  We study modular Hamiltonians corresponding to the vacuum state for deformed
half-spaces in relativistic quantum field theories on $\mathbb{R}^{1,d-1}$. We
show that in addition to the usual boost generator, there is a contribution to
the modular Hamiltonian at first order in the shape deformation, proportional
to the integral of the null components of the stress tensor along the Rindler
horizon. We use this fact along with monotonicity of relative entropy to prove
the averaged null energy condition in Minkowski space-time. This subsequently
gives a new proof of the Hofman-Maldacena bounds on the parameters appearing in
CFT three-point functions. Our main technical advance involves adapting newly
developed perturbative methods for calculating entanglement entropy to the
problem at hand. These methods were recently used to prove certain results on
the shape dependence of entanglement in CFTs and here we generalize these
results to excited states and real time dynamics. We also discuss the AdS/CFT
counterpart of this result, making connection with the recently proposed
gravitational dual for modular Hamiltonians in holographic theories.
