This is the post for general readership. In the mid 70s Hawking merge QFT and GR in a semiclassically approximately to derive Hawking radiation and its spectrum.The entropy of a black hole is given by the equation. “S = \frac{c^{3}kA}{4 \hbar G}” .
One of the interesting discovery of string theory and especially the work of Strominger and Vafa has validated the value of Hawking Bekenstein entropy. There is one disturbing aspect for physicists. We will discuss it in the text to follow.
First according to no hair theorem.
Black hole solutions of the Einstein-Maxwell equations in GR tells us that black holes are characterized by only three externally observable parameters: mass, electric charge, and angular momentum and all the other paramaters are inaccessible beyond the even horizon in a black hole.
How could information be lost in a black hole ?
The bothersome point is that a pure quantum state may evolve into a mixed thermal state. This is what physicist mean when they talk about information loss. This violates the Liouville theorem. In the case of a entangled pure state one part of the entangles system is thrown into the black hole while the other part is kept outside. This results in a mixed state after the partial trace is taken into a black hole. But since within the interior of the black hole will hit the singularity within a fixed time, the party which is traced over partially might disappear and reappear again. It is not clear what goes on at singularities once quantum effects are taken into account.
This violates unitarity and this is one of the most sacrosanct feature of quantum mechanics. Unitarity implies that the operator which describes the progress of a physical system must be a unitary operator. More clearly, its the e^{iHT} operator than one sees in basic quantum mechanics where H is the Hamiltonian. One knows from quantum mechanics and QFT that the S-matrix that describes how the physical system evolves in a scattering process must be a unitary operator. In QFT this is called the optical theorem. The same thing also happens in the case of Schrodinger equation. i.e the probability of finding the states sums to 1.
One of the guess to resolve this apparent paradox was that the information remains in the remnant of a black hole. However, this remnant would carry too much entropy for an extremely small volume and this is not a very favorable conjecture.
Does this means that the fundamental feature of quantum mechanics (unitarity )has to be modified?
The short answer is no. One should not immediately jump into such conclusions as Hawking’s result is just a semiclassical approximation. This resulted in the famous bet between Preskill and Hawking. With Strominger and Vafa the black hole entropy it seems that Hawking’s entropy calculation is coreect. So one needs to think more to understand the apparent paradox. For instance one needs to learn if the information is preserved if one considers more than the semiclassical approximation.
This best way to resolve this paradox is with the use of ADS CFT. Also, if Hawking radiation receives some quantum correction. The other possibility is bizarre and involves allowing non unitary time evolution. This to my mind is a sketchy proposal. Stephen Hawking published a paper and announced a theory that quantum perturbations of the event horizon could allow information to escape from a black hole, which would resolve the information paradox. This assumes unitarity of ADS/CFT correspondence which is dual to a thermal conformal field theory.
