= Axial boundary term in the mass of a cylindrical spacetime
{title2=$\mu_{\rm bulk}=c^2[f'(0^+)-f'(R^-)]/(4G)$}
For $ds^2=-(dx^0)^2+dr^2+f(r)^2d\phi^2+dz^2$, the smooth bulk mass density is $-c^2f''/(8\pi Gf)$. Integrate using proper transverse area $2\pi f\,dr$ to obtain the boundary term. With no shell at the core edge, $f'(R^-)$ equals the exterior conical slope. A regular axis requires $f(0)=0$ and $f'(0)=1$ for angular period $2\pi$. A different printed inner derivative gives a different bulk integral; any singular-axis matter must be specified separately. This prevents silently applying the standard cosmic-string mass-deficit relation to nonregular cores.
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