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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.
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