Non-Static Spherically Symmetric solution of Einstein vacuum Field Equations with $Lambda$

Abstract

The Schwarzschild-deSitter metric is the known solution of Einstein field equations with cosmological constant term for vacuum spherically symmetric space around a point mass M. Recently it has been reported that in a $Lamda$-dominant world the Schwarzschild type coordinate systems are disqualified by redshift-magnitude test as a proper frame of reference(gr-qc/9812092). We derive the solution in a FRW type coordinate system which is qualified according to the mentioned test. Asymptotically it approachs to the non-static form of deSitter metric. The obtained metric is transformable to Schwarzschild-deSitter metric. It is an analytic function of $r$ for all values except $r=0$which is singular. This is carried out with no making use of Eddington-Finkelstein coordinates and without entering any cross term in the metric.