(The subleading term − 1 / 2 ln a hor / ℓ Pl 2 is a quantum gravity correction to Hawking’s semiclassical result. For large black holes, the leading contribution to entropy is proportional to the horizon area, in agreement with quantum field theory in curved spacetimes. The detailed theory of the quantum horizon geometry and the standard statistical mechanical reasoning is then used to calculate the entropy and the temperature.
Finally, the number of quantum states associated with an elementary cell labeled by j I is not 2 but 2 j I + 1. Second, the area of each elementary cell is not a fixed multiple of ℓ Pl 2 but is given by, where I labels the elementary cells and j I can be any half-integer (such that the sum is within a small neighborhood of the classical area of the black hole under consideration).
This requirement is encoded in a certain boundary condition that the canonically conjugate pair A, P must satisfy at the surface and plays a crucial role in the quantum theory. However, the detailed derivation in quantum geometry has several new features.įirst, Wheeler’s argument would apply to any 2-surface, while in quantum geometry the surface must represent a horizon in equilibrium. This qualitative picture is simple and attractive. Thus, apart from a numerical coefficient, the entropy (It) is accounted for by assigning two states (Bit) to each elementary cell. Then the total number of states N is given by N = 2 n, where n = a hor / ℓ Pl 2 is the number of elementary cells, whence entropy is given by S = ln N ∼ a hor.
Divide the black hole horizon into elementary cells, each with one Planck unit of area, ℓ Pl 2, and assign to each cell two microstates. The idea behind the calculation can be heuristically explained using the “It from Bit” argument, put forward by Wheeler in the 1990s. Where does this huge number come from? In loop quantum gravity, this is the number of states of the “quantum horizon geometry.” Furthermore, the number of microscopic states is absolutely huge: some exp 10 77 for a solar mass black hole, a number that completely dwarfs the number of states of systems one normally encounters in statistical mechanics. What about black holes? The microscopic building blocks cannot be gravitons because the discussion involves stationary black holes. For a classical gas, these are carried by molecules for the black body radiation, by photons and for a ferromagnet, by Heisenberg spins. For familiar thermodynamic systems, a statistical mechanical derivation begins with an identification the microscopic degrees of freedom. This immediately raised a challenge to potential quantum gravity theories: give a statistical mechanical derivation of this relation. Seminal advances in fundamentals of black hole physics in the mid-1970s suggested that the entropy of large black holes is given by S BH = a hor / 4 ℓ Pl 2, where a hor is the horizon area. Lewandowski, in Encyclopedia of Mathematical Physics, 2006 Black Holes