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By Root 2057 0

exist. The mathematics underlying his calculations was invariably correct. But he made additional assumptions that, given the intense folding of space and time that would be caused by a black hole, turn out to be too restrictive; in essence, the assumption left out the possibility of matter imploding. The assumptions meant that Einstein’s mathematical formulation did not have the latitude to reveal black holes as possibly real. But this was an artifact of Einstein’s approach, not an indication of whether black holes might actually form. The modern understanding makes clear that general relativity allows for black hole solutions.

2. Once a system reaches a maximal entropy configuration (such as steam, at a fixed temperature, that is uniformly spread throughout a vat), it will have exhausted its capacity for yet further entropic increase. So, the more precise statement is that entropy tends to increase, until it reaches the largest value the system can support.

3. In 1972, James Bardeen, Brandon Carter, and Stephen Hawking worked out the mathematical laws underlying the evolution of black holes, and found that the equations looked just like those of thermodynamics. To translate between the two sets of laws, all one needed to do was substitute “area of black hole’s horizon” for “entropy” (and vice versa), and “gravity at the surface of the black hole” for “temperature.” So, for Bekenstein’s idea to hold—for this similarity to not just be a coincidence, but to reflect the fact that black holes have entropy—black holes would also need to have a nonzero temperature.

4. The reason for the apparent change in energy is far from obvious; it relies on an intimate connection between energy and time. You can think of a particle’s energy as the vibrational speed of its quantum field. Noting that the very meaning of speed invokes the concept of time, a relationship between energy and time becomes apparent. Now, black holes have a profound effect on time. From a distant vantage point, time appears to slow for an object approaching the horizon of a black hole, and comes to a stop at the horizon itself. Upon crossing the horizon, time and space interchange roles—inside the black hole, the radial direction becomes the time direction. This implies that within the black hole, the notion of positive energy coincides with motion in the radial direction toward the black hole’s singularity. When the negative energy member of a particle pair crosses the horizon, it does indeed fall toward the black hole’s center. Thus the negative energy it had from the perspective of someone watching from afar becomes positive energy from the perspective of someone situated within the black hole itself. This makes the interior of the black hole a place where such particles can exist.

5. When a black hole shrinks, the surface area of its event horizon shrinks too, conflicting with Hawking’s pronouncement that total surface area increases. Remember, however, that Hawking’s area theorem is based on classical general relativity. We are now taking account of quantum processes and coming to a more refined conclusion.

6. To be a little more precise, it’s the minimum number of yes-no questions whose answers uniquely specify the microscopic details of the system.

7. Hawking found that the entropy is the area of the event horizon in Planck units, divided by four.

8. For all the insights that will be described as this chapter unfolds, the issue of a black hole’s microscopic makeup has yet to be fully resolved. As I mentioned in Chapter 4, in 1996, Andrew Strominger and Cumrun Vafa discovered that if one (mathematically) gradually turns down the strength of gravity, then certain black holes morph into particular collections of strings and branes. By counting the possible rearrangements of these ingredients, Strominger and Vafa recovered, in the most explicit manner ever achieved, Hawking’s famous black hole entropy formula. Even so, they were not able to describe these ingredients at stronger gravitational strength, i.e., when the black hole actually forms. Other authors,