Weak-Field Gravity of Circular Cosmic Strings
  A weak-field solution of Einstein's equations is constructed. It is generated by a circular cosmic string externally supported against collapse. The solution exhibits a conical singularity, and the corresponding deficit angle is the same as for a straight string of the same linear energy density. This confirms the deficit-angle assumption made in the Frolov-Israel-Unruh derivation of the metric describing a string loop at a moment of time symmetry. 
Shane J. Hughes, Des J. Mc Manus, and Michel A. Vandyck