Interpretation of black hole metric with fractional $\kappa$ instead of the usual $\kappa\in\{-1,0,1\}$ The metric for a black hole can be written:
$$d s^{2}=-\left(\kappa-\frac{2 M}{r}\right) d t^{2}+\left(\kappa-\frac{2 M}{r}\right)^{-1} d r^{2}+r^{2} d \Sigma_{2, k}^{2}$$
where $\kappa=-1,0,1$ corresponds to hyperbolic, planar, and spherical horizon geometries, respectively.
From a purely geometric point of view, what would the interpretation of a metric with fractional $\kappa$, for example $\kappa=\tfrac{1}{3}$, be?
 A: The value of $\kappa$ different from the value of $1$ (which is what we have in the usual form of Schwarzschild solution) corresponds to the deficit (for $\kappa<1$) or excess of solid angle. If  we consider a sphere of constant radius $r=R$ when $R\gg M$, the distance from the central object (black hole) is approximately $\kappa^{-\frac12} R$, while the area of such sphere is $4\pi R^2$.
For $\kappa\ne 1$ the solution is not Ricci-flat but has non-zero components $R^{\theta}_\theta=R^{\phi}_{\phi}\sim r^{-2}$, so this could not be a vacuum solution.
Nevertheless such metric has meaning for $\kappa\ne 1$  as a metric of so-called global monopole [$1$], or black hole with a global monopole charge [$2$] (energy positivity requires then that $0<\kappa<1$). These are solutions of Einstein field equations with matter in the form of triplet scalar field with global $O(3)$ symmetry and Higgs potential with nonzero vev. Outside the core region of the monopole the Higgs field reaches vev in the usual “hedgehog” configuration and stress–energy tensor is determined by $(\nabla \psi^a)^2$ terms.
Such metric is not asymptotically flat and its total energy is divergent. Light ray would be deflected by an angle that would approach constant even at arbitrarily large distances from the center.
Note that the “bare” monopole solution (without black hole horizon) has a negative mass parameter, that does not produce a naked singularity since in the core region, where the Higgs field is different from vev, the solution is different from its asymptotics.
References

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*Barriola, M., & Vilenkin, A. (1989). Gravitational field of a global monopole. Physical Review Letters, 63(4), 341, doi:10.1103/PhysRevLett.63.341.


*Dadhich, N., Narayan, K., & Yajnik, U. A. (1998). Schwarzschild black hole with global monopole charge. Pramana, 50(4), 307-314, doi:10.1007/BF02845552, free pdf.
