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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?

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The missing wedge is telling you that the string (which has a mass!) is causing the spacetime to be curved. Now by using this "Minkowski" like chart, you are trying to cover a curved manifold with a flat chart. You can see an toy example of this by taking a piece of paper, cutting out a wedge and gluing the two edges; you obtain a cone, which is not flat.

Physically, imagine the following setting: ligth_bending 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 enter image description here

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