Complex Metric, Torsion, Spin-Connection Gauge Field, and Gravitomagnetic Monopole, pp. 419-526

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Authors: (Jian Qi Shen, Centre for Optical and Electromagnetic Research, Zhejiang Univ., Hangzhou, The People's Republic of China, Zhejian Institute of Modern Physics, Dept. of Physics, Yuquan Campus, Zhejiang Univ., Hangzhou, The People's Republic of China)

Abstract: The physical and mathematical aspects (e.g., torsion, vierbein field, spin-affine connection and spinor) of general relativity are developed and studied based on the concepts of the complex Hermitian-metric Riemannian spacetime and the gauge field of Yang-Mills type. Four topics relevant to the torsion gravity, the spinor and tensor representations of local Lorentz group, the vierbein formulation of gravity as well as the gravitomagnetic monopole are discussed: (i) the complex Hermitian-metric gravity theory, where we generalize the real and symmetric metric of the Einstein gravity to the case of the complex metric that has a symmetric real part and an antisymmetric imaginary part. It is demonstrated that the gravity of complex metric must have torsion, since the complex contortion is no longer a tensor; (ii) the gravity theory with torsion, where three kinds of dynamics (i.e., the curvature-dominance, torsion-dominance, and torsion-only formalisms) are suggested. It is found that the Hilbert Einstein action of the conventional curvature-only gravity can be rewritten as that of the torsiononly gravity. Though the covariant derivatives (along the four-dimensional spacetime line element) of any vectors in the torsion-only gravity theory differ from that of the curvature-only gravity theory, the geodesic equation of a zero-spin test particle in the former theory can be reduced to that of the latter one, if the gravitational field equation is employed to the geodesic equation; (iii) the reformulation of general relativity as a Yang-Mills type gauge field theory, where the gravitational Lagrangian and the field equation based on the local Lorentz-group gauge invariance with spin-affine connection involved in covariant derivatives (using the vierbein formulation) are constructed. It is shown that the Einstein equation of general relativity is one of the first-integral solutions to the Yang-Mills equation of the spin-connection gauge field (with the Lorentz group being a local gauge symmetry group). It is expected that both the Einstein gravity and the Yang-Mills field can be reformulated as the vielbein fields. Based on the generalized vielbein fields of so-called composite spacetime manifold, it would offer an insight into a possible route to unify these two gauge fields; (iv) the dynamics of gravitomagnetic charge (dual mass), where three theories for gravitomagnetic charge are suggested in the frameworks of curvature-dominance, torsion dominance theories and Yang-Mills version of spin-connection gauge field with torsion, respectively. The common feature of these three theories is that the torsion fields are produced by the gravitomagnetic monopoles. This differs from those torsion-free theories in which the gravitomagnetic monopoles lead to a non-analytic metric.

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