Branched Spacetime Geometries from Harmonic $S^2$ Maps: Energy Matching and Its Limitations

Abstract

We investigate a topological construction of general-relativistic geometries based on harmonic self-maps of the two-sphere, $S^{2}\rightarrow S^{2}$. Such maps are classified by an integer degree $k$. For $k>1$, however, the pullback angular metric is not globally smooth: it defines a branched covering with conical excesses at the north and south poles. We calculate the corresponding distributional curvature and show that the branch points extend over the $(t,r)$ sector as codimension-two defects with negative signed tension. For a branched Schwarzschild geometry, we introduce a smooth Reissner-Nordstr\"{o}m (RN)-like radial deformation and define its finite bulk Killing energy relative to the $Q_{k}=0$ geometry in the same topological sector. An explicit phenomenological matching prescription relates this bulk energy to the scaled excess harmonic-map energy and determines $Q_{k}$. Identifying the matching scale with the outer horizon then yields a discrete sequence approaching an extremal limit. We also examine the associated horizon geometry, conditional entropy, regular curvature scalars, weak-field limit, and horizonless zero-mass sector. The same positive matching prescription does not extend universally. For the Simpson-Visser radial ansatz, the relative bulk Killing energy is negative. In the de Sitter case, the reduction $\Lambda \rightarrow \Lambda /k$ reproduces the excess map energy only through a common-volume reference subtraction, not through the direct energy difference between the physical sectors. The construction therefore provides a classical framework for branched topological sectors while also identifying the assumptions and limitations of the associated energy-matching prescription.