What is the relation betweenΔφandμfor a cosmic string?

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Abstract
We examine the relation between the deficit angle Δφ and the mass per unit length μ of a cosmic string. In the weak-field approximation Δφ=8πμ. In general we define a correction term δ by δ≡(Δφ/8πμ)-1. We show that δ depends on the gravitational strength of the string, but is independent of the thickness of the string. Taking the zero-thickness limit we obtain an infinitely thin string for which Δφ≠8πμ. The implications of this result for the treatment of strings as singular line sources in general relativity is discussed.