Gravitational corner conditions in holography

Gary T Horowitz,Diandian Wang

doi:10.1007/jhep01(2020)155

Abstract

Contrary to popular belief, asymptotically anti-de Sitter solutions of gravitational theories cannot be obtained by taking initial data (satisfying the constraints) on a spacelike surface, and choosing an arbitrary conformal metric on the timelike boundary at infinity. There are an infinite number of corner conditions that also must be satisfied where the initial data surface hits the boundary. These are well known to mathematical relativists, but to make them more widely known we give a simple explanation of why these conditions exist and discuss some of their consequences. An example is given which illustrates their power. Some implications for holography are also mentioned.

Highlights

Where R is the three dimensional scalar curvature, Di is the three dimensional covariant derivative, and k is the trace of kij
If we conformally rescale by an appropriate conformal factor, all time derivatives of the boundary metric at t = 0 are completely fixed
One can discuss the corner conditions in terms of conformally invariant quantities [1,2,3], there are many situations of interest where a preferred conformal frame is picked out by symmetries. (An example is given .) The coordinate freedom in the metric can be dealt with as follows

Summary

Introduction

Where R is the three dimensional scalar curvature, Di is the three dimensional covariant derivative, and k is the trace of kij. This does not contradict the usual picture of the domain of dependence of an asymptotically AdS spacelike surface, which shows that initial data at t = 0 cannot determine the boundary metric at any t > 0. Choose smooth boundary data satisfying the corner conditions

Results

Conclusion