Extra equation of gravity induced by spontaneous local Lorentz symmetry breakdown

Abstract: A model of spontaneous Lorentz violation in four dimension is given, which seems to provide a Lorentz invariant effective theory. An SU(2) Yang-Mills gauge field and an auxiliary U(1) vector field generate gravity and other interactions when they have vacuum expectation values. The emergent gravity is equivalent to conventional general relativity up to the third order terms in the Lagrangian. The coupling to matter, including spin-1/2 fermions, is also given correctly to this level. It remains to be seen whether this formalism reproduces the properties of black holes and other consequences obtained from Einstein's theory.