Cosmological constant preserves null convergence ($R_{ab}{\ell }^a{\ell }^b=8\pi G{T}_{ab}{\ell }^a{\ell }^b$)

Contract the Einstein field equations with a null vector. Both the trace term and the cosmological constant term multiply $g_{ab}{\ell }^a{\ell }^b=0$ and disappear. The null energy condition therefore still implies the null convergence condition, independently of the sign of the cosmological constant. A black-hole area theorem still needs the appropriate global horizon assumptions as well as this local focusing condition.