Holographic Reconstruction and Renormalization in Asymptotically Ricci-flat Spacetimes

TL;DR

This work extends holographic duality to gravity with zero cosmological constant by identifying a dual family of CFTs living on a codimension-two boundary at null infinity for Ricci-flat spacetimes that admit asymptotically hyperbolic leaves. By developing an AdS/CFT–style program for pure gravity, it derives the general spacetime asymptotics, performs holographic renormalization, and shows how the boundary stress tensors encode the bulk metric evolution through Ward identities and the normalizable mode $g_{(d)ij}$. In even dimensions, the holographic Weyl anomaly ties the bulk time coordinate to the spectrum of central charges, with a Planck-length scale setting the characteristic length, and the authors discuss locality constraints on the central-charge spectrum. The analysis also extends to subleading extrinsic-curvature corrections and explores the conditions under which holographic renormalization remains local, outlining future directions for bulk matter reconstruction and the role of bulk diffeomorphisms in connecting the family of boundary theories.

Abstract

In this work we elaborate on an extension of the AdS/CFT framework to a subclass of gravitational theories with vanishing cosmological constant. By building on earlier ideas, we construct a correspondence between Ricci-flat spacetimes admitting asymptotically hyperbolic hypersurfaces and a family of conformal field theories on a codimension two manifold at null infinity. By truncating the gravity theory to the pure gravitational sector, we find the most general spacetime asymptotics, renormalize the gravitational action, reproduce the holographic stress tensors and Ward identities of the family of CFTs and show how the asymptotics is mapped to and reconstructed from conformal field theory data. In even dimensions, the holographic Weyl anomalies identify the bulk time coordinate with the spectrum of central charges with characteristic length the bulk Planck length. Consistency with locality in the bulk time direction requires a notion of locality in this spectrum.

Figures (4)

• Figure 1: Penrose diagram for Minkowski space
• Figure 2: Foliation of Minkowski space
• Figure 3: Ricci-flat embedding near the initial hypersurface $\Sigma$. The dashed lines represent the different time slices obtained by time evolving $\Sigma$ and the generated embedding represents the region between some initial and final slices (which can be taken to be infinitesimally close to $\mathcal{H}$ and to $\Im^{+}$, respectively).
• Figure 4: Conformal embedding near future null infinity. The dashed lines represent different surfaces of constant $t$, while the solid lines are surfaces of constant $z$. The null surface $\mathcal{H} = \{z=0,t=-\infty\}$ is the past Cauchy horizon of the development.