Can a Self-gravitating Thin Cosmic String Obey the Nambu-goto Dynamics ?

We assume that a self-gravitating string is locally described by a thin tube of matter represented by a " smoothed conical metric ". If we impose a specific constraint on the model of string then its central line obeys the Nambu-Goto dynamics in the limit where the radius of the tube tends to zero. If no constraint is added then the world sheet of the central line is totally geodesic. It is admitted that an infinitely thin test cosmic string in a given background spacetime obeys the Nambu-Goto dynamics as it is the case in the Minkowski spacetime. But there is no proof that a self-gravitating cosmic string has the same behaviour. Since the exterior metric generated by a straight cosmic string is a conical metric 1 , some previous works 2− 5 have considered a world sheet whose the points are conical singularities in order to describe a cosmic string of arbitrary shape. They have then found that this world sheet is totally geodesic which is a very particular case of the history of a Nambu-Goto string. The aim of this communication is to examine again the dynamics of a self-gravitating string within a method in which the string is represented by a tube of matter having a thichness. We shall give a short account on the equations of motion of the central line of the tube of matter in the limit where the thichness tends to zero. More complete explanations can be found in our paper 6. As a preliminary, we portray a straight cosmic string as a cylinder of matter in general relativity 7− 10. In the coordinate system (t, z, l, ϕ) with l ≥ 0 and 0 ≤ ϕ < 2π, the interior metric is