Vaidya Solution in Non-Stationary de Sitter Background: Hawking’s Temperature

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In this paper we propose a class of non-stationary
solutions of Einstein’s field equations describing an embedded Vaidya-de Sitter
solution with a cosmological variable function Λ(*u*). Vaidya-de Sitter
solution is interpreted as the radiating Vaidya black hole which is embedded
into the non-stationary de Sitter space with variable Λ(*u*). The energymomentum
tensor of the Vaidya-de Sitter black hole may be expressed as the sum of the
energy-momentum tensor of the Vaidya null fluid and that of the non-stationary
de Sitter field, and satisfies the energy conservation law. We also find that
the equation of state parameter *w*= *p/ρ *= *-*1 of the
non-stationary de Sitter solution holds true in the embedded Vaidya-de Sitter
solution. It is also found that the space-time geometry of non-stationary
Vaidya-de Sitter solution with variable Λ(*u*) is type D in the Petrov
classification of space-times. The surface gravity, temperature and entropy
of the space-time on the cosmological black hole horizon are discussed.

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