Dynamics of a self-gravitating thin cosmic string
  We assume that a self-gravitating thin string can be locally described by what we shall call a smoothed cone. If we impose a specific constraint on the model of the string, then its central line obeys the Nambu-Goto equations. If no constraint is added, then the worldsheet of the central line is a totally geodesic surface. 
B. Boisseau, C. Charmousis and B. Linet