['In', 'the', 'previous', 'work', '\xa0', '[', '6', ']', 'we', 'have', 'applied', 'the', 'asymptotic', 'conditions', 'to', 'derive', 'the', 'Schwarzschild', 'metric', 'in', 'the', 'expanding', 'universe', ',', 'which', 'is', 'exactly', 'the', 'same', 'as', 'that', 'derived', 'by', 'McVittie', 'by', 'solving', 'the', 'full', 'Einstein', 'equations', '.']