Statistical entropies of scalar and spinor fields in Vaidya-de Sitter space-time computed by the thin-layer method

Abstract

The brick-wall method based on thermal equilibrium at a large scale cannot be applied to cases out of equilibrium, such as nonstationary space-time with two horizons, for example, Vaidya-de Sitter space-time. We improve the brick-wall method and propose a thin-layer method. The entropies of scalar and spinor fields in Vaidya-de Sitter space-time are calculated by the thin-layer method. The condition of local equilibrium near the two horizons is used as a working postulate and is maintained for a black hole which evaporates slowly enough and whose mass is far greater than the Planck scale. There are two horizons in Vaidya-de Sitter space-time. We think that the total entropy is mainly attributed to the two layers near the two horizons. The entropy of a scalar field in Vaidya-de Sitter space-time is a linear sum of the area of the black hole horizon and that of the cosmological horizon. Thinking of Dirac equations in the Newman-Penrose formalism, there are four components of the wave functions F 1, F 2, G 1, and G 2. The total entropy is summed up from the entropies corresponding to the four components. On the same condition of the scalar field, the resulting entropy is 7/2 times that of the scalar field, and is also a linear sum of the area of the black hole horizon and that of the cosmological horizon. The difference from the stationary black hole is that the result relies on time-dependent cutoffs. ©2001 The American Physical Society.