Extra equation of gravity induced by spontaneous local Lorentz symmetry breakdown

The Lorentz invariance of the effective theory emergent from spontaneous local Lorentz violation requires a specific quantum geometry to which Einstein gravity is not applicable. A new gravity arising from a broken SU(2) gauge theory of Weyl doublets is described by the quantum vierbein conformal to the gauge field obeying the Yang-Mills action. We show that the quantum geometry by the gauge field is inevitable for the relativistic interactions between Goldstone bosons and quasi particles emergent from Weyl doublets to reproduce QED. In our model, Weyl doublets turn into quasi leptons, while the massive vector bosons transform into the Goldstone photon, the extra graviton, and a massive scalar boson with the Higgs coupling to the quasi leptons. The extra gravity can be extremely strong, since the gravitational constant is characterized by the scale of local Lorentz violation, which will have the same order of a charged lepton mass. The phenomenological implication of the results obtained is discussed.