# How is the deficit angle due to a relativistic cosmic string derived?

Dilatino
The presence of a cosmic string does not lead to gravitational attraction of a particle placed some distance away from it. But it affects the geometry of planes orthogonal to the cosmic string, such that the circumference of a circle traced out when moving around it at a distance r is given by

$$C = r(2\pi -\Delta)$$

which means that the flat plane is deformed to a cone.

How can the deficit angle due to a relativistic string

$$\Delta = \frac{8\pi G T_0}{c^4} = \frac{8\pi G \mu_0}{c^2}$$

be derived from general relativity?

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