Shocks, Superconvergence, and a Stringy Equivalence Principle

TL;DR

This work establishes a deep link between the commutativity of coincident gravitational shocks and high-energy constraints in UV-complete gravity, positing a stringy equivalence principle that demands shock commutativity in flat space, AdS, and dS. It derives flat-space superconvergence sum rules that connect non-minimal gravitational couplings to heavy (stringy) states and shows how Regge/bulk-bounds enforce causality and unitarity, with string theory providing explicit realizations. In AdS/CFT, the authors prove ANEC operators commute on the boundary, derive a broad class of event-shape sum rules, and express bulk calculations in terms of boundary conformal blocks, enabling nonperturbative constraints on CFT data. Collectively, these results offer new consistency conditions that any UV-complete gravitational theory must satisfy and suggest practical bootstrap-like approaches to constrain gravitational couplings via CFT data. The findings have potential implications for constraining beyond-GR physics, informing the structure of holographic theories, and guiding future explorations of higher-spin dynamics and Regge behavior in quantum gravity.

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.

Figures (7)

• Figure 1: The probe geodesic is denoted by a black line and shock waves by red lines. Dashed lines mark time delay associated to each shock. In General Relativity propagation through a pair of closely situated shockwaves is commutative, namely the overall effect does not depend on the order of the shocks. This is no longer true in theories with higher derivative corrections. We argue that commutativity must hold in any UV complete theory.
• Figure 2: An elastic scattering of a probe particle and an on-shell shock state graviton. We will argue that the Regge behavior of this amplitude in consistent theories of gravity is such that gravitational shockwaves always commute. We adopt a CFT correlator-like prescription where the time in the diagram goes from right to left.
• Figure 3: Integration contour for computing the shock amplitude. The integral is along the real axis, rotated by a small positive angle. The dashed lines represent $t$- and $u$-channel cuts. We assume that $s<0$.
• Figure 4: Depending on whether the integrand decays exponentially in the upper or lower half-plane, we can deform the contour in different ways. In both cases, the old contour is shown in gray and the new contour in black. The direction of deforming the contour is indicated with a dotted arrow.
• Figure 5: We consider the causal configuration where $4>x$ and $3<x^+$ with $3$ and $4$ spacelike from each other. The points $1$ and $2$ are integrated over parallel null lines in the same null plane, with nonzero transverse separation. Here, we have suppressed the transverse direction, so the null plane appears as a single diagonal line (blue) from $x$ to $x^+$. (The conformal completion of the null plane also includes the left-moving diagonal line from $x$ to $\infty$ on the left, and then from $\infty$ to $x^+$ on the right.)

Theorems & Definitions (1)

• Conjecture : Stringy equivalence principle