Gravitational and Electroweak Interactions

Abstract

Schrodinger considered the variational principleδ ∫$\sqrt { - g} {\text{ }}d^4 x = 0$, whereg is the determinant of the metricg_μυ, but noted that ifg_μυ is varied, the resultingEuler-Lagrange equations cannot serve as field equations. We writeg_μυ =g_ijh_μ ⁱh_υ ^j, where g_ij = diag(-1, 1,1, 1), and express the vectors of the tetradh_μ ⁱ as derivatives ofnonintegrable functions xⁱ of the typecommonly used for phase factors in gauge theory, i.e.,h_μ ⁱ =x_,μ ⁱ. We have previously shownthat if the xⁱ are varied, the resultingEuler–Lagrange equations serve as field equations which imply the validity of Einstein equationswith a stress-energy tensor for the electroweak fieldand associated currents. In this paper, we express theseEinstein equations into two new forms, and use these forms to derive Lorentz-force-likeequations of motion. The electroweak field appears as aconsequence of the field equations (rather than as a“compensating field” introduced to secure local gauge invariance). There is no need forsymmetry breaking to accommodate mass, because the gaugesymmetry is approximate from the outset.