Quantum fields can sometimes have negative energy density. In gravitational contexts, this threatens to permit exotic phenomena such as traversable wormholes, warp drives, and time machines. I will discuss a new thermodynamics principle that can help to rule out such pathological situations. Despite the gravitational motivation, this principle has led us to an interesting lower bound on the energy, even for field theories in flat spacetime! This bound has since been shown to hold in several kinds of field theories.