Geometrization of electromagnetism in tetrad-spin-connection gravity

Abstract

The metric-affine Lagrangian of Ponomarev and Obukhov for the unified gravitational and electromagnetic field is linear in the Ricci scalar and quadratic in the tensor of homothetic curvature. We apply to this Lagrangian the variational principle with the tetrad and spin connection as dynamical variables and show that, in this approach, the field equations are the Einstein-Maxwell equations if we relate the electromagnetic potential to the trace of the spin connection. We also show that, as in the Ponomarev-Obukhov formulation, the generally covariant Dirac Lagrangian gives rise to the standard spinor source for the Einstein-Maxwell equations, while the spinor field obeys the nonlinear Heisenberg-Ivanenko equation with the electromagnetic coupling. We generalize that formulation to spinors with arbitrary electric charges.