A prototype of an asymptotically non-flat black hole spacetime is that of a Schwarzschild black hole in the background of the Einstein universe, which is a special case of the representation of a black hole in a cosmological background given by Vaidya. Recently, this spacetime has been studied in detail by Nayak et al. They constructed a composite spacetime called the Vaidya–Einstein–Schwarzschild (VES) spacetime. We investigate some of the physical effects inherent to this spacetime. We carry out a background–black hole decomposition of the spacetime in order to separate out the effects due to the background spacetime and the black hole. The physical effects we study include the classical tests—the gravitational redshift, perihelion precession and light bending—and circular geodesics. A detailed classification of geodesics, in general, is also given.
The existence of the Carter constant in the Vaidya-Einstein-Kerr (VEK) spacetime and its relation to the Petrov type is investigated. This spacetime is an example of a black hole in an asymptotically non-flat background. We construct the Carter constant and obtain the Killing tensor in the VEK spacetime. The Newman-Penrose formalism is employed to obtain the spin coefficients. We present a complete (Petrov) classification of the VEK spacetime and the special case of the non-rotating VaidyaEinstein-Schwarzschild spacetime. We demonstrate explicitly that both spacetimes are of type-D.
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