Black holes arise in general relativity, a classical theory of gravity. However, we need to include quantum effects to understand black holes properly.
Roger Penrose and Stephen Hawking showed thirty years ago that, according to general relativity, any object that collapses to form a black hole will go on to collapse to a singularity inside the black hole. This means that there are strong gravitational effects on arbitrarily short distance scales inside a black hole. On short disctance scales, we certainly need to use a quantum theory to describe the collapsing matter. The presence of a singularity in the classical thoery also means that once we go sufficiently far into the black hole, we can no longer predict what will happen. It is hoped that this failure of the classical theory can be cured by quantising gravity as well.
We can try to describe the interaction of some quantum matter with gravity by quantising the matter on a fixed, classical gravitational background. That is, we can try quantising the matter, but not the gravity. This will work only if the gravity is weak. It should work outside a large black hole, but not near the singularity.
Using this approach, Hawking has shown that a black hole will radiate thermally. That is, if we study quantum matter fields on a classical black hole background, we find that, when the matter fields are initially in the vacuum (that is, there is no matter falling into the black hole), there is a steady stream of outgoing radiation, which has a temperature determined by its mass and charge.
This is an extremely startling discovery; classically, no radiation can escape from a black hole, but if we quantise the matter fields, we find there is steady flux of radiation coming out of the black hole! This outgoing radiation decreases the mass of the black holes, so eventually the black hole will disappear. The temperature goes up as the black hole gets smaller (unlike most things, which cool off as they lose energy), so the black hole will disappear abruptly, in a final flash of radiation.
There is an analogy between the classical laws governing black holes, and the laws of thermodynamics. But thermodynamics is just an approximate description of the behaviour of large groups of particles, which works because the particles obey statistical mechanics (a branch of quantum theory). Since black holes have a non-zero temperature, the classical laws of black holes are the laws of thermodynamics applied to black holes, so there must be some more fundamental description of the classical laws governing black holes in terms of statistical mechanics.
Quantising matter fields on a black hole background teaches us a lot about black holes. However, we need a quantum theory of gravity to understand the fundamental principles underlying black hole thermodynamics. We also need a quantum theory to tell us what happens near the singularity. However, quantising gravity is extremely difficult. One theory which offers some hope, particularly for understanding black holes, is string theory.
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