Boundary description of Planckian scattering in curved spacetimes

TL;DR

The paper shows that in the eikonal limit, gravity in curved spacetimes with a nonzero cosmological constant reduces to an effective boundary theory describing shock-wave interactions, highlighting a holographic structure supported by an essential constraint $R_{i\\alpha}=0$ that ties transverse and longitudinal dynamics. Through a controlled rescaling and a twofold expansion in ${\\epsilon}$ and off-diagonal metric components, the authors derive a boundary action for the interacting sector, with strong-curvature and weak-curvature regimes yielding either a rich boundary functional or a quadratic, quantizable theory with noncommuting boundary coordinates. They connect the boundary description to AdS shock-waves and Horowitz–Itzhaki’s shift function, and discuss the AdS/CFT interpretation via conformally coupled scalars, including Delta$_+$ and Delta$_-$ quantizations. The work offers a concrete holographic framework for Planckian scattering in curved backgrounds and points to deeper links with holography, black hole information, and possible extensions to de Sitter spaces and string corrections. Overall, the results provide a principled, semi-classical route to encoding bulk gravitational scattering in boundary degrees of freedom with potential CFT dual descriptions.

Abstract

We show that for an eikonal limit of gravity in a space-time of any dimension with a non-vanishing cosmological constant, the Einstein -- Hilbert action reduces to a boundary action. This boundary action describes the interaction of shock-waves up to the point of evolution at which the forward light-cone of a collision meets the boundary of the space-time. The conclusions are quite general and in particular generalize the previous work of E. and H. Verlinde. The role of the off-diagonal Einstein action in removing the bulk part of the action is emphasised. We discuss the sense in which our result is a particular example of holography and also the relation of our solutions in $AdS$ to those of Horowitz and Itzhaki.