Abelian dominance in Einstein's theory
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- Abelian dominance in Einstein's theory
- Cho, Yongmin; Oh, S. H.; Kim, Sang-Woo
- YANG-MILLS THEORY; FADDEEV-NIEMI DECOMPOSITION; GENERAL-RELATIVITY; GAUGE-THEORY; GRAVITATIONAL FIELD; MONOPOLES; SYMMETRY; TOPOLOGY; MASS; QCD
- Issue Date
- IOP PUBLISHING LTD
- CLASSICAL AND QUANTUM GRAVITY, v.29, no.20, pp. -
- We conjecture the Abelian dominance in Einstein's theory, that is, the Abelian part of the theory plays the central role in the dynamics. Treating Einstein's theory as a gauge theory of the Lorentz group, we show that Einstein's theory can be decomposed into the restricted part made up of the restricted connection which has the full Lorentz gauge invariance and the valence part made up of the valence connection which plays the role of gravitational source of the restricted gravity. In this decomposition, the role of the metric g(mu nu) is replaced by a four-index metric tensor g(mu nu) which transforms covariantly under the Lorentz group, and the metric-compatibility condition del(alpha)g(mu nu) = 0 of the connection is replaced by the gauge and generally covariant condition D(mu)g(mu nu) = 0. We show that there are two different Abelian decompositions, the light-like (or null) decomposition and the non-light-like (or non-null) decomposition, because the Lorentz group has two maximal Abelian subgroups. The decomposition shows the existence of the restricted gravity which has the full general invariance but is much simpler than Einstein's theory. Moreover, it tells us that the restricted gravity can be written as an Abelian gauge theory, which implies that the graviton can be described by a massless spin-1 field. This establishes the Abelian dominance in Einstein's theory.
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