Figure 12.3. The Page curve (black curve). Also shown is the Bekenstein–Hawking result for the entropy of a black hole (dotted line) and Hawking’s result for the entropy of the radiation (dashed line).
This simple jigsaw analogy provides a way to understand Page’s reasoning. The first table is analogous to the black hole and the second table is analogous to the emitted Hawking radiation. If information is conserved in black hole evaporation, we have just learnt that the entanglement entropy between the black hole and the radiation should first rise and then fall, as illustrated in the Page curve of Figure 12.3. The Page time is the time when the entanglement entropy stops rising and begins to fall. It marks the time when the correlations between the Hawking particles start to carry a significant part of the total information content of the original system.
We have also drawn the thermal entropy of the black hole, which gradually falls to zero as the black hole evaporates away, and an ever-rising thermal entropy for the Hawking radiation. This ever-rising curve is the result of Hawking’s original calculation, which appears to show that there are never any quantum correlations in the radiation. Page’s powerful point is that an information-conserving evaporation process must follow the Page curve and not Hawking’s ever-rising curve. And, crucially, the difference between the two curves manifests itself at the Page time, which is when the black hole is not too old and therefore quantum theory and general relativity should both be valid. Viewed this way, solving the information paradox is tantamount to understanding which curve is correct: the Page curve (information comes out) or Hawking’s original curve (information does not come out).
The Page time can also be thought of as the time at which we could begin to decode information contained in the Hawking radiation (if the information comes out). If the radiation is thermal, as it is to a good approximation before the Page time, then its entanglement entropy is equal to its thermal entropy. In this case, no correlations are visible and no information is contained in the radiation. After the Page time, correlations appear and the radiation becomes increasingly information rich. This is the situation we previously described for a quantum book. The initial pages are complete gibberish because the story is encoded in the correlations between the pages. It’s only when we get more than halfway through the book that we can begin to identify the correlations and decipher the meaning.
The fact that we have to wait until the Page time for any correlations to appear leads to another quite bamboozling idea. Because the Page time is roughly halfway through the lifetime of a black hole, which may be in excess of 10100 years, we are claiming that correlations appear between particles whose emission is separated by 10100 years. This illustrates the strangeness of quantum entanglement and perhaps hints that space and time may not be what they seem.
The conclusion we are forced into is that if information is to be conserved, some correction to Hawking’s original calculation must be present not much later than the Page time, which is approximately halfway through the black hole’s lifetime. But to reiterate a crucial point: at the Page time we expect both general relativity and quantum theory to be perfectly adequate in the near-horizon region; we would not expect to need presently unknown physics. And yet Hawking’s calculation, based on quantum theory and general relativity, diverges from expectations if we believe that information should be conserved in black hole evaporation. The challenge is now clear. If we are to show that information is not destroyed by black holes, we must calculate the Page curve.
We now find ourselves in the position of the theoretical physics community around the turn of the millennium. Don Page had laid down the gauntlet because the Page curve should be calculable using the known laws of physics. But the state-of-the-art calculation at the time was Hawking’s, which did not follow the Page curve. There was also complementarity, the crazy idea we met in the previous chapter, which lacked a convincing proof. Complementarity offers a solution to the information paradox in the sense that no information ever actually falls across the horizon and into the black hole as viewed from the outside – but we are also asked to believe that there is another point of view in which information does fall in, and that both points of view are equally valid. Around this time, a bold new idea surfaced that was crucial in convincing many that Hawking must be wrong, and that complementarity has substance. The key idea? The world is a hologram.
* Correlations are commonplace in everyday life: the colour of your left sock is likely to be correlated with the colour of your right sock and living in Manchester is correlated with experiencing drizzle.
† This entangled state is an example of what is known as a ‘Bell state’, named after Northern Irish physicist, John Bell, who pioneered early studies into quantum entanglement.
‡ It is an important feature of quantum mechanics that this ‘link’ between the cakes cannot be used to transmit information faster than light. We could ask for the probability that Lucy finds a good-tasting cake – something that Lucy can measure independent of Ricardo. We will discover that the odds Lucy observes do not depend upon Ricardo’s measurements, which means that Ricardo cannot use his choice of measurement to transmit a message to Lucy. Even though their observations are correlated, it is not a correlation that can be used to transmit information.
§ In quantum mechanics, we use the word determinism to refer to the fact that we can predict the future state of a system if we know its prior state. However, since quantum mechanics is inherently random, knowing the state does not also mean we know the results of experiments. In this regard, quantum mechanics is not deterministic. The more precise terminology is to say that quantum states undergo ‘unitary’ evolution.
¶ For example, if the jigsaw is a 3 x 3 puzzle with square pieces, where the pieces occupy one of 9 possible positions on a grid, the number of possible arrangements is 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 x 49 = 95,126,814,720. The thermal entropy of the jigsaw is the logarithm of this number. The factor of 49 is because each piece can be placed in one of four different orientations.
13
The World as a Hologram
‘Nobody has the slightest idea what is going on.’
Joseph Polchinski
The entropy of a black hole is proportional to its area, which suggests that all the information concerning the stuff that fell into the hole is encoded in tiny bits spread over the surface of the horizon. In time, those bits break free and end up as correlated Hawking particles. These correlations – quantum entanglement in the radiation – encode the information about the stuff that fell in.* From the perspective of someone freely falling through the horizon, they feel nothing and are oblivious to this magical encoding. Moreover, their fate is to be both spaghettified in the singularity (from their own perspective) and burnt up on the horizon (from an outsider’s perspective). But that is no problem for the laws of Nature because no observer can be present at both events. This is the essence of the black hole complementarity resolution to the black hole information paradox. Is it nonsense?