We study propagation of a probe particle through a series of closely situated
gravitational shocks. We argue that in any UV-complete theory of gravity the
result does not depend on the shock ordering - in other words, coincident
gravitational shocks commute. Shock commutativity leads to nontrivial
constraints on low-energy effective theories. In particular, it excludes
non-minimal gravitational couplings unless extra degrees of freedom are
judiciously added. In flat space, these constraints are encoded in the
vanishing of a certain "superconvergence sum rule." In AdS, shock commutativity
becomes the statement that average null energy (ANEC) operators commute in the
dual CFT. We prove commutativity of ANEC operators in any unitary CFT and
establish sufficient conditions for commutativity of more general light-ray
operators. Superconvergence sum rules on CFT data can be obtained by inserting
complete sets of states between light-ray operators. In a planar 4d CFT, these
sum rules express (a-c)/c in terms of the OPE data of single-trace operators.Comment: 93 pages plus appendice
