Flat from anti-de Sitter

TL;DR

This work shows that Ricci-flat four-dimensional spacetimes can be obtained as the flat (k->0) limit of AdS geometries by Laurent-expanding the AdS boundary energy–momentum in k^2 and reconstructing the Ricci-flat bulk via a 1/r expansion. Using a covariant Newman–Unti gauge tailored to a Carrollian boundary, it clarifies how AdS data and the Cotton tensor control the boundary dynamics and how finiteness of the line element enforces a Carrollian evolution with an infinite tower of boundary data. A systematic flat-limit procedure yields flux-balance equations (FBEs) for Carrollian boundary degrees of freedom, introduces Chthonian data that organize at each order into a dynamical tensor F^{ab}_{(n)}, and demonstrates a recursion that underpins the full Ricci-flat solution space, including resummable algebraically special cases. The results illuminate a structured bridge between AdS/CFT-like data and asymptotically flat physics, with potential implications for flat holography and Carrollian boundary theories.

Abstract

Ricci-flat solutions to Einstein's equations in four dimensions are obtained as the flat limit of Einstein spacetimes with negative cosmological constant. In the limiting process, the anti-de Sitter energy--momentum tensor is expanded in Laurent series in powers of the cosmological constant, endowing the system with the infinite number of boundary data, characteristic of the asymptotically flat solution space. The governing flat Einstein dynamics is recovered as the limit of the original energy--momentum conservation law and from the additional requirement of the line-element finiteness, providing at each order the necessary set of flux-balance equations for the boundary data. This analysis is conducted using a covariant version of the Newman--Unti gauge designed for taking advantage of the boundary Carrollian structure emerging at vanishing cosmological constant and its Carrollian attributes such as the Cotton tensor.