The Geometry of Massless Cosmic Strings
  We study the geometry generated by a massless cosmic string. We find that this is given by a Riemann flat spacetime with a conical singularity along the worldsheet of the string. The geometry of such a spacetime is completely fixed by the holonomy of a simple loop wrapping the conical singularity. In the case of a massless cosmic string, this holonomy is a null-rotation/parabolic Lorentz transformation with a parabolic angle given by the linear energy density of the cosmic string. This description explicitly shows that there is no gravitational shockwave accompanying the massless cosmic string as has been suggested in the past. To illustrate the non-singular nature of the surrounding geometry, we construct a metric for the massless cosmic string that is smooth everywhere outside the conical singularity. 
Maarten van de Meent
gr-qc hep-th