Universal Dynamics of Spontaneous Lorentz Violation and a New Spin-Dependent Inverse-Square Law Force

We study the universal low-energy dynamics associated with the spontaneous breaking of Lorentz invariance down to spatial rotations. The effective Lagrangian for the associated Goldstone field can be uniquely determined by the non-linear realization of a broken time diffeomorphism symmetry, up to some overall mass scales. It has previously been shown that this symmetry breaking pattern gives rise to a Higgs phase of gravity, in which gravity is modified in the infrared. In this paper, we study the effects of direct couplings between the Goldstone boson and standard model fermions, which necessarily accompany Lorentz-violating terms in the theory. The leading interaction is the coupling to the axial vector current, which reduces to spin in the non-relativistic limit. A spin moving relative to the "ether" rest frame will emit Goldstone Cerenkov radiation. The Goldstone also induces a long-range inverse-square law force between spin sources with a striking angular dependence, reflecting the underlying Goldstone shockwaves and providing a smoking gun for this theory. We discuss the regime of validity of the effective theory describing these phenomena, and the possibility of probing Lorentz violations through Goldstone boson signals in a way that is complementary to direct tests in some regions of parameter space.