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, $S^2\to S^2$. 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-Nordström (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 $\Lambda \to \Lambda /k$ 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.

Updated version submitted to PLB. This paper is dedicated to Prof. Metin Gürses on the occasion of his completion of sixty fruitful years of contributions to the field of General Relativity