Say we have a straight cosmic string lying along the $z$-axis, with energy-momentum tensor $$T_{\mu\nu}=\mu\delta(x)\delta(y)\operatorname{diag}(1,0,0,-1)\tag{1}\label{1}$$ for some small positive constant $\mu$. In the linearized Einstein theory, this will contribute a small perturbation to the flat metric, and we can write, in cylindrical polar coordinates $$ds^2=-dt^2+dz^2+dr^2+(1-8\mu)r^2d\phi^2\tag{2}\label{2}$$ We can further change the angular coordinate to get $$ds^2=-dt^2+dz^2+dr^2+r^2d\bar{\phi}^2\tag{3}\label{3}$$ Locally, the metric (\ref{3}) looks like Minkowski spacetime, but globally (\ref{2}) has an angular deficiency. The period of $\bar{\phi}$ is $(1-4\mu)2\pi<2\pi$, so that points at $\phi=0$ and $\phi=(1-4\mu)2\pi$ are identified and we're missing a ''wedge''.
Mathematically, this is fine. But physically, what is this wedge, or the lack thereof, and how do we use it to explain gravitational lensing, i.e. the double image of a distant object as observed from behind the cosmic string?
