Okay, these questions aren't completely about the Higgs, but it is a good starting point / explicit example. After spontaneous symmetry breaking occurs, such that the vacuum state itself no longer has the symmetry of the Lagrangian, will there always be something equivalent to a Higgs (a massive goldstone boson)? Because the vacuum state now has a non-zero expectation of these massive bosons, wouldn't that mean there is only one inertial frame in which the Higgs expectation energy-momentum four-vector has zero in the components for momentum? (ie. there exists a "higgs background rest frame"?) I assume the answer is no, but I'd like some explanation of why. Which leads to the third question, is it theoretically possible to have Lorentz symmetry be a symmetry of the Lagrangian, but have some expectation value of the vacuum violate this? Or can only gauge symmetries be affected in this manner, and Lorentz symmetry is not a gauge symmetry?