Juan Maldacena,
Institute for Advanced Study,
School of Natural Sciences Princeton,
New Jersey 08540, USA
Скачать видеозапись (avi): Часть 1 (160 Мб), Часть 2 (170 Мб), Часть 3 (195 Мб)
- 1. Black Holes
- 2. Black Holes and Quantum Mechanics
- 3. Solving These Puzzles
- 4. The Structure of Space-Time
- 5. Bibliography
2. Black Holes and Quantum Mechanics
The next surprise came about when quantum effects were studied. In quantum mechanics the vacuum is not merely the absence of particles. The vacuum is a very interesting state where all the time we have particle pairs being created and destroyed. In flat space we have no net production of particles since energy has to be conserved. All particles that are produced have to annihilate very quickly. In 1974 Steven Hawking showed that when a horizon is present this is no longer the case. What can happen is that a particle with positive energy and one with negative energy are created in the vicinity of the horizon. The negative energy particle falls into the black hole and the positive energy one flies away. In flat space this is not possible because we cannot have negative energy particles. However, on the other side of the horizon, a particle that has negative energy from the point of view of an observer far away can have positive energy from the point of view of an observer inside the horizon. The net effect is that the black hole emits particles. The emitted particles have a thermal distribution with a temperature that is inversely proportional to the black hole mass. For solar-mass black holes this temperature is too tiny for this effect to be measurable. If the black hole were in empty space, it would slowly lose mass and become smaller. Smaller mass black holes could have higher temperatures. A black hole with a mass of the order of 1018 Kg (the mass of a mountain range) would have a temperature of a thousand degrees and it would look white. It would emit as much light as a 1 milli-watt light bulb. As its mass becomes smaller and smaller, its temperature would rise and it would evaporate faster and faster until it, presumably, evaporates completely. In fact if we took a mass of a few kilograms and we collapsed it into a black hole (something that would be very hard to do in practice!) it would evaporate completely in less than a millisecond and it would release more energy than a nuclear bomb.
This thermal property of black holes gives rise to a couple of puzzles: 1) What gives rise to the entropy? and 2) the information paradox. I will try to explain these two puzzles in more detail.
2.1. The Entropy of Black Holes
In ordinary physics, thermal properties always arise from the motion of the constituents. For example, the temperature of the air is related to the average speed of the air molecules. There is a closely related concept, called "entropy." The entropy is the amount of disorder associated to the motion of all the constituents. The entropy is related to the temperature by the laws of thermodynamics, so it can be computed without knowing the microscopic details of the system. Hawking and Bekenstein showed that the entropy of a black hole is the same as the area of the horizon divided by the square of the Planck length, where lPlanck = 10–33 cm. For a macroscopic black hole, this is an enormous entropy. It turns out that the laws of thermodynamics continue to be valid if the black hole contribution to the entropy is included. These are extremely puzzling results since it is not at all clear what the "constituents" of a black hole really are. The black hole is a hole in space-time so finding its constituents is intimately related to finding the most fundamental constituents of space-time geometry.
It is very interesting that the entropy of a black hole is proportional to its area and not its volume. In the early 1990's Hooft and Susskind proposed that in a theory that includes quantum mechanics and gravity, the number of constituents that are necessary to describe a system cannot be bigger than the area of a surface that encloses it. This implies that space-time is rather different from an ordinary solid since in the latter case, the number of constituents (the atoms) grows like the volume. For most practical purposes this entropy bound is not very stringent, but it has interesting theoretical implications, since it suggests that a region of space-time can be described in terms of constituents that live on the boundary of this region.
2.2. The Information Paradox
We have mentioned that we can make a black hole in many different ways, but we always seem to end up with the same black hole. In physics normally if we start with different initial conditions, we get different final states. Sometimes the differences are very subtle, but there are differences. Let me give an example. We start with two plates and on one we write the letter A and on the other the letter B. We then throw each of these plates on the floor so that they break in many small pieces. To a first approximation the end result is the same, lots of broken pieces. However, by examining the pieces in detail we could figure out which letter was written on the plate.
Suppose that we throw one of these plates into a black hole. Apparently, the black hole would eventually evaporate completely through the emission of Hawking radiation. In Hawking's computation this radiation seems perfectly thermal and independent of the initial black hole state. So it looks like we will never recover completely the information about the letter that was on the plate originally.
This seems like a very arcane academic question. We forget things all the time and we don't worry about it! The reason it is a very important question is that quantum mechanics tells us that the laws governing this process should be such that in principle we should be able to recover the information. So solving the information problem is necessary for a consistent theory of quantum gravity. Such a theory must solve the information puzzle.
Many prominent physicists, including S. Hawking, believed that this was impossible. They believed that black holes really destroyed information and that we have to abandon quantum mechanics. They thought that quantum mechanics and gravity are fundamentally incompatible and that the right theory would not obey the principles of quantum mechanics, which imply that information cannot be lost.
Thinking about this question has led to interesting advances in string theory and particle physics.
