Upper and lower bounds on the integrated null energy in gravity

We prove a lower bound on the integrated null energy along achronal geodesic segments using induced gravity on a brane in AdS/CFT. The bound follows from the assumption that bulk causality respects brane causality, and matches a bound recently conjectured by Freivogel and Krommydas for semiclassical gravity. We also prove a more general upper bound on the same quantity that follows simply from achronality. We check that the lower bound is satisfied in recent constructions of traversable wormholes, and demonstrate that the bound is related to causality in the ambient spacetime of the wormhole.