Averaged null energy condition (ANEC)

The averaged null energy condition requiresalong every complete affinely parametrized null geodesic. It is weaker than the pointwise null energy condition.

For the curvature-coupled scalar equation considered here, contraction with an affine null tangent and integration by parts gives, for $\xi <0$,when $\Phi {\Phi }^{\prime }/(1-\xi {\Phi }^2)$ vanishes at the endpoints.