The unification of gravity and the spin-1 field

Abstract

Unifying the massive spin-1 field with gravity requires the implementation of a regular vector field that satisfies the spin-1 Proca equation and is a fundamental part of the spacetime metric. That vector field is one of the pair of vectors in the line element field (\textbf{X},-\textbf{X}), which is paramount to the existence of all Lorentzian metrics and Modified General Relativity (MGR). Symmetrization of the spin-1 Klein-Gordon equation in a curved Lorentzian spacetime introduces the Lie derivative of the metric along the flow of one of the regular vectors in the line element field. The Proca equation in curved spacetime can then be described geometrically in terms of the line element vector, the Lie derivative of the Lorentzian metric, and the Ricci tensor, which unifies gravity and the spin-1 field. Related issues concerning charge conservation and the Lorenz constraint, singularities in a spherically symmetric curved spacetime, and geometrical implications of MGR to quantum theory are discussed. A geometrical unification of gravity with quantum field theory is presented.