Null convergence condition ($R_{ab}{k}^a{k}^b\ge 0$)

The null convergence condition requires $R_{ab}{k}^a{k}^b\ge 0$ for every null vector. Through the Einstein field equations, it follows from the null energy condition, since all terms proportional to the metric tensor vanish on a null vector.

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.