# Shocks, Superconvergence, and a Stringy Equivalence Principle

## TL;DR

This paper explores how gravitational shock interactions constrain low-energy theories and establish a stringy equivalence principle through the commutativity of shock operators and their dual CFT counterparts.

## Contribution

It demonstrates that shock commutativity imposes new constraints on effective theories and proves the commutativity of ANEC operators in any unitary CFT, linking gravitational and conformal field theory principles.

## Key findings

- Shock ordering independence in UV-complete gravity theories.
- Proof of ANEC operator commutativity in all unitary CFTs.
- Derivation of superconvergence sum rules relating CFT data.

## Abstract

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.