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Combing gravitational hair in 2+1 dimensions

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 $[-α,α]$. At linear order in source strength the energy is independent of the combing parameter $α$, but non-linearities cause the full energy to diverge as $α\to 0$. In general, solutions with combed gravitational flux also suffer from what we call displacement from their naive location. For weak sources and large $α$ one may set the displacement to zero by further increasing the energy, though for strong sources and small $α$ 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.