Semiclassical black hole states and entropy

Abstract

We discuss semiclassical states in quantum gravity corresponding to Schwarzschild as well as Reissner-Nordström black holes. We show that reduced quantization of these models is equivalent to Wheeler-DeWitt quantization with a particular factor ordering. We then demonstrate how the entropy of black holes can be consistently calculated from these states. While this leads to the Bekenstein-Hawking entropy in the Schwarzschild and non-extreme Reissner-Nordström cases, the entropy for the extreme Reissner-Nordström case turns out to be zero. ͓S0556-2821͑97͒01904-8͔

The Bohm Interpretation of Quantum Cosmology

“…As shown in the previous section, a non degenerate four-geometry can be attained only if the quantum potential have the specific form (71). In this case, the sole relevant quantum effect will be a change of signature of spacetime, something pointing towards Hawking's ideas.…”

Quantum general relativity and Hawking radiation

“…(For a similar discussion of WKB states for the Schwarzschild black hole see [16] and for two-dimensional dilaton gravity see [17]). We will now show that the wave-functional (11), along with the solutions (13) and (14), yields Hawking radiation when it is applied to a matter distribution that is appropriate to a massive black hole surrounded by dust whose total energy is small compared with the mass of the black hole.…”

Quantum-mechanical model of the Reissner-Nordström black hole

“…when the hole is extreme, the temperature of the hole goes to zero, and no radiation comes out from the hole. Moreover, although an evaluation of the Bekenstein-Hawking entropy of a non-extreme black hole by means of Euclidean methods implies that the entropy is exactly one quarter of the area of the apparent horizon of the hole, the same analysis implies zero entropy for an extreme black hole [2][3][4]. What, then, is the difference between extreme and non-extreme black holes?…”

“…In this equation, the integration is performed over the whole spacetime under consideration, (4) R is the four-dimensional Riemann scalar of spacetime, and…”

“…In this paper we shall see that although the spectrum of the area of the outer horizon of the Reissner-Nordström black hole fails to coincide with Eq. (1.3), the eigenvalues of the sum of the areas of the inner and the outer horizons of the hole -which we shall refer, for the sake of covenience, as the total area of the horizons-are, in the semi-classical limit, of the form: 4) where A ext := 4πQ 2 is the area of an extreme black hole. Hence, our result on the spectrum of the total area of the Reissner-Nordström black hole is closely related to Bekenstein's proposal: according to Bekenstein's proposal, the spectrum of the area of the outer horizon of a black hole is of the form (1.3), whereas our model implies that it is the total area of the hole which is quantized in that manner.…”