Harmonic branches of Einstein--Maxwell--scalar field gravity: exact solutions and regularity

Abstract

We relate black-hole horizons and wormhole throats to the geometry of the harmonic branches of Einstein--Maxwell--dilaton gravity. In the unified Ernst framework, we first solve the harmonic equations prior to fixing the spatial harmonic, and then jointly reconstruct the metric, Maxwell four-potential, and scalar field. The conserved geodesic energy fixes the meridional Weyl quadrature, whereas the initial geodesic data capture the asymptotic charges. Choosing the Schwarzschild harmonic function, regular nonextremal horizons fix the normalization \(H=1\) along with the associated matter-field regularity requirements. This reproduces the Schwarzschild and Reissner-Nordstrom families (including the NUT extension), the Gibbons-Maeda solution, and the dyonic string and Kaluza--Klein solutions, while also preserving their more general branches. If the scalar sector is continued to the phantom sign, one obtains a complementary oblate construction: smooth continuation selects \(H=-1\) and produces the massive and ultrastatic Ellis-Bronnikov wormholes. By comparing with the vacuum ring geometry, we make clear the further importance of the global extension. Altogether, these findings offer a unified black-hole--wormhole geometric reading of the charge relations and regularity constraints for the reconstructed solutions.