A modification of Einstein–Schrödinger theory that contains both general relativity and electrodynamics

Abstract

We modify the Einstein-Schrödinger theory to include a cosmological constant Λ z which multiplies the symmetric metric, and we show how the theory can be easily coupled to additional fields. The cosmological constant Λ z is assumed to be nearly cancelled by Schrödinger's cosmological constant Λ b which multiplies the nonsymmetric fundamental tensor, such that the total Λ = Λ z + Λ b matches measurement. The resulting theory becomes exactly Einstein-Maxwell theory in the limit as |Λ z | → ∞. For |Λ z | ∼ 1/(Planck length) 2 the field equations match the ordinary Einstein and Maxwell equations except for extra terms which are <10 −16 of the usual terms for worst-case field strengths and rates-of-change accessible to measurement. Additional fields can be included in the Lagrangian, and these fields may couple to the symmetric metric and the electromagnetic vector potential, just as in Einstein-Maxwell theory. The ordinary Lorentz force equation is obtained by taking the divergence of the Einstein equations when sources are included. The Einstein-Infeld-Hoffmann (EIH) equations of motion match the equations of motion for Einstein-Maxwell theory to Newtonian/Coulombian order, which proves the existence of a Lorentz force without requiring sources. This fixes a problem of the original Einstein-Schrödinger theory, which failed to predict a Lorentz force. An exact charged solution matches the Reissner-Nordström solution except for additional terms which are ∼10 −66 of the usual terms for worst-case radii accessible to measurement. An exact electromagnetic plane-wave solution is identical to its counterpart in Einstein-Maxwell theory.

A modification of Einstein–Schrödinger theory that contains Einstein–Maxwell–Yang–Mills theory

“…Our theory introduces an additional cosmological termḡΛ z as in [13], and also allows Γ ρ νµ and N νµ to have d×d matrix components [19],…”

“…For Abelian fields the third and fourth terms are the same, and this tensor reduces to the Abelian version in [13]. This tensor reduces to the ordinary Ricci tensor for Γ For our theory the Maxwell-Yang-Mills field tensor f νµ is defined bȳ…”

“…Here (f 3 )Λ Therefore the solution of the connection equations from [13] can again be abbreviated as…”

“…Substituting (53) using (102) shows that as in [13], the non-Abelian Ricci tensor (19) can again be abbreviated as…”

“…At least we have made great efforts to find an experiment in which these extra terms would be evident, and we have been unable to find such an experiment. As shown in [13], the ordinary Lorentz force equation can be derived from the divergence of the Einstein equations for the purely electromagnetic case of this theory. In [13] we also presented an exact electromagnetic plane-wave solution which is identical to its counterpart in Einstein-Maxwell theory.…”