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 frac{a-c}{c} in terms of the OPE data of single-trace operators.
Highlights
In General Relativity (GR), particles follow geodesics regardless of their polarizations or internal composition
As we review below, geodesics are insensitive to the ordering of gravitational shocks
We find that the converse is true: in any UV-complete gravitational theory, soft Regge behavior guarantees that coincident gravitational shocks must commute
Summary
In General Relativity (GR), particles follow geodesics regardless of their polarizations or internal composition. In the presence of non-minimal (higher-derivative) couplings, this principle is no longer true — the path of a particle can depend on its polarization and is not given by a geodesic. Such modifications of GR are known to be in tension with causality and unitarity.. In theories with non-minimal gravitational couplings, there can be gravitational birefringence: depending on the polarization of the probe particle, the effect of the shock can be different. It was argued in [3] that the masses of these higher-spin particles must be related to the scale that enters the modified gravitational coupling
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