Sum Rule for the ADM Mass and Tensions in Planar AdS Spacetimes

TL;DR

This paper establishes a sum rule relating ADM mass, tensions, and a new charge in planar AdS spacetimes, linking bulk gravitational quantities to boundary stress tensor properties in the AdS/CFT correspondence.

Contribution

It introduces an ADM charge associated with the scaling symmetry in AdS, showing it vanishes for solutions and relates to the boundary stress tensor trace in AdS/CFT.

Findings

Q vanishes for Einstein solutions with negative cosmological constant

Sum rule links ADM mass, tensions, and the new charge Q

Perturbative stress-energy tensor must be tracefree in a simple model

Abstract

An asymptotically planar AdS spacetimes is characterized by its ADM mass and tensions. We define an additional ADM charge Q associated with the scaling Killing vector of AdS, show that Q is given by a certain sum over the ADM mass and tensions and that Q vanishes on solutions to the Einstein equation with negative cosmological constant. The sum rule for the mass and tensions thus established corresponds in an AdS/CFT context to the vanishing of the trace of the boundary stress tensor. We also show that an analogous sum rule holds for local planar sources of stress-energy sources in AdS. In a simple model consisting of a static, plane symmetric source we find that the perturbative stress-energy tensor must be tracefree.