gravity, he chose to approach the problem using an essentially classical general relativistic description of the black hole itself, and applied quantum field theory to the curved spacetime around the event horizon. What he found was quite shocking.
As he later explained:
However, when I did the calculation, I found, to my surprise and annoyance, that even nonrotating black holes should apparently create and emit particles at a steady rate. At first I thought that this emission indicated that one of the approximations I had used was not valid. I was afraid that if Bekenstein found out about it, he would use it as a further argument to support his ideas about the entropy of black holes, which I still did not like.9
Hawking found that within the constraints imposed by Heisenberg’s uncertainty relation, virtual particle—anti-particle pairs are produced in the curved spacetime near the black hole’s event horizon. The particles are produced with zero net energy. This means that one particle in the pair may possess positive energy and the other negative energy.
Under normal circumstances, the pair would quickly annihilate. But if the negative energy particle were to be drawn into the black hole before it can be annihilate
d, it can acquire positive energy and become a ‘real’ particle. Another way of looking at this is to think of the negative energy particle as having negative mass (from E = mc2). As the negative mass particle falls into the back hole, it gains mass and becomes a real particle.
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 h
ole 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.