MG13 - Talk detail

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~~~~~Based on the Abelian projection of Einstein's theory proposed recently we show that one can construct the restricted theory of gravitation, $A_2$ gravity and $B_2$ gravity, made of the Abelian connection which has the full general invariance which describes the core dynamics of Einstein's theory. We show that the restricted gravity describes the Einstein-Rosen-Bondi gravitational wave and the Weyl's C-class rotation-boost symmetric metric. Furthermore, we show that classically the restricted gravity and Einstein's gravity have identical topology. In particular they have identical vacuum topology, which can be classified by $\pi_3(S^3)\simeq \pi_3(S^2)$ when the $3$-dimensional space allows a global chart. This establishes the Abelian dominance in Einstein's theory. \\~\\ ~~~~~A remarkable feature of the restricted gravity is that the dynamics is described by Abelian gauge theory with a spin-one gauge potential, which implies that the graviton could also be described by a massless spin-one gauge particle. This sheds a new light to quantum gravity. \\~\\

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~~~~~Based on the view that Einstein's theory can be interpreted as a gauge theory of Lorentz group, we decompose the gravitational connection (the gauge potential of Lorentz group) $\vGm_\mu$ into the restricted connection made of the potential of the maximal Abelian subgroup $H$ of Lorentz group $G$ and the valence connection made of $G/H$ part of the potential which transforms covariantly under Lorentz gauge transformation. With this decomposition we show that the Einstein's theory can be decomposed into the restricted part made of the restricted connection which has the full Lorentz gauge invariance and the valence part made of the valence connection which plays the role of gravitational source of the restricted gravity. ~~~~~We show that there are two different Abelian decompositions of Einstein's theory, the light-like (or null) decomposition and the non light-like (or non-null) decomposition, because Lorentz group has two maximal Abelian subgroups. In this decomposition the role of the metric $g_\mn$ is replaced by a four-index metric tensor $\vg_\mn$ which transforms covariantly under the Lorentz group, and the metric-compatibility condition $\nabla_\alpha g_\mn=0$ of the connection is replaced by the gauge and generally covariant condition ${\mathscr D}_\mu \vg^\mn=0$. ~~~~The decomposition shows the existence of a restricted theory of gravitation which has the full general invariance but is much simpler than Einstein's theory which has only free graviton. Moreover, it tells that the restricted gravity can be written as an Abelian gauge theory, which implies that the graviton can be described by a massless spin-one field. \\~\\

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