  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.
