Kr. Nayak et al., Black holes in nonflat backgrounds: The Schwarzschild black hole in the Einstein universe - art. no. 024020, PHYS REV D, 6302(2), 2001, pp. 4020
As an example of a black hole in a non-flat background a composite static s
pacetime is constructed. It comprises a vacuum Schwarzschild spacetime for
the interior of the block hole across whose horizon it is matched onto the
spacetime of Vaidya representing a black hole in the background of the Eins
tein universe. The scale length of the exterior sets a maximum to the black
hole mass. To obtain a non-singular exterior, the Vaidya metric is matched
to an Einstein universe. The behavior of scalar waves is studied in this c
omposite model.