The Hidden Reality_ Parallel Universes and the Deep Laws of the Cosmos - Brian Greene [142]
Notice that the value of the entropy and the amount of hidden information are equal. That’s no accident. The number of possible heads-tails rearrangements is the number of possible answers to the 1,000 questions—(yes, yes, no, no, yes, …) or (yes, no, yes, yes, no, …) or (no, yes, no, no, no, …), and so on—namely, 21000. With entropy defined as the logarithm of the number of such rearrangements—1,000 in this case—entropy is the number of yes-no questions any one such sequence answers.
I’ve focused on the 1,000 coins so as to offer a specific example, but the link between entropy and information is general. The microscopic details of any system contain information that’s hidden when we take account of only macroscopic, overall features. For instance, you know the temperature, pressure, and volume of a vat of steam, but did an H2O molecule just hit the upper right-hand corner of the box? Did another just hit the midpoint of the lower left edge? As with the dropped dollars, a system’s entropy is the number of yes-no questions that its microscopic details have the capacity to answer, and so the entropy is a measure of the system’s hidden information content.6
Entropy, Hidden Information, and Black Holes
How does this notion of entropy, and its relation to hidden information, apply to black holes? When Hawking worked out the detailed quantum mechanical argument linking a black hole’s entropy to its surface area, he not only brought quantitative precision to Bekenstein’s original suggestion, he also provided an algorithm for calculating it. Take the event horizon of a black hole, Hawking instructed, and divide it into a gridlike pattern in which the sides of each cell are one Planck length (10–33 centimeters) long. Hawking proved mathematically that the black hole’s entropy is the number of such cells needed to cover its event horizon—the black hole’s surface area, that is, as measured in square Planck units (10–66 square centimeters per cell). In the language of hidden information, it’s as if each such cell secretly carries a single bit, a 0 or a 1, that provides the answer to a single yes-no question delineating some aspect of the black hole’s microscopic makeup.7 This is schematically illustrated in Figure 9.2.
Figure 9.2 Stephen Hawking showed mathematically that the entropy of a black hole equals the number of Planck-sized cells that it takes to cover its event horizon. It’s as if each cell carries one bit, one basic unit of information.
Einstein’s general relativity, as well as the black hole no-hair theorems, ignores quantum mechanics and so completely misses this information. Choose values for its mass, its charge, and its angular momentum, and you’ve uniquely specified a black hole, says general relativity. But the most straightforward reading of Bekenstein and Hawking tells us you haven’t. Their work established that there must be many different black holes with the same macroscopic features that, nevertheless, differ microscopically. And much as is the case in more commonplace settings—coins on the floor, steam in a vat—the black hole’s entropy reflects information hidden within the finer details.
Exotic as black holes may be, these developments suggested that, when it comes to entropy, black holes behave much like everything else. But the results also raised puzzles. Although Bekenstein and Hawking tell us how much information is hidden within a black hole, they don’t tell us what that information is. They don’t tell us the specific yes-no questions the information answers, nor do they even specify the microscopic constituents that the information is meant to describe. The mathematical analyses pinned down the quantity of information a given black hole contains, without