AdS/Ricci-flat correspondence

TL;DR

This work establishes a concrete AdS/Ricci-flat correspondence by performing generalized dimensional reductions, mapping AdS solutions with negative cosmological constant to Ricci-flat spacetimes, with the map implemented via $\tilde{g}(r,x;n)=\hat{g}(r,x;-n)$ and $\tilde{\phi}(r,x;n)=\hat{\phi}(r,x;-n)$ and $d\leftrightarrow -n$. It then explores holographic interpretations, showing how AdS boundary data translate into interior $p$-brane data in Ricci-flat spacetimes, and how holographic correlation functions map under the correspondence. A major application is constructing non-homogeneous Ricci-flat black branes from AdS fluids/gravity, yielding explicit slowly varying geometries, their ADM stress tensors, an entropy current with non-negative divergence, and a cubic-order Gregory-Laflamme dispersion relation that aligns with numerical results, especially at large transverse dimensions. The paper also elaborates the AdS/Rindler limit ($n\to-1$), connecting AdS holography to Rindler-fluid dynamics, and discusses generalizations to other internal manifolds, matter couplings, and potential extensions to string theory and nonlocal observables. Overall, the work provides a framework for translating holographic and hydrodynamic structures between AdS and flat-space settings, suggesting a holographic-style structure for a broad class of Ricci-flat spacetimes.

Abstract

We present a comprehensive analysis of the AdS/Ricci-flat correspondence, a map between a class of asymptotically locally AdS spacetimes and a class of Ricci-flat spacetimes. We provide a detailed derivation of the map, discuss a number of extensions and apply it to a number of important examples, such as AdS on a torus, AdS black branes and fluids/gravity metrics. In particular, the correspondence links the hydrodynamic regime of asymptotically flat black $p$-branes or the Rindler fluid with that of AdS. It implies that this class of Ricci-flat spacetimes inherits from AdS a generalized conformal symmetry and has a holographic structure. We initiate the discussion of holography by analyzing how the map acts on boundary conditions and holographic 2-point functions.