Self-gravitating String-like Configurations from Nonlinear Electrodynamics

We consider static, cylindrically symmetric configurations in general relativity coupled to nonlinear electrodynamics (NED) with an arbitrary gauge-invariant Lagrangian of the form Lem = Φ(F ) , F = FαβF αβ . We study electric and magnetic fields with three possible orientations: radial (R), longitudinal (L) and azimuthal (A), and try to find solitonic stringlike solutions, having a regular axis and a flat metric at large r , with a possible angular defect. Assuming that the function Φ(F ) is regular at small F , it is shown that a regular axis is impossible in R-fields if there is a nonzero effective electric charge and in A-fields if there is a nonzero effective electric current along the axis. Thus solitonic solutions are only possible for purely magnetic R-fields and purely electric A-fields, in cases when Φ(F ) tends to a finite limit at large F . For both Rand A-fields it is shown that the desired large r asymptotic is only possible with a non-Maxwell behaviour of Φ(F ) at small F . For L-fields, solutions with a regular axis are easily obtained (and can be found by quadratures) whereas a desired large r asymptotic is only possible in an exceptional solution; the latter gives rise to solitonic configurations in case Φ(F ) = const · √ F . We give an explicit example of such a solution.