  A null line is a complete achronal null geodesic. It is proven that for any
quantum fields minimally coupled to semiclassical Einstein gravity, the
averaged null energy condition (ANEC) on null lines is a consequence of the
generalized second law of thermodynamics for causal horizons. Auxiliary
assumptions include CPT and the existence of a suitable renormalization scheme
for the generalized entropy.
  Although the ANEC can be violated on general geodesics in curved spacetimes,
as long as the ANEC holds on null lines there exist theorems showing that
semiclassical gravity should satisfy positivity of energy, topological
censorship, and should not admit closed timelike curves. It is pointed out that
these theorems fail once the linearized graviton field is quantized, because
then the renormalized shear squared term in the Raychaudhuri equation can be
negative. A "shear-inclusive" generalization of the ANEC is proposed to remedy
this, and is proven under an additional assumption about perturbations to
horizons in classical general relativity.
