Spinning straight cosmic strings with flat exterior solutions generically violate the weak energy condition

Any interior solution for a cylindrically symmetric, stationary cosmic string with flat exterior, spinning around its longitudinal axis, and without internal longitudinal currents ($g_{zz}=1$, $g_{tz}=0$), must somewhere violate the weak energy condition of standard general relativity. Existing interior solutions may be readily fixed by adding mass to the string above that generating its angular deficit, but at the cost of introducing an exterior gravitational field.