A new class of monopole solutions in five-dimensional general relativity and the role of negative scalar field energy in vacuum solutions

TL;DR

This paper introduces new monopole solutions in five-dimensional general relativity, revealing that scalar field energy can be negative and cancel electric field energy, with implications for understanding vacuum solutions.

Contribution

The study presents novel monopole solutions in 5D GR that incorporate scalar fields with negative energy, expanding the understanding of vacuum solutions and their physical interpretations.

Findings

Scalar field energy density is negative of electric field energy density.

Total electric and scalar field energy of monopoles is zero.

Solutions recover Reissner-Nordstr"om and Coulomb limits.

Abstract

Using numerical algebra tools, new classes of monopole solutions are obtained to the static, spherically-symmetric vacuum field equations of five-dimensional general relativity. First proposed by Kaluza, 5D general relativity unites gravity and classical electromagnetism with a scalar field. These monopoles correspond to bodies carrying mass, electric charge, and scalar charge. The Reissner-Nordstr\"om limit allows us to constrain the signature of the fifth component to be spacelike, but valid solutions are obtained for either sign of the scalar field. We find that Kaluza vacuum solutions imply the scalar field energy density is the negative of the electric field energy density, so the total electric and scalar field energy of the monopole is zero. Yet the new solutions provide reasonable Reissner-Nordstr\"om and Coulomb limits in mathematical form, with varying possibilities for the scalar field. The vanishing of the total electric and scalar field energy density for vacuum solutions seems to imply the scalar field can be understood as a negative-energy foundation on which the electric field is built.