Semiclassical Quantization on Black Hole Spacetimes

The microcanonical treatment of black holes as opposed to the canonical formulation is reviewed and some major differences are displayed. In particular the black hole decay rates are compared in the two different pictures. The treatment of black holes as thermodynamical systems has many mathematical and physical inconsistencies and drawbacks. In this approach the specific heat turns out to be negative, which is a clear signal that the thermodynamical analogy fails. A second problem is that the partition function as calculated from the microcanonical density of states is infinite 2 for all temperatures and hence the canonical ensemble is not equivalent to the (more fundamental) microcanonical ensemble, as is required for thermodynamical equilibrium. Black holes can be shown to radiate 1 and in the thermodynamical approach the radiation coming out of the black hole has a Planckian distribution. Since black holes can in principle radiate away completely, this result implies that information can be lost, because pure states can come into the black hole but only mixed states come out. The breakdown of the unitarity principle is one of the most serious drawbacks of the thermodynamical interpretation, since it requires the replacement of quantum mechanics with some new (unspecified) physics. We have investigated 2-5 an alternative description of black holes which is free of the problems encountered in the thermodynamical approach. In our approach black holes are considered to be extended quantum objects (p-branes) with degeneracy proportional to the inverse of the probability to tunnel through the black hole's horizon, σ ≃ c e A/4. Explicit expressions for σ can be obtained for some geometries, e.g. the D-dimensional Schwarzschild black hole with mass M , σ(M) ∼ e C(D) M D−2/D−3. (1) where C(D) is a mass-independent function. An exponentially rising density of states is the clear signal of a non-local field theory 6. Comparing Eq. (1) to those known for non-local field theories, we find that it corresponds to the degeneracy of states for a p-brane of dimension p = (D − 2)/(D − 4). A second example is the Kerr-Newman dilaton black hole 3,4 An analysis of the microcanonical density Ω(M, J, Q; V) of a gas of such black holes in a volume V shows that the most probable configuration is one massive black hole which carries all the charge and angular momentum and is surrounded by n − 1 lighter, Schwarzschild black holes. This is the …