A Classical Unified Field Theory with Solution having Fixed Mass and Charge

We consider a smooth manifold with a non-degenerate (0,2)-tensor as the only field. Defining a symmetric connection, which is compatible with the symmetric part of that tensor, enables covariant differentiation, and identifies the symmetric part of the tensor as the Riemannian metric. The antisymmetric part can be decomposed into harmonic, exact and co-exact differential 2-forms. The electromagnetic field, which is an exact 2-form, must then be associated with a linear combination of the exact portion and the Hodge dual of the co-exact portion. The divergence of the electromagnetic field is “electric current”; the divergence of a second, linearly independent combination can be identified as “magnetic current”.

We consider all possible Lagrangian terms that are at most second order in derivatives of the (0,2)-tensor, and at most quadratic in its antisymmetric part. Gravity (the Einstein-Hilbert Lagrangian) and electromagnetism (the standard electromagnetic field Lagrangian) are contained in these terms, along with new fluid-like terms, and curvature-mediated short-range interactions, some of which change sign under a parity transformation.

An appropriate choice of the coefficients in the linear combination defining the electromagnetic field eliminates all long-range momentum-energy interaction between electric and magnetic current. In this case, exclusively magnetically charged objects interact with exclusively electrically charged objects only through gravity and curvature-mediated, short-range forces. This naturally separates the two charges such that their only long-range interaction is gravity.

For the simplest possible theory that can admit electric current, we find a single, stable, spherical solution. It has a dense core, with density limiting to infinity at the solution's center. Waves of alternating charge surround the core, extending to infinity. Its charge and mass are fixed by the fundamental constants of the theory.