  The Averaged Null Energy Condition (ANEC) states that the integral along a
complete null geodesic of the projection of the stress-energy tensor onto the
tangent vector to the geodesic cannot be negative. ANEC can be used to rule out
spacetimes with exotic phenomena, such as closed timelike curves, superluminal
travel and wormholes. We prove that ANEC is obeyed by a minimally-coupled, free
quantum scalar field on any achronal null geodesic (not two points can be
connected with a timelike curve) surrounded by a tubular neighborhood whose
curvature is produced by a classical source. To prove ANEC we use a
null-projected quantum inequality, which provides constraints on how negative
the weighted average of the renormalized stress-energy tensor of a quantum
field can be. Starting with a general result of Fewster and Smith, we first
derive a timelike projected quantum inequality for a minimally-coupled scalar
field on flat spacetime with a background potential. Using that result we
proceed to find the bound of a quantum inequality on a geodesic in a spacetime
with small curvature, working to first order in the Ricci tensor and its
derivatives. The last step is to derive a bound for the null-projected quantum
inequality on a general timelike path. Finally we use that result to prove
achronal ANEC in spacetimes with small curvature.
