Preliminaries

7.1 Preliminaries

The first and the second laws of black hole mechanics discussed in Section 4,

have a close similarity with the first and second laws of thermodynamics

This suggest that one should assign to black holes an entropy proportional to their area. Indeed, already in the early seventies, using imaginative thought experiments, Bekenstein [42 , 43 ] argued that such an assignment is forced upon us if, in presence of black holes, we are not going to simply abandon the second law of thermodynamics. Specifically, consider an experiment in which a box containing radiation is slowly lowered into a black hole. At the end of the process, the entropy of the exterior region decreases in apparent violation of the second law. Bekenstein argued that the area of the horizon increases in the process so that the total entropy would not decrease if the black hole is assigned entropy equal to an appropriate multiple of the area. Hawking’s celebrated calculation of black hole evaporation [112 ] provided the precise numerical coefficient. For, it suggested that we should assign the black hole a temperature $T = ℏκ ∕2π$ . By comparing the forms of the first laws of black hole mechanics and thermodynamics, one is then led to set

Note that the factor of $ℏ$ is absolutely essential: In the limit $ℏ ↦→ 0$ , the black hole temperature goes to zero and its entropy tends to infinity just as one would expect classically.

The relation (87 ) is striking and deep because it brings together the three pillars of fundamental physics – general relativity, quantum theory, and statistical mechanics. However, the argument itself is a rather hodge-podge mixture of classical and semi-classical ideas, reminiscent of the Bohr theory of atom. A natural question then is: What is the analog of the more fundamental, Pauli–Schrödinger theory of the hydrogen atom? More precisely, what is the statistical mechanical origin of black hole entropy? To answer this question, one has to isolate the microscopic degrees of freedom responsible for this entropy. For a classical ideal gas, these are given by the positions and momenta of all molecules; for a magnet, by the states of each individual spin at lattice sites. How do we represent the analogous microscopic degrees of freedom for a black hole? They can not be described in terms of quantum states of physical gravitons because we are dealing with black holes in equilibrium. In the approach based on weakly isolated horizons, they are captured in the quantum states of the horizon geometry. Just as one would expect from Bekenstein’s thought experiments, these degrees of freedom can interact with the exterior curved geometry, and the resulting notion of black hole entropy is tied to observers in the exterior regions.

A heuristic framework for the calculation of entropy was suggested by John Wheeler in the nineties, which he christened ‘It from bit’. Divide the black hole horizon into elementary cells, each with one Planck unit of area $ℓ2Pl$ and assign to each cell two microstates. Then the total number of states $𝒩$ is given by $𝒩 = 2n$ , where $n = (a∕ℓ2 ) Pl$ is the number of elementary cells, whence entropy is given by $S = ln 𝒩 ∼ a$ . Thus, apart from a numerical coefficient, the entropy (‘It’) is accounted for by assigning two states (‘bit’) to each elementary cell. This qualitative picture is simple and attractive. But it raises at least three central questions:

What is the origin of the ‘elementary cells’ and why is each endowed with an area $2 ℓPl$ ?
What is the origin of the microstates carried by each elementary cell and why are there precisely two microstates?
What does all this have to do with a black hole? Why doesn’t it apply to any 2-surface, including a 2-sphere in Minkowski space-time?

An understanding of geometry of quantum WIHs provides a detailed framework which, in particular, answers these questions. The precise picture, as usual, is much more involved than that envisaged by Wheeler. Eigenvalues of area operator turn out to be discrete in quantum geometry and one can justify dividing the horizon 2-sphere into elementary cells. However, there are many permissible area eigenvalues and cells need not all carry the same area. To identify horizon surface states that are responsible for entropy, one has to crucially use the WIH boundary conditions. However, the number of surface states assigned to each cell is not restricted to two. Nonetheless, Wheeler’s picture turns out to be qualitatively correct.