Flat limit of AdS/CFT from AdS geodesics: scattering amplitudes and antipodal matching of Liénard-Wiechert fields

TL;DR

The paper addresses how flat-space physics, including the S-matrix, can be extracted from AdS/CFT by analyzing AdS geodesics and their boundary hits to define flat-limit boundary operators. It develops a geodesic-based framework to map boundary CFT regions to flat-space infinity regions, computes the flat limit of CFT correlators for both massless and massive scattering, and demonstrates that Liénard-Wiechert fields in AdS exhibit antipodal matching whose flat-limit reduction reproduces the known flat-space antipodal structure. A key result is that spacetime geodesics in AdS encode the boundary data needed to reconstruct flat-space scattering in a way that connects to celestial and Carrollian holography. The work provides a concrete geometric mechanism for flat-space holography within AdS/CFT and offers avenues for extending to black-hole backgrounds and to other asymptotic frameworks.

Abstract

We revisit the flat limit of AdS/CFT from the point of view of geodesics in AdS. We show that the flat space scattering amplitudes can be constructed from operator insertions where the geodesics of the particles corresponding to the operators hit the conformal boundary of AdS. Further, we compute the Liénard-Wiechert solutions in AdS by boosting a static charge using AdS isometries and show that the solutions are antipodally matched between two regions, separated by a global time difference of $Δτ=π$. Going to the boundary of AdS along null geodesics, in the flat limit, this antipodal matching leads to the flat space antipodal matching near spatial infinity.

Figures (7)

• Figure 1: Soup-can picture of AdS: the causal structure of $\mathrm{AdS}_4$.
• Figure 2: Timelike geodesics and null geodesics in AdS.
• Figure 3: Null geodesics in AdS global coordinates: massless particles hit the boundary of AdS at $\tau =\cfrac{\pi }{2}$.
• Figure 4: Timelike geodesics in AdS global coordinates: massive particles hit the boundary of AdS at $\tau =\frac{\pi}{2}+\frac{i}{2}\log(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m})$.
• Figure 5: Mapping of CFT regions to flat space regions. Here, $\partial \mathcal{M}_+$ denotes future timelike infinity, and $\partial \mathcal{M}_-$ denotes past timelike infinity. The blue fringe regions represent future and past null infinity. These regions are described by $\tau = \pm \frac{\pi}{2}$, corresponding to future null infinity $\mathscr{I}^+$ and past null infinity $\mathscr{I}^-$, collectively referred to as $\mathscr{I}^{\pm}$ on the AdS boundary. The pill-shaped regions represent analytic continuations of the boundary CFT in the imaginary direction of global time, acting as future and past timeline infinity $i^{\pm}$.