# Gravitational lensing and cosmic strings

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?

Physically, imagine the following setting: i.e. the string is in between a star and an observer. The metric (3) tells us that the spacetime is flat with a wedge missing, hence in this chart, the light rays move on straight lines. However, we need to identify the sides of the missing wedge, and hence we see that in fact the light rays bend 