= Solution
Outside the core, $f''=0$ and the metric is locally flat: set $\widetilde\phi=(1-\lambda)\phi$. Static test particles have no Newtonian-like attraction because $g_{00}$ is constant and $\Gamma^r{}_{00}=0$. Local trajectories are straight lines in locally unwrapped coordinates, so a very thin dark core is hard to identify through ordinary orbital accelerations or local <tidal forces>.
The global angular range is nevertheless $2\pi(1-\lambda)$ rather than $2\pi$. The missing angle $\delta=2\pi\lambda$ can give <gravitational lensing> with two nearly identical, essentially undistorted images of a background galaxy, and multiple such pairs aligned along the projected object. For a small deficit, a straight string and favourable geometry, their angular separation is of order $\delta D_{ls}/D_s$, with an additional inclination factor for a tilted string. The matching spectra and common line of image pairs distinguish this from an ordinary compact lens.
A moving <cosmic string> can also induce a step in the <cosmic microwave background> <temperature> across its path. That signature needs motion in addition to the static geometry supplied here. These are consequences of the exterior conical geometry; interpreting the inferred deficit as a positive mass per length requires the regular-axis/core assumptions distinguished in part <2/b>.
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