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Gravitational and electroweak interactions. (English) Zbl 0916.53051

Summary: Schrödinger considered the variational principle \(\delta \int\sqrt{-g}d^4x=0\), where \(g\) is the determinant of the metric \(g_{\mu\nu}\), but noted that if \(g_{\mu\nu}\) is varied, the resulting Euler-Lagrange equations cannot serve as field equations. We write \(g_{\mu\nu}= g_{ij} h^i_\mu h^j_\nu\), where \(g_{ij}=\text{diag}(-1, 1, 1,1)\), and express the vectors of the tetrad \(h^i_\mu\) as derivatives of nonintegrable functions \(x^i\) of the type commonly used for phase factors in gauge theory, i.e., \(h^i_\mu= x^i_{,\mu} \). We have previously shown that if the \(x^i\) are varied, the resulting Euler-Lagrange equations serve as field equations which imply the validity of Einstein equations with a stress-energy tensor for the electroweak field and associated currents [D. Pandres jun., ibid. 34, 733-759 (1995; Zbl 0829.53066)].
In this paper, we express these Einstein equations into two new forms and use these forms to derive Lorentz-force-like equations of motion. The electroweak field appears as a consequence of the field equations (rather than as a “compensating field” introduced to secure local gauge invariance). There is no need for symmetry breaking to accommodate mass, because the gauge symmetry is approximate from the outset.

MSC:

53Z05 Applications of differential geometry to physics
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C10 Equations of motion in general relativity and gravitational theory

Citations:

Zbl 0829.53066
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