Black Holes in Non-stationary de Sitter Space with Variable Λ(u)

Abstract

In this paper we discuss a class of non-stationary solutions of Einstein’s field equations based on the non-stationary de Sitter space-time. These solutions include Schwarz-schild-de Sitter and Vaidya-de Sitter black holes with a cosmological variable Λ(u). Schwarzschild-de Sitter solution with variable Λ(u) is regarded as a generalization of Schwarzschild-de Sitter solution with constant Λ. Vaidya-de Sitter black hole with variable Λ(u) is also a generalization of the radiating Vaidya black hole embedded into the stationary de Sitter space with constant Λ. It is shown the interaction of the Vaidya null fluid with the non-stationary de Sitter field expressing in an energy-momentum tensor. The energy-momentum tensor of the embedded de Sitter black holes satisfies the energy conservation law. The energy conditions (like weak, strong and dominant conditions) for the energy-momentum tensor are also studied. The physical properties of the time-like vector fields for both the embedded solutions are discussed. It is also found that the space-time geometry of Schwarzschild-de Sitter and Vaidya-de Sitter solution with variable Λ(u) are type D in the Petrov classifications of space-times. We also discuss the surface gravity, temperature and entropy of the space-time on the cosmological black hole horizons. It is also suggested that the modified Einstein’s field equations associated with a variable cosmological Λ(u) will take the form R_{ab}−(1/2) R g_{ab}+ Λ(u) g_{ab} = −K{T_{ab}+T^(NS)_{ab} } for any type of matter field distribution T_{ab}.