Gravity in the 3+1-Split Formalism I: Holography as an Initial Value Problem

TL;DR

The paper develops a detailed 3+1 split formulation of gravity with a cosmological constant and frames holography as an initial value problem at the boundary. It derives a full Fefferman-Graham expansion within this formalism and shows that the boundary data correspond to initial position and velocity, with the boundary stress tensor encoded in the FG coefficient related to the Weyl tensor. It demonstrates that holographic renormalization and the Kounterterms approach are two canonical transformations on the boundary phase space that yield the same boundary CFT data. This framework unifies bulk holography with a boundary initial-value formulation, providing a robust basis for AdS4/CFT3 applications and insights into self-duality in gravity.

Abstract

We present a detailed analysis of the 3+1-split formalism of gravity in the presence of a cosmological constant. The formalism helps revealing the intimate connection between holography and the initial value formulation of gravity. We show that the various methods of holographic subtraction of divergences correspond just to different transformations of the canonical variables, such that the initial value problem is properly set up at the boundary. The renormalized boundary energy momentum tensor is a component of the Weyl tensor.