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The positive-energy particles shoot outward from just above the black hole’s event horizon, so to someone watching from afar they look like radiation, a form since named Hawking radiation. The negative-energy particles are not directly seen, because they fall into the black hole, but they nevertheless have a detectable impact. Much as a black hole’s mass increases when it absorbs anything that carries positive energy, so its mass decreases when it absorbs anything that carries negative energy. In tandem, these two processes make the black hole resemble a piece of burning coal: the black hole emits a steady outward stream of radiation as its mass gets ever smaller.5 When quantum considerations are included, black holes are thus not completely black. This was Hawking’s bolt from the blue.

Which is not to say that your average black hole is red hot, either. As particles stream from just outside the black hole, they fight an uphill battle to escape the strong gravitational pull. In doing so, they expend energy and, because of this, cool down substantially. Hawking calculated that an observer far from the black hole would find that the temperature for the resulting “tired” radiation was inversely proportional to the black hole’s mass. A huge black hole, like the one at the center of our galaxy, has a temperature that’s less than a trillionth of a degree above absolute zero. A black hole with the mass of the sun would have a temperature less than a millionth of a degree, minuscule even compared with the 2.7-degree cosmic background radiation left to us by the big bang. For a black hole’s temperature to be high enough to barbecue the family dinner, its mass would need to be about a ten-thousandth of the earth’s, extraordinarily small by astrophysical standards.

But the magnitude of a black hole’s temperature is secondary. Although the radiation coming from distant astrophysical black holes won’t light up the night sky, the fact that they do have a temperature, that they do emit radiation, suggests that the experts had too quickly rejected Bekenstein’s suggestion that black holes do have entropy. Hawking then nailed the case. His theoretical calculations determining a given black hole’s temperature and the radiation it emits gave him all the data he needed to determine the amount of entropy the black hole should contain, according to the standard laws of thermodynamics. And the answer he found is proportional to the surface area of the black hole, just as Bekenstein had proposed.

So by the end of 1974, the Second Law was law once again. The insights of Bekenstein and Hawking established that in any situation, total entropy increases, as long as you account for not only the entropy of ordinary matter and radiation but also that contained within black holes, as measured by their total surface area. Rather than being entropy sinks that subvert the Second Law, black holes play an active part in upholding the law’s pronouncement of a universe with ever-increasing disorder.

The conclusion provided a welcome relief. To many physicists, the Second Law, emerging from seemingly unassailable statistical considerations, came as close to sacred as just about anything in science. Its restoration meant that, once again, all was right with the world. But, in time, a vital little detail in the entropy accounting made it clear that the Second Law’s balance sheet was not the deepest issue in play. That honor went to identifying where entropy is stored, a matter whose importance becomes clear when we recognize the deep link between entropy and the central theme of this chapter: information.

Entropy and Hidden Information

So far, I’ve described entropy, loosely, as a measure of disorder and, more quantitatively, as the number of rearrangements of a system’s microscopic constituents that leave its overall macroscopic features unchanged. I’ve left implicit, but will now make explicit, that you can think of entropy as measuring the gap in information between the data you have (those overall macroscopic features)