Combing gravitational hair in 2+1 dimensions

TL;DR

This paper constructs and analyzes classical solutions in 2+1 dimensional AdS gravity with gravitational hair, revealing limitations of gravitational Wilson lines and their energy properties, especially in the flat space limit.

Contribution

It provides a detailed analysis of time-symmetric solutions with gravitational hair in 2+1 AdS gravity, highlighting issues with displacement and the non-existence of certain Wilson line configurations.

Findings

Energy diverges as combing parameter approaches zero.

Solutions with gravitational flux also exhibit displacement issues.

Finite-energy flux solutions exist in the flat space limit.

Abstract

The gravitational Gauss law requires any addition of energy to be accompanied by the addition of gravitational flux. The possible configurations of this flux for a given source may be called gravitational hair, and several recent works discuss gravitational observables (`gravitational Wilson lines') which create this hair in highly-collimated `combed' configurations. We construct and analyze time-symmetric classical solutions of 2+1 Einstein-Hilbert gravity such as might be created by smeared versions of such operators. We focus on the AdS$_3$ case, where this hair is characterized by the profile of the boundary stress tensor; the desired solutions are those where the boundary stress tensor at initial time $t=0$ agrees precisely with its vacuum value outside an angular interval $[-\alpha,\alpha]$. At linear order in source strength the energy is independent of the combing parameter $\alpha$, but non-linearities cause the full energy to diverge as $\alpha \to 0$. In general, solutions with combed gravitational flux also suffer from what we call displacement from their naive location. For weak sources and large $\alpha$ one may set the displacement to zero by further increasing the energy, though for strong sources and small $\alpha$ we find no preferred notion of a zero-displacement solution. In the latter case we conclude that naively-expected gravitational Wilson lines do not exist. In the zero-displacement case, taking the AdS scale $\ell$ to infinity gives finite-energy flux-directed solutions that may be called asymptotically flat.